saturation equilibrium binding experiment

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How to measure and evaluate binding affinities

• Ishraq AlSadhan
• Pavanapuresan P Vaidyanathan
• Department of Biochemistry, Stanford University, United States ;
• Department of Chemical Engineering, Stanford University, United States ;
• Stanford ChEM-H, Stanford University, United States ;
• Open access
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• Daniel Herschlag
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Introduction

Materials and methods, data availability, article and author information.

Quantitative measurements of biomolecule associations are central to biological understanding and are needed to build and test predictive and mechanistic models. Given the advances in high-throughput technologies and the projected increase in the availability of binding data, we found it especially timely to evaluate the current standards for performing and reporting binding measurements. A review of 100 studies revealed that in most cases essential controls for establishing the appropriate incubation time and concentration regime were not documented, making it impossible to determine measurement reliability. Moreover, several reported affinities could be concluded to be incorrect, thereby impacting biological interpretations. Given these challenges, we provide a framework for a broad range of researchers to evaluate, teach about, perform, and clearly document high-quality equilibrium binding measurements. We apply this framework and explain underlying fundamental concepts through experimental examples with the RNA-binding protein Puf4.

Molecular associations lie at the heart of biology. Their thermodynamics provides information critical for deriving a fundamental understanding of molecular functions. In a broader biological context, these associations are linked and interconnected in complex networks that allow sensitive and precise developmental programs and responses to environmental cues, and that are altered in disease states. The outputs of pathways and networks are determined by the quantitative interplay of their many constituent molecules and interactions. Thus, equilibrium constants for association between network components are needed to define, model, predict, and ultimately precisely manipulate biology.

A limitation of traditional biochemical measurements is their low throughput, especially in relation to the large number of cellular interactions. Excitingly, several strategies have recently emerged to obtain high-throughput, quantitative information for intermolecular associations (e.g. Buenrostro et al., 2014 ; Tome et al., 2014 ; Lambert et al., 2014 ; Nutiu et al., 2011 ; Maerkl and Quake, 2007 ; Adams et al., 2016 ; Jain et al., 2017 ). Given these potentially transformative advances, it is especially timely to assess the accuracy of equilibrium binding measurements. We wanted to know whether current practices are sufficient to ensure reliable and accurate measurements, and whether the reliability of these measurements can be readily ascertained from the information provided in published work.

Our survey of 100 literature binding measurements, presented below, uncovered recurring problems with a large majority of studies. Fortunately, there are straightforward procedures, laid out here, that can be followed to ensure that published binding measurements are reliable. The principles underlying these procedures have been discussed and we build on these previous reports ( Pollard, 2010 ; Hulme and Trevethick, 2010 ; Sanders, 2010 ). We focus on a minimal set of critical actionable steps and controls that biologists of any background should be able to implement in their binding measurements. We apply these procedures with experimental examples and also demonstrate the pitfalls of omitting essential controls. To further streamline application of these standard procedures, we provide a convenient checklist that can organize and guide experiments and can be used as an aid in summarizing and presenting results for publication.

Assessing the current state of binding measurements

We evaluated published binding measurements using RNA-protein interactions as an illustrative example. We surveyed 100 studies that reported equilibrium dissociation constants ( K D values) and scored them based on two key criteria for reliable binding measurements: sufficient time to equilibration and proper concentration regime ( Figure 1 ).

Assessment of published K D values for RNA-binding proteins.

We analyzed 100 papers reporting K D or ‘apparent K D ’ values of RNA/protein interactions. Measurements were evaluated based on two criteria: demonstrating equilibration (horizontal axis) and controlling for titration (vertical axis). Detailed criteria are described in Materials and methods, and the source data are provided in Supplementary file 1 . The right column includes predominantly studies that used ITC and SPR, techniques that inherently record binding progress over time (24/30 in this column). The fraction of studies that varied time to demonstrate equilibration in non-ITC/SPR experiments is considerably smaller (6 of the 76 papers that did not exclusively use ITC or SPR, or <10%).

First, we asked if equilibration was demonstrated. By definition, an equilibrium state is invariant with time. So, determining a binding equilibrium constant requires showing that there is no change in the amount of bound complex over time. Of the 100 studies surveyed, 70 did not report varying time for reported equilibrium measurements ( Figure 1 ; Supplementary file 1 ). Of the 30 studies that did vary time, 24 exclusively used techniques with built-in monitoring of progress over time (isothermal titration calorimetry (ITC) and surface plasmon resonance [SPR]). Of the remaining 76 studies—those using approaches such as native gel shifts, nitrocellulose filter binding, and fluorescence anisotropy—less than 10% reported varying time ( Figure 1 , Figure 1—figure supplement 1 ).

We know from individual discussions that some researchers carry out these controls, as we advocate below, but do not report them. Unfortunately, the published record then cannot distinguish between these studies and others that have not demonstrated equilibration.

A second critical control entails demonstrating that the K D is not affected by titration, as artifacts can arise when the concentration of the constant limiting component is too high relative to the dissociation constant ( K D ). Similar to varying time to establish equilibration, systematically varying the concentration of the limiting component provides a definitive control for effects of titration. In our survey, only 5% of studies reported performing this or equivalent control ( Figure 1—figure supplement 2 ). Nevertheless, most authors appeared to be aware of the need to avoid titration, as the majority of studies (~70%) reported using appropriately low concentrations of the limiting component or employed advanced analysis methods. We consider these examples as reasonably titration-controlled for the purpose of the survey, but emphasize the importance of empirical controls in the sections below. Importantly, this leaves, at a minimum, one-fourth of studies at risk for titration ( Figure 1 , Figure 1—figure supplement 2 ).

To what extent do these limitations affect the reported equilibrium binding constants in practice? As an example, for Puf4 binding (see below), not controlling for the factors above gave apparent K D values that were up to seven-fold higher than the actual K D values. A more extreme literature example is discussed in the next section, with discrepancies reaching 1000-fold, and other examples have been previously noted ( Hulme and Trevethick, 2010 ; Strohkendl et al., 2018 ). There is a tendency to be less careful about controls in pursuit of relative affinities (specificity) rather than absolute affinity. However, failing to account for the factors noted above can also underestimate specificity by orders of magnitude (see Figure 4—figure supplement 1 and Figure 5—figure supplement 4 below).

These observations highlight an urgent need to revisit the criteria for reliable binding measurements. There is a parallel need to render these criteria accessible to a broad range of biologists, regardless of background or training, in the form of clear and readily actionable guidelines. To meet these needs, we provide simple, concrete strategies so that any practitioner can carry out reliable binding measurements, clearly communicate their results, and evaluate results from others.

Fortunately, the key requirements for binding measurements can be broken down into a small number of steps. We present two required steps for equilibrium binding measurements—varying the incubation time (see section 'Vary incubation time to test for equilibration') and controlling for titration (see section 'Avoid the titration regime'), and we illustrate these steps for the example of RNA binding to the Saccharomyces cerevisiae Puf4 protein ( Gerber et al., 2004 ; Miller et al., 2008 ). We also present additional steps that can be taken to further increase confidence in K D values and to obtain kinetic information about the binding event under investigation (see sections 'Test K D by an independent approach' and 'Determine the fraction of active protein'). Finally, we describe strategies to address cases where no binding is initially detected and explain why it is often premature to conclude an absence of binding (see section 'The case of no observed binding').

Practical considerations

In principle, one would like to have well-behaved and perfectly controlled measurements in all cases, but biology and biochemistry can be messy. There are many times, working with extracts and partially purified systems where protein concentrations cannot be accurately determined, where proteases and nucleases may limit achievable equilibration times, and where there may be additional interacting components. Regardless of these potential complications, the simple steps indicated below can establish the robustness of measured affinities and can diagnose and help overcome issues like loss of activity over time. Moreover, these controls (and quantitative measurements more generally) can help uncover new features and regulatory mechanisms, based on deviations from ‘ideal’ behavior of simple binding equilibria.

Vary incubation time to test for equilibration

The most basic test for whether a binding reaction has reached equilibrium is that the fraction of complex formed between two molecules does not change over time. Nevertheless, the majority of papers we surveyed that present binding measurements and report apparent affinities or equilibrium dissociation constants do not report that time has been varied ( Figure 1 ). We first describe two related concepts that will help readers develop an intuition for the time scales of binding processes and we then apply these concepts to Puf4 binding.

Binding and other simple kinetic processes, in general, follow exponential curves ( Figure 2 ). The key property of an exponential curve is that it has a constant half-life (t 1/2 )—that is, the time it takes for the reaction to proceed from 0% to 50% complete, 50% to 75% complete, 75% to 87.5% complete, etc. is the same ( Figure 2 ). After three half-lives, an exponential process is almost 90% complete (3t 1/2 = 87.5%; Figure 2 ), which is close enough to equilibration for most applications. Below we adopt the more common standard of taking reactions to five half-lives, or 96.6% completion; this more conservative standard is safer given that there are multiple sources of potential error in practice.

Exponential kinetics used to estimate the time needed for binding equilibration.

Arrows indicate reaction half-life t 1/2 . Fraction bound is defined by the equation 1 -  e -t  ×  ln2/ t 1/2 = 1 -  e -t  ×   k equil .

Equilibration rate constant

The equilibration rate constant is effectively the inverse of the binding half-life ( k equil = ln2 t 1/2 ≈ 1 t 1/2 ) and, importantly, is concentration-dependent. For the binding equilibrium shown in Figure 3 , under conditions where one binding partner (here, the protein, P) is in large excess over the other (RNA), the rate equation for approach to equilibrium, k equil , is described as: 

k on is the association rate constant, [P] is the concentration of protein, or the binding partner in excess, and  k off is the dissociation rate constant ( Pollard, 2010 ). According to Equation 1 , equilibration is the slowest at the lowest protein concentrations. For this reason, equilibration times need to be established from the low end of the concentration range. In practice, it is useful to consider the limiting case with the protein concentration approaching zero ([P] ~ 0), such that Equation 1 simplifies to Equation 2 ( Hulme and Trevethick, 2010 ):

Thus, the more long-lived the complex (i.e. the lower its dissociation rate constant), the longer the incubation time required to reach equilibrium.

Model for one-step, non-cooperative, 1:1 binding between two molecules.

Protein (P) binding to an RNA (R) molecule is shown for illustrative purposes.

What is the range of equilibration times for typical biomolecular interactions? While k off measurements (and, consequently, k equil ) are less common in literature than K D measurements, equilibration times can be readily estimated ( Sanders, 2010 ). Given that K D = k off k on ( Figure 3 ) and assuming that the binding of molecules occurs as fast as diffusional collisions ( k on = 10 8 M −1 s −1 ), we can calculate that an interaction with a K D value of 1 pM would require a 10 hr incubation to reach equilibrium, whereas a 1 µM K D interaction would only require 40 ms ( Table 1 ). Notably, binding rate constants for processes involving macromolecules are often smaller than the diffusion driven limit of ~10 8 M −1 s −1 , for example when additional conformational rearrangements are required for stabilizing binding after two molecules collide ( Karbstein and Herschlag, 2003 ; Peluso et al., 2000 ; Wu et al., 2002 ). As a result, equilibration can take much longer. Thus, equilibration times for two interactions with the same K D value can vary by orders magnitude, and some reactions in the biologically relevant affinity range can require equilibration times of 10s of hr or even longer in vitro ( Table 1 ; Hulme and Trevethick, 2010 ; Sanders, 2010 ). These long times underscore that biology has developed mechanisms to circumvent or utilize such slow processes—for example, rapid association may be facilitated by high intracellular concentrations of binding partners, and cellular factors such as molecular chaperones, helicases, chromatin remodelers, or translation can speed up binding and dissociation.

Equilibration times (t equil ) for different affinities and association rate constants.

*t equil was calculated as five half-lives: t equil = 5t 1/2  = 5 × 0.693/ k equil , where k equil  =  k off  =  K D  ×  k on ( Equation 2 and Figure 3 ).

Implications of insufficient equilibration

Despite the realistic possibility of long equilibration times for biological association events, nearly 90% of the reported incubation times were 1 hr or less ( Figure 1—figure supplement 1B ). As a concrete example, several ‘equilibrium’ dissociation constants reported for CRISPR nucleases, which are well known for tight RNA and/or DNA binding, were determined from incubations of 1 hr or less (e.g. Semenova et al., 2011 ; Westra et al., 2012 ; Westra et al., 2013 ; Sternberg et al., 2014 ; O'Connell et al., 2014 ; Wright et al., 2015 ; Ma et al., 2015 ; Jiang et al., 2015 ; Sternberg et al., 2015 ; Beloglazova et al., 2015 ; Rutkauskas et al., 2015 ; Abudayyeh et al., 2016 ; Supplementary file 2 ). But when target dissociation of these proteins was measured over time, it took many hours ( Strohkendl et al., 2018 ; Richardson et al., 2016 ; Boyle et al., 2017 ; Raper et al., 2018 ), suggesting that equilibration takes much longer than an hour and that the reported K D values based on these short incubation times underestimate the true binding strength. In one striking example, kinetic measurements revealed an equilibration time of >100 hr for the Cas12a complex and an equilibrium constant that was 1000-fold lower than previously reported for the same enzyme at similar conditions after much shorter incubation time ( Strohkendl et al., 2018 ). Insufficient incubation times for tight binders may have also led to underestimation of specificity, a topic of central concern for CRISPR targeting (and for much of biology). Figure 4—figure supplement 1 illustrates how target affinities that differ by two orders of magnitude may appear identical if the incubation time is too short.

An example in which extending the incubation time changed the mechanistic interpretation comes from studies of the signal recognition particle (SRP). Originally, the observation that 4.5S RNA enhanced the assembly of the signal recognition particle (SRP) and SRP receptor led to a proposed mechanism in which the 4.5S RNA stabilized the complex. Subsequently, binding studies extended to longer times revealed that the 4.5S RNA accelerated the otherwise slow SRP/receptor binding and dissociation without affecting the binding affinity ( Peluso et al., 2000 ). Exploring the time dependence of the assembly process changed the mechanistic conclusions: 4.5S RNA could be shown to play a catalytic, rather than stabilizing role in SRP/receptor assembly.

Figure 4—figure supplement 1 illustrates how incubation times that are very far from equilibrium can lead to systematic deviations of the data from the fit to an equilibrium binding equation. While a poor fit is not sufficient to diagnose insufficient equilibration (and, conversely, a good fit does not prove complete equilibration), an inability to fit the data well to a simple binding model provides an important indicator that additional controls are required. Only after simple controls for equilibration and titration (see below) have been performed, should more complex binding models, such as cooperativity, be considered, unless such models are independently supported. Indeed, among the studies in our literature survey omitting one or both key controls, several included poorly fit binding curves. Importantly, graphs of fits of the data to a clearly defined equilibrium binding model should be published along with the K D values when possible, and the quality of the fit over the entire concentration range should always be carefully assessed. In summary, the incubation time must be varied to ensure equilibration, ideally across a range of at least 10-fold. Below we illustrate this control, and the need for it, with experimental results for Puf4 binding to its consensus RNA.

Time dependence of Puf4 binding at 25°C and 0°C

To establish the equilibration time for Puf4 binding to its cognate RNA sequence, Puf4 was mixed, over a series of concentrations, with a trace amount of labeled RNA (in this case, 32 P-labeled; 0.002–0.016 nM) and incubated for a specified time (t 1 ) ( Figure 4A ). The fraction of bound RNA was subsequently determined by non-denaturing gel electrophoresis (see Materials and methods).

Establishing equilibration in affinity measurements.

( A ) Mixing scheme. RNA * : labeled RNA (here—5´-terminally labeled with 32 P). In addition to varying equilibration time t 1 (main text), the time and conditions between adding the loading buffer and loading (t 2 ) are controlled (see Appendix 2—note 2). ( B, C ) Concentration dependence of Puf4 binding at 25°C ( B ) and at 0°C ( C ) after different incubation times. Data were collected at protein concentrations greater than or equal to the concentration of labeled RNA (0.002–0.016 nM, indicating the lower and upper limit of labeled RNA concentration; see section 'Avoid the titration regime' and Appendix 2—note 4).

At 25°C, we observed the same amount of binding with incubations of t 1  = 30 min, 1.5 hr, and 4.5 hr at each protein concentration, providing strong evidence for equilibration even at the shortest time ( Figure 4B ). Consequently, we can proceed to the next key control at this condition, using an incubation time of ≥30 min.

We also present Puf4 binding results at 0°C as these data provide an example of slow equilibration and because many binding studies report incubations on ice to stabilize binding. Indeed, the results at 0°C were very different than those at 25°C. As shown in Figure 4C , Puf4 bound different amounts of RNA in the 30 min, 1.5 hr, and longer incubations. Not until the incubation was extended to 4.5 hr did the extent of binding level off at the lowest Puf4 concentrations—that is, the amount bound was the same after 4.5 and 24 hr. Consequently, equilibration of Puf4–RNA binding on ice requires at least 4.5 hr, and incubation for only 30 min would give an apparent K D value that is seven times higher than after a 24 hr incubation. Moreover, binding at 0°C was so tight that we were only able to obtain part of the binding curve while maintaining the protein concentrations in excess of labeled RNA ( Figure 4C ). The importance of this excess to obtain reliable K D values is described in the next section. In the 0°C case and more generally, it is important to re-assess the equilibration time after establishing that binding is in an appropriate concentration regime, as we demonstrate in later sections. Similarly, changes in conditions, such as salt concentration, temperature or pH, can affect both the affinity and the equilibration time and therefore should be accompanied by confirming that equilibration has occurred.

Avoid the titration regime

The most common approach to measuring affinity is to vary the concentration of one component, while keeping the concentration of the other binding partner constant. However, this experimental design is not always sufficient, as there are two limiting regimes, determined by the concentration of the constant component; only one of these concentration regimes allows the K D to be reliably determined, while the other does not.

In the first, ‘binding’ regime, the concentration of the constant (‘trace’) component, R, is well below the dissociation constant ([R] total  <<  K D for the example in Figure 3 ). In this case, the concentration of the variable component (P in Figure 3 ) that gives half binding is equal to the K D ( Figure 5A ). In the other, ‘titration’ regime, the concentration of the constant component is much greater than the K D ([R] total  >>  K D ) so that essentially all added P is depleted from solution due to binding to R, until there is no more free R left to bind. In this case, the concentration of P that gives half binding does not equal or even approximate the K D . Rather, at high excess of R over the K D , the concentration of P that gives half binding is simply half of the concentration of (active) R molecules—a value that can differ from the sought-after K D by orders of magnitude ( Figure 5B ; Figure 5—figure supplement 1 ).

Two concentration regimes.

( A ) Binding curve for the model in Figure 3 in the ‘binding’ regime—that is, the trace binding partner concentration ([R] total ) is much lower than K D and much lower than [P] total ( Equation 4b ). Here, the K D is simply the protein concentration at which half of the RNA is bound ( K 1/2 , here corresponding to 1 nM). The same simulated binding curve is shown in linear (top) and log (bottom) plots, as both are useful and common in the literature. ( B ) Binding curve in the ‘titration’ regime, simulated for an interaction with a K D value of 0.01 nM and an [R] total of 2 nM. Although the K 1/2 value in this example is identical to the example in Part A, here it does not equal K D , instead exceeding the real K D value by 100-fold.

A potentially useful intermediate regime exists between the two extremes, with limiting component concentrations similar to or in modest excess over the K D . The K D can be determined in this regime by using an appropriate binding equation, although with potential pitfalls (see below).

Distinguishing between concentration regimes

The challenge is that distinguishing between the regimes requires the knowledge of the K D , and consequently it is impossible to know a priori which regime holds. A useful rule of thumb for avoiding the titration regime is to always maintain the concentration of the excess binding partner significantly above that of the trace limiting partner. The reason for this can be gleaned from the equation that describes the fraction of bound RNA for the simple binding scheme of Figure 3 :

Here [P] free is the unbound protein concentration and K D is simply the free protein concentration at which half of the RNA is bound. But while Equation 4a holds universally, in practice we only know the total concentration of P, [P] total —how much we added to the solution—not the free concentration ([P] free ). Therefore, we want to operate under simplifying conditions where [P] free ≈ [P] total so that we can substitute [P] total into Equation 4a to give Equation 4b :

The condition [P] free ≈ [P] total holds true if P is in large excess of RNA across the entire experiment, meaning that only a small fraction of total protein is used up by binding to RNA. Most importantly, this condition must hold for the protein concentration that gives half-saturation to determine the K D ; hence the requirement for the binding regime that the concentration of the limiting component must be << K D . Nevertheless, simply maintaining an excess of protein over the limiting component may not always be sufficient to maintain a binding regime, given the uncertainty often surrounding concentration measurements and even greater uncertainty surrounding active concentrations.

In principle, a more complex quadratic binding equation provides an alternative to working under the [P] free ≈ [P] total assumption, as it explicitly accounts for bound protein:

Indeed, several techniques (most notably ITC) commonly operate outside the binding regime and rely on Equation 5 (or equivalent formulations) for data fitting. Importantly, the quadratic equation is only applicable to the intermediate and binding regimes, but not the titration regime. The reason for this is that at very high concentrations relative to the K D , the contribution of K D in determining the fraction bound ( Equation 5 ) becomes negligible, and as a result a meaningful K D value cannot be extracted from the fit to the binding data. Simulated data in Figure 5—figure supplements 2 and 3 illustrate this limitation. Consequently, even when using Equation 5 , the concentration of the limiting component should be kept to a minimum to avoid the titration regime.

Where does the intermediate regime end and titration begin? The answer depends on the technique and the quality of the data. For ITC measurements, which provide highly precise information for each added binding aliquot, up to 1000-fold excess of the limiting species over the measured K D can be acceptable ( Velázquez-Campoy et al., 2004 ). However, in most other cases, this limit is much lower. Simulations in Figure 5—figure supplement 3 suggest that up to ~10-fold excess consistently allows for reasonably well-defined K D values in the presence of typical binding data, and up to 100-fold excess can be useful for data with minimal noise. In contrast, performing the experiments in the binding regime (fit with Equation 4b ) yields well-defined K D values even with substantial noise in the data ( Figure 5—figure supplement 3 ).

Implications of the titration regime

Of the 100 literature studies we surveyed, most (65%) determined K D values under the assumption of the binding regime, by using Equation 4b or equivalent analysis. Nevertheless, the required condition that the limiting species concentration be << K D was not always supported. One-third of the studies using Equation 4b (n = 21) reported K D values that were comparable to (<10-fold excess) the concentration of the trace component, including nine studies in which the reported K D was indistinguishable from (within ~2-fold) or even below the stated trace component concentration, consistent with an intermediate or even titration regime ( Figure 1—figure supplement 2 ).

The implication in all these cases is that the reported K D values may underestimate the real affinities. Unfortunately, it is difficult to determine the extent of this underestimation post-factum without further experimental controls. To understand why, recall from the example in Figure 5B that in the titration regime the midpoint of the binding curve only reflects ~half the concentration of the limiting species, which sets a lower limit to the apparent K D derived from Equation 4b , even if the real K D is much lower. Conversely, if the midpoint of the binding curve (and the reported K D in the above cases) is approximately the same as the limiting component concentration (allowing for some uncertainty in the concentration), the real K D could be anything below this value, from several-fold to many orders of magnitude less. As with insufficient incubation, systematic deviations of the data from the fit to Equation 4b can be a clear indicator that the apparent K D is limited by titration, but a good fit should not be considered sufficient to prove the binding regime, as experimental uncertainties and other causes can mask deviations.

