Structural fluctuation of proteins induced by thermodynamic perturbation Fumio Hirata and Kazuyuki Akasaka Citation: The Journal of Chemical Physics 142, 044110 (2015); doi: 10.1063/1.4906071 View online: http://dx.doi.org/10.1063/1.4906071 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/142/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Role of filament annealing in the kinetics and thermodynamics of nucleated polymerization J. Chem. Phys. 140, 214904 (2014); 10.1063/1.4880121 Protein structure prediction using global optimization by basin-hopping with NMR shift restraints J. Chem. Phys. 138, 025102 (2013); 10.1063/1.4773406 Communication: Accurate determination of side-chain torsion angle χ1 in proteins: Phenylalanine residues J. Chem. Phys. 134, 061101 (2011); 10.1063/1.3553204 Revisiting the finite temperature string method for the calculation of reaction tubes and free energies J. Chem. Phys. 130, 194103 (2009); 10.1063/1.3130083 Influence of the chain stiffness on the thermodynamics of a Gō-type model for protein folding J. Chem. Phys. 126, 165103 (2007); 10.1063/1.2727465
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THE JOURNAL OF CHEMICAL PHYSICS 142, 044110 (2015)
Structural fluctuation of proteins induced by thermodynamic perturbation Fumio Hirata1 and Kazuyuki Akasaka2
1
College of Life Sciences, Ritsumeikan University, and Molecular Design Frontier Co. Ltd., Shiga 525-8577, Japan 2 High Pressure Protein Research Center, Institute of Advanced Technology, Kinki University, Wakayama 649-6493, Japan
(Received 14 October 2014; accepted 5 January 2015; published online 27 January 2015) A theory to describe structural fluctuations of protein induced by thermodynamic perturbations, pressure, temperature, and denaturant, is proposed. The theory is formulated based on the three methods in the statistical mechanics: the generalized Langevin theory, the linear response theory, and the three dimensional interaction site model (3D-RISM) theory. The theory clarifies how the change in thermodynamic conditions, or a macroscopic perturbation, induces the conformational fluctuation, which is a microscopic property. The theoretical results are applied, on the conceptual basis, to explain the experimental finding by Akasaka et al., concerning the NMR experiment which states that the conformational change induced by pressure corresponds to structural fluctuations occurring in the ambient condition. A method to evaluate the structural fluctuation induced by pressure is also suggested by means of the 3D-RISM and the site-site Kirkwood-Buff theories. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4906071] I. INTRODUCTION
Structural fluctuation of protein around its native state plays essential roles in a variety of process in which a biomolecule expresses its intrinsic function.1 For example, so called “gating” mechanisms of ion channels are often regulated by the structural fluctuation of amino-acid residues consisting the gate region of the channel. Molecular recognition such as the formation of an enzyme-substrate complex in an enzymatic reaction is controlled often by structural fluctuation of protein. Few typical examples of structural fluctuations around a native conformation of protein, related to function, are “breathing,”2 “hinge-bending,”3 and “arm-rotating”4 motions. Those motions are collective in nature, involving many atoms moving in the same direction. It is not surprising that considerable efforts have been devoted to clarify the conformational fluctuation of protein theoretically, which has started at the end of the last century based on the molecular-mechanics (MM) or dynamics. One of the earliest attempts was to relate the structural fluctuation to the normal mode of protein.5,6 Those works have demonstrated the importance of the collective mode in the fluctuation. However, those efforts could not have provided a realistic physical insight into the dynamics of actual biological processes, since they are concerned with a protein in vacuo, which of course cannot describe the fluctuation correlated with that of solvent. In actual biological processes, solvent plays vital roles both in equilibrium and in fluctuation of protein.7,8 It may not be necessary to spend many words for emphasizing the crucial role played by solvent for stabilizing or destabilizing native structure of protein, such as the hydrophobic interaction and hydrogen bonds. Here, let us consider the roles played by water in fluctuation of protein around its native conformation, associated with recognition of a ligand by protein. The process is primarily a thermodynamic process, governed by 0021-9606/2015/142(4)/044110/8/$30.00
the free energy difference between the two states before and after the recognition. It is obvious that water plays crucial role in the thermodynamics, since the equilibrium structures are determined by the free energies including the excess chemical potential or the solvation free energy of water. However, it is not a mere role of water in the process. Water actually regulates the kinetic pathway of the process as well by controlling the structural fluctuation of amino-acid residues consisting the active site. An example of such processes is a mouth-like motion of amino-acid residues and/or domains. The open-andclose motion of the mouth is driven not only by the direct force acting among atoms in protein but also by that originated from the solvent induced force which in turn caused by the fluctuation in the solvation free energy, or the non-equilibrium free energy. An earlier attempt to consider the effect of solvent on structural fluctuation of protein has been made by Kitao et al. using the molecular dynamics simulation.9,10 By diagonalizing the variance-covariance matrix of the fluctuation, obtained from the trajectory, they have characterized the fluctuation in terms of the frequency spectrum. Unfortunately, such analyses are largely limited by the sampling of the configuration space. Recently, three novel methods to characterize the structural fluctuation of protein in solution have been proposed, one due to an experiment, the other two by theories. Using the high pressure NMR, Akasaka and his coworkers have developed a unique method to realize the fluctuation of protein in solution.11–13 The idea is based on their finding that the NMR signals such as the chemical shift of amino acids change upon increasing pressure, and that the changes become more significant as the pressure enhanced. Based on the finding, Akasaka has proposed a hypothesis that an equilibrium structure at higher pressure should correspond to one of fluctuated structures at the ambient pressure. It suggests that the fluctuated states of protein can be realized by changing pressure as
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an order parameter. The idea is conceptually analogous to that by Marcus, who has realized the solvent fluctuation around solutes in polar solvent by changing the charges on solute, as an order parameter.14 Ikeguchi et al. presented a method based on the linear response theory to describe the structural change of protein induced by a perturbation.15,16 The most striking feature of the theory is that the response function consists of the variancecovariance matrix of the structural fluctuation in the unperturbed system. The theoretical characterization concerning the structural fluctuation of protein in solution has been proposed by Kim and Hirata,17 combining two methods in the statistical mechanics, the three dimensional interaction site model (3DRISM)/RISM theory18 and the generalized Langevin theory (GLT).19,20 The theory has proved that the free energy surface around an equilibrium conformation is essentially quadratic, or equivalently that the structural fluctuation is a Gaussian process. The behavior looks similar at one glance to that of the harmonic oscillator in vacuo, treated in the earlier studies by means of the normal mode analysis (NMA). However, the physical origin of the Gaussian behavior revealed by GLT is entirely different from that by NMA. The “Gaussian fluctuation” of a harmonic oscillator originates from the “hamonicity” of the mechanical potential energy, while that of protein in solution comes from the central limiting theorem characteristic to probabilistic processes.31 Unlike NMA, the harmonic behavior of the fluctuation in the Kim-Hirata theory has been derived naturally without truncating the potential energy among atoms in protein and water at the quadratic terms (see the Appendix). The authors have also suggested that the variance-covariance matrix of the fluctuation can be obtained from the 3D-RISM/RISM theory as the second derivative of the free energy surface with respect to the atomic coordinates of protein. It is our purpose in the present paper to combine the three recent developments in order to propose a new theory to characterize the structural fluctuation of protein induced by thermodynamic perturbation. However, before doing so, we would like to review briefly the earlier treatment concerning the structural fluctuation of protein in order to clarify the difference between the two treatments due to NMA and GLT. Section II is devoted to a brief review of the theoretical treatment concerning the structural fluctuation of protein. In Sec. III, the equations for structural fluctuation of protein induced by thermodynamic perturbation are derived. Section IV is devoted to make a contact between the theory and the experiment in a conceptual level. Section V is concerned with a possible scheme to extend the theory to a fluctuation in nonlinear regime.
