Correlation analysis for heat denaturation of Trp‐cage miniprotein with explicit solvent

Abstract

Energetics was analyzed for Trp‐cage miniprotein in water to elucidate the solvation effect in heat denaturation. The solvation free energy was computed for a set of protein structures at room and high temperatures with all‐atom molecular dynamics simulation combined with the solution theory in the energy representation, and its correlations were investigated against the intramolecular (structural) energy of the protein and the average interaction energy of the protein with the solvent water. It was observed both at room and high temperatures that the solvation free energy is anticorrelated to the structural energy and varies in parallel to the electrostatic component of the protein–water interaction energy without correlations to the van der Waals and excluded‐volume components. When the set of folded structures sampled at room temperature was compared with the set of unfolded ones at high temperature, it was found that the preference order of the two sets is in correspondence to the van der Waals and excluded‐volume components in the sum of the protein intramolecular and protein‐water intermolecular interactions and is not distinguished by the electrostatic component.

Abbreviations

LJ

Lennard‐Jones interaction

MD

molecular dynamics

PME

particle‐mesh Ewald

SASA

solvent‐accessible surface area

Introduction

Heat is a common cause of protein denaturation. A protein may denature at elevated temperature through modification of the solvation effect since the protein structure in solution is determined by the balance between the intramolecular interaction within the protein and the intermolecular interaction with solvent. In statistical thermodynamics, the probability of finding a certain configuration (structure) of the protein in solvent (water) is given by the sum of the intramolecular (structural) energy and the free energy of solvation. The solvation free energy is thus of central importance in identifying the governing factor to control the temperature effect on the protein structure, and its analysis is desirably conducted at atomic resolution since the hydrogen bonding and hydrophobic effect play key roles and can be faithfully described with explicit solvent. The focus of the present study is the solvation free energy of Trp‐cage miniprotein (PDB code: 1L2Y)1 over a variety of configurations both at room and high temperatures. We employ the molecular dynamics (MD) simulation in combination with a statistical‐mechanical theory of solutions, and address the response of the protein‐water energetics to the temperature variation.

Trp‐cage is a 20‐residue miniprotein and exhibits a stable, folded structure at room temperature.1 It has been intensively studied both experimentally and computationally to explore the principle of structure formation of protein at atomic level.2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 In the present work, we focus on the energetics of solvation (hydration) of Trp‐cage. MD was performed with explicit solvent to sample a set of folded structures at room temperature and a set of unfolded ones at high temperature, and the correlations of the solvation free energy were examined over the two sets against the intramolecular energy of the protein, the electrostatic and van der Waals components of the protein‐water interaction energy, and the excluded‐volume effect to see how the structural variation of the protein is connected to the intramolecular and intermolecular interactions of the protein.

In a previous work, the correlation between the solvation free energy and the solute‐solvent interaction energy was analyzed to elucidate the urea‐induced effects on amino‐acid analog solutes.14 This analysis was done since the solute‐solvent interaction energy is more straightforward for interpretation and prediction than the free energy. For example, it is a rule of thumb that the electrostatic interaction in polar solvent is more favorable for a more polar solute and that the van der Waals interaction is stronger with a larger solute. The free energy reflects, in contrast, the effects of all the interaction components in nonlinear manner, though it determines the observable properties of solvation in the context of statistical thermodynamics. In Ref. 14, the interaction component which plays the decisive role in the energetics of transfer from pure‐water solvent to urea‐water mixed solvent was identified by examining the correlations of the solvation free energy against a variety of components in the solute–solvent interaction. In this work, the correlation analysis was conducted to see the interaction component which governs the effect of temperature elevation on the protein.

A key quantity in our analyses is the solvation free energy. Its computation requires much demand with explicit solvent, however, when such standard methods as free‐energy perturbation and thermodynamic integration are employed in molecular simulation; these methods are conducted in practice by introducing a number of intermediate states connecting the pure‐solvent and solution systems of interest.15, 16 In the present work, we resort to the method of energy representation to obtain the solvation free energy.17, 18, 19, 20 It is a theory of distribution functions in solution and is formulated by adopting the solute‐solvent pair interaction energy for the coordinate of the distribution functions constituting the free‐energy functional. Among a variety of approximate free‐energy methods,21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34 the method of energy representation is unique in compromising the accuracy, the efficiency, and the range of applicability.14, 18, 19, 35, 36, 37, 38, 39, 40, 41 In the energy‐representation method, the simulations are performed only for the pure‐solvent and solution systems, and a set of energy distribution functions obtained from the simulations provides the solvation free energy through an approximate functional. The computational load is thus reduced, and the solvation free energy can be evaluated for a protein with a few hundred residues.42, 43, 44, 45, 46 It was observed, furthermore, that the error from the approximate functional is not larger than the error due to the use of force field.14, 39, 41

