Redox entropy of plastocyanin: developing a microscopic view of mesoscopic polar solvation

We report applications of analytical formalisms and molecular dynamics (MD) simulations to the calculation of redox entropy 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 280-310 K. 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 electrostatic protein hydration need to incorporate realistic density profile of water at the protein-water interface.     

Simulation studies of the protein-water interface. II. Properties at the mesoscopic resolution

We report molecular dynamics (MD) simulations of three protein-water systems (ubiquitin, apo-calbindin D(9K), and the C-terminal SH2 domain of phospholipase C-gamma1), from which we compute the dielectric properties of the solutions. Since two of the proteins studied have a net charge, we develop the necessary theory to account for the presence of charged species in a form suitable for computer simulations. In order to ensure convergence of the time correlation functions needed for the analysis, the minimum length of the MD simulations was 20 ns. The system sizes (box length, number of waters) were chosen so that the resulting protein concentrations are comparable to experimental conditions. A dielectric component analysis was carried out to analyze the contributions from protein and water to the frequency-dependent dielectric susceptibility chi(omega) of the solutions. Additionally, an even finer decomposition into protein, two solvation shells, and the remaining water (bulk water) was carried out. The results of these dielectric decompositions were used to study protein solvation at mesoscopic resolution, i.e., in terms of protein, first and second solvation layers, and bulk water. This study, therefore, complements the structural and dynamical analyses at molecular resolution that are presented in the companion paper. The dielectric component contributions from the second shell and bulk water are very similar in all three systems. We find that the proteins influence the dielectric properties of water even beyond the second solvation shell, in agreement with what was observed for the mean residence times of water molecules in protein solutions. By contrast, the protein contributions, as well as the contributions of the first solvation shell, are system specific. Most importantly, the protein and the first water shell around ubiquitin and apo-calbindin are anticorrelated, whereas the first water shell around the SH2 domain is positively correlated.     

Dipole solvation in dielectrics

This paper presents an exact solution for the free energy of linear solvation of a dipolar solute in an arbitrary dielectric material with a microscopic spectrum of polarization fluctuations. The solution is given in terms of wave vector-dependent longitudinal and transverse structure factors of the polarization fluctuations in the pure dielectric. Good agreement with computer simulations of dipole solvation in dipolar and dipolar--quadrupolar liquids is achieved.     

Protein-water electrostatics and principles of bioenergetics

Despite its diversity, life universally relies on a simple basic mechanism of energy transfer in its energy chains-hopping electron transport between centers of electron localization on hydrated proteins and redox cofactors. Since many such hops connect the point of energy input with a catalytic site where energy is stored in chemical bonds, the question of energy losses in (nearly activationless) electron hops, i.e., energetic efficiency, becomes central for the understanding of the energetics of life. We show here that standard considerations based on rules of Gibbs thermodynamics are not sufficient, and the dynamics of the protein and the protein-water interface need to be involved. The rate of electronic transitions is primarily sensitive to the electrostatic potential at the center of electron localization. Numerical simulations show that the statistics of the electrostatic potential produced by hydration water are strongly non-Gaussian, with the breadth of the electrostatic noise far exceeding the expectations of the linear response. This phenomenon, which dramatically alters the energetic balance of a charge-transfer chain, is attributed to the formation of ferroelectric domains in the protein's hydration shell. These dynamically emerging and dissipating domains make the shell enveloping the protein highly polar, as gauged by the variance of the shell dipole which correlates with the variance of the protein dipole. The Stokes-shift dynamics of redox-active proteins are dominated by a slow component with the relaxation time of 100-500 ps. This slow relaxation mode is frozen on the time-scale of fast reactions, such as bacterial charge separation, resulting in a dramatically reduced reorganization free energy of fast electronic transitions. The electron transfer activation barrier becomes a function of the corresponding rate, self-consistently calculated from a non-ergodic version of the transition-state theory. The peculiar structure of the protein-water interface thus provides natural systems with two "non's"-non-Gaussian statistics and non-ergodic kinetics-to tune the efficiency of the redox energy transfer. Both act to reduce the amount of free energy released as heat in electronic transitions. These mechanisms are shown to increase the energetic efficiency of protein electron transfer by up to an order of magnitude compared to the "standard picture" based on canonical free energies and the linear response approximation. In other words, the protein-water tandem allows both the formation of a ferroelectric mesophase in the hydration shell and an efficient control of the energetics by manipulating the relaxation times.     

