Effect of protein dynamics on biological electron transfer

Competing quantum effects in heavy-atom tunnelling through conical intersections

Thermally activated chemical reactions are typically understood in terms of overcoming potential-energy barriers. However, standard rate theories break down in the presence of a conical intersection (CI) because these processes are inherently nonadiabatic, invalidating the Born-Oppenheimer approximation. Moreover, CIs give rise to intricate nuclear quantum effects such as tunnelling and the geometric phase, which are neglected by standard trajectory-based simulations and remain largely unexplored in complex molecular systems. We present new semiclassical transition-state theories based on an extension of golden-rule instanton theory to describe nonadiabatic tunnelling through CIs and thus provide an intuitive picture for the reaction mechanism. We apply the method in conjunction with first-principles electronic-structure calculations to the electron transfer in the bis(methylene)-adamantyl cation. Our study reveals a strong competition between heavy-atom tunnelling and geometric-phase effects.

Electron transfer pathways from quantum dynamics simulations

This work explores the possibility of simulating an electron transfer process between a donor and an acceptor in real time using time-dependent density functional theory electron dynamics. To achieve this objective, a central issue to resolve is the definition of the initial state. This must be a non-equilibrium electronic state able to trigger the charge transfer dynamics; here, two schemes are proposed to prepare such states. One is based on the combination of the density matrices of the donor and acceptor converged separately with appropriate charges (for example, -1 for the donor and +1 for the acceptor). The second approach relied on constrained DFT to localize the charge on each fragment. With these schemes, electron transfer processes are simulated in different model systems of increasing complexity: an atomic hydrogen dimer, a polyacetylene chain, and the active site of the T. cruzi hybrid type A heme peroxidase, for which two possible electron transfer paths have been postulated. For the latter system, the present methodology applied in a hybrid Quantum Mechanics - Molecular Mechanics framework allows us to establish the relative probabilities of each path and provides insight into the inhibition of the electron transfer provoked by the substitution of tryptophan by phenylalanine in the W233F mutant.

Ring polymer quantization of the photon field in polariton chemistry

We use the ring polymer (RP) representation to quantize the radiation field inside an optical cavity to investigate polariton quantum dynamics. Using a charge transfer model coupled to an optical cavity, we demonstrate that the RP quantization of the photon field provides accurate rate constants of the polariton mediated electron transfer reaction compared to Fermi's golden rule. Because RP quantization uses extended phase space to describe the photon field, it significantly reduces the computational costs compared to the commonly used Fock state description of the radiation field. Compared to the other quasi-classical descriptions of the photon field, such as the classical Wigner based mean-field Ehrenfest model, the RP representation provides a much more accurate description of the polaritonic quantum dynamics because it alleviates the potential quantum distribution leakage problem associated with the photonic degrees of freedom (DOF). This work demonstrates the possibility of using the ring polymer description to treat the quantized radiation field in polariton chemistry, offering an accurate and efficient approach for future investigations in cavity quantum electrodynamics.

Kinetic Isotope Effects on Electron Transfer Across Self-Assembled Monolayers on Gold

Reactions requiring controlled delivery of protons and electrons are important in storage of energy in small molecules. While control over proton transfer can be achieved by installing appropriate chemical functionality in the catalyst, control of electron-transfer (ET) rates can be achieved by utilizing self-assembled monolayers (SAMs) on electrodes. Thus, a deeper understanding of the ET through SAM to an immobilized or covalently attached redox-active species is desirable. Long-range ET across several SAM-covered Au electrodes to covalently attached ferrocene is investigated using protonated and deuterated thiols (R-SH/R-SD). The rate of tunneling is measured using both chronoamperometry and cyclic voltammetry, and it shows a prominent kinetic isotope effect (KIE). The KIE is ∼2 (normal) for medium-chain-length thiols but ∼0.47 (inverse) for long-chain thiols. These results imply substantial contribution from the classical modes at the Au-(H)SR interface, which shifts substantially upon deuteration of the thiols, to the ET process. The underlying H/D KIE of these exchangeable thiol protons should be considered when analyzing solvent isotope effects in catalysis utilizing SAM.

Tunneling matrix element and tunneling pathways of protein electron transfer calculated with a fragment molecular orbital method

Practical ways to calculate the tunneling matrix elements and analyze the tunneling pathways for protein electron-transfer (ET) reactions with a fragment molecular orbital (FMO) method are presented. The straightforward use of minimal basis sets only for the atoms involved in the covalent bond detachment in FMO can properly describe the ETs through the protein main-chains with the cost-effective two-body corrections (FMO2) without losing the quality of double-zeta basis sets. The current FMO codes have been interfaced with density functional theory, polarizable continuum model, and model core potentials, with which the FMO-based protein ET calculations can consider the effects of electron correlation, solvation, and transition-metal redox centers. The reasonable performance of the FMO-based ET calculations is demonstrated for three different sets of protein-ET model molecules: (1) hole transfer between two tryptophans covalently bridged by a polyalanine linker in the ideal α-helix and β-strand conformations, (2) ET between two plastoquinones covalently bridged by a polyalanine linker in the ideal α-helix and β-strand conformations, and (3) hole transfer between ruthenium (Ru) and copper (Cu) complexes covalently bridged by a stretch of a polyglycine linker as a model for Ru-modified derivatives of azurin.