High-affinity interactions are most susceptible to titration, a corollary of the simple fact that for very low K D values it becomes increasingly difficult to maintain concentrations much lower than K D while still allowing for detection. Since CRISPR nucleases represent some of the most widely studied high-affinity binders, we surveyed a sample of studies to determine the concentration regime under which the reported K D values were measured ( Supplementary file 2 ). Of the 15 studies, the majority (13, or 90%) assumed the binding regime in their analysis, indicated by the use of Equation 4b or equivalent. However, only two of these studies (15%) reported using labeled DNA or RNA concentrations considerably below the apparent K D , and in five cases the lowest reported K D was essentially identical to the labeled RNA or DNA concentration (within ~2-fold), consistent with possible titration.

Importantly, because relative affinities are typically based on the tightest binders, titration effects on the ‘wild-type’ substrate measurements can distort all specificity (relative affinity) values that are based on it. Figure 5—figure supplement 4 illustrates an example, in which two substrates with a 100-fold difference in affinity appear to have identical or near-identical affinities when titration is not controlled for.

Given the impossibility of designing experiments for the binding regime a priori, without knowing the affinity, it is important to rule out titration empirically. Thus, analogously to varying time to establish equilibration, we strongly recommend systematically varying the concentration of the limiting species to establish the binding regime (or, with use of Equation 5 , the intermediate regime). The hallmark of a valid K D is that it is not affected by varying the concentration of the limiting component, whereas a titration regime would result in concentration-dependent apparent K D values. At a minimum, this control should always be performed when the measured K D value is comparable to the concentration of the limiting component ( Equation 4b ), or when Equation 5 yields poorly defined apparent K D values or values much lower than the limiting concentration. Below we demonstrate the titration control for Puf4 affinity measurements.

RNA concentration dependence of Puf4 binding at 25°C and 0°C

We systematically varied the labeled RNA concentration in Puf4 binding experiments at 25°C and 0°C, to illustrate the binding and intermediate regimes, respectively. Figure 5—figure supplement 5 provides a schematic description of the two regimes to help build the reader’s intuition.

At 25°C, the Puf4 binding curves were identical across a nine-fold range of RNA concentrations ( Figure 6A,B ), and the data were well described by Equation 4b . From the constancy of the binding curves in Figure 6B , we can conclude that the binding regime holds for Puf4 at 25°C, and thus that the observed K D value of 120 pM obtained from Equation 4b represents a true equilibrium constant. As expected for the binding regime, the measured K D is higher than the RNA concentrations (120 pM vs 2–18 pM).

Varying the concentration of the 'trace' binding partner.

( A ) Mixing scheme, as in Figure 4A but now with a series of labeled RNA concentrations. ( B ) Puf4 binding to different concentrations of 32 P-labeled RNA at 25°C. For simplicity, only the lower limits of RNA concentration are indicated; the corresponding upper limits were 15–140 pM RNA (see Materials and methods and Appendix 2—note 4). Incubation time t 1 was 0.5 hr, as established in Figure 4B . ( C ) Puf4 binding to different concentrations of 32 P-labeled RNA at 0°C. Lower limits of labeled RNA concentration are indicated. Incubation time t 1 was 40 hr. Note that these data are not fit well by Equation 4b , which assumes [R*] total << K D (solid lines). Quadratic fits, which do not assume negligible RNA concentration, are shown in dashed lines ( Equation 5 ). ( D ) Effect of RNA concentration on apparent K D ( K D app ) at 0°C. Red symbols indicate K D app values from a hyperbolic fit ( Equation 4b and solid lines in C ) and grey symbols indicate K D app values from fits to the quadratic equation ( Equation 5 ). The error bars denote 95% confidence intervals, as determined by fitting the data to the indicated equation in Prism 8.

The situation is different at 0°C ( Figure 6C ). Here, varying the labeled RNA concentration revealed divergent binding curves and a pronounced dependence of apparent affinity (determined by fitting the data to Equation 4b ) on the concentration of RNA, the constant component ( Figure 6C,D ). Moreover, the fits of the data to Equation 4b (solid lines in Figure 6C ), which assumes [P] free ≈ [P] total , were poor, increasingly so for higher RNA concentrations. These data are indicative of protein depletion due to binding to labeled RNA. The apparent K D values vary by five-fold across the 30-fold range of RNA concentrations used ( Figure 6D , red circles), and even greater discrepancies would arise at higher RNA concentrations ( Figure 5—figure supplement 1 ). Consequently, only an upper limit of the real affinity can be extracted from these data ( K D  ≤ 2.3 pM, based on the fit value at the lowest RNA concentration used).

To address the limitation in our 0°C data we could, in principle, lower the concentration of labeled RNA even further, until the labeled RNA concentration is << K D and until an RNA concentration-independent regime is established. But this is difficult when binding is very tight, as a limit is set by the sensitivity of the technique used. In our case, at ~1 pM 32 P-labeled RNA we are already near the limit of reliable detection. If the concentration of the trace component cannot be lowered further, a more sensitive approach can sometimes be found. Kinetic approaches are particularly suitable for tight binders (see Appendix 1), or one can report an upper limit of the K D . In some cases, increasing the salt concentration or other changes to the solution or binding partners can be used to weaken binding to make it easier to obtain affinities at higher concentrations of the labeled species; this approach can be especially valuable if one is primarily interested in the relative affinities of multiple ligands ( Altschuler et al., 2013 ).

As noted earlier, the quadratic binding equation enables K D determination for binding reactions in the intermediate regime. The quadratic equation provides a good fit to the 0°C data ( Figure 6C , dashed lines) and yields uniform and well-defined K D values of ~1.9 pM across the different RNA concentrations, consistent with an intermediate (rather than titration) regime ( Figure 6D , grey circles). The same K D value was obtained from kinetic experiments, providing independent support for and confidence in this determination (Appendix 1).

In summary, we want to use the binding regime whenever possible, as it allows for the most straightforward and reliable K D measurements. It is necessary to avoid the titration regime and caution is required in the intermediate regime. In practice, varying the concentration of both components is an essential control for ruling out titration, ruling out other potential artifacts, and ensuring the measurement of valid dissociation constants.

Re-evaluating the equilibration time at 0°C

In the previous section, we mentioned the need for re-evaluating the equilibration time for Puf4 binding at 0°C after a binding regime was established. In principle, after determining sufficiently low RNA concentration for the binding regime, one could vary the incubation time again, as done in Figure 4 . In our case, we used the shortcut defined in Equation 2 and instead determined the upper limit of the equilibration time by measuring the k off at 0°C (Appendix 1; see also Appendix 2—note 1 for precautions when applying this shortcut). These measurements revealed an equilibration time of 30 hr (five half-lives), far above the typical incubation times of 1 hr or less ( Figure 1—figure supplement 1 ).

Dependence of binding affinity on conditions

The 100-fold difference in Puf4 affinity between 0°C and 25°C underscores the important point that the equilibrium dissociation constant is only a constant value at a given set of conditions, and that the affinity can change dramatically when the conditions (temperature, salt, pH) are changed. This dependence on conditions should always be considered when comparing literature values or when applying in vitro results to biology.

Test K D by an independent approach

Even when no challenges are encountered, as in the case of Puf4 binding at 25°C, it is a good idea to determine the K D by a second approach to ensure that the measurement is not biased by experimental artifacts or idiosyncrasies of a particular technique. This is especially important when using a secondary readout (vs. a direct approach) such as native gel shift or nitrocellulose filter binding, where major loss (or gain) of bound complex can potentially occur between the equilibration and detection steps (see below and Appendix 2—note 2).

Of course, there are many approaches to carrying out equilibrium binding measurements one can choose from (e.g. Velázquez-Campoy et al., 2004 ; Wong and Lohman, 1993 ; Eftink, 1997 ; McDonnell, 2001 ). Here, we used a kinetic approach for independent K D determination for Puf4 at 25°C and 0°C, as described in Appendix 1. Kinetic measurements provide an information-rich alternative and complement to the equilibrium measurements and are often simple to carry out provided they fall within a measurable time range ( Pollard, 2010 ; Hulme and Trevethick, 2010 ; Sanders, 2010 ; Pollard and De La Cruz, 2013 ). In case of Puf4, the affinities determined by kinetic measurements were within two-fold of those from equilibrium determinations, strongly supporting their accuracy.

Determine the fraction of active protein

The amount of bound ligand is determined not by the total protein concentration but by the concentration of total active protein. If 90% of the protein is damaged due to misfolding, aggregation, degradation or, for example, inactivated by phosphorylation at the binding interface, then the observed affinity will be that for only 10% of the total protein present—and will be ten-fold higher than the actual K D value. Moreover, if the binding-competent protein concentration is much lower than the total and therefore much closer to the limiting component concentration than expected, the binding regime may not be maintained, leading to even greater discrepancies between the real and observed K D . As a common cause of non-active or less active protein is aggregation, determining the monodispersity of the protein following purification is advisable ( Altschuler et al., 2013 ).

In addition, we recommend, when possible, a titration experiment to determine the fraction of binding-competent protein ( Altschuler et al., 2013 ). Here, a concentration of ligand that is much greater than the measured K D is intentionally used and the protein concentration is varied by approximately an order of magnitude above and below the ligand concentration. To ensure accurate ligand concentration and to prevent excessive signal (if labeled ligand is used), the trace labeled ligand should be mixed with a large excess of identical unlabeled molecule at a known concentration. Assuming that the stoichiometry of the bound complex is known and that the ligand is 100% active, the breakpoint in fraction bound versus the ratio of protein to ligand indicates the amount of active protein ( Figure 7 ). For example, for a 1:1 complex, a breakpoint at a protein:RNA ratio of 2.0 suggests that half of the protein is active. In Figure 7 , the ratio of 1.3 suggests that the Puf4 preparation is 75% active (0.75 = 1/1.3). Consequently, the apparent K D values determined in the previous sections should be multiplied by the active protein fraction (which ranged from 0.75 to 0.90 for Puf4) to determine the final K D value. In an alternative approach, the titration data could be fit to a quadratic equation, with a coefficient used to represent the active protein fraction ( Figure 7—figure supplement 1 ).

Measuring the fraction of active protein by titration.

The fraction of active protein is derived from the breakpoint, that is, the intersection of linear fits to the low and high-Puf4 concentration data. See Figure 7—figure supplement 1 for an alternative strategy using Equation 5 .

A limitation of the titration experiment is that it assumes the constant component to be 100% active, which may not always be the case, especially in the case of protein-protein interactions. Therefore, one should ensure, to the extent possible, maximum purity of both binding components. Importantly, one should always make clear whether experiments were carried out to determine ‘fraction active’.

The case of no observed binding

Researchers often conclude that there is ‘no binding’—that ‘X does not bind to Y’. Typically, the underlying experimental observation is an absence of observed binding up to a certain protein (or ligand) concentration. Therefore, one should report a lower limit for the dissociation constant ( K D ), rather than draw an absolute conclusion of ‘no binding’. But even an accurate lower limit often requires additional experiments, because the absence of observed binding—say in a gel shift, filter binding, or pull-down experiment—can arise either because there is no significant binding or because the complex does not withstand the assay conditions ( Pollard, 2010 ). While this objection may seem like a technicality, there are many instances where known binders do not give a gel shift or filter binding.

Immuno-precipitation and pull-down assays are pervasive in current biological investigations and are often interpreted in terms of ‘binding’ or ‘no binding’. But the reality of the interpretation of these experiments—and the reality of molecular interactions—is more nuanced ( Pollard, 2010 ). A ligand with the same affinity, slightly lower affinity, or even higher affinity than another ligand with demonstrated binding can incorrectly be concluded to ‘not bind’.

Consider, for example, an RNA pull-down with an RNA binding protein with K D  = 10 −9 M and k on  = 10 8 M −1 s −1 ; this gives k off  = 0.1 s −1 or a half-life for dissociation of ~10 s. If the washing steps following a pull-down take 30 s, only ~10% of the complex is expected to remain. If the affinity is 10-fold weaker ( k off  = 1 s −1 ), then no detectable complex is likely to remain after 30 s of washing (10 −13 of the starting amount). Further, if another RNA ligand binds with the same affinity, but 10-fold slower (and thus also dissociating 10-fold slower; k off  = 0.01 s −1 , half-life of ~100 s), most (~75%) of the complex will remain after the 30 s washing steps despite an identical K D to the first ligand. In addition, the limited dynamic range of visual readouts of gels that are often used to evaluate pull-down experiments increases the danger of misinterpretation or overinterpretation of these experiments.

Overall, observing binding in pull-downs and related experiments is a complex function of the experimental components and conditions. This doesn’t at all mean these experiments should not be done—they often provide critical clues and insights into biology. But, for these and all experiments, we need to keep in mind the nature of the assay, and thus what can and cannot be concluded from the experiment.

Whether binding is absent or not detected can be tested by using approaches that directly report on the equilibrium between bound and unbound components in solution (e.g. ITC, fluorescence anisotropy, and other fluorescence-based techniques), as opposed to indirect approaches like native gel shift and pull-downs that are based on physically separating bound and unbound components, so that unstable complexes may fall apart prior to the detection step. Nevertheless, direct approaches also have limitations. For example, fluorescence intensity or FRET (Förster resonance energy transfer) is limited at high concentrations by inner filter effects, and ITC will miss binding events when the release (or uptake) of heat upon binding is too small (i.e. the binding enthalpy is too small).

A simple way to test whether binding occurs when there is no binding signal is to carry out a competition experiment. If the ligand is bound but not detected in an approach such as native gel shift or filter binding, it will still lessen binding of another ligand for which there is an established signal. The amount lessened depends quantitatively on the K D values and concentrations of each ligand, given sufficient time for equilibration. A competition experiment to obtain the K D value for a weak RNA substrate of Puf4 is shown in Appendix 3, along with the binding scheme and equation to determine the K D value.

Competition binding measurements can also have a practical benefit; after an initial K D is determined for a labeled substrate, K D values for additional substrates can be determined by competition without labeling each substrate ( Hulme and Trevethick, 2010 ; Sanders, 2010 ; Ryder et al., 2008 ).

Given the increasingly multi-disciplinary nature of research, scientists are increasingly venturing into disciplines outside their expertise. Our goal is to support these valuable efforts by enabling both experts and non-experts in thermodynamics to get the most out of their binding experiments, and to help them evaluate work by others, published or under review for publication.

While the number of steps described to obtain reliable equilibrium data may initially seem daunting, the accompanying experimental illustrations and guides can transform an opaque process into one that is readily understandable and can be carried out in a straightforward, stepwise fashion by researchers from varied backgrounds.

We found it useful to develop and use an Equilibrium Binding Checklist to organize our approach and findings. We provide a template of such a checklist, along with completed examples in Appendix 4 ( Appendix 4—figure 1 , 2 , 3 ). We expect that many readers will find these valuable.

There has been much discussion about problems with reproducibility and rigor in the scientific literature ( Landis et al., 2012 ; Plant et al., 2014 ; Nature, 2013 ; Nosek and Errington, 2017 ; Koroshetz et al., 2020 ). Historically, a powerful means to ensure reliability of published data has been to develop community standards. Reporting guidelines have been successfully adopted by journals in a variety of fields, including structural biology ( Berman et al., 2000 ), enzymology ( http://www.beilstein-institut.de/en/projects/strenda/guidelines ), organic synthesis (e.g. http://pubs.acs.org/page/joceah/submission/ccc.html ), and many others, and new standards, guidelines and databases are continually being devised (see https://fairsharing.org/ for a curated list). We encourage journals to adopt analogous standards for reporting binding measurements. Contingent on implementation of such standards, we ultimately envision a well-curated and well-documented quantitative database that is routinely used to build and test models for individual molecular interactions and for cellular and molecular networks.

Survey of published equilibrium binding measurements

We surveyed 100 papers, including 66 papers from the list of quantitative RNA/protein studies assembled by the Liu lab ( Yang et al., 2013 ) and 34 additional studies reporting K D and apparent K D values for RNA/protein interactions ( Supplementary file 1 ). To confirm that our survey was not biased, we also scored 20 publications from a single PubMed search for ‘RNA protein binding dissociation constant’, after confirming that they reported K D values for RNA/protein binding. Four of the 20 papers also appeared in the above list. The fractions of papers controlling for equilibration and/or titration were similar to those in the main survey ( Figure 1 ): 30% of the 20 papers controlled both for equilibration and titration, 15% controlled for neither, 50% only controlled for titration and 5% only controlled for equilibration.

Equilibration was evaluated as follows. If a study reported systematically varying the incubation time, it was counted as controlled for equilibration. If dissociation kinetics were measured in addition to performing equilibrium measurements (n = 3), the study was scored as equilibration-controlled, but only if the reported incubation time was at least three half-lives based on the reported k off , and only if the kinetic and equilibrium experiments were performed at the same conditions (n = 1). Studies exclusively using approaches that intrinsically monitor the binding progress (ITC, SPR, biolayer interferometry [BLI]) also were counted as equilibration controlled. However, if several approaches were used in a given study to determine affinities for distinct binding interactions and/or conditions, and if for at least one approach time was not varied, the study was scored as not equilibration controlled. Some exceptions where equilibration can be reasonably assumed are noted in Supplementary file 1 .

To generate Figure 1—figure supplement 1 , we used the incubation times reported for non-equilibration controlled binding experiments. If a narrow range of times (e.g. 15–20 min, 45–60 min; n = 2) was indicated, this was not counted as systematically varying time and the longer time was used for Figure 1—figure supplement 1 . If only a lower limit of the incubation time was reported (e.g. ‘at least 30 min’; n = 1), this lower limit was used for Figure 1—figure supplement 1 . If two sequential incubations were performed at different temperatures (e.g. ‘10 min at room temperature and 10 min at 4°C’, n = 4), the total incubation time was used for the purposes of the survey. However, since affinity is condition-specific, only equilibration at a constant temperature can yield meaningful K D values, and two-temperature incubations should be avoided.

To evaluate if titration was controlled for, first, we confirmed if the concentration of the limiting species was systematically varied to determine effects on K D (n = 5); these studies were counted as titration controlled. If a study reported a range of concentrations of the limiting species, without stating that the effects on K D were assessed, we did not count this as a titration control, as in practice such a range typically only indicates optimization of radioactive/fluorescent signal to account for radioactive decay and/or varying labeling efficiencies. For the remaining studies, we asked if Equation 4b (which assumes the binding regime) or Equation 5 (which also allows for the intermediate regime) was used to fit the data. If no equation was indicated, or if the midpoint of the binding curve/gel signal was used to determine the K D , or if linear transformation was used in lieu of the hyperbolic fit, we counted the study as using Equation 4b . For studies using Equation 4b , we asked if the lowest apparent K D value was in at least 10-fold excess over the limiting component concentration, in which case we counted the study as titration controlled. If a range of limiting component concentrations was reported, we used the lowest value. If only the amount (not concentration) of the limiting species was reported, the concentration was calculated based on the provided volume or, if not indicated, based on a 10 µL reaction volume; nevertheless, binding equilibria depend on concentrations, not amounts, and concentrations, in units of ‘M’, should always be indicated. If Equation 5 was used (incl. all ITC measurements), we counted the study as titration controlled, unless the reported K D was more than 1000-fold below the limiting species concentration (corresponding to a cutoff typically used in ITC [ Velázquez-Campoy et al., 2004 ]). For simplicity, we assumed that all SPR/BLI measurements (where the concentration of the immobilized species is difficult to estimate and not reported) were titration controlled; nevertheless, we emphasize the importance of explicitly reporting controls for mass transport in SPR measurements ( Myszka, 1999 ). If multiple approaches were used, but at least in one approach titration was not controlled for according to the above criteria, the study was scored as not titration controlled, unless the affected values were corroborated by a titration-controlled approach in the same study.

If no details on the incubation time and/or the concentration of the limiting reagent were provided, but instead a previous study was cited (‘as described’, n = 4), the information for the above evaluation was obtained from the cited study. This included two cases in which the authors had performed rigorous equilibration and titration controls in their previous referenced work.

Puf4 purification

The RNA-binding domain (residues 537–888) of S. cerevisiae Puf4 was cloned into a custom pET28a-based expression vector in frame with an N-terminal 6X His-tag and a C-terminal SNAP tag (New England Biolabs, Ipswich, MA). The construct was transformed into E. coli protein expression strain BL21 (DE3) and protein expression was induced at an OD600 of 0.6 with 1 mM IPTG at 20°C for ~20 hr. Induced cells were harvested by centrifugation at 4500 × g for 20 min. Cell pellets were re-suspended in Buffer A (20 mM HEPES-sodium (HEPES-Na)), pH 7.4, 500 mM potassium acetate (KOAc), 5% glycerol, 0.2% Tween-20, 10 mM imidazole, 2 mM dithiothreitol (DTT), 1 mM phenylmethylsulfonyl fluoride (PMSF) and cOmplete, Mini, protease inhibitor cocktail (Roche Diagnostics GmbH, Mannheim, Germany) and lysed four times using an Emulsiflex (Avestin, Inc, Ottawa, ON, Canada). The lysate was clarified by centrifugation at 20,000 × g for 20 min, nucleic acids were precipitated with polyethylene imine (0.21% final concentration) at 4°C for 30 min with constant stirring and pelleted by centrifugation at 20,000 × g for 20 min. The supernatant was loaded on a Nickel-chelating HisTrap HP column (GE Healthcare, Pittsburgh, PA). Bound protein was washed extensively over a shallow 10–25 mM imidazole gradient and eluted over a linear 25–500 mM gradient of imidazole. Peak Puf4 protein fractions were pooled and desalted into Buffer B (20 mM HEPES-Na, pH 7.4, 50 mM KOAc, 5% glycerol, 0.1% Tween-20, 2 mM DTT) using a desalting column. The His-tag was cleaved by overnight incubation with His-tagged TEV protease at 4°C, and the protein was purified on a HisTrap HP column. The flow-through was desalted into Buffer B and loaded on a HiTrap Q HP column (GE Healthcare) and washed extensively with Buffer B to remove any bound RNA. Protein was eluted over a linear gradient of potassium acetate from 50 to 1000 mM. Protein fractions were pooled and desalted into Buffer C (20 mM HEPES-Na, pH 7.4, 100 mM KOAc, 5% glycerol, 0.1% Tween-20 and 2 mM DTT), concentrated and diluted two-fold with Buffer C containing 80% glycerol for final storage at −20°C. UV absorbance spectra indicated that the protein was free from significant RNA contamination (<1 RNA base per protein).

RNA 5´-end labeling

Puf4_HO RNA (AUGUGUAUAUUAGU; Integrated DNA Technologies (IDT), Coralville, IA; 5 µM) was labeled with equimolar [γ- 32 P] ATP (Perkin Elmer, Inc, Boston, MA) using T4 polynucleotide kinase (Thermo Fisher Scientific, Vilnius, Lithuania) and purified by non-denaturing gel electrophoresis (20% acrylamide). The RNA was eluted into TE buffer (10 mM Tris-HCl, pH 8.0; 1 mM EDTA) at 4°C overnight, and the lower limit of eluted RNA concentration, assuming no unlabeled RNA, was determined by scintillation counting and calibration against the specific activity of the [γ- 32 P] ATP stock used for labeling. The upper limit of RNA concentration was calculated from total RNA input and the elution buffer volume, assuming a 100% yield.

