How to Distinguish Conformational Selection and Induced Fit Based on Chemical Relaxation Rates.

Protein binding often involves conformational changes. Important questions are whether a conformational change occurs prior to a binding event ('conformational selection') or after a binding event ('induced fit'), and how conformational transition rates can be obtained from experiments. In this article, we present general results for the chemical relaxation rates of conformational-selection and induced-fit binding processes that hold for all concentrations of proteins and ligands and, thus, go beyond the standard pseudo-first-order approximation of large ligand concentration. These results allow to distinguish conformational-selection from induced-fit processes-also in cases in which such a distinction is not possible under pseudo-first-order conditions-and to extract conformational transition rates of proteins from chemical relaxation data.

Introduction

Protein function often involves conformational changes during the binding to ligand molecules [1]. Advanced NMR experiments [2-7] and single-molecule spectroscopy [8-10] indicate that these conformational changes can occur without ligand, or with bound ligand and thus point to an intrinsic conformational dynamics of the proteins. An important question is how the conformational dynamics is coupled to the binding events. Two mechanisms for this coupling are ‘conformational selection’ [11] and ‘induced fit’ [12] (see Fig 1(a) and 1(b)). In conformational-selection binding, a conformational change occurs prior to the binding of a ligand molecule, as a conformational excitation from the unbound-ground state conformation of the protein. In this mechanism, the ligand seems to ‘select’ and stabilize a higher-energy conformation for binding. In induced-fit binding, the conformational change occurs after ligand binding and is a conformational relaxation into the bound ground-state conformation that is apparently ‘induced’ by the ligand. These two mechanisms are in particular plausible for small ligand molecules that can quickly ‘hop’ in and out of the protein binding pocket, i.e. that can enter and exit this binding pocket within transition times that are significantly smaller than the residence or dwell times of the proteins in the different conformations [13].

Fig 1

Characteristic chemical relaxation of induced-fit and conformational-selection binding.

(a) In induced-fit binding, the change between the conformations P1 and P2 of the protein occurs after binding of the ligand L. The intermediate state P1L relaxes into the bound ground state P2L with rate k, and is excited from the ground state with rate k. (b) In conformational-selection binding, the conformational change of the protein occurs prior to ligand binding. The intermediate state P2 is excited from the unbound ground state P1 with rate k, and relaxes back into the ground state with rate k. (c) The dominant, smallest relaxation rate kobs of induced-fit binding is minimal at the total ligand concentration where [P]0 is the total protein concentration and K the overall dissociation constant. As a function of [L]0, the dominant rate kobs is symmetric with respect to this minimum. (d) The dominant, smallest relaxation rate kobs of conformational-selection binding has a characteristic minimum as a function of [L]0 for k > k−, but is not symmetric with respect to this minimum. (e) The dominant rate kobs of conformational-selection binding decreases monotonically with [L]0 for k < k−.

A central problem is to identify protein binding mechanisms based on experimental data [13-24]. Advanced NMR experiments and single-molecule spectroscopy can reveal higher-energy conformations that are necessary for conformational-selection or induced-fit binding, but do not directly indicate the binding mechanism because such higher-energy conformations may exist both in the bound and unbound state of the protein [4, 8]. In principle, both conformational-selection or induced-fit binding then are possible. Standard mixing or temperature-jump experiments that probe the chemical relaxation into the binding equilibrium can provide additional information that allows to identify the binding mechanism [22, 25–28]. Of particular interest is the dominant, slowest relaxation rate kobs observed in the experiments, and how this rate depends on the total ligand concentration [L]0 [22, 25, 28]. The chemical relaxation experiments are often performed and analysed under pseudo-first-oder conditions, i.e. at ligand concentrations that greatly exceed the protein concentrations [22, 25, 29–36]. In the case of induced-fit binding, the dominant relaxation rate kobs increases monotonically with the ligand concentration [L]0 under pseudo-first-oder conditions. In the case of conformational-selection binding, kobs decreases monotonically with increasing [L]0 for conformational excitation rates k < k−, and increases monotonically with [L]0 for k > k− where k− is the unbinding rate of the ligand from the bound ground-state conformation of the protein (see Fig 1(b)). A decrease of the dominant relaxation rate kobs with increasing ligand concentration [L]0 thus indicates conformational-selection binding [25]. However, an increase of kobs with [L]0 under pseudo-first-order conditions is possible both for induced-fit binding and conformational-selection binding and does not uniquely point towards a binding mechanism [22].