Dielectric relaxation of aqueous solutions of hydrophilic versus amphiphilic peptides

We report on molecular dynamics simulations of the frequency-dependent dielectric relaxation spectra at room temperature for aqueous solutions of a hydrophilic peptide and an amphiphilic peptide at two concentrations. We find that only the high-concentration amphiphilic peptide solution exhibits an anomalous dielectric increment over that of pure water, while the hydrophilic peptide exhibits a significant dielectric decrement. The dielectric component analysis carried out by decomposing these peptide solutions into peptide, hydration layer, and outer layer(s) of water clearly shows the presence of a unique dipolar component with a relaxation time scale on the order of approximately 25 ps (compared to the bulk water time scale of approximately 11 ps) that originates from the interaction between the hydration layer water and the outer layer(s) of water. Results obtained from the dielectric component analysis further show the emergence of a distinct and much lower frequency relaxation process for the high-concentration amphiphilic peptide compared to the hydrophilic peptide due to strong peptide dipolar couplings to all constituents, accompanied by a slowing of the structural relaxation in all water layers, giving rise to time scales close to approximately 1 ns. We suggest that the molecular origin of the dielectric relaxation anomalies is due to frustration in the water network arising from the amphiphilic chemistry of the peptide that does not allow it to reorient on the picosecond time scale of bulk water motions. This explanation is consistent with the idea of the "slaving" of residue side chain motions to protein surface water, and furthermore offers the possibility that the anomalous dynamics observed from a number of spectroscopies arises at the interface of hydrophobic and hydrophilic domains on the protein surface.     

Hydration Dynamics of Water near an Amphiphilic Model Peptide at Low Hydration Levels: A Dielectric Relaxation Study

A dielectric relaxation study of aqueous solutions of the amphiphilic model peptide N‐acetyl‐leucine amide (NALA) at 298 K over a wide range of hydration levels is presented. The experiments range from states where water builds up several hydration layers to states where single water molecules or small water clusters are shared by several NALA molecules. The dielectric spectra reveal two modes on the 10 and 100 ps timescales. These are largely broadened with regard to the Lorentzian shape caused by simple Debye‐type relaxation, and are well described by the Kohlrausch–Williams–Watts stretched exponential function. The fast mode is assigned to water reorientation comprising bulk water as well as hydration water. Even when all water molecules are in contact with the solute, this fast component is dominant, and its mean relaxation time is retarded by less than a factor of two relative to neat water. The amplitude of the slow process is far higher than expected for the dipolar reorientation of the solute. The observations are consistent with results from molecular dynamics simulations for a similar model peptide reported in the literature. They suggest that the slow relaxation mode is mainly founded in peptide–water dipolar couplings, with some additional contribution from slowly reorienting hydration water molecules. The results are discussed with regard to the hydration dynamics of proteins and the interpretation of dielectric spectra of protein solutions.     

Secondary structure sensitivity of hydrogen bond lifetime dynamics in the protein hydration layer

The heterogeneous nature of a protein surface plays an essential role in its biological activity and molecular recognition, and this role is mediated at least partly through the surrounding water molecules. We have performed atomistic molecular dynamics simulations of an aqueous solution of HP-36 to investigate the correlation between the dynamics of the hydration layer water molecules and the lifetimes of protein-water hydrogen bonds. The nonexponential hydrogen bond lifetime correlation functions have been analyzed by using the formalism of Luzar and Chandler, which allowed identification of the quasi-bound states in the surface and quantification of the dynamic equilibrium between quasi-bound and free water molecules in terms of time-dependent rate of interconversion. It is noticed that, irrespective of the structural heterogeneity of different segments of the protein, namely the three alpha-helices, the positively charged amino acid residues form longer-lived hydrogen bonds with water. The overall relaxation behavior of protein-water hydrogen bonds is found to differ significantly among the three helices of the protein. Study of water number density fluctuation reveals that the hydration layer of helix-3 is much less rigid, which can be correlated with faster structural relaxation of the hydrogen bonds between its residues and water. This also agrees excellently with faster translational and rotational motions of water near helix-3, and hence the lower rigidity of its hydration layer. The lower rigidity of the helix-3 hydration layer also correlates well with the biological activity of the protein, as several of the active-site residues of HP-36 are located in helix-3.     