Equilibrium binding measurements

All reactions were performed in a binding buffer containing 20 mM HEPES-sodium or HEPES-potassium buffer, pH 7.4, 2 mM magnesium chloride (MgCl 2 ), 100 mM KOAc, 2 mM DTT, 0.2% Tween 20, 5% glycerol, 0.1 mg/ml BSA, at 25 or 0°C, as indicated. The protein and labeled RNA dilutions were prepared in binding buffer at two-times the indicated concentration and were kept on ice until the binding reactions were initiated by mixing 10 µL of protein with 10 µL of labeled RNA. The pipette tips used for mixing and aliquoting the 0°C reactions were kept on ice. The labeled RNA concentrations and incubation times are indicated in the individual figure legends. Following the incubation, 7.5 µL aliquots were moved to 5 µL of ice-cold loading buffer containing 6.25% Ficoll PM 400 (Sigma-Aldrich, Saint Louis, MO), 0.075% bromophenol blue (BPB), and 2.5 µM unlabeled Puf4_HO RNA. The unlabeled RNA in the loading buffer prevented additional association to the labeled RNA from occurring during sample loading (Appendix 2—note 2). Control experiments indicated negligible re-equilibration in loading buffer (t 1/2 ≥ 3 hr in three independent measurements), consistent with the slow dissociation rate constant measured in binding buffer at 0°C (Appendix 1). All samples were loaded on the gel within 20 min from mixing with the loading buffer. Non-denaturing acrylamide gels (20%) were pre-run for at least 1 hr at 42 V/cm constant voltage, 4–6°C with 0.5x TBE buffer (50 mM Tris, 42 mM boric acid, 0.5 mM EDTA•Na 2 , pH 8.5–8.6 final) using a circulating cooling system. Aliquots (7.5 µL) were carefully loaded on continuously running gels and separated for 45–90 min. (Extreme caution must be exercised at this step; see, e.g. https://ehs.stanford.edu/reference/electrophoresis-safety for electrical safety hazards.) The gels were dried and exposed to phosphorimager screens, scanned with a Typhoon 9400 Imager and quantified with TotalLab Quant software (TotalLab, Newcastle-Upon-Tyne, UK). Fitting was performed with KaleidaGraph 4.1 (Synergy Software, Reading, PA; RRID: SCR_014980 ).

The K D values in Table 2 indicate the average and standard error from five independent equilibrium experiments (25°C). For 0°C measurements, K D (hyperbolic) indicates the upper limit determined using Equation 4b at the lowest RNA concentration ( Figure 6C,D ); K D (quadratic) indicates the average and standard error of K D values determined with Equation 5 at the four RNA concentrations shown in Figure 6C,D .

Summary of equilibrium and kinetic measurements of Puf4 affinity.

*The values have been normalized by active protein fraction (75–90%). K D (hyperbolic) and K D (quadratic) refer to values derived from fits to Equation 4b and Equation 5 , respectively. Errors are defined in Materials and methods.

Kinetic measurements

Measurements of k off (Appendix 1) were performed by incubating the indicated concentrations of Puf4 with trace concentration of labeled Puf4_HO RNA for 10 min at 25°C or 0°C in the binding buffer described in Equilibrium binding measurements . Labeled RNA concentrations were 0.04–0.5 nM, corresponding to the lower and upper limits, as defined in RNA 5´-end labeling . Dissociation was initiated by transferring the binding reaction to 2.5x volume of unlabeled chase in binding buffer. The chase RNA concentrations in the final reaction were 250 nM and 1000 nM. At various times, 7.5 µL aliquots were moved to 5 µL of ice-cold loading buffer containing 6.25% Ficoll PM 400% and 0.075% BPB, and 7.5 µL aliquots were loaded on a pre-run, continuously running 20% non-denaturing gel at 4–6°C. All pipette tip boxes and solutions used for the 0°C reactions were kept on ice. The chase solution for the 25°C reaction was pre-warmed in a 25°C water bath for 10 min before initiating the dissociation reaction. All time courses were fit to single exponentials using KaleidaGraph 4.1.

The effectiveness of unlabeled Puf4_HO RNA chase was tested by pre-incubating 10 nM Puf4 with 100–1000 nM unlabeled RNA (final concentrations) for 12 min at 25°C before adding trace amount of labeled Puf4_HO RNA (0.04–0.4 nM). The fractions of bound labeled RNA ranged from 0.01 (1000 nM) to 0.1 (100 nM), compared to 0.95 fraction bound in the absence of chase, confirming the effectiveness of the chase.

The k off values reported in Table 2 indicate the average and standard error from two replicate experiments (25°C) or the average and standard error across different concentrations in a single experiment (0°C).

Values of k on were determined by mixing 40 µL each of trace labeled RNA solution (0.004–0.05 nM) and varying dilutions of Puf4. At varying times, 7.5 µL aliquots were transferred to 5 µL of ice-cold loading buffer containing 6.25% Ficoll PM 400, 0.075% BPB, and 2.5 µM unlabeled Puf4_HO RNA and loaded on a 20% gel as above. The protein and RNA solutions were pre-incubated at the reaction temperature (0°C or 25°C) before mixing, and ice-cold tips were used for the 0°C reactions. To control for titration by labeled RNA at the low protein concentrations used, at 0°C, the equilibration rate constants were also measured at three-fold higher labeled RNA concentration, giving consistent rate constants within 1.1–1.3-fold (Appendix 1).

The k on values reported in Table 2 are the slopes and standard errors of linear fits to observed rate constants from two replicate experiments (25°C) or a single experiment (0°C). The k on values were corrected for the active protein fraction.

Measuring the fraction of active protein by titration

Unlabeled Puf4_HO RNA (10 or 100 nM) was incubated for 30 min with varying Puf4 concentrations in the presence of trace labeled Puf4_HO RNA (0.06–0.4 nM); the labeled and unlabeled RNA was pre-mixed before adding Puf4. The fraction bound RNA was determined as described in Equilibrium binding measurements .

Competition measurements

Trace labeled Puf4_HO RNA (0.02–0.19 nM) was equilibrated with 0.4 nM or 1.2 nM Puf4 and diluted two-fold into solutions containing varying concentrations of unlabeled competitor RNA (CGUAUAUUA; IDT). The reactions were incubated at 25°C for the indicated time, followed by transfer of 7.5 µL aliquots to 5 µL ice-cold loading buffer (6.25% Ficoll PM 400, 0.075% BPB, and 2.5 µM unlabeled Puf4_HO RNA). The samples were loaded immediately on a continuously running native acrylamide gel (4–5°C). The curves were fit to Equation 9 , as described in Appendix 3.

Simulations

The simulated data in Figure 5 were generated by using Equation 4b (panel A) and Equation 5 (panel B) to calculate the fraction of bound RNA at each total protein concentration. In Figure 5—figure supplements 1 , 2 , 4 and 5 , Equation 5 was used to calculate fractions bound at each protein and ligand concentration. In Figure 4—figure supplement 1 , Equation 4b was used to determine the fraction of ligand bound at each protein concentration at equilibrium, assuming [P] = [P] total . This equilibrium value was then used as an amplitude (A) term in the single-exponential equation shown in Figure 2 to determine the fraction of bound ligand at each time point t: Fraction bound(t) = A × ( 1   −   e − t   ×   k e q u i l ) = F r a c t i o n   b o u n d ( e q u i l i b r i u m ) × ( 1   −   e − t   ×   ( k o n [ P ]   +   k o f f ) ) .

The simulated data in Figure 5—figure supplement 3 were generated as follows. First, Equation 5 was used to calculate the expected fraction of bound RNA at equilibrium for each [R] total and [P] total indicated in the figure. Two-fold serial dilution of protein was chosen as representative of a typical equilibrium binding experiment. In the case of 0.001 nM R total , Equation 4b was used instead to calculate the expected fraction bound, as this condition satisfies the [P] free = [P] total assumption. Random noise in fraction bound was then generated around each predicted data point by sampling from a normal distribution with the indicated standard deviation, using the scipy and random packages in Python. Ten binding series were generated this way for each condition and each noise level. These datasets were then individually fit to Equation 5 (or Equation 4b in the case of 0.001 nM R total ) in Prism 8 (GraphPad Software, LLC, San Diego, CA; RRID: SCR_002798 ), with the equations modified to include amplitude (A) and y axis offset (O) terms:

To facilitate fitting to Equation 6 , [R] total was constrained to the known value, and the K D was constrained to positive values only, with the real affinity (0.1 nM) used as an initial estimate.

Kinetic approach to affinity determination

An equilibrium dissociation constant is the ratio of dissociation and binding rate constants ( K D = k off k on ), and thus can be determined by directly measuring these rate constants. Because k off is concentration-independent, it is the easiest and most robust parameter to measure. Appendix 1—figure 1 describes the steps for this measurement. After forming the complex between protein and a trace concentration of labeled RNA, a large excess of unlabeled RNA is added to the reaction. The role of the unlabeled 'chase' RNA is to bind any dissociated protein before it can re-bind the labeled RNA. Thus, the chase RNA must be in large excess of the protein concentration and must be a tight binder. The probability of rebinding can be further reduced by diluting the reaction mixture. At specified time points (t 2 ; Appendix 1—figure 1A ), the amount of remaining complex can be determined by native gel electrophoresis or another approach. Although in principle k off can be determined from a single binding reaction, as in any experiment, reliability is best established with several controls (see Appendix 2—note 6).

Appendix 1—figures 1B, C show dissociation of RNA from Puf4 at 25°C and 0°C, respectively. As expected for simple dissociation with an effective chase, the curves are well fit by a single exponential curve with endpoints that approach zero and the rate constant is independent of protein and chase concentrations. A critical control is to test that the k off is not affected by the chase concentration. This is because in some contexts of multi-step dissociation processes, the chase can facilitate dissociation (e.g. Hadizadeh et al., 2016 ).

Kinetics of Puf4 dissociation.

( A ) Mixing scheme for measuring the dissociation rate constant. After equilibration of a saturating or near-saturating concentration of Puf4 protein with a trace concentration of labeled RNA (t 1 ), a large excess of unlabeled RNA is added, with concomitant dilution of the binding reaction to prevent rebinding after dissociation. ( B–C ) Time dependence of Puf4 dissociation from its consensus RNA at 25°C (C; k off = (0.014 ± 0.003) s −1 ) and at 0°C (D; k off = (2.92 ± 0.17) × 10 −5 s −1 ).

To measure the association rate constant one can use a different type of chase experiment that we refer to as a ‘ k on chase’ ( Appendix 1—figure 2A ; Hertel et al., 1994 ). Here the time that the protein and labeled RNA are incubated together is varied (t 1 ) and the amount bound after each time t 1 is determined by native gel shift or another assay. To ensure that the amount bound accurately reflects what has occurred during t 1 and not subsequently, a chase is added to prevent, or quench, additional binding, analogous to the k off experiment above ( Appendix 1—figure 1A ). The time t 2 is kept constant, removing potential variability from dissociation subsequent to the binding reaction during t 1 (see Appendix 2—note 2).

Kinetics of Puf4/RNA association.

( A ) Mixing scheme for measuring association rate constants. ( B, C ) Time dependence of Puf4 association to its consensus RNA at 25°C ( B ) and 0°C ( C ). ( D, E ) Determination of k on from the slope of the Puf4 concentration dependence of equilibration rate constants in parts B and C, respectively (circles). The k off values from Appendix 1—figure 1 are also shown (diamonds) to illustrate the correspondence between the y-intercept and k off ( Equation 1 ). Panels D and E show results from two and one independent experiments, respectively (error bars in E correspond to averages from measurements at two different labeled RNA concentrations).

The observed association rate constant is expected to vary with protein concentration—that is, it is first order in protein ( Figure 3 )—so it is important to carry out these measurements across a wide range of protein concentrations. Appendix 1—figures 2B, C show the data obtained at 25°C and 0°C, respectively. Each individual time course is well fit by an exponential, and Appendix 1—figures 2D, E plot the rate constants obtained from these time courses versus Puf4 concentration, giving the expected linear dependencies, the slopes of which correspond to k on ( Appendix 1—figure 2D ).

The plot in Appendix 1—figure 2D also shows a clear, non-zero intercept. While not intuitive, this intercept arises because the ‘ k on ’ experiment actually measures the rate constant to reach equilibrium, k equil , where k equil equals k on [P] + k off ( Equation 1 ) so that the slope gives k on and the intercept gives k off ( Appendix 1—figure 2D ). There is good agreement between the intercepts and the independently measured k off values in our experiments ( Appendix 1—figures 2D, E , diamonds). It is generally preferable to compare directly-obtained k off values to these intercepts, rather than relying on the intercept for k off determination, as this allows independent tests of data consistency and accuracy.

The K D values obtained in the equilibrium and kinetics experiments agree within two-fold, which is reasonable experimental agreement in our experience ( Table 2 ). Such agreement strongly supports (although does not prove) that both methods are giving correct binding constants.

Additional considerations

Note 1: k off as shortcut for establishing equilibration times.

Measuring k off provides a fast and dependable way to determine the equilibration time needed for simple two-state binding reactions ( Figure 3 and Equation 2 ). However, we still recommend monitoring the time course of complex formation in the presence of ligand, in case binding is more complex than a single step, for example involving an additional slow conformational step (e.g. LeCuyer and Crothers, 1994 ; Smith et al., 2009 ; Bevilacqua et al., 1992 ; Mueller-Planitz and Herschlag, 2008 ; James and Tawfik, 2005 ; Pisareva et al., 2006 ). If there is a slow step preceding binding, the rate of equilibration may become limited by this slow step. For example, the formation of long-lived stable alternative structures is well known for RNA (e.g. Uhlenbeck, 1995 ; Herschlag, 1995 ). Such alternative states can lead to rapid equilibrium binding for a sub-population and then slow binding as the misfolded, alternative state re-equilibrates to give partial binding or non-exponential kinetics. The following are diagnostics for these and related issues:

Association and/or dissociation kinetics do not follow a single exponential. Such more complex kinetics indicate the presence of additional species that must be identified.

Association kinetics are not first order in protein; that is, the binding rate constant is independent of protein concentration instead of the linear dependence seen in Appendix 1—figure 2D, E and predicted by Equation 1 . This behavior indicates additional species in the binding process.

Equilibration rate constant is dependent on protein concentration, but binding does not go to completion even at saturating concentrations. E.g. if only half of a ligand is bound at saturation, that may indicate that 50% of the ligand is trapped in a long-lived conformation or covalently heterogeneous (e.g. protein ligands that are partially or heterogeneously covalently modified). Alternatively, incomplete binding can mean that the ligand dissociates during disruptive sample processing steps (see Note 2). Further tests, such as extending the incubation time, pre-incubating an RNA ligand at increased temperatures, or analysis of the labeled ligand by HPLC, gel, sedimentation, or other methods, can determine if incomplete binding is caused by a slow conformational step or by covalent heterogeneity.

If these situations do not apply, k off is sufficient to determine the required equilibration time.

Note 2: Controls for changes during sample processing in ‘indirect’ binding measurements, and approaches to prevent these changes

Techniques such as native gel shift, nitrocellulose filter binding, and any pull-down-based approaches ( Lambert et al., 2014 ; Wong and Lohman, 1993 ; Ryder et al., 2008 ; Campbell et al., 2012 ) involve sample processing steps between the binding incubation and detection of bound complex, and are thus ‘indirect.’ Changes in conditions during sample handling and analysis can perturb the amount of complex from that present at the end of the initial incubation. It is therefore important to control for and ideally prevent such changes, and additional confirmatory experiments, such as the kinetic experiments described in the main text, are necessary to obtain high-confidence K D values.

In the case of native gel shift experiments, following the incubation (time t 1 in Figure 4A ), the sample is transferred to a gel loading buffer, containing a high-density additive such as Ficoll or glycerol to facilitate gel loading, and typically a dye, and is loaded onto a non-denaturing gel. These steps are accompanied by changes in concentration, solution conditions, and often temperature, all of which have the potential to induce dissociation or additional binding.

Some general strategies to minimize such perturbations include:

Minimizing any changes in conditions between the incubation (t 1 ) and detection (t 2 ) steps. E.g. the loading buffer can be omitted altogether by including sufficient glycerol in the reaction itself and the gel can be run at the same temperature as the binding reaction ( Hellman and Fried, 2007 ). However, this will not always be feasible, e.g. due to rapid dissociation of the complex during room-temperature gel electrophoresis.

Empirically assessing the effects of any changes in conditions on binding and the time scales on which these effects occur. Certain changes will have negligible and consistent effects on binding and will not affect the quantification if samples are handled quickly and consistently.

Varying the original incubation conditions while maintaining the same gel loading and gel running conditions is a worthwhile control to establish that the observed fraction bound does reflect at least some property from the original incubation conditions.

For our Puf4 binding assays, we were able to prevent changes during the native gel shift assay by utilizing certain favorable properties of Puf4/RNA binding. Below we describe the specific steps we took, as some of the Puf4 strategies can be adapted to other systems with similar properties.

Preventing complex dissociation:

During initial exploration, we found that Puf4 dissociates from its consensus RNA extremely slowly at 0°C (Appendix 1). Thus, by keeping our loading buffer on ice and running the gels at 4–5°C we were able to effectively ‘quench’ complex dissociation, with only negligible dissociation (t 1/2 ≥ 3 hr) occurring during the short time (seconds–minutes) the samples spent in loading buffer or the gel running buffer.

While dissociation of weaker Puf4 ligands was non-negligible even at low temperatures (data not shown), loading these samples quickly (within seconds) and with a consistent loading time ensured that dissociation only affected the amplitude and not the shape of the equilibrium binding curve (and thus the K D determined from it). This was confirmed by competition measurements.

Preventing additional complex formation:

In contrast to the slow dissociation, Puf4 association rate constant remains high even at 0°C (Appendix 1). Thus, additional binding can occur during sample loading, and the amount of additional binding will vary with the concentrations of the binding partners and the time prior to loading and gel entry.

To avoid the above complexities, we included a large, saturating excess of unlabeled RNA in the loading buffer (here we used the same oligonucleotide as the labeled RNA; more generally—a tight binder that binds at least as tightly as the labeled RNA should be used). This is equivalent to the k on chase used in the kinetics measurements (Appendix 1) and ensures that the additional binding that occurs before entering the gel is to the unlabeled RNA—such that the fraction of bound labeled RNA still accurately reflects the fraction bound during the original incubation.

If applying an analogous chase approach, it is important to keep in mind that the unlabeled RNA concentration in the loading buffer must be at least 10-fold higher than the protein concentration used. Otherwise, a substantial fraction of labeled RNA can still bind during t 2 , in a manner dependent on protein concentration. Using a chase RNA that is bound more tightly than the ligand being tested (e.g. wild-type RNA sequence vs. a mutant) allows the use of lower excess.

Additional measures to minimize changes during binding measurements by native gel shift:

To maintain consistent loading time for all samples, and to minimize the time in the gel running buffer, the samples should be (carefully!) loaded on a continuously running gel. (DANGER: As there is sufficient current to cause injury or death, extreme caution is required in this step. Always be sure there are no leaks, never touch the gel while running, even with gloves on, and maintain a safe distance as current can arc; see, e.g. https://ehs.stanford.edu/reference/electrophoresis-safety for safety information.)

The ratio of the sample volume to the area of the bottom of the well should be kept as low as possible. This ratio can be optimized by loading different sample volumes and varying the comb size and the gel thickness.

The percentage of acrylamide is another variable that should be adjusted if excessive dissociation on the gel occurs (indicated by smearing), with higher acrylamide percentages recommended to increase complex stability in the gel (see also Altschuler et al., 2013 ).

Using high-density compounds such as Ficoll or glycerol in the loading buffer facilitates rapid gel entry by concentrating the sample at the bottom of the well.

It is advisable to vary the above and other factors (including voltage and temperature) to determine if they influence the results. Factors following sample incubation should not affect the results.

Note 3: Number of time points for establishing equilibration time

Although in principle using two well-separated equilibration times is sufficient, using three or more times is preferable. Using only two incubation times has the potential to give a misleading result—if binding continues to increase while, for example, protein is denaturing, these factors can cancel each other out to give apparently constant binding. It is also critical to use times that span a considerable range, preferably approaching or exceeding 10-fold; here the concern is that if the time interval is narrow, the (inevitable) measurement error can make it difficult to distinguish if measurements are or are not time-independent.

Note 4: Uncertainty in the concentration of labeled trace binding partner

Accurately quantifying trace concentrations of labeled ligands can be a challenge, for instance, when working with radioactively labeled oligonucleotides. The concentration of labeled material can be estimated from specific activity of the isotope used for labeling. However, if labeling is incomplete and the purification procedure used after labeling does not fully separate labeled from unlabeled material, more oligonucleotide will be present than accounted for by specific activity. To be conservative, the upper limit of labeled oligonucleotide concentration should always be calculated based on the total oligonucleotide input in the labeling reaction. The concentration determined by specific activity provides a lower limit.

If not all protein is active or if the protein concentration is inaccurate, one may have false confidence of being in the excess-protein regime described by Equation 4b . Thus, although keeping the trace component at least 10-fold below the dissociation constant is a useful benchmark, we recommend always varying the trace ligand concentration (section 'Avoid the titration regime'). When possible, the concentration of active protein should be determined by titration against a known concentration of the ligand (see section 'Determine the fraction of active protein').

Note 5: Additional factors that can affect binding-competent concentrations of protein and ligand

Depending on the system and technique, other factors can lead to concentration-dependent variation in the observed K D values. Performing the simple controls in the Equilibrium Binding Checklist (Appendix 4) will typically detect these problems and help assess the robustness of measured K D values. We provide a non-exhaustive list below that will be helpful in devising appropriate additional controls, but experimenters should consult references for technique-specific information and advice.

Proteins can denature over time, and can also be subject to time-dependent proteolysis in extracts or partially purified systems.

Apparent weakening of binding with increased incubation time will help detect this behavior (for reactions where protein is the excess binding partner).

Optimizing solution conditions (e.g. lowering temperature, including glycerol, varying pH, etc.) and using protease inhibitors may help extend accessible times.

k off measurements can help identify the shortest feasible time to be used for equilibrium incubation to limit damage (in cases where damage occurs during unnecessarily long incubations).

Proteins can aggregate/form higher-order complexes at very high concentrations.

Active protein concentration (section 'Determine the fraction of active protein') should be assessed at several protein concentrations, including one at the high-end of the concentration range used in equilibrium experiments (see also discussion in Altschuler et al., 2013 ).

If concentration-dependent changes in stoichiometry of the complex are detected (which can be detected with some approaches like gel electrophoresis, certain fluorescence-based methods), models beyond the simple model in Figure 3 should be devised and tested.

Single-stranded oligonucleotides can form intermolecular base-pairs when used at high concentrations (e.g. in competition experiments) or during storage.

Nearest-neighbor base-pairing predictions can be used to estimate if base-pairing may be an issue at the temperature, salt conditions, and oligonucleotide concentrations used. If interactions between oligonucleotides are suspected, one can weaken base-pairing by using higher temperature or lowering salt concentrations in the binding experiments. If such changes are not possible or sufficient, the binding model should be modified to incorporate the oligonucleotide interactions.

For nucleic acid constructs with extensive complementarity that can form long-lived intermolecular interactions during storage, a dilute solution should be heated before use in binding experiments.

Nucleic acids can be covalently damaged by nucleases and other factors.

Care should be taken to remove all potential nuclease contamination by using sterile, high-purity water and reagents, sterile supplies and surface decontaminants. In our experience, some of the most robust RNase contamination has come from contaminated lots of commercial RNase inhibitors ; thus, we recommend testing these products before relying on their efficacy.

UV exposure should be limited or avoided during nucleic acid purification to avoid covalent damage ( Kladwang et al., 2012 ; Greenfeld et al., 2011 ).

Both proteins and nucleic acids can stick to tubes, lowering the concentrations accessible for binding.

Varying the type of reaction tube, and including small amounts of detergent and bovine serum albumin (BSA) can be used to assess and prevent sticking. (NOTE: Some BSA and other protein preparations contain nuclease contaminants.)

Varying the concentration of the labeled trace partner and measuring the dissociation constant by more than one approach (equilibrium vs. kinetics, or different techniques) can control for loss of material due to sticking.

Long-lived misfolded RNA concentrations can reduce binding-competent concentration during short incubation times (see Note 1).