In this article, we present general analytical results for the dominant relaxation rate kobs of induced-fit binding and conformational-selection binding processes that hold for all ligand and protein concentrations. Our general results are based on an expansion of the rate equations for these binding processes around the equilibrium concentrations of ligands and proteins, and include the pseudo-first-order results in the limit of large ligand concentrations. In the case of induced-fit binding, we find that kobs exhibits a minimum at the total ligand concentration for total protein concentrations [P]0 that are larger than the overall dissociation constant K of the binding process. As a characteristic feature, the function kobs([L]0) for induced-fit binding is symmetric with respect to this minimum. At sufficiently large protein concentrations [P]0, the function kobs([L]0) tends to identical values for small ligand concentrations [L]0 ≪ [P]0 and for large ligand concentrations [L]0 ≫ [P]0 because of its symmetry (see Fig 1(c)). In the case of conformational-selection binding, we find that kobs exhibits a minimum for conformational excitation rates k > k− and sufficiently large protein concentrations [P]0 (see Fig 1(d)). The location of this minimum depends on [P]0, K, and the rates k and k− (see Eq (10) below). In contrast to induced-fit binding, the function kobs([L]0) for conformational-selection binding is not symmetric with respect to this minimum. At sufficiently large protein concentrations [P]0, the function kobs([L]0) attains values for small ligand concentrations [L]0 ≪ [P]0 that can greatly exceed the values for large ligand concentrations [L]0 ≫ [P]0 (see Fig 1(d)). For excitation rates k < k− of conformational-selection binding processes, the dominant relaxation rate kobs decreases monotonically with increasing ligand concentration [L]0 (see Fig 1(e)). Our general results for the dominant relaxation rate kobs of induced-fit and conformational-selection binding processes allow to clearly distinguish between these two binding mechanisms for sufficiently large protein concentrations [P]0 (see Figs 2 and 3 below for numerical examples).

Fig 2

Numerical example for conformational-selection binding with the rate constants k = 10 s−1, k = 100 s−1, k+ = 100 μM−1s−1, and k− = 1 s−1.

(a) Relaxation data for the bound complex obtained by numerical integration of the rate equations and subsequent addition of Gaussian noise with amplitude 0.002 μM at the total protein concentration [P]0 = 0.5 μM and exemplary total ligand concentrations [L]0. The black lines represent multi-exponential fits of the data points. (b) to (d) Comparison of kobs values obtained from multi-exponential fits of numerical relaxation data (points) to our theoretical results for kobs (lines) at the three different total protein concentrations [P]0 = 0.5 μM, 1.0 μM, and 1.5 μM and total ligand concentrations [L]0 between 0.1 μM and 10 μM. The full lines represent fits of Eq (6) for conformational-selection binding (blue) and of Eq (1) for induced-fit binding (orange), with fit parameter values specified in the figure. In these fits, the dissociation constant K = 0.11 μM is assumed to be known from equilibrium data. The dashed blue lines are obtained from Eq (6) for the ‘true’ rate constants of the numerical example.

Fig 3

Numerical example for induced-fit binding with the rate constants k+ = 100 μM−1s−1, k− = 100 s−1, k = 1 s−1, and k = 10 s−1.

(a) Relaxation data for the bound complex obtained by numerical integration of the rate equations and subsequent addition of Gaussian noise with amplitude 0.004 μM at the total protein concentration [P]0 = 1 μM and exemplary total ligand concentrations [L]0. The black lines represent multi-exponential fits of the data points. (b) to (d) Comparison of kobs values obtained from multi-exponential fits of numerical relaxation data (points) to our theoretical results for kobs (lines) at the three different total protein concentrations [P]0 = 1 μM, 2 μM, and 3 μM and total ligand concentrations [L]0 between 0.1 μM and 10 μM. The full lines represent fits of Eq (1) for induced-fit binding (blue) and of Eq (6) for conformational-selection binding (orange), with fit parameter values specified in the figure. In these fits, the dissociation constant K = 1/11 μM is assumed to be known from equilibrium data. The dashed blue lines are obtained from Eq (1) for the ‘true’ rate constants of the numerical example.

