Binding Curve Viewer: Visualizing the Equilibrium and Kinetics of Protein–Ligand Binding and Competitive Binding

Understanding the thermodynamics and kinetics of the protein–ligand interaction is essential for biologists and pharmacologists. To visualize the equilibrium and kinetics of the binding reaction with 1:1 stoichiometry and no cooperativity, we obtained the exact relationship of the concentration of the protein–ligand complex and the time in the second-order binding process and numerically simulated the process of competitive binding. First, two common concerns in measuring protein–ligand interactions were focused on how to avoid the titration regime and how to establish the appropriate incubation time. Then, we gave examples of how the commonly used experimental conditions of [L]0 ≫ [P]0 and [I]0 ≫ [P]0 affected the estimation of the kinetic and thermodynamic properties. Theoretical inhibition curves were calculated, and the apparent IC50 and IC50 were estimated accordingly under predefined conditions. Using the estimated apparent IC50, we compared the apparent Ki and Ki calculated by using the Cheng–Prusoff equation, Lin–Riggs equation, and Wang’s group equation. We also applied our tools to simulate high-throughput screening and compare the results of real experiments. The visualization tool for simulating the saturation experiment, kinetic experiments of binding and competitive binding, and inhibition curve, “Binding Curve Viewer,” is available at www.eplatton.net/binding-curve-viewer.

1 Solutions of differential equations and equilibrium concentrations Throughout this section, we describe the binding process initiated by mixing an arbitrary volume of protein with an arbitrary volume of ligand.This meant that there was no protein-ligand complex at the start of the binding process.The binding of the protein and ligand was 1:1 stoichiometry and had no cooperativity.In the dissociation process, we assumed that there was no rebinding.

Kinetics of the second-order binding process
The binding of the protein and ligand is described by the following equation,

PL
The concentration of the protein-ligand complex changes during the binding process and where the [P] and [L] are the concentrations of the unbound protein and ligand, the k on and k off is respectively the association rate constant and dissociation rate constant.The concentration of the total protein and ligand, [P] 0 and [L] 0 , and the concentration of the unbound protein and ligand, [P] and [L] are related by  on d = [PL] 2 -( [P] 0 + [L] 0 +  off  on ) [PL] + [P] 0 [L] 0 The coefficients and discriminant of the right-hand side quadratic equation are =  on d From the expression of [PL] 1 , we know that [PL] 1 > 0. When [PL] increases from 0 to [PL] 1 , decreases from the maximum value to 0, i.e., , the binding process approaches equilibrium, so .Thus, the above equation gives When t = 0, [PL] = 0. Thus , substituting the above equation gives

Thermodynamics of the second-order binding process
The equilibrium state of the binding process is described by the following equation, where the [P] eq , [L] eq , and [PL] eq is the equilibrium concentration of the unbound protein and ligand and the equilibrium concentration of the protein-ligand complex.The total protein and ligand concentration, [P] 0 and [L] 0 , and the equilibrium concentration of the unbound protein and ligand [P] eq and [L] eq are related by Thus, This equation is the same as the right-hand side of equation 1.2, its coefficients and discriminant are From the expressions of [PL] eq1 and [PL] eq2 , we know that 0 > [PL] eq1 > [PL] eq2 .We assume that the [PL] increases from 0 to the first real root to make the quadratic equation zero.So, [PL] eq1 should be the equilibrium concentration of the protein-ligand complex.It should be noted that the thermodynamic process of the binding reaction is independent of the initial concentrations of the binding species.Hence, we can calculate the total concentration of the protein and ligand from the initial binding system and use the approaches presented here to calculate the equilibrium concentration of the protein, ligand, and protein-ligand complex.

Kinetics of the pseudo-first-order binding process
The binding of the protein and ligand is described by the following equation, , the binding process is effectively first-order since the [L] is hardly affected by the [L] 0 ≫ [P] 0 binding process, then the equation can be transformed to

PL
The concentration of the protein-ligand complex changes during the binding process and During the binding process, , then The total protein concentration ([P] 0 ) and the concentration of the unbound protein ([P]) are related by Thus, When [PL] increases from 0 to [PL] eq , decreases from the maximum value to 0. and Because the right-hand side of equation 1.6 is greater than 0, When .Thus, , substituting the above equation gives Substitute equation 1.8 with equation 1.7 gives Define the observation rate constant k obs by

Thermodynamics of the pseudo-first-order binding process
The equilibrium state of the binding process is described by the following equation, [PL] eq where [P] eq , [L] eq , and [PL] eq are the equilibrium concentrations of the unbound protein and ligand and the equilibrium concentration of the protein-ligand complex.If , or strictly speaking, , .Hence, the [L] 0 is hardly affected by the binding process, then the equation can be transformed to The total protein concentration ([P] 0 ) and the equilibrium concentration of the unbound protein ([P] eq ) are related by Thus, In either the second-order binding process or the pseudo-first-order binding process, the [PL] eq is the same calculated by the thermodynamic and kinetic approaches.But the equations calculated from the second-order binding process are different from the equations calculated from the pseudo-first-order binding process.

