Lie Group Analysis of a p53-mdm2 ODE Model

This paper presents a symmetry analysis based on Lie groups of a system of ordinary differential equations (ODEs) modelling the p53mdm2 regulatory pathway. This pathway is being investigated across several research groups as a biological system from which to extract dynamical and algebraic characteristics relevant to the emerging concept of Interaction Computing. After providing a conceptual motivation for the approach and some biological background for the choice of pathway, the paper gives an intuitive introduction to the method of Lie groups for a non-mathematical audience. This is followed by a general statement of the problem of finding the symmetries of a general system of four 1st-order ODEs, and then by the analysis of one such system modelling the p53-mdm2 pathway. The system chosen does not appear to harbour any symmetries, and therefore the effectiveness of the Lie group method cannot be demonstrated on this particular example. The symmetry analysis, however, helped reduce the system to a single Riccati equation for a specific choice of parameters, whose oscillatory behaviour appears to be relevant to the bio-computing perspective being discussed in a companion paper.