[2016 Report]

Statistical mechanics theory to control the structural fluctuation ofprotein in aqueous solution	 Fumio Hirata (Toyota Physical and Chemical Institute)
The fellow	
 has developed a statistical mechanics of liquids (RISM & 3D-RISM) to
explore chemical processes in solution and in life phenomena,	
 including chemical
reactions and the structure-function relation of protein. However, an important problem
is remained unsolved. It is the structural fluctuation and dynamics of the biomolecule inwater. The structural fluctuation plays a crucial role when a protein expresses its
function such as an enzyme and ion channels. It is also important in a process of drug
design to analyse the structural fluctuation of a target protein. A purpose of the researchis to develop a new theory to control the structural fluctuation of protein in aqueous
solutions, combining the two theories in statistical mechanics, the generalized Langevintheory and the RISM/3D-RISM/ theory. 	
 [Introduction]
Needless to say, protein is a main player for living bodies to maintain their life.
Concerning the structural fluctuation of protein, a physical model has been professed
persistently among biophysisists. It is a harmonic oscillator model or normal mode
analysis in vacuum. [1,2] The model, however, is unable to describe a structuralfluctuation of living protein in our body. Why? It is because such a protein will undergo
a non-linear (prastic) deformation by a small perturbation. (If you doubt the statement,
perform a molecular dynamics simulation of a protein in vacuum for, maybe, 100 picoseconds, starting from its X-ray structure.) A protein in a cell makes large
conformational change when it is functioning. However, the protein restores its native
structure after a perturbation associated with its activity is removed, on the contrary to
the case of a portein in vacuum. For examples, an enzyme in a cell makes large
structural change when it is working as a catalyst. However, it recovers its original
structure upon completing its catalytic activity, and becomes ready for next reaction. (Infact, it is the reason why the molecule is called “biocatalyst.”) That is, the structural
response of a protein at work to a perturbation is linear or elastic deformation. The

linearlity of the response is an essential requirement for a living body to maintain its life.
Then, why is the structural fluctuation of living protein linear? The answer to thequestion lies in water. In order to understand the statement intuitively, let’s look at Fig.
1. Illustrated conceptually in Fig. 1 is the structural response of a protein to appliedpressure. Fig. 1(a) illustrates the situation in which pressure is applied to a protein in
vacuum. It is readily imagined that the protein will be crushed by small pressure, andthat the protein will not restore its original structure spontaneously. On the other hand,
illustrated in Fig. 1(b) is a protein in water. In this case, the pressure (or perturbation)
gives influence on the structure of protein through water. The structure will be
deformed completely into a random coil if a large magnitude of pressure is applied.
However, the native strcuture will be restored upon the pressure being removed,
contrary to the case of protein in vacuum, as has been proven by Akasaka experimentaly
by means of the pressure NMR. [3] Namely, water makes the structural response ofprotein to pressure to be linear.

Fig. 1
Why does the structural flucuation of protein in water become linear? The answer to thequestion is that the structure is determined not only by the intramolecular interaction
among atoms in protein, but also by the solvation free energy. For examples, thesecondary structure of protein is supported by hydrogen-bonds among backone atoms.
If the large pressure is applied to such a molecule in vacuun, extensive reorganization ofhydrogen-bonds will take place to make protein structure entangled, and the original
structure will never restore after the presuure is removed. However, if the molecule is inwater, the broken hydrogen-bonds due to the perturbation will be complemented bythose with water molecules. Upon the pressure being removed, recombination ofhydrogen-bonds will take place to recover the native structure.
We have derived a generalized Langevin equation to describe the structuraldynamics of a protein in water. The equation suggests strongly that the structuralfluctuation of protein is linear. If the strutural fluctuation is linear, the linear response
theory should be valid. Therefore, we have proposed a linear reponse theory to describea structural response to a thermodynamic (static) perturbation. In the last year, thetheory was further extended to a structural response to a time dependent perturbation, an
example of which is the photo-induced structural dynamics of a protein. Those works
are briefly reviewed in the present report.
I. Theory of the structural fluctuation of protein [B. Kim & F. Hirata, J. Chem. Phys.,138, 054108 (2012)]
The research starts from the following equation that describes the strutcuralfluctuation of protein in aqueous solutions. (The equation was derived by us in 2012.)

dΔRα (t)= ΔVα (t)
dt

( )

t
dVα (t)kT
1= − B ∑ L−1 αβ ⋅ ΔR β (t) − ∫ ds ∑
Γ αβ (t − s) ⋅ ΔVβ (t) + Wα (t)
0
dt
Mα β
β Mα

