Kramers Theory Research Articles

Thermally activated escape rate for a Brownian particle in a double-well potential for all values of the dissipation

The translational Brownian motion in a (2-4) double-well potential is considered. The escape rate, the position correlation function and correlation time, and the generalized susceptibility are evaluated from the solution of the underlying Langevin equation by using the matrix-continued fraction method. The escape rate and the correlation time are compared with the Kramers theory of the escape rate of a Brownian particle from a potential well as extended by Mel'nikov and Meshkov [J. Chem. Phys. 85, 1018 (1986)]. It is shown that in the low-temperature limit, the universal Mel'nikov and Meshkov expression for the escape rate provides a good estimate of both escape rate and inverse position correlation time for all values of the dissipation including the very low damping (VLD), very high damping (VHD), and turnover regimes. Moreover, for low barriers, where the Mel'nikov and Meshkov method is not applicable, analytic equations for the correlation time in the VLD and VHD limits are derived.

The kinetics of overcoming the “entropy barrier”

The escape of a point particle from a cavity of an arbitrary configuration through a small orifice in a cavity wall is considered as a model of entropy barrier overcoming. The dynamics of the particle is determined by collisions with the wall and the dissipative action of the medium. As in reactions with overcoming energy barriers (the Kramers theory), three characteristic regimes can be identified depending on the intensity of friction: diffusion and intermediate regimes and the region of weak friction. For all of them, analytic dependences of rate constants on problem parameters were obtained. The procedure for sewing solutions together, similar to that employed in the Kramers theory, gives a unified equation for the transmission factor over the whole range of friction values. In the weak friction mode, overcoming the entropy barrier generally leads to a nonexponential kinetics.

Activated barrier crossing of macromolecules at a submicron-size entropic trap

We study the thermally activated barrier crossing by long chain molecules, initially confined to one side of an entropic trap. The entropic barrier is assumed to be of Kramers type. The barrier width is considered to be larger than the chain. The latter is in turn assumed to be long enough, so that a continuum description of the chain is applicable throughout the space. The barrier crossing rate is calculated using multidimensional Kramers theory and the functional integral method. For chains having the same total number of segments, the activation energy itself remains constant. However, the preexponential factor depends on the structure of the polymer. Polymers with the same molecular weight but having longer arms can effect larger fluctuations, thereby increasing its chance to cross the barrier. This leads to an almost exponential increase of the rate prefactor with the radius of gyration. The difference in the barrier crossing rates could be effectively exploited for the separation of molecules having architectural differences, for example, DNA of same length but different degrees of supercoiling. This is illustrated by considering star polymers. The Rouse-Ham model is used to analyze the mechanism of the barrier crossing. We show how the rate expression of the Arrhenius type is affected by the long arms of the star chain.

Thermally activated escape rate for the Brownian motion of a fixed axis rotator in an asymmetrical double-well potential for all values of the dissipation

The Kramers theory of the escape rate of a Brownian particle from a potential well as extended by Mel'nikov and Meshkov is used to evaluate the relaxation times and the dynamic susceptibility for the rotational Brownian motion of fixed axis rotators in an asymmetric double-well potential. An expression for the escape rate valid for all values of the dissipation including the very low damping (VLD), very high damping (VHD), and crossover regimes is derived. It is shown that this expression provides a good asymptotic estimate of the inverse of the smallest nonvanishing eigenvalue lambda(1) of the underlying Fokker-Planck operator calculated by using the matrix-continued fraction method. For low barriers, where the Mel'nikov and Meshkov approach is not applicable, analytic equations for the correlation time tau( parallel) of the longitudinal dipole correlation function in the VLD and VHD limits are derived and a simple extrapolating equation valid for all values of the damping is proposed.

Forced Kramers escape in single-molecule pulling experiments

The pulling-induced rupture of noncovalent bonds is studied by the overdamped Kramers theory with full account of a time-varying barrier. Mechanic pulling reduces the energy barrier and leads to loading-rate dependence of the rupture force F(u)(F(t)). Tested against Langevin dynamics, four distinct regimes are identified, including kinetic dominant, weak pulling, strong pulling, and mechanic pulling dominant. Asymptotic analyses show that F(u) approximately ln F(t) in weak pulling regime and becomes 1-(F(u)F(c)) approximately [ln(F(t))E(b)](23) in strong pulling regime. Kinetic informations such as activation energy E(b) and critical force F(c) were extracted from pulling experiments for biotin-streptavidin complex.