Polariton-Mediated Electron Transfer via Cavity Quantum Electrodynamics

We investigate the polariton-mediated electron transfer reaction in a model system with analytic rate constant theory and direct quantum dynamical simulations. We demonstrate that the photoinduced charge transfer reaction between a bright donor state and dark acceptor state can be significantly enhanced or suppressed by coupling the molecular system to the quantized radiation field inside an optical cavity. This is because the quantum light-matter interaction can influence the effective driving force and electronic couplings between the donor state, which is the hybrid light-matter excitation, and the molecular acceptor state. Under the resonance condition between the photonic and electronic excitations, the effective driving force can be tuned by changing the light-matter coupling strength; for an off-resonant condition, the same effect can be accomplished by changing the molecule-cavity detuning. We further demonstrate that using both the electronic coupling and light-matter coupling helps to extend the effective couplings across the entire system, even for the dark state that carries a zero transition dipole. Theoretically, we find that both the counter-rotating terms and the dipole self-energy in the quantum electrodynamics Hamiltonian are important for obtaining an accurate polariton eigenspectrum as well as the polariton-mediated charge transfer rate constant, especially in the ultrastrong coupling regime.

Formulation of Long-Range Transport Rates through Molecular Bridges: From Unfurling to Hopping

Weak fluctuations about the rigid equilibrium structure of ordered molecular bridges drive charge transfer in donor-bridge-acceptor systems via quantum unfurling, which differs from both hopping and ballistic transfer, yet static disorder (low frequency motions) in the bridge is shown to induce a change of mechanism from unfurling to hopping when local fluctuations along the molecular bridge are uncorrelated. Remarkably, these two different transport mechanisms manifest in similar charge-transfer rates, which are nearly independent of the molecular bridge length. We propose an experimental test for distinguishing unfurling from hopping in DNA models with different helix directionality. A unified formulation explains the apparent similarity in the length dependence of the transfer rate despite the difference in the underlying transport mechanisms.

Fluctuating hydrogen-bond networks govern anomalous electron transfer kinetics in a blue copper protein

We combine experimental and computational methods to address the anomalous kinetics of long-range electron transfer (ET) in mutants of Pseudomonas aeruginosa azurin. ET rates and driving forces for wild type (WT) and three N47X mutants (X = L, S, and D) of Ru(2,2'-bipyridine)₂ (imidazole)(His83) azurin are reported. An enhanced ET rate for the N47L mutant suggests either an increase of the donor-acceptor (DA) electronic coupling or a decrease in the reorganization energy for the reaction. The underlying atomistic features are investigated using a recently developed nonadiabatic molecular dynamics method to simulate ET in each of the azurin mutants, revealing unexpected aspects of DA electronic coupling. In particular, WT azurin and all studied mutants exhibit more DA compression during ET (>2 Å) than previously recognized. Moreover, it is found that DA compression involves an extended network of hydrogen bonds, the fluctuations of which gate the ET reaction, such that DA compression is facilitated by transiently rupturing hydrogen bonds. It is found that the N47L mutant intrinsically disrupts this hydrogen-bond network, enabling particularly facile DA compression. This work, which reveals the surprisingly fluctional nature of ET in azurin, suggests that hydrogen-bond networks can modulate the efficiency of long-range biological ET.

Tunneling Time and the Breakdown of Born-Oppenheimer Approximation

In electron-transfer reactions in proteins and other molecular systems involving long-distance electron tunneling, the tunneling time, i.e., the time that an electron spends in the barrier region between redox centers, can be comparable to vibrational periods of the nuclei. One consequence of this is the breakdown of the Born-Oppenheimer (BO) approximation at the far tails of the tunneling electronic wave functions. These tails define the coupling of redox centers exchanging electrons and hence the rates of electron transfer. We discuss the transition in the distance dependence of the rate of electron transfer that separates the BO and non-BO behavior of the tunneling reaction and show how the crossover is related to tunneling time.

Direct Electron Transfer of Enzymes in a Biologically Assembled Conductive Nanomesh Enzyme Platform

Nondestructive assembly of a nanostructured enzyme platform is developed in combination of the specific biomolecular attraction and electrostatic coupling for highly efficient direct electron transfer (DET) of enzymes with unprecedented applicability and versatility. The biologically assembled conductive nanomesh enzyme platform enables DET-based flexible integrated biosensors and DET of eight different enzyme with various catalytic activities.

Gold nanoparticle-mediated electron transfer of cytochrome c on a self-assembled surface

Gold nanoparticles provide short-range electron transfer between cytochrome c and gold electrode.