II. A REVIEW OF THEORIES FOR STRUCTURAL FLUCTUATION OF PROTEIN A. Earlier treatment of the structural fluctuation of protein in vacuo
The earlier treatment concerning the structural fluctuation of protein assumes either that the interaction among atoms
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in protein is a harmonic potential or that the displacement of atoms from their equilibrium position is small.21,22 Then, the deviation (or “fluctuation”) of atoms in protein from their equilibrium positions follows the Hooke equation of motion Mα
d 2∆Rα (t) = − Hα β · ∆R β (t), dt 2 β ∆Rα (t) = Rα (t) − ⟨Rα ⟩,
(2.1) (2.2)
where Rα (t), ⟨Rα ⟩, and Ma are the position at time t, the equilibrium position, and the mass of atom α in protein, respectively. Hα β is the force constant or the Hessian matrix defined by Hα β =
∂ 2U , ∂∆Rα ∂∆R β
(2.3)
where U(R1, R2, ..., RN) is a harmonic potential, and it has a quadratic form with respect to ∆R, that is, 1 Hα β ∆Rα ∆R β . (2.4) U(∆R1, ∆R2, ..., ∆R N ) = 2 α β If one considers that the protein atoms are in thermal motion, the distribution of ∆Rα becomes “Gaussian,” p(∆R1, ∆R2, ..., ∆R N ) ∥ A∥ 1 = exp − Aα β ∆Rα ∆R β , (2π)3N 2 α β where Aα β is defined by
(2.5)
Hα β , (2.6) kT and ∥ A∥ is the determinant of the matrix {Aab }. The variancecovariance matrix of the “fluctuation” is related to the Hessian through Aα β =
kT ∆Rα ∆R β = . Hα β
(2.7)
Diagonalizing the variance-covariance matrix gives rise to the normal mode vectors and the eigen values. The equation implies that the structural “fluctuation” of a protein can be characterized theoretically by calculating the second derivative of the interaction potential with respect to atomic coordinates. The idea has been employed in the 1980s through the early 1990s to characterize the structural fluctuation of protein by means of the computational science,5,6 and there is a persistent belief that NMA describes the structural fluctuation of protein faithfully as long as the low-frequency or collective modes are concerned with.23,24 However, the “fluctuation” extracted from NMA is different essentially from that taking place in living body, or in aqueous solutions. First, the fluctuation described by NMA is a mechanical oscillation as was clarified in this section, and it has a characteristic peak in the low frequency region of the power spectrum, while such a peak largely disappears in aqueous solution, indicating that the collective motion is not an oscillation but a stochastic process. Second, the protein structure analysed by NMA, taken from Protein Data Bank (PDB), is an equilibrium structure in aqueous solutions, but not in vacuum. In vacuum, the PDB structure is a fluctuated
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structure, and the structure treated by NMA is one of the local minima in high-energy states. One can readily verify it by carrying out a molecular dynamics simulation of a protein in vacuum at ordinary temperature, taking a PDB structure as an initial condition. One will observe that the native conformation quickly breaks down to become a random-coil-like structure. It indicates that the “fluctuation” extracted from NMA is not a real one but merely a “ghost.” B. Structural fluctuation of protein in solution treated by the generalized Langevin theory
The structural fluctuation of protein is not just a mechanical process represented by NMA described above, but conjugated mechanical and thermodynamic processes, in which structure and thermodynamics of water play crucial roles. Therefore, the theory should include both the structures of protein and thermodynamics of water as variables in the equation. The equations describing the time evolution of protein structure and solvent thermodynamics, which are conjugated each other, have been proposed by Kim and Hirata recently based on the GLT.17 The theory is briefly outlined in the Appendix. If one ignores the terms concerning the friction and the random force, Eq. (A14) in the Appendix becomes d 2∆R(t) = −k BTL−1 · ∆R(t), (2.8) dt 2 which is formally identical to Eq. (2.1) with identifying H = −k BTL−1, where L is defined by (A16) in the Appendix. The equation indicates that the structural fluctuation takes place on a quadratic surface, 1 F ({∆R}) = k BT ∆Rα · L−1 α, β · ∆R β , (2.9) 2 α, β and the distribution of the fluctuation is Gaussian. The quadratic surface in (2.9), however, has a physical origin that is entirely different from that in Eq. (2.1). Equation (2.1) was derived by assuming that the interactions among protein atoms are “harmonic,” or the motion is limited to “small oscillation,” while Eq. (2.9) was derived without making any assumption for the interaction among atoms in protein. The equilibrium conformation represented by ⟨R⟩ in Eq. (2.2) is the one which satisfies the condition ∂U ({R}) = 0, ∂Rα where U({R}) is the mechanical interaction potential among atoms in protein. On the other hand, ⟨R⟩ in Eq. (A7) is the conformation which meets the condition ∂F ({∆R}) = 0, ∂∆Rα where F({R}) is the free energy surface of protein in solution, which includes not only the direct interaction among atoms in protein but also the solvation free energy. In short, the average conformation ⟨R⟩ in Eq. (2.2) represents the mechanically most stable structure, while ⟨R⟩ in Eq. (A7) is the thermodynamically most stable conformation. The same argument applies also to the variance-covariance matrix of the Gaussian
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fluctuation. The Hαβ in Eq. (2.1) is related to the second derivative of the mechanical potential energy with respect to the atom position through Eq. (2.3), while L αβ in Eq. (2.8) is related to the second derivative of the free energy through k BT L−1
= αγ
∂ 2F , ∂∆Rα ∂∆Rγ
(2.10)
where (L−1)αγ denotes the αγ element of the inverse matrix of L. C. Linear response theory
The structural fluctuation induced by a perturbation can be described by the linear response theory if the system recovers its equilibrium state upon removing the perturbation, or unless the perturbation causes a non-linear (“plastic”) deformation on the system. The free energy of protein, structure of which is perturbed by a force, can be expressed as 1 ∆Rα · L −1 α, β · ∆R β − ∆Rα · fα . F ({∆R}) = k BT 2 α α, β (2.11) In the equation, the first term describes the free energy surface of an unperturbed system and the second term the perturbation. By applying the variational principle, or ∂F ({∆R}) = 0, ∂∆Rα
(2.12)
one can get the following equation for the structural change due to the perturbation:17
⟨∆Rα ⟩1 = (k BT0)−1 ∆Rα ∆R β 0 · f β . (2.13) β
In the equation, the suffix “0” put to the variance-covariance matrix indicates that the quantity is concerned with the reference state or the unperturbed system. The equation has been derived first by Ikeguchi et al.15 and reformulated by Kim and Hirata based on the variational principle.17 If one considers fβ as a perturbation, the new structure represents a fluctuated state produced by the perturbation.
III. STRUCTURAL FLUCTUATION OF PROTEIN INDUCED BY THERMODYNAMIC PERTURBATION
In the present section, we propose a new theory to describe the structural fluctuation of protein, induced by a thermodynamic perturbation, based on the linear response theory described in Sec. II. Application of the linear response theory to the process under concern is rationalized primarily by the pressure-NMR experiment carried out by Akasaka and his coworkers.11–13 By applying pressure on protein solutions, Akasaka and the coworkers have probed the NMR signals concerning the fluctuated states of protein, ranging from partially denatured to fully denatured, depending on the magnitude of pressure. When they have removed the pressure, the protein reversibly recovered its native conformation irrespective of whether it is partially or fully denatured. The experiment suggests strongly that the process is not a non-linear or plas-
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tic deformation but a linear fluctuation. Although we developed the concept originally for pressure as a perturbation, we conjecture that the same argument applies to the other perturbation, such as temperature and denaturant concentration, as far as the structural response stays in the linear regime. A. Structural fluctuation induced by pressure
We define the force acting on an atom of protein in solution by the derivative of the Helmholtz free energy of the system with respect to the coordinates of the atom ) ( ∂Fs , (3.1) fα = − ∂Rα where Rα and Fs denote the αth atom in protein and the Helmholtz free energy of the entire system, respectively. (Note that the notation Fs is used to distinguish it from the free
( −
∂Fs = ∂Rα )
···
( −∂U )
×
Nu
dRα
α=1
N n
α=1
1 U({R},{r}) ··· exp − kT
(3.6)
where ∆V¯ is the partial molar volume of a protein in the solution. It is resulted from an a priori consideration that the change in the volume due to the structural change is only concerned with the partial molar volume. Combined Eqs. (2.13), (3.1), (3.4), (3.5), and (3.6) together give rise to the following equation: ) (
∂∆V¯ −1 ⟨∆Rα ⟩ = (k BT0) P. (3.7) ∆Rα ∆R β 0 ∂R β P,T β The equation provides a physical explanation concerning how the hydrostatic pressure applied to the system induces the structural change, or fluctuation, in protein. According to
(3.2)
where {R} and {r} denote sets of coordinates of atoms in protein and water, respectively, U({R},{r}) is the interaction energy among the atoms in the system, Rα denotes the coordinate of the αth atom in protein, and ria labels the ath atom in the ith water molecule. Taking the derivative of Fs with respect to the position of an atom in protein, we get the following expression:
Nu N n 1 exp − kT U({R},{r}) dRα drai
where V denotes the volume of the system. The first factor in the right-hand side is related to the pressure in terms of the standard thermodynamics, ( ) ∂Fs − = P, (3.5) ∂V T
drai ,
i=1 a=1
∂Rα
The expression indicates that fα is a mean force acting on the atom α due to all other atoms in protein and in solvent, averaged over the statistical ensemble. It is the thermodynamic work due to pressure that induces the structural change of protein. With the consideration in mind, we decouple the derivative into two steps as follows: ( ) ( ) ( ) ∂Fs ∂Fs ∂V , (3.4) = ∂Rα ∂V T ∂Rα P,T
while the second factor is identified as ( ) ( ) ∂V ∂∆V¯ = , ∂Rα P,T ∂Rα P,T
energy surface of the protein defined by F({∆R}).) In order to clarify the physical meaning of fα, we define the free energy of the entire system in terms of the statistical mechanics by the configurational integral as follows: 1 Fs = −kT log ··· exp − U ({R},{r}) kT
Nu α=1
dRα
i=1 a=1
N n i=1 a=1
(3.3)
drai
the equations, the structural response to pressure is realized through the successive response functions. The first one is the first derivative of the partial molar volume with respect to the atomic coordinates of protein. The second is the variancecovariance matrix of the fluctuation in the unperturbed system. When pressure is applied, the deformation first occurs so that the partial molar volume decreases. Then, the deformation is propagated to the entire protein through the variancecovariance matrix. There are two important points to be remarked concerning (3.7). The first is that the both response functions are intrinsic properties of a protein solution, meaning that the properties are determined just by the interactions among atoms in protein and water. The pressure plays a role of magnifying the intrinsic properties, as anticipated in Akasaka’s experiment.11–13 Second, the equation indicates that the entropy of the system does not change due to the perturbation. It, in turn, implies that the dispersion of the structural fluctuation is not changed by applying pressure. The theoretical finding is in accord with the experimental observation by Akasaka, which shows that conformational variation produced by applying pressure is limited within its original ensemble.26
B. Structural fluctuation induced by denaturant
The structural fluctuation is induced by adding the denaturant such as urea into the system. It is obviously the change in the chemical potential that drives the structural change due to denaturant. The thermodynamic potential which includes the chemical potential as independent variables is (PV), and its change caused by adding denaturant is defined by
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d(PV ) = SdT +
Ni dµi + PdV,
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(3.8)
i
where Ni and µi are the number of molecules and the chemical potential of the species i included in the solution. Water is one of those species, but protein is not included in the species, because the solution is at the infinite dilution limit. We define the thermodynamic perturbation due to the denaturant by ) ( ( ∂ µi ) ∂(PV ) = − Ni . (3.9) fα = − ∂Rα T,V ∂Rα T,V i Combining Eq. (3.9) with Eq. (2.13), one finds ) (
∂ µi ⟨∆Rα ⟩ = −(k BT0)−1 Ni . ∆Rα ∆R β 0 ∂R β β i
(3.10)
Here, it should be noted that µi is an implicit functional of the protein structure {R}, because the direct and solvent-induced interactions between protein and denaturant depend sensitively on the structure. According to the equation, the perturbation of the denaturant on the structure of protein is created first through a response of the structure to the change in the chemical potential of the species included in solution. Then, the structural response is propagated over the entire molecule through the variance-covariance matrix. C. Structural fluctuation induced by heat
The deformation due to heat is the most well-studied property of protein. In this case, the change in the entropy is the perturbation that induces the structural change of protein, and it is the energy that contains the entropy as the independent variable. The energy is expressed by dE = T dS − pdV.