Results and Discussion

In the present work, two types of MD calculations were carried out. In one of them, the protein was treated as flexible. Its structure was allowed to fluctuate naturally in the presence of water, and a set of protein configurations (snapshot structures) were sampled during the course of equilibrium fluctuation. In the other, each protein structure thus sampled was kept rigid and the protein–water energetics was analyzed by allowing only the water molecules to move. The MD simulations were performed at two thermodynamics states of

R: 300 K and the solvent‐water density of 0.997 g/cm³,

H: 400 K and 0.937 g/cm³.

R is a room‐temperature state in which Trp‐cage keeps the folded structure, and H is a high‐temperature state in which the protein is mostly unfolded with the force field employed in this work.9, 10 They correspond to the (experimental) pressure of 1 atm, and the densities were taken from Ref. 47.

The protein structure was sampled from MD simulations at states R and H. In the following, the set of folded structures sampled at the room‐temperature state and the set of mostly unfolded ones at the high‐temperature state are called sets F and U, respectively; the preparation of these sets is described in detail in the subsection of “MD simulations of flexible protein”. Figure 1 shows the distributions of the solvent‐accessible surface area (SASA) for the protein structures in sets F and U. Evidently, the SASA in set U is larger and distributes over wider domain. The protein at high temperature is more extended with structural variations.

Figure 1.

Probability distribution functions of the solvent‐accessible surface area (SASA) for the protein structures in sets F and U. SASA was computed from the protein‐water boundary defined as the surface at contact with the spherical probe which mimics the water molecule and has a radius of 1.4 Å.

The protein–water interaction was analyzed at both of states R and H for each structure of the protein in sets F and U. There are thus four groups of computations: set F at state R, set F at state H, set U at state R, and set U at state H, which will be hereafter called FatR, FatH, UatR, and UatH, respectively, for brevity. The (relative) stabilities of the sets of folded and unfolded structures can then be examined by comparing FatR and UatR at room temperature and UatH and FatH at high temperature.

When the (snapshot) structure of protein is denoted as X, the probability distribution function $P(\mathbf{X})$ of finding the structure X in solution is expressed as

$$-k_{\mathrm{B}}T\hspace{0.17em}\log \hspace{0.17em}P(\mathbf{X})=E^{\text{intra}}(\mathbf{X})+\Delta \mu (\mathbf{X})+\text{constant},$$ (1)

where k _B is the Boltzmann constant, T is the temperature, $E^{\text{intra}}(\mathbf{X})$ is the intramolecular (structural) energy of the protein solute, and $\Delta \mu (\mathbf{X})$ is the free energy of solvation (hydration). $E^{\text{intra}}$ consists of the bonded terms for the bond‐stretching, bending, and proper and improper torsions and the nonbonded terms of Coulombic and Lennard‐Jones forms; the nonbonded terms in $E^{\text{intra}}$ will be examined later in the subsection of “Protein structure and interaction component”. The relative stabilities of sets F and U of protein structures are determined by the difference in the free energy given by

$${\displaystyle {\int }_{\mathrm{S}}d}\mathbf{X}P(\mathbf{X})E^{\text{intra}}(\mathbf{X})+{\displaystyle {\int }_{\mathrm{S}}d}\mathbf{X}P(\mathbf{X})\Delta \mu (\mathbf{X})+k_{\mathrm{B}}T{\displaystyle {\int }_{\mathrm{S}}d}\mathbf{X}P(\mathbf{X})\log \hspace{0.17em}P(\mathbf{X}),$$ (2)

where S denotes either of sets F and U and specifies the domain of integration over the protein configuration (structure) X. The first term of Eq. (2) is the average intramolecular energy of the protein solute within set S, and the second term is the averaged free energy of solvation in set S. The third term corresponds to the conformational entropy and quantifies the structural variety of the protein. It is more negative (more favorable) when the protein covers a wider range of the configuration space.48, 49, 50, 51, 52, 53, 54, 55, 56, 57 Actually, $P(\mathbf{X})$ is not a dimensionless quantity and the value of the third term of Eq. (2) depends on the choice of the unit of X. The term carrying the unit is common between sets F and U, however, and does not play any role in discussing the relative stability. We base our analysis on Eqs. (1) and (2) and examine the correlations among a variety of interaction components.

