Protein electron transfer: is biology (thermo)dynamic?

Protein Medium Facilitates Electron Transfer in Photosynthetic Heliobacterial Reaction Center

This computational study addresses the question of how large membrane-bound proteins of electron transport chains facilitate fast vector-based charge transport. We study electron transfer reactions following ultrafast initial charge separation induced by absorption of light by P800 primary pair and leading to the electron localization at the A0 cofactor. Two subsequent, much slower reactions, electron transfer to the iron-sulfur cluster Fx and reduction of the menaquinone (MQ) cofactor, are studied by combining molecular dynamics simulations, electronic structure calculations, and theoretical modeling. The low value of the electronic coupling between A0 and Fx brings this reaction to the microsecond time scale even at the zero activation barrier. In contrast, A0-MQ electron transfer occurs on a subnanosecond time scale and might become the preferred route for charge transport. We elucidate mechanistic properties of the protein medium allowing fast, vectorial charge transfer. The electric field is high and inhomogeneous inside the protein and is coupled to high polarizabilities of cofactors to significantly lower the reaction barrier. The A0-MQ separation puts this reaction at the edge between the plateau characterizing the reaction dynamical control and exponential falloff due to electronic tunneling. A strong separation in relaxation times of the medium dynamics for the forward and backward reactions promotes vectorial charge transfer.

Elementary Reactions in the Functional Triads of the Blue-Light Photoreceptor BLUF Domain

The blue light using the flavin (BLUF) domain is one of the smallest photoreceptors in nature, which consists of a unique bidirectional electron-coupled proton relay process in its photoactivation reaction cycle. This perspective summarizes our recent efforts in dissecting the photocycle into three elementary processes, including proton-coupled electron transfer (PCET), proton rocking, and proton relay. Using ultrafast spectroscopy, we have determined the temporal sequence, rates, kinetic isotope effects (KIEs), and concertedness of these elementary steps. Our findings provide important implications for illuminating the photoactivation mechanism of the BLUF domain and suggest an engineering platform to characterize intricate reactions involving proton motions that are ubiquitous in nonphotosensitive protein machines.

Long-Range Conductivity in Proteins Mediated by Aromatic Residues

Single-molecule measurements show that many proteins, lacking any redox cofactors, nonetheless exhibit electrical conductance on the order of a nanosiemen over 10 nm distances, implying that electrons can transit an entire protein in less than a nanosecond when subject to a potential difference of less than 1 V. This is puzzling because, for fast transport (i.e., a free energy barrier of zero), the hopping rate is determined by the reorganization energy of approximately 0.8 eV, and this sets the time scale of a single hop to at least 1 μs. Furthermore, the Fermi energies of typical metal electrodes are far removed from the energies required for sequential oxidation and reduction of the aromatic residues of the protein, which should further reduce the hopping current. Here, we combine all-atom molecular dynamics (MD) simulations of non-redox-active proteins (consensus tetratricopeptide repeats) with an electron transfer theory to demonstrate a molecular mechanism that can account for the unexpectedly fast electron transport. According to our MD simulations, the reorganization energy produced by the energy shift on charging (the Stokes shift) is close to the conventional value of 0.8 eV. However, the non-ergodic sampling of molecular configurations by the protein results in reaction-reorganization energies, extracted directly from the distribution of the electrostatic energy fluctuations, that are only ∼0.2 eV, which is small enough to enable long-range conductivity, without invoking quantum coherent transport. Using the MD values of the reorganization energies, we calculate a current decay with distance that is in agreement with experiment.

Electron Tunneling in Biology: When Does it Matter?

Electrons can tunnel between cofactor molecules positioned along biological electron transport chains up to a distance of ≃ 20 Å on the millisecond time scale of enzymatic turnover. This tunneling range determines the design of biological energy chains facilitating the cross-membrane transport of electrons. Tunneling distance and cofactors' redox potentials become the main physical parameters affecting the rate of electron transport. In addition, universal charge-transport properties are assigned to all proteins, making protein identity, flexibility, and dynamics insignificant. This paradigm is challenged by dynamical models of electron transfer, showing that the electron hopping rate is constant within the crossover distance R* ≃ 12 Å, followed with an exponential falloff at longer distances. If this hypothesis is fully confirmed, natural and man-made energy chains for electron transport should be best designed by placing redox cofactors near the crossover distance R*. Protein flexibility and dynamics affect the magnitude of the maximum hopping rate within the crossover distance. Changes in protein flexibility between forward and backward transitions contribute to vectorial charge transport. For biological energy chains, charge transport through proteins is not defined by universal parameters, and protein identity matters.