(3.11)
Then, the perturbation produced on the protein structure by heat is defined as ) ( ) ( ) ( dE dS d∆ S¯ dE fα = − = −T , (3.12) =− dRα dS V dRα dRα
to temperature is not considered, in general, as a linear process. Nevertheless, the theory may be applied to the non-linear regime, employing the idea of analytical continuation, which is suggested in Sec. V.
IV. CORRESPONDENCE BETWEEN THE THEORY AND THE PRESSURE NMR EXPERIMENT
The theory presented above can be applied to provide a physical explanation for the experimental finding concerning the pressure NMR by Akasaka and coworkers.11–13 In Fig. 1, illustrated conceptually is the free energy surface of protein for two different pressures, which can be deduced from the general features of Eqs. (2.9) and (3.7): the both curves should be quadratic with the same curvature, but with the different equilibrium structures. In the figure, the x-axis represents the structure of protein which is denoted by {R}, and the y-axis is the free energy surface F({R}) of the structure. The curves marked by P1 and P2 (P1 < P2) describe the surface corresponding to the two pressures, and the minimum of the each curve represents the equilibrium conformation corresponding to the respective pressure. The figure illustrates that the conformation is thermally fluctuating around an equilibrium structure, producing a Gaussian distribution. The vertical arrow indicates that the pressure is abruptly increased from P1 to P2 so that the equilibrium structure in the state denoted by P1 becomes a fluctuated or non-equilibrium conformation in the P2 state, as is indicated by the end point of the arrow landing on the P2 surface. Then, the conformation relaxes to the equilibrium point along the P2 surface corresponding to the new thermodynamic condition, which is illustrated by the winding arrow. The fluctuated conformations are distributed around the new minimum to consist a conformational ensemble. The equilibrium structure in the P2 surface corresponds to a fluctuated conformation in the P1 surface, which is indicated by the vertical dashed-line.
where ∆ S¯ denotes the partial molar entropy of protein, which is also a functional of protein structure. Combining Eq. (3.12) with Eq. (2.13), one finds ) (
d∆ S¯ −1 ⟨∆Rα ⟩ = −(k BT0) T, (3.13) ∆Rα ∆R β 0 dR β β where T0 denotes the temperature concerned with the reference system. The expression tells us how the structural fluctuation of protein due to heat or temperature is coupled with the change in the partial molar entropy of protein. The change in the partial molar entropy presumably consists of two major contributions: one from the solvent, i.e., the hydration, the other from the structural entropy. The last statement concerning the structural entropy suggests us a caution to be made in the actual application of (2.13). If the structural entropy increases upon rising temperature, the variance-covariance matrix should also increase, since the quantity is a measure of variation or dispersion of protein structure. Therefore, the response of protein structure
FIG. 1. Schematic picture of the free energy surface of a protein in solution, drawn against the structure of protein: {R} represents the structure of protein; the labels P1 and P2 indicate that the surfaces correspond to the two pressures; {R1} and {R2} represent the equilibrium structure corresponding to the two pressures.
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The picture drawn here is in complete accord with the hypothesis proposed by Akasaka and his coworkers based on the high pressure NMR. The hypothesis states that different conformations induced by pressure are the fluctuating states of the protein when the solution is in the native condition, and that pressure would not create a new state, but simply creates a new population distribution among the existing states. In order to show the consistency of the two concepts, let us suppose that P1 in Fig. 1 is the ambient pressure, thereby the P1 surface represents the conformational fluctuation taking place around the native structure, while the surface labeled by P2 depicts the surface of higher pressure. The minimum of the P2 surface represents the equilibrium conformation at the pressure P2, but the structure corresponds to a fluctuated state on the P1 surface, which is indicated by the vertical dashed line. Since the argument applies to any higher pressure within the linear regime, the conformation corresponding to the completely denatured state, produced by pressure, should exist in the ambient condition as a fluctuated state, though in much lower probability. Basically, the same argument applies to the reverse process, or, “refolding.” Suppose that the P2 surface in Fig. 1 represents the unfolded state of a protein. If one lowers the pressure to the ambient pressure P1, abruptly, as is practiced in the ordinary experiment, the system jumps up vertically to a fluctuated state of the P1 surface and relaxes to the native conformation along the surface. The relaxation process is nothing but the refolding. It would be worthwhile to make a further comment on Fig. 1. The surface drawn in Fig. 1 is just a conceptual illustration presenting the global nature of the free energy surface. An actual free energy surface is not monotonical but has a lot of local minima corresponding to conformational substates that have a hierarchical structure. The rate of fluctuation and relaxation is determined by the detailed structure of the free energy surface. For example, if the surface has two major substates, i.e., the native and denatured states, then the rate will be determined by the height of the barrier which separates the two substates. An important implication of the new theory is that the fluctuated states of protein can be reproduced theoretically, which are identical in its physics to those found by Akasaka et al. It has been already shown by Kim and Hirata17 that the variance-covariance matrix in Eq. (3.7) is related to the second derivative of the free energy surface with respect to the atomic coordinates of protein (see Eq. (2.10)). The free energy surface, which is an explicit function of the atomic coordinates, can be calculated from the 3D-RISM/RISM theory.18 The first derivative of the partial molar volume can be also calculated based on the 3D-RISM/RISM theory combined with the site-site Kirkwood-Buff theory27,28 ∆V¯ ({R}) = k BT χT0 *.1 − ρ ,
γ
V
cγ (r;{R})dr+/ . -
(4.1)
In the equation, cγ (r;{R}) is the direct correlation function of the water molecules around protein, which is a functional of the protein structure {R}.