Theory of solvation in polar nematics

We develop a linear response theory of solvation of ionic and dipolar solutes in anisotropic, axially symmetric polar solvents. The theory is applied to solvation in polar nematic liquid crystals. The formal theory constructs the solvation response function from projections of the solvent dipolar susceptibility on rotational invariants. These projections are obtained from Monte Carlo simulations of a fluid of dipolar spherocylinders which can exist both in the isotropic and nematic phases. Based on the properties of the solvent susceptibility from simulations and the formal solution, we have obtained a formula for the solvation free energy which incorporates the experimentally available properties of nematics and the length of correlation between the dipoles in the liquid crystal. The theory provides a quantitative framework for analyzing the steady-state and time-resolved optical spectra and makes several experimentally testable predictions. The equilibrium free energy of solvation, anisotropic in the nematic phase, is given by a quadratic function of cosine of the angle between the solute dipole and the solvent nematic director. The sign of solvation anisotropy is determined by the sign of dielectric anisotropy of the solvent: solvation anisotropy is negative in solvents with positive dielectric anisotropy and vice versa. The solvation free energy is discontinuous at the point of isotropic-nematic phase transition. The amplitude of this discontinuity is strongly affected by the size of the solute becoming less pronounced for larger solutes. The discontinuity itself and the magnitude of the splitting of the solvation free energy in the nematic phase are mostly affected by microscopic dipolar correlations in the nematic solvent. Illustrative calculations are presented for the equilibrium Stokes shift and the Stokes shift time correlation function of coumarin-153 in 4-n-pentyl-4'-cyanobiphenyl and 4,4-n-heptyl-cyanopiphenyl solvents as a function of temperature in both the nematic and isotropic phases.     

Protein dielectrophoresis: Key dielectric parameters and evolving theory

Globular proteins exhibit dielectrophoresis (DEP) responses in experiments where the applied field gradient factor ∇E² appears far too small, according to standard DEP theory, to overcome dispersive forces associated with the thermal energy kT of disorder. To address this a DEP force equation is proposed that replaces a previous empirical relationship between the macroscopic and microscopic forms of the Clausius–Mossotti factor. This equation relates the DEP response of a protein directly to the dielectric increment δε⁺ and decrement δε⁻ that characterize its β-dispersion at radio frequencies, and also indirectly to its intrinsic dipole moment by way of providing a measure of the protein's effective volume. A parameter Γ_pw, taken as a measure of cross-correlated dipole interactions between the protein and its water molecules of hydration, is included in this equation. For 9 of the 12 proteins, for which an evaluation can presently be made, Γ_pw has a value of ≈4600 ± 120. These conclusions follow an analysis of the failure of macroscopic dielectric mixture (effective medium) theories to predict the dielectric properties of solvated proteins. The implication of a polarizability greatly exceeding the intrinsic value for a protein might reflect the formation of relaxor ferroelectric nanodomains in its hydration shell.     

CHARMM fluctuating charge force field for proteins: I parameterization and application to bulk organic liquid simulations