Note 6: Controls and considerations for dissociation rate constant measurements

We recommend the following steps to ensure accurate k off measurements.

Establish that the chase is effective. Mixing the chase (in large excess over the protein concentration) with labeled ligand before addition of protein to form the complex should lead to no detectable protein binding to labeled ligand. If this is not the case, a higher chase concentration and/or a higher-affinity chase ligand is needed.

Establish independence of k off from the chase ligand concentration. Multiple chase ligand concentrations should be used, preferably spanning at least an order of magnitude. It is expected that k off will be constant, but variation can indicate an experimental artifact (such as a chase component affecting k off ) or, more interestingly, an ability of one ligand to facilitate dissociation of another. Here, dissociation by dilution becomes an important control—i.e. the bound complex formed with protein concentration near the K D is diluted by varying amounts until full or near-full dissociation is observed. The dissociation rate constant should be consistent across different dilution factors that give full dissociation. Incomplete dissociation (with the dissociation curve plateauing substantially above zero) most simply suggests insufficient chase (and/or dilution), which is usually readily resolved by increasing the chase concentration. Less commonly, incomplete dissociation can indicate heterogeneity of the bound complex, with a slowly dissociating sub-population remaining bound on the time scale of the experiment. In this case, increasing the chase concentration will not lead to complete dissociation, and the origins of complex heterogeneity should be investigated. Indeed, the more slowly dissociating fraction is more likely to represent a functional form, as it is more tightly associated.

Establish independence of protein concentration. While the starting fraction bound may vary depending on how far above the K D the protein concentration is, the dissociation rate constant should always be the same. Changes in k off with protein concentration can indicate a contaminant in the protein solution or, more interestingly, the formation of a protein multimer that increases or decreases the RNA dissociation rate. Thus, k off measurements at several protein concentrations (ideally three or more spanning at least an order of magnitude), in addition to serving as controls, can help discover new complexes and pathways.

Considerations for competition measurements

Protein concentrations ~2–5 times above the K D for the labeled ligand ( K D * in Appendix 3—figure 1A ) should typically be used. It is important to be near or above the K D * to have sufficient signal, but at the same time not too far above the K D * so that the protein concentration is sufficiently below the K D of unlabeled competitor ( K D,comp ) for most of the competitor to remain unbound (analogous to the ‘trace’ condition recommended for direct binding measurements; see section 'Avoid the titration regime'). Under these conditions ([P] total  <<  K D,comp ), the quadratic solution to the 'Lin and Riggs’ equation can be used to determine K D,comp ( Equation 9 , Appendix 3—figure 1C ; Lin and Riggs, 1972 ). Protein concentrations below K D * can be used as controls for competitive binding (analogous to varying the labeled ligand concentration to rule out titration in direct binding measurements).

If the unlabeled competitor binds with a K D very similar to, or lower than that of the labeled ligand, the condition [P] total  <<  K D,comp for Equation 9 will not be satisfied. In such case, the cubic equation, which accounts for the depletion of the competitor, can be used ( Wang, 1995 ). For the high-affinity competitors, it may be more reliable to measure binding directly (rather than by competition); alternatively, a higher-affinity labeled species could be used.

The competitive binding equation includes labeled ligand concentration ( Appendix 3—figure 1C ). In a typical experiment where labeled ligand is used in trace ([R*] total << K D * in Appendix 3—figure 1C ) this value is negligible and does not contribute substantially to the K D,comp . Nevertheless, it is still important to establish whether the labeled ligand concentration affects K D,comp in a given experiment by fitting the data with lower and upper limits of labeled ligand concentration used in the equation (see Appendix 2—note 4). In case of the Puf4 experiment, using the upper and lower limits of labeled RNA concentration resulted in similar K D,comp values (230 nM and 204 nM, respectively), but using the upper limit led to a poorly fit amplitude due to expected slight protein depletion. This observation, along with others, suggests that the real labeled RNA concentration was more closely approximated by the lower limit (based on 32 P quantification) than the upper limit (based on total RNA used for R* preparation), and that negligible co-purified unlabeled RNA was present in our R* preparation (Appendix 2—note 4).

Measuring binding affinity by competition.

( A ) Competitive binding reaction scheme. R*: labeled RNA ligand; R comp : unlabeled competitor RNA; K D * : protein affinity for R*;  K D,comp : protein affinity for R comp . ( B ) Mixing scheme for a competition measurement. ( C ) Competition between the U1C point mutant of the Puf4 consensus (R comp = C GUAUAUUA) and the labeled consensus RNA (R*= 32 P-AUG UGUAUAUUA GU). The data were fit to the following equation ( Lin and Riggs, 1972 ; Weeks and Crothers, 1992 ):

A indicates the maximum amplitude, constrained to the fit amplitude of the R* binding curve that is measured in parallel by a direct binding experiment (A = 0.89 for Puf4 binding to R*). O is the y axis offset (background). [R*] total was constrained to the lower limit of the labeled RNA concentration. K D * was constrained to Puf4 affinity for the labeled RNA, as determined by direct measurement in the same experiment (0.105 nM, after accounting for active protein fraction of 75%). [P] total was 0.45 nM, after accounting for active protein fraction. The fit  K D,comp value was 204 nM. Incubation times of 10, 30, and 110 min gave consistent K D,comp values (190–210 nM), as did lowering the protein concentration by three-fold (180 nM). Equation 9 is applicable only for K D,comp   >>   K D * . For other cases see Wang, 1995 .

Equilibrium binding checklist

Equilibrium binding checklist template.

Example of a completed equilibrium binding checklist based on Puf4/RNA binding at 25°C.

Example of a completed equilibrium binding checklist based on Puf4/RNA binding at 0°C.

No datasets were generated in this work. The figures include all data, or, where most appropriate for clarity, representative data from a single experiment for every type of experiment performed.

• Abudayyeh OO
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Author details

Contribution, contributed equally with, for correspondence, competing interests.

Present address

• Department of Biochemistry, Stanford University, Stanford, United States
• Department of Chemical Engineering, Stanford University, Stanford, United States
• Stanford ChEM-H, Stanford University, Stanford, United States

National Institutes of Health (R01 GM132899)

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

Acknowledgements

We thank Geeta Narlikar, Enrique De La Cruz, and members of the Herschlag lab for discussions and comments. We are also grateful to Hashim Al-Hashimi, Tom Cech, Katrin Karbstein, Olke Uhlenbeck, Chris Walsh, and Deborah Wuttke for critical feedback and suggestions. This work was funded by a grant from the US National Institutes of Health to DH (R01 GM132899).

Version history

• Received: March 26, 2020
• Accepted: August 5, 2020
• Accepted Manuscript published: August 6, 2020 (version 1)
• Version of Record published: August 27, 2020 (version 2)
• Version of Record updated: September 5, 2023 (version 3)

© 2020, Jarmoskaite et al.

This article is distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use and redistribution provided that the original author and source are credited.

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Further reading

• Structural Biology and Molecular Biophysics

Rapid, DNA-induced interface swapping by DNA gyrase

DNA gyrase, a ubiquitous bacterial enzyme, is a type IIA topoisomerase formed by heterotetramerisation of 2 GyrA subunits and 2 GyrB subunits, to form the active complex. DNA gyrase can loop DNA around the C-terminal domains (CTDs) of GyrA and pass one DNA duplex through a transient double-strand break (DSB) established in another duplex. This results in the conversion from a positive (+1) to a negative (–1) supercoil, thereby introducing negative supercoiling into the bacterial genome by steps of 2, an activity essential for DNA replication and transcription. The strong protein interface in the GyrA dimer must be broken to allow passage of the transported DNA segment and it is generally assumed that the interface is usually stable and only opens when DNA is transported, to prevent the introduction of deleterious DSBs in the genome. In this paper, we show that DNA gyrase can exchange its DNA-cleaving interfaces between two active heterotetramers. This so-called interface ‘swapping’ (IS) can occur within a few minutes in solution. We also show that bending of DNA by gyrase is essential for cleavage but not for DNA binding per se and favors IS. Interface swapping is also favored by DNA wrapping and an excess of GyrB. We suggest that proximity, promoted by GyrB oligomerization and binding and wrapping along a length of DNA, between two heterotetramers favors rapid interface swapping. This swapping does not require ATP, occurs in the presence of fluoroquinolones, and raises the possibility of non-homologous recombination solely through gyrase activity. The ability of gyrase to undergo interface swapping explains how gyrase heterodimers, containing a single active-site tyrosine, can carry out double-strand passage reactions and therefore suggests an alternative explanation to the recently proposed ‘swivelling’ mechanism for DNA gyrase (Gubaev et al., 2016).

7,8-Dihydroxyflavone is a direct inhibitor of human and murine pyridoxal phosphatase

Vitamin B6 deficiency has been linked to cognitive impairment in human brain disorders for decades. Still, the molecular mechanisms linking vitamin B6 to these pathologies remain poorly understood, and whether vitamin B6 supplementation improves cognition is unclear as well. Pyridoxal 5’-phosphate phosphatase (PDXP), an enzyme that controls levels of pyridoxal 5’-phosphate (PLP), the co-enzymatically active form of vitamin B6, may represent an alternative therapeutic entry point into vitamin B6-associated pathologies. However, pharmacological PDXP inhibitors to test this concept are lacking. We now identify a PDXP and age-dependent decline of PLP levels in the murine hippocampus that provides a rationale for the development of PDXP inhibitors. Using a combination of small-molecule screening, protein crystallography, and biolayer interferometry, we discover, visualize, and analyze 7,8-dihydroxyflavone (7,8-DHF) as a direct and potent PDXP inhibitor. 7,8-DHF binds and reversibly inhibits PDXP with low micromolar affinity and sub-micromolar potency. In mouse hippocampal neurons, 7,8-DHF increases PLP in a PDXP-dependent manner. These findings validate PDXP as a druggable target. Of note, 7,8-DHF is a well-studied molecule in brain disorder models, although its mechanism of action is actively debated. Our discovery of 7,8-DHF as a PDXP inhibitor offers novel mechanistic insights into the controversy surrounding 7,8-DHF-mediated effects in the brain.

• Stem Cells and Regenerative Medicine

Targeted protein degradation systems to enhance Wnt signaling

Molecules that facilitate targeted protein degradation (TPD) offer great promise as novel therapeutics. The human hepatic lectin asialoglycoprotein receptor (ASGR) is selectively expressed on hepatocytes. We have previously engineered an anti-ASGR1 antibody-mutant RSPO2 (RSPO2RA) fusion protein (called SWEETS) to drive tissue-specific degradation of ZNRF3/RNF43 E3 ubiquitin ligases, which achieved hepatocyte-specific enhanced Wnt signaling, proliferation, and restored liver function in mouse models, and an antibody–RSPO2RA fusion molecule is currently in human clinical trials. In the current study, we identified two new ASGR1- and ASGR1/2-specific antibodies, 8M24 and 8G8. High-resolution crystal structures of ASGR1:8M24 and ASGR2:8G8 complexes revealed that these antibodies bind to distinct epitopes on opposing sides of ASGR, away from the substrate-binding site. Both antibodies enhanced Wnt activity when assembled as SWEETS molecules with RSPO2RA through specific effects sequestering E3 ligases. In addition, 8M24-RSPO2RA and 8G8-RSPO2RA efficiently downregulate ASGR1 through TPD mechanisms. These results demonstrate the possibility of combining different therapeutic effects and degradation mechanisms in a single molecule.

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• Published: 27 May 2019

Probe dependency in the determination of ligand binding kinetics at a prototypical G protein-coupled receptor

• Reggie Bosma 1 ,
• Leigh A. Stoddart 2 , 3 ,
• Victoria Georgi 4 ,
• Monica Bouzo-Lorenzo 2 , 3 ,
• Nick Bushby   ORCID: orcid.org/0000-0002-9025-7015 5 ,
• Loretta Inkoom 1 ,
• Michael J. Waring 6   nAff8 ,
• Stephen J. Briddon 2 , 3 ,
• Henry F. Vischer 1 ,
• Robert J. Sheppard 7 ,
• Amaury Fernández-Montalván   ORCID: orcid.org/0000-0001-9156-0000 4   nAff9 ,
• Stephen J. Hill 2 , 3 &
• Rob Leurs 1  

Scientific Reports volume  9 , Article number:  7906 ( 2019 ) Cite this article

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• Biochemical assays
• G protein-coupled receptors
• Receptor pharmacology

Drug-target binding kinetics are suggested to be important parameters for the prediction of in vivo drug-efficacy. For G protein-coupled receptors (GPCRs), the binding kinetics of ligands are typically determined using association binding experiments in competition with radiolabelled probes, followed by analysis with the widely used competitive binding kinetics theory developed by Motulsky and Mahan. Despite this, the influence of the radioligand binding kinetics on the kinetic parameters derived for the ligands tested is often overlooked. To address this, binding rate constants for a series of histamine H 1 receptor (H 1 R) antagonists were determined using radioligands with either slow (low k off ) or fast (high k off ) dissociation characteristics. A correlation was observed between the probe-specific datasets for the kinetic binding affinities, association rate constants and dissociation rate constants. However, the magnitude and accuracy of the binding rate constant-values was highly dependent on the used radioligand probe. Further analysis using recently developed fluorescent binding methods corroborates the finding that the Motulsky-Mahan methodology is limited by the employed assay conditions. The presented data suggest that kinetic parameters of GPCR ligands depend largely on the characteristics of the probe used and results should therefore be viewed within the experimental context and limitations of the applied methodology.

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G protein-coupled receptors (GPCRs): advances in structures, mechanisms, and drug discovery

Analyzing kinetic signaling data for G-protein-coupled receptors

G protein-coupled receptors: structure- and function-based drug discovery

Introduction.

The pharmacodynamics of a drug are often related to the half-maximal modulation of target function (IC 50 , EC 50 ), which typically depends on the concentration required to obtain half-maximal target binding (K i , K d ). However, it is increasingly debated whether these pharmacological parameters provides sufficient information to predict the in vivo effectiveness of a ligand 1 , 2 , 3 , 4 . Drug-target binding kinetics have therefore received increased interest in the last decade, and the drug-target residence time has been linked to the in vivo efficacy of a number of important target classes, including the large family of membrane-bound G protein-coupled receptors (GPCRs) 3 , 5 , 6 , 7 , 8 , 9 . Radioligand binding is routinely used to determine ligand binding affinity and kinetics to GPCR targets 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 . To determine the binding kinetics of unlabeled ligands, the competitive effect on the association binding of a GPCR radioligand is analyzed using the theoretical model derived by Motulsky and Mahan 19 . Despite the wide use of this methodology in the GPCR-field, it is not known to which extent the calculated binding rate constants of unlabeled ligands depend on the binding kinetics of the radiolabeled probe used.

The histamine H 1 receptor (H 1 R) is a prototypical Family A GPCR which is therapeutically targeted by several 2 nd generation antagonists in the treatment of allergic conditions such as allergic rhinitis and urticaria 20 . The therapeutic success of the 2 nd generation H 1 R antagonists is generally attributed to their reduced brain penetration compared to 1 st generation H 1 R antagonists, which results in a decrease of on-target side effects such as sedation. Interestingly, the binding kinetics of several H 1 R antagonists have been investigated using the Motulsky-Mahan methodology 13 , 21 , 22 , 23 , 24 and were found to have a long residence time at the H 1 R 25 . In one study the prolonged residence time of levocetirizine was linked to the presence of a carboxylic acid group, which is a frequently occurring chemical moiety for 2 nd generation antihistamines 13 .

The success of the H 1 R as a drug target has resulted in a rich repertoire of antagonists that can bind the receptor, including different radiolabeled versions of commonly studied compounds 20 , 21 , 25 , 26 , 27 . Several radioligands ([ 3 H]mepyramine, [ 3 H]levocetirizine and [ 3 H]olopatadine) have previously been characterized for their kinetic binding profile at the H 1 R. Interestingly, [ 3 H]mepyramine and [ 3 H]levocetirizine show similar binding affinities at the H 1 R, but markedly different binding kinetics 21 . Recently, methodologies which utilize fluorescent ligands in place of radioligands have been introduced to characterize the binding kinetics of GPCR ligands and these newer methods have advantages over radioligand binding in terms of throughput and kinetic resolution 28 . Both bioluminescence (BRET 29 ) and time-resolved (HTRF 30 ) resonance energy transfer techniques have been applied to study binding kinetics at the H 1 R.

Due to the wide range of radioactive and fluorescently labelled ligands available for H 1 R, we used this GPCR as a model system to investigate if the measured binding rate constants of unlabeled ligands are influenced by the binding kinetics of the employed labelled probe. To this end, [ 3 H]mepyramine and [ 3 H]levocetirizine were used to characterize the binding kinetics of a set of unlabeled H 1 R ligands by the Motulsky-Mahan methodology. This was followed by the determination of the binding kinetics of H 1 R ligands via competitive association binding using two different non-radioactive H 1 R binding assays (BRET-based 29 or HTRF based 30 approaches). The k on and K i values, obtained from kinetic and steady-state experiments, respectively, were correlated between the various datasets employing either fluorescent ligands or radioligands as probes. However, it was found that k off -values are in part dependent on the used assay methodology. Therefore, both probe-dependent and assay-dependent factors are important contributors to the accurate determination of binding kinetics of unlabeled ligands.

The radioligand [ 3 H]mepyramine (20 Ci/mmol) was purchased from Perkin Elmer (Waltham, MA, USA). The mepyramine based fluorescent HTRF ligand (Gmep) was purchased from Cisbio (Codolet, France). Other employed, commercially available ligands were: triprolidine hydrochloride monohydrate (Tocris Bioscience, Bristol, United Kingdom), doxepin hydrochloride (Tocris Bioscience, E/Z mixture with a ~85:15 ratio), Olopatadine hydrochloride (BOC Sciences, Shirley, NY, USA), acrivastine (BOC Sciences), levocetirizine dihydrochloride (Biotrend, Cologne, Germany), S-cetirizine dihydrochloride (TLC PharmaChem, Mississauga, Canada), Mepyramine maleate (Research Biochemicals International, Natick, MA, USA), R-fexofenadine (Sepracor Inc., Marlborough, MA, USA), S-fexofenadine (Sepracor Inc.), desloratadine (HaiHang Industry, Jinan City, China), Terfenadine (MP biomedicals, Santa Ana, CA, USA). VUF14454, VUF14544, VUF14506, VUF14493 and mianserin were synthesized at the Vrije Universiteit Amsterdam and were fully characterized with respect to purity and identity 22 , 23 . [ 3 H]levocetirizine (25.9 Ci/mmol) was synthesized at AstraZeneca and was fully characterized with respect to purity and identity. AV082 (mepyramine-ala-ala-BY630) was synthesized at the University of Nottingham as described previously 29 . All other chemicals and reagents were obtained from Sigma Aldrich and Fisher, unless specified otherwise in the text.

Synthesis of [ 3 H]olopatadine

The synthesis of [ 3 H]olopatadine is schematically depicted in Fig.  1 . Column chromatography was carried out using pre-packed silica gel cartridges (SiliCycle, Quebec, Canada) on an Isco Companion (Teledyne Isco, NE, USA). 1 H NMR spectra were recorded on a Bruker (600 MHz or 400 MHz) using the stated solvent. Chemical shifts (δ) in ppm are quoted relative to CDCl 3 (δ 7.26 ppm) and DMSO-d 6 (δ 2.50 ppm). Liquid chromatography-mass spectrometry (LC-MS) data was collected using a Waters Alliance LC (Waters Corporation, MA, USA) with Waters ZQ mass detector. Analytical HPLC data was recorded using Agilent 1200 HPLC system with a β-Ram Flow Scintillation Analyser, using the following conditions: Waters Sunfire C 18 , 3.5 µm, 4.6 × 100 mm column at 40 °C, eluting with 5% acetonitrile/water +0.1% TFA to 95% acetonitrile/water +0.1% TFA over a 32 minute gradient. Specific activities were determined gravimetrically with a Packard TriCarb 2100CA Liquid Scintillation Analyser (Packard Instrument Company Inc., IL, USA) using Ultima Gold TM cocktail. Reactions with tritium gas were carried out on a steel manifold obtained from RC Tritec AG (Teufen, Switzerland). Specific activity was calculated by comparison of the ratio of tritium/hydrogen or carbon-14/carbon-12 for the tracer against the unlabelled reference. [ 3 H]Methyl nosylate was obtained from Quotient Bioresearch as a solution in toluene at 3150 GBq mmol −1 . Tritium gas was supplied absorbed onto a depleted uranium bed by RC Tritec AG (Teufen, Switzerland).

Synthesis of [ 3 H]olopatadine. Key: ( a ) 4-toluenesulfonic acid, EtOH, reflux, 2 h, 89%; ( b ) (i), 1-chloroethyl chloroformate, DCE, reflux, 4 h; (ii) MeOH, reflux, 2 h, 10% over two steps; ( c ) [ 3 H]methyl nosylate, DMF, 50 °C, 1 h; ( d ) NaOH, EtOH/H 2 O, r.t., 2 h.

Ethyl (Z)-2-(11-(3-(dimethylamino)propylidene)-6,11-dihydrodibenzo[b,e]oxepin-2-yl)acetate

( Z )-2-(11-(3-(dimethylamino)propylidene)-6,11-dihydrodibenzo[ b , e ]oxepin-2-yl)acetic acid hydrochloride (150 mg, 0.40 mmol), ethanol (3 mL, 0.40 mmol) and 4-toluenesulfonic acid (23 mg, 0.12 mmol) were stirred at reflux under Dean-Stark conditions for 2 h. Triethylamine (73 µL, 0.52 mmol) was added and the mixture evaporated under reduced pressure, the residue was partitioned between water (5 mL) and ethyl acetate (15 mL). The organic phase was washed with NaHCO 3 (satd. aq, 5 mL), brine (5 mL), dried (MgSO 4 ), filtered and evaporated to give ethyl ( Z )-2-(11-(3-(dimethylamino)propylidene)-6,11-dihydrodibenzo[ b , e ]oxepin-2-yl)acetate (131 mg, 0.358 mmol, 89%) as a colourless oil. LCMS (ESI) m/z 366 [M + H] + .

Ethyl (Z)-2-(11-(3-(methylamino)propylidene)-6,11-dihydrodibenzo[b,e]oxepin-2-yl)acetate

1-Chloroethyl chloroformate (38.2 µl, 0.35 mmol) was added to a stirred solution of ethyl ( Z )-2-(11-(3-(dimethylamino)propylidene)-6,11-dihydrodibenzo[ b , e ]oxepin-2-yl)acetate (128 mg, 0.35 mmol) in 1,2-dichloroethane (3500 µL) and the mixture heated to reflux. After 2 h additional 1-chloroethyl chloroformate (38.2 µL, 0.35 mmol) was added, and after a further 2 h the solvent was evaporated under reduced pressure. Methanol (3 mL) was added and the mixture heated to reflux for 2 h. The mixture was purified by preparative HPLC (Waters XBridge Prep C18 OBD column, 5 µ silica, 19 mm diameter, 100 mm length), using decreasingly polar mixtures of water (containing 0.1% TFA) and MeCN as eluents. Fractions containing the desired compound were combined, concentrated under vacuum, adjusted to pH 9 with NaHCO 3 , extracted with DCM (2 × 20 mL), then further purified by flash silica chromatography, elution gradient 0 to 6% NH 3 -MeOH (7 M) in DCM to afford ethyl ( Z )-2-(11-(3-(methylamino)propylidene)-6,11-dihydrodibenzo[b,e]oxepin-2-yl)acetate (12 mg, 0.034 mmol, 10%) as a colourless gum. LCMS (ESI) m/z 352 [M + H] + . 1 H NMR (600 MHz, DMSO-d 6 ) 1.17 (t, J  = 7.0 Hz, 3H), 2.24 (s, 3H), 2.45–2.5 (m, 2H), 2.62 (t, J  = 6.8 Hz, 2H), 3.57 (s, 2H), 4.06 (q, J  = 7.0 Hz, 2H), 5.16 (s, 2H), 5.70 (t, J  = 7.2 Hz, 1H), 6.76 (d, J  = 8.3 Hz, 1H), 7.05 (dd, J  = 2.3, 8.3 Hz, 1H), 7.08 (d, J  = 2.3 Hz, 1H), 7.26 (d, J  = 7.6 Hz, 1H), 7.28–7.32 (m, 1H), 7.35 (t, J  = 7.4 Hz, 2H).

