Time series expansion to find solutions of nonlinear systems: an application to enzymatic reactions

Mathematical modeling of enzymatic reactions is among the most known and celebrated results provided by mathematicians to the biological field, and dates back to the early twentieth century. From that point ahead many steps have been done, and nowadays several results, achieved ad- dressing the modeling problem from both the deterministic and stochastic approaches, are available, paving the way to fruitful applications in pharmacokinetics and pharmacodynamics. The common denominator has been the search for analytical, approximate or numerical solutions of the underlying systems, since the intrinsic double time scale of enzymatic reactions prevents the use of standard computational methods. In this note, we exploit recent results allowing to write the solution of an Ordinary Differential Equation (ODE) in terms of a time-series expansion. Here we apply these results to the basic enzymatic reaction scheme and prove its efficacy compared to standard numerical tools. To this end, no approximations are exploited, such as the popular Quasi Steady-State Approxima- tion. Simulations are promising and underline the benefits when standard methods seem to fail.