Structural and Entropic Allosteric Signal Transduction Strength via

Letter pubs.acs.org/JPCL

Structural and Entropic Allosteric Signal Transduction Strength via Correlated Motions Dong Long and Rafael Brüschweiler* Department of Chemistry and Biochemistry and National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32306, United States S Supporting Information *

ABSTRACT: Allosteric signal transduction in biomacromolecules can play an essential role in their function. Internal motional correlations in proteins provide a possible communication mechanism, but the quantitative relationship between statistical correlations and allostery is unknown. Quantitative relationships between internal motional correlations and the efficiency of propagation of allosteric structural and entropic effects are introduced and validated against conformational ensembles obtained from molecular dynamics simulations. This framework can explain a range of phenomena, such as the occurrence of an allosteric entropy change in the absence of any noticeable structural change.

SECTION: Biophysical Chemistry and Biomolecules

A

locations. At the same time, these polypeptides systems also display a significant number of medium-to-long-range dynamics correlations of backbone ϕ and ψ dihedral angles, which were found to be closely associated with the capability of allosteric signal propagation. A prerequisite for allosteric communication between two protein regions A and B is the existence of an energetic coupling VAB between them, whereby a structural change in one part alters the potential energy in the other part and vice versa. Such a coupling is typically manifested in statistical dynamics correlations, which can be expressed by the Pearson correlation coefficient R between degrees of freedom ξA and ξB that belong to protein regions A and B, respectively, whereby R = 0 in the absence of a coupling between regions A and B (see Supporting Information (SI)). Quantitative relationships between dynamics correlations and allosteric signal propagation by population shift can be established for different model potentials. We first consider a system with two dihedral angles x = (ξ1, ξ2) that move in a harmonic potential V(x) = (Aξ21 + 2Cξ1ξ2 + Bξ22)/2, defined by the force constants A, B, and C. At thermal equilibrium, the dihedral angles display a bivariate Gaussian distribution:

distinctive property of many proteins is their inherent capability to communicate structural and dynamic changes over a broad range of distances to allosterically control protein−protein and protein−ligand interactions for the regulation of many cellular processes.1−4 Advances in the characterization of protein structure and dynamics by experiment, computation, and theory have led to an increasingly detailed picture of the Boltzmann-weighted protein ensembles at physiological conditions.5−9 In the “population shift” framework, a binding event of a ligand to an effector (or allosteric) site of a protein causes a redistribution of the conformer populations.2,4,9−12 However, the atomic-detail mechanism of a population redistribution amounting to a significant structural, dynamic, or thermodynamic change at the active site of the protein is not well understood. Here, we propose analytical relationships between allosteric population shifts, conformational entropy changes, and the correlation coefficients (R) of dihedral angle distributions, which were successfully tested and applied to protein and peptide systems. In cases where the apo state of a protein, i.e., in the absence of an allosteric ligand or substrate, can be adequately represented by a molecular dynamics (MD) trajectory, a kinetic network model can be derived from the trajectory,13−18 providing a means to directly assess protein allosteric properties. The effect of ligand binding can be studied by locally biasing the conformational equilibrium at the effector binding site to the bound state, and the propagation of this perturbation to other protein parts can be described using a Master-equation based approach (MAPS).18 Application of MAPS to polypeptides revealed how local perturbations can cause significant structural and dynamic changes at distant © 2012 American Chemical Society

p(x) = c·exp{−V (x)/kBT }

(1)

where c is a normalization constant. When ξ1 and ξ2 are correlated with each other with a Pearson correlation coefficient R12, the presence of a local perturbation on ξ2, Received: April 20, 2012 Accepted: June 5, 2012 Published: June 5, 2012 1722