Kinetics of the dissociation process
The dissociation of the protein-ligand complex with no rebinding is described by the following equation, The concentration of the protein-ligand complex changes during the dissociation process and . Thus, , substituting the above equation gives

association and dissociation
We simulated the competitive binding of association and dissociation with the denominator of the time step being 200,000 and 400,000 respectively in duration of shorter t 0.99 plus 10×longer t 0.99 (see Methods).The theoretical [PL] eq-theo was calculated by Wang equation 1 .In the competitive binding of dissociation, the volume ratio was 1, i.e., the volume of the protein and ligand was the same as the volume of the inhibitor.For example, in exp.No. 1d, after mixing, the total concentration of the protein, ligand, and inhibitor is 0.1, 1, and 10, respectively.The experiment conditions and results are shown in the Table S1 and S2.In all experiments, the difference between the [PL] eq-theo and [PL] eq-sim was less than 0.1% of the [PL] eq-theo .
Table S1.Comparison of the simulated [PL] eq and theoretical [PL] eq under four experiment conditions Exp.No.

Hypothetical association and dissociation kinetic experiments
We first demonstrated two ligands with the same thermodynamic property but different kinetic properties (Table S3).The ligand in experiment No. (hereafter referred to as exp.No.) 1 was fast-onfast-off, its k on was 1e6 M -1 s -1 and k off was 0. approached the [PL] eq .When the [L] 0 was the largest in exp.No. 5, the [PL] pseudo-eq and [PL] eq was the most approximate to each other.This was consistent with the condition "[L] 0 >> [P] 0 " of the pseudofirst-order binding process.As shown by exp.No. 5, 6, 7, and 8, increasing either the k on or k off by 10fold accelerated the binding process to 99% equilibrium, whereas increasing k on had a more obvious effect on the acceleration of the binding process.
For the ligand in exp.No. 1, the k off was 0.1 s -1 , the half-life for dissociation (i.e., ln(2)/k off ) equaled 7 seconds and the time to 99% complete dissociation equaled 46 seconds.For the ligand in exp.No. 2, the k off was 0.01 s -1 , the half-life for dissociation equaled 69 seconds and the time to 99% complete dissociation equaled 461 seconds.

Table S3. The association kinetic experiments in different conditions
Exp. No.
[P] 0 (nM) [L] 0 (nM) k on (M -1 s - 1 ) [PL] eq (nM) [PL] pseudo- eq (nM) t  S3 and S4).The consistent t 0.99 suggested that the competitive binding process could be accurately approximated by numerical simulation.The simulations of four competitive binding experiments also showed that the errors between the simulated [PL] eq and theoretical [PL] eq were less than 0.1% of the theoretical [PL] eq in our default setting (Table S1 and S2).
We changed the concentrations and kinetic parameters in the competitive binding of association to compare the numerical simulation of the second-order process and the analytical integration of the pseudo-first-order process (Table S4).As shown in exp.No. 10a, we decreased the [P] 0 from 100 nM to 10 nM, which was less than the [L] 0 and [I] 0 , with the kinetic properties being constant, the equilibrium concentration ([PL] pseudo-eq ) of the protein-ligand complex of the pseudo-first-order binding process approached that of the second-order binding process (the theoretical [PL] eq ).The t pseudo-0.99 of the pseudo-first-order binding process also approached the t 0.99 of the second-order binding process.In exp.No. 11a and 12a, with the increment of the [L] 0 from 75 nM to 150 and 300 nM, and other parameters being constant, the t 0.99 decreased and the IC 50 increased.In exp.No. 13a and 14a, with the increment of the [I] 0 from 75 nM to 150 and 300 nM, and other parameters being constant, the t 0.99 also decreased.Compared with exp.No. 13a and 14a, the t 0.99 of the slow-on-slow-off inhibitors in exp.No.
15a and 16a increased.Furthermore, the larger the [I] 0 was, the less the t 0.99 was.The t 0.99 of the fast- No. 19d, 20d, 21d, and 22d.The t 0.99 being 0 meant that the instantaneous concentration of the proteinligand complex after 1:1 mixing with the inhibitor had already been in the range of [0.99×theoretical [PL] eq , 1.01×theoretical [PL] eq ].

