Protein charge transfer far from equilibrium: a theoretical perspective.

Potential differences for protein-assisted electron transfer across lipid bilayers or in bio-nano setups can amount to several 100 mV; they lie far outside the range of linear response theory. We describe these situations by Pauli-master equations that are based on Marcus theory of charge transfer between self-trapped electrons and that obey Kirchhoff's current law. In addition, we take on-site blockade effects and a full non-linear response of the local potentials into account. We present analytical and numerical current-potential curves and electron populations for multi-site model systems and biological electron transfer chains. Based on these, we provide empirical rules for electron populations and chemical potentials along the chain. The Pauli-master mean-field results are validated by kinetic Monte Carlo simulations. We briefly discuss the biochemical and evolutionary aspects of our findings.

A trick of the tail: computing the entropic contribution to the energetics of quinone-protein unbindung.

We estimate the entropic contributions to the free energy of quinone unbinding in bacterial and mitochondrial respiratory chains using molecular dynamics (MD) and Monte Carlo (MC) computer simulations. For a varying length of the isoprenoid side chain, MD simulations in lipid bilayers and in unpolar solvents are used to assess the dihedral angle distributions along the chain. These form the basis of a MC estimate of the number of molecular structures that do not exhibit steric self-overlap and that are confined to the bilayer. We obtain an entropy drive of TΔS = 1.4 kcal mol-1 for each isoprene unit, which in sum is comparable to the redox potential differences involved in respiratory chain electron transfer. We postulate an entropy-driven zipper for quinone unbinding and discuss it in the context of the bioenergetics and the structure of complex I, and we indicate possible consequences of our findings for MD-based free energy computations.

Water purification modeling by functionalized hourglass-shape multilayer nano-channel.

In this study, inspired by the overall structure and operation of the aquaporin channel, graphene-based nanochannels are proposed to be used as potential membranes for the water purification process. To this end, an hourglass-shaped channel has been designed using the three-layer porous graphene sheets and the effects of some main channel's elements, such as the channel bending angle and attached functional groups to it, on the filtration performance have been examined by using molecular dynamics simulations. We find that a suitable bending channel shape can improve the channel efficiency, i.e. both the water permeability and the ion rejection rate of the suitable bent channels were more than for the straight channels. In addition, regarding the different functionalized channels, the half-functionalized channels were more efficient than the completed functionalized ones. Furthermore, by monitoring the dynamics of water molecules as they pass through the narrowest part of the channels, it was found that water molecule rotation assists water transport.

The Unified Redox Scale for All Solvents: Consistency and Gibbs Transfer Energies of Electrolytes from their Constituent Single Ions.

We have devised the unified redox scale Eabs H2O , which is valid for all solvents. The necessary single ion Gibbs transfer energy between two different solvents, which only can be determined with extra-thermodynamic assumptions so far, must clearly satisfy two essential conditions: First, the sum of the independent cation and anion values must give the Gibbs transfer energy of the salt they form. The latter is an observable and measurable without extra-thermodynamic assumptions. Second, the values must be consistent for different solvent combinations. With this work, potentiometric measurements on silver ions and on chloride ions show that both conditions are fulfilled using a salt bridge filled with the ionic liquid [N2225 ][NTf2 ]: if compared to the values resulting from known pKL values, the silver and chloride single ion magnitudes combine within a uncertainty of 1.5 kJ mol-1 to the directly measurable transfer magnitudes of the salt AgCl from water to the solvents acetonitrile, propylene carbonate, dimethylformamide, ethanol, and methanol. The resulting values are used to further develop the consistent unified redox potential scale Eabs H2O that now allows to assess and compare redox potentials in and over six different solvents. We elaborate on its implications.

Protonation and orientation: a computational approach to cocaine diffusion through a model membrane.

We study the diffusion of cocaine through a DMPC lipid bilayer as an example of a protonable, amphiphilic molecule passing a biological membrane. Using classical molecular dynamics simulations, the free energy surfaces are computed applying the umbrella sampling technique for the protonated and the neutral molecule. For the combined surface, we numerically solve the diffusion equation at constant flow and for time-dependent concentrations. We find a potential of mean force dominated by a barrier of 3.5 kcal mol-1 within the membrane, and a pH-dependent entry and exit barrier of 2.0 kcal mol-1 and 4.1 kcal mol-1, respectively. This behaviour can be rationalized chemically by the amphiphilic nature of the molecule and the change of its protonation state while passing the membrane. Diffusion through the barriers is 3.5 times slower than along the membrane, and the typical time scale of passage amounts to 0.1 ms. We discuss biochemical and medical implications of our findings, and comment on the mechanism of the drug passing the blood-brain barrier.