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which fully restrains the dihedral at a value Δξ2 = ξ2 restrained − ⟨ξ2⟩ξ2 free away from its equilibrium position, will induce a conformational shift in the thermal average of ξ1: Δξ1 = ⟨ξ1⟩ξ2 restrained − ⟨ξ1⟩ξ2 free. It can be shown that the relationship between the allosteric population shift and correlation coefficient follows in good approximation: Δξ1 = R12·Δξ2

correlated internal dynamics with R12 as the transmission efficiency. While unimodal distributions dominate the backbone dihedral angle fluctuations of globular proteins in their native energy basins, larger conformational changes often involve multimodal distributions characterized by jumps between a finite number of substates separated by energy barriers.18 In a bimodally distributed system, ξ1 and ξ2 can be represented by two discrete states α and β, with equilibrium populations P(j) ξ1 , P(k) (j,k = α,β). This model adequately describes, for example, ξ2 the ψ angle dynamics of the Ala5-pentapeptide18 and FiP3519 observed in extended MD simulations. Upon restraining ξ2 to one of its two states, which is designated here as the α state, i.e., P(α) ξ2 = 1, the coupling between ξ1 and ξ2 induces a shift of population ΔPξ1 of the α state of ξ1. For two coupled dihedrals, numerical calculations show that the induced population change ΔPξ1 follows in good approximation:

(2)

This statistical dihedral angle relationship, which is valid as long as (A/B)1/2 ≈ 1 (see also eq S17 (SI)), represents a lowdimensional, alternative formulation to a linear response theoretical approach in Cartesian space.12 Equation 2 is illustrated in Figure 1A, where the relationship between

ΔPξ1 = R12·ΔPξ 2

(3)

as shown in Figure 1C. The high linearity of ΔPξ1/ΔPξ2 versus R12, which takes the same type of closed-form expression as eq 2, was tested on the bimodally distributed ψ angles of Ala5 and FiP3 displaying excellent agreement (Figure 1D). The specific forms of eqs 2 and 3 suggest a generalization to long-range signal transmission along a chain of sequentially coupled dihedral angle pairs. For example, for three sequential dihedrals, each exhibiting a discrete bimodal distribution, where direct dynamic couplings only occur for consecutive pairs, an original perturbation of ξ3 by ΔPξ3 will induce a population shift for ξ2 by ΔPξ2 with transmission strength R23, given by ΔPξ2 = R23·ΔPξ3, which will be relayed to site 1 via the coupling R12, ΔPξ1 = R12·ΔPξ 2 = R12·R 23·ΔPξ3

(4)

Equation 4 explains the product relationship, R13 ≈ R12·R23, which was empirically found for ubiquitin,20 where the apparent coupling R13 mediates the population shift between “remote site” ξ3 and site ξ1. Generalization of eq 4 for the successive signal propagation over N dihedral angles yields ΔPξ1 ≈ R12R23...RN−1,NΔPξN. With increasing N, it leads to an exponential attenuation of the allosteric signal, which makes the sequential long-range transmission over many medium or weakly correlated dihedral pairs rather inefficient. For instance, a strong perturbation inducing a local population change of 60% at the effector site, after transmission over four successive dihedral angle pairs with pairwise correlation coefficients of 0.5 each, will result in only a 3.8% change of population at the active site. Although eq 4 does not apply for nonlinear networks of coupled dihedral angles, such systems can be studied by explicit numerical calculations. It is shown in Figure 2A,C that restraining of a second effector site can lead to a significant enhancement of the population shift at the active site for both unimodal Gaussian distributions and bimodal distributions if the correlations between the active site and both effector sites have the same sign. Counterexamples do occur, as can be seen in Figure 2C, for bimodal distributions where restraining of the second site occasionally lowers ΔP or even inverses its sign, i.e., the population changes in the opposite direction to that of the effector sites (Figure 2C). This is in part because exhaustive numerical exploration of the relationship depicted in this figure also covers distributions that are hardly ever encountered in biomolecular systems. Therefore, we examined the possible

Figure 1. Dependence of coordinate shift and population shift of a dihedral angle ξ1 upon perturbation of dihedral angle ξ2 on the Pearson correlation coefficient R12 between the two dihedral angles for (A,B) unimodal and (C,D) bimodal distributions. (A,C) The black dots represent numerical model calculations for 104 randomly generated data sets. (B) The blue dots represent the results observed in a 1 μs MD trajectory of ubiquitin. (D) The red and blue dots represent the results observed in extended MD trajectories of Ala5pentapeptide and FiP35, respectively. Diagonals are indicated as solid black lines. The insets show snapshots of the polypeptides.

