Coarse graining of master equations with fast and slow states

We propose a general method for simplifying master equations by eliminating from the description rapidly evolving states. The physical recipe we impose is the suppression of these states and a renormalization of the rates of all the surviving states. In some cases, this decimation procedure can be analytically carried out and is consistent with other analytical approaches, like in the problem of the random walk in a double-well potential. We discuss the application of our method to nontrivial examples: diffusion in a lattice with defects and a model of an enzymatic reaction outside the steady state regime.Comment: 9 pages, 9 figures, final version (new subsection and many minor improvements

Spectroscopy of drums and quantum billiards: perturbative and non-perturbative results

We develop powerful numerical and analytical techniques for the solution of the Helmholtz equation on general domains. We prove two theorems: the first theorem provides an exact formula for the ground state of an arbirtrary membrane, while the second theorem generalizes this result to any excited state of the membrane. We also develop a systematic perturbative scheme which can be used to study the small deformations of a membrane of circular or square shapes. We discuss several applications, obtaining numerical and analytical results.Comment: 29 pages, 12 figures, 7 tabl

Rigorous elimination of fast stochastic variables from the linear noise approximation using projection operators

The linear noise approximation (LNA) offers a simple means by which one can study intrinsic noise in monostable biochemical networks. Using simple physical arguments, we have recently introduced the slow-scale LNA (ssLNA) which is a reduced version of the LNA under conditions of timescale separation. In this paper, we present the first rigorous derivation of the ssLNA using the projection operator technique and show that the ssLNA follows uniquely from the standard LNA under the same conditions of timescale separation as those required for the deterministic quasi-steady state approximation. We also show that the large molecule number limit of several common stochastic model reduction techniques under timescale separation conditions constitutes a special case of the ssLNA.Comment: 10 pages, 1 figure, submitted to Physical Review E; see also BMC Systems Biology 6, 39 (2012

Passive Scalar: Scaling Exponents and Realizability

An isotropic passive scalar field $T$ advected by a rapidly-varying velocity field is studied. The tail of the probability distribution $P(\theta,r)$ for the difference $\theta$ in $T$ across an inertial-range distance $r$ is found to be Gaussian. Scaling exponents of moments of $\theta$ increase as $\sqrt{n}$ or faster at large order $n$, if a mean dissipation conditioned on $\theta$ is a nondecreasing function of $|\theta|$. The $P(\theta,r)$ computed numerically under the so-called linear ansatz is found to be realizable. Some classes of gentle modifications of the linear ansatz are not realizable.Comment: Substantially revised to conform with published version. Revtex (4 pages) with 2 postscript figures. Send email to [email protected]

Evaluation of rate law approximations in bottom-up kinetic models of metabolism.

BackgroundThe mechanistic description of enzyme kinetics in a dynamic model of metabolism requires specifying the numerical values of a large number of kinetic parameters. The parameterization challenge is often addressed through the use of simplifying approximations to form reaction rate laws with reduced numbers of parameters. Whether such simplified models can reproduce dynamic characteristics of the full system is an important question.ResultsIn this work, we compared the local transient response properties of dynamic models constructed using rate laws with varying levels of approximation. These approximate rate laws were: 1) a Michaelis-Menten rate law with measured enzyme parameters, 2) a Michaelis-Menten rate law with approximated parameters, using the convenience kinetics convention, 3) a thermodynamic rate law resulting from a metabolite saturation assumption, and 4) a pure chemical reaction mass action rate law that removes the role of the enzyme from the reaction kinetics. We utilized in vivo data for the human red blood cell to compare the effect of rate law choices against the backdrop of physiological flux and concentration differences. We found that the Michaelis-Menten rate law with measured enzyme parameters yields an excellent approximation of the full system dynamics, while other assumptions cause greater discrepancies in system dynamic behavior. However, iteratively replacing mechanistic rate laws with approximations resulted in a model that retains a high correlation with the true model behavior. Investigating this consistency, we determined that the order of magnitude differences among fluxes and concentrations in the network were greatly influential on the network dynamics. We further identified reaction features such as thermodynamic reversibility, high substrate concentration, and lack of allosteric regulation, which make certain reactions more suitable for rate law approximations.ConclusionsOverall, our work generally supports the use of approximate rate laws when building large scale kinetic models, due to the key role that physiologically meaningful flux and concentration ranges play in determining network dynamics. However, we also showed that detailed mechanistic models show a clear benefit in prediction accuracy when data is available. The work here should help to provide guidance to future kinetic modeling efforts on the choice of rate law and parameterization approaches

Preparations for Recoil Detection System at the Cooler T-Site

This research was sponsored by the National Science Foundation Grant NSF PHY-931478

Overview of mathematical approaches used to model bacterial chemotaxis II: bacterial populations

We review the application of mathematical modeling to understanding the behavior of populations of chemotactic bacteria. The application of continuum mathematical models, in particular generalized Keller–Segel models, is discussed along with attempts to incorporate the microscale (individual) behavior on the macroscale, modeling the interaction between different species of bacteria, the interaction of bacteria with their environment, and methods used to obtain experimentally verified parameter values. We allude briefly to the role of modeling pattern formation in understanding collective behavior within bacterial populations. Various aspects of each model are discussed and areas for possible future research are postulated