Results

Solving the rate equations of the induced-fit and conformational-selection binding models shown in Fig 1(a) and 1(b) is complicated by the fact that the binding steps in these models are second-order reactions that depend on the product of the time-dependent concentrations of unbound proteins and unbound ligands. In the standard pseudo-first-order approximation, the rate equations are simplified by assuming that the total ligand concentration greatly exceeds the total protein concentration, so that the amount of ligand consumed during binding is negligible compared to the total amount of ligand. The concentration of the unbound ligand then can be taken to be constant, and the rate equations only contain terms that are linear in the time-dependent concentration of the protein, which makes them solvable. In our more general approach, a linearization of the rate equations is achieved by expanding around the equilibrium concentrations of the bound and unbound proteins and ligands (see Methods). This expansion captures the final relaxation into equilibrium, which is governed by the smallest, dominant relaxation rate kobs, for all concentrations of proteins and ligands, and leads to general results for kobs that include the results from the pseudo-first-order approximation in the limit of large ligand concentrations.

Dominant relaxation rate of induced-fit binding

Expanding the rate equations of the induced-fit binding mechanism shown in Fig 1(a) around the equilibrium concentrations of proteins and ligands leads to the dominant, smallest relaxation rate (see Methods) with and with the overall dissociation constant of induced-fit binding. This general result for kobs holds for all total ligand concentrations [L]0 and protein concentrations [P]0. In the limit of large ligand concentrations [L]0 ≫ [P]0, we obtain δ ≃ [L]0 + K and γ ≃ − k − k + k− + k+[L]0 from Eqs (2) and (3), which agrees with results derived in pseudo-first-order approximation [21, 22].

As a function of the total ligand concentration [L]0, the dominant relaxation rate kobs exhibits a minimum at for total protein concentrations [P]0 > K. The function kobs([L]0) is symmetric with respect to (see (Fig 1(c)). This symmetry and the location of the minimum result from the fact that kobs depends on [L]0 only via the term δ, which is minimal at and symmetric with respect to . The dominant relaxation rate kobs is minimal when δ is minimal. For large ligand concentrations [L]0, kobs tends towards the maximum value k + k as in pseudo-first-order approximation. The location of the minimum and the symmetry of the function kobs([L]0) with respect to this minimum are properties that the induced-fit binding model appears to ‘inherit’ from the elementary binding model P + L ⇌ PL (see Eq (46) in Methods section). However, the function kobs([L]0) of the elementary binding model is V-shaped and does not tend to a constant maximum value for large ligand concentrations [L]0.

Dominant relaxation rate of conformational-selection binding

For the conformational-selection binding mechanism shown in Fig 1(b), an expansion of the rate equations around the equilibrium concentrations of proteins and ligands leads to the dominant, smallest relaxation rate (see Methods) with and δ as in Eq (3), and with the overall dissociation constant of conformational-selection binding. This general result for kobs holds for all total ligand concentrations [L]0 and protein concentrations [P]0. In the limit of large ligand concentrations [L]0 ≫ [P]0, we obtain α ≃ − k + k + k− + k+[L]0 and β ≃ 4k(k − k−) from Eqs (3), (7) and (8), in agreement with results derived in pseudo-first-order approximation [21, 22].

For conformational-selection binding, the shape of the function kobs([L]0) depends on the values of the conformational excitation rate k and the unbinding rate k− (see Fig 1(d) and 1(e)). For k < k−, the dominant relaxation rate kobs decreases monotonically with increasing total ligand concentration [L]0. For k > k−, the dominant relaxation rate kobs exhibits a minimum as a function of [L]0 at sufficiently large total protein concentrations [P]0. The minimum is located at (see Methods) if the conformational relaxation rate k is much larger than the excitation rate k, which typically holds for the conformational exchange between ground-state and excited-state conformations of proteins. In contrast to induced-fit binding, the function kobs([L]0) is not symmetric with respect to this minimum. For large ligand concentrations, the limiting value of the dominant relaxation rate is kobs(∞) = k as in pseudo-first-order approximation. For vanishing ligand concentrations [L]0 → 0, the limiting value is kobs(0) = k + k for total protein concentrations [P]0 > K(k + k − k−)/k− and kobs(0) = k−([P]0 + K)/K for [P]0 < K(k + k − k−)/k−.