Intramolecular energy and solvation

To see the relationship between the protein structure and the solvation (hydration) effect on the energetic basis, in Figure 2 we present the correlation plot between the intramolecular energy $E^{\text{intra}}$ and the solvation free energy $\Delta \mu$. As noted above, there are four groups of data from FatR, FatH, UatR, and UatH. Over each of the data groups, $E^{\text{intra}}$ and $\Delta \mu$ are anticorrelated to each other; the anticorrelation was also seen for cytochrome c.43 When sets F and U are compared, the ranges of variations of $E^{\text{intra}}$ and $\Delta \mu$ are larger for UatR and UatH than for FatR and FatH. This reflects the property that the configuration space of the protein is broader for set U than for set F, as illustrated in Figure 1.

Figure 2.

Correlation plot between the intramolecular (structural) energy $E^{\text{intra}}$ and the solvation (hydration) free energy $\Delta \mu$, where each point in the figure refers to a (snapshot) structure in FatR, FatH, UatR, and UatH introduced at the beginning of the section of “Results and Discussion”. The correlation coefficient is −0.83, −0.84, −0.93, and −0.90 for FatR, FatH, UatR, and UatH, respectively. The slope in the linear regression with the least‐square fit of $E^{\text{intra}}$ to $\Delta \mu$ is −1.1 for FatR and FatH and is −1.0 for UatR and UatH.

According to Figure 2, it is also seen that the average value of $\left(E^{\text{intra}}+\Delta \mu \right)$ is more favorable (more negative) for FatR than for UatR. This shows that the sum of the first and second terms of Eq. (2) is more favorable for set F than for set U at state R. The third term will be more favorable for set U, on the other hand, since set U is an ensemble of (mostly) unfolded structures of the protein. Thus, the protein remains folded at state R because of the energetic effect represented by $\left(E^{\text{intra}}+\Delta \mu \right)$.

The argument is opposite at state H. Even when the temperature is elevated to 400 K, $\left(E^{\text{intra}}+\Delta \mu \right)$ is still more favorable for set F than for set U. Set U is then more stable than set F due to the presence of the third term in Eq. (2). The conformational entropy plays a key role in heat denaturation.

Solvation free energy and solute–solvent interaction

In the previous subsection, we saw that the solvation free energy $\Delta \mu$ varies on the order of 100 kcal/mol over the sets of protein structures examined. It is then of interest to investigate which component of the intermolecular interaction between the protein solute and the solvent water drives the variation of $\Delta \mu$. To do so, we correlate $\Delta \mu$ against the average sum of the solute–solvent interaction energy $<v>$ in the solution system (protein–water system) and its electrostatic and van der Waals components denoted as $<v{>}_{\text{elec}}$ and $<v{>}_{\text{vdW}}$, respectively. In the present work, $\Delta \mu$ was calculated for fixed structure X of the protein and $<v>$ was similarly computed at fixed structure; both of $\Delta \mu$ and $<v>$ are functions of X. The intermolecular interaction potential is expressed as a sum of electrostatic and van der Waals components, furthermore, and $<v>$ was correspondingly given by the sum of the electrostatic component $<v{>}_{\text{elec}}$ and the van der Waals component $<v{>}_{\text{vdW}}$. It should be noted that while $\Delta \mu$ in this study is calculated with an approximate functional, $<v>$ and its electrostatic and van der Waals components are exact (within statistical error) under the used set of force fields. The decomposition scheme of $<v>$ into $<v{>}_{\text{elec}}$ and $<v{>}_{\text{vdW}}$ is described in more detail in the subsection of “Free‐energy calculation”.

In Figure 3, $\Delta \mu$ is correlated against $<v>,<v{>}_{\text{elec}}$, and $<v{>}_{\text{vdW}}$ in their dimensionless forms (divided by $k_{\mathrm{B}}T$; $\beta =1/k_{\mathrm{B}}T$). The correlation is clearly present for the total interaction energy $\beta <v>$ and the electrostatic component $\beta <v{>}_{\text{elec}}$. In fact, $\beta \Delta \mu$ can be fit to a common straight line against $\beta <v{>}_{\text{elec}}$ over the four groups of data from FatR, FatH, UatR, and UatH. In a previous work,43 we saw that the equilibrium fluctuation of protein structure in water is driven by the electrostatic interaction between protein and water. Figure 3 shows that the correlation between $\beta \Delta \mu$ and $\beta <v{>}_{\text{elec}}$ persists even over nonequilibrium ensembles of FatH and UatR. The slope of the least‐square fit of $\beta <v{>}_{\text{elec}}$ against $\beta \Delta \mu$ is 2.2 with a correlation coefficient of 1.0, and this strong correlation is reminiscent of the linear response.21, 23 The van der Waals component $<v{>}_{\text{vdW}}$ is essentially invariant, on the other hand, over each of FatR, FatH, UatR, and UatH. It plays a minor role in determining the response of the solvation energetics to the protein structure.