Measurements and Utilization of Consistent Gibbs Energies of Transfer of Single Ions: Towards a Unified Redox Potential Scale for All Solvents.

Utilizing the "ideal" ionic liquid salt bridge to measure Gibbs energies of transfer of silver ions between the solvents water, acetonitrile, propylene carbonate and dimethylformamide results in a consistent data set with a precision of 0.6 kJ mol-1 over 87 measurements in 10 half-cells. This forms the basis for a coherent experimental thermodynamic framework of ion solvation chemistry. In addition, we define the solvent independent pe abs H 2 O - and the E abs H 2 O values that account for the electronating potential of any redox system similar to the pH abs H 2 O value of a medium that accounts for its protonating potential. This E abs H 2 O scale is thermodynamically well-defined enabling a straightforward comparison of the redox potentials (reducities) of all media with respect to the aqueous redox potential scale, hence unifying all conventional solvents' redox potential scales. Thus, using the Gibbs energy of transfer of the silver ion published herein, one can convert and unify all hitherto published redox potentials measured, for example, against ferrocene, to the E abs H 2 O scale.

Predicting Ion Channel Conductance via Dissipation-Corrected Targeted Molecular Dynamics and Langevin Equation Simulations.

Ion channels are important proteins for physiological information transfer and functional control. To predict the microscopic origins of their voltage-conductance characteristics, here we applied dissipation-corrected targeted molecular dynamics in combination with Langevin equation simulations to potassium diffusion through the gramicidin A channel as a test system. Performing a nonequilibrium principal component analysis on backbone dihedral angles, we find coupled protein-ion dynamics to occur during ion transfer. The dissipation-corrected free energy profiles correspond well to predictions from other biased simulation methods. The incorporation of an external electric field in Langevin simulations enables the prediction of macroscopic observables in the form of I-V characteristics.

Protein vibrations and their localization behaviour. A numerical scaling analysis.

Using a classical force field, we investigate the localization properties of protein normal modes. For a set of eighteen proteins that cover five classes of increasing size, we compute the participation ratio as a measure of the spatial extent of protein vibrations. In this scaling analysis, we find extended low-frequency far-infrared and Terahertz modes, in contrast to localized high-frequency near-infrared vibrations. These regimes are separated by a broad crossover around a wave number of 260 cm-1. Biophysical and biochemical implications are discussed, and the vibrational localization properties are compared to those of amorphous solids.

Monte Carlo simulation and thermodynamic integration applied to protein charge transfer.

We introduce a combination of Monte Carlo simulation and thermodynamic integration methods to address a model problem in free energy computations, electron transfer in proteins. The feasibility of this approach is tested using the ferredoxin protein from Clostridium acidurici. The results are compared to numerical solutions of the Poisson-Boltzmann equation and data from recent molecular dynamics simulations on charge transfer in a protein complex, the NrfHA nitrite reductase of Desulfovibrio vulgaris. Despite the conceptual and computational simplicity of the Monte Carlo approach, the data agree well with those obtained by other methods. A link to experiments is established via the cytochrome subunit of the bacterial photosynthetic reaction center of Rhodopseudomonas viridis.

Long-range electron-electron interaction and charge transfer in protein complexes: a numerical approach.

With application to the nitrite reductase hexameric protein complex of Desulfovibrio vulgaris, NrfH2A4, we suggest a strategy to compute the energy landscape of electron transfer in large systems of biochemical interest. For small complexes, the energy of all electronic configurations can be scanned completely on the level of a numerical solution of the Poisson-Boltzmann equation. In contrast, larger systems have to be treated using a pair approximation, which is verified here. Effective Coulomb interactions between neighbouring sites of excess electron localization may become as large as 200 meV, and they depend in a nontrivial manner on the intersite distance. We discuss the implications of strong Coulomb interactions on the thermodynamics and kinetics of charging and decharging a protein complex. Finally, we turn to the effect of embedding the system into a biomembrane.

Thermodynamic integration network approach to ion transport through protein channels: Perspectives and limits.