Theory of Protein Charge Transfer: Electron Transfer between Tryptophan Residue and Active Site of Azurin

One reaction step in the conductivity relay of azurin, electron transfer between the Cu-based active site and the tryptophan residue, is studied theoretically and by classical molecular dynamics simulations. Oxidation of tryptophan results in electrowetting of this residue. This structural change makes the free energy surfaces of electron transfer nonparabolic as described by the Q-model of electron transfer. We analyze the medium dynamical effect on protein electron transfer produced by coupled Stokes-shift dynamics and the dynamics of the donor-acceptor distance modulating electron tunneling. The equilibrium donor-acceptor distance falls in the plateau region of the rate constant, where it is determined by the protein-water dynamics, and the probability of electron tunneling does not affect the rate. The crossover distance found here puts most intraprotein electron-transfer reactions under the umbrella of dynamical control. The crossover between the medium-controlled and tunneling-controlled kinetics is combined with the effect of the protein-water medium on the activation barrier to formulate principles of tunability of protein-based charge-transfer chains. The main principle in optimizing the activation barrier is the departure from the Gaussian-Gibbsian statistics of fluctuations promoting activated transitions. This is achieved either by incomplete (nonergodic) sampling, breaking the link between the Stokes-shift and variance reorganization energies, or through wetting-induced structural changes of the enzyme's active site.

Assessing Thermal Response of Redox Conduction for Anti-Arrhenius Kinetics in a Microbial Cytochrome Nanowire

A micrometers-long helical homopolymer of the outer-membrane cytochrome type S (OmcS) from Geobacter sulfurreducens is proposed to transport electrons to extracellular acceptors in an ancient respiratory strategy of biogeochemical and technological significance. OmcS surprisingly exhibits higher conductivity upon cooling (anti-Arrhenius kinetics), an effect previously attributed to H-bond restructuring and heme redox potential shifts. Herein, the temperature sensitivity of redox conductivity is more thoroughly examined with conventional and constant-redox and -pH molecular dynamics and quantum mechanics/molecular mechanics. A 30 K drop in temperature constituted a weak perturbation to electron transfer energetics, changing electronic couplings (⟨H_mn⟩), reaction free energies (ΔG_mn), reorganization energies (λ_mn), and activation energies (E_a) by at most |0.002|, |0.050|, |0.120|, and |0.045| eV, respectively. Changes in ΔG_mn reflected -0.07 ± 0.03 V shifts in redox potentials that were caused in roughly equal measure by altered electrostatic interactions with the solvent and protein. Changes in intraprotein H-bonding reproduced the earlier observations. Single-particle diffusion and multiparticle steady-state flux models, parametrized with Marcus theory rates, showed that biologically relevant incoherent hopping cannot qualitatively or quantitatively describe electrical conductivity measured by atomic force microscopy in filamentous OmcS. The discrepancy is attributed to differences between solution-phase simulations and solid-state measurements and the need to model intra- and intermolecular vibrations explicitly.

Conformational dynamics modulating electron transfer

Diffusional dynamics of the donor-acceptor distance are responsible for the appearance of a new time scale of diffusion over the distance of electronic tunneling in electron-transfer reactions. The distance dynamics compete with the medium polarization dynamics in the dynamics-controlled electron-transfer kinetics. The pre-exponential factor of the electron-transfer rate constant switches, at the crossover distance, between a distance-independent, dynamics-controlled plateau and exponential distance decay. The crossover between two regimes is controlled by an effective relaxation time slowed down by a factor exponentially depending on the variance of the donor-acceptor displacement. Flexible donor-acceptor complexes must show a greater tendency for dynamics-controlled electron transfer. Energy chains based on electron transport are best designed by placing the redox cofactors near the crossover distance.

The nature of proton-coupled electron transfer in a blue light using flavin domain

Proton-coupled electron transfer (PCET) is key to the activation of the blue light using flavin (BLUF) domain photoreceptors. Here, to elucidate the photocycle of the central FMN-Gln-Tyr motif in the BLUF domain of OaPAC, we eliminated the intrinsic interfering W90 in the mutant design. We integrated the stretched exponential function into the target analysis to account for the dynamic heterogeneity arising from the active-site solvation relaxation and the flexible H-bonding network as shown in the molecular dynamics simulation results, facilitating a simplified expression of the kinetics model. We find that, in both the functional wild-type (WT) and the nonfunctional Q48E and Q48A, forward PCET happens in the range of 105 ps to 344 ps, with a kinetic isotope effect (KIE) measured to be ∼1.8 to 2.4, suggesting that the nature of the forward PCET is concerted. Remarkably, only WT proceeds with an ultrafast reverse PCET process (31 ps, KIE = 4.0), characterized by an inverted kinetics of the intermediate FMNH˙. Our results reveal that the reverse PCET is driven by proton transfer via an intervening imidic Gln.

