Fluctuations In Chemical Reactions Research Articles

Equilibrium fluctuations in chemical reactions: a viable source of random data (numbers, maps and sequences)

Equilibrium fluctuations in chemical reactions: a viable source of random data (numbers, maps and sequences)

Extracting signal from noise: kinetic mechanisms from a Michaelis–Menten‐like expression for enzymatic fluctuations

Enzyme-catalyzed reactions are naturally stochastic, and precision measurements of these fluctuations, made possible by single-molecule methods, promise to provide fundamentally new constraints on the possible mechanisms underlying these reactions. We review some aspects of statistical kinetics: a new field with the goal of extracting mechanistic information from statistical measures of fluctuations in chemical reactions. We focus on a widespread and important statistical measure known as the randomness parameter. This parameter is remarkably simple in that it is the squared coefficient of variation of the cycle completion times, although it places significant limits on the minimal complexity of possible enzymatic mechanisms. Recently, a general expression has been introduced for the substrate dependence of the randomness parameter that is for rate fluctuations what the Michaelis-Menten expression is for the mean rate of product generation. We discuss the information provided by the new kinetic parameters introduced by this expression and demonstrate that this expression can simplify the vast majority of published models.

Non-invasive estimation of dissipation from non-equilibrium fluctuations in chemical reactions

We show how to extract an estimate of the entropy production from a sufficiently long time series of stationary fluctuations of chemical reactions. This method, which is based on recent work on fluctuation theorems, is direct, non-invasive, does not require any knowledge about the underlying dynamics and is applicable even when only partial information is available. We apply it to simple stochastic models of chemical reactions involving a finite number of states, and for this case, we study how the estimate of dissipation is affected by the degree of coarse-graining present in the input data.

Robustness of circadian rhythms in the presence of molecular fluctuations: An investigation based on a mechanistic, statistical theory and a simulation algorithm

After a very brief introduction to a mechanistic and statistical theory of molecular fluctuations in chemical reactions developed by Joel Keizer, we explore the robustness of a circadian rhythm model by using the theory and the exact stochastic simulation (ESS). The comparative study shows that the theory reflects the effects of the dynamics of the model on the robustness more than ESS does. Even though the theory is a macroscopic one, the robustness of the model compares well with that computed from the ESS when the system size is larger than 50. The robustness increases nonlinearly with the system size and it reaches an asymptotic value at higher system sizes. As we can expect from the dynamics of the system, the robustness is minimum near the bifurcation point and as the most sensitive parameter increases away from the bifurcation point the robustness according to the theory as well as the ESS increases and then reaches to a steady value.

Analysis of fluctuations in chemical reactions with several steady states

The effect of fluctuations in nonequilibrium systems is treated in terms of the stochastic theory. A solution of the “fundamental” equation is analyzed using the Monte-Carlo method /1, 2/. The results are compared with the conclusions following from the equations corresponding to a phenomenological model.

Master equations, Langevin equations, and the effect of diffusion on concentration fluctuations

An anomalous prediction of the master equation formalism is not obtained when concentration fluctuations in chemical reactions are described by the fluctuation-dissipation theory. The known connection between these two theories shows that the discrepancy may be eliminated by taking the properly scaled thermodynamic limit of the master equation prior to achieving steady state. The effect of diffusion on fluctuations is also considered, and exact results are obtained for several reaction mechanisms. At steady states away from equilibrium nonlocal correlations of the Ornstein–Zernike form are found. This agrees with the continuum limit of the master equation formalism used by Gardiner et al.

The irreversible thermodynamic approach to the fluctuations in chemical reactions

Irreversible thermodynamics is used to obtain the variance matrix of a chemical kinetic system at equilibrium. In contrast to the stochastic master-equation approach, the variance matrix of the present approach does not depend on the rate constants of the reactions. It is thus possible to obtain a general formula for the variances σij=〈 (Ni−Nie) (Nj−Nje) 〉, where Nje is the equilibrium concentration of species i, which applies to linear systems as well as to some nonlinear systems.

Concentration fluctuations in chemical reactions

A theory of microscopic fluctuations, which is based on generalized fluctuation–dissipation assumptions, is used to describe concentration fluctuations in a uniform system undergoing chemical reactions. The theory is shown to agree with the linear Langevin-type theory of concentration fluctuations near equilibrium, and provides a method for calculating the time evolution of an arbitrary initial probability distribution for a system far from equilibrium. These results are easily compared to the birth and death (master equation) theory of chemical reactions, and recent work on the asymptotic solutions of these master equations indicates that in the limit of a macroscopic system, the two approaches are identical.

Statistical physics of bifurcation, spatial structures, and fluctuations of chemical reactions

We present a fully nonlinear stochastic theory of chemical reactions closely below, at and above instability points. As explicit example we treat the Prigogine-Lefever-Nicolis model of two interacting kinds of molecules, generalizing it to two and three dimensions, to a mode continuum, and taking into account fluctuations. Adopting a description by means of birth and death processes we establish the master-equation, and proceed to the Fokker-Planck equation. This is transformed to new coordinates connected with unstable and stable modes. After adiabatic elimination of the stable modes, we obtain a functional Fokker-Planck equation for a continuous set of “unstable” modes. This final equation can now be treated by standard methods. In one dimension our results reveal striking analogies to the Ginzburg-Landau theory of superconductivity, to the continuous mode laser and to small-band excitations at hydrodynamical instabilities, while in three dimensions, in thin layers a hexagonal structure similar to Benard cells occurs.

Erratum: Fluctuations in chemical reactions around the steady states

First Page

On the discrepancy between results of Nicolis and Saitô concerning fluctuations in chemical reactions

On the discrepancy between results of Nicolis and Saitô concerning fluctuations in chemical reactions

Fluctuations in chemical reactions around the steady states

Birth-and-death type master equations for linear chemical reactions in closed and open systems are solved to discuss the properties of fluctuations. For nonlinear chemical reactions the generating functions of the probability of number fluctuation are usually second order differential equations, and a method for obtaining the moments of fluctuations is presented with applications to simple examples. The deterministic kinetic equation is valid for large systems and the second moments are the same as those of Poisson distributions in open systems treated here, while for the higher moments this is not always the case.