Insights into Solid-State Electron Transport through Proteins from Inelastic Tunneling Spectroscopy: The Case of Azurin

Surprisingly efficient solid-state electron transport has recently been demonstrated through "dry" proteins (with only structural, tightly bound H2O left), suggesting proteins as promising candidates for molecular (bio)electronics. Using inelastic electron tunneling spectroscopy (IETS), we explored electron-phonon interaction in metal/protein/metal junctions, to help understand solid-state electronic transport across the redox protein azurin. To that end an oriented azurin monolayer on Au is contacted by soft Au electrodes. Characteristic vibrational modes of amide and amino acid side groups as well as of the azurin-electrode contact were observed, revealing the azurin native conformation in the junction and the critical role of side groups in the charge transport. The lack of abrupt changes in the conductance and the line shape of IETS point to far off-resonance tunneling as the dominant transport mechanism across azurin, in line with previously reported (and herein confirmed) azurin junctions. The inelastic current and hence electron-phonon interaction appear to be rather weak and comparable in magnitude with the inelastic fraction of tunneling current via alkyl chains, which may reflect the known structural rigidity of azurin.

Progress and challenges in simulating and understanding electron transfer in proteins

This Review presents an overview of the most common numerical simulation approaches for the investigation of electron transfer (ET) in proteins. We try to highlight the merits of the different approaches but also the current limitations and challenges. The article is organized into three sections. Section 2 deals with direct simulation algorithms of charge migration in proteins. Section 3 summarizes the methods for testing the applicability of the Marcus theory for ET in proteins and for evaluating key thermodynamic quantities entering the reaction rates (reorganization energies and driving force). Recent studies interrogating the validity of the theory due to the presence of non-ergodic effects or of non-linear responses are also described. Section 4 focuses on the tunneling aspects of electron transfer. How can the electronic coupling between charge transfer states be evaluated by quantum chemistry approaches and rationalized? What interesting physics regarding the impact of protein dynamics on tunneling can be addressed? We will illustrate the different sections with examples taken from the literature to show what types of system are currently manageable with current methodologies. We also take care to recall what has been learned on the biophysics of ET within proteins thanks to the advent of atomistic simulations.

Electron transfer, decoherence, and protein dynamics: insights from atomistic simulations

Electron transfer in biological systems drives the processes of life. From cellular respiration to photosynthesis and enzymatic catalysis, electron transfers (ET) are chemical processes on which essential biological functions rely. Over the last 40 years, scientists have sought understanding of how these essential processes function in biology. One important breakthrough was the discovery that Marcus theory (MT) of electron transfer is applicable to biological systems. Chemists have experimentally collected both the reorganization energies (λ) and the driving forces (ΔG°), two parameters of Marcus theory, for a large variety of ET processes in proteins. At the same time, theoretical chemists have developed computational approaches that rely on molecular dynamics and quantum chemistry calculations to access numerical estimates of λ and ΔG°. Yet another crucial piece in determining the rate of an electron transfer is the electronic coupling between the initial and final electronic wave functions. This is an important prefactor in the nonadiabatic rate expression, since it reflects the probability that an electron tunnels from the electron donor to the acceptor through the intervening medium. The fact that a protein matrix supports electron tunneling much more efficiently than vacuum is now well documented, both experimentally and theoretically. Meanwhile, many chemists have provided examples of the rich physical chemistry that can be induced by protein dynamics. This Account describes our studies of the dynamical effects on electron tunneling. We present our analysis of two examples of natural biological systems through MD simulations and tunneling pathway analyses. Through these examples, we show that protein dynamics sustain efficient tunneling. Second, we introduce two time scales: τcoh and τFC. The former characterizes how fast the electronic coupling varies with nuclear vibrations (which cause dephasing). The latter reflects the time taken by the system to leave the crossing region. In the framework of open quantum systems, τFC is a short time approximation of the characteristic decoherence time of the electronic subsystem in interaction with its nuclear environment. The comparison of the respective values of τcoh and τFC allows us to probe the occurrence of non-Condon effects. We use ab initio MD simulations to analyze how decoherence appears in several biological cofactors. We conclude that we cannot account for its order of magnitude by considering only the atoms or bonds directly concerned with the transfer. Decoherence results from contributions from all atoms of the system appearing with a time delay that increases with the distance from the primarily concerned atoms or bonds. The delay and magnitude of the contributions depend on the chemical nature of the system. Finally, we present recent developments based on constrained DFT for efficient and accurate evaluations of the electronic coupling in ab initio MD simulations. These are promising methods to study the subtle fluctuations of the electronic coupling and the mechanisms of electronic decoherence in biological systems.