Table S4. Competitive binding experiments of association and dissociation in different conditions a
Exp. No.
[P]0 (nM) [L]0 (nM) kon-inhibitor (M -1 s - a For easy comparison, the [P]0, [L]0, and [I]0 in the competitive binding of dissociation have been converted to the total concentration of the protein, ligand, and inhibitor after mixing in the volume ratio of 1.

[PL] curves in "Competitive Binding Kinetics -Dissociation"
showed S-shape in increasing concentrations of free protein The volume ratio of the protein and ligand to the inhibitor was 1, 10, and 100 in (A), (B), and (C).From (A) to (C), at the time of mixing, the concentration of the free protein increased.the inhibitor could bind more free proteins, so the concentration of the protein-ligand complex decreased more slowly.
In the competitive binding experiment, the binding of the protein and ligand and the binding of the protein and inhibitor are described by the following equations, The K d is the dissociation constant of the protein and ligand.The K i is the dissociation constant of the protein and inhibitor.At equilibrium, the total concentration of the protein ([P] 0 ) is the sum of the equilibrium concentration of the protein ([P] eq ), the equilibrium concentration of the protein-ligand complex ([PL] eq ), and the equilibrium concentration of the protein-inhibitor complex ([PI] eq ). (8.1) At equilibrium, [PL] eq (8.2)  i = [P] eq [I] eq [PI] eq Equation 8.2 can be written as We substitute equation 8.3 into equation 8.1 and obtain [P] 0 = P eq + [PL] eq + [P] eq [I] eq  i (8.4) We multiple on both sides of equation 8.4 and obtain Equation 8.5 can be transformed into equation 8.6 When 50% of the "initial binding" of the protein and ligand is inhibited, the concentration of the free competing inhibitor ([I] eq ) is defined as the IC 50 and the concentration of the total competing inhibitor ([I] 0 ) is defined as the apparent IC 50 .The initial binding means the equilibrium concentration of the protein-ligand complex in the blank control in the competitive binding of either association or dissociation.In association, the blank control was the mixture of the protein and the solution with ligand and no inhibitor.In dissociation, the blank control was the mixture of the equilibrated protein and ligand and the solution without inhibitor.The blank control is important to eliminate the equilibrium concentration change of the protein-ligand complex upon the volume change after mixing.At the equilibrium state of 50% inhibition of the protein-ligand binding, the concentration of the protein-ligand complex ([PI] eq-50 ) can be written as (8.7) [PL] eq -50 = [P] 0 [L] eq -50 In this work, we used the Wang equation 1 to simulate the theoretical inhibition curve, and estimated the IC 50 and apparent IC 50 (see Methods), without the restrictions of concentrations.[PL] eq -50 ≈ [P] 0 [L] 0 Thus, IC 50 is almost equal to [I] 0 .From equation 1.13, we know that in the absence of the inhibitor and when , the equilibrium concentration of the protein-ligand complex ([PL] eq-0 ) equals At the time of 50% inhibition of the protein-ligand binding, , so [PL] eq -0 = 2 × [PL] eq -50 K i in higher numerical precision (Figure S10).9 Protein and ligand with optimal K d showed the highest sensitivity in primary screen To theoretically access the sensitivity of the competitive binding assay in the primary screen, we conducted simulation experiments using different combination of the K d and K i .In experimental groups, after mixing, the initial working concentration of the inhibitor was 10 μM.The equilibrium concentration of the protein-ligand complex in blank controls ([PL] 0 ) was 3.0, 5.0 nM, and 7.0 nM, which was respectively 30%, 50%, and 70% of the total concentration of the ligand (10 nM).The K d was from 1 to 5,000 nM and the K i was from 1 to 10,000 nM.Both K d and K i were increased by a step size of 1, which resulted in a 5000×10000 K d -K i matrix.
Under the conditions of [PL] 0 = 3.0, 5.0, and 7.0 nM, the ratio of the equilibrium concentrations of the protein-ligand complex in experimental groups ([PL] eq ) to [PL] 0 are shown in Figure S11A-C.With the K d being constant, inhibitors possessing larger K i showed lower levels of inhibition (i.e., larger [PL] eq /[PL] 0 ).As the K d increased, the corresponding K i on each contour line first increased and then decreased.The increasing and decreasing trends were more obvious in the case of [PL] 0 = 7.0 nM.At a preferred level of inhibition, the K d corresponding to the largest K i is expected to represent the most sensitive configuration in the primary screen.We graphed the most sensitive K d at all inhibition levels (Figure S11D-F).We simulated a series of competitive binding experiments under different conditions.We kept the [PL] eq in blank controls 5.0 nM, half of the total concentration of the ligand (10 nM).The relationship between the equilibrium concentration of the protein-ligand complex in experimental groups and the concentration of the inhibitor under different K d and K i is shown in Figure S12.At lower K d values, the apparent IC 50 was closer to the IC 50 .When the [P] 0 was larger, the more inhibitors were needed to inhibit the binding of the protein and ligand.

