Coupling of protein motion to electron transfer: Molecular dynamics and stochastic quantum mechanics study of photosynthetic reaction centers

Inverted region in the reaction of the quinone reduction in the A1-site of photosystem I from cyanobacteria

Photosystem I from the menB strain of Synechocystis sp. PCC 6803 containing foreign quinones in the A1 sites was used for studying the primary steps of electron transfer by pump-probe femtosecond laser spectroscopy. The free energy gap (- ΔG) of electron transfer between the reduced primary acceptor A0 and the quinones bound in the A1 site varied from 0.12 eV for the low-potential 1,2-diamino-anthraquinone to 0.88 eV for the high-potential 2,3-dichloro-1,4-naphthoquinone, compared to 0.5 eV for the native phylloquinone. It was shown that the kinetics of charge separation between the special pair chlorophyll P700 and the primary acceptor A0 was not affected by quinone substitutions, whereas the rate of A0 → A1 electron transfer was sensitive to the redox-potential of quinones: the decrease of - ΔG by 400 meV compared to the native phylloquinone resulted in a ~ fivefold slowing of the reaction The presence of the asymmetric inverted region in the ΔG dependence of the reaction rate indicates that the electron transfer in photosystem I is controlled by nuclear tunneling and should be treated in terms of quantum electron-phonon interactions. A three-mode implementation of the multiphonon model, which includes modes around 240 cm-1 (large-scale protein vibrations), 930 cm-1 (out-of-plane bending of macrocycles and protein backbone vibrations), and 1600 cm-1 (double bonds vibrations) was applied to rationalize the observed dependence. The modes with a frequency of at least 1600 cm-1 make the predominant contribution to the reorganization energy, while the contribution of the "classical" low-frequency modes is only 4%.

Electron Transfer-Induced Active Site Structural Relaxation in 64-Photolyase of Drosophila melanogaster

While catalytic electron flow and photoreactivation of CPD-photolyases are increasingly understood, the microscopic details of the 64-photolyase repair mechanism are perpetually debated. Here, we investigate in long-time (μs) molecular dynamics simulations combined with extensive quantum mechanical/molecular mechanical (QM/MM) simulations the primary electron transfer (ET) reactions in 64-photolyase of Drosophila melanogaster (D. melanogaster). The characterization of the relative energetics of locally excited and charge separated states in the (6-4) photoproduct enzyme repair complex reveals a charge-separated state involving the adenine moiety of the FADH^- cofactor that facilitates reduction of the photoproduct. Microscopic details of the collective reaction coordinate of ET reactions are identified that involve the reorganization of the hydrogen bond network and structural relaxation of the active site. The simulations reveal complex active site relaxation dynamics involving distinguished amino acids (Lys246, His365, and His369), conformational reorganization of the hydroxyl group of the (6-4) photoproduct, and a strengthening of hydrogen bonds with immobilized water molecules. In particular, rotation of the Lys246 side chain is found to impose a double-well character along the reaction coordinate of the ET reaction. Our findings suggest that the primary ET reactions in the (6-4) photoproduct enzyme repair complex of D. melanogaster are governed by a complex multi-minima active site relaxation dynamics and potentially precede the equilibration of the protein. ET pathways mediated by the adenine moiety and the 5' side of the photoproduct are proposed to be relevant for triggering the catalytic (6-4) photoproduct reactivation.