(1)

where ΔRα (t) represents the displacement of atom α of protein from its equilibrium
position, that is, “fluctuation.” The equation has a form of the Langevin equation: thefirst, second and third terms in the right hand side, are a restoring force proportional tothe displacement, a frictional force proportinal to the velocity, and a random force
exerted by solvent, respectively. If one ignores the second and third terms, it reduces toan equation which looks like a “harmonic oscillator.” The most intriguing aspect of theequation lies in the first term which has a restoring force proportional to thedisplacement. If it is the case in a harmonic oscillator in “vacuum,” the restoring force isderived from the Taylor expansion of the potential energy with respect to the atom
position, by ignoring the terms higher than the second order, based on an assumption of

“small oscillation.”On the contrary, our equation does not include such anassumption. The restoring force was natually derived as a result of the ensemble
average of the mechanical variables over the entire phase-space including both protein
and solvent. And, the “force constant” gets the following expression in terms of thevariance-covariance (or dsipersion) matrix ( ΔRΔR ) of the fluctuation.

kBTL−1 ≡ kBT ΔRΔR

−1

(2)

The ensemble average in the variance-covariance matirx involves not only thecoordinates of protein but also those of water molecules. From an analogy to the case ofa harmonic oscillator in vacuum, we gave the following definition to thevarinace-covariance matrix.

( )kBT L−1 αβ =∂ 2 F ({R})∂ΔRα ∂ΔR β(3)In the equation, F ({R}) is the free energy surface of protein, that can be defined by
the following equation;

F ({R}) = U ({R}) + Δµ ({R})(4)

where U ({R}) and Δµ ({R}) are the interaction energy among protein atoms andthe solvation free energy, respectively. Considering that the equilibrium structure of aprotein is a minimum point of the free energy surface, the definition for the force
constant is rational. The free energy surface can be obtained from the RISM and
3D-RISM theories, to which the author has made significant contributions. Therefore,
the force constant of the free energy surface can be calculated from Eq. (3).
II. Linear response theory
If the force constant in Eq. (1) is expressed in terms of the free energy by Eq. (3),the free energy should be expressed reversely in the quadratic function of ΔR as

F ({ΔR}) =( )1
kBT ∑ ΔRα ⋅ L−1 α , β ⋅ΔR β .
2
α ,β

(5)

Suppose a perturbation is applied to the system. The structure of protein as well as its
free energy should be changed due to the perturbation. The structural change can bepredicted based on the variational principle as follows. Firstly, we add a energy change,
proportinal to the perturbation (f), to the free energy surface.
F ({ΔR}) =( )1
kBT ∑ ΔRα ⋅ L−1 α , β ⋅ΔR β − ∑ ΔRα ⋅ fα
2
α ,β
α

(6)

Then, apply the variational priciple to Eq. (6) to find a linear responce expression
describing structural change induced by the perturabtion. (The expression has beenoriginally derived by Ikeguchi et. al. in a different way.)

ΔRα 1 = ( kBT0 )−1∑ ΔR ΔR
α

β

β 0

⋅ fβ

(7)

III. Structural response of protein to a thermodynamic perturbation [F. Hirata & K.
Akasaka,J. Chem. Phys., 142, 044110 (2015)]
The (static) linear response theory, Eq. (7), can be applied to the structural change
induced by a thermodynamic pertubation such as pressure, temperature, and denaturant.
For examples, the structural change due to pressure can be described by the following
equation.
ΔRα = ( kBT0 )−1∑ ΔR ΔRβαβ 0⎛ ∂ΔV ⎞
⎜ ∂R ⎟ P⎝
β ⎠ P,T(8)

In the equation, P, ΔV , ΔRα ΔR β , and ΔRα denote pressure (change), the partial
0
molar volume of protein in water, the variance-covariance matrix of the structuralfluctuation of protein concerning the unperturbed system, and the displacement or
fluctuation of protein atom α from its equilibrium position. Eq. (8) implies thefollowing physics for the pressure induced structural change of protein. The pressure(P) first induces the displacement in the position (or structure) of an atom β of protein
so as to reduce the partial molar volume (Le Chatelier’s law). The displacement is
propagated to an atom α through the variance-covariance matrix

ΔRα ΔR β0toinduce the displacment of α . Integrating the displacement over the contributions fromall β atoms gives rise to

ΔRα .

The theory can be apllied to the computer-aided-drug-design (CADD) as a tool. A
central issue in CADD is so called compound screening. It is a process to find acompound among as many as few hundred thousand compounds, that has the highest
affinity to a target protein. A pin of the neck, that has been bothering scientists in thecommunity, is the structural fluctuation of protein. As is emphasized in this report,
conformation of a protein is fluctuating in our body to make a statistical ensemble
around an equilibrium structure, the distribution of which is most likely a Gaussian. It is
not nccessarily the equilibrium structure that has the highest affinity to a drug
compound. It is very likely that one of fluctuated structures has higher affinity to a drug
compound. Therefore, such a fluctuated structure should be regarded as a target forcompound screening. However, it is not an easy task for conventional method. It is
becuase a fluctuated structure is a “rare event” by definition, greater the fluctuation,
harder for a coventinal method such as MD to sample the conformational space. The
method proposed here can produce a structural fluctuation with any magnitude bycontroling an applied perturbation.
IV. Dynamic linear response theory [F. Hirata, J. Chem. Phys., 145, 234106 (2016)]
In the previous section, conformational changes of protein induced by a static
pertubation, such as changes of thermodynamic variables, were described as anapplication of the theory of structural fluctuation. Here, a theory to describe thestructural response of protein to a time-dependent perturbation is reveiewed briefly.
The time-dependent perturbation is defined by{}

