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The Role of Conformational Changes in Molecular Recognition Mazen Ahmad,*,† Volkhard Helms,‡ Olga V. Kalinina,† and Thomas Lengauer† †

Department for Computational Biology and Applied Algorithmics, Max Planck Institute for Informatics, Campus E1 4, 66123 Saarbrücken, Germany ‡ Center for Bioinformatics, Saarland University, Campus E2 1, 66123 Saarbrücken, Germany S Supporting Information *

ABSTRACT: Conformational changes of molecules are crucial elements in many biochemical processes, and also in molecular recognition. Here, we present a novel exact mathematical equation for the binding free energy of a receptor−ligand pair. It shows that the energetic contribution due to conformational changes upon molecular recognition is defined by the so-called Kullback− Leibler (KL) divergence between the probability distributions of the conformational ensemble of the biomolecule in the bound and free states. We show that conformational changes always contribute positively to the change in free energy and therefore disfavor the association process. Using the example of ligands binding to a flexible cavity of T4 lysozyme, we illustrate that, due to enthalpy−entropy compensation, the conformational entropy is a misleading quantity for assessing the conformational contribution to the binding free energy, in contrast to the KL divergence, which is the correct quantity to use in this context.

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activator protein.6 This correlation is usually interpreted as an indication of the significant role of the conformational entropy in the recognition process.6,8 However, both increases and decreases in the conformational mobility of biomolecules and consequently in their conformational entropy were reported.16 For example, NMR suggested a thermodynamic role of change in conformational entropy favoring the association between the carbohydrate recognition domain of galectin-3 and its ligands18 and between the mouse major urinary binding protein I and its hydrophobic ligand.19 Changes in conformational entropy were also suggested to favorably influence the association of DNA to some variants of the catabolite activator proteins and to disfavor the association of other mutations.6 Despite the apparent additivity of the change in conformational entropy in binding free energy, in thermodynamic measurements, one typically observes enthalpy−entropy compensation (EEC).20−23 This phenomenon entails the cancelation of a significant portion of the enthalpic and entropic changes when summed up in the free energy. EEC is known to impose difficulties in thermodynamically guided lead optimization.24,25 Moreover, EEC is expected to be a problem for interpreting the thermodynamic data of protein−ligand interactions as substantiated by the strong correlation between the enthalpic and entropic changes provided by ITC for more than 400 receptor−ligand complexes with a slope close to room temperature (ΔH = (282.89 ± 4.71)ΔS − 822 ± 0.13 kcal/ mol).26 We recently introduced the general theoretical background for understanding the compensation between

INTRODUCTION Conformational changes play a critical role in many biological processes such as molecular recognition, enzymatic activity, and allosteric regulation. The “lock and key” interaction model1 was the first mechanistic model for explaining molecular recognition regarding ligand and receptor as rigid molecules. The increasing understanding of the role of conformational changes in biomolecular functions then stimulated the formulation of other mechanistic models such as the “induced fit”2 and the “conformational selection”3 models. Very recently, the biological importance of locally unfolded proteins and of intrinsically disordered proteins became apparent.4,5 Hence, it appeared worthwhile to us to reconsider how the ensemble nature of biomolecular conformations is accounted for in current theoretical models of molecular recognition. Commonly, the thermodynamic contribution of conformational changes to the change in free energy in the course of a biochemical reaction is thought to be represented by the conformational entropy of the involved biomolecules.4−17 Unfortunately, experimental quantification of the change in the molecular conformational entropy is not straightforward. Isothermal calorimetric titration (ITC) is a popular method, but it yields the total entropic change of the system comprising the conformational change of the biomolecules and contributions from the solvent. More useful in this respect is NMR spectroscopy, whereby the detected changes of the dynamics of the biomolecules are usually used to estimate the change in the conformational entropy of the biomolecular system.8,9,11,17 The change in conformational entropy estimated via NMR was found to correlate well with the total entropic change measured by ITC, e.g., upon binding of Calmodulin to its targets8 and upon binding of DNA to various mutants of the catabolite © XXXX American Chemical Society