Determining the atomic charge of calcium ion requires the information of its coordination geometry in an EF-hand motif

It is challenging to parameterize the force field for calcium ions (Ca²⁺) in calcium-binding proteins because of their unique coordination chemistry that involves the surrounding atoms required for stability. In this work, we observed a wide variation in Ca²⁺ binding loop conformations of the Ca²⁺-binding protein calmodulin, which adopts the most populated ternary structures determined from the molecular dynamics simulations, followed by ab initio quantum mechanical (QM) calculations on all 12 amino acids in the loop that coordinate Ca²⁺ in aqueous solution. Ca²⁺ charges were derived by fitting to the electrostatic potential in the context of a classical or polarizable force field (PFF). We discovered that the atomic radius of Ca²⁺ in conventional force fields is too large for the QM calculation to capture the variation in the coordination geometry of Ca²⁺ in its ionic form, leading to unphysical charges. Specifically, we found that the fitted atomic charges of Ca²⁺ in the context of PFF depend on the coordinating geometry of electronegative atoms from the amino acids in the loop. Although nearby water molecules do not influence the atomic charge of Ca²⁺, they are crucial for compensating for the coordination of Ca²⁺ due to the conformational flexibility in the EF-hand loop. Our method advances the development of force fields for metal ions and protein binding sites in dynamic environments.

Molecular dynamics simulations in photosynthesis

Photosynthesis is regulated by a dynamic interplay between proteins, enzymes, pigments, lipids, and cofactors that takes place on a large spatio-temporal scale. Molecular dynamics (MD) simulations provide a powerful toolkit to investigate dynamical processes in (bio)molecular ensembles from the (sub)picosecond to the (sub)millisecond regime and from the Å to hundreds of nm length scale. Therefore, MD is well suited to address a variety of questions arising in the field of photosynthesis research. In this review, we provide an introduction to the basic concepts of MD simulations, at atomistic and coarse-grained level of resolution. Furthermore, we discuss applications of MD simulations to model photosynthetic systems of different sizes and complexity and their connection to experimental observables. Finally, we provide a brief glance on which methods provide opportunities to capture phenomena beyond the applicability of classical MD.

Modeling to Understand Plant Protein Structure-Function Relationships-Implications for Seed Storage Proteins

Proteins are among the most important molecules on Earth. Their structure and aggregation behavior are key to their functionality in living organisms and in protein-rich products. Innovations, such as increased computer size and power, together with novel simulation tools have improved our understanding of protein structure-function relationships. This review focuses on various proteins present in plants and modeling tools that can be applied to better understand protein structures and their relationship to functionality, with particular emphasis on plant storage proteins. Modeling of plant proteins is increasing, but less than 9% of deposits in the Research Collaboratory for Structural Bioinformatics Protein Data Bank come from plant proteins. Although, similar tools are applied as in other proteins, modeling of plant proteins is lagging behind and innovative methods are rarely used. Molecular dynamics and molecular docking are commonly used to evaluate differences in forms or mutants, and the impact on functionality. Modeling tools have also been used to describe the photosynthetic machinery and its electron transfer reactions. Storage proteins, especially in large and intrinsically disordered prolamins and glutelins, have been significantly less well-described using modeling. These proteins aggregate during processing and form large polymers that correlate with functionality. The resulting structure-function relationships are important for processed storage proteins, so modeling and simulation studies, using up-to-date models, algorithms, and computer tools are essential for obtaining a better understanding of these relationships.

Microsecond Simulation of Electron Transfer in DNA: Bottom-Up Parametrization of an Efficient Electron Transfer Model Based on Atomistic Details

The transfer of electrons over long distances in complex molecular systems is a phenomenon of significance in both biochemistry and technology. In recent years, we have been developing efficient models to study ET in complex systems, including DNA as a prominent example. Ab initio and model approaches have been combined in an "on-the-fly" calculation of ET parameters, which can be used to propagate nuclear and electronic degrees of freedom simultaneously. These previous efforts have aimed at deriving an efficient nonadiabatic quantum mechanical-molecular mechanical (QM/MM) simulation scheme for ET, making nanosecond simulations of ET in realistic systems possible. This, however, is still insufficient for the treatment of large donor-bridge-acceptor systems, like the ET in DNA, overcoming long adenine bridges. Therefore, we have constructed a theoretical model in a bottom-up manner. All quantum-chemical as well as force-field calculations are substituted by theoretical models of the involved phenomena on a molecular level, including polarization and relaxation of the molecular environment, which are often omitted in other recently developed theoretical models of ET. A nonadiabatic simulation scheme is employed, and no assumptions regarding the ET mechanism are needed. Thus, the predictive power of the simulations is preserved, while pushing the limits of the accessible time scales beyond microseconds. This model-based simulation scheme is applied to ET in various DNA species. Good agreement with the "full" atomistic nonadiabatic QM/MM scheme is observed for the archetypal DNA ET systems, the polyA sequence, as well as the sequences GT_nGGG, containing adenines as bridge sites. Furthermore, ET in larger, more complex DNA sequences is simulated, and the results are discussed.