V. EXTENDING THE THEORY TO A LARGE FLUCTUATION
It is likely that the variance-covariance matrix in Eq. (2.13) changes significantly due to a perturbation, even though the distribution still remains Gaussian or approximately Gaussian. One such example is the thermal denaturation processes described by Eq. (3.13). The variancecovariance matrix for the denatured state is apparently much greater than that of the native state, since the conformational fluctuation for the denatured state should be greater than the native state due to the increased structural entropy. The linear response theory, Eq. (2.13), in its original form may not be applied to those processes. However, we may be able to extend the theory to such a case by means of the analytical continuation. We divide the perturbation and the response into several steps for which the linear response theory is valid. We refer to such a process as a “quasi-linear” process. The process can be written as ⟨∆Rα ⟩ j+1 = (k BT j )−1 ⟨∆Rα ∆R β ⟩ j · f βj , (5.1) β
where j denotes the each quasi-linear step, starting from zero. In the equation, the variance-covariance matrix at the ( j + 1)th step should be calculated based on the protein coordinates in the jth step. With a proper care made for “scheduling” of increasing the order parameter, we may be able to get successive curves similar to Fig. 1, but with different values for the curvature, or the variance-covariance matrix. VI. CONCLUDING REMARKS
In the present article, we have presented a new theory to realize structural fluctuation of protein using thermodynamic variables as the order parameters. The new development is motivated by the two recent findings, both indicating that the fluctuation is a Gaussian or linear process: the finding by Akasaka and coworkers based on the pressure NMR experiment, and that by Kim and Hirata based on the generalized Langevin theory. The following question might be raised. Why does the structural fluctuation against a thermodynamic perturbation stay in the linear regime, or in another word, why is the free energy surface quadratic with respect to the structural fluctuation, or equivalently why is the distribution of structure Gaussian? First of all, we have excluded any perturbation that might cause plastic deformation, for example, aggregation among proteins, breaking of chemical bonds, complete evaporating of solvent, and so on. In another word, we are considering the thermodynamic condition which is not far from that in which a living body survives. In such a relatively mild condition, the central limiting theorem31 governs the stochastic processes. As is well regarded, the Gaussian distribution is a manifestation of the central limiting theorem. After all, protein structure is designed to be consistent with the general law of nature. Of course, the general law does not exclude the existence of specific sub-states which play key roles in the function of protein. Rather, the fact that
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the structural fluctuation follows the general law assures the robustness of a protein to maintain our life. Characterizing the structural fluctuation of protein is important not only in the analyses of bio-functions but also in the computer-aided drug discovery, because the affinity of a drug compound to a target protein is significantly affected by the structural fluctuation. Currently, the molecular dynamic simulation has been exclusively employed to realize the structural fluctuation. However, the fluctuation produced by the simulation is largely limited within a small dispersion, since a large fluctuation is a rare event. The method proposed in the present article will provide the best alternative to the molecular simulation.
∆Rα ≡ Rα (t) − ⟨Rα ⟩, ˙ α. Pα = Mα R
(A7) (A8)
On the other hand, the collective density fluctuation of atom, δρa, and the momentum density J are chosen for solvent, defined as follows: δ r − rai (t) − ⟨ρa ⟩, δ ρa = (A9) i
APPENDIX: GENERALIZED LANGEVIN THEORY FOR STRUCTURAL FLUCTUATION OF PROTEIN
Ja =
Here, we briefly review the theory for the structural fluctuation of protein in solution, presented by Kim and Hirata.17 The time evolution of a system in the phase space can be described by the Liouville equation which reads20 dB(t) = iLB(t), (A1) dt where B(t) represents time dependent variables defined in the phase space, and L is the Liouville operator defined in terms of the Hamiltonian of the system under concern as ( ∂H ) ( ∂ ) ( ∂H ) ( ∂ ) L ≡i · − · . (A2) ∂ri ∂pi ∂pi ∂ri i Here, we are concerned with a system of a single protein immersed in a large number of solvent molecules. We define the Hamiltonian of such a system in terms of the MM type potential function H = H0 + H1 + H2,
time evolution is rewritten in terms of the dynamic variables. A crucial step of developing the theory is how to choose the dynamic variables, which should be able to describe the structural and thermodynamic fluctuations of both protein and water. We chose the deviation or fluctuation of atomic positions from its equilibrium states, ∆Rα , and their conjugated momentum, Pα , as the dynamic variables for protein, defined as
(A3)
where H0, H1, and H2 are, respectively, the Hamiltonian concerning solvent, protein, and solute-solvent interaction, defined by the following equations: N n pa · pa ( ) i i b a H0 = + U0 ri − r j , (A4) 2ma i=1 a=1 j,1 b,a Nu Pα · Pα H1 = + U1 Rα − R β , (A5) 2M α β,α α=1 Nu N n (A6) H2 = Uint Rα − rai . α=1 i=1 a=1
In the equations, pia and ria denote the momentum and position of the atom a in solvent, respectively, U0 the interaction energy between a pair of atoms in solvent, Pα and Rα the momentum and position of the atom α of protein, respectively, U1 the interaction energy between a pair of atoms in the protein, Uint the interaction energy between an atom of solvent and an atom of the protein. Following the general recipe of the GLT, all the variables in the phase space are projected onto few variables under concern, called the “dynamic variables,” and the equation of
ma δ r − r˙ ai (t) .
(A10)
i
Constructing a vector C from the dynamic variables, we project a variable B(t) in the Liouville equation onto the vector of the dynamic-variables through C · (C,B(t)) (C,C) where P is the projection operator defined by PB(t) =
(A11)
C · (C, ) , (A12) (C,C) which satisfies the idempotence. The scalar product in the equations is defined by
˜ (C, B) = CB˜ = dΓ f eq (Γ)C(Γ)B(Γ), (A13) P=
where ⟨···⟩ indicates the canonical ensemble average. After a lengthy mathematical manipulation which is omitted here, we obtain the two equations describing the time evolution of the dynamic variables, concerning the protein structure and the solvent density, around an equilibrium state d 2∆Rα (t) L−1 α β · ∆R β (t) Mα = −k BT 2 dt β t P β (S) − ds Γα β (t − s) · +Wα (t), (A14) Mβ 0 β d 2δ ρak (t) 2 b = −k Jac(k) χ−1 cb (k)δ ρk (t) dt 2 b,c t ik −1 Jac (k) dsMkbc (t − s) · Jkb (s) +ik · Ξka (t). − N 0 (A15) In the equations, Γα β (t), Wα (t), Jac(k), Mkbc (t − s), and Ξka (t) have the following physical meaning. Γα β (t − s): the frictional force due to solvent, acting on protein atoms. Wα (t): the random force acting on protein atoms. Jac(k): the correlation of momentum density between a pair of atoms in solvent. χcb(k): the density correlation function between a pair of solvent atoms under influence of protein.
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J. Chem. Phys. 142, 044110 (2015)
Mkbc (t − s): the frictional force acting on solvent molecules. Ξka (t): the random force acting on solvent atoms. The second equation describes the time evolution of the density fluctuation of water (or solution). The equation looks quite similar to the one derived by one of the authors for dynamics of pure water.25 The main difference is that the new equation describes the dynamics of solvent in inhomogeneous field produced by protein, while the old equation is valid just for dynamics in a homogeneous environment. The first equation, (A14), concerns the dynamics of protein atoms in solvent. The equation has a general form of the Langevin equation, including the friction and randomforce terms. The matrix L included in Eq. (A14) is defined by L α β = ⟨∆R∆R⟩α β ,
(A16)
where ∆R is the displacement, or fluctuation, of atoms in protein from their equilibrium position, defined by Eq. (A7). Therefore, the matrix represents a correlation function of structural fluctuation, or the variance-covariance matrix. The structural fluctuation and dynamics of protein in solution can be explored by solving Eqs. (A14) and (A15) coupled together. It will be worthwhile to make a remark on Eq. (A14) in order to clarify the essential physics involved in the equation. The equation looks similar to the Langevin harmonicoscillator treated by Lamm and Szabo,29 and by Wang and Uhlenbeck.30 However, the two equations are essentially different in their physics. The earlier treatments assume that the oscillation is infinitesimally small, or equivalently that the interaction potential among atoms is harmonic. Such an assumption allows to represent the Hessian in terms of the second derivative of the potential energy with respect to atom positions. On the other hand, no such assumption concerning the interaction potential was made to derive Eq. (A14). The harmonic-oscillator-like equation has been derived from the Liouville equation through a projection of solvent degrees of freedom on to those of protein in the phase-space. The apparent linearity of the dynamics in Eq. (A14), therefore, is not due to making the assumption on the potential energy of protein but originates from the projection that makes the
process stochastic. The linearity of the stochastic process is a manifestation of the central limiting theorem.31 1K.