Distinguishing induced fit and conformational selection

The general results for the dominant relaxation rate kobs presented in the previous sections allow to clearly distinguish induced-fit from conformational-selection binding processes. In Fig 2, we consider a conformational-selection binding process with the rate constants k = 10 s−1, k = 100 s−1, k+ = 100 μM−1s−1, and k− = 1 s−1 as a numerical example. The data points in Fig 2(a) represent relaxation curves for the bound complex that have been generated by numerical integration of the rate equations and subsequent addition of Gaussian noise to mimic measurement errors. The black lines in Fig 2(a) are multi-exponential fits of the data points. The number of exponentials in these fits has been determined with the Akaike information criterion (AIC), which is a standard criterion for the trade-off between quality of fit and number of fit parameters, and ranges from 2 to 4. The data points in Fig 2(b) to 2(d) represent the dominant relaxation rates kobs that are obtained from multi-exponential fits of relaxation curves for different total ligand concentrations [L]0 and total protein concentrations [P]0. The dominant relaxation rate kobs here is identified as the smallest relaxation rate of a multi-exponential fit. The full blue lines in Fig 2(b) to 2(d) result from fitting our general result Eq (6) for conformational-selection binding to the kobs data points. The full orange lines represent fits of our general result Eq (1) for induced-fit binding. For all fits, we assume that the dissociation constant K = 0.11 μM is known from equilibrium data, and use k, k, and k− as fit parameters. Finally, the blue dashed lines in Fig 2(b) to 2(d) are the kobs curves obtained from Eq (6) for the ‘true’ rate constants of the conformational-selection binding process given above. These dashed lines agree with the data points, which indicates that the kobs values from multi-exponential fits as in Fig 2(a) are identical to the values obtained from Eq (6) within the statistical errors of the numerical example.

The fits in Fig 2(b) to 2(d) clearly identify conformational selection as the correct binding mechanism in this example. The blue fit curves for conformational selection agree with the data points within statistical errors, while the orange fit curves for induced fit deviate from the data. For conformational-selection binding, the fit values of the conformational transition rates k and k and of the unbinding rate k− specified in the figure agree with the correct values k = 10 s−1, k = 100 s−1, and k− = 1 s−1 of the numerical example within statistical errors.

In Fig 3, we consider an induced-fit binding process with rate constants k+ = 100 μM−1s−1, k− = 100 s−1, k = 1 s−1, and k = 10 s−1 as a second numerical example. The kobs data points in Fig 3(b) to 3(d) are again obtained from multi-exponential fits of relaxation curves that have been generated by numerical integration of the rate equations and subsequent addition of Gaussian noise (see Fig 3(a)). The fits in Fig 3(b) to 3(d) clearly identify induced-fit binding as the correct mechanism in this example. The full blue curves that represent fits of Eq (1) for induced-fit binding are in overall agreement with the kobs points, while the orange fit curves of Eq (6) for conformational-selection binding deviate from the data. The fit values of the conformational transition rates k and k for the induced-fit binding model are in good agreement with the correct values k = 1 s−1, and k = 10 s−1 of the example. The dashed blue curves in Fig 3(b) to 3(d), which are obtained from Eq (1) for the ‘true’ rate constants of the induced-fit binding process, are in overall agreement with the data points. Slight deviations result from the fact that the amplitude of the slow relaxation mode with rate kobs is rather small compared to the amplitude of the fast modes (see Fig 3(a)), which can lead to numerical inaccuracies.

In both numerical examples of Figs 2 and 3, the correct binding mechanism cannot be identified under pseudo-first-order conditions because kobs is monotonically increasing with [L]0 for ligand concentrations that greatly exceed the protein concentration [P]0 [22].