Targeted energy transfer between a Rotor and a Morse oscillator: A model for selective chemical dissociation

Standard Kramers theory of chemical reactions involves a coupling with a Langevin thermal bath which intrinsically forbids the possible existence of Discrete Breathers (DBs) (i.e. local modes). However, it is now known that in complex systems, that energy may focus for long time as Discrete Breathers (local mode). In very special systems, targeted energy transfer may occur subsequently to another selected site and induces an ultraselective chemical reaction operating at low temperature. The dynamics of the reaction is non brownian but highly coherent along a specific path in the phase space where the system is nearly integrable ( chemical expressway). A simple toy model illustrating this idea is reduced to a Rotor weakly coupled to a Morse oscillator (supposed to represent two specific local modes in a complex system) which are appropriately tuned for targeted energy transfer. When the Rotor is initially rotating with a frequency resonant with those of the Morse oscillator at rest, the energy of the Rotor is almost completely transferred to the Morse oscillator and induces chemical dissociation. The periodic oscillations of the Rotor and Morse oscillator remain coherent and their frequencies simultaneously vary, but always remain resonant. This process is analytically described within an integrable approximation. Numerical investigations of this model confirm that in the appropriate conditions, the particle in the Morse oscillator is indeed promptly ejected at infinity with a finite velocity (chemical dissociation) despite some chaotic transient manifesting imperfect integrability.

Determination of Barrier Heights and Prefactors from Protein Folding Rate Data

We determine both barrier heights and prefactors for protein folding by applying constraints determined from experimental rate measurements to a Kramers theory for folding rate. The theoretical values are required to match the experimental values at two conditions of temperature and denaturant that induce the same stability. Several expressions for the prefactor in the Kramers rate equation are examined: a random energy approximation, a correlated energy approximation, and an approximation using a single Arrhenius activation energy. Barriers and prefactors are generally found to be large as a result of implementing this recipe, i.e., the folding landscape is cooperative and smooth. Interestingly, a prefactor with a single Arrhenius activation energy admits no formal solution.

Variation of the Resonant Transfer Rate When Passing from Nonadiabatic to Adiabatic Electron Transfer

Two competing theories are used for bridging the gap between the nonadiabatic and the deeply adiabatic electron transfer between symmetric parabolic wells. For the high friction limit, a simple analytic interpolation is proposed as a reasonable alternative to them, well-fitted to the results of numerical simulations. It provides a continuous description of the electron transfer rate in the whole range of variation of the nonadiabatic coupling between the diabatic states. For lower friction, the original theories are used for the same goal. With an increase in coupling, the cusped barrier transforms into the parabolic one. Correspondingly, the pre-exponent of the Arrhenius transfer rate first increases with coupling, then levels off approaching the "dynamic solvent effect" plateau but finally reduces reaching the limit of the adiabatic Kramers theory for the parabolic barrier. These changes proceeding with a reduction in the particle separation affect significantly the spatial dependence of the total transfer rate. When approaching the contact distance, the exact rate becomes smaller than in the theory of dynamical solvent effects and much smaller than predicted by perturbation theory (golden rule), conventionally used in photochemistry and electrochemistry.

The Effect of the Atomic Dynamics in a Crystalline Catalyst on the Rate Constant for a Catalyzed Chemical Reaction

Transition-state theory is often used to calculate the rate constant of a chemical reaction. Usually, it gives quite good results for gas-phase reactions but not for reactions in solution since the interactions of the solvent molecules with the reactants are not included in the theory. Kramers formulated an extension of the transition-state theory that included these interactions. They are represented by a friction coefficient that often is related to the viscosity of the solvent by Stoke's law. For reactions on the surface of a crystalline catalyst, there will also be an effect on the rate constant from the interaction between the reactants and the oscillating substrate atoms. In contrast to reactions in solution, where the friction coefficient is related to the viscosity of the solvent, there exists no simple property of the solid that may be directly related to a friction coefficient for the motion of the adsorbed molecule on the surface. In this paper, we propose a method that may be used to calculate a friction coefficient that may be used in Kramers theory to calculate the rate constant for a chemical reaction on the surface of a crystalline solid.

A Novel Thermodynamic Relationship Based on Kramers Theory for Studying Enzyme Kinetics under High Viscosity

In most studies of enzyme kinetics it has been found sufficient to use the classical Transition State Theory (TST) of Eyring and others. This theory was based on the solvent being an ideal dilute substance treated as a heat bath. However, enzymes found in organisms adapted to very low (psychrophiles) and very high (thermophiles) temperatures are also subjected to variable solute concentrations and viscosities. Therefore, the TST may not always be applicable to enzyme reactions carried out in various solvents with viscosities ranging from moderate to very high. There have been numerous advances in the theory of chemical reactions in realistic non-ideal solvents such as Kramers Theory. In this paper we wish to propose a modified thermodynamic equation, which have contributions from kcat, Km and the viscosity of the medium in which the enzyme reaction is occurring. These could be very useful for determining the thermodynamics of enzymes catalyzing reactions at temperature extremes in the presence of substrate solutions of different compositions and viscosities.

