Coupling of the overall molecular motion with the conformational transitions. II. The full rotational problem

The method proposed in the companion paper for analysing the coupling between overall and internal dynamics is applied to the study of the full rotational motion of a molecule with one internal degree of freedom. For systems characterized by a finite set of stable conformers determined by the minima of the intramolecular potential, a simplified time evolution operator of mixed type is derived, with the continuous diffusion equation and the generalized random walk operator representing the overall rotation and the internal dynamics, respectively. The dependence on the conformational state of the rotational diffusion tensor is one source of coupling between these two types of motion. Another source is represented by the recoil rotations acting on each subunit during a conformational transition. Both conformational-dependent rotational diffusion tensors and recoil rotations can be calculated from a model for the friction exerted by the solvent. Some applications of the theory are presented in relation to the butane molecule and the molecules having the structure of biphenyl, with particular emphasis on the calculation of the experimental observables in NMR and dielectric relaxation measurements.     

Computer simulation of polymeric materials. I. Stress-strain behavior

A model of polymeric materials has been developed that includes many of the features of condensed-phase polymer chain dynamics, central among them chain relaxation via conformational motion. The model consists of a number of chains of particles that are connected by bonds with multiwelled potentials to approximate the energetics of conformational motion. Interactions between particles on adjacent chains are modeled by short range repulsive potentials. We have examined the stress-strain behavior of the model using molecular dynamics simulations and find qualitative agreement with the observed experimental behavior of polymeric materials.     

Non-exponential non-Arrhenius relaxation in the course of CO rebinding to heme proteins

The dispersive transport model for relaxation of photolyzed heme proteins has been improved to take into account the coupling of the ligand-heme geminate recombination and the non-Gaussian diffusive dynamics of conformational changes in heme proteins. Contrary to the earlier deterministic version of the model, the present more rigorous formulation is based on the stochastic approach to the problem. This implies that the time evolution of protein conformations should be described in terms of the transient distribution which satisfies the Smoluchowski-type differential equation with a time-dependent diffusion coefficient. The obtained analytical solution of this equation enables us to relate main kinetic parameters of the geminate recombination and quantities characterizing the ligand-heme interaction. The derived expressions demonstrate that the reaction barrier shifts with time towards higher values following the near-stretched exponential behavior in agreement with experiment. Such a behavior is governed by the non-exponential non-Arrhenius conformational relaxation. The latter process can be identified by the characteristics “footprint” left on the experimental rebinding curve and is shown to be responsible for some kinetically different phases of the ligand-heme geminate recombination observed within distinct temperature ranges.     

Rigorous coarse-graining for the dynamics of linear systems with applications to relaxation dynamics in proteins

Reduced-dimensionality, coarse-grained models are commonly employed to describe the structure and dynamics of large molecular systems. In those models, the dynamics is often described by Langevin equations of motion with phenomenological parameters. This paper presents a rigorous coarse-graining method for the dynamics of linear systems. In this method, as usual, the conformational space of the original atomistic system is divided into master and slave degrees of freedom. Under the assumption that the characteristic timescales of the masters are slower than those of the slaves, the method results in Langevin-type equations of motion governed by an effective potential of mean force. In addition, coarse-graining introduces hydrodynamic-like coupling among the masters as well as non-trivial inertial effects. Application of our method to the long-timescale part of the relaxation spectra of proteins shows that such dynamic coupling is essential for reproducing their relaxation rates and modes.     

Theory of molecular motions in flexible nematogens

Liquid crystal phases are typically formed by molecules having several degrees of internal freedom. These systems exhibit, therefore, complex dynamics, with internal motions superimposed on the rotational diffusion of the whole molecule. The problem of the internal transitions has been treated in terms of a master equation for jumps between configurational sites, derived by projecting the multidimensional diffusion equation for the torsional variables on a suitable set of site functions. The coupling with the overall diffusion has been taken into account explicitly, by considering the conformational dependence of both the mean field torque and the molecular diffusion tensor. A Marcelja-like potential acting on the various molecular moieties has been used, and the frictional effects have been calculated for the different chain conformations. In this way, the rates for the internal transitions are orientation dependent, and the solution of the diffusional problem requires a matrix representation in the full space of angular and site functions. The nematogen 4-n-pentyl-4'-cyanobiphenyl, for which a large amount of experimental data is available from detailed NMR relaxation measurements, is taken as a reference system. The spectral densities of the relevant correlation functions for the deuterons in the various positions of the molecule have been calculated, for different degrees of ordering and different choices of the energetic and hydrodynamic parameters.     

