What is binding affinity

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Events and training Live events Recorded Events User training events. About Us. Products by Measurement type Binding affinity Binding affinity. Label free protein ligand interaction analysis solutions using ITC. What is Binding Affinity? 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.

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. Therefore, one should ensure, to the extent possible, maximum purity of both binding components.

Typically, the underlying experimental observation is an absence of observed binding up to a certain protein or ligand concentration. 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, While this objection may seem like a technicality, there are many instances where known binders do not give a gel shift or filter binding.

But the reality of the interpretation of these experiments—and the reality of molecular interactions—is more nuanced Pollard, 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.

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.

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. 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, ; Sanders, ; Ryder et al. 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.

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. 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.

Four of the 20 papers also appeared in the above list. Equilibration was evaluated as follows. If a study reported systematically varying the incubation time, it was counted as controlled for equilibration. Studies exclusively using approaches that intrinsically monitor the binding progress ITC, SPR, biolayer interferometry [BLI] also were counted as 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. If only a lower limit of the incubation time was reported e. If two sequential incubations were performed at different temperatures e. 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.

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. For studies using Equation 4b , we asked if the lowest apparent K D value was in at least 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 Equation 5 was used incl.

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.

This included two cases in which the authors had performed rigorous equilibration and titration controls in their previous referenced work. The RNA-binding domain residues — of S. The construct was transformed into E. Bound protein was washed extensively over a shallow 10—25 mM imidazole gradient and eluted over a linear 25— mM gradient of imidazole. Protein was eluted over a linear gradient of potassium acetate from 50 to mM. The labeled RNA concentrations and incubation times are indicated in the individual figure legends.

Following the incubation, 7. The unlabeled RNA in the loading buffer prevented additional association to the labeled RNA from occurring during sample loading Appendix 2—note 2. All samples were loaded on the gel within 20 min from mixing with the loading buffer. Aliquots 7. Extreme caution must be exercised at this step; see, e. Fitting was performed with KaleidaGraph 4. 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. Labeled RNA concentrations were 0. Dissociation was initiated by transferring the binding reaction to 2. At various times, 7. All time courses were fit to single exponentials using KaleidaGraph 4.

The fractions of bound labeled RNA ranged from 0. At varying times, 7. The k on values were corrected for the active protein fraction. The fraction bound RNA was determined as described in Equilibrium binding measurements. The curves were fit to Equation 9 , as described in Appendix 3. 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. 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. 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. 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.

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. 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.

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. Hadizadeh et al. 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.

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. A Mixing scheme for measuring association rate constants. 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.

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. 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.

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.

LeCuyer and Crothers, ; Smith et al. 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.

Uhlenbeck, ; Herschlag, 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:.

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. 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.

Buenrostro et al. 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 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, ; Hulme and Trevethick, ; Sanders, 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.

We evaluated published binding measurements using RNA-protein interactions as an illustrative example. We surveyed 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.

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. A Percentages of publications that did or did not report and vary the incubation time.

The light gray portion of the first column indicates the studies using SPR and ITC, techniques in which time is varied by default. B Incubation times in papers that reported a single time. A Percentages of publications that did blue or did not red control for titration effects. The first category includes studies that systematically varied the limiting component concentration to rule out titration.

Studies that reported using an appropriate concentration regime or analysis methods to minimize the effects of titration second and third column, respectively were considered titration controlled; nevertheless, we emphasize the importance of performing and reporting the control experiments described herein, instead of relying on concentrations alone see section 'Avoid the titration regime'. B Breakdown of studies that did not report controlling for titration.

The first three columns denote studies that assumed negligible concentration of the limiting component in their analysis; however, the reported concentrations and K D values were inconsistent with this assumption, with the ratio of the lowest measured K D value to the limiting component concentration indicated.

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 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]. 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.

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 fold, and other examples have been previously noted Hulme and Trevethick, ; Strohkendl et al. 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.

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'.

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.

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. Below we adopt the more common standard of taking reactions to five half-lives, or 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:.

