Molecular Systems with Infinite and Finite Degrees of Freedom. Part I: Multi-Scale Analysis

Molecular Systems with Infinite and Finite Degrees of Freedom. Part I: Multi-Scale Analysis Luca Sbano, Mathematics Institute, University of Warwick sbano@maths.warwick.ac.uk and Markus Kirkilionis Mathematics Institute, University of Warwick mak@maths.warwick.ac.uk Abstract The paper analyses stochastic systems describing reacting molecular systems with a combination of two types of state spaces, a finite-dimensional, and an in- finite dimenional part. As a typical situation consider the interaction of larger macro-molecules, finite and small in numbers per cell (like protein complexes), with smaller, very abundant molecules, for example metabolites. We study the construc- tion of the continuum approximation of the associated Master Equation (ME) by using the Trotter approximation [27]. The continuum limit shows regimes where the finite degrees of freedom evolve faster than the infinite ones. Then we develop a rigourous asymptotic adiabatic theory upon the condition that the jump process arising from the finite degrees of freedom of the Markov Chain (MC, typically de- scribing conformational changes of the macro-molecules) occurs with large frequency. In a second part of this work, the theory is applied to derive typical enzyme kinetics in an alternative way and interpretation within this framework. 1 Introduction Think of a typical situation in Cell Biology, the interaction of macro-molecules in the cell. In most cases there will be a small number of macro-molecular machines, like enzymes, ion-channels, polymerases, ribosomes etc. which are essential for cellular func- tion, but which will not be very abundant in numbers per cell. Moreover this number will typically not change over time of observation. These machines will have different states of operation, like an ion channel can be closed or open. The states of operation of such machines can in general be described by finitely many different discrete states. These discrete states can be associated with meta-stable conformations of proteins (see for example [14]). Smaller molecules like ions, or metabolites like sugars, will interact with these macro-molecules. The most typical and best studied situation are enzymes catalysing metabolic reactions. The classical way to describe the resulting kinetics is given in [24], see also [25]. The number of these smaller molecules clustered in different species will change typically over time of observation. Assuming no inherent spatial 1 arXiv:0802.4259v2 [q-bio.BM] 29 Feb 2008 structure of the process, this gives rise to coupled systems of two well studied mathe- matical objects, Markov chains describing the transitions between the different modes of operation of the macro-molecules, and birth-death processes with reactions describing the change in numbers of the smaller abundant molecules. In this paper we will study both mathematical objects simultaneously as one system, giving a rigourous derivation of the continuum limit. With other interpretations the theory can also applied in various other fields of sciences where interaction of different finite state ’machines’ will occur, like epidemiology, manufacturing or economy. Figure 1: Different typical interactions of small and large molecules in a typical cell. (A): Interaction of enzymes with metabolites. The product is a molecule consisting of two elementary species. Enzymes react as catalysators. (B): Genetic interactions. For example a repressor can bind to the DNA only in the case it is in a conformation characterised by the absence of smaller ’inducing’ molecules. (C): Membrane proteins act in various ways as molecular machines, for example as ion channels. As a concrete example from Genetics assume mRNA is transcribed depending on whether a specific DNA binding site is bound to a transcription factor A. There are two possible limit regimes: either no A molecules are binding causing no mRNA tran- scription, or molecules A are binding to the DNA implying a maximum transcription rate for the mRNA. Usually such binding/unbinding events occur at large frequencies, and they are proportional to the concentration of the transcription factor. This leads to an effective transcription rate resulting from an ”effective average” of the binding/un- buinding event depending on the concentration of A. This effect is usually modelled by a Hill-type kinetics ([3]): K([A]) = K1 [A]n 1 + K2 [A]n This kinetics describes an effective reaction rate with saturation behaviour for large concentrations [A] of the transcription factor A. Here n is a positive integer, an exponent controlling the slope of the sigmoidal K([A]). In this paper we consider as an illustrative example the case n = 1, in [23] we shall describe the general case. We will follow this particular example throughout this paper (part I) starting from microscopic assumptions, and derive the above deterministic limit in part II. 2 Birth-death processes and the density assumption We will assume that at time t the state of the subsystem d

…(Full text truncated)…

This content is AI-processed based on ArXiv data.