Modeling treatment and drug effects at the molecular level using hybrid system theory

In this paper, we propose to study the treatment and drug effects at the molecular level using a hybrid system model. Specifically, we propose a generic piecewise linear model to analyze drug effects on the state of the genes in a genetic regulatory network. We intend to answer the following question: given an initial state, would a treatment or drug (control input) drive the target gene to a new desired state that are not reachable without the treatment or drug? assuming that the concentration level of the drug remains constant. In other words, we try to identify whether there is a chance that the treatment or drug will be effective for changing gene expressions at all. We provide detailed analysis for two cases. In the first case, there is only one target gene; while in the second case, there is also another gene interacting with the target gene. The relationships between various parameters (of the genetic regulatory network and the design of the drug) and the convergence and the steady state of the controlled genes are derived analytically and discussed in detail. Simulations are performed using MATLAB/SIMULINK and the results confirmed our analytical findings.