Thermally activated escape rate for the Brownian motion of a fixed axis rotator in a double well potential for all values of the dissipation.

The extension of the Kramers theory of the escape rate of a Brownian particle from a potential well to the entire range of damping proposed by Mel'nikov and Meshkov [J. Chem, Phys. 85, 1018 (1986)] is applied to the rotational Brownian motion of fixed axis rotators in a double well cosine potential. The procedure yields an expression for the Kramers escape rate valid for all values of the dissipation including the very low damping (VLD), very high damping (VHD), and crossover regimes. This equation provides a good asymptotic estimate of the correlation time tau per pendicular of the longitudinal dipole moment correlation function calculated by solving the underlying Langevin equation using the matrix-continued fraction method. Moreover, for low barriers, where the Mel'nikov and Meshkov approach is not applicable, analytic equations for tau in the VLD and VHD limits are derived and a simple extrapolating equation that is valid for all values of the damping is proposed.

A Limiting Speed for Protein Folding at Low Solvent Viscosity

Because protein folding dynamics are heavily overdamped, Kramers theory predicts the rate of folding to scale inversely with the reaction friction, which is usually interpreted to mean the solvent viscosity. This does not mean, however, that the speed of folding can increase without limit as solvent viscosity decreases. We show that, in a sufficiently fast-folding protein, the folding speed approaches a finite limit at low solvent viscosity, indicating a reaction controlled by internal friction.

Trehalose-enzyme interactions result in structure stabilization and activity inhibition. The role of viscosity.

Stress resistance is essential for survival. The mechanisms of molecule stabilization during stress are of interest for biotechnology, where many enzymes and other biomolecules are increasingly used at high temperatures and/or salt concentrations. Diverse organisms, exhibit rapid synthesis and accumulation of the disaccharide trehalose in response to stress. Trehalose is also rapidly hydrolyzed as soon as stress ends. In isolated enzymes, trehalose stabilizes both, structure and activity. In contrast, at optimal assay conditions, trehalose inhibits enzyme activity. A general mechanism underlying the trehalose effects observed at all temperatures probably is the trehalose-mediated increase in solution viscosity that leads to protein domain motion inhibition. This may be analyzed using Kramer's theory. The role of viscosity in the effects of trehalose is analyzed in examples from the literature and in studies on the plasma membrane H(+)-ATPase from Kluyveromyces lactis. In the cell, it may be proposed that the large concentration of trehalose reached during stress stabilizes structures through viscosity. However, once stress ends trehalose has to be rapidly hydrolyzed in order to avoid the viscosity-mediated inhibition of enzymes.

Thermal escape from zero-voltage states in hysteretic superconducting interferometers

We analyze the thermal escape behavior of the metastable zero-voltage state in biased symmetric hysteretic interferometers in external magnetic fields. An effective single degree of freedom equation of motion for the system with small loop inductance is derived and used to evaluate the two necessary parameters, energy barrier and attempt frequency, for the successful application of Kramers theory for thermal escape from metastable states. The slightly revised theory, resulting in a double pendulum model of the interferometer, is outlined and numerical simulation results are used to validate the approach. Recent experimental reports of excessive thermal escape temperatures for interferometers are discussed and interpreted.

Measuring Solution Viscosity and its Effect on Enzyme Activity.

In proteins, some processes require conformational changes involving structural domain diffusion. Among these processes are protein folding, unfolding and enzyme catalysis. During catalysis some enzymes undergo large conformational changes as they progress through the catalytic cycle. According to Kramers theory, solvent viscosity results in friction against proteins in solution, and this should result in decreased motion, inhibiting catalysis in motile enzymes. Solution viscosity was increased by adding increasing concentrations of glycerol, sucrose and trehalose, resulting in a decrease in the reaction rate of the H+-ATPase from the plasma membrane of Kluyveromyces lactis. A direct correlation was found between viscosity (η) and the inhibition of the maximum rate of catalysis (V max). The protocol used to measure viscosity by means of a falling ball type viscometer is described, together with the determination of enzyme kinetics and the application of Kramers’ equation to evaluate the effect of viscosity on the rate of ATP hydrolysis by the H+-ATPase.

Trehalose-mediated inhibition of the plasma membrane H+-ATPase from Kluyveromyces lactis: dependence on viscosity and temperature.