Analysis of chemical relaxation rates for recoverin binding

Chakrabarti et al. [28] have recently investigated the conformational dynamics and binding kinetics of the protein recoverin with chemical relaxation and advanced NMR experiments. Recoverin exhibits a conformational change during binding of its ligand, which is a rhodopsin kinase peptide fused to the B1 domain of immunoglobulin protein G in the experiments of Chakrabarti et al. [28]. The data points in Fig 4 represent the dominant relaxation rates kobs obtained by Chakrabarti et al. from relaxation experiments at the temperatures 30°C and 10°C for a recoverin concentration of 10 μM. The lines in Fig 4 result from fitting our general results Eqs (1) and (6) for the dominant relaxation rate kobs of induced-fit and conformational-selection binding processes. In these fits, we have used the values K = 1.0 ± 0.2 μM and K = 1.8 ± 0.2 μM obtained by Chakrabarti et al. from isothermal titration calometry experiments at 30°C and 10°C, which reduces the parameters to k, k, and k−. The fits of our general result Eq (6) for conformational-selection binding are rather insensitive to the relaxation rate k, which is illustrated in Fig 4 by nearly identical fits for k = 100 s−1 and k = 1000 s−1 (see dashed and full blue lines). Our fit values for the conformational excitation rate k specified in the figure caption agree with the values k = 33 ± 5 s−1 and k = 23 ± 5 s−1 obtained by Chakrabarti et al. from advanced NMR experiments at 30°C and 10°C, respectively. From these experiments, Chakrabarti et al. obtain the values k = 990 ± 100 s−1 and k = 920 ± 200 s−1 at 30°C and 10°C, which cannot be deduced from our fits of the kobs data because these fits are insensitive to k. The NMR experiments indicate that the higher-energy conformation of unbound recoverin resembles the ground-state conformation of bound recoverin [28] as required for the conformational-selection binding mechanism illustrated in Fig 1(b), and that the excited-state conformation of unbound recoverin has the equilibrium occupancy P = k/(k + k) = 3.2% ± 0.5% at 30°C and P = 2.4% ± 0.7% at 10°C, relative to the ground-state conformation.

Fig 4

Analysis of experimentally determined relaxation rates kobs for the binding of recoverin to a rhodopsin kinase peptide ligand.

The data points represent results of Chakrabarti et al. [28] obtained from chemical relaxation experiments at the temperatures 30°C and 10°C for a recoverin concentration of 10 μM. The blue lines result from fits of Eq (6) for conformational-selection binding with the values k = 1000 s−1 (full) and k = 100 s−1 (dashed) of the conformational relaxation rate. At 30°C, the parameter values obtained from fitting are k = 31.5 ± 0.8 s−1 and k− = 5.1 ± 0.4 s−1 for k = 1000 s−1, and k = 31.1 ± 0.8 s−1 and k− = 5.0 ± 0.4 s−1 for k = 100 s−1. At 10°C, the fit parameter values are k = 19.3 ± 1.4 s−1 and k− = 3.9 ± 0.7 s−1 for k = 1000 s−1, and k = 19.0 ± 1.3 s−1 and k− = 3.8 ± 0.7 s−1 for k = 100s−1. The yellow lines represent fits of Eq (1) for induced-fit binding with constraints on the conformational excitation and relaxation rates k and k. At 30°C, the obtained fit values for the conformational exchange rates are k = k = 15 ± 10 s−1 for the constraint k > k, k = 3.1 ± 1.9 s−1 and k = 31 ± 4s−1 for the constraint k > 10k, and k = 1.1 ± 0.8 s−1 and k = 44 ± 8 s−1 for k > 40k. At 10°C, the fit values are k = 4.5 ± 4.0 s−1 and k = 14 ± 10 s−1 for the constraint k > k, k = 1.9 ± 1.5 s−1 and k = 19 ± 5 s−1 for k > 10k, and k = 0.7 ± 0.5 s−1 and k = 28 ± 11 s−1 for k > 40k. In all fits of Eq (1) for induced-fit binding, we obtain k− ≫ k, i.e. the fit values of the unbinding rate k− are much larger than the conformational relaxation rate k and cannot be specified.

