Simulating Vibrational Energy Flow in Proteins: Relaxation Rate and Mechanism for Heme Cooling in Cytochrome c

Thermal Energy Transport through Nonbonded Native Contacts in Protein

Within the protein interior, where we observe various types of interactions, nonuniform flow of thermal energy occurs along the polypeptide chain and through nonbonded native contacts, leading to inhomogeneous transport efficiencies from one site to another. The folded native protein serves not merely as thermal transfer medium but, more importantly, as sophisticated molecular nanomachines in cells. Therefore, we are particularly interested in what sort of "communication" is mediated through native contacts in the folded proteins and how such features are quantitatively depicted in terms of local transport coefficients of heat currents. To address the issue, we introduced a concept of inter-residue thermal conductivity and characterized the nonuniform thermal transport properties of a small globular protein, HP36, using equilibrium molecular dynamics simulation and the Green-Kubo formula. We observed that the thermal transport of the protein was dominated by that along the polypeptide chain, while the local thermal conductivity of nonbonded native contacts decreased in the order of H-bonding > π-stacking > electrostatic > hydrophobic contacts. Furthermore, we applied machine learning techniques to analyze the molecular mechanism of protein thermal transport. As a result, the contact distance, variance in contact distance, and H-bonding occurrence probability during MD simulations are found to be the top three important determinants for predicting local thermal transport coefficients.

Vibrational Energy Landscapes and Energy Flow in GPCRs

We construct and analyze disconnectivity graphs to provide the first graphical representation of the vibrational energy landscape of a protein, in this study β2AR, a G-protein coupled receptor (GPCR), in active and inactive states. The graphs, which indicate the relative free energy of each residue and the minimum free energy barriers for energy transfer between them, reveal important composition, structural and dynamic properties that mediate the flow of energy. Prolines and glycines, which contribute to GPCR plasticity and function, are identified as bottlenecks to energy transport along the backbone from which alternative pathways for energy transport via nearby noncovalent contacts emerge, seen also in the analysis of first passage time (FPT) distributions presented here. Striking differences between the disconnectivity graphs and FPT distributions for the inactive and active states of β2AR are found where structural and dynamic changes occur upon activation, contributing to allosteric regulation.

Origin of Protein Quake: Energy Waves Conducted by a Precise Mechanical Machine

A long-standing challenge in protein biophysics is to understand protein quake in myoglobin─the structural dynamics responsible for redistributing the excess heme energy after photolysis. Despite extensive efforts, the molecular mechanism of this process remains elusive. Using the energy flow theory, we uncovered a fundamental new phenomenon: the heme energy is redistributed by sinusoidal waves with a ubiquitous fundamental frequency and two overtones. The energy waves emanate from the heme into the myoglobin backbone via a conduit of five consecutive dihedrals of the proximal histidine and then travel quickly along the backbone to reach sidechains across the protein. This mechanism is far more effective than the diffusion-based mechanism from previous studies because waves are systematic while diffusion is random. To propagate energy waves, coordinates must cooperate, resulting in collective modes that are singular vectors of the generalized work functional. These modes show task partitioning: a handful of high-energy modes generate large-scale breathing motion, which loosens up the protein matrix to enable hundreds of low-energy vibrational modes for energy transduction.

Computational Study on the Thermal Conductivity of a Protein

Protein molecules are thermally fluctuating and tightly packed amino acid residues strongly interact with each other. Such interactions are characterized in terms of heat current at the atomic level. We calculated the thermal conductivity of a small globular protein, villin headpiece subdomain, based on the linear response theory using equilibrium molecular dynamics simulation. The value of its thermal conductivity was 0.3 ± 0.01 [W m^-1 K^-1], which is in good agreement with experimental and computational studies on the other proteins in the literature. Heat current along the main chain was dominated by local vibrations in the polypeptide bonds, with amide I, II, III, and A bands on the Fourier transform of the heat current autocorrelation function.

