Endotoxin Tolerance: Mathematical Models

摘要

Endotoxin tolerance is an important phenomenon of innate immunity. It is usually defined as a reduced responsiveness to a lipopolysaccharide (LPS) challenge following a first encounter with endotoxin. The actors of the endotoxin tolerance are a Gram-negative bacterial lipopolysaccharide (LPS), the proinflammatory cytokines (the best marker of the inflammatory process is considered to be the TNF-α) and the downregulating factors. It has to be noted that there are two possible outcomes at a secondary challenge with endotoxin: 1) the first and the second responses are similar or 2) the second response is significantly reduced (close to 0). This last outcome is the usual case and it is known as the typical endotoxin tolerance phenomenon. This is the subject of this paper. There are many reasons which explain the interest in understanding the endotoxin tolerance (for example, the connections with sepsis). It would be of great help to have a simple but good enough mathematical model for testing and simulating endotoxin tolerance in various reported situations and also for a better understanding of the factors acting during this complex phenomenon. In some previous works the authors introduced an original mathematical (ODE) model of endotoxin tolerance. The aim of this lecture is to present this model and some new improvements (the Petri net associated to the endotoxin tolerance phenomenon) together with applications (mainly in sepsis). Our original mathematical model of the endotoxin tolerance is based on a generalized version of the Michaelis - Menten - Hill equations for enzymatic reactions. This is a nonlinear and non autonomous ODE system with LPS as an input. We also tried to keep our model as simple as possible; the model could be, of course, developed to a more sophisticated one. In order to test our model we considered several typical scenarios for the input (LPS challenge) such as: in vivo, in vitro, immune paralysis (clinical sepsis). In each case, the mathematical simulation fits well-enough with the reported experimental data.