Model Reduction for Kinetic Models of Biological Systems

Abstract

High dimensionality continues to be a challenge in computational systems biology. The kinetic models of many phenomena of interest are high-dimensional and complex, resulting in large computational effort in the simulation. Model order reduction (MOR) is a mathematical technique that is used to reduce the computational complexity of high-dimensional systems by approximation with lower dimensional systems, while retaining the important information and properties of the full order system. Proper orthogonal decomposition (POD) is a method based on Galerkin projection that can be used for reducing the model order. POD is considered an optimal linear approach since it obtains the minimum squared distance between the original model and its reduced representation. However, POD may represent a restriction for nonlinear systems. By applying the POD method for nonlinear systems, the complexity to solve the nonlinear term still remains that of the full order model. To overcome the complexity for nonlinear terms in the dynamical system, an approach called the discrete empirical interpolation method (DEIM) can be used. In this paper, we discuss model reduction by POD and DEIM to reduce the order of kinetic models of biological systems and illustrate the approaches on some examples. Additional computational costs for setting up the reduced order system pay off for large-scale systems. In general, a reduced model should not be expected to yield good approximations if different initial conditions are used from that used to produce the reduced order model. We used the POD method of a kinetic model with different initial conditions to compute the reduced model. This reduced order model is able to predict the full order model for a variety of different initial conditions.