We present a molecular dynamics simulation study of alkali metal cation transport through the double-helical and the head-to-head conformers of the gramicidin ion channel. Our approach is based on a thermodynamic integration network, which consists of a sequence of transport reactions, absolute free energies of solvation and cycles of alchemical transmutations of the ions. In this manner, we can reliably estimate free energies and their statistical errors via a least-squares method without imposing external forces on the system. Within the double helical channel, we find a free energy surface typical for hopping transport between isoenergetic sites of ion localization, separated by comparatively large activation barriers. For fast transport through the head-to-head conformation, the thermodynamic network scheme starts to break down. © 2018 Wiley Periodicals, Inc.

Charge transfer through a fragment of the respiratory complex I and its regulation: an atomistic simulation approach.

We simulate electron transfer within a fragment of the NADH:ubiquinone oxidoreductase (respiratory complex I) of the hyperthermophilic bacterium Aquifex aeolicus. We apply molecular dynamics simulations, thermodynamic integration, and a thermodynamic network least squares analysis to compute two key parameters of Marcus' theory of charge transfer, the thermodynamic driving force and the reorganization energy. Intramolecular contributions to the Gibbs free energy differences of electron and hydrogen transfer processes, ΔG, are accessed by calibrating against experimental redox titration data. This approach permits the computation of the interactions between the species NAD+, FMNH2, N1a-, and N3-, and the construction of a free energy surface for the flow of electrons within the fragment. We find NAD+ to be a strong candidate for the regulation of charge transfer.

Simulating biological charge transfer: Continuum dielectric theory or molecular dynamics?

We study the thermodynamic parameters of Marcus's theory of charge transfer, the driving forces and the reorganization energies, using two widely applied approaches to bioenergetic problems that seem to be radically different: continuum dielectric theory via a numerical solution of Poisson's equation, and the thermodynamic integration approach based upon classical Newtonian molecular dynamics, as perfomed by Na et al., PCCP 19, 18,938 (2017). With application to a nitrite reductase NrfHA protein heterodimer, we obtain an excellent agreement between the respective driving forces with an r.m.s. deviation of 1.7 kcal/mol, and a lower limit to the reorganization energies. The computational methods turn out to be mutually supportive: molecular dynamics can be used to determine the parameters of a dielectric theory computation, which on the other hand can be used to properly rescale the reorganization energies and partition them into aqueous and protein contributions. In addition, we use the electrostatic approach to study the influence of Ca2+ ions on the free energy landscape of charge transfer.

The Ideal Ionic Liquid Salt Bridge for the Direct Determination of Gibbs Energies of Transfer of Single Ions, Part I: The Concept.

Described is a procedure for the thermodynamically rigorous, experimental determination of the Gibbs energy of transfer of single ions between solvents. The method is based on potential difference measurements between two electrochemical half cells with different solvents connected by an ideal ionic liquid salt bridge (ILSB). Discussed are the specific requirements for the IL with regard to the procedure, thus ensuring that the liquid junction potentials (LJP) at both ends of the ILSB are mostly canceled. The remaining parts of the LJPs can be determined by separate electromotive force measurements. No extra-thermodynamic assumptions are necessary for this procedure. The accuracy of the measurements depends, amongst others, on the ideality of the IL used, as shown in our companion paper Part II.

The Ideal Ionic Liquid Salt Bridge for Direct Determination of Gibbs Energies of Transfer of Single Ions, Part II: Evaluation of the Role of Ion Solvation and Ion Mobilities.

An important intermediate goal to evaluate our concept for the assumption-free determination of single-ion Gibbs transfer energies Δtr G°(i, S1 →S2 ) is presented. We executed the crucial steps a) and b) of the methodology, described in Part I of this treatise, exemplarily for Ag+ and Cl- with S1 being water and S2 being acetonitrile. The experiments showed that virtually all parts of the liquid junction potentials (LJPs) at both ends of a salt bridge cancel, if the bridge electrolyte is an "ideal" ionic liquid, that is, one with nearly identical diffusion of anion and cation. This ideality holds for [N2225 ]+ [NTf2 ]- in the pure IL, but also in water and acetonitrile solution. Electromotive force measurements of solvation cells between S1 and S2 demonstrated Nernstian behavior for Ag+ concentration cells and constant like cell potentials for solutions with five tested Ag+ counterions.

Thermodynamic integration network study of electron transfer: from proteins to aggregates.