Origin of thiocyanate spectral shifts in water and organic solvents

Vibrational spectroscopy is a useful technique for probing chemical environments. The development of models that can reproduce the spectra of nitriles and azides is valuable because these probes are uniquely suited for investigating complex systems. Empirical vibrational spectroscopic maps are commonly employed to obtain the instantaneous vibrational frequencies during molecular dynamics simulations but often fail to adequately describe the behavior of these probes, especially in its transferability to a diverse range of environments. In this paper, we demonstrate several reasons for the difficulty in constructing a general-purpose vibrational map for methyl thiocyanate (MeSCN), a model for cyanylated biological probes. In particular, we found that electrostatics alone are not a sufficient metric to categorize the environments of different solvents, and the dominant features in intermolecular interactions in the energy landscape vary from solvent to solvent. Consequently, common vibrational mapping schemes do not cover all essential interaction terms adequately, especially in the treatment of van der Waals interactions. Quantum vibrational perturbation (QVP) theory, along with a combined quantum mechanical and molecular mechanical potential for solute-solvent interactions, is an alternative and efficient modeling technique, which is compared in this paper, to yield spectroscopic results in good agreement with experimental FTIR. QVP has been used to analyze the computational data, revealing the shortcomings of the vibrational maps for MeSCN in different solvents. The results indicate that insights from QVP analysis can be used to enhance the transferability of vibrational maps in future studies.

Energy Transfer across Nonpolar and Polar Contacts in Proteins: Role of Contact Fluctuations

Molecular dynamics simulations of the villin headpiece subdomain HP36 have been carried out to examine relations between rates of vibrational energy transfer across non-covalently bonded contacts and equilibrium structural fluctuations, with focus on van der Waals contacts. Rates of energy transfer across van der Waals contacts vary inversely with the variance of the contact length, with the same constant of proportionality for all nonpolar contacts of HP36. A similar relation is observed for hydrogen bonds, but the proportionality depends on contact pairs, with hydrogen bonds stabilizing the α-helices all exhibiting the same constant of proportionality, one that is distinct from those computed for other polar contacts. Rates of energy transfer across van der Waals contacts are found to be up to 2 orders of magnitude smaller than rates of energy transfer across polar contacts.

Primary Electronic and Vibrational Dynamics of Cytochrome c Observed by Sub-10 fs NUV Laser Pulses

The primary reaction mechanism of cytochrome c (Cyt c) was elucidated for two redox forms of ferric (oxidized) and ferrous (reduced) Cyt c by measuring their transient absorption (TA) spectra using a homemade sub-10 fs broadband NUV laser pulses system. The TA traces measured in the broad probe wavelength region were analyzed by the global analysis method to study the electronic dynamics. The difference of relaxation dynamics dependent on the excitation bandwidth enabled us to elucidate that the 2.5 ps component in ferrous Cyt c can be assigned to intramolecular vibration energy redistribution and not to vibrational cooling, which was not clear until this work. The temporal resolution of 10 fs observes TA signal modulation caused by the molecular vibration in the time domain, which can be used to calculate the instantaneous frequency of the molecular vibration mode. The observed vibrational dynamics has visualized that the heme structure changes in 0.8 ps for ferric Cyt c and in >1.0 ps for ferrous Cyt c. These estimated lifetimes of vibrational dynamics reflect vibrational relaxation in the ground state of ferric Cyt c and electronic transition from the S2 state to the S1 state in ferrous Cyt c, respectively.

Energy flow and intersubunit signalling in GSAM: A non-equilibrium molecular dynamics study

Non-equilibrium molecular dynamics simulations of vibrational energy flow induced by the imposition of a thermal gradient have been performed on the μ2-dimeric enzyme glutamate-1-semialdehyde aminomutase (GSAM), the key enzyme in the biosynthesis of chlorophyll, in order to identify energy transport pathways and to elucidate their role as potential allosteric communication networks for coordinating functional dynamics, specifically the negative cooperativity observed in the motion of the two active site gating loops. Fully atomistic MD simulations of thermal diffusion were executed with a GROMACS simulation package on a fully solvated GSAM enzyme by heating various active site target ligands (initially, catalytic intermediates and cofactors) to 300K while holding the remainder of the protein and the solvent bath at 10K and monitoring the temperature T(t) of all the enzyme residues as a function of time over a 1ns observation window. Energy is observed to be deposited in a relatively small number of discrete chains of residues most of which contribute to specific structural or biochemical functionality. Thermal linkages between all thermally active chains were established by isolating a specific pair of chains and performing a thermal diffusion simulation on the pair, one held at 300K and the other at 10K, with the rest of the protein frozen in its initial atomic configuration and thus thermally unresponsive. Proceeding in this way, it was possible to map out multiple pathways of vibrational energy flow leading from one of the active sites through a network of contiguous residues, many of which were evolutionarily conserved and linked by hydrogen bonds, into the other active site and ultimately to the other gating loop.