Fits of our general result Eq (1) for the dominant relaxation rate kobs of induced-fit binding with unconstrained parameters k, k, and k− lead to fit values for the conformational exchange rates k and k with k ≫ k. For such values of k and k, the conformation 1 of the induced-fit binding model illustrated in Fig 1(a) is the ground-state conformation both for the unbound state and the bound state of recoverin, which contradicts the experimental observation that recoverin changes its conformation during binding [28]. Distinct ground-state conformations for the unbound and bound state of recoverin can be enforced by constraining k to values larger than k. The yellow lines in Fig 4 result from fits with the constraints k > k, k > 10k, and k > 40k. These constraints correspond to equilibrium occupancies P of the excited-state conformation of bound recoverin with P < 50%, P < 9.1%, and P < 2.4%, respectively. The fits of Eq (1) for induced-fit binding with the constraints k > 10k and k > 40k deviate rather strongly from the two data points with the smallest ligand concentrations [L]0 = 3 μM and 5 μM, in contrast to fits of Eq (6) for conformational-selection binding (blue lines). A Bayesian model comparison of conformational-selection binding and induced-fit binding based on Eqs (1) and (6) leads to Bayes factors of 9.8 ⋅ 1013 and 1.5 ⋅ 1023 at 30°C for the constraints k > 10k and k > 40k, and to Bayes factors of 4.2 ⋅ 103 and 9.6 ⋅ 109 at 10°C for k > 10k, and k > 40k, respectively (see Methods for details). These Bayes factors indicate that the kobs data of Fig 4 strongly point towards conformational-selection binding. Bayes factors larger than 102 are generally considered to be decisive [37]. For the bound recoverin complex, Chakrabarti et al. did not observe an excited-state conformation in NMR experiments, which limits the excited-state occupancy P to undetectable values smaller than 1% for a conformational exchange that is fast compared to the NMR timescale as in the case of unbound recoverin. The analysis of the experimental data for the dominant relaxation rate kobs of recoverin binding based on our general results Eqs (1) and (6) thus indicates a conformational-selection binding mechanism, in agreement with a numerical analysis of Chakrabarti et al. [28]. In this numerical analysis, Chakrabarti et al. include the chemical relaxation data for recoverin binding, additional relaxation data from dilution experiments, the values for the conformational exchange rates k and k obtained from NMR experiments, and the K values deduced from isothermal titration calometry [28]. In contrast, our analysis of the kobs data in Fig 4 from the chemical relaxation experiments of recoverin binding only includes the K values from isothermal titration calometry as additional input.

Discussion

We have shown here that the dominant rate kobs of chemical relaxation experiments with total protein and ligand concentrations of comparable magnitude conveys information on the binding mechanism and conformational transition rates of proteins. For sufficiently large protein concentrations [P]0, the function kobs([L]0) obtained from such experiments has characteristic features that are clearly distinct for induced-fit binding and conformational-selection binding. The function kobs([L]0) of induced-fit binding exhibits a characteristic symmetry around a minimum and tends to identical values for small and large ligand concentrations [L]0 as in Fig 1(c) if the protein concentration [P]0, which determines the location of the minimum, is sufficiently large. In contrast, the function kobs([L]0) of conformational-selection binding is either monotonically decreasing for k < k−, or asymmetric around a minimum for k > k−. In both cases, kobs([L]0) tends for small ligand concentrations [L]0 to values that exceed the values for large ligand concentrations as in Fig 1(d) and 1(e) if the protein concentration [P]0 is sufficiently large.

Our general results for the dominant rate kobs of chemical relaxation experiments thus provide a transparent route to distinguish induced-fit binding from conformational-selection binding based on the shape of the function kobs([L]0), and to infer conformational transition rates from fitting. Alternatively, these binding mechanisms can be identified from a numerical analysis of time-dependent relaxation curves [26-28], based on steric effects that may prohibit ligand entry and exit in the bound ground-state conformation of the protein and, thus, rule out conformational-selection binding [15], from a comparison of conformational excitation rates to overall, effective binding and unbinding rates [4, 13], or from the effect of distal mutations that mainly affect the conformational exchange, but not the binding kinetics in different protein conformations [13, 16, 21, 38]. Of particular interest is how such mutations change the overall binding and unbinding rates. If both conformational-selection and induced-fit binding are viable, increasing the ligand concentration may shift the binding mechanism from conformational selection to induced fit [16, 18, 26, 39, 40].

