A Guide to Numerical Modelling in Systems Biology

This chapter presents basics of mathematical modelling in systems biology. In Sect. 1.1, a brief introduction to the topic is given, mainly in terms of examples such as problems from population dynamics or from drug administration. In Sect. 1.2, the assembly of large ODE networks from simple chemical and physiological mechanisms, given in terms of chemical reaction modules, is described. Reasons are given, why the so-called Michaelis-Menten kinetics is no longer needed in the numerical simulation of such systems. For reaction diagram parts, where only the properties “stimulating” or “inhibiting” are known, the formulation in terms of Hill functions is presented. Finally, in Sect. 1.3, necessary mathematical background material is collected as far as it seems important for the class of applications in question. Main topics are the uniqueness and sensitivity of solutions as well as asymptotic stability. Mathematical contents are typically explained by examples rather than by theorems, while emphasis is laid on consequences for practical calculations.

In the preceding chapter we had worked out how to establish possibly large ODE models for systems biological networks. In the present chapter, we deal with their numerical simulation. For this purpose, we describe various numerical integrators for initial value problems in necessary detail. In Sect. 2.1, we present basic concepts to characterize different discretization methods. We start with local versus global discretization errors, first in theory, then in algorithmic realization. Stability concepts for discretizations lead to an elementary pragmatic understanding of the term “stiffness” of ODE systems. In the remaining part of the chapter, different families of integrators such as Runge-Kutta methods, extrapolation methods, and multistep methods are characterized. From a practical point of view they are divided into explicit methods (Sect. 2.2), implicit methods (Sect. 2.3), and linearly implicit methods (Sect. 2.4), to be discussed in terms of their structural strengths and weaknesses. Finally, in Sect. 2.5, a roadmap of numerical methods is given together with two moderate problems that look rather similar, but require different numerical integrators. Moreover, we present a more elaborate example concerning the dynamics of tumor cells; therein we show, what kind of algorithmic decisions may influence the speed of computations.

In most systems biological models, a series of parameters enters that need to be discussed. In the preceding Sect. 2.​5.​3, we presented an example, where simulations for a large set of parameters have been performed and analyzed. This is the situation, when no measurements are available. The present chapter deals with the case, when measurements are available and can be used, in principle, to identify at least part of the parameters. Such parameter identification problems in ODE models typically arise as nonlinear least squares problems, see Sect. 3.1. They are solved by Gauss-Newton methods, which require the numerical solution of linear least squares problems within each iteration. For pedagogical reasons, the order of these three topics is reversed in our presentation. Therefore, in Sect. 3.2, linear least squares problems are discussed first including the important issue of automatic detection of rank deficiencies in matrix factorization. Clearly, not all data sets are equally well suited to fit all unknown parameters of a given model. Next, in Sect. 3.3, the class of “adequate” nonlinear least squares problems is defined, both theoretically and computationally, for which the local Gauss-Newton method converges. Globalization via a damping strategy is presented. The case of possible non-convergence is treated in detail to find out which part originates from an insufficient model and which one from “bad” initial guesses for the Gauss-Newton iteration. In Sect. 3.4, all pieces of the text presented so far are glued together to apply to the ODE models, which are the general topic of the book. Finally, in Sect. 3.5, three examples with increasing complexity are presented. First, the notorious predator-prey problem is revisited, which turns out to be quite standard. Next, in order to connect the advocated computational ideas with modelling intuition, a simple illustrative example is worked out in algorithmic detail. Last, a more complex parameter identification problem related to a model of the human menstrual cycle is discussed in detail.