Fisher equations, information, and irreversible disorder

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 exclusive) present in the system. This inequality is a reconstruction of the Cauchy-Schwartz Inequality and is therefore a mathematical truth describing the system of interest. To construct the inequality any first order irreversible decay process can be analyzed, experimentally the survival function provides all the measurements necessary to construct the inequality, therefore only the time dependent concentration throughout the reaction is necessary to complete the analysis.

This inequality relates two separate terms the Statistical length and the Fisher Divergence where \(J\geq L^2\). The length is best understood as an integral over the rate coefficient which is then finally squared. The divergence squares each rate coefficient and then integrates. These definitions consider the effect of fluctuations on the rate coefficient trajectories. Therefore a comparison of not only the rate coefficient but its square encapsulates the nature of these fluctuations which provide disorder in these systems. The inequality is measuring the fluctuations within the population of rate coefficients, it should therefore encapsulate a type of variance capturing the time dependent parameter k(t).

Next we define an absolute rate which is the mixed formulation. \[k_{abs}\equiv \frac{-d}{dt}[\ln S_i(t)]\]

Likewise we define the “mixed formulation” which is truly the absolute formulation. \[I_a=\sum_i \gamma_i k_{abs}=\sum_i\gamma_i\frac{-d}{dt}[\ln S_i(t)]\]

For the absolute formulation the GFE is defined as follows \[GFE_a=\frac{d}{dt}\sum [\gamma_i(t)k_a(t)]\] Simplifying we find \[GFE_a=\sum[\gamma_i\frac{d}{dt}[k_a]]+\sum[-k_a\gamma_i \frac{d}{dt}[\ln (\Sigma S_i)]]+\sum[\frac{k_a}{\Sigma S_i}\frac{d}{dt}[S_i]]\] \[GFE_a=\sum[\gamma_i\frac{d}{dt}[k_a]]-(\sum k_a\gamma_i)^2+\sum[k_a^2\gamma_i]=\sum[\gamma_i\frac{d}{dt}[k_a]]-(\overline{k})^2+\overline{k_a^2}\] We can now take the \(GFE_a\) and analyze the simplest model, two different states A and A’ with associated survival functions \(e^{-\omega t}\) and \(e^{-\omega't}\). \[\frac{e^{-\omega t}}{e^{-\omega t}+e^{-\omega't}}\frac{d}{dt}[e^{-\omega t}]+\frac{e^{-\omega't}}{e^{-\omega t}+e^{-\omega' t}}\frac{d}{dt}[e^{-\omega't}]-(\frac{e^{-\omega t}}{e^{-\omega t}+e^{-\omega't}}\frac{d}{dt}[e^{-\omega t}]+\frac{e^{-\omega't}}{e^{-\omega t}+e^{-\omega' t}}\frac{d}{dt}[e^{-\omega't}])^2+\frac{e^{-\omega t}}{e^{-\omega t}+e^{-\omega't}}(\frac{d}{dt}[e^{-\omega t}])^2+\frac{e^{-\omega't}}{e^{-\omega t}+e^{-\omega' t}}(\frac{d}{dt}[e^{-\omega't}])^2\] If we now take the limiting case of no disorder and only 1 A state where \(\omega=\omega'\equiv u\) we can see the most simplified application of this formulation. \[GFE_a=-ue^{-ut}\]

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