Ethyl (Z)-2-(11-(3-(([ 3 H]methyl)(methyl)amino)propylidene)-6,11-dihydrodibenzo[b,e]oxepin-2-yl)acetate

[ 3 H]Methyl nosylate in toluene (2.5 mL, 1080 MBq) was concentrated under a stream of nitrogen. To this was added a solution of ethyl ( Z )-2-(11-(3-(methylamino)propylidene)-6,11-dihydrodibenzo[ b , e ]oxepin-2-yl)acetate (0.86 mg, 2.4 µmol) in DMF (0.5 mL) and the mixture stirred at 50 °C for 1 h. After lyophilisation, a solution of di- tert -butyldicarbonate (1.1 mg, 4.9 µmol) in DCM (1 mL) was added to the residue and the mixture stirred for 1 h then purified by silica chromatography eluting with 0 to 6% NH 3 -MeOH (7 M) in DCM. Fractions containing product were evaporated and dissolved in ethanol (1 ml) to give ethyl ( Z )-2-(11-(3-(([ 3 H]methyl)(methyl)amino)propylidene)-6,11-dihydrodibenzo[ b , e ]oxepin-2-yl)acetate solution. LCMS (ESI) m/z 372 [M + H] + .

(Z)-2-(11-(3-(([ 3 H]methyl)(methyl)amino)propylidene)-6,11-dihydrodibenzo[b,e]oxepin-2-yl)acetic acid ([ 3 H]Olopatadine)

Sodium hydroxide (2 M aq, 200 µL) was added to the ethanol solution (1 mL) of ethyl ( Z )-2-(11-(3-(([ 3 H]methyl)(methyl)amino)propylidene)-6,11-dihydrodibenzo[ b , e ]oxepin-2-yl)acetate (0.11 mg, 0.28 µmol) and the mixture stirred for 2 h. The ethanol was evaporated and water (1 mL) was added. The pH was adjusted to 9 by addition of HCl (2 M) and the mixture concentrated then purified on a Waters Oasis HLB cartridge, washing with water (5 mL), drying under a flow of nitrogen and then eluting with acetonitrile (5 mL). Purification by preparative HPLC (Waters XBridge C18 column, 4.6 × 150 mm) using decreasingly polar mixtures of water (containing 0.1% NH 3 ) and MeCN as eluents afforded [ 3 H]olopatadine (107 MBq) which was dissolved in ethanol (2 mL) for storage as a colourless solution. Radiochemical purity >98%. LCMS (ESI) m/z 344 [M + H] + . 3 H NMR (640 MHz, DMSO-d 6 ) 2.00 (s). Specific activity by mass spectrometry: 2920 GBq mmol −1 .

Cell culture

Human embryonic kidney cells transformed with large T antigen (HEK293T) and stably expressing Nluc-H 1 were generated as described elsewhere 29 , as is the transient transfection of these HEK293T cells with the N-terminally HA-tagged H 1 R 31 . Both native and transfected HEK293T cells were maintained in Dulbecco’s Modified Eagles medium supplemented with 10% fetal calf serum at 37 °C, 5% CO 2 . Cell pellets of transiently transfected HEK293T cells were prepared and stored at −20 °C until used in radioligand binding experiments, as previously described 31 . Frozen aliquots of TagLite® cells expressing the Tb-labeled SNAP-H 1 R were acquired from Cisbio.

Radioligand binding assays

Radioligand binding experiments were performed as described before with minor alterations as summarized below 31 . Frozen cell pellets of HEK293T cells transiently expressing the H 1 R were thawed, resuspended in radioligand binding buffer (50 mM Na 2 HPO 4 and 50 mM KH 2 PO 4 , pH 7.4) and homogenized with a Branson sonifier 250 (Branson Ultrasonics, Danbury, CT, USA). Homogenates (0.5–3 mg/well) were then incubated with the respective ligands at 25 °C under gentle agitation. For equilibrium saturation binding, increasing concentrations [ 3 H]mepyramine or [ 3 H]levocetirizine were incubated for 4 h in the absence or presence of mianserin (10 µM). Mianserin has a pK i at the H 1 R of 9.4 ± 0.1 (not shown) and the used concentration should therefore prevent any specific binding of the radioligands. For equilibrium competition binding, [ 3 H]mepyramine (3 nM) was used in the presence of increasing concentrations unlabeled ligands. In radioligand association binding experiments, four concentrations [ 3 H]mepyramine (0.2–10 nM), [ 3 H]levocetirizine (1–60 nM) or [ 3 H]olopatadine (9–19 nM) were used. Moreover, radioligand association binding was performed at 37 °C as well as 25 °C. For competitive association binding experiments 1–100x K i concentrations of the respective unlabeled ligand were co-incubated with a single concentration of radioligand ranging between 1.5–12 nM for [ 3 H]mepyramine or 5–15 nM for [ 3 H]levocetirizine. Kinetic ligand binding was performed for the depicted incubation times.

For dissociation experiments a single concentration of [ 3 H]mepyramine (3–13 nM), [ 3 H]levocetirizine (2–50 nM) or [ 3 H]olopatadine (10–15 nM) was pre-incubated with cell homogenate for 2 h (0.5–3 mg/well), after which a saturating concentration mianserin (10 µM) was added for various incubation times (triplicate binding reactions per time point). Non-specific binding was determined by the presence of mianserin (10 µM) during the pre-incubation step. Dissociation experiments were performed at both 37 °C and 25 °C.

Binding reactions were terminated using a cell harvester (Perkin Elmer) by rapid filtration and wash steps over PEI-coated GF/C filter plates. Filter bound radioligand was then quantified by scintillation counting using Microscint-O and a Wallac Microbeta counter (Perkin Elmer).

HTRF binding assays

HTRF based binding assays were performed as described before 30 with minor changes as summarized below. The mepyramine based fluorescent ligand (Gmep, Cisbio; time-resolved fluorescence resonance energy transfer (TR-FRET) acceptor) and unlabeled ligands were predispensed in 384-well plates. Binding reactions were started upon addition of TagLite® cells expressing the Tb-labeled SNAP-H 1 R (Cisbio) (1:8 predilution and 1:2.5 dilution in well; TR-FRET donor). TR-FRET signals arising from Gmep binding were measured at room temperature with an excitation wavelength of 337 nm and emission wavelengths of 490 ± 10 and 520 ± 10 nm, using a PHERAstar FS plate reader (BMG Labtech) with syringes for sample injection. The ratio values (520 nm/490 nm * 10000) were calculated as defined in the instrument software.

The steady-state affinity of the probe was determined by saturation binding experiments. Increasing concentrations of Gmep (2 fold serial dilution; 3.66 × 10 −10  M–3 × 10 −6  M; and 0 M) were incubated for 2.5 h in the absence and presence of doxepin hydrochloride (1 µM). Kinetic rate constants of Gmep binding were obtained with “association then dissociation” experiments 32 . Briefly, H 1 R expressing cells were added to increasing concentrations of Gmep in the absence and presence of doxepin hydrochloride (1 µM) and association was measured for 25.6 min with kinetic intervals of 26 s. Dissociation of Gmep was immediately initiated by addition of doxepin hydrochloride (1.1 µM) and detected for a further 40 min with kinetic intervals of 100 s. Competitive binding experiments were performed to quantify the affinities and kinetic rate constants of ligand binding: Gmep (100 nM) was co-incubated with increasing concentrations unlabeled ligands for 3 h as an endpoint measurement (11-point 3.5 fold serial dilutions of ligand, and 0 nM) or with a kinetic interval of 60 s for competitive association experiments (kPCA; 4-point 10 fold serial dilutions of ligand).

NanoBRET binding assays

For NanoBRET assays, HEK293Tcells stably expressing Nluc-H 1 were seeded 24 h before experimentation in white Thermo Scientific 96-well microplates in normal growth medium. For saturation and competition experiments, the medium was removed and replaced with HEPES-buffered saline solution (HBSS; 25 mM HEPES, 10 mM glucose, 146 mM NaCl, 5 mM KCl, 1 mM MgSO 4 , 2 mM sodium pyruvate, 1.3 mM CaCl 2 ) with the required concentration of AV082 and competing ligand. Cells were then incubated for 1 h at 37 °C (no CO 2 ). The Nluc substrate furimazine (Promega) was then added to each well at a final concentration of 10 µM and allowed to equilibrate for 5 min prior to measurement of fluorescence and luminescence. For association kinetic and competitive association kinetic experiments, medium was replaced by HBSS containing furimazine (10 µM) and incubated at room temperature in the dark for 15 min to allow the luminescence signal to reach equilibrium. For association kinetic experiments, the required concentration of AV082 in the presence and absence of doxepin (10 µM) was then added simultaneously. Immediately after, all wells of the microplate were read once per minute for 60 min. For competitive association experiments, AV082 (10 nM) was added simultaneously with the required concentration of unlabeled ligand or doxepin (10 µM) and read once per minute for 60 min. For all experiments fluorescence and luminescence was read sequentially using the PHERAstar FS plate reader (BMG Labtech) at room temperature. Filtered light emissions were measured at 460 nm (80-nm bandpass) and at >610 nm (longpass) and the raw BRET ratio was calculated by dividing the >610-nm emission by the 460-nm emission.

Data analysis

Analysis of saturation binding experiments, competition binding experiments and association binding experiments are fully described elsewhere 29 , 30 , 31 . The kinetic experiments were analyzed by GraphPad Prism (GraphPad Software, Inc., La Jolla, CA, USA) using non-linear regression of the data to pharmacological models that assume a one-step binding of the ligand to the receptor.

Competitive association – Motulsky-Mahan model

The baseline signal was subtracted from the total signal obtained in competitive association experiments. In the case of radioligand binding experiments and NanoBRET experiments equation ( 1 ) was employed to fit the data. In the case of HTRF experiments the adapted equation ( 2 ) is used to account for the observed signal drift (using k drift as a fitting constant). RL* is the baseline corrected signal that corresponds to the level of receptor binding by the labeled ligands. B max is the theoretical RL* in the case that all receptors would be occupied by the labeled ligand. [L*] and [I] stand for the concentrations labeled ligand and unlabeled ligand, respectively. Association rate constants are denoted by k 1 or k 3 and the dissociation rate constants by k 2 or k 4 for labeled ligand or cold ligand respectively. For the kinetics of competitive binding model, binding rate constants of the labeled ligands are required to fit the binding rate constants of the unlabeled ligand. The relative error was calculated for k 3 and k 4 values by dividing the reported error of the non-linear regression (SE) by the fitted mean value.

Dissociation experiments

Non-specific binding was subtracted from the total bound radioligand and the resulting specific radioligand binding over time was analyzed with a one-phase dissociation model (Graphpad prism: ‘Dissociation – One phase exponential decay’). When the radioligand was not fully dissociated within the timespan of the experiment, the final steady-state radioligand binding was constrained to baseline during analysis.

Characterization of radioligand probes

To explore how the binding kinetics of a radioligand affects the Motulsky-Mahan analysis of radioligand association in competition with unlabeled GPCR ligands, three different radioligands and two fluorescence-based probes for the H 1 R were investigated in this study. In addition to the widely used and commercially available [ 3 H]mepyramine (fast off rate), radiolabeled versions of the 2 nd generation antihistamine radioligands olopatadine and levocetirizine (slow off rate) 21 , 33 were synthesized as described in the method section and structures are depicted in Fig.  2 . Equilibrium binding of increasing concentrations of [ 3 H]mepyramine (Fig.  3a ), [ 3 H]levocetirizine (Fig.  3b ) or [ 3 H]olopatadine (Fig.  3c ) to H 1 R-expressing HEK293T cell homogenates, revealed that all radioligands saturably bind to the H 1 R with high affinities, resulting in pK d values of 8.6 ± 0.1, 8.1 ± 0.1 and 8.7 ± 0.1, respectively (Table  1 ). Moreover, saturation binding experiments revealed similar B max values (25–31 pmol receptor per mg protein) for all three radioligands, indicating that these three radioligands interact with the same H 1 R population. To determine the binding rate constants of the radioligands at the H 1 R, four different concentrations of either [ 3 H]mepyramine (Fig.  3d ), [ 3 H]levocetirizine (Fig.  3e ) or [ 3 H]olopatadine (Fig.  3f ) were incubated with cell homogenate for increasing incubation times and the data was fitted using a one-step binding model. The rate constants (k on and k off ) of [ 3 H]levocetirizine binding to the H 1 R are 30- to 100-fold lower than the binding rate constants of [ 3 H]mepyramine binding (Table  1 ), as was previously described 21 . The binding rate constants of the 2 nd generation antihistamines [ 3 H]levocetirizine and [ 3 H]olopatadine were comparable (<2 fold differences). Moreover, equilibrium dissociation constants calculated from the binding rate constants (pK d,kin  = k off /k on ) were in good agreement with equilibrium dissociation constants determined by saturation binding experiments (pK d ) (Table  1 ).

Structures of the used probes. T indicates a tritium atom.

Binding of [ 3 H]mepyramine, [ 3 H]levocetirizine and [ 3 H]olopatadine to the H 1 R. Increasing concentrations of [ 3 H]mepyramine ( a ), [ 3 H]levocetirizine ( b ) or [ 3 H]olopatadine ( c ) were incubated with H 1 R-expressing cell homogenates for 4 h at 25 °C. Indicated concentrations of [ 3 H]mepyramine ( d ), [ 3 H]levocetirizine ( e ) or [ 3 H]olopatadine ( f ) were incubated with cell homogenate for several incubation times at 25 °C. Representative graphs are shown of ≥3 experiments and the depicted data points represent the mean ± SEM of triplicate values ( a – c ) or depict individual measurements with duplicate values per time point ( d – f ). Extracted binding constants and statistical information are shown in Table  1 and Supplementary Table  2 .

The k off values of the radioligand probes were verified by radioligand displacement experiments, in which pre-bound radioligand is forced to dissociate by a high concentration of the unlabeled competitor mianserin (Supplementary Fig.  1 ). The k off values of [ 3 H]mepyramine and [ 3 H]levocetirizine deviated less than 1.5-fold between the association and dissociation experiments (Supplementary Table  2 ). However, there was very little dissociation of [ 3 H]olopatadine within the 6 hour time frame measured (Supplementary Fig.  1c ). To accelerate the association and dissociation of the radioligands and, thereby obtain a more robust quantification of the binding kinetics for the three radioligands, experiments were also performed at 37 °C (Supplementary Fig.  1 ) 21 . As expected, both the k on and the k off values of [ 3 H]mepyramine and [ 3 H]levocetirizine increased by 3–10 fold (Supplementary Table  2 ). However, even at 37 °C there was still limited dissociation of [ 3 H]olopatadine within the 6 h time span. Interestingly, at 37 °C the association of [ 3 H]olopatadine was not described well by a mono-exponential increase in binding as expected for a one-step binding mechanism. Consequently, [ 3 H]olopatadine was therefore excluded as probe, as the Motulsky-Mahan model that is used to describe competitive association ligand binding is based a one-step binding mechanism.

Quantifying the binding characteristics of unlabeled H1R antagonist

A chemically diverse set of unlabeled H 1 R ligands (structures are depicted in Supplementary Fig.  2 ), including reference molecules with known differences in their H 1 R binding kinetics, was selected for characterization of their H 1 R binding kinetics using either [ 3 H]mepyramine or [ 3 H]levocetirizine 13 , 23 , 31 . To guide the design of competitive association experiments, binding affinities (K i ) of the unlabeled ligands were first determined by equilibrium competition binding. Cell homogenates were therefore co-incubated with [ 3 H]mepyramine and increasing concentrations of the unlabeled ligands (Supplementary Fig.  3 ). Binding affinities (K i ) for H 1 R were calculated from the determined IC 50 values using the Cheng-Prusoff equation 34 and are depicted in Supplementary Table  3 .

The binding rate constants of the unlabeled ligands at the H 1 R were determined by competitive association binding experiments, in which H 1 R binding of the radioligand probes is quantified over time in the absence or presence of unlabeled ligands at three different concentrations. Concentrations of unlabeled ligand were varied ten-fold between the lowest and highest used concentration which was within an equipotent range of 1–100 times the respective K i of the ligands at the H 1 R. From the resulting radioligand association binding curves, the binding rate constants of unlabeled ligands can be determined by Motulsky-Mahan analysis (Fig.  4 , Table  2 ) 19 . As each time point requires a new binding reaction, the kinetic resolution for quantifying radioligand binding is limited and dependent on the number of parallel incubations. Therefore, incubation times were adjusted for the individual radioligands to best capture their kinetic profile. For the rapidly binding radioligand [ 3 H]mepyramine a relatively short 80 min incubation time was chosen (Fig.  4a–c ), whereas a 360 min incubation time was employed for the slowly binding probe [ 3 H]levocetirizine (Fig.  4d–f ). The association of the radiolabeled probes to the H 1 R in the presence and absence of three competing unlabeled ligands with (from left to right) fast, intermediate, and slow binding kinetics, is depicted in Fig.  4 and covers the diversity in binding kinetics observed within the full set of unlabeled ligands. Binding of [ 3 H]mepyramine in the presence of unlabeled mepyramine leads to a gradual increase in radioligand binding until binding equilibrium has been established after approximately 10 min (Fig.  4a ). In the presence of doxepin and levocetirizine there is first a transient overshoot in the binding of [ 3 H]mepyramine which results from the relatively slow dissociation (lower k off value) of both unlabeled ligands compared to the rapid binding of [ 3 H]mepyramine (Fig.  4b,c ). Conversely, since [ 3 H]levocetirizine binds much slower than [ 3 H]mepyramine (Fig.  3d,e ), no overshoot pattern is observed for [ 3 H]levocetirizine binding to the H 1 R in the presence of the same three unlabeled ligands (Fig.  4d–f ). The selected time points and length of incubation depended on the employed radioligand (Fig.  4 ), which might also affect the resulting binding rate constants in competitive association experiments.

Association binding of [ 3 H]mepyramine and [ 3 H]levocetirizine in the presence of competing unlabeled ligands at the H 1 R at 25 °C. The kinetic binding of [ 3 H]mepyramine to H 1 R-expressing cell homogenates was measured with various concentrations of either mepyramine ( a ) doxepin ( b ) or levocetirizine ( c ). Similarly, the kinetic binding of [ 3 H]levocetirizine to H 1 R-expressing cell homogenates was measured in competition with various concentrations of either mepyramine ( d ) doxepin ( e ) or levocetirizine ( f ). Representative graphs are shown of ≥3 experiments and each condition was measured in duplicate. Extracted binding constants and statistical information are shown in Table  2 and Supplementary Table  2 .

Probe dependent differences in binding characteristics

The observed association binding data of both radioligands (Fig.  4a–f ) agreed well with the fitted non-linear regression lines based on the Motulsky-Mahan model from which binding rate constants (k on and k off ) of the unlabeled ligands could be calculated (Table  2 ). Thus, two datasets were obtained with the binding rate constants of unlabeled ligands that were determined by using either [ 3 H]mepyramine or [ 3 H]levocetirizine as competitive probe. The measured binding rate constants correlated well between datasets as is depicted in Fig.  5a (k on values: R 2  = 0.80, P < 0.001) and in Fig.  5b (k off values: R 2  = 0.77, P = 0.002). However, the regression lines (solid lines) deviate from unity (dashed line) and some unlabeled ligands showed larger differences in binding kinetics between the two datasets than others. For example, more than 10-fold differences in the k on and k off values were observed for VUF14454 and VUF14544 between both datasets (with the K d,kin , calculated as the k off /k on , deviating less than 2-fold). The differences in the k on values between datasets were largest for ligands with a relatively high k off value (Table  2 ). Additionally, a probe-dependent difference for the range in k off values was observed, with [ 3 H]mepyramine discriminating unlabeled ligands over a range with higher k off -values (Fig.  5b , logk off −2.2 and 0.1) and [ 3 H]levocetirizine discriminating unlabeled ligands over a range with lower k off -values (Fig.  5b , logk off −3.2 and −0.7). These data suggest that the [ 3 H]mepyramine-based assay better distinguishes fast dissociating unlabeled ligands(high k off values), whereas the [ 3 H]levocetirizine-based assay better distinguishes slow dissociating unlabeled ligands (low k off values). From the determined binding rate constants, the binding affinity (pK d,kin  = k off /k on ) and the residence time (RT = 1/k off ), a proposed metric to relate binding kinetics to in vivo drug efficacy 3 , 5 , 6 , 7 , 8 , 9 , were calculated (Table  2 ). The pK d,kin values correspond well with the respective pK i values (Fig.  5c ), with a good correlation for both the [ 3 H]mepyramine-dataset (R 2  = 0.93, P < 0.0001) and [ 3 H]levocetirizine-dataset (R 2  = 0.87, P < 0.0001). Furthermore, the pK d,kin values correlate nicely between the probe specific datasets as well (R 2  = 0.87, P < 0.0001, data not shown). Since the K d,kin value is directly derived from the binding rate constants (k off /k on ), these correlations highlight the reciprocal changes in k on - and k off values in both binding assays and suggests that the ratio between the binding rate constants (e.g., reflected by the steady state level of radioligand binding, Fig.  4 ) is more robustly determined by the Motulsky-Mahan model than the binding rate constants themselves.

Comparison of the binding properties of unlabeled ligands as measured by using two different probes at 25 °C. Binding rate constants of unlabeled ligands were determined in radioligand binding studies. A correlation plot is depicted for the logk on ( a ) and logk off ( b ) as determined from competitive association experiments using either [ 3 H]mepyramine (x-axis) or [ 3 H]levocetirizine (y-axis) as competitive probe. ( c ) The correlation plot between the affinity calculated from the kinetic binding rate constants (pK d,kin ) and the affinity from competition binding experiments (pK i ) is depicted. Errors represent SEM values. Dashed lines represent a perfect correlation respective to the X-axis values and solid lines represent the linear regression lines.

To investigate the accuracy of the obtained k on and k off values, the relative errors of the fitted binding rate constants were calculated for each individual experiment for both data sets. The relative errors of the k on and the k off values were plotted against the corresponding mean k off value for each individual competitive association experiment, as depicted in Fig.  6 . It is shown that the relative error on the determined binding rate constants depends on the fitted mean k off values of unlabeled ligands at the H 1 R (Fig.  6 ). For both binding-rate-constants a decrease in accuracy, i.e. an increase in the relative error, is observed for unlabeled ligands with high mean k off -values at the H 1 R (Fig.  6a,b ) as is apparent for, e.g., −logk off values > −1. Additionally, an increase in the relative error on the k off -values (Fig.  6b ), but not on the k on -values (Fig.  6a ), is observed for unlabeled ligands with low mean k off -values as is apparent for, e.g., −logk off values < −2. Interestingly, Fig.  6 clearly shows a probe-dependent accuracy for the determination of the k on and the k off - values of unlabeled ligands. The k on value for H 1 R binding is generally more accurate when determined in a [ 3 H]mepyramine binding experiment (blue curve, Fig.  6a ), whereas the k off value is more accurate for the [ 3 H]levocetirizine dataset in the case of unlabeled ligands with a logk off  < −2 and less accurate for ligands with a logk off  > −2 (red curve, Fig.  6b ).