Henzler-Widman and D. Kern, Nature 450, 964 (2007). Makowski, D. J. Rodi, S. Mandava, D. Minh, D. B. Gore, and R. F. Fischetti, J. Mol. Biol. 375, 529 (2008). 3H. S. Mchaourab, K. J. Oh, C. J. Fang, and W. L. Hubbell, Biochemistry 36, 307 (1997). 4T. P. Burghardt, M. P. Josephson, and K. Ajtai, Biochemistry 50, 7809 (2011). 5N. Go, T. Nouguchi, and T. Nishikawa, Proc. Natl. Acad. Sci. U. S. A. 80, 3696 (1983); N. Go, Biophys. Chem. 35, 105 (1990); T. Horiuchi and N. Go, Proteins 10, 106 (1991). 6B. Brooks and M. Karplus, Proc. Natl. Acad. Sci. U. S. A. 80, 6571 (1983). 7Water and Biomolecules, edited by Kuwajima et al. (Springer, 2009). 8Water: The Forgotten Biological Molecule, edited by D. Le Bihan and H. Fukuyama (Pan Stanford Publishing, 2010). 9A. Kitao, F. Hirata, and N. Go, Chem. Phys. 158, 447 (1991). 10S. Hayward, A. Kitao, F. Hirata, and N. Go, J. Mol. Biol. 234, 1207 (1993). 11K. Akasaka, Biochemistry 42, 10877 (2003). 12K. Akasaka, Chem. Rev. 106, 1814 (2006). 13K. Akasaka, “Pressure and protein dynamism,” High Pressure Res. 34, 222 (2014). 14R. A. Marcus, J. Chem. Phys. 24, 966–979 (1956). 15M. Ikeguchi, J. Ueno, M. Sato, and A. Kidera, Phys. Rev. Lett. 94, 078102 (2005). 16R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957). 17B. Kim and F. Hirata, J. Chem. Phys. 138, 054108 (2013). 18Molecular Theory of Solvation, edited by F. Hirata (Kluwer Academic Publishers, 2003). 19H. Mori, Phys. Rev. 112, 1829 (1958); Prog. Theor. Phys. 28, 763 (1962). 20U. Balucani and M. Zoppi, Dynamics of Liquid State (Oxford Univesity Press, 2001). 21H. Goldstein, Classical Mechanics (Addison-Wesley Publishing Company, 1974). 22L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon Press, 1976). 23E. Lindahl, C. Azuara, P. Koehl, and M. Delarue, Nucleic Acids Res. 34, W52 (2006). 24I. Bahar, T. R. Lezon, A. Bakan, and I. H. Shrivastava, Chem. Rev. 110(3), 1463 (2010). 25F. Hirata, J. Chem. Phys. 96, 4619 (1992). 26K. Akasaka, H. Li, H. Yamada, R. Li, T. Thoresen, and C. K. Woodward, Protein Sci. 8, 1946 (1999). 27J. Kirkwood and F. P. Buff, J. Chem. Phys. 19, 774 (1951). 28T. Imai, A. Kovalenko, and F. Hirata, Chem. Phys. Lett. 395, 1 (2004). 29G. Lamm and A. Szabo, “Langevin modes of macromolecules,” J. Chem. Phys. 85, 7334–7348 (1986). 30M. C. Wang and G. E. Uhlenbeck, “On the theory of the Brownian motion II,” Rev. Mod. Phys. 17, 323–342 (1945). 31S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943). 2L.
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THE JOURNAL OF CHEMICAL PHYSICS 142, 044110 (2015)
Structural fluctuation of proteins induced by thermodynamic perturbation Fumio Hirata1 and Kazuyuki Akasaka2
1
College of Life Sciences, Ritsumeikan University, and Molecular Design Frontier Co. Ltd., Shiga 525-8577, Japan 2 High Pressure Protein Research Center, Institute of Advanced Technology, Kinki University, Wakayama 649-6493, Japan
(Received 14 October 2014; accepted 5 January 2015; published online 27 January 2015) A theory to describe structural fluctuations of protein induced by thermodynamic perturbations, pressure, temperature, and denaturant, is proposed. The theory is formulated based on the three methods in the statistical mechanics: the generalized Langevin theory, the linear response theory, and the three dimensional interaction site model (3D-RISM) theory. The theory clarifies how the change in thermodynamic conditions, or a macroscopic perturbation, induces the conformational fluctuation, which is a microscopic property. The theoretical results are applied, on the conceptual basis, to explain the experimental finding by Akasaka et al., concerning the NMR experiment which states that the conformational change induced by pressure corresponds to structural fluctuations occurring in the ambient condition. A method to evaluate the structural fluctuation induced by pressure is also suggested by means of the 3D-RISM and the site-site Kirkwood-Buff theories. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4906071] I. INTRODUCTION
Structural fluctuation of protein around its native state plays essential roles in a variety of process in which a biomolecule expresses its intrinsic function.1 For example, so called “gating” mechanisms of ion channels are often regulated by the structural fluctuation of amino-acid residues consisting the gate region of the channel. Molecular recognition such as the formation of an enzyme-substrate complex in an enzymatic reaction is controlled often by structural fluctuation of protein. Few typical examples of structural fluctuations around a native conformation of protein, related to function, are “breathing,”2 “hinge-bending,”3 and “arm-rotating”4 motions. Those motions are collective in nature, involving many atoms moving in the same direction. It is not surprising that considerable efforts have been devoted to clarify the conformational fluctuation of protein theoretically, which has started at the end of the last century based on the molecular-mechanics (MM) or dynamics. One of the earliest attempts was to relate the structural fluctuation to the normal mode of protein.5,6 Those works have demonstrated the importance of the collective mode in the fluctuation. However, those efforts could not have provided a realistic physical insight into the dynamics of actual biological processes, since they are concerned with a protein in vacuo, which of course cannot describe the fluctuation correlated with that of solvent. In actual biological processes, solvent plays vital roles both in equilibrium and in fluctuation of protein.7,8 It may not be necessary to spend many words for emphasizing the crucial role played by solvent for stabilizing or destabilizing native structure of protein, such as the hydrophobic interaction and hydrogen bonds. Here, let us consider the roles played by water in fluctuation of protein around its native conformation, associated with recognition of a ligand by protein. The process is primarily a thermodynamic process, governed by 0021-9606/2015/142(4)/044110/8/$30.00
the free energy difference between the two states before and after the recognition. It is obvious that water plays crucial role in the thermodynamics, since the equilibrium structures are determined by the free energies including the excess chemical potential or the solvation free energy of water. However, it is not a mere role of water in the process. Water actually regulates the kinetic pathway of the process as well by controlling the structural fluctuation of amino-acid residues consisting the active site. An example of such processes is a mouth-like motion of amino-acid residues and/or domains. The open-andclose motion of the mouth is driven not only by the direct force acting among atoms in protein but also by that originated from the solvent induced force which in turn caused by the fluctuation in the solvation free energy, or the non-equilibrium free energy. An earlier attempt to consider the effect of solvent on structural fluctuation of protein has been made by Kitao et al. using the molecular dynamics simulation.9,10 By diagonalizing the variance-covariance matrix of the fluctuation, obtained from the trajectory, they have characterized the fluctuation in terms of the frequency spectrum. Unfortunately, such analyses are largely limited by the sampling of the configuration space. Recently, three novel methods to characterize the structural fluctuation of protein in solution have been proposed, one due to an experiment, the other two by theories. Using the high pressure NMR, Akasaka and his coworkers have developed a unique method to realize the fluctuation of protein in solution.11–13 The idea is based on their finding that the NMR signals such as the chemical shift of amino acids change upon increasing pressure, and that the changes become more significant as the pressure enhanced. Based on the finding, Akasaka has proposed a hypothesis that an equilibrium structure at higher pressure should correspond to one of fluctuated structures at the ambient pressure. It suggests that the fluctuated states of protein can be realized by changing pressure as
142, 044110-1
© 2015 AIP Publishing LLC
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an order parameter. The idea is conceptually analogous to that by Marcus, who has realized the solvent fluctuation around solutes in polar solvent by changing the charges on solute, as an order parameter.14 Ikeguchi et al. presented a method based on the linear response theory to describe the structural change of protein induced by a perturbation.15,16 The most striking feature of the theory is that the response function consists of the variancecovariance matrix of the structural fluctuation in the unperturbed system. The theoretical characterization concerning the structural fluctuation of protein in solution has been proposed by Kim and Hirata,17 combining two methods in the statistical mechanics, the three dimensional interaction site model (3DRISM)/RISM theory18 and the generalized Langevin theory (GLT).19,20 The theory has proved that the free energy surface around an equilibrium conformation is essentially quadratic, or equivalently that the structural fluctuation is a Gaussian process. The behavior looks similar at one glance to that of the harmonic oscillator in vacuo, treated in the earlier studies by means of the normal mode analysis (NMA). However, the physical origin of the Gaussian behavior revealed by GLT is entirely different from that by NMA. The “Gaussian fluctuation” of a harmonic oscillator originates from the “hamonicity” of the mechanical potential energy, while that of protein in solution comes from the central limiting theorem characteristic to probabilistic processes.31 Unlike NMA, the harmonic behavior of the fluctuation in the Kim-Hirata theory has been derived naturally without truncating the potential energy among atoms in protein and water at the quadratic terms (see the Appendix). The authors have also suggested that the variance-covariance matrix of the fluctuation can be obtained from the 3D-RISM/RISM theory as the second derivative of the free energy surface with respect to the atomic coordinates of protein. It is our purpose in the present paper to combine the three recent developments in order to propose a new theory to characterize the structural fluctuation of protein induced by thermodynamic perturbation. However, before doing so, we would like to review briefly the earlier treatment concerning the structural fluctuation of protein in order to clarify the difference between the two treatments due to NMA and GLT. Section II is devoted to a brief review of the theoretical treatment concerning the structural fluctuation of protein. In Sec. III, the equations for structural fluctuation of protein induced by thermodynamic perturbation are derived. Section IV is devoted to make a contact between the theory and the experiment in a conceptual level. Section V is concerned with a possible scheme to extend the theory to a fluctuation in nonlinear regime.