Methods

Near-equilibrium relaxation of induced-fit binding

The induced-fit binding model of Fig 1(a) leads to the four rate equations that describe the time-dependent evolution of the concentration [P1] of the unbound protein, the concentration [L] of the unbound ligand, and the concentrations [P1L] and [P2L] of the bound complexes. These four rate equations are redundant because the total concentrations [P]0 and [L]0 of proteins and ligands are conserved: With Eqs (15) and (16), the concentrations [P1] and [P1L] can be expressed in terms of [L] and [P2L], which results in the two non-redundant rate equations These rate equations can be written in the vectorial form with The two components of the vector F(c) in Eq (19) are the right-hand sides of the Eqs (17) and (18). The rate equations describe the temporal evolution of the concentrations [L] and [P2L] towards equilibrium, and are nonlinear because of the quadratic term in [L] on the right-hand side of Eq (17).

To obtain linearized versions of the rate equations that describe the slow processes corresponding to the final relaxation into equilibrium, we expand the vector F(c) in Eq (19) around the equilibrium concentrations ceq: Here, J is the Jacobian matrix of F with elements J = ∂F/∂c. The right-hand side of Eq (21) follows from F(ceq) = 0. Inserting the expansion (21) into Eq (19) and making use of leads to the linearized rate equations with and the equilibrium concentration of the unbound ligand The overall dissociation constant K of the induced-fit binding process is given in Eq (4). The relaxation rates of the linearized rate Eq (22) are the two eigenvalues of the matrix −J(ceq). These eigenvalues are kobs given in Eq (1) and with γ and δ given in Eqs (2) and (3). The relaxation rate kobs is smaller than k2 and, thus, dominates the final relaxation into equilibrium.

Near-equilibrium relaxation of conformational-selection binding

The four rate equations of the conformational-selection binding model of Fig 1(b) are The total concentrations [L]0 and [P]0 of the ligands and proteins are conserved: With these equations, the concentrations [P1] and [P2L] can be expressed in terms of [L] and [P2], which leads to the two rate equations These rate equations can be written in the vectorial form of Eq (19) with and with a vector F(c) that contains the right-hand sides of the Eqs (32) and (33) as components. An expansion of the vector F(c) around the equilibrium concentrations ceq leads to Eq (22) with the Jacobian matrix and the equilibrium concentrations The overall dissociation constant K of the conformational-selection binding process is given in Eq (9). The relaxation rates of the linearized rate equations are the two eigenvalues of the matrix −J(ceq). These eigenvalues are kobs given in Eq (6) and with α and β given in Eqs (7) and (8). The relaxation rate kobs is smaller than k2 and therefore dominates the final relaxation into equilibrium.

To derive Eq (10) for the location of the minimum of kobs as a function of the total ligand concentration [L]0, we now consider the near-equilibrium relaxation of the conformational-selection model in quasi-steady-state approximation (qssa), which assumes that the concentration of the intermediate [P2] does not change in time. The left-hand side of Eq (32) then is equal to zero, and the two Eqs (32) and (33) reduce to the single equation An expansion of the function f([L]) around the equilibrium concentration [L]eq leads to the linear equation with and δ and [L]eq given in Eqs (3) and (37). The derivative of is zero at with given in Eq (10). In general, the quasi-steady-state result is a good approximation of kobs if the rates for the transitions out of the intermediate state P2 of conformational-selection binding are much larger than the rates for the transitions to P2. A numerical analysis shows that the location of the minimum of is in good agreement with the location of the minimum of kobs([L]) for conformational transitions rates with k ≫ k.

Multi-exponential relaxation

In the numerical examples illustrated in Figs 2 and 3, chemical relaxation curves for conformational-selection and induced-fit binding are fitted with a multi-exponential model. Such multi-exponential models are an adequate description for the time evolution of concentrations in first-order chemical reactions. However, the binding steps of the induced-fit and conformational-selection models of Fig 1(a) and 1(b) are of second order. To justify that multi-exponential models can also be used to approximate the chemical relaxation of second-order reactions, we consider here the elementary binding model of a protein P and ligand L. For the initial condition [PL](0) = 0, the rate equation of the elementary binding model can be written as and has the analytical solution [38] with where K = k−/k+ is the dissociation constant of the elementary binding model.