A 300-fold conductivity increase in microbial cytochrome nanowires due to temperature-induced restructuring of hydrogen bonding networks

Although proteins are considered as nonconductors that transfer electrons only up to 1 to 2 nanometers via tunneling, Geobacter sulfurreducens transports respiratory electrons over micrometers, to insoluble acceptors or syntrophic partner cells, via nanowires composed of polymerized cytochrome OmcS. However, the mechanism enabling this long-range conduction is unclear. Here, we demonstrate that individual nanowires exhibit theoretically predicted hopping conductance, at rate (>10¹⁰ s^-1) comparable to synthetic molecular wires, with negligible carrier loss over micrometers. Unexpectedly, nanowires show a 300-fold increase in their intrinsic conductance upon cooling, which vanishes upon deuteration. Computations show that cooling causes a massive rearrangement of hydrogen bonding networks in nanowires. Cooling makes hemes more planar, as revealed by Raman spectroscopy and simulations, and lowers their reduction potential. We find that the protein surrounding the hemes acts as a temperature-sensitive switch that controls charge transport by sensing environmental perturbations. Rational engineering of heme environments could enable systematic tuning of extracellular respiration.

Reorganization Free Energy of Copper Proteins in Solution, in Vacuum, and on Metal Surfaces

Metalloproteins, known to efficiently transfer electronic charge in biological systems, recently found their utilization in nanobiotechnological devices where the protein is placed into direct contact with metal surfaces. The feasibility of oxidation/reduction of the protein redox sites is affected by the reorganization free energies, one of the key parameters determining the transfer rates. While their values have been measured and computed for proteins in their native environments, i.e., in aqueous solution, the reorganization free energies of dry proteins or proteins adsorbed to metal surfaces remain unknown. Here, we investigate the redox properties of blue copper protein azurin, a prototypical redox-active metalloprotein previously probed by various experimental techniques both in solution and on metal/vacuum interfaces. We used a hybrid QM/MM computational technique based on DFT to explore protein dynamics, flexibility, and corresponding reorganization free energies in aqueous solution, vacuum, and on vacuum gold interfaces. Somewhat surprisingly, the reorganization free energy only slightly decreases when azurin is dried because the loss of the hydration shell leads to larger flexibility of the protein near its redox site. At the vacuum gold surfaces, the energetics of the structure relaxation depends on the adsorption geometry, however, significant reduction of the reorganization free energy was not observed. These findings have important consequences for the charge transport mechanism in vacuum devices, showing that the free energy barriers for protein oxidation remain significant even under ultra-high vacuum conditions.

General principles of secondary active transporter function

Transport of ions and small molecules across the cell membrane against electrochemical gradients is catalyzed by integral membrane proteins that use a source of free energy to drive the energetically uphill flux of the transported substrate. Secondary active transporters couple the spontaneous influx of a "driving" ion such as Na+ or H+ to the flux of the substrate. The thermodynamics of such cyclical non-equilibrium systems are well understood, and recent work has focused on the molecular mechanism of secondary active transport. The fact that these transporters change their conformation between an inward-facing and outward-facing conformation in a cyclical fashion, called the alternating access model, is broadly recognized as the molecular framework in which to describe transporter function. However, only with the advent of high resolution crystal structures and detailed computer simulations, it has become possible to recognize common molecular-level principles between disparate transporter families. Inverted repeat symmetry in secondary active transporters has shed light onto how protein structures can encode a bi-stable two-state system. Based on structural data, three broad classes of alternating access transitions have been described as rocker-switch, rocking-bundle, and elevator mechanisms. More detailed analysis indicates that transporters can be understood as gated pores with at least two coupled gates. These gates are not just a convenient cartoon element to illustrate a putative mechanism but map to distinct parts of the transporter protein. Enumerating all distinct gate states naturally includes occluded states in the alternating access picture and also suggests what kind of protein conformations might be observable. By connecting the possible conformational states and ion/substrate bound states in a kinetic model, a unified picture emerges in which the symporter, antiporter, and uniporter functions are extremes in a continuum of functionality. As usual with biological systems, few principles and rules are absolute and exceptions are discussed as well as how biological complexity may be integrated in quantitative kinetic models that may provide a bridge from the structure to function.

Electronic Transport in Molecular Wires of Precisely Controlled Length Built from Modular Proteins

DNA molecular wires have been studied extensively because of the ease with which molecules of controlled length and composition can be synthesized. The same has not been true for proteins. Here, we have synthesized and studied a series of consensus tetratricopeptide repeat (CTPR) proteins, spanning 4 to 20 nm in length, in increments of 4 nm. For lengths in excess of 6 nm, their conductance exceeds that of the canonical molecular wire, oligo(phenylene-ethylenene), because of the more gradual decay of conductance with length in the protein. We show that, while the conductance decay fits an exponential (characteristic of quantum tunneling) and not a linear increase of resistance with length (characteristic of hopping transport), it is also accounted for by a square-law dependence on length (characteristic of weakly driven hopping). Measurements of the energy dependence of the decay length rule out the quantum tunneling case. A resonance in the carrier injection energy shows that allowed states in the protein align with the Fermi energy of the electrodes. Both the energy of these states and the long-range of hopping suggest that the reorganization induced by hole formation is greatly reduced inside the protein. We outline a model for calculating the molecular-electronic properties of proteins.

