Methods of model reduction for large-scale biological systems: a survey of current methods and trends

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 decomposition-based approaches. Methods are reviewed in the context of large-scale systems biology type models, and future areas of research are briefly discussed

Reduction of dynamical biochemical reaction networks in computational biology

Biochemical networks are used in computational biology, to model the static and dynamical details of systems involved in cell signaling, metabolism, and regulation of gene expression. Parametric and structural uncertainty, as well as combinatorial explosion are strong obstacles against analyzing the dynamics of large models of this type. Multi-scaleness is another property of these networks, that can be used to get past some of these obstacles. Networks with many well separated time scales, can be reduced to simpler networks, in a way that depends only on the orders of magnitude and not on the exact values of the kinetic parameters. The main idea used for such robust simplifications of networks is the concept of dominance among model elements, allowing hierarchical organization of these elements according to their effects on the network dynamics. This concept finds a natural formulation in tropical geometry. We revisit, in the light of these new ideas, the main approaches to model reduction of reaction networks, such as quasi-steady state and quasi-equilibrium approximations, and provide practical recipes for model reduction of linear and nonlinear networks. We also discuss the application of model reduction to backward pruning machine learning techniques

Robust simplifications of multiscale biochemical networks

<p>Abstract</p> <p>Background</p> <p>Cellular processes such as metabolism, decision making in development and differentiation, signalling, etc., can be modeled as large networks of biochemical reactions. In order to understand the functioning of these systems, there is a strong need for general model reduction techniques allowing to simplify models without loosing their main properties. In systems biology we also need to compare models or to couple them as parts of larger models. In these situations reduction to a common level of complexity is needed.</p> <p>Results</p> <p>We propose a systematic treatment of model reduction of multiscale biochemical networks. First, we consider linear kinetic models, which appear as "pseudo-monomolecular" subsystems of multiscale nonlinear reaction networks. For such linear models, we propose a reduction algorithm which is based on a generalized theory of the limiting step that we have developed in <abbrgrp><abbr bid="B1">1</abbr></abbrgrp>. Second, for non-linear systems we develop an algorithm based on dominant solutions of quasi-stationarity equations. For oscillating systems, quasi-stationarity and averaging are combined to eliminate time scales much faster and much slower than the period of the oscillations. In all cases, we obtain robust simplifications and also identify the critical parameters of the model. The methods are demonstrated for simple examples and for a more complex model of NF-<it>κ</it>B pathway.</p> <p>Conclusion</p> <p>Our approach allows critical parameter identification and produces hierarchies of models. Hierarchical modeling is important in "middle-out" approaches when there is need to zoom in and out several levels of complexity. Critical parameter identification is an important issue in systems biology with potential applications to biological control and therapeutics. Our approach also deals naturally with the presence of multiple time scales, which is a general property of systems biology models.</p

Efficient Finite-difference Methods for Sensitivity Analysis of Stiff Stochastic Discrete Models of Biochemical Systems

In the study of Systems Biology it is necessary to simulate cellular processes and chemical reactions that comprise biochemical systems. This is achieved through a range of mathematical modeling approaches. Standard methods use deterministic differential equations, but because many biological processes are inherently probabilistic, stochastic models must be used to capture the random fluctuations observed in these systems. The presence of noise in a system can be a significant factor in determining its behavior. The Chemical Master Equation is a valuable stochastic model of biochemical kinetics. Models based on this formalism rely on physically motivated parameters, but often these parameters are not well constrained by experiments. One important tool in the study of biochemical systems is sensitivity analysis, which aims to quantify the dependence of a system's dynamics on model parameters. Several approaches to sensitivity analysis of these models have been developed. We proposed novel methods for estimating sensitivities of discrete stochastic models of biochemical reaction systems. We used finite-difference approximations and adaptive tau-leaping strategies to estimate the sensitivities for stiff stochastic biochemical kinetics models, resulting in significant speed-up in comparison with previously published approaches for a similar accuracy. We also developed an approach for estimating sensitivity coefficients involving adaptive implicit tau-leaping strategies. We provide a comparison of these methodologies in order to identify which approach is most efficient depending of the features of the model. These results can facilitate efficient sensitivity analysis, which can serve as a foundation for the formulation, characterization, verification and reduction of models as well as further applications to identifiability analysis

