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The function of a protein often requires dynamic changes between different three-dimensional conformations. Conformational changes that occur when a protein binds a ligand molecule are well documented by experimentally determined structures of proteins in the unbound state and the ligand-bound state.1, 2 In 1958, Koshland3 suggested that the binding of the ligand may induce the conformational change of the protein. More recently, Tsai et al.4, 5 suggested an alternative mechanism in which the ligand selects and stabilizes a complementary protein conformation from an equilibrium of low-energy and higher-energy conformations. This selected-fit, or conformational selection mechanism is based on the energy-landscape picture of protein folding.6, 7
How can we distinguish whether a protein binds its ligand in an induced-fit or selected-fit mechanism? Recent single-molecule fluorescence8–13 and NMR relaxation experiments14–18 provide dynamic insights into conformational transitions. NMR relaxation experiments of the enzyme DHFR, which binds two ligands, reveal excited-state conformations that resemble ground-state conformations of adjacent ligand-bound or ligand-free states on the catalytic cycle.16 These experiments provide evidence for a pre-existing equilibrium of conformations that is shifted by the binding or unbinding of a ligand, as suggested in the selected-fit mechanism of protein binding.4, 5 For the binding of myosin to actin, in contrast, an induced-fit mechanism has been proposed2 based on cryo-microscopic images of the actin-myosin complex.19
We consider here the selected-fit and induced-fit binding kinetics in a simple four-state model of protein-ligand binding (see Fig. 1). In this model, the protein, or enzyme, has two dominant conformations E1 and E2. The conformation E1 is the ground-state conformation in the unbound state of the protein, while E2 is the ground-state conformation in the ligand-bound state. Two routes connect the unbound ground state E1 and bound ground state E2L. On the induced-fit route E1 → E1L → E2L, the protein first binds the ligand in conformation E1, which causes the transition into conformation E2. On the selected-fit route E1 → E2 → E2L, the protein binds the ligand in the higher-energy conformation E2.
We find a characteristic difference between the selected-fit and induced-fit binding kinetics. If the conformational relaxation into the ground state is fast, the selected-fit on-rate depends on the equilibrium constant of the conformations E1 and E2, while the selected-fit off-rate is independent of the conformational equilibrium. The induced-fit on-rate, in contrast, is independent of the conformational equilibrium between E1L and E2L, whereas the induced-fit off-rate depends on this equilibrium. Mutations or other perturbations that shift the conformational equilibrium without affecting the shape or free energies of the binding site thus may help to identify whether a protein binds its ligand in a selected-fit or induced-fit mechanism.
The four states of our model are connected by four transitions with equilibrium constants K1, K2, Ku, and Kb (see Fig. 1). In the model, the constants Ku and Kb for the conformational equilibrium in the unbound and the bound state obey
since E1 is the ground-state conformation in the unbound state of the protein, and E2L is the ground state when the ligand is bound. We assume here that the excited-state conformations are significantly higher in free energy than the ground states. The binding equilibrium of the two conformations is governed by the constants
From the two equalities [E2L] = K2[E2][L] = K2Ku[E1][L] and [E2L] = Kb[E1L] = KbK1[E1][L] that follow directly from these definitions, we obtain the general relation
between the four equilibrium constants. Thus, only three of the equilibrium constants are independent.
The selected-fit route E1 → E2 → E2L in our model dominates over the induced-fit route E1 → E1L → E2L if the concentration [E2] of the selected-fit intermediate is larger than the concentration [E1L] of the induced-fit intermediate. Since [E2] = Ku[E1] and [E1L] = K1[E1][L], the selected-fit mechanism thus is dominant for small ligand concentrations
while the induced-fit mechanism is dominant for large ligand concentrations [L] > Ku/K1.
Selected-fit binding kinetics
We first consider the binding kinetics along the selected-fit route of our model (see Figs. 1 and 2). The selected-fit binding rate is the dominant relaxation rate of the process
from E1 to E2L, with E2L as an “absorbing state” without backflow into E2. Here, s21 is the transition rate from state E1 to E2, s12 is the rate for the reverse transition, and sb is the binding rate per mole ligand in state E2. In the appendix, we calculate the exact relaxation rates for a process of the form (5). Since E2 is the excited state, we have s21/s12 = Ku ≪ 1, and the on-rate of the selected-fit process (5) is approximately given by
[see Eq. (A8) in the Appendix]. For small ligand concentrations [L], or fast conformational relaxation s12 into the ground-state conformation E1, we have sb[L] ≪ s12, and the selected-fit on-rate per mole ligand is
The selected-fit on-rate (7) just depends on the equilibrium constant Ku of the conformations E1 and E2 in the unbound state, and the binding rate sb of the conformation E2. Since the equilibrium probability P(E2) = [E2]/([E1] +[E2]) of conformation E2 is approximately P(E2) ≈ [E2]/[E1] = Ku for [E2] ≪ [E1], the selected-fit on-rate (7) can also be directly understood as the product of the probability that the protein is in conformation E2 and the binding rate sb of this conformation.