Ewald sum corrections in simulations of ion and dipole solvation and electron transfer

Periodic boundary conditions and Ewald sums used in standard simulation protocols require finite-size corrections when the total charge of the simulated system is nonzero. Corrections for ion solvation were introduced by Hummer, Pratt, and García, [J. Chem. Phys. 107, 9275 (1997)]. The latter approach is extended here to derive finite-size correction for the Stokes-shift and reorganization energy applied to electron-transfer reactions. The same correction term, scaling inversely with the box size, adds to the reorganization energy from the energy-gap variance but is subtracted from the reorganization energy calculated from the Stokes shift. Finite-size corrections thus widen the gap between these two quantities, which were recently found to diverge for protein electron transfer. Corrections to the free energy of dipole solvation and the variance of the electric field scale as m²/L³ with the solute dipole m and the box size L.

Exact eigenenergies of a model of vibronically coupled electron transfer reactions

We present the first exact solution to the time-independent Schrödinger equation of a model Hamiltonian consisting of a vibrational mode coupled to three electronic states. This Hamiltonian serves as a generic model for photo-induced electronic transfer reactions. The solution is non-perturbative and can be applied to ET reactions with weak and strong electronic and vibrational coupling strengths. This work suggests a new direction towards understanding the vibronic effects in ET dynamics beyond the non-adiabatic limit and Condon approximation.

Mechanistic Insights on Heme-to-Heme Transmembrane Electron Transfer Within NADPH Oxydases From Atomistic Simulations

NOX5 is a member of the NADPH oxidase family which is dedicated to the production of reactive oxygen species. The molecular mechanisms governing transmembrane electron transfer (ET) that permits to shuttle electrons over the biological membrane have remained elusive for a long time. Using computer simulations, we report conformational dynamics of NOX5 embedded within a realistic membrane environment. We assess the stability of the protein within the membrane and monitor the existence of cavities that could accommodate dioxygen molecules. We investigate the heme-to-heme electron transfer. We find a reaction free energy of a few tenths of eV (ca. −0.3 eV) and a reorganization free energy of around 1.1 eV (0.8 eV after including electrostatic induction corrections). The former indicates thermodynamically favorable ET, while the latter falls in the expected values for transmembrane inter-heme ET. We estimate the electronic coupling to fall in the range of the μeV. We identify electron tunneling pathways showing that not only the W378 residue is playing a central role, but also F348. Finally, we reveal the existence of two connected O₂₋binding pockets near the outer heme with fast exchange between the two sites on the nanosecond timescale. We show that when the terminal heme is reduced, O₂ binds closer to it, affording a more efficient tunneling pathway than when the terminal heme is oxidized, thereby providing an efficient mechanism to catalyze superoxide production in the final step. Overall, our study reveals some key molecular mechanisms permitting reactive oxygen species production by NOX5 and paves the road for further investigation of ET processes in the wide family of NADPH oxidases by computer simulations.

The modified Q-cycle: A look back at its development and forward to a functional model

On looking back at a lifetime of research, it is interesting to see, in the light of current progress, how things came to be, and to speculate on how things might be. I am delighted in the context of the Mitchell prize to have that excuse to present this necessarily personal view of developments in areas of my interests. I have focused on the Q-cycle and a few examples showing wider ramifications, since that had been the main interest of the lab in the 20 years since structures became available, - a watershed event in determining our molecular perspective. I have reviewed the evidence for our model for the mechanism of the first electron transfer of the bifurcated reaction at the Q_o-site, which I think is compelling. In reviewing progress in understanding the second electron transfer, I have revisited some controversies to justify important conclusions which appear, from the literature, not to have been taken seriously. I hope this does not come over as nitpicking. The conclusions are important to the final section in which I develop an internally consistent mechanism for turnovers of the complex leading to a state similar to that observed in recent rapid-mix/freeze-quench experiments, reported three years ago. The final model is necessarily speculative but is open to test.

Why are Vibrational Lines Narrow in Proteins?

The vibrational Stark effect in proteins yields line shifts indicative of strong internal electric fields up to a few volts per angstrom. These values are supported by numerical simulations of proteins. The simulations also show a significant breadth of field fluctuations translating to inhomogeneous broadening of vibrational lines. According to fluctuation-dissipation arguments, strong internal fields should lead to broad lines. Experimentally reported vibrational lines in proteins are, however, very narrow. This disconnect is explained here in terms of the insufficient (nonergodic) sampling of the protein's configurations on the lifetime of the vibrational probe. The slow component of the electric field fluctuations in proteins relaxes on the time scale of tens of nanoseconds and is dynamically frozen on the vibrational lifetime.

