^{1}and Dmitry V. Matyushov

^{1,a)}

### Abstract

We report applications of analytical formalisms and molecular dynamics (MD) simulations to the calculation of redoxentropy of plastocyanin metalloprotein in aqueous solution. The goal of our analysis is to establish critical components of the theory required to describe polar solvation at the mesoscopic scale. The analytical techniques include a microscopic formalism based on structure factors of the solvent dipolar orientations and density and continuum dielectric theories. The microscopic theory employs the atomistic structure of the protein with force-field atomic charges and solvent structure factors obtained from separate MD simulations of the homogeneous solvent. The MD simulations provide linear response solvation free energies and reorganization energies of electron transfer in the temperature range of . We found that continuum models universally underestimate solvation entropies, and a more favorable agreement is reported between the microscopic calculations and MD simulations. The analysis of simulations also suggests that difficulties of extending standard formalisms to protein solvation are related to the inhomogeneous structure of the solvation shell at the protein-water interface combining islands of highly structured water around ionized residues along with partial dewetting of hydrophobic patches. Quantitative theories of electrostaticprotein hydration need to incorporate realistic density profile of water at the protein-water interface.

This research was supported by the National Science Foundation (CHE-0616646). The code used for the NRFT calculations is available at http://theochemlab.asu.edu/codes.html. We are grateful to Marco Sola for introducing us to the problem of protein redox entropy.

I. INTRODUCTION

II. MICROSCOPIC SOLVATION MODEL

III. COMPUTATIONAL ALGORITHM

A. Solute

B. Charging scheme

C. Solvent

IV. SIMULATIONS PROTOCOL

V. RESULTS

A. Redox thermodynamics

B. Solute-solvent average energy

C. Solvent Gibbs and reorganization free energies

VI. DISCUSSION

## Figures

Structure of plastocyanin: the active site includes copper ion (green), 2 histidines (blue), methionine (red), and cysteine (orange) residues.

Structure of plastocyanin: the active site includes copper ion (green), 2 histidines (blue), methionine (red), and cysteine (orange) residues.

Distribution of the positive and negative charge on the surface of the protein. The positively and negatively charged residues are shown, respectively, in red and blue. The copper ion is shown in green.

Distribution of the positive and negative charge on the surface of the protein. The positively and negatively charged residues are shown, respectively, in red and blue. The copper ion is shown in green.

Diagram of the computational algorithm.

Diagram of the computational algorithm.

Separation of real space into regions for the calculation of the Fourier transform of the solute electric field [Eq. (24)]. The Fourier transform is calculated numerically in region 2 and analytically [Eq. (27)] in region 3. The field is set equal to zero within the hard repulsive core of the solute (region 1).

Separation of real space into regions for the calculation of the Fourier transform of the solute electric field [Eq. (24)]. The Fourier transform is calculated numerically in region 2 and analytically [Eq. (27)] in region 3. The field is set equal to zero within the hard repulsive core of the solute (region 1).

Longitudinal and transverse polarization structure factors of TIP3P water [Eq. (28)] calculated at different temperatures from MD trajectories. Also shown is the PPSF calculation at .

Longitudinal and transverse polarization structure factors of TIP3P water [Eq. (28)] calculated at different temperatures from MD trajectories. Also shown is the PPSF calculation at .

Contact of a redox pair with the metal electrode. and show the fluctuating energy gaps for reduction and oxidation electron transfer, respectively. The equilibrium electrochemical potential of the electrode is established when the equilibrium energy gaps are equal for the reduction and oxidation reactions [Eq. (33)]. The Marcus electron transfer parabolas, shown by the dependence of free energy on the energy gap coordinate , are symmetric in this case producing equal oxidation and reduction currents [Eq. (35)].

Contact of a redox pair with the metal electrode. and show the fluctuating energy gaps for reduction and oxidation electron transfer, respectively. The equilibrium electrochemical potential of the electrode is established when the equilibrium energy gaps are equal for the reduction and oxidation reactions [Eq. (33)]. The Marcus electron transfer parabolas, shown by the dependence of free energy on the energy gap coordinate , are symmetric in this case producing equal oxidation and reduction currents [Eq. (35)].

Average solute-solvent interaction energy obtained from MD simulations (closed circles), NRFT (diamonds), and DELPHI continuum calculations (vdW cavity, triangles). The closed diamonds refer to the total average energy including the polarization and density components, while the open diamonds denote the polarization component only. The dashed lines represent linear regressions through the points.

Average solute-solvent interaction energy obtained from MD simulations (closed circles), NRFT (diamonds), and DELPHI continuum calculations (vdW cavity, triangles). The closed diamonds refer to the total average energy including the polarization and density components, while the open diamonds denote the polarization component only. The dashed lines represent linear regressions through the points.

Radial distribution functions between surface residues of PC and oxygens of water. The upper panel shows ionized residues and the lower panel refers to nonpolar residues. The legends in the figure list. aspartic acid (ASP), the probe atom is the oxygen at the first position; lysine (LYS), the probe atom is nitrogen at the position; proline (PRO), the probe atom is the carbon; tyrosine (TYR), with the first carbon as the probe atom.