Accuracy of the measured binding rate constants depend on the fitted mean k off of unlabeled ligands at the H 1 R at 25 °C. The accuracy in which the Motulsky-Mahan model fitted the k on ( a ) and k off ( b ) by non-linear regression was examined for the different experimental conditions that were employed in this study. To compare the accuracy of the fitted mean k on and k off values over a broad range, the relative magnitude of the error (SE), as derived from non-linear regression, was calculated for each individual replicate experiment and pooled for all ligands. The relative error was calculated by normalizing the SE by the mean (relative error = SE/mean). Subsequently, the relative error for the k on and k off were plotted against the corresponding mean k off determined from the same competitive association curve. Data points derived from competitive association experiments that employed [ 3 H]levocetirizine are depicted in red. The arrows depict the k off of the used probes with [ 3 H]levocetirizine in red and [ 3 H]mepyramine in blue as reported in Table  1 . Dashed lines represent a relative error of 1 (mean = SE).

Cross-comparison between fluorescent-ligand and radioligand binding experiments

Recently, promising advances using resonance energy transfer techniques have been made in the use of fluorescent ligands as probes to characterize the binding kinetics of unlabeled ligands to GPCRs such as the H 1 R 29 , 30 . Since the binding of a fluorescent ligand can be measured continuously using a HTRF or NanoBRET-based approach, the kinetic resolution and throughput of such assays are much higher than conventional radioligand binding kinetics experiments. The availability of these assays for H 1 R, allows them to be compared to traditional radioligand binding assays for measuring the binding kinetics of unlabeled ligands. We initially sought to characterize the binding kinetics of the fluorescent probes in both the NanoBRET and HTRF assays. For the NanoBRET binding assay a BODIPY630/650-labeled mepyramine analog which emits in the red range (AV082; formally described as compound 10 in Stoddart et al . 29 and depicted in Fig.  2 ) is used as fluorescent probe and is used with HEK293T cells expressing H 1 R tagged on the N-terminus with NanoLuc which are grown in a mono-layer. For the HTRF binding assay, a commercially available fluorescent analog of mepyramine (structure unknown) which emits light in the green range (Gmep) was employed alongside TagLite® cells expressing an N-terminally SNAP-tagged H 1 R and labelled with terbium cryptate. Characterization of both fluorescent probes was as previously described (Table  1 ) 29 , 30 . Although the relative large size of the attached fluorophore is likely to affect the binding properties of the unlabeled ligand 29 , the k off value for the binding of mepyramine-analogs AV082 and Gmep to the H 1 R resembled those of [ 3 H]mepyramine (<2-fold difference), albeit with differences in their k on values (2–100 fold) (Table  1 ).

Both non-radioactive assays were used to characterize the H 1 R binding properties of the set of unlabeled ligands depicted in Supplementary Fig.  2 . Equilibrium competition binding experiments were performed to obtain pK i values of the unlabeled ligands as described before 29 , 30 (Supplementary Fig.  3 , Supplementary Table  3 ) and ligands were further characterized in kinetic competition association experiments (Fig.  7 , Supplementary Table  4 ). For the binding of AV082 measured by NanoBRET, in line with the competitive association experiments using [ 3 H]mepyramine (Fig.  7a–c ), an overshoot was apparent when co-incubated with doxepin and levocetirizine but not with mepyramine. Kinetic binding rate constants were determined by fitting the NanoBRET signal over time to the Motulsky-Mahan model.

Association binding of AV082 and Gmep in the presence of competing unlabeled ligands at the H 1 R at 25 °C. The kinetic binding of AV082 to the H 1 R, stably expressed on adherent cells, was measured in the presence of various concentrations mepyramine ( a ) doxepin ( b ) or levocetirizine ( c ). AV082 binding was measured continuously by NanoBRET for 60 min. The kinetic binding of Gmep to the H 1 R, stably expressed on freshly thawed cells in suspension, was measured in the presence of various concentrations mepyramine ( d ) doxepin ( e ) or levocetirizine ( f ). Gmep binding was measured continuously by HTRF for 180 min. Representative graphs are shown of ≥3 experiments ( a – f ).

For the competitive association experiments with Gmep (Fig.  7d–f ), a signal drift was observed in the absence of unlabeled ligand. To allow robust fitting of the HTRF signal (including the signal drift), an additional one-phase decay function was incorporated into the Motulsky-Mahan model, as described before 30 . Despite the signal drift, an overshoot in Gmep binding is apparent in competition association experiments with unlabeled ligands with slow binding kinetics. After an initial rapid increase of Gmep binding, the HTRF-signal decreases much faster when co-incubated with 25 nM or 250 nM levocetirizine than in the absence of any competitor (Fig.  7f ). In contrast, in the presence of mepyramine (Fig.  7d ), the HTRF-signal never decreased faster than was observed for Gmep in the absence of unlabeled ligand.

A comparison of the binding constants that were obtained with [ 3 H]mepyramine binding (x-axis) and the two fluorescent binding assays (y-axis) are depicted in Fig.  8a–c and Supplementary Table  3 . Interestingly, a good correlation was observed between assays for the relative binding affinities (pK i ) (Fig.  8a ; (AV082: R 2  = 0.88, P < 0.0001; Gmep: R 2  = 0.94, P < 0.0001) and logk on values (Fig.  8b ; AV082: R 2  = 0.84, P < 0.0001; Gmep: R 2  = 0.92, P < 0.0001). Although, the use of AV082 in probe-displacement experiments resulted additionally in a log-unit lower pK i -values compared to values obtained in the orthogonal assays (Fig.  8a ). Moreover, the logk off values determined with Gmep as probe (HTRF assay) also correlated with those determined using [ 3 H]mepyramine as probe (Fig.  8c ; R 2  = 0.96, P < 0.0001). However, when employing AV082 (NanoBRET assay) in competitive association experiments, the relative differences in the k off values between the unlabeled ligands differ from those observed in the orthogonal assays (Fig.  8c ). Since both the K i -values and k on values correlate between orthogonal assays, the k off can be calculated (K i  × k on ) for the NanoBRET assay in order to estimate the relative differences in the k off between unlabeled ligands. As expected, these calculated values correlate better with the observed values in the orthogonal assays (Supplementary Fig.  4 ).

Comparison of the binding properties of unlabeled ligands as measured by using orthogonal assays at 25 °C. The binding rate constants of unlabeled ligands that were determined using the fluorescent probes AV082 and Gmep were compared with the binding constants obtained with [ 3 H]mepyramine. A comparison of the pK i values ( a ) (from equilibrium competition binding experiments) is depicted as well as the comparison of k on ( b ) and k off ( c ) values (from kinetic competition association experiments). Dashes lines represent a perfect correlation respective to the X-axis values and solid lines represent the linear regression lines.

In GPCR drug discovery, drug-receptor binding kinetics are often quantified using competition association experiments with a radioligand probe. Despite the increased use of this methodology, it is unclear whether the kinetic properties of the probe influence the obtained kinetic binding parameters of unlabeled ligands. Therefore, in this study we employed two radioligand probes, that differ in their H 1 R binding kinetics ([ 3 H]mepyramine and [ 3 H]levocetirizine), to measure the binding rate constants of a diverse set of unlabeled antagonists. The analysis shows that the k on and k off values obtained with each probe correlate between both probe-specific datasets (Fig.  5a,b ). However, large differences in the binding rate constants are observed for some compounds, e.g., VUF14454 and VUF14544. Moreover, although more than 10-fold differences in the k off are observed among (S)-cetirizine, triprolidine, mepyramine, VUF14454, VUF14493 and VUF14544 when using [ 3 H]mepyramine as probe, no difference is observed when the k off are measured for the same set of unlabeled ligands with [ 3 H]levocetirizine as probe. The comparison of these two datasets therefore suggests a probe-dependent limit to discriminate binding rate constants of unlabeled ligands. A related probe-dependent effect is apparent in the relative errors of the determined k on (Fig.  6a ) and k off values (Fig.  6b ) of the unlabeled ligands. In our competitive binding experiments, only the binding of the probe can be directly observed. When unlabeled ligands reach a binding equilibrium rapidly, before noticeable binding of the probe, binding kinetics of these unlabeled ligands is masked by the slow onset of probe binding at each time point. Since the onset of a receptor-binding equilibrium is faster with increasing k off of the respective ligands 19 , it seems logical that at some point, when the unlabeled ligands have an increasingly high k off compared to that of the probe, kinetic binding of the unlabeled ligands can no longer be distinguished. This is in line with the observation that the relative error in the fitted binding rate constants increases when the corresponding mean k off value (i.e. from the same competitive association curve) increases (Fig.  6 ). Moreover, a pronounced increase in the relative error is observed when the unlabeled ligands dissociate faster than the respective radioligand (k off unlabeled > k off radioligand, see arrows Fig.  6a,b ). This implies that [ 3 H]mepyramine, which has a 100-fold higher k off than [ 3 H]levocetirizine, is better suited to discriminate the binding kinetics of fast dissociating (high k off ) unlabeled ligands at the H 1 R. Moreover, in our dataset the k on (Fig.  7a ) and k off (Fig.  7b ) are fitted with a higher accuracy using [ 3 H]mepyramine as probe for unlabeled ligands with a residence time less than 100 min (log k off  > −2).

In contrast to the determined k on -values, which show a growing inaccuracy upon increases of the linked k off -value (Fig.  6a ), the k off -values are increasingly inaccurate at the lower end of the spectrum as well, i.e. both with low k off as well as high k off values (Fig.  6b ). Interestingly, a probe dependent difference in inaccuracy in these determinations is again apparent. However, for slowly binding unlabeled ligands (residence time of more than 100 min; fitted log k off  < −2) a moderately better accuracy is obtained when using [ 3 H]levocetirizine (and not [ 3 H]mepyramine) as probe in competition association experiments (Fig.  6b ).

Taken together, analysis of the competitive association radioligand binding data show probe-dependent differences in the measured binding rate constants of unlabeled ligands, which result (at least to some extent) from the accuracy with which these binding rate constants can be fitted to the competitive association curves of the radioligand probes. Considering that the accuracy of the measured binding rate constants decreased most extensively for unlabeled ligands that bind the receptor faster than the probe (k off unlabeled > k off radioligand), it is recommended to use a fast binding probe for the GPCR of interest (like [ 3 H]mepyramine for the H 1 R).

To avoid that the kinetics of the probe will mask the binding kinetics of the unlabeled ligands, the k off of the probe should ideally be higher than that of the unlabeled ligands. In radioligand binding experiments, the bound radioligand should not dissociate during the wash steps. A minimum residence time is therefore required and the k off should probably not go beyond 1 min −1 (at room temperature). For probes in fluorescent binding experiments, which do not require wash steps, probes could be designed to have a very high k off . There are not yet sufficient structure-kinetics-relationships available that allow clear cut optimization of the k off . However, reducing the binding affinity sufficiently will in most cases increase the k off 35 . Introducing subtle steric clashes in the binding pocket 22 might therefore be a way to fine-tune the k off of fluorescent probes.

Comparing the binding kinetics of unlabeled H 1 R antagonists determined in experiments using the fluorescently labelled H 1 R probes AV082 and Gmep with those obtained using [ 3 H]mepyramine, again reveals that the determined pK i values and the binding rate constants are highly correlated (Fig.  8 ). This might be explained (partially) by the fact that all three H 1 R probes have quite similar k off values (Table  1 ). In fact, the correlations for the measured binding rate constants were stronger when comparing the different assays (Fig.  8a,b ) than when comparing the datasets obtained with the different radioligands (Fig.  6a,b ). The notable exception were the k off -values determined with AV082 competition association binding experiments (measured by NanoBRET), which deviated only slightly among unlabeled ligands, suggesting again a probe-dependent limitation for discriminating the binding kinetics of unlabeled ligands. However, this effect is most likely not explained by the binding kinetics of the H 1 R probe as the binding rate constants of AV082 and [ 3 H]mepyramine differed only <3-fold. Assay-dependent differences might therefore also underlie the observed disconnect between k off values. It has been described that the measurement of drug-target binding kinetics can be additionally convoluted in pharmacological assays by rebinding 36 , 37 , the mechanism of ligand binding 38 , 39 , and the differences in local ligand concentration 40 , all of which cannot be accounted for during this analysis. Interestingly, in silico docking suggests that the hydrophobic fluorophore of large fluorescent ligands, like AV082, can protrude out of the GPCR 7TM pocket and might incorporate in the membrane 29 , which is distinct from the binding mode of [ 3 H]mepyramine, which is deeply buried within the transmembrane region of the H 1 R 41 , 42 , 43 . One can speculate that the simple one-step binding mechanism that is the conceptual basis of the Motulsky-Mahan model for probe binding to the H 1 R, is not a valid approximation in the case of AV082. Besides the probe, the extracellular N-terminal tag of the employed H 1 R was different between the NanoBRET-based detection of AV082 binding and the HTRF-based detection of Gmep binding to the H 1 R protein. Moreover, since AV082 was the only probe that was employed in binding experiments on adherent living cells, the extracellular environment might shape a unique exosite that promotes ligand-rebinding. For example, the epithelial layer of human lungs in organ bath perfusion experiments proved crucial for the insurmountable antagonism of the H 1 R imposed by azelastine 44 . Since the insurmountable antagonism depends on the length of the receptor-occupancy by azelastine 23 , 44 , 45 , the extracellular environment may in some cases contribute to the observed ligand binding kinetics.

In conclusion, the Motulsky-Mahan approach is a useful approach to quantify the binding rate constants of unlabeled GPCR ligands, especially with the high throughput and kinetic resolution that can be obtained with fluorescent ligand binding experiments. However, it should be taken into account that probe-dependent and assay-dependent factors can have an impact on the measured binding kinetics of the unlabeled ligands. It is recommended to use orthogonal approaches to confirm the binding kinetics of a set of reference compounds, for example, by studying the kinetics by which ligands functionally modulate GPCR activity. Previously, we found the ligand binding kinetics at the H 1 R of a set of H 1 R antagonists to correlate well with their kinetic effects on functional H 1 R responses 23 , 31 . Furthermore, the use of benchmark ligands will also allow comparison between different methodologies and will allow the selection of the best method to reach highest confidence for discriminating the target-binding kinetics of unlabeled ligands by the Motulsky-Mahan method. Considering that the Motulsky-Mahan model is by far the most frequently used way to derive the residence time of GPCR ligands, this study provides important considerations for the study of drug-target binding kinetics at GPCRs.

Data Availability

The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

This research was financially supported by the EU/EFPIA Innovative Medicines Initiative (IMI) Joint Undertaking, K4DD (Grant No. 115366).

Author information

Michael J. Waring

Present address: Northern Institute for Cancer Research, School of Natural and Environmental Sciences, Bedson Building, Newcastle University, Newcastle upon Tyne, NE1 7RU, United Kingdom

Amaury Fernández-Montalván

Present address: Servier Research Institute 125, Chemin de Ronde, 78290, Croissy-sur-Seine, France

Authors and Affiliations

Amsterdam Institute for Molecules, Medicines and Systems (AIMMS), Division of Medicinal Chemistry, Faculty of Science, Vrije Universiteit Amsterdam, De Boelelaan 1108, 1081 HZ, Amsterdam, The Netherlands

Reggie Bosma, Loretta Inkoom, Henry F. Vischer & Rob Leurs

Division of Physiology, Pharmacology and Neuroscience, School of Life Sciences, University of Nottingham, Nottingham, NG7 2UH, UK

Leigh A. Stoddart, Monica Bouzo-Lorenzo, Stephen J. Briddon & Stephen J. Hill

Centre of Membrane Proteins and Receptors, University of Birmingham and University of Nottingham, Midlands, UK

Drug Discovery, Bayer AG, Berlin, Germany

Victoria Georgi & Amaury Fernández-Montalván

IMED Operations, IMED Biotech Unit, AstraZeneca, Alderley Park, United Kingdom

Nick Bushby

Medicinal Chemistry, Oncology, IMED Biotech Unit, AstraZeneca, Alderley Park, United Kingdom

Medicinal Chemistry, Cardiovascular, Renal, and Metabolic Diseases, IMED Biotech Unit, AstraZeneca, Gothenburg, Sweden

Robert J. Sheppard

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R.B., L.I., H.V. and R.L. contributed the data and analysis for the “radioligand binding assays” methodology. L.S., M.B., S.B. and S.H. contributed the data and analysis for the “NanoBRET binding assays” methodology. V.G. and A.F. contributed the data and analysis for the “HTRF binding assays” methodology. Nick Bushby contributed to the synthetic procedures. All authors participated in writing of the manuscript.

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Bosma, R., Stoddart, L.A., Georgi, V. et al. Probe dependency in the determination of ligand binding kinetics at a prototypical G protein-coupled receptor. Sci Rep 9 , 7906 (2019). https://doi.org/10.1038/s41598-019-44025-5

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In a saturation binding experiment, you vary the concentration of radioligand and measure binding at equilibrium. The goal is to determine the Kd (ligand concentration that binds to half the receptor sites at equilibrium) and Bmax (maximum number of binding sites).

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Saturation Behavior: a general relationship described by a simple second-order differential equation

• Gordon R Kepner 1  

Theoretical Biology and Medical Modelling volume  7 , Article number:  11 ( 2010 ) Cite this article

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The numerous natural phenomena that exhibit saturation behavior, e.g ., ligand binding and enzyme kinetics, have been approached, to date, via empirical and particular analyses. This paper presents a mechanism-free, and assumption-free, second-order differential equation, designed only to describe a typical relationship between the variables governing these phenomena. It develops a mathematical model for this relation, based solely on the analysis of the typical experimental data plot and its saturation characteristics. Its utility complements the traditional empirical approaches.

For the general saturation curve, described in terms of its independent ( x ) and dependent ( y ) variables, a second-order differential equation is obtained that applies to any saturation phenomena. It shows that the driving factor for the basic saturation behavior is the probability of the interactive site being free, which is described quantitatively. Solving the equation relates the variables in terms of the two empirical constants common to all these phenomena, the initial slope of the data plot and the limiting value at saturation. A first-order differential equation for the slope emerged that led to the concept of the effective binding rate at the active site and its dependence on the calculable probability the interactive site is free. These results are illustrated using specific cases, including ligand binding and enzyme kinetics. This leads to a revised understanding of how to interpret the empirical constants, in terms of the variables pertinent to the phenomenon under study.

Conclusions

The second-order differential equation revealed the basic underlying relations that describe these saturation phenomena, and the basic mathematical properties of the standard experimental data plot. It was shown how to integrate this differential equation, and define the common basic properties of these phenomena. The results regarding the importance of the slope and the new perspectives on the empirical constants governing the behavior of these phenomena led to an alternative perspective on saturation behavior kinetics. Their essential commonality was revealed by this analysis, based on the second-order differential equation.

This paper answers the question: is there a general mathematical model common to the numerous natural phenomena that display identical saturation behavior? Examples include ligand binding, enzyme kinetics, facilitated diffusion, predator-prey behavior, bacterial culture growth rate, infection transmission, surface adsorption, and many more. The mathematical model developed here is based on a general second-order differential equation (D.E.), free of empirical constants, that describes the basic relation underlying these saturation phenomena [ 1 ].

A common and productive way to analyze a specific saturation phenomenon uses a model for the proposed mechanism. This leads to an algebraic relation that describes the experimental observations, and helps interpret features of the mechanism. Where the phenomenon involves chemical reactions, for example, the models rely on assumptions about reaction mechanisms, dissociation constants, and mass action rate constants [ 2 – 7 ]. Note that such mechanisms cannot be proved definitively by standard kinetic studies [ 8 ].

In view of the ubiquity of saturation phenomena, it seems useful to seek one mathematical model that describes all such phenomena. The model presented here relies solely on the basic mathematical properties of the experimentally observed data plot for these phenomena--the independent variable versus the dependent variable. It is free of mechanism and therefore applies uniformly to all these phenomena. The analysis starts with a second-order differential equation, free of constants, that offers a general way of describing them. This equation is then integrated and applied to illustrative examples.

Basic saturation behavior case

The general nature of the initial extensive mathematical analysis suggests using familiar mathematical symbols-- x , y , dy , dx , dy / dx , d 2 y / dx 2 , etc.--instead of using the symbols and notation particular to a specific saturation phenomenon, such as ligand binding where x would be A (free ligand), and y would be A b (bound ligand). One can then substitute any phenomenon's particular symbols into the key equations.

A typical experimental data plot for these natural phenomena that exhibit saturation behavior is shown in Figure 1 . Its essential feature is that each successive incremental increase, dx , in x is less effective at increasing dy . At very large values of x (saturation), the plot approaches its limiting value, the asymptote. As x increases: the fractional changes ( dx / x and dy / y ) decrease; the slope ( dy / dx ) is positive, steadily decreasing, and continuous; the second derivative ( d 2 y / dx 2 ) is steadily decreasing, and negative--because the tangent at P is above the curve. Thus, ( d 2 y / dx 2 ) = -| d 2 y / dx 2 |.

Typical idealized experimental data plot for those natural phenomena showing saturation behavior . The black dashed line is the initial slope, ( dy / dx ) 0 , and the red dashed line is the tangent at point P , ( dy / dx ) P .

The following generalized D.E. leads to many different mathematical relations, depending on the particular integer values of N and M. These describe, collectively, numerous natural phenomena.

Note that each term takes the fractional change form. It will be shown here, for N = M = 2, that this yields the second-order D.E., free of empirical constants, that gives the mathematical relation y = a · x /( b + x ). This relation describes the saturation plot of Figure 1 . Integration and analysis then lead to the definitions of the basic empirical constants that describe all saturation plots. Setting κ = dy / dx = slope gives

where dκ / κ is the fractional change in the slope.

Integrating and taking anti-logarithms gives the first-order D.E. for the slope,

Integrating again and rearranging gives

This algebraic relation, when substituted into equation ( 1 ), satisfies the second-order D.E. Therefore, it is a general solution. The system constants are determined by forcing the general solution to fit the physical boundary conditions ( x → 0 and x → ∞), giving a unique solution.

Evaluate C 1 and C 2 using equation ( 4 ). Let x → 0, so that C 1 >> C 2 · x , and therefore

Rearranging so y = 1/[( C 1 / x ) + C 2 ], let x → ∞, then C 2 = 1/ y ∞ = 1/ Υ sat , where Υ sat is the limiting value as y approaches the asymptote (saturation). Thus

This equation defines the roles of the two directly measurable and independent empirical constants of the experimental system, κ 0 and Y sat . Rearranging equation ( 6 ) gives

the general form of the standard algebraic relation used to describe the data plot in Figure 1 , [ 2 – 7 , 9 – 12 ].

These saturation phenomena are typified by the binding of a substance (e.g., a ligand or substrate) to a binding site. This can be analyzed in terms of random interactions between x and the binding site. In general, y x / Υ sat = Γ bd , the fraction bound, which can be equated to the probability the site is occupied, for a given value of x . Thus Γ bd = x /( K + x ) = κ 0 · x /[ Y sat + ( κ 0 · x )]. The probability the site is free is Γ fr = 1 - Γ bd , so

Thus, as x → 0, Γ fr → 1, and as x → ∞, Γ fr → 0.

Define Δ to mean the change in . Then the (change in slope)/(slope) equals Δ ( dy / dx )/( dy / dx ). Let Δ ( dy / dx ) = (1/2)·( d 2 y / dx 2 )· dx . The average slope is ( y / x ). Thus, Δ ( y / x ) = d ( y / x ), where d ( y / x )/( y / x ) = ( dy / y ) - ( dx / x ). Rearranging equation ( 1 ) with N = M = 2, and substituting equations ( 7 ) and ( 8 ) into it, yields

Thus, the change in the slope ( dy / dx ) divided by the change in the average slope ( y / x ) is determined by Γ fr .