Femtosecond dynamics of short-range protein electron transfer in flavodoxin

Intraprotein electron transfer (ET) in flavoproteins is important for understanding the correlation of their redox, configuration, and reactivity at the active site. Here, we used oxidized flavodoxin as a model system and report our complete characterization of a photoinduced redox cycle from the initial charge separation in 135-340 fs to subsequent charge recombination in 0.95-1.6 ps and to the final cooling relaxation of the product(s) in 2.5-4.3 ps. With 11 mutations at the active site, we observed that these ultrafast ET dynamics, much faster than active-site relaxation, mainly depend on the reduction potentials of the electron donors with minor changes caused by mutations, reflecting a highly localized ET reaction between the stacked donor and acceptor at a van der Waals distance and leading to a gas-phase type of bimolecular ET reaction confined in the active-site nanospace. Significantly, these ultrafast ET reactions ensure our direct observation of vibrationally excited reaction product(s), suggesting that the back ET barrier is effectively reduced because of the decrease in the total free energy in the Marcus inverted region, leading to the accelerated charge recombination. Such vibrationally coupled charge recombination should be a general feature of flavoproteins with similar configurations and interactions between the cofactor flavin and neighboring aromatic residues.

Redox state dependence of axial ligand dynamics in Nitrosomonas europaea cytochrome c552

Analysis of NMR spectra reveals that the heme axial Met ligand orientation and dynamics in Nitrosomonas europaea cytochrome c552 (Ne cyt c) are dependent on the heme redox state. In the oxidized state, the heme axial Met is fluxional, interconverting between two conformers related to each other by inversion through the Met δS atom. In the reduced state, there is no evidence of fluxionality, with the Met occupying one conformation similar to that seen in the homologous Pseudomonas aeruginosa cytochrome c551. Comparison of the observed and calculated pseudocontact shifts for oxidized Ne cyt c using the reduced protein structure as a reference structure reveals a redox-dependent change in the structure of the loop bearing the axial Met (loop 3). Analysis of nuclear Overhauser effects (NOEs) and existing structural data provides further support for the redox state dependence of the loop 3 structure. Implications for electron transfer function are discussed.

Unique mechanisms of excitation energy transfer, electron transfer and photoisomerization in biological systems

We discuss unique mechanisms typical in the elementary processes ofbiological functions. We focus on three topics. Excitation energytransfer in the light-harvesting antenna systems of photosyntheticbacteria is unique in its structure and the energy transfer mechanism. Inthe case of LH2 of Rhodopseudomonas acidophila, the B850 intra-ringenergy transfer and the inter-ring energy transfer between B800 and B850take place by the intermediate coupling mechanism of energy transfer. Theexcitonic coherent domain shows a wave-like movement along the ring, andthis property is expected to play a significant role in the inter-ringenergy transfer between LH2's. The electron transfer in biological systemsis mostly long-range electron transfer that occurs by the electrontunneling through the protein media. There is a long-standing problem thatwhich part of protein media is used for the electron tunneling root. As aresult of our detailed analysis, we found that the global electron tunnelingroot is a little winded with a width of a few angstrom, reflecting theproperty of tertiary and secondary structures of the protein and it isaffected by the thermal fluctuation of protein structure. Photoisomerizationof rhodopsin is very unique: The cis-transphotoisomerization ofrhodopsin occurs only around the C11 = C12 bond in the counterclockwisedirection. Its molecular mechanism is resolved by our MD simulation studyusing the structure of rhodopsin which was recently obtained by the X-raycrystallographic analysis.

Excitation energy transfer modulated by oscillating electronic coupling of a dimeric system embedded in a molecular environment

We have developed a theoretical formulation for excitation energy transfer between structurally fluctuating dimer molecules in surrounding environments. On the basis of a generalized master equation in which a memory function plays a vital role, the temporal evolutions of the population densities of exciton at the donor and acceptor sites are described. By employing an ansatz form for the memory function, the competitive effects of dimeric coupling and bath modes are analyzed quantitatively, where the roles of oscillating electronic coupling are highlighted.

Electron tunneling pathways in respiratory complex I. The role of the internal water between the enzyme subunits

Recently, the atomistic details of the electronic wiring of seven Fe/S clusters (N3, N1b, N4, N5, N6a, N6b, N2) of respiratory complex I, along which electrons are injected into the electron transport chain, have been revealed; the tunneling pathways between the clusters and the contributing key residues were identified [1]. In this study, the sensitivity of the electron tunneling pathways to the internal water at the protein subunit boundaries is investigated by simulating tunneling pathways of N3→N1b and N6b→N2 with and without the internal water. It is found that the hydrogen bonding networks formed along the internal water can provide efficient tunneling pathways. In N3→N1b, the tunneling pathway with the internal water is drastically different with significantly shorter (3.4 Å) total tunneling distance along the trajectory. In N6b→N2, the internal water contributes to the tunneling as a bridge between N6b and 9Ile99 with two shorter through-space jumps instead of one longer jump. The resulting enhancement of the rates of the individual electron tunneling process is two to three orders of magnitude. This study demonstrates that the tunneling pathways and tunneling rates are sensitive to the internal water, which suggests that the tunneling pathways can change dynamically due to the diffusion of the internal water, and that the efficient electron tunneling occurs at some specific optimal positions of the internal water.