Ergodicity Breaking in Thermal Biological Electron Transfer? Cytochrome C Revisited II

It was recently suggested that cytochrome c operates in an ergodicity-breaking regime characterized by unusually large energy gap thermal fluctuations and associated reorganization free energies for heme oxidation of up to 3.0 eV. The large fluctuations were reported to lower activation free energy for oxidation of the heme cofactor by almost a factor of 2 compared to the case where ergodicity is maintained. Our group has recently investigated this claim computationally at several levels of theory and found no evidence for such large energy gap fluctuations. Here we address the points of our earlier work that have raised criticism and we also extend our previous investigation by considering a simple linear polarizability model for cytochrome c oxidation. We find very consistent results among all our computational approaches, ranging from classical molecular dynamics, to the linear polarizability model to QM(PMM)/MM to full QM(DFT)/MM electrostatic emdedding. None of them support the notion of very large energy gap fluctuations or ergodicity breaking. The deviation between our simulations and the ones reported in [ J. Phys. Chem. B 2017, 121, 4958] is traced back to rather large electric fields at the Fe site of the heme c cofactor in that study, not seen in our simulations, neither with the AMBER nor with the CHARMM force field. While ergodicity breaking effects may well occur in other biological ET, our numerical evidence suggests that this is not the case for cytochrome c.

Nonequilibrium dynamics of photoinduced forward and backward electron transfer reactions

The biological functions of photoenzymes are often triggered by photoinduced electron transfer (ET) reactions. An ultrafast backward ET (BET) reaction follows the initial photoinduced forward ET (FET), which dissipates the energy of absorbed photons and terminates the biological function in vain. Based upon our previous works, we reasoned that the dynamics of the BET is coupled with that of the FET and other local motions. In this work, the dynamics of the FET and BET is modeled as the master equation of the reduced density operator of a three-state system coupled with a classical harmonic reservoir. The coupling of the FET and BET is reflected in the time-evolution of the charge-transfer state's population, which is generated by a source, the reaction flux for the FET, and annihilated by a sink, the reaction flux for the BET. Surprisingly, numerical simulations show that when the BET is in the Marcus normal region, the BET can be accelerated by nonequilibrium local motions and becomes faster than what is predicted by the Marcus theory. The experimental confirmation of this novel dynamics would provide qualitative evidence for nonequilibrium effects on ultrafast ET dynamics. Additionally, the effects of quantum vibrational modes on the dynamics are discussed. This work can help understand the dynamical interactions between the chain of ultrafast reactions and the complex local environmental motions, revealing the physical nature underlying biological functions.

The Use of Pulsed Electromagnetic Field to Modulate Inflammation and Improve Tissue Regeneration: A Review

Pulsed electromagnetic field (PEMF) is emerging as innovative treatment for regulation of inflammation, which could have significant effects on tissue regeneration. PEMF modulates inflammatory processes through the regulation of pro- and anti-inflammatory cytokine secretion during different stages of inflammatory response. Consistent outcomes in studies involving animal and human tissue have shown promise for the use of PEMF as an alternative or complementary treatment to pharmaceutical therapies. Thus, PEMF treatment could provide a novel nonpharmaceutical means of modulating inflammation in injured tissues resulting in enhanced functional recovery. This review examines the effect of PEMF on immunomodulatory cells (e.g., mesenchymal stem/stromal cells [MSCs] and macrophages [MΦ]) to better understand the potential for PEMF therapy to modulate inflammatory signaling pathways and improve tissue regeneration. This review cites published data that support the use of PEMF to improve tissue regeneration. Our studies included herein confirm anti-inflammatory effects of PEMF on MSCs and MΦ.

2'-Deoxy-2'-fluoro-arabinonucleic acid: a valid alternative to DNA for biotechnological applications using charge transport

The non-biological 2'-deoxy-2'-fluoro-arabinonucleic acid (2'F-ANA) may be used as a valid alternative to DNA in biomedical and electronic applications because of its higher resistance to hydrolysis and nuclease degradation. However, the advantage of using 2'F-ANA in such applications also depends on its charge-transfer properties compared to DNA. In this study, we compare the charge conduction properties of model 2'F-ANA and DNA double-strands, using structural snapshots from MD simulations to calculate the electronic couplings and reorganization energies associated with the hole transfer steps between adjacent nucleobase pairs. Inserting these charge-transfer parameters into a kinetic model for charge conduction, we find similar conductive properties for DNA and 2'F-ANA. Moreover, we find that 2'F-ANA's enhanced chemical stability does not correspond to a reduction in the nucleobase π-stack structural flexibility relevant to both electronic couplings and reorganization free energies. Our results promote the use of 2'F-ANA in applications that can be based on charge transport, such as biosensing and chip technology, where its chemical stability and conductivity can advantageously combine.