Broad-Band Pump-Probe Spectroscopy Quantifies Ultrafast Solvation Dynamics of Proteins and Molecules

In this work, we demonstrate the use of broad-band pump-probe spectroscopy to measure femtosecond solvation dynamics. We report studies of a rhodamine dye in methanol and cryptophyte algae light-harvesting proteins in aqueous suspension. Broad-band impulsive excitation generates a vibrational wavepacket that oscillates on the excited-state potential energy surface, destructively interfering with itself at the minimum of the surface. This destructive interference gives rise to a node at a certain probe wavelength that varies with time. This reveals the Gibbs free-energy changes of the excited-state potential energy surface, which equates to the solvation time correlation function. This method captures the inertial solvent response of water (∼40 fs) and the bimodal inertial response of methanol (∼40 and ∼150 fs) and reveals how protein-buried chromophores are sensitive to the solvent dynamics inside and outside of the protein environment.

Unraveling the Mechanism of Photoinduced Charge Transfer in Carotenoid-Porphyrin-C60 Molecular Triad

Photoinduced charge transfer (CT) plays a central role in biologically significant systems and in applications that harvest solar energy. We investigate the relationship of CT kinetics and conformation in a molecular triad. The triad, consisting of carotenoid, porphyrin, and fullerene is structurally flexible and able to acquire significantly varied conformations under ambient conditions. With an integrated approach of quantum calculations and molecular dynamics simulations, we compute the rate of CT at two distinctive conformations. The linearly extended conformation, in which the donor (carotenoid) and the acceptor (fullerene) are separated by nearly 50 Å, enables charge separation through a sequential CT process. A representative bent conformation that is entropically dominant, however, attenuates the CT, although the donor and the acceptor are spatially closer. Our computed rate of CT at the linear conformation is in good agreement with measured values. Our work provides unique fundamental understanding of the photoinduced CT process in the molecular triad.

Temperature and detection-wavelength dependence of the electron transfer rates in initial stages of photosynthesis

Unusual temperature behavior, observed in the initial electron transfer stages in the photosynthetic reaction centers of the purple bacteria, and a strong probing pulse wavelength dependence of transfer rates, determined in transient absorption spectroscopy, can easily be explained on assuming that the transfer takes place from dynamically unrelaxed states of protein environment. The transitions from the primary special pair (P) to a single bacteriochlorophyll (B) and next to a bacteriopheophytin (H) are controlled by diffusion down the energy value of underdamped vibrational modes of frequency 200 K, probably determining distances between the succeeding cofactors. The subsequent transition to the quinone A (Q) is controlled by diffusion in the position value of an overdamped conformational mode, probably corresponding to the local polarization. From the fit of available experimental data to simple theoretical formulas, the important physical conclusion arises that the very electronic transitions are fast as compared to the relaxation processes and, in the first approximation, only the latter contribute to the overall times of the initial electron transfer stages in photosynthesis.

