Generalized thermodynamic description of complex biological systems

Solving the complex ODEs systems is a problematic aspect of the QE model. Therefore, it has been necessary to make further investigation, in order to identify the best and fastest numerical scheme.
Matlab is provided with a set of ODE solvers that solves initial value problems for ordinary differential equations. Particularly, it is possible to classify the different algorithms according to the order of accuracy and the type of systems (stiff or nonstiff) that are to be solved.
It is now time to have a look at the characteristics of the algorithms which have been compared, in order to understand what the best for our purposes is. We will directly quote the documentation provided by the software. First of all, the algorithms to solve generic nonstiff systems are:
- ode45 is based on an explicit Runge-Kutta (4,5) formula, the Dormand-Prince pair. It is a one-step solver in computing y(t_n), it only needs the solution at the immediately preceding time point, y(t_n-1). Generally speaking, it is recommended as the function to apply as a first try for most of the problems;
- ode23 is an implementation of an explicit Runge-Kutta (2,3) pair of Bogacki and Shampine. It may be more efficient than ode45 at crude tolerances and in the presence of moderate stiffness. Like ode45, ode23 is a one-step solver;
- ode113 is a variable order Adams-Bashforth-Moulton PECE solver. It may be more efficient than ode45 at stringent tolerances and when the ODE file function is particularly expensive to evaluate. ode113 is a multistep solver: it normally needs the solutions at several preceding time points to compute the current solution.

Secondly, the most suitable algorithms for solving stiff problems (such as those introduced most of the time by the analyzed biological systems):
- ode23t is an implementation of the trapezoidal rule using a "free" interpolating. This solver is usually used if the problem is only moderately stiff and it needs a solution without numerical damping;
- ode23tb is an implementation of TR-BDF2, an implicit Runge-Kutta formula with a first stage that is a trapezoidal rule step, and a second stage that is a backward differentiation formula of order two. By construction, the same iteration matrix is used to evaluate both stages. This solver may be more efficient at crude tolerances.

Thereafter, some tests has been performed: running a few times the QE_Tuning function for the previous system has made it possible to determine the average computational time and the mean residual error, depending on the ODE solver tested.