Received: November 27, 2015 Revised: February 22, 2016

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DOI: 10.1021/acs.jpcb.5b11593 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B enthalpy and entropy and delineated how conformational changes affect the change in free energy.20,27 Specifically, we showed that the concept for EEC due to solvent reorganization28,29 can be extended to all conformational changes occurring in the solutes and in the solvent.20 Furthermore, we showed that the conformational changes in biomolecules cause changes in enthalpy due to the changes in the average of a part of the potential energy (termed the unperturbed interactions). At the same time, the entropy due to the unperturbed interactions changes by the same quantity such that exactly cancels the corresponding enthalpic changes when summed in the free energy change. Thus, these two canceling quantities can be regarded as manifesting a relaxation in the system.20 We also showed that the two compensating terms may vary in magnitude and sign, posing a serious difficulty for interpreting the experimentally measured total change in enthalpy and entropy.20 Examples of EEC were presented due to conformational changes inside ligands upon their solvation and due to conformational changes inside a protein upon ligand association.20 Additionally,27 we showed that the energetic contributions of conformational changes to the change in free energy upon perturbing a thermodynamic system are quantified by the Kullback−Leibler (KL) divergence (also known as relative entropy); see below. Here, we will apply these findings to the field of biomolecular recognition, in order to illustrate the role of the conformational changes in this area.

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THEORETICAL BASIS AND CALCULATIONS To understand the contribution of conformational changes to the binding free energy, we will consider the difference between the full binding free energy where all conformational changes take place and a conceptual “fixed-conformation” binding free energy in which conformational changes are not allowed. For this purpose, we need to define the physical meaning of a conformational ensemble and of the conformational changes. We consider the association process between a receptor and a ligand in solution. Both the receptor and the ligand populate conformational ensembles, R and L, respectively. Each ensemble is characterized by the respective probability distribution of the conformations. A particular conformation rl ∈ RL of the complex is defined by the conformation of the receptor r and the ligand l. We distinguish between the bound state of the system in which the complex is formed and the free state before the association in which the receptor and the ligand are in different compartments of the solution and their conformations are independent. The conformational changes upon association are defined by the differences between the probability distributions of the conformations between the free and bound states (Figure 1). Now we address the question of how the binding free energy differs if the conformational changes in the biomolecule already take place before the association. To answer this question, we split the association process into two steps (Figure 1). In the first step, we change the probability distribution of the conformations of the participating molecules in the free state to be the same distribution as in the bound state. In the second step, we let the association process take place between the rigid conformations of the molecules. The relationship between the true binding free energy ΔG° and the fixed conformation binding free energies ΔG°rl for the association between a (rigid) conformation of the receptor r and a rigid conformation of the ligand l (without allowing conformational changes) is given by (see the Supporting Information)

Figure 1. Energetic contribution of conformational changes to the association process. The change of the binding free energy is split into two steps: the first step at the top contributes to the free energy by the KL divergence which accounts for the difference between the conformational ensemble of the free (unbound) state (red curve) and the conformational ensemble of the bound state (blue curve). The changes in the sizes of the symbols represent the changes in the probabilities of observing the corresponding conformations. The association of the ligand (yellow disks) to the new conformational ensemble in the second step contributes a quantity of ∑rl∈RL Pbound(rl) × ΔG°rl to the free energy of the system; see eq 1.

ΔG° =

∑

Pbound(rl) × ΔGrl° rl ∈ RL  energetic contributions not due to conformational changes ⎡P (rl) ⎤ + kT ∑ Pbound(rl) ln⎢ bound ⎥ ⎣ Pfree(rl) ⎦ rl ∈ RL    contributions due to conformational changes

(1)

An analogue of this eq 1 was recently derived by us and verified by explicit calculations for the case of solvation free energies of small organic molecules.27 Here, Pbound(rl) and B

DOI: 10.1021/acs.jpcb.5b11593 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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Although the calculation of ΔG° is beyond the scope of this work, it is interesting to show that an analogue of eq 1 can be derived for the reverse direction of dissociation from bound to free state (see the Supporting Information):