Charge transfer in E. coli DNA photolyase: understanding polarization and stabilization effects via QM/MM simulations

We study fast hole transfer events in E. coli DNA photolyase, a key step in the photoactivation process, using a multiscale computational method that combines nonadiabatic propagation schemes and linear-scaling quantum chemical methods with molecular mechanics force fields. This scheme allows us to follow the time-dependent evolution of the electron hole in an unbiased fashion; that is, no assumptions about hole wave function localization, time scale separation, or adiabaticity of the process have to be made beforehand. DNA photolyase facilitates an efficient long-range charge transport between its flavin adenine dinucleotide (FAD) cofactor and the protein surface via a chain of evolutionary conserved Trp residues on the sub-nanosecond time scale despite the existence of multiple potential trap states. By including a large number of aromatic residues along the charge transfer pathway into the quantum description, we are able to identify the main pathway among alternative possible routes. The simulations show that charge transfer, which is extremely fast in this protein, occurs on the same time scale as the protein response to the electrostatic changes; that is, time-scale separation as often presupposed in charge transfer studies seems to be inappropriate for this system. Therefore, coupled equations of motion, which propagate electrons and nuclei simultaneously, appear to be necessary. The applied computational model is shown to capture the essentials of the reaction kinetics and thermodynamics while allowing direct simulations of charge transfer events on their natural time scale.

A hybrid approach to simulation of electron transfer in complex molecular systems

Electron transfer (ET) reactions in biomolecular systems represent an important class of processes at the interface of physics, chemistry and biology. The theoretical description of these reactions constitutes a huge challenge because extensive systems require a quantum-mechanical treatment and a broad range of time scales are involved. Thus, only small model systems may be investigated with the modern density functional theory techniques combined with non-adiabatic dynamics algorithms. On the other hand, model calculations based on Marcus's seminal theory describe the ET involving several assumptions that may not always be met. We review a multi-scale method that combines a non-adiabatic propagation scheme and a linear scaling quantum-chemical method with a molecular mechanics force field in such a way that an unbiased description of the dynamics of excess electron is achieved and the number of degrees of freedom is reduced effectively at the same time. ET reactions taking nanoseconds in systems with hundreds of quantum atoms can be simulated, bridging the gap between non-adiabatic ab initio simulations and model approaches such as the Marcus theory. A major recent application is hole transfer in DNA, which represents an archetypal ET reaction in a polarizable medium. Ongoing work focuses on hole transfer in proteins, peptides and organic semi-conductors.

Efficient algorithms for the simulation of non-adiabatic electron transfer in complex molecular systems: application to DNA

In this work, a fragment-orbital density functional theory-based method is combined with two different non-adiabatic schemes for the propagation of the electronic degrees of freedom. This allows us to perform unbiased simulations of electron transfer processes in complex media, and the computational scheme is applied to the transfer of a hole in solvated DNA. It turns out that the mean-field approach, where the wave function of the hole is driven into a superposition of adiabatic states, leads to over-delocalization of the hole charge. This problem is avoided using a surface hopping scheme, resulting in a smaller rate of hole transfer. The method is highly efficient due to the on-the-fly computation of the coarse-grained DFT Hamiltonian for the nucleobases, which is coupled to the environment using a QM/MM approach. The computational efficiency and partial parallel character of the methodology make it possible to simulate electron transfer in systems of relevant biochemical size on a nanosecond time scale. Since standard non-polarizable force fields are applied in the molecular-mechanics part of the calculation, a simple scaling scheme was introduced into the electrostatic potential in order to simulate the effect of electronic polarization. It is shown that electronic polarization has an important effect on the features of charge transfer. The methodology is applied to two kinds of DNA sequences, illustrating the features of transfer along a flat energy landscape as well as over an energy barrier. The performance and relative merit of the mean-field scheme and the surface hopping for this application are discussed.