Ultrafast Dynamics and Vibrational Relaxation in Six-Coordinate Heme Proteins Revealed by Femtosecond Stimulated Raman Spectroscopy

Identifying the structural rearrangements during photoinduced reactions is a fundamental challenge for understanding from a microscopic perspective the dynamics underlying the functional mechanisms of heme proteins. Here, femtosecond stimulated Raman spectroscopy is applied to follow the ultrafast evolution of two different proteins, each bearing a six-coordinate heme with two amino acid axial ligands. By exploiting the sensitivity of Raman spectra to the structural configuration, we investigate the effects of photolysis and the binding of amino acid residues in cytochrome c and neuroglobin. By comparing the system response for different time delays and Raman pump resonances, we show how detailed properties of atomic motions and energy redistribution can be unveiled. In particular, we demonstrate substantially faster energy flow from the dissociated heme to the protein moiety in cytochrome c, which we assign to the presence of covalent heme–protein bonds.

Mechanism of Solvent Control of Protein Dynamics

We find that the coupled interactions between protein and water polarization fluctuations play a dominant role in driving the configuration space random walk of solvated proteins. We perform atomistic molecular dynamics simulations on five proteins. Owing to a very low dielectric constant of protein, its dipolar groups experience forces from water along with local forces due to protein atoms. Energy fluctuations reveal a pronounced anticorrelation between protein and water contributions. The protein energy spectrum shows bimodal 1/f noise, which can be attributed to the influence of water on the dynamics of protein.

Temperature-dependent conformational dynamics of cytochrome c: Implications in apoptosis

Heat, electric shock, and burn injuries induce apoptosis by releasing cytochrome c (cyt-c) from mitochondria and by subsequently activating the death protease, caspases-3. During apoptosis, cyt-c undergoes changes in the secondary structure that have been suggested to increase its peroxidase activity. Information about these structural changes will provide better understanding of the apoptotic mechanism. Hence, temperature-dependent conformational dynamics of cyt-c has been investigated through molecular dynamics (MD) simulations to explain the structural changes and to correlate them with its apoptotic behavior. We observe that, at lower temperatures (223, 248, and 300K), the secondary structure of cyt-c, remains stable, while at higher temperatures (323, 373, 423, and 473K), the secondary structural regions change significantly. Further, our MD results indicate that these structural changes are mainly localized on α-helices, turns, β-sheets, and important loops that were involved in the stabilization of the heme conformation. This conformational transition between specific regions of secondary structure of cyt-c directly affects the electron tunneling properties of the proteins as observed experimentally. We quantify and compare these changes and explain that the temperature plays a vital role in assuring the structural stability of cyt-c and thus its functions. Our findings from this MD study reproduce experimental results at high temperatures and provide evidence for the alteration of the heme through the disruption of the H-bonding interactions between specific regions of cyt-c, thereby enhancing its peroxidase activity which plays a crucial role in the apoptotic process.

Perturbation Approach for Computing Infrared Spectra of the Local Mode of Probe Molecules

Linear and two-dimensional infrared (IR) spectroscopy of site-specific probe molecules provides an opportunity to gain a molecular-level understanding of the local hydrogen-bonding network, conformational dynamics, and long-range electrostatic interactions in condensed-phase and biological systems. A challenge in computation is to determine the time-dependent vibrational frequencies that incorporate explicitly both nuclear quantum effects of vibrational motions and an electronic structural representation of the potential energy surface. In this paper, a nuclear quantum vibrational perturbation (QVP) method is described for efficiently determining the instantaneous vibrational frequency of a chromophore in molecular dynamics simulations. Computational efficiency is achieved through the use of (a) discrete variable representation of the vibrational wave functions, (b) a perturbation theory to evaluate the vibrational energy shifts due to solvent dynamic fluctuations, and (c) a combined QM/MM potential for the systems. It was found that first-order perturbation is sufficiently accurate, enabling time-dependent vibrational frequencies to be obtained on the fly in molecular dynamics. The QVP method is illustrated in the mode-specific linear and 2D-IR spectra of the H-Cl stretching frequency in the HCl-water clusters and the carbonyl stretching vibration of acetone in aqueous solution. To further reduce computational cost, a hybrid strategy was proposed, and it was found that the computed vibrational spectral peak position and line shape are in agreement with experimental results. In addition, it was found that anharmonicity is significant in the H-Cl stretching mode, and hydrogen-bonding interactions further enhance anharmonic effects. The present QVP method complements other computational approaches, including path integral-based molecular dynamics, and represents a major improvement over the electrostatics-based spectroscopic mapping procedures.