II. A REVIEW OF THEORIES FOR STRUCTURAL FLUCTUATION OF PROTEIN A. Earlier treatment of the structural fluctuation of protein in vacuo
The earlier treatment concerning the structural fluctuation of protein assumes either that the interaction among atoms
J. Chem. Phys. 142, 044110 (2015)
in protein is a harmonic potential or that the displacement of atoms from their equilibrium position is small.21,22 Then, the deviation (or “fluctuation”) of atoms in protein from their equilibrium positions follows the Hooke equation of motion Mα
d 2∆Rα (t) = − Hα β · ∆R β (t), dt 2 β ∆Rα (t) = Rα (t) − ⟨Rα ⟩,
(2.1) (2.2)
where Rα (t), ⟨Rα ⟩, and Ma are the position at time t, the equilibrium position, and the mass of atom α in protein, respectively. Hα β is the force constant or the Hessian matrix defined by Hα β =
∂ 2U , ∂∆Rα ∂∆R β
(2.3)
where U(R1, R2, ..., RN) is a harmonic potential, and it has a quadratic form with respect to ∆R, that is, 1 Hα β ∆Rα ∆R β . (2.4) U(∆R1, ∆R2, ..., ∆R N ) = 2 α β If one considers that the protein atoms are in thermal motion, the distribution of ∆Rα becomes “Gaussian,” p(∆R1, ∆R2, ..., ∆R N ) ∥ A∥ 1 = exp − Aα β ∆Rα ∆R β , (2π)3N 2 α β where Aα β is defined by
(2.5)
Hα β , (2.6) kT and ∥ A∥ is the determinant of the matrix {Aab }. The variancecovariance matrix of the “fluctuation” is related to the Hessian through Aα β =
kT ∆Rα ∆R β = . Hα β
(2.7)
Diagonalizing the variance-covariance matrix gives rise to the normal mode vectors and the eigen values. The equation implies that the structural “fluctuation” of a protein can be characterized theoretically by calculating the second derivative of the interaction potential with respect to atomic coordinates. The idea has been employed in the 1980s through the early 1990s to characterize the structural fluctuation of protein by means of the computational science,5,6 and there is a persistent belief that NMA describes the structural fluctuation of protein faithfully as long as the low-frequency or collective modes are concerned with.23,24 However, the “fluctuation” extracted from NMA is different essentially from that taking place in living body, or in aqueous solutions. First, the fluctuation described by NMA is a mechanical oscillation as was clarified in this section, and it has a characteristic peak in the low frequency region of the power spectrum, while such a peak largely disappears in aqueous solution, indicating that the collective motion is not an oscillation but a stochastic process. Second, the protein structure analysed by NMA, taken from Protein Data Bank (PDB), is an equilibrium structure in aqueous solutions, but not in vacuum. In vacuum, the PDB structure is a fluctuated
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structure, and the structure treated by NMA is one of the local minima in high-energy states. One can readily verify it by carrying out a molecular dynamics simulation of a protein in vacuum at ordinary temperature, taking a PDB structure as an initial condition. One will observe that the native conformation quickly breaks down to become a random-coil-like structure. It indicates that the “fluctuation” extracted from NMA is not a real one but merely a “ghost.” B. Structural fluctuation of protein in solution treated by the generalized Langevin theory
The structural fluctuation of protein is not just a mechanical process represented by NMA described above, but conjugated mechanical and thermodynamic processes, in which structure and thermodynamics of water play crucial roles. Therefore, the theory should include both the structures of protein and thermodynamics of water as variables in the equation. The equations describing the time evolution of protein structure and solvent thermodynamics, which are conjugated each other, have been proposed by Kim and Hirata recently based on the GLT.17 The theory is briefly outlined in the Appendix. If one ignores the terms concerning the friction and the random force, Eq. (A14) in the Appendix becomes d 2∆R(t) = −k BTL−1 · ∆R(t), (2.8) dt 2 which is formally identical to Eq. (2.1) with identifying H = −k BTL−1, where L is defined by (A16) in the Appendix. The equation indicates that the structural fluctuation takes place on a quadratic surface, 1 F ({∆R}) = k BT ∆Rα · L−1 α, β · ∆R β , (2.9) 2 α, β and the distribution of the fluctuation is Gaussian. The quadratic surface in (2.9), however, has a physical origin that is entirely different from that in Eq. (2.1). Equation (2.1) was derived by assuming that the interactions among protein atoms are “harmonic,” or the motion is limited to “small oscillation,” while Eq. (2.9) was derived without making any assumption for the interaction among atoms in protein. The equilibrium conformation represented by ⟨R⟩ in Eq. (2.2) is the one which satisfies the condition ∂U ({R}) = 0, ∂Rα where U({R}) is the mechanical interaction potential among atoms in protein. On the other hand, ⟨R⟩ in Eq. (A7) is the conformation which meets the condition ∂F ({∆R}) = 0, ∂∆Rα where F({R}) is the free energy surface of protein in solution, which includes not only the direct interaction among atoms in protein but also the solvation free energy. In short, the average conformation ⟨R⟩ in Eq. (2.2) represents the mechanically most stable structure, while ⟨R⟩ in Eq. (A7) is the thermodynamically most stable conformation. The same argument applies also to the variance-covariance matrix of the Gaussian
J. Chem. Phys. 142, 044110 (2015)
fluctuation. The Hαβ in Eq. (2.1) is related to the second derivative of the mechanical potential energy with respect to the atom position through Eq. (2.3), while L αβ in Eq. (2.8) is related to the second derivative of the free energy through k BT L−1
= αγ
∂ 2F , ∂∆Rα ∂∆Rγ
(2.10)
where (L−1)αγ denotes the αγ element of the inverse matrix of L. C. Linear response theory
The structural fluctuation induced by a perturbation can be described by the linear response theory if the system recovers its equilibrium state upon removing the perturbation, or unless the perturbation causes a non-linear (“plastic”) deformation on the system. The free energy of protein, structure of which is perturbed by a force, can be expressed as 1 ∆Rα · L −1 α, β · ∆R β − ∆Rα · fα . F ({∆R}) = k BT 2 α α, β (2.11) In the equation, the first term describes the free energy surface of an unperturbed system and the second term the perturbation. By applying the variational principle, or ∂F ({∆R}) = 0, ∂∆Rα
(2.12)
one can get the following equation for the structural change due to the perturbation:17
⟨∆Rα ⟩1 = (k BT0)−1 ∆Rα ∆R β 0 · f β . (2.13) β
In the equation, the suffix “0” put to the variance-covariance matrix indicates that the quantity is concerned with the reference state or the unperturbed system. The equation has been derived first by Ikeguchi et al.15 and reformulated by Kim and Hirata based on the variational principle.17 If one considers fβ as a perturbation, the new structure represents a fluctuated state produced by the perturbation.