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. Thus, the more long-lived the complex i. 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, 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, ; Sanders, 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.

Semenova et al. But when target dissociation of these proteins was measured over time, it took many hours Strohkendl et al. 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.

Subsequently, binding studies extended to longer times revealed that the 4. Exploring the time dependence of the assembly process changed the mechanistic conclusions: 4. 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 fold. Below we illustrate this control, and the need for it, with experimental results for Puf4 binding to its consensus RNA.

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. The fraction of bound RNA was subsequently determined by non-denaturing gel electrophoresis see Materials and methods.

A Mixing scheme. 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. Data were collected at protein concentrations greater than or equal to the concentration of labeled RNA 0.

A Binding parameters for protein P interactions with two ligands, L1 and L2. The dissociation rate constant k off for L1 is fold lower than for L2, such that L1 requires much longer to equilibrate than L2 Equation 2. B Simulated binding data for L1 and L2 with varying incubation times t 1. The binding to each ligand is measured individually with trace amounts of L1 blue or L2 red. Solid lines are fits to an equilibrium binding equation Equation 4b , with dashed lines indicating the protein concentration at which half of the ligand is bound.

Arrows and numbers indicate K D app rel values at each time point. Note the systematic deviations of the simulated data from the fit curve in cases where equilibrium has not been reached. The presence of such deviations in experimental data indicates the need for additional controls to establish equilibration and rule out titration.

Not until the incubation was extended to 4. Consequently, equilibration of Puf4—RNA binding on ice requires at least 4. The importance of this excess to obtain reliable K D values is described in the next section. 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.

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 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 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. The same simulated binding curve is shown in linear top and log bottom plots, as both are useful and common in the literature. Curves indicate fits of the simulated data to a hyperbolic equation Equation 4b.

Ten datasets were simulated per condition and noise level and were individually fit to Equation 4b leftmost column or Equation 5 the remaining columns to determine the K D. The binding curves are shown as black lines, and the overlaid white circles indicate the expected fractions bound if the data were not affected by noise, with error bars indicating the standard deviation.

Gray bars indicate that the K D could not be determined from a quadratic fit. CIs that extend beyond the axis limits indicate that the lower limit of the K D was not defined. Note that with increasing noise and increasing RNA concentration the K D values derived from the quadratic fits become increasingly poorly constrained, particularly the lower CIs.

By contrast, using the binding regime and Equation 4b to fit the data leftmost column consistently yields well-defined K D values, even with substantial noise. A Affinities of protein P for ligands L1 and L2. B Simulated equilibrium binding curves.

Solid lines are fits to Eq. There is a pronounced dependence of apparent relative affinity on ligand concentration if [L] is not much lower than the K D for the most tightly bound ligand among the ligands being compared. If sufficiently low ligand concentrations are not accessible, Equation 5 should be used and results may be less reliable see section 'Avoid the titration regime' of main text.

In each case, protein concentration is varied 6, 18, 54, arbitrary units , and K D equals 18 in the same units. C Protein concentration dependence of binding in each of the above regimes. The discrepancy would further increase with higher RNA concentrations, as shown in Figure 5—figure supplement 1. We can understand the origin of this discrepancy as follows. In part A , the RNA concentration red is below the K D value and below the protein concentration blue , such that the free concentration of the protein is essentially unchanged after RNA binding at both saturating complete binding of RNA and sub-saturating protein concentrations.

Thus, the free protein concentration, which determines the extent of binding according to Equation 4a , is depleted and can no longer be approximated by the total protein concentration in Equation 4b to obtain an accurate K D value. On the molecular scale, the lowered free protein results in less binding. Consequently, for a given K D , more protein is required to achieve half-saturation at higher RNA concentration than with a trace concentration of RNA.

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. 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. 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.

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. 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. However, in most other cases, this limit is much lower. 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. 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.

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.

Figure 5—figure supplement 4 illustrates an example, in which two substrates with a 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. For simplicity, only the lower limits of RNA concentration are indicated; the corresponding upper limits were 15— pM RNA see Materials and methods and Appendix 2—note 4.

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