A first-generation fluctuating charge (FQ) force field to be ultimately applied for protein simulations is presented. The electrostatic model parameters, the atomic hardnesses, and electronegativities, are parameterized by fitting to DFT-based charge responses of small molecules perturbed by a dipolar probe mimicking a water dipole. The nonbonded parameters for atoms based on the CHARMM atom-typing scheme are determined via simultaneously optimizing vacuum water-solute geometries and energies (for a set of small organic molecules) and condensed phase properties (densities and vaporization enthalpies) for pure bulk liquids. Vacuum solute-water geometries, specifically hydrogen bond distances, are fit to 0.19 A r.m.s. error, while dimerization energies are fit to 0.98 kcal/mol r.m.s. error. Properties of the liquids studied include bulk liquid structure and polarization. The FQ model does indeed show a condensed phase effect in the shifting of molecular dipole moments to higher values relative to the gas phase. The FQ liquids also appear to be more strongly associated, in the case of hydrogen bonding liquids, due to the enhanced dipolar interactions as evidenced by shifts toward lower energies in pair energy distributions. We present results from a short simulation of NMA in bulk TIP4P-FQ water as a step towards simulating solvated peptide/protein systems. As expected, there is a nontrivial dipole moment enhancement of the NMA (although the quantitative accuracy is difficult to assess). Furthermore, the distribution of dipole moments of water molecules in the vicinity of the solutes is shifted towards larger values by 0.1-0.2 Debye in keeping with previously reported work.     

Electric field inside a "Rossky cavity" in uniformly polarized water

Electric field produced inside a solute by a uniformly polarized liquid is strongly affected by dipolar polarization of the liquid at the interface. We show, by numerical simulations, that the electric "cavity" field inside a hydrated non-polar solute does not follow the predictions of standard Maxwell's electrostatics of dielectrics. Instead, the field inside the solute tends, with increasing solute size, to the limit predicted by the Lorentz virtual cavity. The standard paradigm fails because of its reliance on the surface charge density at the dielectric interface determined by the boundary conditions of the Maxwell dielectric. The interface of a polar liquid instead carries a preferential in-plane orientation of the surface dipoles thus producing virtually no surface charge. The resulting boundary conditions for electrostatic problems differ from the traditional recipes, affecting the microscopic and macroscopic fields based on them. We show that relatively small differences in cavity fields propagate into significant differences in the dielectric constant of an ideal mixture. The slope of the dielectric increment of the mixture versus the solute concentration depends strongly on which polarization scenario at the interface is realized. A much steeper slope found in the case of Lorentz interfacial polarization also implies a higher free energy penalty for polarizing such mixtures.     

The effect of surface dipoles and of the field generated by a polarization gradient on the repulsive force

Double-layer and hydration interactions have been coupled into a single set of equations because both are dependent on the polarization of the water molecules. The coupled equations involve the electric fields generated by the surface charge and surface dipoles, as well as the field due to the neighboring dipoles in water. The dipoles on the surface are generated through the counterions' binding to sites of opposite charge. The equations obtained were employed to explain the restabilization observed experimentally at large ionic strengths for colloidal particles on which protein molecules were adsorbed. Polar molecules adsorbed on a charged surface of colloidal particle can generate a field either in the same direction as that generated by the charge or in the opposite direction. The effect of the sign of the dipole of the adsorbed polar molecules on the interaction between surfaces was also examined.     

A molecular dynamics study of the dielectric properties of aqueous solutions of alanine and alanine dipeptide

Molecular dynamics simulations were used to compute the frequency-dependent dielectric susceptibility of aqueous solutions of alanine and alanine dipeptide. We studied four alanine solutions, ranging in concentration from 0.13-0.55 mol/liter, and two solutions of alanine dipeptide (0.13 and 0.27 mol/liter). In accord with experiment we find a strong dielectric increment for both solutes, whose molecular origin is shown to be the zwitterionic nature of the solutes. The dynamic properties were analyzed based on a dielectric component analysis into solute, a first hydration shell, and all remaining (bulk) waters. The results of this three component decomposition were interpreted directly, as well as by uniting the solute and hydration shell component to a "suprasolute" component. In both approaches three contributions to the frequency-dependent dielectric properties can be discerned. The quantitatively largest and fastest component arises from bulk water [i.e., water not influenced by the solute(s)]. The interaction between waters surrounding the solute(s) (the hydration shell) and bulk water molecules leads to a relaxation process occurring on an intermediate time scale. The slowest relaxation process originates from the solute(s) and the interaction of the solute(s) with the first hydration shell and bulk water. The primary importance of the hydration shell is the exchange of shell and bulk waters; the self-contribution from bound water molecules is comparatively small. While in the alanine solutions the solute-water cross-terms are more important than the solute self-term, the solute contribution is larger in the dipeptide solutions. In the latter systems a much clearer separation of time scales between water and alanine dipeptide related properties is observed. The similarities and differences of the dielectric properties of the amino acid/peptide solutions studied in this work and of solutions of mono- and disaccharides and of the protein ubiquitin are discussed.     