Figure 3.

Correlation plots for the solvation free energy $\Delta \mu$ against the average sum of solute–solvent interaction energy $<v>$ in the solution system, the electrostatic component $<v{>}_{\text{elec}}$, the van der Waals component $<v{>}_{\text{vdW}}$, and the excluded‐volume component $\Delta {\mu }_{\text{excl}}$ in their dimensionless forms (divided by $k_{B}T$; $\beta =1/k_{B}T$), where each point in the figure refers to a (snapshot) structure in FatR, FatH, UatR, and UatH introduced at the beginning of the section of “Results and Discussion”. The solid line represents the linear regression from the least‐square fit of $\beta <v{>}_{\text{elec}}$ against $\beta \Delta \mu$.

When a solute is introduced into solvent, a number of solvent molecules need to be removed from the region to be occupied by the solute. The free‐energy penalty corresponding to this removal is denoted as the excluded‐volume effect. It is a major part of repulsive component of solute–solvent interaction and is pointed out as a driving force for structure formation of large molecule.56, 58, 59, 60 The excluded‐volume component is a nonobservable part of the solvation free energy, though, and its treatment requires a model of solvation. We examine the excluded‐volume effect within the framework of the approximate functional in the energy‐representation method.

The excluded volume is the region of solute‐solvent configuration where the solute molecule overlaps with the solvent and the interaction energy is prohibitively large. Within the formalism of the energy‐representation method, the solvation free energy $\Delta \mu$ (actually, the intermolecular part $\Delta {\mu }_{\text{inter}}$ described in the subsection of “Free‐energy calculation”) is written as

$$\Delta \mu =<v>+{\displaystyle \int d}ϵf(ϵ)={\displaystyle \int d}ϵϵ\rho (ϵ)+{\displaystyle \int d}ϵf(ϵ),$$ (3)

where ϵ is the pair interaction energy between solute and solvent, $\rho (ϵ)$ denotes the average distribution (histogram) of ϵ in the solution system (protein–water system), and $f(ϵ)$ describes the effect of solvent reorganization including the excluded‐volume effect. $\Delta \mu$ is expressed as an integral over ϵ in Eq. (3), and its excluded‐volume component $\Delta {\mu }_{\text{excl}}$ is obtained by restricting the integral to high‐energy domain ranging from ϵ^c to infinity. ϵ^c is the threshold for specifying the excluded‐volume domain, and its value can be set somewhat arbitrarily within a requirement that the domain of $ϵ>{ϵ}^c$ corresponds to the solute–solvent overlap and is inaccessible with vanishing $\rho (ϵ)$ in the solution system. In this work, we adopted ${ϵ}^c=20$ kcal/mol, while the following discussion is valid irrespective of the (reasonable) choice of the ϵ^c value.

The correlation of the excluded‐volume component $\Delta {\mu }_{\text{excl}}$ is examined in Figure 3 against the (total) solvation free energy $\Delta \mu$ in their dimensionless forms (divided by $k_{\mathrm{B}}T$). It is seen that the correlation is absent over each of FatR, FatH, UatR, and UatH. The role of the excluded‐volume effect is minor in describing the dependence of the solvation energetics on the protein structure.

Protein structure and interaction component

As described at the beginning of the section of “Results and Discussion”, the stability of a (snapshot) structure X of the protein is determined by $\left(E^{\text{intra}}(\mathbf{X})+\Delta \mu (\mathbf{X})\right)$. In the present subsection, we pursue the interaction component which is reflected in the protein stability. To do so, the relationships of $\left(E^{\text{intra}}+\Delta \mu \right)$ are analyzed against the electrostatic, van der Waals, and excluded‐volume components in the intramolecular (structural) energy of the protein solute and the intermolecular interaction between the solute and the solvent water. Figure 4 shows these interaction components against $\left(E^{\text{intra}}+\Delta \mu \right)$, where $E_{\text{elec}}^{\text{intra}}$ and $E_{\text{vdW}}^{\text{intra}}$ are the electrostatic and van der Waals components in the protein intramolecular energy $E^{\text{intra}}$, respectively, and $<v{>}_{\text{elec}}$ and $<v{>}_{\text{vdW}}$ are the electrostatic and van der Waals components of the solute–solvent interaction energy, respectively. The excluded‐volume effect is not present in the intramolecular energy of the protein solute, and is expressed only with the intermolecular contribution $\Delta {\mu }_{\text{excl}}$.