Free energy calculations on snake venom metalloproteinase BaP1

BaP1 is a snake venom metalloproteinase from the venom of Bothrops asper, showing high structural homology with the catalytic domain of human adamalysins and matrix metalloproteinases. It induces the release of cytokines, like interleukin-1 and tumor necrosis factor alpha. Recently, the high-resolution crystal structure of BaP1 with a bound inhibitor became available, representing an interesting model concerning inhibitor design for medicinally important metalloproteinases such as tumor necrosis factor alpha-converting enzyme and MMP13. We here use computational modeling to gain a better understanding about the binding properties of various ligands to BaP1, with a focus on computing ligand binding free energies. The obtained results should be of general significance for future research on medicinally important metalloproteinases. We have investigated the binding of the original inhibitor in detail and calculated its binding strength using MMP/GBSA free energy calculations. Additionally, the binding strengths of alternative ligands have been computed, and two of them are predicted and experimentally verified to strongly inhibit the enzyme. A suggestion for chemical modifications of BaP1 inhibitors could be made to guide future synthesis efforts. Furthermore, a contribution to the proteolytic reaction mechanism of metzincins is given. The pK value of the catalytically active glutamic acid residue 143 has been found to be significantly raised when compared with a free glutamate side chain. Calculations on other matrix metalloproteinases confirmed that this is not confined to BaP1, but seems to be a common feature of metzincins.

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.

Molecular dynamics simulations of porphyrin-dendrimer systems: toward modeling electron transfer in solution

We have performed computational simulations of porphyrin-dendrimer systems--a cationic porphyrin electrostatically associated to a negatively charged dendrimer--using the method of classical molecular dynamics (MD) with an atomistic force field. Previous experimental studies have shown a strong quenching effect of the porphyrin fluorescence that was assigned to electron transfer (ET) from the dendrimer's tertiary amines (Paulo, P. M. R.; Costa, S. M. B. J. Phys. Chem. B 2005, 109, 13928). In the present contribution, we evaluate computationally the role of the porphyrin-dendrimer conformation in the development of a statistical distribution of ET rates through its dependence on the donor-acceptor distance. We started from simulations without explicit solvent to obtain trajectories of the donor-acceptor distance and the respective time-averaged distributions for two dendrimer sizes and different initial configurations of the porphyrin-dendrimer pair. By introducing explicit solvent (water) in our simulations, we were able to estimate the reorganization energy of the medium for the systems with the dendrimer of smaller size. The values obtained are in the range 0.6-1.5 eV and show a linear dependence with the inverse of the donor-acceptor distance, which can be explained by a two-phase dielectric continuum model taking into account the medium heterogeneity provided by the dendrimer organic core. Dielectric relaxation accompanying ET was evaluated from the simulations with explicit solvent showing fast decay times of some tens of femtoseconds and slow decay times in the range of hundreds of femtoseconds to a few picoseconds. The variations of the slow relaxation times reflect the heterogeneity of the dendrimer donor sites which add to the complexity of ET kinetics as inferred from the experimental fluorescence decays.

Structural model and excitonic properties of the dimeric RC-LH1-PufX complex from Rhodobacter sphaeroides

The light-harvesting apparatus of the purple bacterial photosynthetic unit consists of a pool of peripheral light-harvesting complexes that transfer excitation energy to a reaction center (RC) via the surrounding pigment-protein complex LH1. Recent electron microscopy and atomic force microscopy studies have revealed that RC-LH1 units of Rhodobacter sphaeroides form membrane-bending dimeric complexes together with the polypeptide PufX. We present a structural model for these RC-LH1-PufX dimeric complexes constructed using the molecular dynamics flexible fitting method based on an EM density map. The arrangement of the LH1 BChls displays a distortion near the proposed location of the PufX polypeptide. The resulting atomic model for BChl arrays is used to compute the excitonic properties of the dimeric RC-LH1 complex. A comparison is presented between the structural and excitonic features of the S-shaped dimeric BChl array of Rhodobacter sphaeroides and the circular BChl arrangement found in other purple bacteria.