Polaronic quantum master equation theory of inelastic and coherent resonance energy transfer for soft systems

This work extends the theory of coherent resonance energy transfer [S. Jang, J. Chem. Phys. 131, 164101 (2009)] by including quantum mechanical inelastic effects due to modulation of donor-acceptor electronic coupling. Within the approach of the second order time local quantum master equation (QME) in the polaron picture and under the assumption that the bath degrees of freedom modulating the electronic coupling are independent of other modes, a general time evolution equation for the reduced system density operator is derived. Detailed expressions for the relaxation operators and inhomogeneous terms of the QME are then derived for three specific models of modulation in distance, axial angle, and dihedral angle, which are all approximated by harmonic oscillators. Numerical tests are conducted for a set of model parameters. Model calculation shows that the torsional modulation can make significant contribution to the relaxation and dephasing mechanisms.     

Study of internal ring rotation and molecular reorientation in the liquid crystal 5O.7 by deuterium N.M.R. spectroscopy

The spectral densities of motion were determined by deuterium N.M.R. relaxation measurements in the nematic, smectic A and smectic C phases of 4-n-pentyloxybenzylidene-d₁-4'-heptylaniline and 4-n-pentyloxybenzylidene-4'-heptylaniline-2,3,5,6-d₄. By examining two atomic sites on a 5O.7 molecule, we were able to gain information on the reorientation motion and internal rotation of the aniline ring. It was also found that director fluctuations make some contribution to the spectral density J₁ (ω). We use the superimposed rotations model to account for the internal ring motion and the small step rotational diffusion model for the molecular reorientation. The derived rotational diffusion constants for the spinning and tumbling motions appear to give physically plausible activation energies in the mesophases of 5O.7.     

Simulation of the brownian motion of macromolecular chains. II. Chain dynamics

A simulation of the Brownian motion of polyethers and polyethylene is reported. The chains under consideration are confined to a tetrahedral lattice and subjected to conformational energy conditions. The dynamic behavior of the whole chain is studied. The diffusion of individual chain atoms reflects the chemical nature of the chain, and the diffusion of the center of mass, in connection with experimental results, affords a time scale for the simulation. The relaxation of bond orientation, which is connected with fluorescence depolarization, is found to be consistent with the theory of Valeur and co-workers. The relaxation of the total dipole is interpreted in terms of conformational features of the motions and correlations between neighboring dipoles. Finally, relations between chemical structure and dynamic behavior are established. Three classes of polyether chains are to be distinguished: the rigid chain poly(methylene oxide), the highly flexible poly(ethylene oxide) and an intermediate class typified by polyethylene.     

Model for proton transport coupled to protein conformational change: application to proton pumping in the bacteriorhodopsin photocycle

A modeling method is presented for protein systems in which proton transport is coupled to conformational change, as in proton pumps and in motors driven by the proton-motive force. Previously developed methods for calculating pKa values in proteins using a macroscopic dielectric model are extended beyond the equilibrium case to a master-equation model for the time evolution of the system through states defined by ionization microstate and a discrete set of conformers. The macroscopic dielectric model supplies free energy changes for changes of protonation microstate, while the method for obtaining the energetics of conformational change and the relaxation rates, the other ingredients needed for the master equation, are system dependent. The method is applied to the photoactivated proton pump, bacteriorhodopsin, using conformational free energy differences from experiment and treating relaxation rates through three adjustable parameters. The model is found to pump protons with an efficiency relatively insensitive to parameter choice over a wide range of parameter values, and most of the main features of the known photocycle from very early M to the return to the resting state are reproduced. The boundaries of these parameter ranges are such that short-range proton transfers are faster than longer-range ones, which in turn are faster than conformational changes. No relaxation rates depend on conformation. The results suggest that an "accessibility switch", while not ruled out, is not required and that vectorial proton transport can be achieved through the coupling of the energetics of ionization and conformational states.     