Figure 4.

Correlation plots for $\left(E^{\text{intra}}+\Delta \mu \right)$ against the electrostatic component in the sum of the intramolecular and intermolecular energies $\left(E_{\text{elec}}^{\text{intra}}+<v{>}_{\text{elec}}\right)$, the corresponding van der Waals component $\left(E_{\text{vdW}}^{\text{intra}}+<v{>}_{\text{vdW}}\right)$, and the excluded‐volume component $\Delta {\mu }_{\text{excl}}$ at (a) state R (FatR and UatR) and (b) state H (FatH and UatH), where each point in the figure refers to a (snapshot) structure in FatR, FatH, UatR, and UatH introduced at the beginning of the section of “Results and Discussion”.

When the protein structure varies within set F, $\left(E^{\text{intra}}+\Delta \mu \right)$ does not correlate to any of $\left(E_{\text{elec}}^{\text{intra}}+<v{>}_{\text{elec}}\right)$, $\left(E_{\text{vdW}}^{\text{intra}}+<v{>}_{\text{vdW}}\right)$, and $\Delta {\mu }_{\text{excl}}$. If taken alone, none of the electrostatic, van der Waals, and excluded‐volume components plays a decisive role in determining the preference of the protein structure in FatR and FatH. The structural variation extends over a larger scale in set U, on the other hand, as shown in Figure 1. The correlation is then observed over set U (for UatR and UatH) between $\left(E^{\text{intra}}+\Delta \mu \right)$ and $\Delta {\mu }_{\text{excl}}$. This indicates that the structural preference of the protein in a large‐scale variation reflects the excluded‐volume effect.

When sets F and U are compared, it is observed in Figure 4 that the van der Waals and excluded‐volume components are more favorable (more negative) in set F than in set U, in correspondence to the relative stabilities of sets F and U shown in Figure 2 [when the conformational entropy expressed as the third term of Eq. (2) is disregarded]. To see this point more clearly, in Table 1 we list the averages and standard deviations of $\left(E_{\text{elec}}^{\text{intra}}+<v{>}_{\text{elec}}\right)$, $\left(E_{\text{vdW}}^{\text{intra}}+<v{>}_{\text{vdW}}\right)$, and $\Delta {\mu }_{\text{excl}}$ in FatR, FatH, UatR, and UatH. According to Table 1, $\left(E_{\text{vdW}}^{\text{intra}}+<v{>}_{\text{vdW}}\right)$ and $\Delta {\mu }_{\text{excl}}$ are distinct between FatR and UatR and between FatH and UatH with the margins of their standard deviations. As a consequence, when the Gaussian distribution is assumed for the values of the interaction components in Table 1, the van der Waals and excluded‐volume components provide the correct order of preference between sets F and U with a probability of more than 0.7 ( $={0.84}^2$) when single structures are sampled from both sets and their interactions are compared. Detailed arguments are as follows. When m and σ are the average and standard deviation of a Gaussian distribution of a stochastic variable x, respectively, x is sampled with a probability of 0.84 either in $x<m+\sigma$ or in $x>m-\sigma$. Let x _F be the value of the van der Waals or excluded‐volume component for a protein structure sampled from set F and x _U be the value from set U. The correct order of structural preference is then given with single samplings from sets F and U if x _F < x _U. Table 1 shows, in turn, that, $m_{\mathrm{F}}+{\sigma }_{\mathrm{F}}$ for set F is similar to $m_{\mathrm{U}}-{\sigma }_{\mathrm{U}}$ for set U, where m _F, σ _F, m _U, and σ _U refer to the statistical quantities in sets F and U. The probability of x _F < x _U is thus larger than that satisfying $x_{\mathrm{F}}<m_{\mathrm{F}}+{\sigma }_{\mathrm{F}}$ and $x_{\mathrm{U}}>m_{\mathrm{U}}-{\sigma }_{\mathrm{U}}$ since the latter is a sufficient condition of the former, and is more than 0.7 given that each of $x_{\mathrm{F}}<m_{\mathrm{F}}+{\sigma }_{\mathrm{F}}$ and $x_{\mathrm{U}}>m_{\mathrm{U}}-{\sigma }_{\mathrm{U}}$ is realized independently with a probability of 0.84. A similar argument does not hold for the electrostatic component $\left(E_{\text{elec}}^{\text{intra}}+<v{>}_{\text{elec}}\right)$ since its distribution overlaps significantly between FatR and UatR and between FatH and UatH.