Nucleic Acid Charge Transfer: Black, White and Gray

Theoretical studies of charge transport in deoxyribonucleic acid (DNA) and peptide nucleic acid (PNA) indicate that structure and dynamics modulate the charge transfer rates, and that different members of a structural ensemble support different charge transport mechanisms. Here, we review the influences of nucleobase geometry, electronic structure, solvent environment, and thermal conformational fluctuations on the charge transfer mechanism. We describe an emerging framework for understanding the diversity of charge transport mechanisms seen in nucleic acids.

Electron tunneling in respiratory complex I

NADH:ubiquinone oxidoreductase (complex I) plays a central role in the respiratory electron transport chain by coupling the transfer of electrons from NADH to ubiquinone to the creation of the proton gradient across the membrane necessary for ATP synthesis. Here the atomistic details of electronic wiring of all Fe/S clusters in complex I are revealed by using the tunneling current theory and computer simulations; both density functional theory and semiempirical electronic structure methods were used to examine antiferromagnetically coupled spin states and corresponding tunneling wave functions. Distinct electron tunneling pathways between neighboring Fe/S clusters are identified; the pathways primarily consist of two cysteine ligands and one additional key residue. Internal water between protein subunits is identified as an essential mediator enhancing the overall electron transfer rate by almost three orders of magnitude to achieve a physiologically significant value. The identified key residues are further characterized by sensitivity of electron transfer rates to their mutations, examined in simulations, and their conservation among complex I homologues. The unusual electronic structure properties of Fe(4)S(4) clusters in complex I explain their remarkable efficiency of electron transfer.

The diabatic picture of electron transfer, reaction barriers, and molecular dynamics

Diabatic states have a long history in chemistry, beginning with early valence bond pictures of molecular bonding and extending through the construction of model potential energy surfaces to the modern proliferation of methods for computing these elusive states. In this review, we summarize the basic principles that define the diabatic basis and demonstrate how they can be applied in the specific context of constrained density functional theory. Using illustrative examples from electron transfer and chemical reactions, we show how the diabatic picture can be used to extract qualitative insight and quantitative predictions about energy landscapes. The review closes with a brief summary of the challenges and prospects for the further application of diabatic states in chemistry.

Modeling Cu(II) binding to peptides using the extensible systematic force field

The utility of the extensible systematic force field (ESFF) was tested for copper(II) binding to a 34-amino-acid Cu(II) peptide, which includes five histidine residues and is the putative copper-binding site of lysyl oxidase. To improve computational efficiency, distance geometry calculations were used to constrain all combinations of three histidine ligands to be within bonding distance of the copper and the best results were utilized as starting structures for the ESFF computations. All likely copper geometries were modeled, but the results showed only a small dependence on the geometrical model in that all resulted in a distorted square pyramidal geometry about the copper, some of the imidazole rings were poorly oriented for ligation to the Cu(II), and the copper-nitrogen bond distances were too long. The results suggest that ESFF should be used with caution for Cu(II) complexes where the copper-ligand bonds have significant covalency and when the ligands are not geometrically constrained to be planar.

Long-Distance Electron Tunneling in Proteins: A New Challenge for Time-Resolved Spectroscopy

Long-distance electron tunneling is a fundamental process which is involved in energy generation in cells. The tunneling occurs between the metal centers in the respiratory enzymes, typically over distances up to 20 or 30 Å. For such distances, the tunneling time-i.e., the time during which an electron passes through the body of the protein molecule from one metal center to another-is of the order of 10 fs. Here the process of electron tunneling in proteins is reviewed, and a possibility of experimental observation of real-time electron tunneling in a single protein molecule is discussed.

Fluctuations in biological and bioinspired electron-transfer reactions

Central to theories of electron transfer (ET) is the idea that nuclear motion generates a transition state that enables electron flow to proceed, but nuclear motion also induces fluctuations in the donor-acceptor (DA) electronic coupling that is the rate-limiting parameter for nonadiabatic ET. The interplay between the DA energy gap and DA coupling fluctuations is particularly noteworthy in biological ET, where flexible protein and mobile water bridges take center stage. Here, we discuss the critical timescales at play for ET reactions in fluctuating media, highlighting issues of the Condon approximation, average medium versus fluctuation-controlled electron tunneling, gated and solvent relaxation controlled electron transfer, and the influence of inelastic tunneling on electronic coupling pathway interferences. Taken together, one may use this framework to establish principles to describe how macromolecular structure and structural fluctuations influence ET reactions. This framework deepens our understanding of ET chemistry in fluctuating media. Moreover, it provides a unifying perspective for biophysical charge-transfer processes and helps to frame new questions associated with energy harvesting and transduction in fluctuating media.