Δξ1/Δξ2 and R12 is plotted for randomly generated Gaussian probability distributions of eq 1. The same relationship was then tested using all 124 ϕ and ψ dihedral angles of ubiquitin that are unimodally distributed during the course of a 1 μs MD simulation.8 For this purpose, a “conditional ensemble” was generated from the original MD ensemble by selecting only those MD snapshots for which a given dihedral angle ξ2 was shifted by Δξ2 = 11−13° from its unrestrained average, followed by analyzing the resulting shifts Δξ1 of all other dihedrals in relationship to their pairwise correlation coefficients R12. As shown in Figure 1B, they follow eq 2 in very good approximation, where deviations can be attributed to differential fluctuation amplitudes and deviations from Gaussian behavior (see SI). The coordinate shift of eq 2 represents the transmission of an allosteric signal from ξ2 to ξ1 enabled by 1723

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although the physical situation here is different (for a more detailed discussion, see the SI). It is interesting to note that the local entropy change ΔS at the active site is independent of the restrained value ⟨ξ2⟩ at the effector site. Therefore, an allosteric entropy change absent of a structural change can emerge if the effector site is motionally restrained at the original equilibrium value. We tested for the occurrence of such “pure” entropy changes in the 124 unimodally distributed ϕ and ψ dihedral angles of ubiquitin by restraining individual dihedral angles to their equilibrium values. After selecting the conditional ensemble with a minimal average structural change by confining Δξ2 within ±1° of its equilibrium value, the resulting shifts of the average coordinates (Δξ1) and change of local entropy (ΔS) of all other dihedrals were determined. As can be seen in Figure 3, while allosteric

Figure 2. Allosteric population shifts at the active site upon restraining one effector site (y-axes) versus two effector sites (x-axes), assuming zero or positive correlations between the active site and both effector sites. (A,B) Δξ1′ is the relative shift of the equilibrium value of dihedral angle ξ1 (see eq S28). (C,D) ΔPξ1(ξ2fixed) and ΔPξ1(ξ2,ξ3fixed) are the change of population at the active site ξ1 for 2-site jump distributions. (A,C) Numerical model calculations of 104 randomly generated data sets (black dots). (B) Results of all unimodally distributed backbone dihedral angles during MD simulation of ubiquitin and (D) two-site jump distributed ψ angles of Ala5pentapeptide (red dots) and FiP35 (blue dots) during MD simulations. The points in the off-white shaded regions reflect population shift enhancement when restraining two effector sites over restraining only one effector site. Figure 3. Changes of local entropy (A) and average structure (B) of each of the 124 unimodally distributed backbone dihedral angles of ubiquitin upon fixing every other dihedral angle, one at a time, at its original equilibrium position. (A) The red dots represent the dependence of entropy changes on correlation coefficients extracted from the 1 μs MD trajectory of ubiquitin. The black solid curve depicts the theoretical dependence of eq 5. (B) Distribution of the coordinate shift Δξ1 associated with panel A at the active site showing that its average structure remains essentially unaffected.

enhancement of the perturbation at the active site based on the MD ensembles of Ala5 and FiP35 for bimodal distributions and ubiquitin for unimodal distributions.8,18,19 As can be seen in Figure 2B,D, enhanced perturbation at the active site upon restraining the second effector site is found most of the time for both unimodal and bimodal distributions where the off-white shaded areas reflect regions of enhanced perturbation when both effector sites are restrained. Because protein allostery involves the population redistribution of the conformational ensemble, a change in conformational dynamics can lead to a change of the configurational entropy. We illustrate the origin of allosteric entropy changes21−23 in correlated systems with either a Gaussian unimodal distribution or a discrete bimodal distribution. For a bivariate Gaussian distribution in ξ1 and ξ2, the restraining of the effector site distribution (ξ2) will generally result in a decrease of entropy at the active site (ξ1). In the limiting case where the variance of ξ2 is entirely restrained, the local entropy change (ΔS) of ξ1 is given by 1 ΔS = kB log(1 − R12 2) (5) 2 (for a derivation, see the SI). Equation 5 has the same mathematical form as expressions reported previously24−26 for the mutual information of bivariate Gaussian functions,

