Computing Optimal Properties of Drugs Using Mathematical Models of Single Channel Dynamics

Abstract

Single channel dynamics can be modeled using stochastic differential equations, and the dynamics of the state of the channel (e.g. open, closed, inactivated) can be represented using Markov models. Such models can also be used to represent the effect of mutations as well as the effect of drugs used to alleviate deleterious effects of mutations. Based on the Markov model and the stochastic models of the single channel, it is possible to derive deterministic partial differential equations (PDEs) giving the probability density functions (PDFs) of the states of the Markov model. In this study, we have analyzed PDEs modeling wild type (WT) channels, mutant channels (MT) and mutant channels for which a drug has been applied (MTD). Our aim is to show that it is possible to optimize the parameters of a given drug such that the solution of theMTD model is very close to that of the WT: the mutation’s effect is, theoretically, reduced significantly.We will present the mathematical framework underpinning this methodology and apply it to several examples. In particular, we will show that it is possible to use the method to, theoretically, improve the properties of some well-known existing drugs.