Substituting into equation ( 3 ) for the slope gives

Ligand binding

Consider a small molecule, the ligand, that is present in either the free form, A , or the bound form, A b . For the simplest case, assume that each ligand binds to a single specific binding site (bs). This could be on a macromolecule, M bs , such as a protein. These sites are presumed to be independent and to have the same binding constant. The details of the experimental conditions required for these binding studies are found in standard reference texts [ 2 , 5 , 7 , 9 – 12 ].

The basic overall binding reaction is defined to be

The necessary and sufficient condition for this analysis is the experimental data plot of A b versus A . In Figure 1 , set y = A b and x = A . Then substitute into the key equations, for example, equation ( 6 ).

The total number of binding sites in the experimental system ( M bs , A , A b ) is ( A b ) sat . It is the limiting amount of ligand binding observed at saturation with A . The initial slope is κ 0 , the system's limiting binding rate when A → 0, and Γ fr → 1. Thus, ( A b ) sat and κ 0 are the empirical constants of the ligand binding system. The conventional models of the binding mechanism identify K d as the dissociation constant in mol L -1 [ 2 , 5 , 6 , 10 – 12 ]. Equation ( 11 ) is often referred to as the Langmuir adsorption isotherm, or the Hill binding equation. It is sometimes written using the binding fraction, Γ bd = A b /( A b ) sat .

The units of κ 0 = k bind ·( A b ) sat , where k bind = 1/ K d , are

Thus, k bind is the binding rate constant for one mole of binding sites evaluated at A → 0, where Γ fr → 1. It characterizes the binding strength of the ligand for the binding site. Therefore, a high value of k bind means a high value of the initial slope of the system, κ 0 , and so the value of K d is decreased.

The complete expression for the units of κ 0 illustrates how descriptive information could be lost when units are cancelled. Thus, the units [mol L -1 of ( dA b ) bound/mol L -1 of ( dA ) added] 0 describe a useful aspect of the binding process--the fraction of the added ( dA ) that is bound [( dA b )/ dA ], as A → 0. Taken over one minute, this yields the binding rate constant for one mole of binding sites.

The slope is given by

where Γ fr = ( A b ) sat /[( A b ) sat + ( κ 0 · A )]. Thus, κ A is defined as the system's effective binding rate at any value of A --to distinguish it from the highest value of κ , when A → 0, giving κ 0 , the system's limiting binding rate. Thus, if κ 0 is increased, then the ligand binding increases and (Γ fr ) A decreases, for a given value of A , because now more of the sites are occupied at A . If ( A b ) sat is doubled, for example, Γ fr will be increased--but not proportionately, see equation ( 11 ).

Typical plot of idealized experimental data to facilitate calculation of the empirical constants, ( A b ) sat and κ 0 .

Other examples

The term, binding site, is used for convenience as a general way of identifying the interactive locus of many saturation phenomena. For example: ligand binds to a macromolecule; a nutrient molecule binds to a receptor on a bacterial membrane and is transported inside; a prey is bound to a predator's jaws; a substrate binds to an enzyme's catalytic site; a molecule is adsorbed at sites on a surface (Langmuir's adsorption). Some saturation phenomena are less well-suited to this binding site characterization--e.g., the stock-recruitment model for producing new fish biomass from spawning stock [ 13 ].

where r is the experimentally measured bacterial growth rate ( g ·L -1 min -1 ), at a given concentration of nutrient, A ( g ·L -1 ). R sat is the limiting growth rate at saturation with nutrient ( g ·L -1 min -1 ). So, K = R sat / κ 0 in g ·L -1 , where the initial slope is κ 0 (grams bacteria·L -1 min -1 /grams nutrient·L -1 ), evaluated at A → 0. It measures the effectiveness of the specific bacteria's ability to convert a specific nutrient to bacterial growth--when all the receptor sites on the bacterial membrane are available. Thus, different bacteria using the same nutrient would have different values of κ 0 , reflecting the relative effectiveness of nutrient binding to the different receptor sites.

where n is the number of prey attacked over unit time by the predators present, and N sat is the limiting rate of attack at saturation with prey. Set A equal to the prey density (e.g., number of prey per square kilometer). Then K = N sat / κ 0 , where the initial slope, κ 0 , measures the effectiveness of the predator attacking the prey, as A → 0. Thus, a predator attacking two different prey yields different values of κ 0 .

Michaelis-Menten (M-M) enzyme kinetics

The basic overall enzymatic reaction is the conversion of one substrate molecule, A , to one product molecule, P , by an enzyme molecule, E , that catalyzes this conversion at its catalytic site (cs).

The necessary and sufficient condition for this analysis is the experimental data plot of ( dP/dt ) = p , versus A . See Figure 1 , where p = y and A = x . The experimental conditions required for measuring p and A are described in standard reference texts [ 2 – 4 , 7 , 9 – 12 ]. The use here of p , instead of the conventional v , focuses attention on the actual measured quantity and how it relates to A , in terms of dp / dA and d 2 p / dA 2 .

Equation ( 9 ) becomes Δ ( dp / dA )/Δ ( p / A ) = Γ fr , and equation ( 1 ) gives

Thus, the slope at any point, A , is κ A = κ 0 ·(Γ fr ) 2 , where, from equation ( 8 ), Γ fr = P sat /[ P sat + ( κ 0 · A )]. Note that κ A equals [( dA )converted/( dA )added] A = ( dp / dA ) A . This could be viewed as a measure of how effectively the system is converting substrate to product at A . It decreases rapidly as (1/ A 2 ).

Equation ( 6 ) becomes

The initial slope is κ 0 , and P sat is the limiting rate of enzyme catalysis when saturated with A . Increasing κ 0 will increase the binding of A to the catalytic site. These are the two independent empirical constants of the experimental system ( E t , A , P ). Equating ( P sat / κ 0 ) to K m , the Michaelis constant, and with p = v , and P sat = V , gives

the standard form for the M-M equation of enzyme kinetics.

Note that P sat ≡ k cat · E t , where E t is the total enzyme concentration present experimentally, which may not be known. The catalytic constant, k cat , is the limiting catalytic rate at which one mole of enzyme molecule could operate if completely saturated with substrate. Similarly, κ 0 ≡ k bind · E t , where k bind is here defined to be the binding rate constant of the substrate for the catalytic sites on one mole of enzyme (min -1 mol -1 ) -- evaluated when A → 0, where the catalytic site is maximally available, because Γ fr → 1.

Equation ( 8 ) can be rewritten to give

Thus, if k bind is increased, Γ fr is decreased, because there are fewer free sites available at a given value of A . Whereas, if k cat is increased, Γ fr is increased. The increased turnover rate means more free sites are available at a given value of A . As expected, Γ fr is independent of E t , because Γ fr depends only on the basic properties of the enzyme's catalytic function, k bind and k cat .

Enzyme kinetics differs from ligand binding because there is also a conversion step. The binding step is much faster than the conversion step, where the catalytic site converts the bound substrate to product and releases it. This is commonly assumed to involve a simple 1:1 stoichiometric relation between substrate bound and product released [ 19 ]. The binding rate constant for one mole of enzyme is defined here to be = κ 0 / E t = k cat / K m . The ratio, k cat / K m , is often referred to as the specificity constant [ 19 , 20 ]. Thus, k bind indicates the strength of the mutual interaction between a specific substrate and a specific enzyme, at the catalytic site, measured when A → 0. It defines a collective property for each particular combination of substrate and enzyme. For example, let A and E cs → k bind , while A ' and E ' cs → k ' bind , where k bind most probably differs from k ' bind , but might not. Therefore, the higher the value of k bind , the more effectively does the substrate bind to the enzyme's catalytic site. The enzyme and substrate, taken together, perform better at higher values of k bind [ 20 ].

Thus, k bind and k cat can be considered the basic properties of this single enzyme molecule's catalytic function. So

Therefore, K m is defined by the ratio of the experimental system's empirical constants, which depend on the enzyme's basic properties. When E t is known, one can obtain values for k cat and k bind . Whereas, although P sat and κ 0 often are measured experimentally where E t is not known, their ratio still gives K m . Doubling E t will double both P sat and κ 0 , so the ratio, K m , remains unchanged.

For clarity and convenience, the definitions and units of the various constants are explicitly stated here.

There are various standard linear transformations of equations ( 15 ) and ( 16 ) that aid in the initial analysis of the data plot in Figure 1 , [ 4 – 6 ]. Equation ( 19 ) is one.

This gives a linear plot of ( A / p ) versus A . The slope is (1/ P sat ) and the ordinate intercept is (1/ κ 0 ), recall Figure 2 . This provides direct evaluation of the system's empirical constants, κ 0 , and P sat , from the experimental data. Using C 1 = 1/ κ 0 , obtained from equation ( 19 ), one can calculate κ A , at any value of A , using the equation for κ A .

The mathematical model presented here is based solely on the observed experimental data plot for these phenomena, as shown in Figure 1 . This analysis of the second-order D.E. offers an alternative approach, free of mechanism, that describes the common process underlying all natural phenomena exhibiting saturation behavior. It provides a general mathematical description of these phenomena. The D.E. approach takes a path of discovery that reveals the salient features of these phenomena on the way to reaching y = x /[(1/ κ 0 ) + (1/ Y sat )· x ]. It complements approaches that model each specific saturation phenomenon separately, in terms of a proposed mechanism.

The D.E. analysis provided two general integration constants, C 1 and C 2 , evaluated at the known boundary conditions, x → 0 and x → ∞. This gave the two empirical constants, κ 0 and Y sat , that defined the relation between the variables of any saturation phenomenon -- see equation ( 6 ), the general algebraic description of these saturation phenomena. The empirical constant, κ 0 , the initial slope, and its practical significance, have not been recognized previously.

Applying the quantitative relation for Γ fr clarified the functioning of the interactive site. It showed that the underlying relation describing these phenomena, equation ( 1 ), became Δ( dy / dx )/Δ( y / x ) = Γ fr , see equation ( 9 ). The slope, equation ( 3 ), became κ = κ 0 ·(Γ fr ) 2 . Its strong dependence on (1/ x 2 ) was shown. As x increases, each added increment, dx , sees a lower Γ fr , because a greater fraction of the sites are occupied at the instant of adding dx . This leaves fewer sites free to attend to the conversion of this additional dx . This behavior is the essence of how these saturation phenomena function in response to increased x .

Ligand binding, bacterial growth, predator-prey

The response of these saturation phenomena to increased A is driven by Γ fr , see equations ( 9 ) and ( 10 ). The independent empirical constants for each phenomenon relate the variables of each and define the K that characterizes each one, see equations ( 11 ), (13a) and (13b). This mathematical model defines K , in general, as the ratio of the limiting rate/initial slope. Figure 2 shows how to obtain their values from the data. Other applications of this general approach include surface adsorption, facilitated transport, and transmission of infection. It emphasizes the utility of the initial slope, κ 0 .

Michaelis-Menten enzyme kinetics

Equation ( 10 ) shows that the slope, κ A , depends on (Γ fr ) 2 and (1/ A 2 ). Thus, Γ fr drives the experimental system's behavior and accounts, quantitatively, for the decrease in the slope with increasing A . This leads to the concepts of:

■ the system's effective binding rate, for E t moles of enzyme, at A .

■ the binding rate constant for one mole of enzyme, at A → 0.

For E t moles, ( dp / dA ) 0 = κ 0 , and for one mole, ( dp / dA ) 0 / E t = κ 0 / E t = k bind . Note that κ A = ( dp / dA ) A can be calculated using the equation for the slope and equation ( 19 ). Thus, the (slope) A /(slope) 0 = (Γ fr ) 2 , where Γ fr = P sat /[ P sat + ( κ 0 · A )].

The D.E. analysis defined the two independent empirical constants of this experimental system as κ 0 and P sat . Equation ( 15 ) is the general algebraic relation for illustrating their independent roles. Equation ( 18 ) ties together these empirical constants and the basic properties, k cat and k bind , to relate them to K m . Thus, k bind and k cat , taken together, can expand the ability to characterize and compare the interaction of enzymes and their substrates.

The usual model for the M-M enzyme reaction mechanism defines K m as a constant derived from the reaction rate constants. Such models are essential in pursuing the details of a proposed mechanism for M-M enzyme reactions, or for any saturation phenomenon. Yet, numerous different interpretations of what K m means have arisen in the literature, based on the standard model and mechanism. Some examples include: parameter, kinetic constant, not an independent kinetic constant, empirical quantity, a constant for the steady-state, measures affinity in the steady-state, should not be used as a measure of substrate affinity, most useful fundamental constant of enzyme chemistry, not a true equilibrium constant, dubious assertion that K m reflects an enzyme's affinity for its substrate [ 2 – 12 , 21 ]. According to Riggs, "Notice that the Michaelis constant is not a rate constant, nor an affinity constant, nor a dissociation constant, but is merely a constant of convenience" [ 22 ]. The interpretation presented here, based on the mathematical model, is rooted in equation ( 18 ). It showed that K m = k cat / k bind , the ratio of the enzyme's basic properties. Thus, this model viewed K m as a derived quantity, and not as an independent basic property of the enzyme molecule's catalytic function.

The action of enzyme inhibitors offers additional perspective on interpreting K m = k cat / k bind . Consider five basic cases of enzyme inhibition: Competitive, Uncompetitive, Pure Non-Competitive, Predominantly Competitive, Predominantly Uncompetitive [ 19 ]. In no case does the inhibitor ( I ) cause the limiting rate, P sat , or the initial slope, κ 0 , of the observed data plot for the experimental system ( E t , I , A , P ) to increase . The value of K m = k cat / k bind , however, is observed to increase , remain unchanged, or decrease--depending on the relative effects of the inhibitor on k cat and k bind . Any basic property of an enzyme molecule's catalytic function should never increase in the presence of an inhibitor. Thus, k cat and k bind meet this condition. Their ratio, k cat / k bind = K m , does not. Therefore, K m is not one of the two basic properties. Changing from K m to A M , the Michaelis concentration, would be consistent with this interpretation [ 19 ].

The ratio of these observable empirical constants, P sat / κ 0 , defines K m . Thus, the mathematical analysis offers an operational definition of K m , independent of any interpretations [ 19 ]. This approach to defining K m is consistent with all the known factors. "Definitions based on what is actually observed are therefore on a sounder and more lasting basis than those that depend on an assumed mechanism" [ 19 ]. Numerous mechanisms can generate M-M kinetics; "Consequently there is no general definition of any of the kinetic parameters... in terms of the rate constants for the elementary steps of a reaction's mechanism" [ 19 ].

The algebraic relation, p = P sat · A /( K m + A ), describes the data plot of an enzyme kinetic study. Its validity is independent of any mechanism. The mechanism-free approach derives this algebraic relation directly from the second-order D.E. This general analysis also reveals the underlying factors, such as Γ fr , that govern the basic behavior of these saturation phenomena.

The enzyme's catalytic function involves two distinct processes, binding the substrate and converting it to product. This mathematical analysis demonstrated that the two empirical constants of the D.E., κ 0 = E t · k bind and P sat = E t · k cat , define these two processes--binding and catalysis--in terms of the basic properties, k bind and k cat .

The first-order D.E. derived here introduces the concept of the effective binding rate. It is directly related to the slope of the experimental data plot. The initial slope is where it is highest, the binding rate constant of the ligand for the binding site. The analysis revealed the significance of the initial slope as an independent empirical constant for these systems exhibiting saturation behavior, and its role in determining the probability that the active site is free.

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Kepner, G.R. Saturation Behavior: a general relationship described by a simple second-order differential equation. Theor Biol Med Model 7 , 11 (2010). https://doi.org/10.1186/1742-4682-7-11

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DOI : https://doi.org/10.1186/1742-4682-7-11

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Saturation binding

These experiments provide information about the concentration of a receptor. They are solicited to compare the concentrations of different receptors in a given tissue and to monitor variations in receptor concentration as a result of normal physiological regulation, medication and pathophysiological conditions.

For saturation binding experiments, constant amounts of membrane suspension are incubated with increasing concentrations of radioligand. Obviously, both total and non-specific binding should be measured at each concentration of radioligand (Figure 31). In the example shown, binding is expressed as a function of the free concentration of radioligand by a saturation binding plot. Obviously, only the specific binding is of interest.

To analyze these saturation binding data, it is necessary to advance a relevant molecular model for the radioligand-receptor interaction. In the simple (and fortunately the most common) situation, the interaction of the radioligand (L) with the

Figure 31 Saturation binding of the a2-adrenergic antagonist [3H]RX 821002 to a2 adrenergic receptors in membranes from the human frontal cortex. Reprinted from Neurochemistry International, 17, Vauquelin G., De Vos H., De Backer J.-P. and Ebinger G., Identification of a2 adrenergic receptors in human frontal cortex membranes by binding of [3H]RX 821002, the 2-methoxy analog of [3H]idazoxan, 537-546. Copyright (1990), with permission from Elsevier.

receptor (R) can be expressed as a reversible bimolecular reaction that obeys the law of mass action: i.e.

Where k1 and are the association and dissociation rate constants, respectively. The equilibrium dissociation constant (KD) is given by:

Where [R] is the amount of free receptors, [L] the amount of free ligand and [L-R] the amount of bound ligand/receptors.

The relationship between the amount of occupied receptors and the free radioligand concentration (i.e. the saturation binding plot) is as follows:

Where [Rtot] is the total number of receptors.

'B' and 'Bmax' (which stand for binding and maximum binding and are often expressed in fmol/mg protein) usually replace [L - R] and [Rtot]. Equation (3) then becomes:

This equation is analogous to the Michaelis-Menten equation of enzyme kinetics and describes a rectangular hyperbola. Initially, B increases almost linearly with L. Then

B tends to level off when L is further increased. The limit value is Bmax (Figure 32). It is important to notice that this that Bmax will be attained only at infinite concentrations of L. Thus, one will never observe Bmax experimentally; Bmax may be approached but never attained. Half-maximal binding is obtained when L = KD (since Equation (4) becomes B = Bmax/2). In other words, the KD of a radioligand corresponds to its concentration for which half of the receptors are occupied. The KD value is thus an 'inverse' measure of the radioligand's affinity for the receptor: a low KD corresponds to high affinity and a high KD to low affinity.

Bmax and KD cannot be easily determined by graphical analysis of the saturation binding plot (Figure 32) since Equation (4) is a non-linear relationship and since Bmax is only reached when L = This equation can, however, be transformed mathematically to yield a linear 'Scatchardplot (Figure 33) corresponding to the following equation:

SCATCHARD PLOT

C : positive cooperativity Analysis: computer-assisted

A : 1 site or > 1 site with the same affinity

Analysis: linear regression

B : > 1 site or negative cooperativity Analysis: computer-assisted bound

Figure 34 Scatchard plots: different possibilities.

The Scatchard plot of the above saturation binding data reveals a linear relationship between B/[L] (the ordinate) and B (the abscissa). KD corresponds to the negative reciprocal of the line. The intercept of the line with the abscissa (i.e. when B/[L] = 0) is Bmax. Thus, it is relatively easy to calculate KD and Bmax values by linear regression analysis of the Scatchard plot.

The relationship described by Equation (5) is for the simplest case; i.e. a single class of non-interacting receptor sites. However, it is possible that the radioligand binds to two different receptors with different affinities or even that one receptor is present in two or more (non-interconverting) affinity states for the radioligand. This situation will result in a non-linear Scatchard plot: i.e. showing downward concavity (Figure 34 curve B).

Moreover, certain receptors (e.g. ion channel-gating receptors which make part of a larger structure) possess multiple binding which influence each other's binding characteristics. This may result in either negative or positive co-operative interactions among the binding sites. In other words, binding of the radioligand to one site decreases (negative co-operativity) or increases (positive co-operativity) the affinity of the radioligand for other sites. This will also result in non-linear Scatchard plots with, respectively, downward concavity (negative co-operativity, Figure 34 curve B) or upward concavity (positive co-operativity, Figure 34 curve C).

A more sensitive method to ascertain whether radioligand binding obeys the law of mass action is to analyse the 'Hill plot' of the saturation binding data (Figure 35). The Hill equation is, in fact, a logarithmic transformation of Equation (4).

Log(B/(Bmax — B)) is the ordinate and Log([L]) is the abscissa of the Hill plot. The slope corresponds to the Hill coefficient: 'nH'. The law of mass action is obeyed if nH = 1 (in practice, values between 0.8 and 1.2 will do). This means that the radioligand binds with the same affinity to all the sites. nH < 1 is indicative of either negative co-opera-tivity or of the existence of binding sites with different affinity. nH > 1 is indicative of positive co-operativity, i.e. where radioligand binding to one site increases the affinity of the radioligand for other sites.

[3H]RX821002 concentration (Log)

Figure 35 Hill plot of the saturation binding data from Figure 31 (B in fmol/mg protein, F in nM).

A disadvantage of a Scatchard plot is that the extrapolation of data obtained over an insufficient concentration range of L may give an artificial impression of the lack of complexity of the radioligand-receptor interactions. This may result in inaccurate Hill plots since they rely on a correct estimation of the Bmax value. Although Scatchard and Hill plots are still sometimes shown in publications for the sake of clarity (e.g. it is easy to visualize differences in KD and Bmax values of one or more radioligands with a Scatchard plot), radioligand binding parameters are now almost always calculated by computer programmes which are based on non-linear regression analysis of the saturation binding data. In the case of two non-interconverting binding sites, these programmes even allow the calculation of the concentration of each site and its respective affinity for the radioligand.

Finally, certain important considerations need to be taken into account before correctly analyzing saturation binding data; they include:

• The data must represent an equilibrium situation. In practice, this means that the incubation must have occurred long enough for equilibrium to be reached. Investigating binding of a given concentration of radioligand as a function of the incubation time can check this. This binding will increase time-wise until a plateau value (corresponding to the equilibrium situation) is reached. Equilibrium binding is often obtained within minutes at usual incubation temperatures (20-37 °C), but that it may become considerably longer when the temperature is lowered to (0-4 °C).

• Binding is expressed as a function of the free concentration of radioligand. This concentration may be set equal to the concentration of radioligand added (i.e. [L] = [Linit]) if only a small fraction of the added radioligand is bound (in other words, if most of the added radioligand still remains free). If a more substantial amount of radioligand is bound (e.g. >5%), then [L] is smaller than [Linit], and its correct value should be calculated by the equation: [L] = [Linit] — [L — R].

• The ligand must not aggregate, at higher concentrations, to a dimer or multimer.

Continue reading here: Kinetic experiments

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Readers' Questions

How to describe saturation binding?

Saturation binding is the process of binding molecules to a surface until no further binding is possible. It occurs when the amount of free sites on a surface is limited and the number of molecules that need to bind to it is greater than the number of sites. This causes the binding to become saturated and more molecules are unable to bind. Saturation binding is a common phenomenon when studying the binding affinity of macromolecules such as proteins and nucleic acids. It is also used as a tool to measure binding constants, which can be used to compare different molecules and their interactions.

What does bmax value on saturation binding of ans and bsa?

The Bmax value on saturation binding of ANS and BSA is the total number of binding sites present on the protein surface that can bind the respective ligand molecules. It is usually determined by fitting a binding curve to the data obtained when both molecules are present in increasing concentrations.

How to use kd and bmax to describe binding affinity?