Dynamics of water in the hydration layer of a partially unfolded structure of the protein HP-36

Atomistic molecular dynamics simulations of the folded native structure and a partially unfolded molten globule structure of the protein villin headpiece subdomain or HP-36 have been carried out with explicit solvent to explore the effects of unfolding on the dynamical behavior of water present in the hydration layers of different segments (three alpha-helices) of the protein. The calculations revealed that the unfolding of helix-2 influences the translational and rotational motions of water present in the hydration layers of the three helices in a heterogeneous manner. It is observed that a correlation exists between the unfolding of helix-2 and the microscopic kinetics of protein-water hydrogen bonds formed by its residues. This in turn has an influence on the rigidity of the hydration layers of the helices in the unfolded structure versus that in the folded native structure. These results should provide a microscopic explanation to recent solvation dynamics experiments on folded native and unfolded structures of proteins.     

Charge asymmetries in hydration of polar solutes

We study the solvation of polar molecules in water. The center of water's dipole moment is offset from its steric center. In common water models, the Lennard-Jones center is closer to the negatively charged oxygen than to the positively charged hydrogens. This asymmetry of water's charge sites leads to different hydration free energies of positive versus negative ions of the same size. Here, we explore these hydration effects for some hypothetical neutral solutes, and two real solutes, with molecular dynamics simulations using several different water models. We find that, like ions, polar solutes are solvated differently in water depending on the sign of the partial charges. Solutes having a large negative charge balancing diffuse positive charges are preferentially solvated relative to those having a large positive charge balancing diffuse negative charges. Asymmetries in hydration free energies can be as large as 10 kcal/mol for neutral benzene-sized solutes. These asymmetries are mainly enthalpic, arising primarily from the first solvation shell water structure. Such effects are not readily captured by implicit solvent models, which respond symmetrically with respect to charge.     

Dipoles-dipole association of mesogenic molecules in solution

The dielectric polarization has been used to study dipolar association of 4-n-pentyl-4'-cyanobiphenyl in benzene solution. The results have been interpreted with the assumption of a monomer-dimer equilibrium. To explain the relatively high effective dipole moment of the dimers, a new structure has been proposed for them.     

Atomistic mechanism of protein denaturation by urea

Effects of urea on protein stability have been studied from all-atom molecular dynamics simulations of ubiquitin, G311 protein, and immunoglobulin binding domain (B1) of streptococcal protein G (GB1) in water and 8 M aqueous urea solution. The mechanism of the change in the solvent environment and the early events in protein unfolding by urea have been identified with emphasis on the change in the interactions of hydrophilic and hydrophobic parts of the protein by calculating the potential of mean force (PMF). Urea replaces the protein-protein and protein-water contacts by forming stronger contacts with the protein, which is indicated by the longer survival times of the protein-urea hydrogen bonds.     

Dipolar Relaxation Dynamics at the Active Site of an ATPase Regulated by Membrane Lateral Pressure

The active transport of ions across biological membranes requires their hydration shell to interact with the interior of membrane proteins. However, the influence of the external lipid phase on internal dielectric dynamics is hard to access by experiment. Using the octahelical transmembrane architecture of the copper‐transporting P_1B‐type ATPase from Legionella pneumophila as a model structure, we have established the site‐specific labeling of internal cysteines with a polarity‐sensitive fluorophore. This enabled dipolar relaxation studies in a solubilized form of the protein and in its lipid‐embedded state in nanodiscs. Time‐dependent fluorescence shifts revealed the site‐specific hydration and dipole mobility around the conserved ion‐binding motif. The spatial distribution of both features is shaped significantly and independently of each other by membrane lateral pressure.