Steering electrons on moving pathways

Electron transfer (ET) reactions provide a nexus among chemistry, biochemistry, and physics. These reactions underpin the "power plants" and "power grids" of bioenergetics, and they challenge us to understand how evolution manipulates structure to control ET kinetics. Ball-and-stick models for the machinery of electron transfer, however, fail to capture the rich electronic and nuclear dynamics of ET molecules: these static representations disguise, for example, the range of thermally accessible molecular conformations. The influence of structural fluctuations on electron-transfer kinetics is amplified by the exponential decay of electron tunneling probabilities with distance, as well as the delicate interference among coupling pathways. Fluctuations in the surrounding medium can also switch transport between coherent and incoherent ET mechanisms--and may gate ET so that its kinetics is limited by conformational interconversion times, rather than by the intrinsic ET time scale. Moreover, preparation of a charge-polarized donor state or of a donor state with linear or angular momentum can have profound dynamical and kinetic consequences. In this Account, we establish a vocabulary to describe how the conformational ensemble and the prepared donor state influence ET kinetics in macromolecules. This framework is helping to unravel the richness of functional biological ET pathways, which have evolved within fluctuating macromolecular structures. The conceptual framework for describing nonadiabatic ET seems disarmingly simple: compute the ensemble-averaged (mean-squared) donor-acceptor (DA) tunneling interaction, <H(DA)(2)>, and the Franck-Condon weighted density of states, rho(FC), to describe the rate, (2pi/variant Planck's over 2pi)<H(DA)(2)>rho(FC). Modern descriptions of the thermally averaged electronic coupling and of the Franck-Condon factor establish a useful predictive framework in biology, chemistry, and nanoscience. Describing the influence of geometric and energetic fluctuations on ET allows us to address a rich array of mechanistic and kinetic puzzles. How strongly is a protein's fold imprinted on the ET kinetics, and might thermal fluctuations "wash out" signatures of structure? What is the influence of thermal fluctuations on ET kinetics beyond averaging of the tunneling barrier structure? Do electronic coupling mechanisms change as donor and acceptor reposition in a protein, and what are the consequences for the ET kinetics? Do fluctuations access minority species that dominate tunneling? Can energy exchanges between the electron and bridge vibrations generate vibronic signatures that label some of the D-to-A pathways traversed by the electron, thus eliminating unmarked pathways that would otherwise contribute to the DA coupling (as in other "which way" or double-slit experiments)? Might medium fluctuations drive tunneling-hopping mechanistic transitions? How does the donor-state preparation, in particular, its polarization toward the acceptor and its momentum characteristics (which may introduce complex rather than pure real relationships among donor orbital amplitudes), influence the electronic dynamics? In this Account, we describe our recent studies that address puzzling questions of how conformational distributions, excited-state polarization, and electronic and nuclear dynamical effects influence ET in macromolecules. Indeed, conformational and dynamical effects arise in all transport regimes, including the tunneling, resonant transport, and hopping regimes. Importantly, these effects can induce switching among ET mechanisms.

Molecular dynamics of amicyanin reveals a conserved dynamical core for blue copper proteins

Molecular dynamics simulation has been carried out for the blue copper protein amicyanin from two different sources, Paracoccus denitrificans and Paraccocus versutus, to investigate the structural and dynamical properties common to the two molecules and to identify prominent features shared with proteins of the same family, the monomeric cupredoxins. The two amicyanins have almost identical secondary and tertiary structure. In the simulation, they differ for the number of hydrogen bonds in the main chain and the conformation of some beta-strands. However, they strictly maintain the arrangement of the portions of the beta-barrel that are conserved in the folding architecture of the blue copper proteins. Paracoccus versutus amicyanin equilibrates more rapidly, shows lower atomic deviation values, and is less rigid with respect to Paracoccus denitrificans amicyanin. Principal component analysis reveals that the conformational subspaces corresponding to eigenvectors with the same index for each of the two molecules are not necessarily equivalent. Nevertheless, a core scaffold with constrained dynamics exist for both amicyanins. In addition, two fairly flexible regions that are located on the opposite side with respect to the interaction sites with the partner molecules in the redox process have been evidenced in the protein structure. This description of amicyanin, with a few mobile regions remote from the active site and a rigid scaffold including most of the protein beta-barrel, has a close similarity with that of azurin and plastocyanin, two other cupredoxins previously investigated in simulation. Furthermore, similarities in the distribution of the atomic fluctuations indicate that amicyanin, azurin, and plastocyanin possess common dynamical features, in spite of differences in their structure. On the basis of these findings, we suggest that topological constraints imposed by the folding in correspondence of protein regions that are the most conserved determine the protein dynamics of the cupredoxin family. The dynamical properties of the cupredoxins might be controlled for functional advantages that include the binding mechanism with the biological partners and the collective inner motions of the protein matrix required for the electron transfer, whereas long-range conformational changes in the redox reaction should be excluded.