III. STRUCTURAL FLUCTUATION OF PROTEIN INDUCED BY THERMODYNAMIC PERTURBATION
In the present section, we propose a new theory to describe the structural fluctuation of protein, induced by a thermodynamic perturbation, based on the linear response theory described in Sec. II. Application of the linear response theory to the process under concern is rationalized primarily by the pressure-NMR experiment carried out by Akasaka and his coworkers.11–13 By applying pressure on protein solutions, Akasaka and the coworkers have probed the NMR signals concerning the fluctuated states of protein, ranging from partially denatured to fully denatured, depending on the magnitude of pressure. When they have removed the pressure, the protein reversibly recovered its native conformation irrespective of whether it is partially or fully denatured. The experiment suggests strongly that the process is not a non-linear or plas-
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tic deformation but a linear fluctuation. Although we developed the concept originally for pressure as a perturbation, we conjecture that the same argument applies to the other perturbation, such as temperature and denaturant concentration, as far as the structural response stays in the linear regime. A. Structural fluctuation induced by pressure
We define the force acting on an atom of protein in solution by the derivative of the Helmholtz free energy of the system with respect to the coordinates of the atom ) ( ∂Fs , (3.1) fα = − ∂Rα where Rα and Fs denote the αth atom in protein and the Helmholtz free energy of the entire system, respectively. (Note that the notation Fs is used to distinguish it from the free
( −
∂Fs = ∂Rα )
···
( −∂U )
×
Nu
dRα
α=1
N n
α=1
1 U({R},{r}) ··· exp − kT
(3.6)
where ∆V¯ is the partial molar volume of a protein in the solution. It is resulted from an a priori consideration that the change in the volume due to the structural change is only concerned with the partial molar volume. Combined Eqs. (2.13), (3.1), (3.4), (3.5), and (3.6) together give rise to the following equation: ) (
∂∆V¯ −1 ⟨∆Rα ⟩ = (k BT0) P. (3.7) ∆Rα ∆R β 0 ∂R β P,T β The equation provides a physical explanation concerning how the hydrostatic pressure applied to the system induces the structural change, or fluctuation, in protein. According to
(3.2)
where {R} and {r} denote sets of coordinates of atoms in protein and water, respectively, U({R},{r}) is the interaction energy among the atoms in the system, Rα denotes the coordinate of the αth atom in protein, and ria labels the ath atom in the ith water molecule. Taking the derivative of Fs with respect to the position of an atom in protein, we get the following expression:
Nu N n 1 exp − kT U({R},{r}) dRα drai
where V denotes the volume of the system. The first factor in the right-hand side is related to the pressure in terms of the standard thermodynamics, ( ) ∂Fs − = P, (3.5) ∂V T
drai ,
i=1 a=1
∂Rα
The expression indicates that fα is a mean force acting on the atom α due to all other atoms in protein and in solvent, averaged over the statistical ensemble. It is the thermodynamic work due to pressure that induces the structural change of protein. With the consideration in mind, we decouple the derivative into two steps as follows: ( ) ( ) ( ) ∂Fs ∂Fs ∂V , (3.4) = ∂Rα ∂V T ∂Rα P,T
while the second factor is identified as ( ) ( ) ∂V ∂∆V¯ = , ∂Rα P,T ∂Rα P,T
energy surface of the protein defined by F({∆R}).) In order to clarify the physical meaning of fα, we define the free energy of the entire system in terms of the statistical mechanics by the configurational integral as follows: 1 Fs = −kT log ··· exp − U ({R},{r}) kT
Nu α=1
dRα
i=1 a=1
N n i=1 a=1
(3.3)
drai
the equations, the structural response to pressure is realized through the successive response functions. The first one is the first derivative of the partial molar volume with respect to the atomic coordinates of protein. The second is the variancecovariance matrix of the fluctuation in the unperturbed system. When pressure is applied, the deformation first occurs so that the partial molar volume decreases. Then, the deformation is propagated to the entire protein through the variancecovariance matrix. There are two important points to be remarked concerning (3.7). The first is that the both response functions are intrinsic properties of a protein solution, meaning that the properties are determined just by the interactions among atoms in protein and water. The pressure plays a role of magnifying the intrinsic properties, as anticipated in Akasaka’s experiment.11–13 Second, the equation indicates that the entropy of the system does not change due to the perturbation. It, in turn, implies that the dispersion of the structural fluctuation is not changed by applying pressure. The theoretical finding is in accord with the experimental observation by Akasaka, which shows that conformational variation produced by applying pressure is limited within its original ensemble.26
B. Structural fluctuation induced by denaturant
The structural fluctuation is induced by adding the denaturant such as urea into the system. It is obviously the change in the chemical potential that drives the structural change due to denaturant. The thermodynamic potential which includes the chemical potential as independent variables is (PV), and its change caused by adding denaturant is defined by
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044110-5
F. Hirata and K. Akasaka
d(PV ) = SdT +
Ni dµi + PdV,
J. Chem. Phys. 142, 044110 (2015)
(3.8)
i
where Ni and µi are the number of molecules and the chemical potential of the species i included in the solution. Water is one of those species, but protein is not included in the species, because the solution is at the infinite dilution limit. We define the thermodynamic perturbation due to the denaturant by ) ( ( ∂ µi ) ∂(PV ) = − Ni . (3.9) fα = − ∂Rα T,V ∂Rα T,V i Combining Eq. (3.9) with Eq. (2.13), one finds ) (
∂ µi ⟨∆Rα ⟩ = −(k BT0)−1 Ni . ∆Rα ∆R β 0 ∂R β β i
(3.10)
Here, it should be noted that µi is an implicit functional of the protein structure {R}, because the direct and solvent-induced interactions between protein and denaturant depend sensitively on the structure. According to the equation, the perturbation of the denaturant on the structure of protein is created first through a response of the structure to the change in the chemical potential of the species included in solution. Then, the structural response is propagated over the entire molecule through the variance-covariance matrix. C. Structural fluctuation induced by heat
The deformation due to heat is the most well-studied property of protein. In this case, the change in the entropy is the perturbation that induces the structural change of protein, and it is the energy that contains the entropy as the independent variable. The energy is expressed by dE = T dS − pdV.
(3.11)
Then, the perturbation produced on the protein structure by heat is defined as ) ( ) ( ) ( dE dS d∆ S¯ dE fα = − = −T , (3.12) =− dRα dS V dRα dRα
to temperature is not considered, in general, as a linear process. Nevertheless, the theory may be applied to the non-linear regime, employing the idea of analytical continuation, which is suggested in Sec. V.
IV. CORRESPONDENCE BETWEEN THE THEORY AND THE PRESSURE NMR EXPERIMENT
The theory presented above can be applied to provide a physical explanation for the experimental finding concerning the pressure NMR by Akasaka and coworkers.11–13 In Fig. 1, illustrated conceptually is the free energy surface of protein for two different pressures, which can be deduced from the general features of Eqs. (2.9) and (3.7): the both curves should be quadratic with the same curvature, but with the different equilibrium structures. In the figure, the x-axis represents the structure of protein which is denoted by {R}, and the y-axis is the free energy surface F({R}) of the structure. The curves marked by P1 and P2 (P1 < P2) describe the surface corresponding to the two pressures, and the minimum of the each curve represents the equilibrium conformation corresponding to the respective pressure. The figure illustrates that the conformation is thermally fluctuating around an equilibrium structure, producing a Gaussian distribution. The vertical arrow indicates that the pressure is abruptly increased from P1 to P2 so that the equilibrium structure in the state denoted by P1 becomes a fluctuated or non-equilibrium conformation in the P2 state, as is indicated by the end point of the arrow landing on the P2 surface. Then, the conformation relaxes to the equilibrium point along the P2 surface corresponding to the new thermodynamic condition, which is illustrated by the winding arrow. The fluctuated conformations are distributed around the new minimum to consist a conformational ensemble. The equilibrium structure in the P2 surface corresponds to a fluctuated conformation in the P1 surface, which is indicated by the vertical dashed-line.
where ∆ S¯ denotes the partial molar entropy of protein, which is also a functional of protein structure. Combining Eq. (3.12) with Eq. (2.13), one finds ) (
d∆ S¯ −1 ⟨∆Rα ⟩ = −(k BT0) T, (3.13) ∆Rα ∆R β 0 dR β β where T0 denotes the temperature concerned with the reference system. The expression tells us how the structural fluctuation of protein due to heat or temperature is coupled with the change in the partial molar entropy of protein. The change in the partial molar entropy presumably consists of two major contributions: one from the solvent, i.e., the hydration, the other from the structural entropy. The last statement concerning the structural entropy suggests us a caution to be made in the actual application of (2.13). If the structural entropy increases upon rising temperature, the variance-covariance matrix should also increase, since the quantity is a measure of variation or dispersion of protein structure. Therefore, the response of protein structure
FIG. 1. Schematic picture of the free energy surface of a protein in solution, drawn against the structure of protein: {R} represents the structure of protein; the labels P1 and P2 indicate that the surfaces correspond to the two pressures; {R1} and {R2} represent the equilibrium structure corresponding to the two pressures.
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044110-6
F. Hirata and K. Akasaka
J. Chem. Phys. 142, 044110 (2015)
The picture drawn here is in complete accord with the hypothesis proposed by Akasaka and his coworkers based on the high pressure NMR. The hypothesis states that different conformations induced by pressure are the fluctuating states of the protein when the solution is in the native condition, and that pressure would not create a new state, but simply creates a new population distribution among the existing states. In order to show the consistency of the two concepts, let us suppose that P1 in Fig. 1 is the ambient pressure, thereby the P1 surface represents the conformational fluctuation taking place around the native structure, while the surface labeled by P2 depicts the surface of higher pressure. The minimum of the P2 surface represents the equilibrium conformation at the pressure P2, but the structure corresponds to a fluctuated state on the P1 surface, which is indicated by the vertical dashed line. Since the argument applies to any higher pressure within the linear regime, the conformation corresponding to the completely denatured state, produced by pressure, should exist in the ambient condition as a fluctuated state, though in much lower probability. Basically, the same argument applies to the reverse process, or, “refolding.” Suppose that the P2 surface in Fig. 1 represents the unfolded state of a protein. If one lowers the pressure to the ambient pressure P1, abruptly, as is practiced in the ordinary experiment, the system jumps up vertically to a fluctuated state of the P1 surface and relaxes to the native conformation along the surface. The relaxation process is nothing but the refolding. It would be worthwhile to make a further comment on Fig. 1. The surface drawn in Fig. 1 is just a conceptual illustration presenting the global nature of the free energy surface. An actual free energy surface is not monotonical but has a lot of local minima corresponding to conformational substates that have a hierarchical structure. The rate of fluctuation and relaxation is determined by the detailed structure of the free energy surface. For example, if the surface has two major substates, i.e., the native and denatured states, then the rate will be determined by the height of the barrier which separates the two substates. An important implication of the new theory is that the fluctuated states of protein can be reproduced theoretically, which are identical in its physics to those found by Akasaka et al. It has been already shown by Kim and Hirata17 that the variance-covariance matrix in Eq. (3.7) is related to the second derivative of the free energy surface with respect to the atomic coordinates of protein (see Eq. (2.10)). The free energy surface, which is an explicit function of the atomic coordinates, can be calculated from the 3D-RISM/RISM theory.18 The first derivative of the partial molar volume can be also calculated based on the 3D-RISM/RISM theory combined with the site-site Kirkwood-Buff theory27,28 ∆V¯ ({R}) = k BT χT0 *.1 − ρ ,
γ
V
cγ (r;{R})dr+/ . -
(4.1)
In the equation, cγ (r;{R}) is the direct correlation function of the water molecules around protein, which is a functional of the protein structure {R}.
