Quantum and classical dynamics in biochemical reactions

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.

Charge transfer in model peptides: obtaining Marcus parameters from molecular simulation

Charge transfer within and between biomolecules remains a highly active field of biophysics. Due to the complexities of real systems, model compounds are a useful alternative to study the mechanistic fundamentals of charge transfer. In recent years, such model experiments have been underpinned by molecular simulation methods as well. In this work, we study electron hole transfer in helical model peptides by means of molecular dynamics simulations. A theoretical framework to extract Marcus parameters of charge transfer from simulations is presented. We find that the peptides form stable helical structures with sequence dependent small deviations from ideal PPII helices. We identify direct exposure of charged side chains to solvent as a cause of high reorganization energies, significantly larger than typical for electron transfer in proteins. This, together with small direct couplings, makes long-range superexchange electron transport in this system very slow. In good agreement with experiment, direct transfer between the terminal amino acid side chains can be dicounted in favor of a two-step hopping process if appropriate bridging groups exist.

Characterization of azido-NAD+ to assess its potential as a two-dimensional infrared probe of enzyme dynamics

Enzyme active-site dynamics at femtosecond to picosecond time scales are of great biochemical importance, but remain relatively unexplored due to the lack of appropriate analytical methods. Two-dimensional infrared (2D IR) spectroscopy is one of the few methods that can examine chemical biological motions at this time scale, but all the IR probes used so far were specific to a few unique enzymes. The lack of IR probes of broader specificity is a major limitation to further 2D IR studies of enzyme dynamics. Here we describe the synthesis of a general IR probe for nicotinamide-dependent enzymes. This azido analog of the ubiquitous cofactor nicotinamide adenine dinucleotide is found to be stable and bind to several dehydrogenases with dissociation constants similar to that for the native cofactor. The infrared absorption spectra of this probe bound to several enzymes indicate that it has significant potential as a 2D IR probe to investigate femtosecond dynamics of nicotinamide-dependent enzymes.

Electron transfer in proteins: nonorthogonal projections onto donor-acceptor subspace of the Hilbert space

We develop nonorthogonal projectors, called Löwdin projectors, to construct an effective donor-acceptor system composed of localized donor (D) and acceptor (A) states of a long-distance electron transfer problem. When these states have a nonvanishing overlap with the bridge states these projectors are non-Hermitian and there are various possible effective two-level systems that can be built. We show how these can be constructed directly from the Schrödinger or Dyson equation projected onto the D-A subspace of the Hilbert space and explore these equations to determine the connection between Hamiltonian and Green function partitioning. We illustrate the use of these effective two-level systems in estimating the electron transfer rate in the context of a simple electron transfer model.

Protein dynamics and electron transfer: electronic decoherence and non-Condon effects

We compute the autocorrelation function of the donor-acceptor tunneling matrix element <T(DA)(t)T(DA)(0)> for six Ru-azurin derivatives. Comparison of this decay time to the decay time of the time-dependent Franck-Condon factor {computed by Rossky and coworkers [Lockwood, D. M., Cheng, Y.-K. & Rossky, P. J. (2001) Chem. Phys. Lett. 345, 159-165]} reveals the extent to which non-Condon effects influence the electron-transfer rate. <T(DA)(t)T(DA)(0)> is studied as a function of donor-acceptor distance, tunneling pathway structure, tunneling energy, and temperature to explore the structural and dynamical origins of non-Condon effects. For azurin, the correlation function is remarkably insensitive to tunneling pathway structure. The decay time is only slightly shorter than it is for solvent-mediated electron transfer in small organic molecules and originates, largely, from fluctuations of valence angles rather than bond lengths.

Clarification and application of an ion parametric resonance model for magnetic field interactions with biological systems

Theoretical models proposed to date have been unable to clearly predict biological results from exposure to low-intensity electric and magnetic fields (EMF). Recently a predictive ionic resonance model was proposed by Lednev, based on an earlier atomic spectroscopy theory described by Podgoretskii and Podgoretskii and Khrustalev. The ion parametric resonance (IPR) model developed in this paper corrects mathematical errors in the earlier Lednev model and extends that model to give explicit predictions of biological responses to parallel AC and DC magnetic fields caused by field-induced changes in combinations of ions within the biological system. Distinct response forms predicted by the IPR model depend explicitly on the experimentally controlled variables: magnetic flux densities of the AC and DC magnetic fields (Bac and Bdc, respectively); AC frequency (fac); and, implicitly, charge to mass ratio of target-ions. After clarifying the IPR model and extending it to combinations of different resonant ions, this paper proposes a basic set of experiments to test the IPR model directly which do not rely on the choice of a particular specimen or endpoint. While the fundamental bases of the model are supported by a variety of other studies, the IPR model is necessarily heuristic when applied to biological systems, because it is based on the premise that the magnitude and form of magnetic field interactions with unhydrated resonant ions in critical biological structures alter ion-associated biological activities that may in turn be correlated with observable effects in living systems.