We describe electron transfer through the NrfHA nitrite reductase heterodimer using a thermodynamic integration scheme based upon molecular dynamics simulations. From the simulation data, we estimate two of the characteristic energies of electron transfer, the thermodynamic driving forces, ΔG, and the reorganization energies, λ. Using a thermodynamic network analysis, the statistical accuracy of the ΔG values can be enhanced significantly. Although the reaction free energies and activation barriers are hardly affected by protein aggregation, the complete reaction mechanism only emerges from the simulations of the dimer rather than focussing on the individual protein chains: it involves an equienergetic transprotein element of electron storage and conductivity.

Thermodynamic Integration Networks and Their Application to Charge Transfer Reactions within the AauDyPI Fungal Peroxidase.

We present a computer simulation study of the thermodynamics and kinetics of charge transfer reactions within the fungal peroxidase AauDyPI from Auricularia auriculae-judae. Driving forces and reorganization energies are obtained from a thermodynamic integration scheme based upon molecular dynamics simulations. To enhance the numerical accuracy, the free energies are analyzed within a least-squares scheme of a closely knit thermodynamic network. We identify Tyr147, Tyr229, and Trp105 as oxidative agents, and find Trp377 to be a long-lived reaction intermediate. The results are compared to recent experimental findings.

Storage, transport, release: heme versatility in nitrite reductase electron transfer studied by molecular dynamics simulations.

Using molecular dynamics simulations of the thermodynamic integration type, we study the energetics and kinetics of electron transfer through the nitrite reductase enzyme of Sulfurospirillum deleyianum, Wolinella succinogenes and Campylobacter jejuni. In all of these five-heme proteins, the storage of an even number of electrons within a monomeric chain is thermodynamically favoured. Kinetically, two of these electrons are usually transferred almost simultaneously towards the active site. Although the free energy landscape for charge transfer varies significantly from organism to organism, the heme cofactor closest to the interface of a protein dimer always exhibits a particularly low free energy, suggesting that protein dimerization is functional. Interheme electron interaction effects do not play a significant role.

Size matters! On the way to ionic liquid systems without ion pairing.

Several, partly new, ionic liquids (ILs) containing imidazolium and ammonium cations as well as the medium-sized [NTf2 ](-) (0.230 nm(3) ; Tf=CF3 SO3 (-) ) and the large [Al(hfip)4 ](-) (0.581 nm(3) ; hfip=OC(H)(CF3 )2 ) anions were synthesized and characterized. Their temperature-dependent viscosities and conductivities between 25 and 80 °C showed typical Vogel-Fulcher-Tammann (VFT) behavior. Ion-specific self-diffusion constants were measured at room temperature by pulsed-gradient stimulated-echo (PGSTE) NMR experiments. In general, self-diffusion constants of both cations and anions in [Al(hfip)4 ](-) -based ILs were higher than in [NTf2 ](-) -based ILs. Ionicities were calculated from self-diffusion constants and measured bulk conductivities, and showed that [Al(hfip)4 ](-) -based ILs yield higher ionicities than their [NTf2 ](-) analogues, the former of which reach values of virtually 100 % in some cases.From these observations it was concluded that [Al(hfip)4 ](-) -based ILs come close to systems without any interactions, and this hypothesis is underlined with a Hirshfeld analysis. Additionally, a robust, modified Marcus theory quantitatively accounted for the differences between the two anions and yielded a minimum of the activation energy for ion movement at an anion diameter of slightly greater than 1 nm, which fits almost perfectly the size of [Al(hfip)4 ](-) . Shallow Coulomb potential wells are responsible for the high mobility of ILs with such anions.

Charge transfer in strongly correlated systems: an exact diagonalization approach to model Hamiltonians.

We study charge transfer in bridged di- and triruthenium complexes from a theoretical and computational point of view. Ab initio computations are interpreted from the perspective of a simple empirical Hamiltonian, a chemically specific Mott-Hubbard model of the complexes' π electron systems. This Hamiltonian is coupled to classical harmonic oscillators mimicking a polarizable dielectric environment. The model can be solved without further approximations in a valence bond picture using the method of exact diagonalization and permits the computation of charge transfer reaction rates in the framework of Marcus' theory. In comparison to the exact solution, the Hartree-Fock mean field theory overestimates both the activation barrier and the magnitude of charge-transfer excitations significantly. For triruthenium complexes, we are able to directly access the interruthenium antiferromagnetic coupling strengths.