Kd and Bmax are parameters used to describe the binding affinity between two molecules. The Kd parameter measures the affinity of the binding event and is defined as the concentration of a ligand at which half of the maximum achievable binding sites are occupied. The Bmax parameter describes the total number of binding sites available for a ligand to bind. When determining the binding affinity, it is important to measure the Kd and Bmax values of the molecules. This can be done experimentally by measuring the amount of free ligand available at various concentrations. When plotting the data, a sigmoidal curve is often used to represent the binding affinity. The Kd and Bmax parameters can then be obtained by fitting the data to the sigmoid curve and extracting the corresponding parameters.

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Parameters for Saturation Equilibrium Binding Assay

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Analyzing ligand depletion in a saturation equilibrium binding experiment

Affiliation.

• 1 Department of Biochemistry and Molecular Biology and Institute of Neurosciences, Universitat Autònoma de Barcelona (UAB), E-08193 Bellaterra, Spain. [email protected].
• PMID: 21638740
• DOI: 10.1002/bmb.2006.494034062677

I present a proposal for a laboratory practice to generate and analyze data from a saturation equilibrium binding experiment addressed to advanced undergraduate students. [(3) H]Quinuclidinyl benzilate is a nonselective muscarinic ligand with very high affinity and very low nonspecific binding to brain membranes, which contain a high density of muscarinic receptors. These features allow the instructor to devote especial emphasis to evaluate ligand depletion, and therefore, stress the subtle but fundamental difference between total (added) ligand and free ligand concentration at equilibrium.

Copyright © 2006 International Union of Biochemistry and Molecular Biology, Inc.

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Calculation and Visualization of Binding Equilibria in Protein Studies

Associated data.

A set of simulation applets has been developed for visualizing the behavior of the association and dissociation reactions in protein studies. These reactions are simple equilibrium reactions, and the equilibrium constants, most often dissociation constant K D , are useful measures of affinity. Equilibria, even in simple systems, may not behave intuitively, which can cause misconceptions and mistakes. These applets can be utilized for planning experiments, for verifying experimental results, and for visualization of the equilibria in education. The considered reactions include protein homodimerization, ligand binding to a receptor (or heterodimerization), and competitive ligand binding. The latter one can be considered as either a ligand binding to two receptors or a binding of two ligands to a single receptor. In general, the user is required to input the total concentrations of all proteins and ligands and the dissociation constants of all complexes, and the applets output the equilibrium concentrations of all protein species graphically as functions of concentration and as numerical values at a specified point. Also, a curve fitting tool is provided which roughly estimates the concentrations or the dissociation constants based on the experimental data. The applets are freely available online (URL: https://protsim.github.io/protsim ) and readily hackable for custom purposes if necessary.

1. Introduction

Biomolecular complex formation—the association of protein with other proteins or ligands as well as self-association—is an essential property of protein function. Association can be expressed with simple mathematical models describing chemical equilibria which were originally based on the law of mass action. The history for these models is long: Hüfner applied already in 1890 a chemical equilibrium model for describing dioxygen binding to hemoglobin. 1 The equilibrium model can be utilized successfully in many kinds of systems describing protein association. Amounts of different species in equilibrium can be calculated if the total concentrations of components and equilibrium constants (usually dissociation constants K D ) are known. However, the calculations become complicated when the number of system components increases, hampering the use of equilibrium models in protein studies.

Understanding reaction thermodynamics is an important skill for all chemists. Equilibrium states do not always behave intuitively when the conditions are changed, and people sometimes misinterpret their data and draw incorrect conclusions. Especially when working with reaction equilibria, for instance with proteins, ligands, and protein complexes, knowing the thermodynamic laws and their consequences is crucial. This knowledge helps with both planning experiments and interpreting results obtained therefrom.

In this work, a set of simulation applets which can be used to simulate complex formation of proteins has been developed. These tools can be used in designing experiments and interpreting results concerning the association behavior of proteins as well as for educational purposes. Visualization of equilibrium concentrations is especially useful in getting a quick grasp of how the concentrations behave when conditions are altered. The tools presented in this article allow such visualizations with simple controls and illustrative graphics.

Good planning of experiments saves time and materials. It is wasteful to prepare and measure samples that give no useful information, for instance, if one species is so dominant that others cannot be seen. In the typical case of figuring out dissociation constants, if a rough estimate of the values can be given, suitable conditions to measure can be determined, but it is not trivial. The simulation applets allow that by showing how the equilibrium concentrations behave when the initial concentrations are input.

Few examples of similar previously published work exist. Shave et al. 2 have developed a Python package with which homo- and heterodimerization inhibition can be modeled and similar graphs can be generated. While powerful, it requires that the user be able to write program code, the learning curve of which is generally prohibitive. The applets presented in this paper have purposefully been made simplistic so that anyone could use them with ease without needing to learn any programming skills. Regardless, a sufficiently skilled programmer could customize the applets for any needs not already covered.

2.1. Definitions

For a general association reaction

the equilibrium constant is called the association constant K A and is defined as

The inverse of this, the dissociation constant K D , has the same unit as the concentration, which makes it useful since it has a physically and chemically meaningful value:

The Gibbs free energy of the reaction Δ G is defined as

where R is the molar gas constant (∼8.314 J K –1 mol –1 ) and T is temperature (typically 298.15 K = 25 °C). The logarithm is defined only for dimensionless numbers, and in this case K D always has a concentration unit. However, the convention is to effectively take the concentration value in moles per liter and to discard the unit. 3

2.2. Homodimerization

Homodimerization or self-association of protein monomers is described by an equilibrium between dimer (P 2 ) and monomer (P):

There are two adjustable parameters in the model: one total concentration and one dissociation constant K D , which results in a simple quadratic equation ( Supporting Information , section 1.2). Equilibrium concentrations of free monomer P and dimer P 2 are calculated. This model is useful especially if the protein forms a weak or transient dimer with relatively high K D , 4 in which case the amount of dimer is highly dependent on the protein concentration used in experiments ( Figure ​ Figure1 1 ). On the other hand, the model can be used to estimate an approximate K D if the amounts of monomer and dimer in equilibrium are known.

Dimer formation as a function of total protein concentration c P . The pie diagram depicts the proportions of the dimeric (P 2 , red, 80.0%) and monomeric (P, blue, 20.0%) forms at the set value of c P = 1.0 × 10 –4 mol L –1 . The dissociation constant K D is set as 1.0 × 10 –5 mol L –1 , and it corresponds to the total protein concentration where the protein is half (50%) dimerized.

2.3. Ligand Binding to a Receptor

In this case, there is an equilibrium between the protein–ligand complex (PL) and the free uncomplexed species (P and L):

There are three adjustable parameters in this model: two total concentrations and one dissociation constant K D , which again results in a quadratic equation ( Supporting Information , section 2.2). Equilibrium concentrations of PL, P, and L are calculated, though only the species containing protein P are shown in the visualization. This model can be used to describe many kinds of important associations such as drug binding to a receptor, metal binding to a binding site, 5 substrate or inhibitor binding to an enzyme, and protein heterodimerization ( Figure ​ Figure2 2 ).

Linear ligand binding isotherm shows the ligand bound fraction of protein as a function of free ligand concentration [L]. The pie diagram depicts the proportions of the complex (PL, red, 90.8%) and free protein (P, blue, 9.2%) at the free ligand concentration which results from total protein concentration c P = 4.0 × 10 –5 mol L –1 , total ligand concentration c L = 1.5 × 10 –4 mol L –1 and dissociation constant K D = 1.0 × 10 –5 mol L –1 . In this representation, the dissociation constant corresponds to the free ligand concentration in which the protein is half (50%) saturated.

2.4. Competitive Binding of Two Ligands to One Receptor

In this model, two different ligands L and L′ compete with each other in binding to the same site of a receptor protein P.

There are five adjustable parameters in the model: three total concentrations (for protein P and two ligands L and L′) and dissociation constants ( K D and K D ′ ) for two complexes. This produces a cubic equation which is solved numerically, and equilibrium concentrations of all PL, PL′, P, L, and L′ are calculated. Receptor occupancy is calculated as a function of either total protein concentration or total ligand L concentration ( Figure ​ Figure3 3 ). The latter one is useful in designing and analysis of, for example, radioligand displacement or inhibition curves which can be further used to estimate IC 50 values.

Receptor protein occupancies as functions of the total concentration of competing ligand L. The pie diagram depicts the proportions of the complexes PL (red, 87.3%) and PL′ (blue, 10.6%) and the free protein P (green, 2.1%) at the set value of c L = 5.0 × 10 –6 mol L –1 . In this case, the concentration of receptor protein P is c P = 1.0 × 10 –6 mol L –1 , and the concentration of reference ligand L′ is c L′ = 1.0 × 10 –4 mol L –1 . The dissociation constants are K D = 1.0 × 10 –7 mol L –1 for the PL complex and K D ′ = 2.0 × 10 –5 mol L –1 for the PL′ complex.

2.5. Competitive Binding of a Ligand to Two Receptors

The configuration resembles the preceding one, and the same thermodynamic laws apply. The ligand L binds competitively to two different receptors P and P′.

as defined by Eaton et al. 8 It shows higher values of specificity at low ligand concentration but decreases when the ligand concentration increases, indicating loss of specificity ( Figure ​ Figure4 4 ).

Specificity α s as a function of ligand L concentration. The high affinity receptor ( K D = 1.0 × 10 –7 mol L –1 ) has a concentration of c P = 1.0 × 10 –6 mol L –1 . The low affinity receptor ( K D ′ = 8.0 × 10 –5 mol L –1 ) has a concentration of c P′ = 4.5 × 10 –5 mol L –1 . The specificity gets the value 2.2 at the set value of c L = 2.0 × 10 –6 mol L –1 .

The model can be also used in investigating the effect of nonspecific binding. The binding to a high-affinity receptor can be represented by low K D and low receptor P concentration c P . The nonspecific binding can be considered to follow similar rules of binding but the affinity is much weaker and the number of binding sites are much higher. This nonspecific binding can be represented by using a high K D and a high receptor P′ concentration c P′ . Receptor P′ can be in the same protein or in a different protein. In this model, the binding curve for the high-affinity receptor P has a hyperbolic shape and a saturation limit. In principle, the binding curve for the low-affinity receptor would also have a saturation limit but at much higher ligand concentration. In the concentrations in which the binding curve for the binding to P shows a hyperbolic shape, the binding curve for P′ is approximately linear ( Figure ​ Figure5 5 ).

Ligand binding isotherms in the competing receptors simulation applet. The same parameters have been used as in Figure ​ Figure4 4 . The binding curve of the high-affinity receptor (P, red) is hyperbolic and nearly reaches the saturation limit, but the binding curve of the low-affinity receptor (P′, blue) appears to be linear in this concentration range. At the set value of c L = 2.0 × 10 –6 mol L –1 , receptors P and P′ are 87.8% and 0.9% saturated, respectively, and the specificity α s = 2.2. At [L] ≈ 1.7 × 10 –6 mol L –1 , the amount of low-affinity complexes P′L reaches the amount of high-affinity complexes PL.

3. Results and Discussion

3.1. simulation applets.

A set of four simulation applets have been developed for visualizing equilibrium concentrations. The applets have been written in HTML, JavaScript, and CSS code and implemented as web pages that run in a web browser. The mathematics have been worked out on paper and programmed as functions in JavaScript. All numbers are stored internally as double-precision floating-point numbers which are fast but suffer from rounding errors and loss of precision in some cases. However, as long as the adjustable parameters are kept in the ranges of the slider elements, these effects are insignificant and almost unnoticeable. The internal calculations use mostly simple arithmetics; cubic equations are solved using Newton’s method and the bisection method as fallback, and a least-squares fit algorithm has been written for the curve fitting. In addition to text and links, an applet page consists of input elements, an output graph, and an output table. Adjusting the input triggers JavaScript code to update the output. The graph is implemented as a Scalable Vector Graphics (SVG) object which can be zoomed in indefinitely in the browser and resized by dragging the bottom-right corner. The graph can also be exported as an SVG file by clicking the “Export graphic (SVG)” button, and the file can be opened in a graphics editor, for example in Inkscape, 9 CorelDRAW, 10 GIMP, 11 or Photoshop. 12 Screenshots of the user interfaces of each applet are shown in Figures S1–S4 .

The simulation applets visualize the equilibria of the aforementioned reaction cases. The user is able to set total concentrations of the species and the dissociation constants of the complexes using sliders or, alternatively, manually by double-clicking the value label and editing the value in the appearing text box. In the latter case, any value in the range 10 –18 to 10 3 is allowed, though values outside the range of the sliders 10 –9 to 10 –1 may cause numerical and graphical glitches. This is indicated by the changing color of the text box: disallowed values are red, values in the range of the sliders are green, and others are dark yellow. The association constants K A and the Gibbs free energies Δ G corresponding to the set dissociation constants are calculated and displayed as well. The x -axis is configurable to total concentrations or, in the homodimerization, ligand binding and competing receptors simulations, equilibrium concentrations [P], [L], and [L], respectively. The y -axis depicts either the absolute equilibrium concentrations of each species in a logarithmic scale or, in all but the competing receptors simulation, the relative amounts of the species in a linear scale. In the competing receptors simulation, instead of the relative amounts, absolute concentrations are shown in a linear scale, which was deemed to be a more useful and less confusing option. In addition to the curves, the applet shows the equilibrium concentrations at the specified point on the x -axis (the value of the parameter set as the x -axis) in a table under the graph and the proportional amounts as a pie diagram. All the adjustable parameters and calculated equilibrium concentrations in each applet are shown in Table 1 .

There is also a curve-fitting tool that can be used to approximate unknown initial concentrations or dissociation constants using data points. The interface is shown in Figure S5 . The data are input in the text box as pairs of numbers row by row, expressed in plain text as decimal numbers or in the E notation (e.g., “0.00028” or “2.8e-4”). Once input, the data points are drawn in the graph as crosses. One can then choose any of the equilibrium concentration curves to be fitted to the data points. The unknown parameters must be set as free parameters by ticking the corresponding checkboxes, and the calculation attempts to find the values for them resulting in the best possible fit. There are three methods of finding the fit:

• Two-pass search first goes through the ranges of the sliders coarsely and picks the values where the sum of squared residuals is the smallest. Then, a fine search around that point is done to find the best possible solution. This is usually quick but can potentially converge to wrong results in extreme cases where the coarse search results in an incorrect point. This is the default option.
• Single-pass search simply does a full search of the whole input space, in other words, it checks all possible combinations of slider positions. This will always find the best possible solution lying in the defined intervals, but it is slow when more than two parameters are searched.
• Iterative search is like the second step of the two-step search, but the starting values (initial guess) are input with the sliders first, and steps are done iteratively until the search converges on a single point. If the initial guess is close to a solution (a local minimum of the sum of squared residuals), the search will find it quickly.

It is important to realize that the curve-fitting tool is not a proper analytical tool. Being constrained to the possible slider positions, it does not give the most accurate values nor, most importantly, uncertainties of any kind. More accurate estimates of K D values can be obtained by nonlinear curve fitting (for example, in MATLAB 13 or GNU Octave 14 ) provided that an analytic solution for binding equilibria or an appropriate numerical fitting algorithm is available.

The applet source files are freely available via GitHub 15 under the GNU General Public License, version 2. 16 There is also a GitHub page 17 (URL: https://protsim.github.io/protsim ) with which the simulation pages can be directly accessed using a desktop or mobile web browser. The code can be freely downloaded, used, and modified by anyone provided that the authors are acknowledged by citing this paper and that any modifications contain all source code under the same or compatible license.

3.2. Examples of Use

3.2.1. planning a measurement of homodimerization affinity.

Consider a case in which the approximate dissociation constant of a protein dimer is known. For example, Haka et al. 18 estimated a K D of 0.8 mol L –1 for the Triple 3 variant of the Equ c 1 allergen using a single native mass spectrum. If the K D was to be determined more accurately with multiple data points, at which protein concentrations should the new measurements be done? The method of measurement directly or indirectly returns the proportions of monomer and dimer at each tested total protein concentration.

In the protein homodimerization simulation, K D is set to 8.0 × 10 –4 mol L –1 , vertical scale to relative protein concentration and horizontal axis to total protein concentration. The curves of [P 2 ] and [P] show the behavior of the equilibrium when the total protein concentration changes. The range useful for the determination of K D will be near the set value where both species are present in significant amounts. Say, if proportions less than 20% cannot be reliably observed, then it is not necessary to prepare and measure samples for which either proportion is less than 20%. When c P is set to 1.2 × 10 –4 mol L –1 , the proportion of the dimer is 19.5% ( Figure ​ Figure6 6 ), and when it is set to 8.0 × 10 –3 mol L –1 , the proportion of the monomer is 20%. Therefore, the useful protein concentration range will be roughly 0.1–8 mmol L –1 . Of course, if the initial estimate of K D is inaccurate, the experiments may yield unexpected results, but this is the range worth checking first.

Relative amounts of protein monomer (P, blue) and dimer (P 2 , red) as concentrations of total protein concentration when the dissociation constant K D = 8.0 × 10 –4 mol L –1 . At c P = 1.2 × 10 –4 mol L –1 , the relative amounts of monomer and dimer are 80.5% and 19.5%, respectively.

3.2.2. Competitive Binding of Oxygen and Carbon Monoxide to Hemoglobin

To demonstrate the competing ligands simulation, consider how hemoglobin can bind oxygen and carbon monoxide competitively with different affinities. In reality, hemoglobin exists as a homotetramer in biological conditions, and O 2 or CO binding is associated with allosteric regulation (positive co-operativity). For simplicity, in this example, hemoglobin is treated as a monomeric protein with a single binding site for O 2 or CO. The association constants of the complexes with carbon monoxide (L) and oxygen (L′) are K D = 7.5 × 10 8 L mol –1 and K A ′ = 3.2 × 10 6 L mol –1 , respectively, 19 and the corresponding dissociation constants are K D = 1.3 × 10 –9 mol L –1 and K D ′ = 3.1 × 10 –7 mol L –1 , respectively. Let us use the value c P = 9.0 × 10 –3 mol L –1 (14.5 g/dl) for a normal hemoglobin concentration in human blood 20 and set vertical scale to relative and horizontal axis to total ligand L concentration. First, to find out at which oxygen concentration the hemoglobin is 95% saturated (a healthy oxygen saturation), set c L to the minimum slider value of 1.0 × 10 –9 mol L –1 and find a value for c L ′ such that the proportion of PL′ is 95%. This occurs roughly at c L ′ = 8.6 × 10 –3 mol L –1 ( Figure ​ Figure7 7 ). Then, increasing c L , the concentration of PL gradually increases and causes the proportion of PL′ to decrease. At c L = 9.0 × 10 –4 mol L –1 (10% of c P and 10.5% of c L ′ ), the proportion of PL is 10%, a level indicative of carbon monoxide poisoning. 21 This shows how carbon monoxide displaces oxygen in hemoglobin when its concentration in blood increases, though in reality the situation is more complex: the dissolution of gases in blood is ignored, and because carbon monoxide tends to accumulate in the body, much lower concentrations in air than 10.5% of oxygen are sufficient for causing carbon monoxide poisoning over time.

Competitive binding of two ligands with parameters c P = 9.0 × 10 –3 mol L –1 , c L ′ = 8.6 × 10 –3 mol L –1 , K D = 1.3 × 10 –9 mol L –1 , and K D ′ = 3.1 × 10 –7 mol L –1 , modeling the competitive binding of carbon monoxide (L) and oxygen (L′) to hemoglobin (P). When the total concentration of carbon monoxide ( c L ) increases to 9.0 × 10 –4 mol L –1 , the relative amount of carboxyhemoglobin (complex PL) is increased to 10%.

3.2.3. Dissociation Constant of Protein–Metal Complex

As an example, let us recreate the calculation of the dissociation constant of the xylonolactonase–iron complex reported by Pääkkönen et al. 5 The raw data used in the original calculations are presented in Table 2 . The data are the relative amounts of the complex against free iron concentration, so the vertical scale will be set as relative, and the horizontal axis will be set as the free ligand concentration. In this situation, the concentration of the protein c P does not affect the shapes of the curves, but it can still be input as 1.2 × 10 –6 mol L –1 using the slider. The setting of the free ligand concentration [L] only affects the pie diagram and the table of concentrations, so it can be set as any arbitrary value. The dissociation constant K D can also be at any value at this point.

The choice of the calculation method does not matter in this case since they will all give the correct answer reasonably fast. When the curve [PL] is set to be calculated, K D is chosen as the free parameter, and “Calculate” is clicked, the algorithm finds the least-squares fit and sets the K D as 4.5 × 10 –7 mol L –1 . The corresponding values of the association constant K A = 2.2 × 10 6 L mol –1 and the Gibbs free energy Δ G = −36.2 kJ mol –1 ( T = 25 °C) are also displayed. The resulting graph, when c L is set as 10 K D = 4.5 × 10 –6 mol L –1 , is shown in Figure ​ Figure8 8 .

Least squares fit of the data in Table 2 in the ligand binding simulation. The concentration c P is set as 1.2 × 10 –6 mol L –1 , and the dissociation constant K D = 4.5 × 10 –7 mol L –1 has been determined by the fitting algorithm. The pie diagram depicts the proportions of PL (88.4%) and P (11.6%) at the set value of c L = 10 K D = 4.5 × 10 –6 mol L –1 .

The reported value for K D , calculated using unweighted orthogonal distance regression, is (5.0 ± 1.3) × 10 –7 mol L –1 . The values are different because the regression methods are different and because the model in the publication 5 also accounts for the maximum saturation level, which this model assumes as unity. In any case, this result would be a reasonable approximation of the correct value of K D , though only to the precision of one significant digit despite the applet displaying two.

3.2.4. Dissociation Constant of Protein Dimer

As another example, let us recreate the calculation of the dissociation constant of the wild-type Equ c 1 allergen dimer reported by Haka et al. 18 The raw data used in the original calculations are presented in Table 3 . The data are the concentrations of free monomer [P] against total protein concentration c P , so the vertical scale will be absolute, and the horizontal axis will be total protein concentration. The values of c P and K D can be set as any arbitrary value at this point.

The data are input as described above, and the calculation method can be left as the default value. When the curve [P] is set to be calculated and “Calculate” is clicked, the calculation yields K D = 1.6 × 10 –8 mol L –1 , K A = 6.2 × 10 7 L mol –1 and Δ G = −44.5 kJ mol –1 . The resulting graph, when c P is set as 4.0 × 10 –5 mol L –1 (as in the measurement of the mass spectrum in Figure 2B of the publication 18 ), is shown in Figure ​ Figure9 9 . The reported value for the K D is (1.56 ± 0.08) × 10 –8 mol L –1 , very close to this result. The authors have done a similar least-squares fit, and this calculation would have converged to the same value if it were not restricted to the discrete values of the slider.

Least squares fit of the data in Table 3 in the homodimerization simulation. The dissociation constant K D = 1.6 × 10 –8 mol L –1 has been determined by the fitting algorithm. The pie diagram depicts the proportions of P 2 (98.6%) and P (1.4%) at the set value of c P = 4.0 × 10 –5 mol L –1 .

4. Conclusions

The presented simulation applets are useful for visualizing the behavior of equilibrium reactions as shown in all figures and the examples of use. As demonstrated, the applets can be used for planning experiments, for predicting the behavior of systems to which they are applicable and for estimating unknown parameters based on experimental data. Similarly, experimental results can be verified by calculating the theoretical behavior and comparing it to the experimental behavior. Anyone with sufficient programming skills can download the applets and modify them according to their needs and preferences. The hope is that these applets will be useful for researchers who work with equilibrium reactions like these. In the referenced works, such tools have not been available, and while alternative methods for calculation and visualization are perfectly valid, using these simple applets would speed up the workflow and reduce errors.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.2c00560 .

• Detailed mathematical models, data of the curve-fitting examples in copy-pastable format, figures of applet interfaces ( PDF )

The authors declare no competing financial interest.

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