On the internal rotations in p-cresol in its ground and first electronically excited states

The overall rotation and internal rotation of p-cresol (4-methyl-phenol) has been studied by comparison of the microwave spectrum with accurate ab initio calculations using the principal axis method in the electronic ground state. Both internal rotations, the torsions of the methyl and the hydroxyl groups relative to the aromatic ring, have been investigated. The internal rotation of the hydroxyl group can be approximately described as the motion of a symmetrical rotor on an asymmetric frame. For the methyl group it has been found that the potential barrier hindering its internal rotation is very small with the first two nonvanishing Fourier coefficients of the potential V(3) and V(6) in the same order of magnitude. Different splittings of b-type transitions for the A and E species of the methyl torsion indicate a top-top interaction between both internal rotors through the benzene ring. An effective coupling potential for the top-top interaction could be estimated. The hindering barriers of the hydroxyl and methyl rotation have been calculated using second-order Moller-Plesset perturbation theory and the approximate coupled-cluster singles-and-doubles model (CC2) in the ground state and using CC2 and the algebraic diagrammatic construction through second order in the first electronically excited state. The results are in excellent agreement with the experimental values.     

The centrifugally decoupled exponential distorted wave (CDEDW) approximation for the calculation of rotationally inelastic molecular collision cross se

A new approximate method is presented for the rapid calculation of rotationally inelastic molecular collision cross sections. The method is called the centrifugally decoupled exponential distorted wave (CDEDW) approximation and involves the combination of two well known approximations. The first approximation is the neglect of the off-diagonal coupling terms which arise from the orbital angular momentum operator in the coupled differential equations in the body-fixed axis system. The second approximation is to treat the remaining coupling terms, which arise from the interaction potential, using a unitary perturbation approximation. The CDEDW method is applied to the calculation of total and partial rotationally inelastic cross sections in the ArN₂ system, and detailed comparisons are made with exact and several other types of approximate calculations. Agreement with exact calculations is good and often comparable with the coupled states and p-helicity decoupled approximations. The CDEDW method requires a similar amount of computational effort to the infinite order sudden (IOS) approximation, and we show that for the present system the CDEDW method gives more reliable results.     

Electron transfer between cofactors in protein domains linked by a flexible tether

Some key electron-transfer (ET) proteins have domains containing redox cofactors connected by a flexible tether. The relative motion of the domains is essential for reaction because of the strong ET rate dependence on distance. The constrained conformational flexibility produces a kinetic regime intermediate between “unimolecular” and “bimolecular”. We used a simple model for ET coupled to conformational diffusion to explore the structure dependence of the ET kinetics. The model describes the evolution of an initially prepared reduced donor state, and recovers the diffusion and electron tunneling-limited regimes. The restriction of the conformational space imposed by the tether introduces an entropic component to the effective donor–acceptor interaction potential. As such, the tether length may control the transition between the electron tunneling and diffusion-limited regimes.     

反转区电子转移速率的理论研究

利用级联方程方法研究了反转区的电子转移速率与电子态耦合常数以及溶剂弛豫时间的关系.结果表随着电子态间耦合强度的增加,反应速率先增加然后减小,而随着溶剂弛豫时间的减小反应速率增加.我们将这些结果与近似理论费米黄金规则、Landau-Zener公式进行了比较,由于费米金规则基于一阶微扰论,所以只适用于耦合常数较小的情况.而Landau-Zener公式假设电子在跃迁区域弹道运动,在电子态耦合大的情况下也存在问题.     

Molecular simulations of aqueous electrolyte solubility: 1. The expanded-ensemble osmotic molecular dynamics method for the solution phase