The selected-fit off-rate is the dominant relaxation rate of the process
from E2L to the E1. This process follows the same general reaction scheme as the binding process (5). A reasonable, simplifying assumption here is that the conformational relaxation process from the excited state E2 to the ground state E1 is significantly faster than the binding and unbinding process, which implies s12 ≫ su and s12 ≫ sb[L]. The selected-fit off-rate then simply is
[see Eq. (A9) in the Appendix]. The off-rate soff thus is independent of the conformational transition rates between E1 and E2. The off-rate is identical with the rate su for the bottleneck step, the unbinding process from E2L to E2.
Induced-fit binding kinetics
The on-rate along the induced-fit binding route of our model (see Figs. 1 and 3) is the dominant relaxation rate of the process
Here, rb is the binding rate of conformation E1 per mole ligand, ru is the unbinding rate, and r21 is the rate for the conformational transition into the bound ground state E2L. The induced-fit binding process (10) is similar to the selected-fit unbinding process (8). As before, we assume that the conformational transition into the ground state is much faster than the binding and unbinding processes, i.e. we assume r21 ≫ rb[L] and r21 ≫ ru. The induced-fit on-rate per mole ligand then is (see Appendix)
The induced-fit unbinding rate is the dominant relaxation rate of the process
which is similar to the selected-fit binding process (5). Since E2L is the ground-state conformation in the bound state of the protein, we have Kb = r21/r12 ≫ 1. The dominant relaxation rate of (12) then is
[see Eq. (A8) in the Appendix]. For fast conformational relaxation into the bound ground state with rate r21 ≫ ru, the induced-fit off-rate is approximately
The off-rate ru/Kb can again be understood as the product of the unbinding rate ru in the excited state E1L and the equilibrium probability P(E1L) = [E1L]/([E1L] + [E2L]) of this state, which is approximately P(E1L) ≈ ≈ [E1L]/[E2L] = 1/Kb.
Mutational analysis of the binding kinetics
A mutational analysis of the binding kinetics may reveal the characteristic differences between the selected-fit and induced-fit on-rates (7) and (11) and off-rates (9) and (14). Of particular interest here are mutations of residues far away from the ligand-binding site that affect the rate constants for transitions between the conformations, but not the binding rates sb and rb and unbinding rates su and ru of the two conformations. The mutations will change the free-energy differences ΔGu = G(E2) − G(E1) and ΔGb = G(E2L) − G(E1L) between the two conformations in the unbound and the bound state. In terms of these free-energy differences, the equilibrium constants Ku and Kb for the conformational transitions are
From Eq. (7), we find that the ratio of the selected-fit on-rates for wildtype and mutant
depends on the mutation-induced shift ΔΔGu = ΔG ′ u − ΔGu of the free energy difference between the conformations in the unbound state. The prime here indicates the conformational free-energy difference for the mutant. From Eq. (14), we obtain the relation
between the induced-fit off-rates of wildtype and mutant. Here, ΔΔGb = ΔG′ b − ΔGb is the mutation-induced shift of the conformational free-energy difference in the bound state. The selected-fit off-rate (9) and induced-fit on-rate (11), in contrast, are the same for wildtype and mutant if the mutations do not affect the binding and unbinding rates of the two conformations.
DISCUSSION AND CONCLUSIONS
Empirical methods to calculate mutation-induced stability changes of proteins have been investigated intensively in the past years.20–29 These methods can be used to calculate the changes ΔΔGu of the conformational free-energy difference in the unbound state, provided that experimental protein structures are available both for the unbound and ligand-bound state of the protein. The experimental structure of the protein in the ligand-bound state then corresponds to the excited-state conformation in the unbound state of our model. Calculations of mutation-induced changes ΔΔGb of conformational free-energy differences in the bound state are more difficult and require a construction of a ligand-bound excited-state conformation by docking the ligand to the experimental structure of the unbound protein.