Figure S12.
The inhibition curves under different experimental conditions.The total concentration of the ligand was10 nM.In blank controls, the [PL] eq equaled 5.0 nM.The units of all parameters in the legends were nM.
In this section, we reanalyzed some of the data from Jarmoskaite et al.'s radioactive binding assays 5 .In their affinity measurements, the labeled RNA was used as the fixed component, the concentrations of the RNA-binding protein Puf4 were varied.We used the kinetic parameters (in Table 2 of their paper) and related concentrations to calculate the time to reach 99% equilibrium at 25℃ and 0°C (Figure S13 and Table S5).We used the lower limit of the labeled RNA concentration (0.002 nM, see Figure 4 of their paper) as the initial concentration of the fixed component.The gradient concentrations of the Puf4 were 0.001, 0.01, 0.1, 1, 10, and 100 nM.The longest time to reach 99% equilibrium should be used as the incubation time.At 25℃ and 0°C, across all the concentrations of Puf4, the longest time was 3.03e+2 and 1.64e+4 s (i.e., 0.1 and 4.6 h) respectively.Our results of the analysis agree with their results in Section "Time dependence of Puf4 binding at 25°C and 0°C" (see Figure 4 of their paper).
Next, we analyzed the RNA concentration dependence of Puf4 binding at 25°C and 0°C.We used the lower limit of the labeled RNA concentration shown in Figure 6 of their paper as the initial concentration of the fixed component.The K d calculated by using the quadratic equation (in Table 2 of their paper) was used in our analysis.At 25°C and 0°C, the K d was 120 and 1 pM, respectively.Under different labeled RNA concentrations, the apparent K d s calculated by using Binding Curve Viewer were shown in Table S6.Users can easily compare K d and apparent K d with the tool.S6.

Figure S1 .
Figure S1.The relationship between k obs _fit/k obs and [L] 0 in the condition of [P] 0 = 100 nM, k off = 0.01 s -1 , k on = 1e5 M -1 s -1 and in different pre-equilibrium termination states of the kinetic experiments.

Figure S2 .
Figure S2.The relationship between k off _fit/k off and [L] 0 in the condition of [P] 0 = 100 nM, k off = 0.01 s -1 , k on = 1e5 M -1 s -1 and in different pre-equilibrium termination states of the kinetic experiments.

Figure S3 .
Figure S3.The relationship between k on _fit/k on and [L] 0 in the condition of [P] 0 = 100 nM, k off = 0.01 s -1 , k on = 1e5 M -1 s -1 and in different pre-equilibrium termination states of the kinetic experiments.

Figure S4 .
Figure S4.The relationship between k off _fit/k off and [L] 0 in the condition of [P] 0 = 100 nM, k off = 0.01 s -1 , k on = 1e5 M -1 s -1 and in different [L] 0 between two measurements of the kinetic experiments.The end-point measurement was 99% of the equilibrium.

Figure S5 .
Figure S5.The relationship between k on _fit/k on and [L] 0 in the condition of [P] 0 = 100 nM, k off = 0.01 s -1 , k on = 1e5 M -1 s -1 and in different [L] 0 between two measurements of the kinetic experiments.The end-point measurement was 99% of the equilibrium.

Figure S7 .
Figure S7.The binding processes in the same K d .

Figure S8 .
Figure S8.The binding processes in constant k off and changing k on .

Figure S9 .
Figure S9.Three screenshots of the customized webpage of Competitive Binding Kinetics -Dissociation.The concentrations of the protein and ligand were 200 and 150 nM, respectively.The k off of the ligand and inhibitor was 0.01 s -1 , the k on of the ligand and inhibitor was 1e5 M -1 s -1 .In each screenshot, the concentration of the inhibitor was 200, 2000, and 20000 nM in (A), (B), and (C).The volume ratio of the protein and ligand to the inhibitor was 1, 10, and 100 in (A), (B), and (C).From (A) to (C), at the time of mixing, the concentration of the free protein increased.the inhibitor could bind more free proteins, so the concentration of the protein-ligand complex decreased more slowly.

Figure S10 .
Figure S10.The relationship of the K i calculated by using the Wang's group equation and the [L] 0 in higher precision.