We have developed a molecular-level simulation technique called the expanded-ensemble osmotic molecular dynamics (EEOMD) method, for studying electrolyte solution systems. The EEOMD method performs simulations at a fixed number of solvent molecules, pressure, temperature, and overall electrolyte chemical potential. The method combines elements of constant pressure-constant temperature molecular dynamics and expanded-ensemble grand canonical Monte Carlo. The simulated electrolyte solution systems contain, in addition to solvent molecules, full and fractional ions and undissociated electrolyte molecular units. The fractional particles are coupled to the system via a coupling parameter that varies between 0 (no interaction between the fractional particle and the other particles in the system) and 1 (full interaction between the fractional particle and the other particles in the system). The time evolution of the system is governed by the constant pressure-constant temperature equations of motion and accompanied by random changes in the coupling parameter. The coupling-parameter changes are accepted with a probability derived from the expanded-ensemble osmotic partition function corresponding to the prescribed electrolyte chemical potential. The coupling-parameter changes mimic insertion/deletion of particles as in a crude grand canonical Monte Carlo simulation; if the coupling parameter becomes 0, the fractional particles disappear from the system, and as the coupling parameter reaches unity, the fractional particles become full particles. The method is demonstrated for a model of NaCl in water at ambient conditions. To test our approach, we first determine the chemical potential of NaCl in water by the thermodynamic integration technique and by the expanded-ensemble method. Then, we carry out EEOMD simulations for different specified values of the overall NaCl chemical potential and measure the concentration of ions resulting from the simulations. Both computations give consistent results, validating the EEOMD methodology.     

Study of internal ring rotation and molecular reorientation in the liquid crystal 5O.7 by deuterium N.M.R. spectroscopy

Abstract

The spectral densities of motion were determined by deuterium N.M.R. relaxation measurements in the nematic, smectic A and smectic C phases of 4-n-pentyloxybenzylidene-d ₁-4′-heptylaniline and 4-n-pentyloxybenzylidene-4′-heptylaniline-2,3,5,6-d ₄. By examining two atomic sites on a 5O.7 molecule, we were able to gain information on the reorientation motion and internal rotation of the aniline ring. It was also found that director fluctuations make some contribution to the spectral density J ₁ (ω). We use the superimposed rotations model to account for the internal ring motion and the small step rotational diffusion model for the molecular reorientation. The derived rotational diffusion constants for the spinning and tumbling motions appear to give physically plausible activation energies in the mesophases of 5O.7.     

A structural mode-coupling approach to 15N NMR relaxation in proteins.

The two-body Slowly Relaxing Local Structure (SRLS) model was applied to (15)N NMR spin relaxation in proteins and compared with the commonly used original and extended model-free (MF) approaches. In MF, the dynamic modes are assumed to be decoupled, local ordering at the N-H sites is represented by generalized order parameters, and internal motions are described by effective correlation times. SRLS accounts for dynamical coupling between the global diffusion of the protein and the internal motion of the N-H bond vector. The local ordering associated with the coupling potential and the internal N-H diffusion are tensors with orientations that may be tilted relative to the global diffusion and magnetic frames. SRLS generates spectral density functions that differ from the MF formulas. The MF spectral densities can be regarded as limiting cases of the SRLS spectral density. SRLS-based model-fitting and model-selection schemes similar to the currently used MF-based ones were devised, and a correspondence between analogous SRLS and model-free parameters was established. It was found that experimental NMR data are sensitive to the presence of mixed modes. Our results showed that MF can significantly overestimate order parameters and underestimate local motion correlation times in proteins. The extent of these digressions in the derived microdynamic parameters is estimated in the various parameter ranges, and correlated with the time scale separation between local and global motions. The SRLS-based analysis was tested extensively on (15)N relaxation data from several isotropically tumbling proteins. The results of SRLS-based fitting are illustrated with RNase H from E. coli, a protein extensively studied previously with MF.     

Dynamics of macromolecular chains. VI. Carbon 13 and proton nuclear magnetic relaxation of polystyrene in solution

Spin-lattice ¹H and ¹³C nuclear magnetic relaxation (NMR) times T₁ have been measured for solutions of polystyrene in hexachlorobutadiene at two different frequencies. Some nuclear Overhauser enhancements and linewidths have also been determined. At 15 and 25 MHz the relaxation times T₁ of the ortho and meta carbons show two different dependences on temperature. These measurements indicate internal motion of phenyl groups around the C_α—C_para axis. A single isotropic correlation time is inadequate to explain the relaxation data for the para carbon. Use of a diamond-lattice motional model reveals that segmental reorientation of the chain backbone of polystyrene can be described in terms of two correlation times, ρ characterizing the three-bond motion process, and θ reflecting either isotropic motions of subchains or departure from an ideal lattice. Data on low-molecular-weight polystyrene indicate the participation of overall rotatory diffusion in the relaxation process. This motion is no longer efficient in high-molecular-weight polymers, where relaxation is due to segmental reorientation.