We first show that λ2 − λ1 is identical to the dominant relaxation rate kobs obtained from a linear expansion around equilibrium. An expansion of the right-hand side of Eq (42) around the equilibrium concentration leads to the linear equation d[PL]/dt ≃ −kobs([PL] − [PL]eq) with This dominant relaxation rate kobs is identical to λ2 − λ1. As a function of [L]0, the dominant rate kobs of the elementary binding model exhibits a minimum at and is symmetric with respect to this minimum.

We next use the limit of the geometric series with q = e− λ2/λ1 to rewrite Eq (43) as which shows that the chemical relaxation of the elementary binding model can be described as an infinite sum of exponential functions. The exponents of these functions are integer multiples of k, which is reminiscent of the higher harmonics in oscillatory phenomena. The prefactors (λ2/λ1) in Eq (47) decay exponentially with the order n of the harmonic because of λ2/λ1 < 1. The infinite sum of Eq (47) therefore can be truncated in practical situations. Under pseudo-first-order conditions, Eq (47) reduces to a single-exponential relaxation.

In analogy to the elementary binding model, we propose that the time evolution of the concentrations in the induced-fit and conformational-selection models can be represented as a sum of exponentials where the exponents are integer combinations −ikobs − jk2 with i, j = 0, 1, 2, 3, … of the relaxation rates kobs and k2 obtained from a linear expansion around the equilibrium concentrations. Under pseudo-first-order conditions, the chemical relaxation reduces to a double-exponential relaxation [16, 21, 22].

In the numerical examples of Figs 2 and 3, the chemical relaxation of the bound complexes is fitted with a multi-exponential model with k > 0 for all n. We have used the routine NonlinearModelFit of the software Mathematica [41] with the differential evolution algorithm [42], which was repeatedly run with different values of its F parameter ranging from 0.1 to 1 for a given number of exponentials N. Among different runs, we have selected fit results based on the residual sum of squares, after discarding fits with singular results in which two rates k coincide within 95% confidence intervals, or in which one or more rates k are identical to 0 within 95% confidence intervals. We have then determined the number of exponentials N based on the small-sample-size corrected version of Akaike’s information criterion (AIC) [43].

Bayes factors

The Bayes factor K is as measure for how plausible one model is relatively to an alternative model, given experimental data [44]. The Bayes factor for the plausibility of the conformational-selection binding model relative to induced-fit binding model is Here, p(data ∣ M, θ) is the probability that the data were produced by the model M with given parameters θ, where M either stands for conformational-selection binding or induced-fit binding, and p(θ) is the prior distribution on the parameter values, which encodes any prior knowledge that we have about the parameters. The integrals of Eq (49) are taken over all parameter values and result in the probability p(data ∣ M) that the data were produced by the model, regardless of specific parameter values. The data here consist of the slowest relaxation rates with i = 1, 2, …, N obtained from multi-exponential fits of the N time series with ligand concentrations , and the errors σ of these rates. Following standard approaches [44], the probability that the data were generated by the model M with parameters θ = (k, k, k−, K, [P]0) is for k > nk, and 0 otherwise. The inequality k > nk reflects constraints on the conformational relaxation rate k and excitation rate k of the models (see section “Analysis of chemical relaxation rates for recoverin binding”). Eq (50) implies that the errors are independently and normally distributed random variables with standard deviations σ. Depending on the model M, we either use Eqs (1) or (6) to determine . For simplicity, K and [P]0 are kept fixed at the experimentally measured values. We choose a prior p(θ) that is uniform in the logarithm of the rates k, k, k−. Taking the logarithm of the rates is not crucial, as a uniform prior on the rates gives similar results in the analysis of recoverin binding and, thus, leads to the same conclusions. The prior p(θ) here can be chosen to be uniform because it is identical for both the induced-fit and conformational-selection binding models due to the equivalent parameters of the models [45].