Modeling the Bacteriophage Lambda Site Specific Recombination through a Perturbed System of Linear Differential Equations
Authors:Steven Esquivel, Yi-An Lai, Ismael Perez
- Peter Salamon, Professor of Mathematics, San Diego State University
- Anca Mara Segall, Professor of Biology, San Diego State University
Perturbation methods have been traditionally used for finding approximate solutions to otherwise unsolvable physical problems. We perturbed a system of linear differential equations that describe the chemical kinetics of site-specific recombination by the bacteriophage lambda. This is a mechanism that viruses use to insert their genome into the genome of their Escherichia Coli host. We've yet to yield a stable identification of the rate constants when using the Nelder-Mead (Simplex) Algorithm to minimize the sum of square errors between the data and the model. This is due to the fact that one of the steps in the reaction relaxes on a faster time-scale than the time-scale of our observations. If this fast reaction is always in equilibrium, we cannot see the rate constants for this equilibration. In fact however, a weak signal from the fast reaction can be seen in the first few seconds. To focus in on this signal, we perturbed the linear system to obtain a better identification of the rate constants. Using the Hessian of the sum squared error as a function of the parameters shows this to be a good technique on synthetic data. The low precision possible for real data limits the use of this technique for the original experiments. We've shown that perturbation theory can be used in a new context, perturbing linear systems. The technique gives improved identification of parameter values, but not enough to overcome the lack of precision in experimental data from site-specific recombination.