In combination with Eqs. (16) and (17) of our model, calculated free-energy changes may help to identify selected-fit or induced-fit mechanisms of protein binding. A promising candidate is the protein DHFR, which has been studied extensively as a model enzyme to understand the relations between conformational dynamics, binding, and function. The enzymatic mechanism and conformational changes of DHFR have been investigated by mutational analyses,30–34 NMR relaxation,16, 17, 35 single-molecule fluorescence experiments,10 and simulation.36–38 The strong effect of point mutations far from the ligand binding site on catalytic rates and binding rates suggests large conformational changes during the catalytic cycle,30, 32–34, 39 in agreement with experimental structures of DHFR in the unbound and several ligand-bound states.40 Mutations far from the binding sites of DHFR affect the binding kinetics of the two ligands,31–33 and NMR relaxation experiments point towards a selected-fit mechanism of ligand binding.16 However, the mutations seem to affect both the binding and the unbinding rates.31–33 This precludes a direct comparison with our model, either because the conformational dynamics is more complex, or because the mutations have an indirect effect on the binding free energies.
We have considered here a simple four-state model of protein-ligand binding. The kinetic and mutational analysis of this model is partly inspired by previous models of the protein folding kinetics.41, 42 We have found characteristic kinetic differences between the selected-fit and induced-fit binding routes of this model, which should be observable also in more detailed models of conformational changes and binding. Although the timescale of the relevant conformational transitions are in general inaccessible to standard molecular dynamics simulations with atomistic protein models,43 conformational selection during binding has been recently studied by combining atomistic simulations and docking.44–46 Large conformational transitions are also intensively studied by normal mode analysis,47, 48 in particular with coarse-grained elastic network models of proteins.49–54
APPENDIX: RELAXATION RATES
The binding and unbinding processes of our model have a general reaction scheme of the form
For the selected-fit or induced-fit binding process, state A corresponds to E1, and state C to E2L. For the unbinding processes, A corresponds to E2L, and C to E1. We are interested in the dominant, slowest relaxation rate into state C. Therefore, the state C is an “absorbing” state, that is the rate from C back to B is 0. The probability evolution PA(t), PB(t), and PC(t) of the three states is then governed by the set of master equations
where kij denotes the rate from state j to state i. The three equations can be written in the matrix form
where Yo, Y1, and Y2 are the three eigenvectors of the matrix W, and λ1 and λ2 are the two positive eigenvalues of the eigenvectors Y1 and Y2. The coefficients co, c1, and c2 depend on the initial conditions at time t = 0. The eigenvalue of the eigenvector Yo is 0, which ensures that P(t) relaxes towards a finite equilibrium probability distribution P(t → ∞) = coYo for t → ∞. The two nonzero eigenvalues are
These eigenvalues, and their eigenvectors Y1 and Y2, capture the time-dependent relaxation process into equilibrium [see Eq. (A7)].
Selected-Fit Binding and Induced-Fit Unbinding
In selected-fit binding (5), state A corresponds to E1, state B to E2, and C to E2L (see fig. 1). In induced-fit unbinding (12), A corresponds to E2L, B to E1L and C to E1. In our model, we have [E2] ≪ [E1] and [E2L] ≫ [E1L] in equilibrium, which implies kBA ≪ kAB in both cases. The nonzero eigenvalues then are approximately
For kBA ≪ kAB, we have λ1 ≪ λ2. The smaller rate λ1 is the dominant relaxation rate for the initial conditions P(0) = (1,0,0) (only state A populated) or P(0) = (1 −kBA/kAB, kBA/kAB, 0) (‘pre-equilibration’ of states A and B). For both initial conditions, the probability evolution of state C is approximately given by PC(t) ≈ 1 − exp [−λ1t]. The dominant relaxation rate λ1 of Eq. (A8) can also be obtained from Eqs. (A2) to (A4) in a steady-state approximation, i.e. under the assumption that the variation dPB(t)/dt of the intermediate state B is negligibly small.
Selected-Fit Unbinding and Induced-Fit Binding
In selected-fit unbinding (8), the rate kCB corresponds to s12, the rate from the excited state E2 in the unbound ground state E1. In induced-fit binding (10), kCB corresponds to r21, the rate from E1L in the bound ground state E2L. We assume here that the rates for these conformational transitions into the ground state are much larger than the binding and unbinding rates. This implies kAB ≪ kCB and kBA ≪ kCB. The nonzero eigenvalues then simply are
with the clearly smaller eigenvalue λ1 as the dominant rate of the relaxation process from state A to C.