N-Doped Graphene Oxide Decorated with PtCo Nanoparticles for Immobilization of Double-Stranded Deoxyribonucleic Acid and Investigation of Clenbuterol-Induced DNA Damage

We demonstrate here a facile hydrothermal-assisted formation of PtCo alloy nanoparticles (NPs) and their simultaneous anchoring on the graphitic surface of N-doped graphene oxide (NGO). Doping induced nanopores in the carbon surface to facilitate the uniform and homogeneous anchoring of alloy nanoparticles. It was revealed that the formation of PtCo NPs on an NGO interface plodded excellent tendency toward double-stranded deoxyribonucleic acid (dsDNA). The dsDNA immobilization was enabled by the presence of several oxidation states of Pt and Co. The same property was further used to monitor the direct detection of dsDNA damage induced by clenbuterol via screen-printed carbon electrodes. Cyclic voltammetric and electrochemical impedance spectroscopic characterization traced well the dsDNA attachment on the modified electrode surface. Differential pulsed voltammetry was further used as a tool to monitor the characteristic guanine peak before and after incubating with clenbuterol used as a damage probe for the dsDNA. The findings can further be appurtenant in exploring dsDNA immobilization protocols and developing analytical methods for determination of various dsDNA damaging agents.

Ergodicity-Breaking in Thermal Biological Electron Transfer? Cytochrome C Revisited

It was recently suggested that certain redox proteins operate in an ergodicity-breaking regime to facilitate biological electron transfer (ET). A signature for this is a large variance reorganization free energy (several electronvolts) but a significantly smaller Stokes reorganization free energy due to incomplete protein relaxation on the time scale of the ET event. Here we investigate whether this picture holds for oxidation of cytochrome c in aqueous solution, at various levels of theory including classical molecular dynamics with two additive and one electronically polarizable force field, and QM/MM calculations with the QM region treated by full electrostatic DFT embedding and by the perturbed matrix method. Sampling the protein and energy gap dynamics over more than 250 ns, we find no evidence for ergodicity-breaking effects. In particular, the inclusion of electronic polarizability of the heme group at QM/MM levels did not induce nonergodic effects, contrary to previous reports by Matyushov et al. The well-known problem of overestimation of reorganization free energies with additive force fields is cured when the protein and solvent are treated as electronically polarizable. Ergodicity-breaking effects may occur in other redox proteins, and our results suggest that long simulations, ideally on the ET time scale, with electronically polarizable force fields are required to obtain strong numerical evidence for them.

Fractional electron transfer kinetics and a quantum breaking of ergodicity

The dissipative curve-crossing problem provides a paradigm for electron-transfer (ET) processes in condensed media. It establishes the simplest conceptual test bed to study the influence of the medium's dynamics on ET kinetics both on the ensemble level, and on the level of single particles. Single electron description is particularly important for nanoscaled systems like proteins, or molecular wires. Especially insightful is this framework in the semiclassical limit, where the environment can be treated classically, and an exact analytical treatment becomes feasible. Slow medium's dynamics is capable of enslaving ET and bringing it on the ensemble level from a quantum regime of nonadiabatic tunneling to the classical adiabatic regime, where electrons follow the nuclei rearrangements. This classical adiabatic textbook picture contradicts, however, in a very spectacular fashion to the statistics of single electron transitions, even in the Debye, memoryless media, also named Ohmic in the parlance of the famed spin-boson model. On the single particle level, ET always remains quantum, and this was named a quantum breaking of ergodicity in the adiabatic ET regime. What happens in the case of subdiffusive, fractional, or sub-Ohmic medium's dynamics, which is featured by power-law decaying dynamical memory effects typical, e.g., for protein macromolecules, and other viscoelastic media? Such a memory is vividly manifested by anomalous Cole-Cole dielectric response in such media. We address this question based both on accurate numerics and analytical theory. The ensemble theory remarkably agrees with the numerical dynamics of electronic populations, revealing a power-law relaxation tail even in a profoundly nonadiabatic electron transfer regime. In other words, ET in such media should typically display fractional kinetics. However, a profound difference with the numerically accurate results occurs for the distribution of residence times in the electronic states, both on the ensemble level and the level of single trajectories. Ergodicity is broken dynamically even in a more spectacular way than in the memoryless case. Our results question the applicability of all the existing and widely accepted ensemble theories of electron transfer in fractional, sub-Ohmic environments, on the level of single molecules, and provide a real challenge to face, both for theorists and experimentalists.

Energy Medicine: Current Status and Future Perspectives

Current practices in allopathic medicine measure different types of energy in the human body by using quantum field dynamics involved in nuclear medicine, radiology, and imaging diagnostics. Once diagnosed, current treatments revert to biochemistry instead of using biophysics therapies to treat the disturbances in subtle energies detected and used for diagnostics. Quantum physics teaches us there is no difference between energy and matter. All systems in the human being, from the atomic to the molecular level, are constantly in motion-creating resonance. This resonance is important to understanding how subtle energy directs and maintains health and wellness in the human being. Energy medicine (EM), whether human touch or device-based, is the use of known subtle energy fields to therapeutically assess and treat energetic imbalances, bringing the body's systems back to homeostasis (balance). The future of EM depends on the ability of allopathic medicine to merge physics with biochemistry. Biophoton emissions as well as signal transduction and cell signaling communication systems are widely accepted in today's medicine. This technology needs to be expanded to include the existence of the human biofield (or human energy field) to better understand that disturbances in the coherence of energy patterns are indications of disease and aging. Future perspectives include understanding cellular voltage potentials and how they relate to health and wellness, understanding the overlap between the endocrine and chakra systems, and understanding how EM therapeutically enhances psychoneuroimmunology (mind-body) medicine.