The effect of increasing trehalose concentrations on the kinetics of the plasma membrane H+-ATPase from Kluyveromyces lactis was studied at different temperatures. At 20 degrees C, increasing concentrations of trehalose (0.2 to 0.8 M) decreased V(max) and increased S(0.5) (substrate concentration when initial velocity equals 0.5 V(max)), mainly at high trehalose concentrations (0.6 to 0.8 M). The quotient V(max)/S(0.5) decreased from 5.76 micromol of ATP mg of protein(-1) x min(-1) x mM(-1) in the absence of trehalose to 1.63 micromol of ATP mg of protein(-1) x min(-1) x mM(-1) in the presence of 0.8 M trehalose. The decrease in V(max) was linearly dependent on solution viscosity (eta), suggesting that inhibition was due to hindering of protein domain diffusional motion during catalysis and in accordance with Kramer's theory for reactions in solution. In this regard, two other viscosity-increasing agents, sucrose and glycerol, behaved similarly, exhibiting the same viscosity-enzyme inhibition correlation predicted. In the absence of trehalose, increasing the temperature up to 40 degrees C resulted in an exponential increase in V(max) and a decrease in enzyme cooperativity (n), while S(0.5) was not modified. As temperature increased, the effect of trehalose on V(max) decreased to become negligible at 40 degrees C, in good correlation with the temperature-mediated decrease in viscosity. The trehalose-mediated increase in S(0.5) was similar at all temperatures tested, and thus, trehalose effects on V(max)/S(0.5) were always observed. Trehalose increased the activation energy for ATP hydrolysis. Trehalose-mediated inhibition of enzymes may explain why yeast rapidly hydrolyzes trehalose when exiting heat shock.

Prediction of ion channel transport from Grote—Hynes and Kramers theories

Generalized Langevin equation based Grote—Hynes (GH) theory and Langevin equation based Kramers theory are used to calculate the transmission coefficient for K+ diffusion through a model of the biological potassium ion channel IRK1, which contains a high potential barrier in the selectivity filter. The ion friction kernel is determined from a molecular dynamics (MD) simulation of the force on a stationary ion at the barrier top. The GH and Kramers estimates of the transmission coefficient are compared with those obtained from MD simulations of ion diffusion at the barrier top of the IRK1 channel. It is found that the GH estimate agrees with the value determined by rigorous MD, but the Kramers estimate is about 40% too small. The success or failure of GH and Kramers theories for various other systems is discussed and compared with these results.

Simulations of adiabatic bond-breaking electron transfer reactions on metal electrodes

Adiabatic electron transfer at metal electrodes accompanied by bond breaking is studied by molecular dynamic simulations in a two-dimensional model potential. In the simulations the redox system is coupled to a thermal bath that ensures proper statistical mechanics and provides the energy fluctuations required for the system to cross the activation barrier. The rate constants and transmission factors obtained from the simulations are analysed for various system parameters such as the applied overpotential and the strength of the interaction between the reactants and the electrode. A dependency of the rate constant on the friction parameter is also tested considering both isotropy and anisotropy effects. The Kramers turnover region is found in both cases, but the rate constant is much lower than predicted by the Kramers theory. In all simulations a strong saddle-point avoidance was observed.

The Puzzling Persistence of Process-Based Federalism Theories

The theory of the political safeguards of federalism has made a recent comeback, appearing in Supreme Court dissents by Justices Breyer and Souter and in prominent articles by Professor Larry Kramer and Professor Brad Clark. We argue that the idea that the political process can wholly replace judicial review in policing the boundaries between federal and state power is inconsistent with the text, structure, and original understanding of the Constitution. None of these sources allows the federal courts to exercise judicial review while simultaneously excluding entire subject matters from its protections - especially one as central to the constitutional structure as federalism. The political-safeguards theory treats judicial review as purely functional and almost discretionary, while we believe that the constitutional text, structure, and history impose judicial review as a mandatory duty on the courts. We show that political safeguards theory creates severe distortions in the constitutional structure, as does Kramer's provocative theory that the extra-constitutional role of the national political parties can serve as a complete substitute for judicial review of federalism questions. Finally, we show that the original understanding of the Constitution cannot support any theory that insists that the national political process's presence as a possible safeguard permits the exclusion of judicial review over federalism questions.

Effect of Viscosity on the Kinetics of α-Helix and β-Hairpin Formation

Nanosecond laser temperature jumps and fluorescence detection were used to measure the effect of viscosity on the kinetics of α-helix and β-hairpin formation in peptides containing 21 and 16 amino acids, respectively. The solvent viscosity was changed by adding glucose and sucrose at concentrations that do not significantly alter the equilibrium unfolding curves. Analysis of the temperature dependence at fixed solvent compositions shows that the activation energies are also unaffected, indicating that the slowing of the relaxation rates upon addition of the sugars results entirely from their dynamical effect of increasing the solvent viscosity (η). The rate of β-hairpin formation was found to vary nearly as 1/η, as predicted by Kramers theory in the high friction limit. In contrast, the rate of α-helix formation exhibited a fractional viscosity dependence (k = η-α) with an exponent α ≈ 0.6. The intrinsic activation energy for the elementary kinetic step of rotating a pair of backbone dihedral angles, obta...