Simulating Protein Dynamics in Cell Signaling Pathways: A Mathematical Model Approach Incorporating Negative Interaction Mechanisms

Abstract

This study presents an improved mathematical model that incorporates negative interaction mechanisms to predict the dynamics of cell signaling pathways. By employing stochastic differential equations and the Euler–Maruyama method, we simulate the responses of proteins within the mitogen-activated protein kinase and oxytocin signaling pathways over time. Conventional signaling models that consider only positive interactions often lead to unrealistic signal over-amplification, the absence of oscillatory dynamics, and an inability to reproduce compensatory responses following targeted inhibition. To address these limitations, our model explicitly incorporates inhibitory interactions through a sign-changing characteristic function and bounds protein activity using a hyperbolic-tangent transfer function, ensuring biologically plausible saturation behavior. Our findings indicate that the inhibition of upstream proteins such as MEK1/2 leads to a rapid decrease in ERK1/2 activation while causing a compensatory increase in other proteins such as SOS, RAS, and RAF. Furthermore, we explore the synergistic effects of combination therapies, demonstrating that targeting multiple signaling pathways can enhance therapeutic efficacy. Through the application of the Bliss Independence Index, we assess the effectiveness of these therapeutic combinations. Additionally, we investigate the effects of abnormal activation increases caused by gain-of-function mutations on downstream proteins and the resulting changes in balance induced by negative interactions. Overall, our enhanced mathematical model serves as a valuable tool for simulating signaling dynamics with inhibitory crosstalk and for generating mechanistic hypotheses relevant to targeted and combination therapies.