Figure S11 .
Figure S11.Competitive binding assays in the primary screen under different conditions.The initial working concentration of the inhibitor was 10 μM.(A-C) The contour lines and filled contours of the ratio of the [PL] eq in experimental groups to [PL] 0 under the experimental condition of the equilibrium concentration of the protein-ligand complex in blank controls [PL] 0 = 3.0, 5.0, and 7.0 nM in the K d -K i matrix.The points of the largest K i on each [PL] eq /[PL] 0 contour line were shown in blue, orange, and green line, which are magnified in Figure D, E, and F. (D-F) The relationship of the K d and the [PL] eq /[PL] 0 with the largest K i in the K d -K i matrix under the conditions of [PL] 0 = 3.0, 5.0, and 7.0 nM.It should be emphasized that each [PL] eq /[PL] 0 value corresponds to a K d value.

Figure S13 .
Figure S13.The screenshot of the Binding Curve Viewer -Kinetics of Association and Dissociation.The concentration of the labeled RNA was 0.002 nM.At 25°C (A) and 0°C (B), the time to reach 99% equilibrium in 2nd-order binding reaction was recorded in TableS5when the gradient concentrations of the Puf4 were 0.001, 0.01, 0.1, 1, 10, and 100 nM.

Figure S14 .
Figure S14.The screenshot of the Binding Curve Viewer -Determination of Dissociation Constant (K d ).At 25°C (A) and 0°C (B), the K d was 120 and 1 pM, respectively.The labeled RNA was used as the fixed component.The concentrations of the labeled RNA and apparent K d s were recorded in TableS6.

Table S2 .
Comparison of the simulated [PL] eq and theoretical [PL] eq under four experiment conditions Exp. No.
1 s -1 .Compared with exp.No. 1, the ligand in exp.No.2was slow-on-slow-off, its k on was 1e5 M -1 s -1 and k off was 0.01 s -1 .The K d of both ligands equaled 100 nM the same as the "K d in the K d and apparent K d can be shown in theoretical saturation curves" section in the article.In exp.No. 1 and 2, the [P] 0 equaled 100 nM, and the [L] 0 equaled 150 nM.In both experiments, the equilibrium concentration of the free ligand equaled the K d , i.e., [L] eq = 100 nM and [PL] eq = 50 nM.The time to reach 99% equilibrium in the second-order binding process (t 0.99 ) was 18 seconds and 177 seconds for exp.No. 1 and 2 respectively.If we calculated by using the pseudo-first-order binding process, the time to reach 99% equilibrium (t pseudo-0.99 ) was 18 seconds and 184 seconds for exp.No. 1 and 2 respectively.The ratio of the t pseudo-0.99 to t 0.99 (t pseudo- 0.99 /t 0.99 =1.06) was 1.04 for both exp.No. 1 and 2.In TableS3, as shown by exp.No. 2, 3, 4, and 5, increasing the [L] 0 with the other parameters being constant accelerated the binding process to 99% equilibrium, the t 0.99 decreased from 177 seconds to 30 seconds.From exp.No. 2 to 5, both [PL] pseudo-eq and [PL] eq increased and the [PL] pseudo-eq We conducted competitive binding experiments of association and dissociation by simulation (see TableS4).The association and dissociation experiments were specified by #a and #d, using a and d as suffix, respectively.In exp.No. 9a, the total concentration of the ligand and inhibitor ([L] 0 + [I] 0 ) was the same as the [L] 0 in exp.No. 2. Additionally, the kinetic and thermodynamic properties of the ligand and inhibitor in exp.No. 9a and the ligand in exp.No. 2 were the same.The time to 99% equilibrium (t 0.99 ) of the competitive binding in exp.No. 9a estimated by numerical simulation was the same as the t 0.99 of the association process in exp.No. 2 calculated by analytical integration (Table ) but was larger than the t 0.99 in exp.No. 13a and 14a (moderate-on-moderate-off).In contrast to exp.No. 15a and 16a, in exp.No. 17a and 18a, the larger the [I] 0 was, the larger the t 0.99 was.This suggested that although both ligand and inhibitor had the same thermodynamic property, the different kinetic properties (both fast or slow on and off) could prolong the equilibrium process of the competitive binding.Compared with exp.No. 19a, if we decreased the K d (exp.No. 21a) or K i (exp.No. 22a) tenfold or increased the [L] 0 tenfold (exp.No. 20a), the IC 50 increased similarly tenfold in each of the three experimental changes.From exp.No. 9a and 9d to No. 22a and 22d, most competitive bindings of dissociation had a larger t 0.99 than the corresponding competitive bindings of association, except exp.

Table S6 .
The apparent K d calculated by using Binding Curve Viewer under different experimental conditions.