Table 1.

Averages and Standard Deviations of $\left(E_{\text{elec}}^{\text{intra}}+<v{>}_{\text{elec}}\right)$, $\left(E_{\text{vdW}}^{\text{intra}}+<v{>}_{\text{vdW}}\right)$, and $\Delta {\mu }_{\text{excl}}$ in FatR, FatH, UatR, and UatH

	State R		State H
	FatR	UatR	FatH	UatH
Electrostatic	−1255 (18)	−1249 (52)	−1209 (17)	−1198 (48)
van der Waals	−110 (6)	−94 (10)	−103 (6)	−86 (10)
Excluded‐volume	155 (1)	161 (3)	179 (2)	186 (4)

Each entry contains the average and standard deviation and the standard deviation is in parenthesis.

In Figure 3, we saw that the solvation free energy $\Delta \mu$ correlates strongly to the electrostatic component $<v{>}_{\text{elec}}$ of the solute‐solvent interaction energy. Figure 4 and Table 1 shows, on the other hand, that the electrostatic component in the sum of the intramolecular and intermolecular energies $\left(E_{\text{elec}}^{\text{intra}}+<v{>}_{\text{elec}}\right)$ does not vary with the protein structure in correspondence to $\left(E^{\text{intra}}+\Delta \mu \right)$. Actually, it is seen in Figure 5 that the (total) intramolecular energy $E^{\text{intra}}$ is also correlated well to its electrostatic component $E_{\text{elec}}^{\text{intra}}$ when $E^{\text{intra}}$ and $E_{\text{elec}}^{\text{intra}}$ are not large. Still, the correlations to the electrostatic components are canceled when the intramolecular and intermolecular effects are summed into $\left(E^{\text{intra}}+\Delta \mu \right)$, and within the context of correlation, the relative stabilities of sets F and U of protein structures are not governed by the electrostatic interaction at both of states R and H.

Figure 5.

Correlation plots for the protein intramolecular energy $E^{\text{intra}}$ against its electrostatic component $E_{\text{elec}}^{\text{intra}}$ and the van der Waals component $E_{\text{vdW}}^{\text{intra}}$, where each point in the figure refers to a (snapshot) structure in sets F and U introduced at the beginning of the section of “Results and Discussion”.

The opposite argument holds for the van der Waals component. Figure 3 shows that the solvation free energy $\Delta \mu$ does not correlate to the van der Waals component $<v{>}_{\text{vdW}}$ of the solute‐solvent energy, and according to Figure 5, the correlation is absent between the intramolecular energy $E^{\text{intra}}$ and its van der Waals component $E_{\text{vdW}}^{\text{intra}}$. When the intramolecular and intermolecular effects are summed, in contrast, the van der Waals component is in correspondence to the preference order of the protein structure in the large‐scale variation between sets F and U.

As noted above for the correlation within set U and the comparison between sets F and U, the preference of the protein structure is in correspondence to the excluded‐volume effect in a large‐scale variation of the structure. This is actually consistent with Kinoshita et al.'s argument that the protein structure is determined to maximize the “water entropy”.58, 59, 61 Indeed, the excluded‐volume component in the solvation free energy quantifies the free‐energy penalty due to the solute‐solvent overlap, and the solvent water can “move” over larger domain when the configuration space is smaller for the overlap.

Conclusions

Energetics was analyzed for Trp‐cage miniprotein in water at room and high temperatures over a set of folded structures and a set of mostly unfolded ones. It was seen that the sum of the intramolecular energy and the solvation free energy is more favorable (more negative) for the set of folded structures than for the set of unfolded ones at both the room and high temperatures. This shows that the protein is folded due to the energetic effect at room temperature, while it becomes unfolded by heat due to the conformational entropy. When the protein structure varies either at room or high temperature, the solvation free energy changes in parallel to the electrostatic component in the solute‐solvent interaction energy and does not correlate to the van der Waals and excluded‐volume components. When a large‐scale variation of the protein structure was examined through comparison of the sets of folded and unfolded structures, it was observed that the preference order of the two sets is provided by the van der Waals or excluded‐volume component in the sum of the protein intramolecular and protein–water intermolecular effects. The electrostatic component was not distinct between the sets of folded and unfolded structures.

