Validity of transfer-function representation of input-output relation in allosteric models

Statistical mechanics of Monod-Wyman-Changeux (MWC) models

The 50th anniversary of the classic Monod-Wyman-Changeux (MWC) model provides an opportunity to survey the broader conceptual and quantitative implications of this quintessential biophysical model. With the use of statistical mechanics, the mathematical implementation of the MWC concept links problems that seem otherwise to have no ostensible biological connection including ligand-receptor binding, ligand-gated ion channels, chemotaxis, chromatin structure and gene regulation. Hence, a thorough mathematical analysis of the MWC model can illuminate the performance limits of a number of unrelated biological systems in one stroke. The goal of our review is twofold. First, we describe in detail the general physical principles that are used to derive the activity of MWC molecules as a function of their regulatory ligands. Second, we illustrate the power of ideas from information theory and dynamical systems for quantifying how well the output of MWC molecules tracks their sensory input, giving a sense of the "design" constraints faced by these receptors.

A transfer-function representation for the input-output relation in consecutive Michaelis-Menten-type reactions

A transfer-function representation is devised to describe analytically and approximately the reaction velocity (output) in response to the substrate influx rate (input) in single and two consecutive Michaelis-Menten-type reactions in an open and homogeneous system. The transfer function for single reactions has an expression of first-order system, the time constant of which is dependent on the kinetic parameters and flow rate (steady-state value of the input and output), but independent of the magnitude of the input change. The transfer function for the two-reaction system can be formed with successive multiplication of the transfer functions for the first and second reactions. The validity of the representation is examined with variation in the kinetic parameters and flow rate by comparing the output of the transfer function with the actual response obtained from the computer simulation, that is, numerical integration of the rate equation. The analysis of the indicial responses indicates that the transfer functions for single and two consecutive reactions are valid within a certain error for the response around a steady state.

A transfer-function representation for regulatory responses of a controlled metabolic pathway

A transfer-function representation for the response of a controlled metabolic pathway to the changes in influx and efflux rates of metabolites is formulated to describe analytically and approximately the regulatory behavior of the pathway around a steady state. The pathway model analyzed is an open and homogeneous system which consists of two consecutive enzymatic reactions catalyzed by an allosteric enzyme of Monod-Wyman-Changeux (MWC) dimeric model and a Michaelis-Menten-type enzyme, respectively, and undergoes the feedback inhibition by the end product. The rate equation for the system (a system of ordinary differential equations) is linearized about a steady state, so that the responses of the reaction rates to the changes in influx rate of the substrate and efflux rate of the end product are expressed in a form of transfer function. The formulation leads to the transfer function for the response of production rate of the end product to the change in its efflux rate to clarify the regulatory response of feedback mechanism in controlled metabolic pathways. The relationship among the chemical species in the system at steady states also supports a reasonable assumption that the regulatory mechanisms in metabolic pathways are to control the production of end product against the change in its demand from the cellular environments.