Water-Mediated Energy Dynamics in a Homodimeric Hemoglobin

We examine energy dynamics in the unliganded and liganded states of the homodimeric hemoglobin from Scapharca inaequivalvis (HbI), which exhibits cooperativity mediated by the cluster of water molecules at the interface upon ligand binding and dissociation. We construct and analyze a dynamic network in which nodes representing the residues, hemes, and water cluster are connected by edges that represent energy transport times, as well as a nonbonded network (NBN) indicating regions that respond rapidly to local strain within the protein via nonbonded interactions. One of the two largest NBNs includes the Lys30-Asp89 salt bridge critical for stabilizing the dimer. The other includes the hemes and surrounding residues, as well as, in the unliganded state, the cluster of water molecules between the globules. Energy transport in the protein appears to be controlled by the Lys30-Asp89 salt bridge critical for stabilizing the dimer, as well as the interface water cluster in the unliganded state. Possible connections between energy transport dynamics in response to local strain identified here and allosteric transitions in HbI are discussed.

Communication maps of vibrational energy transport through Photoactive Yellow Protein

We calculate communication maps for Photoactive Yellow Protein (PYP) from the purple phototropic eubacterium Halorhodospira halophile and use them to elucidate energy transfer pathways from the chromophore through the rest of the protein in the ground and excited state. The calculations reveal that in PYP excess energy from the chromophore flows mainly to regions of the surrounding residues that hydrogen bond to the chromophore. In addition, quantum mechanics/molecular mechanics and molecular dynamics (MD) simulations of the dielectric response of the protein and solvent environment due to charge rearrangement on the chromophore following photoexcitation are also presented, with both approaches yielding similar time constants for the response. Results of MD simulations indicate that the residues hydrogen bonding to the chromophore make the largest contribution to the response.

Multiphoton absorption of myoglobin-nitric oxide complex: relaxation by D-NEMD of a stationary state

The photodissociation and geminate recombination of nitric oxide in myoglobin, under continuous illumination, is modeled computationally. The relaxation of the photon energy into the protein matrix is also considered in a single simulation scheme that mimics a complete experimental setup. The dynamic approach to non-equilibrium molecular dynamics is used, starting from a steady state, to compute its relaxation to equilibrium. Simulations are conducted for the native form of sperm whale myoglobin and for two other mutants, V68W and L29F, illustrating a fair diversity of spatial and temporal geminate recombination processes. Energy flow to the heme and immediate protein environment provide hints to allostery. In particular, a pathway of energy flow between the heme and the FG loop is illustrated. Although the simulations were conducted for myoglobin only, the thermal fluctuations of the FG corner are in agreement with the large structural shifts of FG during the allosteric transition of tetrameric hemoglobin.

Ultrafast proteinquake dynamics in cytochrome c

We report here our systematic studies of the heme dynamics and induced protein conformational relaxations in two redox states of ferric and ferrous cytochrome c upon femtosecond excitation. With a wide range of probing wavelengths from the visible to the UV and a site-directed mutation we unambiguously determined that the protein dynamics in the two states are drastically different. For the ferrous state the heme transforms from 6-fold to 5-fold coordination with ultrafast ligand dissociation in less than 100 fs, followed by vibrational cooling within several picoseconds, but then recombining back to its original 6-fold coordination in 7 ps. Such impulsive bond breaking and late rebinding generate proteinquakes and strongly perturb the local heme site and shake global protein conformation, which were found to completely recover in 13 and 42 ps, respectively. For the ferric state the heme however maintains its 6-fold coordination. The dynamics mainly occur at the local site, including ultrafast internal conversion in hundreds of femtoseconds, vibrational cooling on the similar picosecond time scale, and complete ground-state recovery in 10 ps, and no global conformation relaxation was observed.