In the present work, the energetics in heat denaturation was investigated in terms of the correlations among a variety of interaction components. It should be noted that although intuitively appealing, the electrostatic, van der Waals, and excluded‐volume interactions are not observable and cannot be separately tuned for real molecules. Still, these notions are useful for interpreting and predicting a large number of observable phenomena, and the correlation analysis may provide a route to justifying or even finding a concept of practical use.

Materials and Methods

MD setups

The solvent was pure water and the solute was Trp‐cage (PDB code: 1L2Y). Its N‐ and C‐termini were represented as ${\text{NH}}_3^{+}$ and COO⁻, respectively, and the protein has 304 atoms with a total charge of +1. The original TIP3P model (without the Lennard‐Jones term for the hydrogen atom) was used for water, and the protein was treated with the Amber99sb force field.62, 63

All the molecular dynamics (MD) simulations were conducted in the canonical (NVT) ensemble using GROMACS 4.5.5.64 The MD unit cell was cubic, and the periodic boundary condition was employed with the minimum image convention. A single solute and 20000 water molecules were located in the unit cell. The edge length of the unit cell was 84.4 Å when the solvent density is 0.997 g/cm³ (state R listed at the beginning of Results and Discussion), and was 86.1 Å when the solvent density is 0.937 g/cm³ (state H). The equation of motion was integrated with the leap‐frog algorithm at a time step of 2 fs, and the stochastic dynamics integrator was used at a coupling constant of 2 ps to keep the system at the temperatures described in Results and Discussion.65 The lengths were fixed with LINCS for all the bonds in the protein, and the water molecules were kept rigid with SETTLE.66, 67

The electrostatic interaction was handled by the smooth particle‐mesh Ewald (PME) method with a real‐space cutoff of 13.5 Å, a spline order of 4, a relative tolerance of ${10}^{-5}$ (inverse decay length of 0.23 Å^– 1), and a reciprocal‐space mesh size of 81 for each of the x, y, and z directions.68 The Lennard‐Jones (LJ) interaction was truncated by applying the switching function in the switching range of 10–12 Å. The truncation was done on atom–atom basis both for the real‐space part of PME interaction and the LJ interaction, and the long‐range correction of LJ interaction was not included since it has no effect on the correlation analysis described in Results and Discussion.

MD simulations of flexible protein

First, the PDB structure of Trp‐cage was energy‐minimized in vacuum by the steepest descent method. The energy‐minimized protein was then solvated by water, and the water configuration was equilibrated over 1 ns at state R by freezing the protein structure. An MD simulation at state R was subsequently performed for 30 ns by treating the protein as flexible. The first 10 ns of the MD was discarded as the equilibration period, and after that, the solvent‐accessible surface area was calculated by sampling the protein structure every 50 ps and defining the protein‐water boundary as the surface at contact with a spherical probe of a radius of 1.4 Å. The snapshot structure of the protein employed for the analyses of energetics was saved every 1 ns between 11 and 30 ns, and set F defined at the beginning of Results and Discussion consists of 20 structures in total.

To prepare set U, the initial configuration to simulate state R described above was employed and MD was conducted at a temperature of 500 K with the MD cell size given in the preceding subsection for state H. The 500 K MD was done over 70 ns, and the protein‐water configurations at 30, 40, 50, 60, and 70 ns were used as initial configurations for the subsequent MD at state H. Each MD at state H was carried out over 35 ns, and the SASA was evaluated every 50 ps by discarding the first 20 ns of the 35 ns MD. The analyses of energetics were conducted over the snapshot structures of the protein saved every 5 ns between 20 and 35 ns. Set U thus consists of 20 structures of the protein.

It should be noted that in each simulation with the flexible protein, the protein center of mass was fixed at the center of the MD unit cell. The protein was kept at the cell center only to prevent possible complications in handling the simulation data due to the periodic boundary condition, and this procedure does not affect the statistical quantities of interest.