H (1) ( t ) = θ (t) δ F S ({R}) − δ F P ({R})(9)

In the equation, δ F P ({R}) and δ F S ({R}) are the free energy of protein before andafter the perturbation being applied, respectively. ( δ F S ({R}) is a non-equilibrium freeenergy surface) θ (t) is a step function defined by⎧ 0 t<0θ (t) = ⎨⎩ 1 t≥0(10)That is, a perturbation corresponding to the non-equilibrium free energy is applied tothe aqueous solution including protein at time t=0. Based on the linear response theory,
the author has derived an equation to describe the structural response to the perturbation
as follows.
Rα ( t ) = Rα ( t = ∞ ) −(0)
1
ΔRα (t)ΔR γ⋅ fγ (0)∑eqk BT γ(11)where fγ (0) is a force exerted on the protein due to the perturbation, defined byfγ (t) = −∂H (1) ({R};t ) .
∂R γ

(12)

Eq. (11) describes the time evolution of the structural relaxation of protein toward
Rα (t = ∞) , which was Rα (0) before t=0.

In the equation, the time evoulution of the variance-covariance matrix of thefluctuation ( ΔRα (t)ΔRγ (0) ) is included as a response function. This is concerned with theeqreference system, and it can be evaluated by the theory described above. Multipling
both sides Eq. (1) by ΔRγ (0) and taking a statistcial average, one gets the following
equation.

d ⎛ C(t) ⎞ ⎛ 0
1 ⎞ ⎛ C(t) ⎞
⎜ 
⎟ =⎜
⎟
⎟⎜ 
dt ⎝ C(t) ⎠ ⎝ −K −ζ ⎠ ⎝ C(t)⎠(13)

where K and ζ correspond, respectively, to the “force constant” (Eq. (3)) and friction
cefficient matrices, and C(t) and C(t)are defined by the following equations.

Cαγ (t) ≡ ΔRα (t)ΔR γ (0) , αγ (t) ≡ ΔVα (t)ΔR γ (0)C

The formal solution of Eq. (13) can be obtained readily to giveC(t) = exp ( At ) C(∞)
where A is defined by the following equation.
⎛ 0
1 ⎞
A=⎜
⎟
⎝ −K −ζ ⎠(14)

The problem is thus reduced to calculate the parameters K and ζ . K can beobtained from the free energy surface of protein as the second derivative with respect tothe atomic coordinates, based on Eq. (3). On the other hand, the author devised thefollowing model for the friction coefficient matrix.

⎧⎪
0 (α ≠ β )ζαβ = ⎨(α =β )⎪⎩ fαζα ,bulk
in which ζα,bulk is the friction exerted on atom α in bulk solvent, and fα is the fraction of
the atom contacting with solvent. We define fα by the following equation based on theradial distribution function (RDF) of solvent around the atom,(56)
where gw(σ;bulk) is the RDF of water in bulk at the contact separation between the atom
and a solvent molecule, and gw(σ;protein) is that corresponding to the atom in protein.
If the atom is at surface of protein, fα is close to unity, because the atom is well exposed
to solvent, and gw(σ;protein) will become close to gw(σ;bulk). On the other hand,gw(σ;protein) will become small or zero if the atom is buried inside the protein, since
there are no or few water molecules around such an atom. Both gw(σ;bulk) andgw(σ;protein) can be readily calculated based on the 3D-RISM/KH theory by making anappropriate definition for the contact separation σ between the protein atom and solvent.
A typical choice of σ can be the position of the first peak in RDF in the bulk solvent.
It is a non-trivial problem to determine ζα,bulk by experimental means, because an
atom in protein has a partial charge in general, which is not the case if an atom isisolated by itself in solution: a charged atom in solution exists just in the form of an"ion" which of course has full charges, such as monovalent and divalent ions. Such an
atom with a partial charge in solution is just an "imaginary" atom. We have proposed arecipe to determine the friction of such an imaginary atom in solution based on thesite-site mode coupling theory for the dynamics of ions [7]. (Alternatively, the quantity
can be estimated readily by the standard MD simulation for an ion with hypotheticalpartial-charges in solution.)