Direct evidence for mode-specific vibrational energy relaxation from quantum time-dependent perturbation theory. III. The nu(4) and nu(7) modes of nonplanar nickel porphyrin models

The time scales and pathways of vibrational energy relaxation (VER) of the nu(4) and nu(7) modes of three nickel porphyrin models, nickel porphine (NiP), nickel protoporphyrin IX (Ni-heme), and nickel octaethylporphyrin (NiOEP), were studied using a non-Markovian time-dependent perturbation theory at the B3LYP/6-31G(d) level. When NiP is calculated with D(4h) symmetry, it has the planar structure and the same VER properties as ferrous iron porphine (FeP). The porphine cores of both Ni-heme and NiOEP were distorted from a planar geometry, assuming a nonplanar structure, similar to that of the heme structure in cytochrome c. The VER time scales of Ni-heme are found to be similar to those predicted for a planar iron heme, but the derived pathways have distinctly different features. In particular, the strong coupling between the nu(7) mode and the overtone of the approximately 350 cm(-1) gamma(7) mode, observed for planar porphyrins, is absent in both nonplanar nickel porphyrins. Direct energy exchange between the nu(4) and nu(7) modes is not observed in NiOEP, but is found to play an essential role in the VER of the nu(4) mode in Ni-heme. The Ni-heme isopropionate groups are involved in the dominant VER pathways of both the nu(4) and nu(7) modes of Ni-heme. However, in contrast with VER pathways derived in planar iron heme, the isopropionate groups are not observed to play an essential role relative to other side chains in spatially directing the vibrational energy flow.

Diversity of solvent dependent energy transfer pathways in heme proteins

The time scales and pathways of heme cooling in both reduced cytochrome c and oxidized cytochrome c following heme photoexcitation were studied using molecular dynamics simulation. Five different solvent models, including normal water, heavy water, normal glycerol, deuterated glycerol, and a nonpolar solvent, were used in the simulation. Single exponential decay of the excess kinetic energy of the heme following photoexcitation was observed in all systems studied. The simulated time scale for heme cooling in normal water agrees with recent experimental results. In contrast to heme cooling in myoglobin, no solvent dependence was observed for the time scale for heme cooling in cytochrome c. The diversity of solvent dependence results from the different local heme environments in the two proteins. In myoglobin, it has been established that the dominant mechanism for heme cooling is direct energy transfer from the heme to the solvent. In cytochrome c, direct interaction between heme and protein residues forms the dominant energy transfer pathway. This distinction is dictated by protein topology and linked to function.

Resilience of the iron environment in heme proteins

Conformational flexibility is essential to the functional behavior of proteins. We use an effective force constant introduced by Zaccai, the resilience, to quantify this flexibility. Site-selective experimental and computational methods allow us to determine the resilience of heme protein active sites. The vibrational density of states of the heme Fe determined using nuclear resonance vibrational spectroscopy provides a direct experimental measure of the resilience of the Fe environment, which we compare quantitatively with values derived from the temperature dependence of atomic mean-squared displacements in molecular dynamics simulations. Vibrational normal modes in the THz frequency range dominate the resilience. Both experimental and computational methods find a higher resilience for cytochrome c than for myoglobin, which we attribute to the increased number of covalent links to the peptide in the former protein. For myoglobin, the resilience of the iron environment is larger than the average resilience previously determined for hydrogen sites using neutron scattering. Experimental results suggest a slightly reduced resilience for cytochrome c upon oxidation, although the change is smaller than reported in previous Mössbauer investigations on a bacterial cytochrome c, and is not reproduced by the simulations. Oxidation state also has no significant influence on the compressibility calculated for cyt c, although a slightly larger compressibility is predicted for myoglobin.

