Methods of Model Reduction for Large-Scale Biological Systems: A Survey of Current Methods and Trends

Bull Math Biol (2017) 79:1449–1486 DOI 10.1007/s11538-017-0277-2 REVIEW ARTICLE Methods of Model Reduction for Large-Scale Biological Systems: A Survey of Current Methods and Trends Thomas J. Snowden1,2 Marcus J. Tindall1,4 · Piet H. van der Graaf2,3 · Received: 10 April 2016 / Accepted: 30 March 2017 / Published online: 27 June 2017 © The Author(s) 2017. This article is an open access publication Abstract Complex models of biochemical reaction systems have become increasingly common in the systems biology literature. The complexity of such models can present a number of obstacles for their practical use, often making problems difficult to intuit or computationally intractable. Methods of model reduction can be employed to alleviate the issue of complexity by seeking to eliminate those portions of a reaction network that have little or no effect upon the outcomes of interest, hence yielding simplified systems that retain an accurate predictive capacity. This review paper seeks to provide a brief overview of a range of such methods and their application in the context of biochemical reaction network models. To achieve this, we provide a brief mathematical account of the main methods including timescale exploitation approaches, reduction via sensitivity analysis, optimisation methods, lumping, and singular value Electronic supplementary material The online version of this article (doi:10.1007/s11538-017-0277-2) contains supplementary material, which is available to authorized users. B Marcus J. Tindall Thomas J. Snowden Piet H. van der Graaf 1 Department of Mathematics and Statistics, University of Reading, Reading RG6 6AX, UK 2 Certara QSP, University of Kent Innovation Centre, Canterbury CT2 7FG, UK 3 Leiden Academic Centre for Drug Research, Universiteit Leiden, Leiden 2333 CC, Netherlands 4 The Institute for Cardiovascular and Metabolic Research (ICMR), University of Reading, Reading RG6 6AX, UK 123 1450 T. J. Snowden et al. decomposition-based approaches. Methods are reviewed in the context of large-scale systems biology type models, and future areas of research are briefly discussed. Keywords Model reduction · Complexity · Systems biology · Mathematical modelling Mathematics Subject Classification 34A34 · 37N25 · 65Y20 · 92-08 Abbreviations CSP DQSSA ENVA GA LASCO ILDM MPVA PCA QSSA REA SVD ZDP Computational singular perturbation Delay quasi-steady-state approximation Elimination of nonessential variables Genetic algorithm Lumping and subsequent optimisation Intrinsic low-dimensional manifold method Multiparametric variability analysis Principle component analysis Quasi-steady-state approximation Rapid equilibrium approximation Singular value decomposition Zero-derivative principle 1 Introduction Model complexity can be used to refer to a number of specific properties of mathematical models occurring in a range of scientific contexts. It can, for example, be used to refer to models that are overparameterised relative to the volume of collectable data, models that are unintuitable due to their scale, or models that are computationally intractable in magnitude. In each case, complexity presents a barrier to standard tools of model analysis. Methods of model reduction offer one possible approach for dealing with the perennial issue of model complexity by seeking to approximate the behaviour of a model by constructing a simplified dynamical system that retains some degree of the predictive power of the original. Model reduction has a long history in the mathematical modelling of biological systems; perhaps the most famous example is Briggs and Haldane’s application of the quasi-steady-state approximation (QSSA) for the simplification of a model of the enzyme–substrate reaction (Briggs and Haldane 1925). They demonstrated that a simplifying assumption could take the unsolvable, nonlinear, four-dimensional system of coupled ordinary differential equations (ODEs) that constituted the model, to a single ODE whilst still providing an accurate description of the dynamics for a wide range of possible parameterisations. The mathematical modelling of biological processes often leads to highly complex systems involving many state-variables and reactions. The relatively recent advent of systems biology, which seeks to model such systems in detail and hence yield a high 123 Methods of Model Reduction for Large-Scale Biological… 1451 degree of mechanistic exploratory power, has greatly increased this complexity such that it is now common to encounter models containing hundreds or even thousands of variables (Li et al. 2010). Even given this rapid increase in complexity, however, concurrent advances in computing power and simulation algorithms may appear to make model reduction a less essential process than it was in the past—it is now possible to accurately and efficiently compute numerical simulations of even highly complex systems where previously some degree of reduction was necessary to understand even the basic dynamical behaviour of many models. Ease of simulation, however, does not necessarily lead to depth of understanding; for a wide range of analyses model complexity can present an insurmountable barrier. Methods of model reduction therefore remain a vital topic and a widely applicable tool in the analysis and modelling of biochemical systems. The methods that will be discussed throughout this paper have been employed for a wide range of purposes in the literature, including to obtain more intuitively understood models, to reduce the number of parameters so as to obtain an identifiable model, to lessen the computational burden of parameter fitting, and to enable the embedding of such systems within agent-based modelling approaches. Here, for example, a researcher may be interested in concurrently modelling a large number of cells comprising a tissue—by employing a reduced description of the individual cells, such a problem may be made more computationally feasible. Despite the utility of model reduction methods, familiarity is often limited to a small range of methods that can be found in the literature. This review therefore seeks to give an overview of the use and application of model reduction methods in this context. Such methods are commonly applied within the fields of engineering and control theory, and a number of reviews of methods within these contexts exist (Okino and Mavrovouniotis 1998; Antoulas 2005). Additionally, Radulescu et al. (2012) have reviewed timescale exploitation methods for the reduction of computational biology models, but their work mostly focuses on the fundamental basis of such methods and the potential applicability of model tropicalisation in this context. The aim of this review is therefore to provide a more contextualised and up-to-date overview of such methods, as well as a survey of the current state of the literature, so as to better assess the possible utility of particular model reduction methodologie (...truncated)