Theoretical study of ultrafast heterogeneous electron transfer reactions at dye-semiconductor interfaces: coumarin 343 at titanium oxide

A theoretical study of photoinduced heterogeneous electron transfer in the dye-semiconductor system coumarin 343-TiO(2) is presented. The study is based on a generic model for heterogeneous electron transfer reactions, which takes into account the coupling of the electronic states to the nuclear degrees of freedom of coumarin 343 as well as to the surrounding solvent. The quantum dynamics of the electron injection process is simulated employing the recently proposed multilayer formulation of the multiconfiguration time-dependent Hartree method. The results reveal an ultrafast injection dynamics of the electron from the photoexcited donor state into the conduction band of the semiconductor. Furthermore, the mutual influence of electronic injection dynamics and nuclear motion is analyzed in some detail. The analysis shows that--depending on the time scale of nuclear motion--electronic vibrational coupling can result in electron transfer driven by coherent vibrational motion or vibrational motion induced by ultrafast electron transfer.

Theoretical prediction of spectral and optical properties of bacteriochlorophylls in thermally disordered LH2 antenna complexes

A general approach for calculating spectral and optical properties of pigment-protein complexes of known atomic structure is presented. The method, that combines molecular dynamics simulations, quantum chemistry calculations, and statistical mechanical modeling, is demonstrated by calculating the absorption and circular dichroism spectra of the B800-B850 bacteriochlorophylls of the LH2 antenna complex from Rs. molischianum at room temperature. The calculated spectra are found to be in good agreement with the available experimental results. The calculations reveal that the broadening of the B800 band is mainly caused by the interactions with the polar protein environment, while the broadening of the B850 band is due to the excitonic interactions. Since it contains no fitting parameters, in principle, the proposed method can be used to predict optical spectra of arbitrary pigment-protein complexes of known structure.

A Molecular Dynamics Study of Lys-Trp-Lys: Structure and Dynamics in Solution Following Photoexcitation

We report studies of the structure and dynamics of a tripeptide Lys-Trp-Lys (KWK) in aqueous solution following photoexcitation by molecular dynamics simulations. For ground-state KWK, we observe three stable conformations with free energy differences of less than 5.2 kJ/mol. Each conformer is stabilized by a pi-cation interaction between one of three protonated amino groups and the indole moiety. For the excited state of tryptophan in KWK, the simulated molecular dynamics of the three isomers are similar, all in good agreement with recent femtosecond experiments (J. Phys. Chem. B 2005, 109, 16901). Specifically, we observe: (1) the fluorescence anisotropy is dominated by a single-exponential component and decays in approximately 130 ps, (2) the total dynamic Stokes shift reaches approximately 2700 cm(-1), and (3) the excited state relaxation dynamics occurs on several time scales ranging from femtoseconds to tens of picoseconds. The relaxation dynamics involve rapid initial response of neighboring water, followed by local motions of flexible peptide chains. These processes drive global restructuring of the tripeptide on a rather flat energy surface, inducing slower dynamics evident in both the water and protein contributions to the stabilization energy of the photoexcited chromophore. The water and protein dynamics are strongly correlated. On a still longer time scale, we observe isomerization of two excited state conformers to the other most stable one, an analogue for evolution of trajectories along the funnel on the rugged free energy landscape to the final "native" state. Our studies suggest new experiments to detect this unique dynamics.

The First Picoseconds in Bacterial Photosynthesis?Ultrafast Electron Transfer for the Efficient Conversion of Light Energy

In this Minireview, we describe the function of the bacterial reaction centre (RC) as the central photosynthetic energy-conversion unit by ultrafast spectroscopy combined with structural analysis, site-directed mutagenesis, pigment exchange and theoretical modelling. We show that primary energy conversion is a stepwise process in which an electron is transferred via neighbouring chromophores of the RC. A well-defined chromophore arrangement in a rigid protein matrix, combined with optimised energetics of the different electron carriers, allows a highly efficient charge-separation process. The individual molecular reactions at room temperature are well described by conventional electron-transfer theory.