Free‐energy calculation

For each of the protein structures in sets F and U, the solvation free energy was computed at both of states R and H. The protein structure was kept rigid in the free‐energy calculation, and the method of energy representation was employed for the calculation of the solvation free energy.18, 20, 37 In this method, a set of energy distribution functions are obtained from simulation and are substituted into a functional for the solvation free energy. For each structure of the protein solute, MD was done for two systems. One is the solution system of interest, with the position and orientation of the solute frozen. In the solution system, the solute interacts with the solvent at full coupling, and the distribution function of solute–solvent pair energy is to be determined. The solution MD was conducted over 2 ns, and the instantaneous configuration (snapshot) was sampled every 100 fs. The other is of pure water, for which an MD is performed only with water for 500 ps at a sampling interval of 100 fs. The simulation setups including the MD cell size were identical between the solution and pure‐solvent systems, except for the simulation length. The protein solute was inserted into the pure‐solvent system as a test particle to obtain the density of states of solute–solvent pair interaction and the solvent–solvent pair correlation. Upon insertion, the solute almost always overlaps with solvent molecules, and the overlapping configurations account for the excluded‐volume term of the solvation free energy in the energy‐representation method.18, 20, 37 In the present study, the test‐particle insertion was carried out at random position and orientation with the fixed structure of the solute for 200 times per pure‐water configuration sampled, leading to the generation of 10⁶ solute‐solvent configurations in total. An exception is the protein structure at 30 ns in set F, with which the convergence of the free‐energy calculation was examined by prolonging both the solution and pure‐water MD to 10 ns; the results are illustrated in the next paragraph. The solvation free energy was obtained through an approximate functional given in previous papers, where the methodological details are described.18, 20, 37

As described above, the convergence of the free‐energy calculation was examined at state R with the 30 ns structure in set F by conducting 10 ns MD for both the solution and pure water. Figure 6 shows the solvation free energy $\Delta \mu$ as function of the MD length. It is seen that the $\Delta \mu$ convergence within 0.2 kcal/mol is reached in 2 ns and 500 ps of the solution and pure‐solvent calculations, respectively, and thus the MD lengths were set to the ones given in the preceding paragraph.

Figure 6.

Convergence of the solvation free energy $\Delta \mu$ of the 30 ns structure in set F as function of the MD length at state R. The solid line refers to the dependence on the MD length of the solution system with keeping the pure‐solvent MD length 10 ns. The dashed line is for the pure water at fixed 10‐ns length of the solution MD. The cumulative number of insertions is also shown as the upper abscissa; 500 ps corresponds to 10⁶ insertions since pure water was sampled every 100 fs and the number of insertions was 200 per solvent configuration sampled. The upper abscissa is meaningful only for the pure‐solvent plot.

Throughout the present study, no counterion was placed against the charged protein. The effect of the apparent non‐neutrality was corrected by adding the self‐energy term of electrostatic interaction to the solvation free energy.69, 70, 71 As described in the subsection of “MD setups”, the particle‐mesh Ewald method was adopted in our simulations to handle the electrostatic interaction,68 and its self‐energy term refers to the interaction energy of the solute with its own images and neutralizing background. When the self energy is denoted as $\Delta {\mu }_{\text{self}}$, the solvation free energy $\Delta \mu$ is computed as

$$\Delta \mu =\Delta {\mu }_{\text{self}}+\Delta {\mu }_{\text{inter}},$$ (4)

where $\Delta {\mu }_{\text{inter}}$ is the free‐energy change of turning on the solute‐solvent intermolecular interaction. In our calculations, $\Delta {\mu }_{\text{self}}$ was evaluated exactly and $\Delta {\mu }_{\text{inter}}$ was obtained approximately by the energy‐representation method. Actually, $\Delta {\mu }_{\text{self}}$ was found to stay in the range of −5 to −7 kcal/mol over the whole sets of protein structures. Thus, $\Delta {\mu }_{\text{self}}$ in Eq. (4) plays no role in the correlation analysis in Results and Discussion, though the $\Delta \mu$ values presented in this paper are those after adding $\Delta {\mu }_{\text{self}}$. In addition, the intermolecular interaction potential is typically expressed as a sum of electrostatic and van der Waals components, as is so in the present work. The ensemble average of the total interaction energy between solute and solvent is then written as the sum of the electrostatic component $<v{>}_{\text{elec}}$ and the van der Waals component $<v{>}_{\text{vdW}}$. In our treatment, the self energy $\Delta {\mu }_{\text{self}}$ is not added to $<v{>}_{\text{elec}}$ and $<v{>}_{\text{vdW}}$ and is incorporated to the (total) solute–solvent energy $<v>$ through

$$<v>=\Delta {\mu }_{\text{self}}+<v{>}_{\text{elec}}+<v{>}_{\text{vdW}}$$ (5)

in correspondence to Eq. (4). In any case, the correlation results reflect only the intermolecular effect since the self‐energy contribution is essentially constant.