Thermodynamics of Reactions Affected by Medium Reorganization

We present a thermodynamic analysis of the activation barrier for reactions which can be monitored through the difference in the energies of reactants and products defined as the reaction coordinate (electron and atom transfer, enzyme catalysis, etc.). The free-energy surfaces along the reaction coordinate are separated into the enthalpy and entropy surfaces. For the Gaussian statistics of the reaction coordinate, the free-energy surfaces are parabolas, and the entropy surface is an inverted parabola. Its maximum coincides with the transition state for reactions with zero value of the reaction free energy. Maximum entropic depression of the activation barrier, anticipated by the concept of transition-state ensembles, can be achieved for such reactions. From Onsager's reversibility, the entropy of equilibrium fluctuations encodes the entropic component of the activation barrier. The reorganization entropy thus becomes the critical parameter of the theory reducing the problem of activation entropy to the problem of reorganization entropy. Standard solvation theories do not allow reorganization entropy sufficient for the barrier depression. Complex media, characterized by many relaxation processes, need to be involved. Proteins provide several routes for achieving large entropic effects through incomplete (nonergodic) sampling of the complex energy landscape and by facilitating an active role of water in the reaction mechanism.

Wetting of the Protein Active Site Leads to Non-Marcusian Reaction Kinetics

Enzymes exist in continuously fluctuating water bath dramatically affecting their function. Water not only forms the solvation shell but also penetrates into the protein interior. Changing the wetting pattern of the protein's active site in response to altering redox state initiates a highly nonlinear structural change and non-Gaussian electrostatic fluctuations at the active site. The free-energy surfaces of electron transfer are highly nonparabolic (non-Marcusian), as shown by atomistic molecular dynamics simulations of hydrated ferredoxin protein and by an analytical model in agreement with simulations. The reorganization energy of electron transfer passes through a spike marking equal probabilities of the wet and dry states of the active site. The activation thermodynamics affected by wetting leads to a non-Arrhenius, passing through a maximum, plot for the reaction rate vs the inverse temperature.

Direct Measurement of Charge Regulation in Metalloprotein Electron Transfer

Determining whether a protein regulates its net electrostatic charge during electron transfer (ET) will deepen our mechanistic understanding of how polypeptides tune rates and free energies of ET (e.g., by affecting reorganization energy, and/or redox potential). Charge regulation during ET has never been measured for proteins because few tools exist to measure the net charge of a folded protein in solution at different oxidation states. Herein, we used a niche analytical tool (protein charge ladders analyzed with capillary electrophoresis) to determine that the net charges of myoglobin, cytochrome c, and azurin change by 0.62±0.06, 1.19±0.02, and 0.51±0.04 units upon single ET. Computational analysis predicts that these fluctuations in charge arise from changes in the pKa  values of multiple non-coordinating residues (predominantly histidine) that involve between 0.42-0.90 eV. These results suggest that ionizable residues can tune the reactivity of redox centers by regulating the net charge of the entire protein-cofactor-solvent complex.

Termination of Biological Function at Low Temperatures: Glass or Structural Transition?

Energy of life is produced by electron transfer in energy chains of respiration or photosynthesis. A small input of free energy available to biology puts significant restrictions on how much free energy can be lost in each electron-transfer reaction. We advocate the view that breaking ergodicity, leading to violation of the fluctuation-dissipation theorem (FDT), is how proteins achieve high reaction rates without sacrificing the reaction free energy. Here we show that a significant level of nonergodicity, represented by a large extent of the configurational temperature over the kinetic temperature, is maintained in the entire physiological range for the cytochrome c electron transfer protein. The protein returns to the state consistent with the FDT below the crossover temperature close to the temperature of the protein glass transition. This crossover leads to a sharp increase in the activation barrier of electron transfer and is displayed by a kink in the Arrhenius plot for the reaction rate constant.

Electron-transfer chain in respiratory complex I

Complex I is a part of the respiration energy chain converting the redox energy into the cross-membrane proton gradient. The electron-transfer chain of iron-sulfur cofactors within the water-soluble peripheral part of the complex is responsible for the delivery of electrons to the proton pumping subunit. The protein is porous to water penetration and the hydration level of the cofactors changes when the electron is transferred along the chain. High reaction barriers and trapping of the electrons at the iron-sulfur cofactors are prevented by the combination of intense electrostatic noise produced by the protein-water interface with the high density of quantum states in the iron-sulfur clusters caused by spin interactions between paramagnetic iron atoms. The combination of these factors substantially lowers the activation barrier for electron transfer compared to the prediction of the Marcus theory, bringing the rate to the experimentally established range. The unique role of iron-sulfur clusters as electron-transfer cofactors is in merging protein-water fluctuations with quantum-state multiplicity to allow low activation barriers and robust operation. Water plays a vital role in electron transport energetics by electrowetting the cofactors in the chain upon arrival of the electron. A general property of a protein is to violate the fluctuation-dissipation relation through nonergodic sampling of its landscape. High functional efficiency of redox enzymes is a direct consequence of nonergodicity.