Radial distribution functions between surface residues of PC and oxygens of water. The upper panel shows ionized residues and the lower panel refers to nonpolar residues. The legends in the figure list. aspartic acid (ASP), the probe atom is the oxygen at the first position; lysine (LYS), the probe atom is nitrogen at the position; proline (PRO), the probe atom is the carbon; tyrosine (TYR), with the first carbon as the probe atom.

Solvation Gibbs energy from MD simulations (closed circles), NRFT calculations (closed diamonds), continuum DELPHI calculation with the standard cavity definition (up triangles), and the cavity surface augmented by the solvent radius (down triangles). The dashed lines are linear regressions through the points.

Solvation Gibbs energy from MD simulations (closed circles), NRFT calculations (closed diamonds), continuum DELPHI calculation with the standard cavity definition (up triangles), and the cavity surface augmented by the solvent radius (down triangles). The dashed lines are linear regressions through the points.

Reorganization energy of PC vs temperature calculated from MD simulations (closed circles), from NRFT (diamonds), and from dielectric continuum using solvent-accessible cavity definition (triangles). The dashed lines are linear regressions through the points. The filled diamonds refer to the full NRFT calculation and the open diamonds denote the polarization response only.

Reorganization energy of PC vs temperature calculated from MD simulations (closed circles), from NRFT (diamonds), and from dielectric continuum using solvent-accessible cavity definition (triangles). The dashed lines are linear regressions through the points. The filled diamonds refer to the full NRFT calculation and the open diamonds denote the polarization response only.

Pair distribution function between oxygen of water and Cu of PC in the reduced (Red) and oxidized (Ox) sates.

Pair distribution function between oxygen of water and Cu of PC in the reduced (Red) and oxidized (Ox) sates.

Reorganization energy of charge-transfer transition in -nitroaniline dissolved in SPC/E water. The results are obtained by MD simulations (Ref. 16) (closed circles) and NRFT calculations (diamonds). Closed and open diamonds refer to the full and polarization response, respectively. Triangles denote half of the Stokes shift from MD simulations; in the LRA, . The dielectric constants of SPC/E water at different temperatures, required for the NRFT input, were taken from MD simulations (Ref. 16).

Reorganization energy of charge-transfer transition in -nitroaniline dissolved in SPC/E water. The results are obtained by MD simulations (Ref. 16) (closed circles) and NRFT calculations (diamonds). Closed and open diamonds refer to the full and polarization response, respectively. Triangles denote half of the Stokes shift from MD simulations; in the LRA, . The dielectric constants of SPC/E water at different temperatures, required for the NRFT input, were taken from MD simulations (Ref. 16).

The distribution of binding energies of water molecules in the first solvation shell of PC(Ox) defined by adding the water diameter to all protein atoms exposed to solution. The binding energy is defined as the total interaction energy of a given water molecule with all atoms of the protein. The inset shows the distribution of distances of charged residues at the protein surface.

The distribution of binding energies of water molecules in the first solvation shell of PC(Ox) defined by adding the water diameter to all protein atoms exposed to solution. The binding energy is defined as the total interaction energy of a given water molecule with all atoms of the protein. The inset shows the distribution of distances of charged residues at the protein surface.

Normalized time self-correlation functions for Cu-water minimum distance, (dotted line), and for the number of particles in the first solvation shell, (solid line). The dashed lines refer to the fits of correlation functions obtained from MD simulations to Eq. (53). The parameters of the fits are listed in the plot.

Normalized time self-correlation functions for Cu-water minimum distance, (dotted line), and for the number of particles in the first solvation shell, (solid line). The dashed lines refer to the fits of correlation functions obtained from MD simulations to Eq. (53). The parameters of the fits are listed in the plot.

## Tables

Atomic partial charges for copper and its four ligands in the reduced (Red) and oxidized (Ox) states of PC.

Atomic partial charges for copper and its four ligands in the reduced (Red) and oxidized (Ox) states of PC.

Temperature-dependent (eV) for PC(Ox). The results are obtained from MD simulations, DELPHI calculations (with vdW and SS cavities), and from NRFT calculations. The calculations were done with three charge distributions of the active site (I–III, see text for description).

Temperature-dependent (eV) for PC(Ox). The results are obtained from MD simulations, DELPHI calculations (with vdW and SS cavities), and from NRFT calculations. The calculations were done with three charge distributions of the active site (I–III, see text for description).

Redox thermodynamics of PC (eV).

Redox thermodynamics of PC (eV).

Redox solvation free energy and redox entropy in for the Red/Ox states of PC. Also listed are the reorganization energy and reorganization entropy, . All energies are in eV and entropies are in meV/K, .

Redox solvation free energy and redox entropy in for the Red/Ox states of PC. Also listed are the reorganization energy and reorganization entropy, . All energies are in eV and entropies are in meV/K, .

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