State Preparation and Excited Electronic and Vibrational Behavior in Hemes

The temporally overlapping, ultrafast electronic and vibrational dynamics of a model five-coordinate, high-spin heme in a nominally isotropic solvent environment has been studied for the first time with three complementary ultrafast techniques: transient absorption, time-resolved resonance Raman Stokes, and time-resolved resonance Raman anti-Stokes spectroscopies. Vibrational dynamics associated with an evolving ground-state species dominate the observations. Excitation into the blue side of the Soret band led to very rapid S2 --> S1 decay (sub-100 fs), followed by somewhat slower (800 fs) S1 --> S0 nonradiative decay. The initial vibrationally excited, non-Boltzmann S0 state was modeled as shifted to lower energy by 300 cm(-1) and broadened by 20%. On a approximately 10 ps time scale, the S0 state evolved into its room-temperature, thermal distribution S0 profile largely through VER. Anti-Stokes signals disappear very rapidly, indicating that the vibrational energy redistributes internally in about 1-3 ps from the initial accepting modes associated with S1 --> S0 internal conversion to the rest of the macrocycle. Comparisons of anti-Stokes mode intensities and lifetimes from TRARRS studies in which the initial excited state was prepared by ligand photolysis [Mizutani, T.; Kitagawa, T. Science 1997, 278, 443, and Chem. Rec. 2001, 1, 258] suggest that, while transient absorption studies appear to be relatively insensitive to initial preparation of the electronic excited state, the subsequent vibrational dynamics are not. Direct, time-resolved evaluation of vibrational lifetimes provides insight into fast internal conversion in hemes and the pathways of subsequent vibrational energy flow in the ground state. The overall similarity of the model heme electronic dynamics to those of biological systems may be a sign that the protein's influence upon the dynamics of the heme active site is rather subtle.

Dynamics and Kinetics of Heat Transfer at the Interface of Model Diamond {111} Nanosurfaces

A molecular dynamics simulation was performed to study the effect of an applied force on heat transfer at the interface of model diamond [111] nanosurfaces. The force was applied to a small, hot nanosurface at 800, 1000, or 1200 K brought into contact with a larger, colder nanosurface at 300 K. The relaxation of the initial nonequilibrium interfacial force occurs on a subpicosecond time scale, much shorter than that required for heat transfer. Heat transfer occurs with exponential kinetics and a rate constant that increases linearly with the interfacial force according to 7 x 10(-4) ps(-1)/nN. This rate constant only increases by at most 10% as the temperature of the hot surface is increased from 800 to 1200 K. Replacing the interfacial H-atoms on both surfaces by D atoms also has a very small effect on the heat transfer. However, if one nanosurface has H atoms on its interface and the other nanosurface's interface has D atoms, then there is a marked 25% decrease in the rate constant for heat transfer. Increasing the size of the hot surface, and, thus, the interfacial contact area, increases the rate of heat transfer but not the rate constant. For the same interfacial force, different anharmonic models for the nanosurfaces' potential energy function give the same heat transfer rate constant. The possibility of quantum effects for heat transfer across the diamond interface is considered.

Thermal transport coefficients for liquid and glassy water computed from a harmonic aqueous glass

We compute thermal transport coefficients for liquid and glassy water in terms of the vibrations of the quenched liquid. The thermal conductivity and thermal diffusivity are computed for H(2)O and D(2)O at densities from 0.93 to 1.2 g cm(-3). The computed thermal diffusivity of liquid water is in reasonable agreement with measured values and is found to increase with increasing temperature due largely to the thermal accessibility of delocalized librational modes. The influence of structure and density on the thermal conductivity of amorphous ices is investigated. The calculations reveal that density alone is unable to explain the measured thermal conductivity of amorphous ices, particularly low-density amorphous ices, for which the thermal conductivity decreases with increasing temperature near 100 K. To investigate the influence of structure on thermal transport in amorphous ices we have computed the thermal transport coefficients for low-density amorphous ices prepared in two different ways, one formed by quenching the liquid at 0.93 g cm(-3) and the other by distortion of cubic ice at the same density. The computed thermal conductivity of the latter is higher, but the structures of both forms are too disordered for the thermal conductivity to exhibit the unusual variation observed experimentally.

Vibrational energy relaxation in proteins

An overview of theories related to vibrational energy relaxation (VER) in proteins is presented. VER of a selected mode in cytochrome c is studied by using two theoretical approaches. One approach is the equilibrium simulation approach with quantum correction factors, and the other is the reduced model approach, which describes the protein as an ensemble of normal modes interacting through nonlinear coupling elements. Both methods result in similar estimates of the VER time (subpicoseconds) for a CD stretching mode in the protein at room temperature. The theoretical predictions are in accord with previous experimental data. A perspective on directions for the detailed study of time scales and mechanisms of VER in proteins is presented.