Quantitative evaluations of mechanisms of radiofrequency interactions with biological molecules and processes

The complexity of interactions of electromagnetic fields up to 10(12) Hz with the ions, atoms, and molecules of biological systems has given rise to a large number of established and proposed biophysical mechanisms applicable over a wide range of time and distance scales, field amplitudes, frequencies, and waveforms. This review focuses on the physical principles that guide quantitative assessment of mechanisms applicable for exposures at or below the level of endogenous electric fields associated with development, wound healing, and excitation of muscles and the nervous system (generally, 1 to 10(2) V m(-1)), with emphasis on conditions where temperature increases are insignificant (<<1 K). Experiment and theory demonstrate possible demodulation at membrane barriers for frequencies < or =10 MHz, but not at higher frequencies. Although signal levels somewhat below system noise can be detected, signal-to-noise ratios substantially less than 0.1 cannot be overcome by cooperativity, signal averaging, coherent detection, or by nonlinear dynamical systems. Sensory systems and possible effects on biological magnetite suggest paradigms for extreme sensitivity at lower frequencies, but there are no known radiofrequency (RF) analogues. At the molecular level, vibrational modes are so overdamped by water molecules that excitation of molecular modes below the far infrared cannot occur. Two RF mechanisms plausibly may affect biological matter under common exposure conditions. For frequencies below approximately 150 MHz, shifts in the rate of chemical reactions can be mediated by radical pairs and, at all frequencies, dielectric and resistive heating can raise temperature and increase the entropy of the affected biological system.

Average electron tunneling route of the electron transfer in protein media

We present a new theoretical method to determine and visualize the average tunneling route of the electron transfer (ET) in protein media. In this, we properly took into account the fluctuation of the tunneling currents and the quantum-interference effect. The route was correlated with the electronic factor <TDA(2)> in the case of ET by the elastic tunneling mechanism. We expanded <TDA(2)> by the interatomic tunneling currents <Jab(2)>'s. Incorporating the quantum-interference effect into the mean-square interatomic tunneling currents, denoted as <Jab(2)>, we could express <TDA(2)> as a sum of variant Planck's over 2pi(2)<Jab(2)>. Drawing the distribution of <Jab(2)> on the protein structure, we obtain the <Jab(2)> map which visually represents which parts of bonds and spaces most significantly contribute to <TDA(2)>. We applied this method to the ET from the bacteriopheophytin anion to the primary quinone in the bacterial photosynthetic reaction center of Rhodobacter sphaeroides. We obtained <Jab(2)>'s by a combined method of molecular dynamics simulations and quantum chemical calculations. In calculating <Jab(2)>, we found that much destructive interference works among the interatomic tunneling currents even after taking the average. We drew the <Jab(2)> map by a pipe model where atoms a and b are connected by a pipe with width proportional to the magnitude of <Jab(2)>. We found that two groups of <Jab(2)>'s, which are mutually coupled with high correlation in each group, have broad pipes and form the average tunneling routes, called Trp route and Met route. Each of the two average tunneling routes is composed of a few major pathways in the Pathways model which are fused at considerable part to each other. We also analyzed the average tunneling route for the ET by the inelastic tunneling mechanism.

Stress tensor analysis of the protein quake of photoactive yellow protein

Immediately after photon absorption, the photoenergy is converted to local stress energy via the ultrafast photoisomerization reaction of the p-coumaric acid (pCA) chromophore in a small water-soluble blue light receptor, photoactive yellow protein (PYP), derived from the halophilic bacterium, Halorhodospira halophila. A series of conformational changes are then induced, which are intimately related with the relaxation process on the energy landscape of PYP. In order to understand the signaling function of PYP in atomic detail, the characterization of the physical mechanism of the protein quake of PYP is important, as is the atomic description of the series of conformational changes associated with the photocycle. Here, we report a theoretical/computational study for the analysis of the intramolecular stress tensor for the dark state and three intermediate states, pR, pB1 and pB2, of PYP. As a result, we found that the magnitude of the stress released during the change from the pR to the pB1 state is significantly large at the hydroxyl oxygen atom of Tyr42, suggesting that this atom is the focus of the protein quake of PYP. This is consistent with previous experimental observations.

Discrimination of class I cyclobutane pyrimidine dimer photolyase from blue light photoreceptors by single methionine residue

DNA photolyase recognizes ultraviolet-damaged DNA and breaks improperly formed covalent bonds within the cyclobutane pyrimidine dimer by a light-activated electron transfer reaction between the flavin adenine dinucleotide, the electron donor, and cyclobutane pyrimidine dimer, the electron acceptor. Theoretical analysis of the electron-tunneling pathways of the DNA photolyase derived from Anacystis nidulans can reveal the active role of the protein environment in the electron transfer reaction. Here, we report the unexpectedly important role of the single methionine residue, Met-353, where busy trafficking of electron-tunneling currents is observed. The amino acid conservation pattern of Met-353 in the homologous sequences perfectly correlates with experimentally verified annotation as photolyases. The bioinformatics sequence analysis also suggests that the residue plays a pivotal role in biological function. Consistent findings from different disciplines of computational biology strongly suggest the pivotal role of Met-353 in the biological function of DNA photolyase.