Pfree(rl) are the probabilities of observing the conformations r of the receptor and l of the ligand in the bound and free states, respectively, k is the Boltzmann constant, and T is the temperature. The free energy ΔG°rl of binding of a rigid conformation of the ligand l to the rigid receptor r is not affected by the conformational changes. It is important to notice that the remaining factorsother than the conformational changeswhich also contribute to the free binding energy change are buried inside the binding free energies for particular fixed conformations, ΔGrl°. Such factors comprise protein−ligand interactions, changes in the solvation energies, and the loss of the translational and rotational degrees of freedom. The discussion of these factors and a more detailed analysis of ΔGrl° is beyond the scope of this work which concentrates on the energetic role of the conformational changes. The contribution to the binding free energy due to the association process (the second step in Figure 1) is the weighted average of the binding free energies of all combinations of rigid (fixed) conformations ∑rl Pbound(rl) × ΔG°rl. Here the conformational probability of each participating molecule is taken from the distribution pertaining to the bound state. The energetic contribution due to the conformational changes (the first step in Figure 1) is the difference between the total free energy ΔG° and the contributions ∑rl Pbound(rl) × ΔG°rl from the second association step and is shown in eq 1 to be Dconf (rl) = kT

ΔG° =

− kT

Pfree(rl) × ΔGrl° ⎡ P (rl) ⎤ Pfree(rl) ln⎢ free ⎥ ⎣ Pbound(rl) ⎦ rl ∈ RL

∑

(3)

Although the computation of ΔG° is possible in both directions using either eq 1 or eq 3, the energetic terms in these equations are asymmetric. This asymmetry is analogous to the asymmetry of the energetic term of a simple thermodynamic reaction when the formula of free energy perturbation is expanded to yield the average of the perturbation and the KL divergence of the conformations of the system.27 The role of the asymmetry in KL divergence is well-known in alchemical free energy calculations, in which the calculation converges faster in the direction of lower KL divergence.27,31,32 The KL divergence and its asymmetric properties received attention in studies of nonequilibrium free energy changes and of the irreversibility of physical processes.32−36 The role of the KL divergence was also discussed in a study of single molecule conformational fluctuations37 and of quantifying the “population shifts” in proteins38,39and for prediction of protein functional sites.40 Computational Methods. The dihedral angle χ1 of Val111 in the binding pocket of T4 lysozyme was found to undergo conformational changes upon ligand binding.41 Here, we compute the corresponding KL divergence (Dconf) which quantifies the contributions to the binding free energy due to these conformational changes. The probability distributions of the dihedral angle χ1 of Val111 were computed for both of the bound and unbound (free) states using biased umbrella sampling molecular dynamics simulations. The simulations were started using the crystal structures of the complexes of T4 lysozyme with benzene, benzofuran, p-xylene, indene, indole, isobutylbenzene, ethylbenzene, n-butylbenzene, and o-xylene taken from the pdb files: 181L, 182L, 187L, 183L, 185L, 184L, 1NHB, 186L, and 188L, respectively. The force field Amber99sb42 was used for the protein, while the general Amber force field43 was used for the ligands. Antechamber44 was used to obtain the force field parameters, and AM1-BCC was used to determine the partial charges. The solvent was represented using the TIP3P water model.45 The systems were solvated in a rectangular box and the water extended at least 1.2 nm beyond the solute surface. Periodic boundary conditions were used. The production simulations were performed using a leapfrog integrator46 and a time step of 2.0 fs. Simulations were performed in the isothermal−isobaric ensemble (1 atm and 300 K) by maintaining the temperature and the pressure through a Berendsen bath.47 The long-range electrostatic interactions were computed using the particle-mesh Ewald method.48 The van der Waals interactions and short-range electrostatic interactions were computed using a cutoff of 1.2 nm. Each simulation was performed for 5 ns using the Gromacs 4.6.5 simulation package.49 The data were collected every 25 steps. The first 100 ps were omitted from the analysis. Umbrella sampling50 simulations were used to obtain the probability distributions. 36 windows were used for each system to sample the dihedral angle of the side chain of VAL111. The centers of the umbrella potentials were spaced by 10°. A harmonic

⎡P (rl) ⎤ Pbound(rl) ln⎢ bound ⎥ ⎣ Pfree(rl) ⎦ rl ∈ RL

∑

The expression Dconf is also known as the KL divergence (the relative entropy) between the probability distributions of the conformations of the biomolecular system in the bound and free states. The KL divergence is a fundamental measure of the difference between two probability distributions in information theory. It describes the additional information needed to describe a probability distribution given the knowledge of another probability distribution.30 The KL divergence is always nonnegative:30