structural changes are essentially absent (Δξ1 ≈ 0), the allosterically induced local entropy changes for ξ1 can be significant, following closely eq 5 (black line). If the distribution of the effector site is only weakly restrained, eq 5 represents an upper limit of the absolute entropy change. For discrete bimodal distributions, the local entropy at the active site can also be altered upon perturbation of the effector site. However, the sign and absolute value of ΔS (eqs S35− S37) depend not only on R but also on the original populations at the active site and the population change at the effector site. In a recent in silico study of alanine-pentapeptide, a local harmonic restraint applied to residue Ala5, confining its conformation to the coil state (Figure S3J), was found to also perturb the backbone ϕ/ψ distributions of the other four residues (Figure S3A−H). In two-dimensional ϕ/ψ dihedral angle subspace, this external perturbation results in a decrease 1724

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of the local backbone entropy of residue Ala5, ΔS/kB = −3.04. At the same time, the local backbone entropies for residues Ala1−Ala4 noticeably increase, with ΔS/kB = 0.57, 0.93, 1.02, and 0.99, respectively, representing an instructive case of allosteric entropy−entropy compensation. Such favorable allosteric entropy changes may have far-reaching implications for a wide range of molecular interaction mechanisms. As recently shown by experiments in the mutant of catabolite activator protein (CAP-S62F), the protein−DNA interaction is driven by a large increase of allosteric conformational entropy, which “rescues” the inactive conformational state.27 In this work, we combine model potentials with MD-derived conformational ensembles to relate correlated dihedral angle dynamics to the inherent ability for signal propagation. Although model potentials represent a coarse simplification of real systems, their analytical tractability helps develop the conceptual framework to assess the strength and the potential biological significance of allosteric signal transmission in biomolecules via correlated motions. While for quantitative predictions, an explicit ensemble analysis of a molecular state is often preferable, the correlation information can provide useful insights into the inner workings of a biomolecule. We found that the efficiency of signal propagation in model systems mediated by motional correlations is directly given by the Pearson correlation coefficient - the most widely used quantitative descriptor of correlation between two variables. Moreover, the Pearson correlation coefficient also determines the local configurational entropy change at the active site when an effector site is dynamically constrained. These results were confirmed using conditional MD ensembles of polypeptides. Internal motional correlations in Cartesian or dihedral angle space are often used in computational studies as indicators for dynamical coupling and communication.20,28−35 The present work corroborates the empirical interpretation of the role of motional correlations, and suggests that quantitative correlation coefficients are useful proxies for the inherent allosteric propensity of a protein system. For systems with two coupled dihedral angles, the relationships between ΔP (or Δξ) and R is both rigorous and surprisingly simple (eqs 2 and 3). It also reveals that the efficiency of signal transmission via multiple successive steps is limited. Conditional ensemble calculations demonstrate how population changes at the active site can be enhanced by simultaneous perturbations at more than one effector site, which suggests a route for allosteric signal amplification by utilizing a parallel network of relatively weak correlations. Purely entropic allosteric effects, which have been identified previously,21−23 can be rationalized by the analytical relationship between ΔS and R (eq 5) providing an estimate of the entropy change at the active site when the fluctuation amplitude at the effector site is restrained. Correlation coefficients and conditional ensembles can be efficiently extracted from Boltzmann-weighted MD ensembles to help elucidate intramolecular communication networks. Together with the analytical relationships introduced here, they permit the prediction of the efficiency of allosteric signal transduction by population shift in biomolecules, providing new quantitative insights into this fundamental aspect of biomolecular function.

Letter

ASSOCIATED CONTENT

S Supporting Information *

Details of analytical theory, numerical simulations, and three supporting figures. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank D. E. Shaw Research for making their MD trajectory of FiP35 publicly available. This work was supported by the National Science Foundation (Grant MCB-0918362).

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