Charge recombination and protein dynamics in bacterial photosynthetic reaction centers entrapped in a sol-gel matrix

Many proteins can be immobilized in silica hydrogel matrices without compromising their function, making this a suitable technique for biosensor applications. Immobilization will in general affect protein structure and dynamics. To study these effects, we have measured the P(+)Q(A)(-) charge recombination kinetics after laser excitation of Q(B)-depleted wild-type photosynthetic reaction centers from Rhodobacter sphaeroides in a tetramethoxysilane (TMOS) sol-gel matrix and, for comparison, also in cryosolvent. The nonexponential electron transfer kinetics observed between 10 and 300 K were analyzed quantitatively using the spin boson model for the intrinsic temperature dependence of the electron transfer and an adiabatic change of the energy gap and electronic coupling caused by protein motions in response to the altered charge distributions. The analysis reveals similarities and differences in the TMOS-matrix and bulk-solvent samples. In both preparations, electron transfer is coupled to the same spectrum of low frequency phonons. As in bulk solvent, charge-solvating protein motions are present in the TMOS matrix. Large-scale conformational changes are arrested in the hydrogel, as evident from the nonexponential kinetics even at room temperature. The altered dynamics is likely responsible for the observed changes in the electronic coupling matrix element.

Mobile polaron solutions and nonlinear electron transfer in helical protein models

We consider the electron transfer along helical forms of proteins. The spatial structure of the protein helices is modeled by three-dimensional oscillator networks whose constituents represent peptide groups. Covalent and hydrogen bonds between the peptide units are modeled by point-point interaction potentials. The electronic degree of freedom is described by a tight-binding system including besides the nearest-neighbor exchange interactions between covalently connected units also third- or fourth-nearest neighbor interactions between hydrogen-bonded sites. In addition each peptide unit possesses an internal vibrational degree of freedom. The various dynamical degrees of freedom are coupled to each other making the exchange of electronic, intramolecular, and bond-vibrational energy possible. In the first part of the paper we investigate the static polaron formation resulting from strong interactions between the electron and the intramolecular vibrations. The 3-10 helix and the alpha helix are investigated. Polaron states are constructed analytically on the basis of a variational approach. Compared to the alpha helix the 3-10 helix supports stronger localized polarons. In the second part of the paper we take the coupling of the polaron with the vibrations of the three-dimensional protein matrix into account focusing interest on the bond-assisted initiation of polaron motion. In detail it is demonstrated that the interplay of the protein matrix and the polaron dynamics conspire to activate not only the polaron motion but also to maintain a long-lived coherently traveling localized pattern along the lattice of peptide units. Starting from a nonequilibrium state it is shown that coexisting electron and bond-vibration breathers assist the relaxation dynamics towards energy equilibration and the attainment of a stationary regime.

Effects of medium polarization and pre-existing field on activation energy of enzymatic charge-transfer reactions

The highly organized spatial structure of proteins' polar groups results in the existence of a permanent intraprotein electric field and in protein's weak dielectric response, i.e. its low dielectric constant. The first factor affects equilibrium free energy gap of a charge-transfer reaction, the second (medium polarization effect) influences both equilibrium and non-equilibrium (reorganization) energies, decreasing the latter substantially. In the framework of the rigorous 'fixed-charge-density' formalism, the medium polarization component of the reaction activation energy has been calculated, both for the activation energy of the elementary act proper, and the effective activation energy accounting for the charges' transfer from water into a low-dielectric structureless medium. In all typical cases of reactions, the energy spent for charge transfer from water into structureless 'protein' is larger than the gain in activation energy due to the protein's low reorganization energy. Therefore, the low dielectric constant of proteins is not sufficient to ensure their high catalytic activity, and an additional effect of the pre-existing intraprotein electric field, compensating for an excessive charging energy, is necessary. Only a combined action of low reorganization energy and pre-existing electric field provides proteins with their high catalytic activity. The dependence of activation energy on the globule geometry has been analyzed. It is shown that, for each reaction, an optimum set of geometric parameters exists. For five hydrolytic enzymes, the optimum globule radii have been calculated using the experimental geometry of their active sites. The calculated radii agree satisfactorily with the real sizes of these macromolecules, both by absolute and by relative values.