Dconf (rl) ≥ 0 Therefore, we call the energetic contribution due to the conformational changes “dissipation”, because it disfavors the association process via a positive contribution to the change in free energy change. The probability distributions in the equations above are joint distributions of all the degrees of freedom in the biomolecular system (e.g., dihedral angles), and the divergence between them accounts for the global conformational change in the biomolecular system (protein and ligand). Although the contributions of the individual local conformational changes to the total dissipation are not additive, the KL divergence of the conformational changes considering a subset of the degrees of freedom of the biomolecular system (e.g., one or more dihedral angles) will account for the dissipation due to the conformational changes pertaining to these degrees of freedom. Moreover, the dissipation due to local conformational changes in a subset of the degrees of freedom θ (e.g., one or more dihedral angles) is a lower bound for the total dissipation according to the following minimum dissipation theorem (see the Supporting Information): Dconf (rl) ≥ Dconf (θ )

∑ rl ∈ RL

(2) C

DOI: 10.1021/acs.jpcb.5b11593 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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Figure 2. Conformational changes upon the association of ligands of different sizes to T4 lysozyme. (a) Probability distribution of the dihedral angle of the side chain of Val111 in the free state (red lines) and in the bound state (blue lines), respectively. The numbers colored green are the computed values of the dissipation Dconf(θ) due to the described conformational change. The corresponding conformational entropy contributions −TΔSconf(θ) are in brown. The values are in kcal/mol. The panels are ordered according to increasing dissipation values (from benzene to o-xylene). (b) Surface representation of the conformational changes in the pocket due to reorientation of the side chain of Val111 (colored red). The nine ligands are shown after superimposing their crystal structures of the complexes. The figures show the two main conformations which correspond to the two peaks in the distributions above. The ligands vary in their induced conformational change due to the clashes with the side chain of VAL111 (the red surface) in the unbound state (left).

potential with a force constant of 200 kJ/mol/rad2 was used to apply the dihedral restraints. The unbiased distributions were obtained through the weighted histogram analysis method (WHAM)51,52 using a bin size of 2°. Using a bin size of 5° gave similar results.

compared to the distribution in the free (unbound) state (red line). The distributions pertaining to the bound state vary depending on the conformational changes taking place to accommodate each ligand. The computed values of the dissipation Dconf(θ) due to the conformational changes of this dihedral angle (θ) show that some ligands (e.g., benzene) perturb the probability distribution only slightly so that the corresponding dissipation is small. On the other hand, larger ligands (e.g., in the second row) significantly change the probability distributions and the corresponding dissipation is larger, namely, around 2 kcal/mol (Figure 2a). For comparison, we also computed the corresponding changes of the conformational entropy (−TΔSconf) from the difference of the computed entropies (−∑P(θ) ln P(θ)) for the free and bound distributions. Interestingly, the entropic changes apparently favor the association, whereas the real contributions (Dconf) to the free energies are positive and significantly disfavor the association. The increases of the entropy for the ligands in the top row in Figure 2a are due to the increased variability in the distributions, where two peaks

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RESULTS AND DISCUSSION Using the example of nine different ligands that bind to the protein T4 lysozyme, we will now illustrate the difference between the KL divergence (Dconf) which quantifies the conformational entropic changes that, in fact, contribute to the binding free energy and the changes in conformational entropy (−TΔSconf). This system is well studied and was used before as a model system for free energy calculations.53,54 The dihedral angle χ1 of Val111 in the binding pocket was found to experience conformational changes depending on the bound ligand (Figure 2b). Figure 2a shows the probability distributions of the dihedral angle χ1 of Val111 monitored during molecular dynamics simulations when the protein (T4 lysozyme) binds to the nine different ligands (blue lines) D