Concluding Remarks
Although the author proposed a concept concerning the structural fluctuation ofprotein, the concept is not specific just to a biomolecule, but applies more universally tothe elasiticity of materials. There have been two physical concepts well regarded for theelasiticity of materials. [8] One of those is the reponse (deformation or strain) of a solidmaterial to a perturbation (stress), in which the restoring force is proportional to thedeformation, and the proportional constant, or force constant, is given by the second
derivative of the potential energy with respect to the deformation. Such an elasiticity isreferred to as “energy elasiticity.” The other elasiticity is the “rubber elasiticity.” In thiscase, too, the restoring force is proportional to the magnitude of deformation. However,
the origin of the force is not energy, but entropy. A rubber, which is a polymer, can takea large number of possible conformations with different dihedral angles among atoms,
or a state of larger entropy, when it is a shrinked state. But, it can take only one
conformation, or a state of the least entropy, when it is fully stretched. So, a rubber
tends to restore spontaneously the state of larger entropy (a shrinked state) from that ofless entropy (a stretched sate). Such an elasiticity is called “entropy elesiticity.”
The concept described in this report is different from either of the conventional
physics concerning elesiticity. It is something like an elasiticity (or linearity) induced bysolvent. The author refers to the elasiticty as “solvent induced elasiticity” or “freeenergy elasiticity.”
Recently, the author has found an interesting phenomenon, which may or may not
be related to the new concept of elasiticity. The author wears contact-lenses everyday,that are supposed to be stocked in saline solution over night. One night, he just threw
them away unconsciously into a garbage can, and found them next morning inmiserable shape, which was entirely dried out. Unfortunately, he did not have extra
lenses at that time, thereby he immersed the lenses in saline solution, hoping that it
would recover the original shape and function. After few minutes, the lenses recovered
their shape and function completely. If one folds a contact lense with fingers in water,
the lense is largely distorted. However, the lense restores its original shape completely
after the pressure is removed. On the other hand, when the lense is dried out, it will beclashed if one folds it with fingers, and the original shape will never be restored. The
phenomenon is akin to the case of protein discussed in the review, suggesting that asame natural law is govering the both phenomena.
References[1] N. Go, T. Noguchi, and T. Nishikawa, Proc. Natl. Acad. Scie., USA, 80, 3696
(1983).
[2] B. Brooks and M. Karplus, Proc. Natl. Acad. Sci. USA, 80, 6571 (1983).
[3] K. Akasaka, Chem. Rev., 106, 1814 (2006).
[4] M. Ikeguchi, J. Ueno, M. Sato, and . Kidera, Phys. Rev. Lett., 94, 078102 (2005).
[5] R. Kubo, J. Phys. Soc. Jpn, 12, 570 (1957).
[6] H. L. Friedman, “A course in Statistical Mechanics,” (Prentice Hall, New Jearsey,1985).
[7] S-H. Chong and F. Hirata, J. Chem. Phys., 108, 7339 (1998); 111, 3654 (1999).
[8] T. Nakagawa, “Rheology,” Iwanamizensho. (Japanese).

［Pulication list in 2016］
1. Jiraphorn Phanich, Thanyada Rungrotmongkol, Daniel Sindhikara, Saree
Phongphanphanee, Norio Yoshida, Fumio Hirata, Nawee Kungwan, and Spot
Hannongbua, “A 3D-RISM/RISM Study of the Oseltamivir Binding Efficiency with the
Wild-type and Resistance-Associated Mutant Forms of the Viral Influenza B
Neuraminidase,” Protein Science, 25, 147 – 158 (2016)
2. Fumio Hirata, Bogsoo Kim, “Multi-scale dynamics simulation of protein based on
the generalized Langevin equation combined with 3D-RISM theory,” J. Mol. Liquids,
217 23-38 (2016).
3. Myroslav Holovko, Andriy Kovalenko, and Fumio Hirata, “Partial molar volume ofnonionic surfactants in aqueous solution studied by the KB/3D-RISM-KH theory,” J.
Mol. Liquids, 217, 103 – 111 (2016)
4. Masatake Sugita and Fumio Hirata, “Predicting the binding free energy of the
inclusion process of 2-hydroxypropyl-b-cyclodextrin and small molecules by means of
the MM/3D-RISM method,” J. Phys.:Condens. Matter, 28, 384002-384013 (2016)

5. Fumio Hirata, “A molecular theory of the structural dynamics of protein induced by aperturbation,” J. Chem. Phys., 145, 234106 (2016); doi: 10.1063/1.4971799［Book Chapter］
Fumio Hirata, Norio Yoshida, and Bongsoo Kim, “Theory of Molecular Recognition
and Structural Fluctuation of Biomolecules,” in “Molecular Science of Fluctuations
Toward Biological Functions,” eds., M. Terazima, M. Kataoka, R. Ueoka, and Y.
Okamoto, Springer, 2016.