The Role of Stochastic Models in Interpreting the Origins of Biological Chirality

Abstract

This review summarizes recent stochastic modeling efforts in the theoretical research aimed at interpreting the origins of biological chirality. Stochastic kinetic models, especially those based on the continuous time discrete state approach, have great potential in modeling absolute asymmetric reactions, experimental examples of which have been reported in the past decade. An overview of the relevant mathematical background is given and several examples are presented to show how the significant numerical problems characteristic of the use of stochastic models can be overcome by non-trivial, but elementary algebra. In these stochastic models, a particulate view of matter is used rather than the concentration-based view of traditional chemical kinetics using continuous functions to describe the properties system. This has the advantage of giving adequate description of single-molecule events, which were probably important in the origin of biological chirality. The presented models can interpret and predict the random distribution of enantiomeric excess among repetitive experiments, which is the most striking feature of absolute asymmetric reactions. It is argued that the use of the stochastic kinetic approach should be much more widespread in the relevant literature.

Stochastic mapping of the Michaelis-Menten mechanism

“…However, the phenomenon of volume dependence is well known from earlier stochastic literature. 30,33,39 In the map of Fig. 2, it is necessary to show these limits because the description of the system depends on the number of substrate molecules present and not only the concentration.…”

“…30,33 In this work, however, all rate constants are used with the deterministic values for internal consistency (e.g., k 1 has the units of M −1 s −1 ), which explains the origins of the appearance of volume (V) and the Avogadro constant (N A ) in the master equation. Equation (5) is a system of linear differential equations, which can be solved exactly.…”

“…Equation (5) is a system of linear differential equations, which can be solved exactly. For direct computations at relatively low particle numbers (m ≤ 10 4 ), the use of an enumerating function 30 is necessary, which gives a unique integer number between 1 and m to every possible state. In this case, a suitable enumerating function is…”

“…30,33 A state of the system is identified by giving the number of molecules present. For the present scheme, one can give the actual number of free enzyme molecules (e) and the number of uncomplexed substrate molecules (s) to identify a given state.…”

“…12,22 Recent advance in the stochastic modeling of absolute asymmetric reactions shows that it is often possible to obtain meaningful analytical solutions for systems exhibiting moderate levels of complexity, and give reasonably complete general mathematical descriptions of those chemical schemes. [29][30][31] This paper attempts to carry out a similar analysis for the Michaelis-Menten mechanism.…”