Long Distance Electron-Transfer Mechanism in Peptidylglycine α-Hydroxylating Monooxygenase: A Perfect Fitting for a Water Bridge

The active sites of copper enzymes have been the subject of many theoretical and experimental investigations from a number of years. Such studies have embraced topics devoted to the modeling of the first coordination sphere at the metallic cations up to the development of biomimetic, or bioinspired, catalytic systems. At least from the theoretical viewpoint, fewer efforts have been dedicated to elucidate how the two copper cations act concertedly in noncoupled dicopper enzymes such as peptidylglycine alpha-hydroxylating monooxygenase (PHM) and dopamine beta-monooxygenase (DbetaM). In these metalloenzymes, an electronic transfer is assumed between the two distant copper cations (11 A). Recent experimental results suggest that this transfer occurs through water molecules, a phenomenon which has been theoretically evidenced to be of high efficiency in the case of cytochrome b5 (Science, 2005, 310, 1311). In the present contribution dedicated to PHM, we overpass the common theoretical approaches dedicated to the electronic and geometrical structures of sites CuM or CuH restricted to their first coordination spheres and aim at directly comparing theoretical results to the experimentally measured activity of the PHM enzyme. To achieve this goal, molecular dynamics simulations were performed on wild-type and various mutants of PHM. More precisely, we provide an estimate of the electron-transfer efficiency between the CuM and CuH sites by means of such molecular dynamics simulations coupled to Marcus theory joined to the Beratan model to approximate the required coupling matrix elements. The theoretical results are compared to the kinetics measurements performed on wild and mutated PHM. The present work, the dynamic aspects of which are essential, accounts for the experimental results issued from mutagenesis. It supports the conclusion that an electronic transfer can occur between two copper(I) sites along a bridge involving a set of hydrogen and chemical bonds. Residue Gln170 is evidenced to be the keystone of this water-mediated pathway.

Interference, fluctuation, and alternation of electron tunneling in protein media. 2. Non-condon theory for the energy gap dependence of electron transfer rate

Developing the quantum transition rate theory of Prezhdo and Rossky (J. Chem. Phys. 1997, 107, 5863), we produced a new non-Condon theory of the rate of electron transfer (ET) which happens through a protein medium with conformational fluctuation. The new theory is expressed by a convolution form of the power spectrum for the autocorrelation function of the electronic tunneling matrix element T(DA)(t) with quantum correction and the ordinary Franck-Condon factor. The new theory satisfies the detailed balance condition for the forward and backward ET rates. The ET rate formula is divided into two terms of elastic and inelastic tunneling mechanisms on the mathematical basis. The present theory is applied to the ET from Bph(-) to Q(A) in the reaction center of Rhodobacter sphaeroides. Numerical calculations of T(DA)(t) were made by a combined method of molecular dynamics simulations and quantum chemistry calculations. We showed that the normalized autocorrelation function of T(DA)(t) is almost expressed by exponential forms. The calculated energy gap law of the ET rate is nearly Marcus' parabola in most of the normal region and around the maximum region, but it does not decay substantially in the inverted region, which is called the anomalous inverted region. We also showed that the energy gap law at the high uphill energy gap in the normal region is elevated considerably from the Marcus' parabola, which is called the anomalous normal region. Those anomalous energy gap laws are due to the inelastic tunneling mechanism which works actively at the energy gap far from zero. We presented an empirical formula for easily calculating the non-Condon ET rate, which is usable by many researchers. We provided experimental evidence for the anomalous inverted region which was basically reproduced by the present theory. The present theory was extensively compared with the previous non-Condon theories.

Ab Initio Prediction of Tryptophan Fluorescence Quenching by Protein Electric Field Enabled Electron Transfer

We report quantum mechanical-molecular mechanical (QM-MM) predictions of fluorescence quantum yields for 20 tryptophans in 17 proteins, whose yields span the range from 0.01 to 0.3, using ab initio computed coupling matrix elements for photoinduced electron transfer from the 1La excited indole ring to a local backbone amide. The average coupling elements span the range 140-1000 cm-1, depending on tryptophan rotamer conformation. The matrix elements were from the singles configuration interaction matrix, and were largely insensitive to which of the three basis sets was used. Large fluctuations were seen on the time scale of tens of femtoseconds, caused primarily by side chain and backbone torsional variations for 150 ps of dynamics at 300 K. The largest coupling occurs for the chi1 = -60 degrees rotamer and is purely through-bond. There is no apparent correlation between the coupling magnitude and quantum yield, which is still dominated by energy gap and reorganization energy. The source of error bars for predicted quenching rates using the weak coupling golden rule may be due to inaccurate averaged Franck-Condon weighted densities because of inadequate simulation times and parameters and/or to failure of the weak coupled golden rule used in these predictions because of the broad distribution of Landau-Zener probabilities arising from the large variable coupling.

Conformationally averaged score functions for electronic propagation in proteins

We explore the influence of conformational dynamics on protein-mediated electron donor-acceptor interactions. We introduce a thermally averaged score function to characterize electronic propagation from redox cofactors into the protein and solvent. The score function is explored for myoglobin at the extended-Hückel level, and the results are compared with those of simpler models. The conformationally averaged quantum results are consistent with the empirical analysis of the Pathways model. Notably, subtle effects of quantum interference among multiple coupling pathways that arise in static structures are largely averaged out when protein thermal motion is included. Propagation through bulk water near the single-protein interface decays rapidly with distance.