Theory and Electrochemistry of Cytochrome c

Extensive simulations of cytochrome c in solution are performed to address the apparent contradiction between large reorganization energies of protein electron transfer typically reported by atomistic simulations and much smaller values produced by protein electrochemistry. The two sets of data are reconciled by deriving the activation barrier for electrochemical reaction in terms of an effective reorganization energy composed of half the Stokes shift (characterizing the medium polarization in response to electron transfer) and the variance reorganization energy (characterizing the breadth of electrostatic fluctuations). This effective reorganization energy is much smaller than each of the two components contributing to it and is fully consistent with electrochemical measurements. Calculations in the range of temperatures between 280 and 360 K combine long, classical molecular dynamics simulations with quantum calculations of the protein active site. The results agree with the Arrhenius plots for the reaction rates and with cyclic voltammetry of cytochrome c immobilized on self-assembled monolayers. Small effective reorganization energy, and the resulting small activation barrier, is a general phenomenology of protein electron transfer allowing fast electron transport within biological energy chains.

Dynamical and orientational structural crossovers in low-temperature glycerol

Mean-square displacements of hydrogen atoms in glass-forming materials and proteins, as reported by incoherent elastic neutron scattering, show kinks in their temperature dependence. This crossover, known as the dynamical transition, connects two approximately linear regimes. It is often assigned to the dynamical freezing of subsets of molecular modes at the point of equality between their corresponding relaxation times and the instrumental observation window. The origin of the dynamical transition in glass-forming glycerol is studied here by extensive molecular dynamics simulations. We find the dynamical transition to occur for both the center-of-mass translations and the molecular rotations at the same temperature, insensitive to changes of the observation window. Both the translational and rotational dynamics of glycerol show a dynamic crossover from the structural to a secondary relaxation at the temperature of the dynamical transition. A significant and discontinuous increase in the orientational Kirkwood factor and in the dielectric constant is observed in the same range of temperatures. No indication is found of a true thermodynamic transition to an ordered low-temperature phase. We therefore suggest that all observed crossovers are dynamic in character. The increase in the dielectric constant is related to the dynamic freezing of dipolar domains on the time scale of simulations.

Polarizability of the active site of cytochrome c reduces the activation barrier for electron transfer

Enzymes in biology’s energy chains operate with low energy input distributed through multiple electron transfer steps between protein active sites. The general challenge of biological design is how to lower the activation barrier without sacrificing a large negative reaction free energy. We show that this goal is achieved through a large polarizability of the active site. It is polarized by allowing a large number of excited states, which are populated quantum mechanically by electrostatic fluctuations of the protein and hydration water shells. This perspective is achieved by extensive mixed quantum mechanical/molecular dynamics simulations of the half reaction of reduction of cytochrome c. The barrier for electron transfer is consistently lowered by increasing the number of excited states included in the Hamiltonian of the active site diagonalized along the classical trajectory. We suggest that molecular polarizability, in addition to much studied electrostatics of permanent charges, is a key parameter to consider in order to understand how enzymes work.

Ultrafast Electron Transfer Kinetics in the LM Dimer of Bacterial Photosynthetic Reaction Center from Rhodobacter sphaeroides

It has become increasingly clear that dynamics plays a major role in the function of many protein systems. One system that has proven particularly facile for studying the effects of dynamics on protein-mediated chemistry is the bacterial photosynthetic reaction center from Rhodobacter sphaeroides. Previous experimental and computational analysis have suggested that the dynamics of the protein matrix surrounding the primary quinone acceptor, QA, may be particularly important in electron transfer involving this cofactor. One can substantially increase the flexibility of this region by removing one of the reaction center subunits, the H-subunit. Even with this large change in structure, photoinduced electron transfer to the quinone still takes place. To evaluate the effect of H-subunit removal on electron transfer to QA, we have compared the kinetics of electron transfer and associated spectral evolution for the LM dimer with that of the intact reaction center complex on picosecond to millisecond time scales. The transient absorption spectra associated with all measured electron transfer reactions are similar, with the exception of a broadening in the QX transition and a blue-shift in the QY transition bands of the special pair of bacteriochlorophylls (P) in the LM dimer. The kinetics of the electron transfer reactions not involving quinones are unaffected. There is, however, a 4-fold decrease in the electron transfer rate from the reduced bacteriopheophytin to QA in the LM dimer compared to the intact reaction center and a similar decrease in the recombination rate of the resulting charge-separated state (P(+)QA(-)). These results are consistent with the concept that the removal of the H-subunit results in increased flexibility in the region around the quinone and an associated shift in the reorganization energy associated with charge separation and recombination.