V. EXTENDING THE THEORY TO A LARGE FLUCTUATION
It is likely that the variance-covariance matrix in Eq. (2.13) changes significantly due to a perturbation, even though the distribution still remains Gaussian or approximately Gaussian. One such example is the thermal denaturation processes described by Eq. (3.13). The variancecovariance matrix for the denatured state is apparently much greater than that of the native state, since the conformational fluctuation for the denatured state should be greater than the native state due to the increased structural entropy. The linear response theory, Eq. (2.13), in its original form may not be applied to those processes. However, we may be able to extend the theory to such a case by means of the analytical continuation. We divide the perturbation and the response into several steps for which the linear response theory is valid. We refer to such a process as a “quasi-linear” process. The process can be written as ⟨∆Rα ⟩ j+1 = (k BT j )−1 ⟨∆Rα ∆R β ⟩ j · f βj , (5.1) β
where j denotes the each quasi-linear step, starting from zero. In the equation, the variance-covariance matrix at the ( j + 1)th step should be calculated based on the protein coordinates in the jth step. With a proper care made for “scheduling” of increasing the order parameter, we may be able to get successive curves similar to Fig. 1, but with different values for the curvature, or the variance-covariance matrix. VI. CONCLUDING REMARKS
In the present article, we have presented a new theory to realize structural fluctuation of protein using thermodynamic variables as the order parameters. The new development is motivated by the two recent findings, both indicating that the fluctuation is a Gaussian or linear process: the finding by Akasaka and coworkers based on the pressure NMR experiment, and that by Kim and Hirata based on the generalized Langevin theory. The following question might be raised. Why does the structural fluctuation against a thermodynamic perturbation stay in the linear regime, or in another word, why is the free energy surface quadratic with respect to the structural fluctuation, or equivalently why is the distribution of structure Gaussian? First of all, we have excluded any perturbation that might cause plastic deformation, for example, aggregation among proteins, breaking of chemical bonds, complete evaporating of solvent, and so on. In another word, we are considering the thermodynamic condition which is not far from that in which a living body survives. In such a relatively mild condition, the central limiting theorem31 governs the stochastic processes. As is well regarded, the Gaussian distribution is a manifestation of the central limiting theorem. After all, protein structure is designed to be consistent with the general law of nature. Of course, the general law does not exclude the existence of specific sub-states which play key roles in the function of protein. Rather, the fact that
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044110-7
F. Hirata and K. Akasaka
J. Chem. Phys. 142, 044110 (2015)
the structural fluctuation follows the general law assures the robustness of a protein to maintain our life. Characterizing the structural fluctuation of protein is important not only in the analyses of bio-functions but also in the computer-aided drug discovery, because the affinity of a drug compound to a target protein is significantly affected by the structural fluctuation. Currently, the molecular dynamic simulation has been exclusively employed to realize the structural fluctuation. However, the fluctuation produced by the simulation is largely limited within a small dispersion, since a large fluctuation is a rare event. The method proposed in the present article will provide the best alternative to the molecular simulation.
∆Rα ≡ Rα (t) − ⟨Rα ⟩, ˙ α. Pα = Mα R
(A7) (A8)
On the other hand, the collective density fluctuation of atom, δρa, and the momentum density J are chosen for solvent, defined as follows: δ r − rai (t) − ⟨ρa ⟩, δ ρa = (A9) i
APPENDIX: GENERALIZED LANGEVIN THEORY FOR STRUCTURAL FLUCTUATION OF PROTEIN
Ja =
Here, we briefly review the theory for the structural fluctuation of protein in solution, presented by Kim and Hirata.17 The time evolution of a system in the phase space can be described by the Liouville equation which reads20 dB(t) = iLB(t), (A1) dt where B(t) represents time dependent variables defined in the phase space, and L is the Liouville operator defined in terms of the Hamiltonian of the system under concern as ( ∂H ) ( ∂ ) ( ∂H ) ( ∂ ) L ≡i · − · . (A2) ∂ri ∂pi ∂pi ∂ri i Here, we are concerned with a system of a single protein immersed in a large number of solvent molecules. We define the Hamiltonian of such a system in terms of the MM type potential function H = H0 + H1 + H2,
time evolution is rewritten in terms of the dynamic variables. A crucial step of developing the theory is how to choose the dynamic variables, which should be able to describe the structural and thermodynamic fluctuations of both protein and water. We chose the deviation or fluctuation of atomic positions from its equilibrium states, ∆Rα , and their conjugated momentum, Pα , as the dynamic variables for protein, defined as
(A3)
where H0, H1, and H2 are, respectively, the Hamiltonian concerning solvent, protein, and solute-solvent interaction, defined by the following equations: N n pa · pa ( ) i i b a H0 = + U0 ri − r j , (A4) 2ma i=1 a=1 j,1 b,a Nu Pα · Pα H1 = + U1 Rα − R β , (A5) 2M α β,α α=1 Nu N n (A6) H2 = Uint Rα − rai . α=1 i=1 a=1
In the equations, pia and ria denote the momentum and position of the atom a in solvent, respectively, U0 the interaction energy between a pair of atoms in solvent, Pα and Rα the momentum and position of the atom α of protein, respectively, U1 the interaction energy between a pair of atoms in the protein, Uint the interaction energy between an atom of solvent and an atom of the protein. Following the general recipe of the GLT, all the variables in the phase space are projected onto few variables under concern, called the “dynamic variables,” and the equation of
ma δ r − r˙ ai (t) .
(A10)
i
Constructing a vector C from the dynamic variables, we project a variable B(t) in the Liouville equation onto the vector of the dynamic-variables through C · (C,B(t)) (C,C) where P is the projection operator defined by PB(t) =
(A11)
C · (C, ) , (A12) (C,C) which satisfies the idempotence. The scalar product in the equations is defined by
˜ (C, B) = CB˜ = dΓ f eq (Γ)C(Γ)B(Γ), (A13) P=
where ⟨···⟩ indicates the canonical ensemble average. After a lengthy mathematical manipulation which is omitted here, we obtain the two equations describing the time evolution of the dynamic variables, concerning the protein structure and the solvent density, around an equilibrium state d 2∆Rα (t) L−1 α β · ∆R β (t) Mα = −k BT 2 dt β t P β (S) − ds Γα β (t − s) · +Wα (t), (A14) Mβ 0 β d 2δ ρak (t) 2 b = −k Jac(k) χ−1 cb (k)δ ρk (t) dt 2 b,c t ik −1 Jac (k) dsMkbc (t − s) · Jkb (s) +ik · Ξka (t). − N 0 (A15) In the equations, Γα β (t), Wα (t), Jac(k), Mkbc (t − s), and Ξka (t) have the following physical meaning. Γα β (t − s): the frictional force due to solvent, acting on protein atoms. Wα (t): the random force acting on protein atoms. Jac(k): the correlation of momentum density between a pair of atoms in solvent. χcb(k): the density correlation function between a pair of solvent atoms under influence of protein.
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044110-8
F. Hirata and K. Akasaka
J. Chem. Phys. 142, 044110 (2015)
Mkbc (t − s): the frictional force acting on solvent molecules. Ξka (t): the random force acting on solvent atoms. The second equation describes the time evolution of the density fluctuation of water (or solution). The equation looks quite similar to the one derived by one of the authors for dynamics of pure water.25 The main difference is that the new equation describes the dynamics of solvent in inhomogeneous field produced by protein, while the old equation is valid just for dynamics in a homogeneous environment. The first equation, (A14), concerns the dynamics of protein atoms in solvent. The equation has a general form of the Langevin equation, including the friction and randomforce terms. The matrix L included in Eq. (A14) is defined by L α β = ⟨∆R∆R⟩α β ,
(A16)
where ∆R is the displacement, or fluctuation, of atoms in protein from their equilibrium position, defined by Eq. (A7). Therefore, the matrix represents a correlation function of structural fluctuation, or the variance-covariance matrix. The structural fluctuation and dynamics of protein in solution can be explored by solving Eqs. (A14) and (A15) coupled together. It will be worthwhile to make a remark on Eq. (A14) in order to clarify the essential physics involved in the equation. The equation looks similar to the Langevin harmonicoscillator treated by Lamm and Szabo,29 and by Wang and Uhlenbeck.30 However, the two equations are essentially different in their physics. The earlier treatments assume that the oscillation is infinitesimally small, or equivalently that the interaction potential among atoms is harmonic. Such an assumption allows to represent the Hessian in terms of the second derivative of the potential energy with respect to atom positions. On the other hand, no such assumption concerning the interaction potential was made to derive Eq. (A14). The harmonic-oscillator-like equation has been derived from the Liouville equation through a projection of solvent degrees of freedom on to those of protein in the phase-space. The apparent linearity of the dynamics in Eq. (A14), therefore, is not due to making the assumption on the potential energy of protein but originates from the projection that makes the
process stochastic. The linearity of the stochastic process is a manifestation of the central limiting theorem.31 1K.
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