Abstract

Proposed by R J fisher in 1930, the Generalized Fisher Equation provides a statistical relationship relating the time derivative of an average rate for a population to its variance and average time derivative of their rate. The beauty of this relationship comes from its generality, given a population and a rate one can write a powerful mathematical relationship as Fisher proposed in the field of populations biology.

Ross and others have taken the generality of Fisher’s statement and shown its application to chemical kinetics, a field understood in terms of rate coefficients, with populations of molecules undergoing reactions. When exactly is only one rate coefficient sufficient for explaining the evolution of a First Order Exponential? Chemical reactions occur in complex environments, the energetics of the environment surrounding the event can generate fluctuations in the rate of the reaction which generates a non-exponential rate. These disordered processes are generally described by two (potentially inclusive) processes, Dynamic and Static Disorder. Briefly static disorder can be understood as a distribution of rate coefficients all decaying at separate rates (such as a solvation effect in solution) which produces non-exponential decay. Dynamic disorder refers to processes with an implicitly time-dependent rate coefficient such as a double well with a barrier height that changes in time. These types of processes occur in a wide range of scenarios such as in enzyme catalyzed reactions, and single molecule pulling experiments. The kinetics of these processes must therefore be treated uniquely, and we have developed a method for quantitative analysis of these systems. Focusing our analysis to first order irreversible decay processes we were able to show an inequality relating the statistical length and the Fisher Divergence can be utilized in understanding the effects of disorder on kinetic processes.

We now take that framework and make connections to the Generalized Fisher Equations. Specifically we develop an understanding of the absolute, relative, and survival formulation of the GFE’s, and synthesize their connections to the Fisher Information. From the Fisher information we gain an understanding of the rate coefficients dictating these processes and show the inequality to measure the effect of disorder in these chemical systems.

The Generalized Fisher Equation can be understood as a mathematical relationship between the time derivative of an average rate and its variance and an average time derivative of the rate. \[\frac{d\overline{r(t)}}{dt} = + \sigma^2(t) + \overline{\frac{dr}{dt}}\]

The definition and meaning of the Fisher Information is a rather subtle but powerful mathematical concept. It should be thought of as a way to statistically measure a parameter. Based on our definition this information is again motivated as a type of variance, however it measures the fluctuations in the survival function itself. The Fisher information was proposed by Fisher as follows. \[I(t)=\sum_i\gamma_i\left(\frac{d}{dt}\ln\gamma_i\right)^2\]

The main result of our previous work was an inequality, which becomes an equality only when there is no disorder acting on the system. This inequality therefore provides a method for quantifying the amount of disorder (static and/or dynamic not exclu