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hand, can be compensated for by the entropic changes at other locations and do not provide any bound on the global entropic changes.15,56

are observed in the bound state distributions (blue lines) in comparison with one peak in the free state distribution (red lines). The entropic changes are very small for the ligands in the second row of Figure 2a because the distributions of the bound and free states are similar. In contrast to the entropic changes, the dissipation values for the ligands in the second row have significant magnitude, due to the divergence of the distributions. In other words, the entropic change just accounts for the information content in the present conformational ensemble, whereas the KL divergence keeps track of the history and measures the deviation of the bound probability distribution relative to the unbound one. Importantly, Figure 2a demonstrates that the conformational entropy misleadingly favors the associations, whereas the real energetic contributions due to the conformational changes (the KL divergence) disfavor the associations. According to the EEC theorem mentioned above,20 the apparently increased entropy in the examples studied here is exactly canceled by the correspondingly decreased enthalpy which is due to the change in the average of the internal potential energy because of the changes in the distribution.20 The analysis given here illustrates the ambivalent role of the conformational entropy in association processes.16 This is closely related to the well-known difficulties in using thermodynamic changes to guide the lead optimization process.25 The important new insight provided by this work pertains to the relationship between the dynamics at the molecular scale and the thermodynamics of biomolecular interactions. Methods incorporating dynamical aspects of biomolecules (dynamic methods) such as NMR are essential in many highly important fields such as allosteric regulation and the molecular recognition of locally or fully disordered proteins. Due to EEC, conformational entropy measures capture a misleading picture of the valuable information provided by dynamic methods. Here we introduce the exact fundamental relation (eq 1) which accounts for the relationship between the change in free energy change and the change of the conformational ensemble. The ingredients required for the new eq 1 are the probability of the conformations which can be obtained by interpreting NMR relaxation data; e.g., the chemical shifts upon ligand association may be interpreted as population shifts.55 The newly established connection between molecular thermodynamics and information theory contributes to a better understanding of biochemical reactions. The fact that the KL divergence is always nonnegative implies that the energetic contributions due to conformational changes generally disfavor any association process. (That fact that molecules bind to each other, nonetheless, is due to the overriding contributions of molecular interactions and changes in solvation energies.) Moreover, the influence of the local dynamics on the free energy change is provided by the minimum dissipation theorem in eq 2. This relationship is highly relevant when studying local conformational changes or when the conformational changes are viewed collectively, e.g., by two-state models (active/ inactive, open/closed). When viewing such conformational changes at lower resolution (less detailed), the corresponding dissipation provides a lower bound on the total conformational dissipation (fully detailed). We previously showed that the difference between the dissipation pertaining to a subset of the involved degrees of freedom and the total dissipation is absorbed in the accompanied energetic changes (here ΔGrl°).27 The local changes of conformational entropy, on the other

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CONCLUSIONS AND PROSPECTS We have presented a novel exact equation for the binding free energy of a receptor−ligand pair. It shows that the energetic contribution due to conformational changes upon molecular recognition is defined by the Kullback−Leibler (KL) divergence between the probability distributions of the conformational ensemble of the biomolecule in the bound and free states. Moreover, we have shown that, due to enthalpy−entropy compensation, the conformational entropy is a misleading quantity for assessing the contributions from the conformational changes to the binding free energy. The energetic quantification of the changes in the conformational ensemble provides a simple thermodynamic explanation of the allosteric regulation of biochemical interactions. Interestingly, the specificity of the binding of ubiquitin to its different targets was recently shown to be controlled by the conformational equilibrium of its unbound state.57 The new insight is also beneficial for computational molecular design methods such as docking, where finding an optimal configuration poses a tradeoff between the interactions and the dissipation which we showed to be quantified by the KL divergence. So far, the energetics of conformational changes usually have been ignored in scoring functions. Equation 1 introduces an exact way of handling multiple conformations or binding poses by computational methods. Although we have studied only simple examples of conformational changes in protein side chains, the sizable energetic contributions (∼2 kcal/mol) show that conformational changes present a major driving force of molecular recognition. For example, freezing a rotor in supramolecular complexes was reported in an experiment to cause a change in energy of more than 1 kcal/mol.58 The changes in side-chain rotamers upon ligand binding are very common,59 and introducing their energetic contributions in the scoring function promises to be highly beneficial.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.5b11593.

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The derivation of the equations (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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DOI: 10.1021/acs.jpcb.5b11593 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.jpcb.5b11593 J. Phys. Chem. B XXXX, XXX, XXX−XXX