A model reduction method for biochemical reaction networks

Background: In this paper we propose a model reduction method for biochemical reaction networks governed by a variety of reversible and irreversible enzyme kinetic rate laws, including reversible Michaelis-Menten and Hill kinetics. The method proceeds by a stepwise reduction in the number of complexes, defined as the left and right-hand sides of the reactions in the network. It is based on the Kron reduction of the weighted Laplacian matrix, which describes the graph structure of the complexes and reactions in the network. It does not rely on prior knowledge of the dynamic behaviour of the network and hence can be automated, as we demonstrate. The reduced network has fewer complexes, reactions, variables and parameters as compared to the original network, and yet the behaviour of a preselected set of significant metabolites in the reduced network resembles that of the original network. Moreover the reduced network largely retains the structure and kinetics of the original model. Results: We apply our method to a yeast glycolysis model and a rat liver fatty acid beta-oxidation model. When the number of state variables in the yeast model is reduced from 12 to 7, the difference between metabolite concentrations in the reduced and the full model, averaged over time and species, is only 8%. Likewise, when the number of state variables in the rat-liver beta-oxidation model is reduced from 42 to 29, the difference between the reduced model and the full model is 7.5%. Conclusions: The method has improved our understanding of the dynamics of the two networks. We found that, contrary to the general disposition, the first few metabolites which were deleted from the network during our stepwise reduction approach, are not those with the shortest convergence times. It shows that our reduction approach performs differently from other approaches that are based on time-scale separation. The method can be used to facilitate fitting of the parameters or to embed a detailed model of interest in a more coarse-grained yet realistic environment

On Projection-Based Model Reduction of Biochemical Networks-- Part I: The Deterministic Case

This paper addresses the problem of model reduction for dynamical system models that describe biochemical reaction networks. Inherent in such models are properties such as stability, positivity and network structure. Ideally these properties should be preserved by model reduction procedures, although traditional projection based approaches struggle to do this. We propose a projection based model reduction algorithm which uses generalised block diagonal Gramians to preserve structure and positivity. Two algorithms are presented, one provides more accurate reduced order models, the second provides easier to simulate reduced order models. The results are illustrated through numerical examples.Comment: Submitted to 53rd IEEE CD

Use and abuse of the quasi-steady-state approximation

The transient kinetic behaviour of an open single enzyme, single substrate reaction is examined. The reaction follows the Van Slyke–Cullen mechanism, a spacial case of the Michaelis–Menten reaction. The analysis is performed both with and without applying the quasi-steady-state approximation. The analysis of the full system shows conditions for biochemical pathway coupling, which yield sustained oscillatory behaviour in the enzyme reaction. The reduced model does not demonstrate this behaviour. The results have important implications in the analysis of open biochemical reactions and the modelling of metabolic systems

A combined model reduction algorithm for controlled biochemical systems

Background: Systems Biology continues to produce increasingly large models of complex biochemical reaction networks. In applications requiring, for example, parameter estimation, the use of agent-based modelling approaches, or real-time simulation, this growing model complexity can present a significant hurdle. Often, however, not all portions of a model are of equal interest in a given setting. In such situations methods of model reduction offer one possible approach for addressing the issue of complexity by seeking to eliminate those portions of a pathway that can be shown to have the least effect upon the properties of interest. Methods: In this paper a model reduction algorithm bringing together the complementary aspects of proper lumping and empirical balanced truncation is presented. Additional contributions include the development of a criterion for the selection of state-variable elimination via conservation analysis and use of an ‘averaged’ lumping inverse. This combined algorithm is highly automatable and of particular applicability in the context of ‘controlled’ biochemical networks. Results: The algorithm is demonstrated here via application to two examples; an 11 dimensional model of bacterial chemotaxis in Escherichia coli and a 99 dimensional model of extracellular regulatory kinase activation (ERK) mediated via the epidermal growth factor (EGF) and nerve growth factor (NGF) receptor pathways. In the case of the chemotaxis model the algorithm was able to reduce the model to 2 state-variables producing a maximal relative error between the dynamics of the original and reduced models of only 2.8% whilst yielding a 26 fold speed up in simulation time. For the ERK activation model the algorithm was able to reduce the system to 7 state-variables, incurring a maximal relative error of 4.8%, and producing an approximately 10 fold speed up in the rate of simulation. Indices of controllability and observability are additionally developed and demonstrated throughout the paper. These provide insight into the relative importance of individual reactants in mediating a biochemical system’s input-output response even for highly complex networks. Conclusions: Through application, this paper demonstrates that combined model reduction methods can produce a significant simplification of complex Systems Biology models whilst retaining a high degree of predictive accuracy. In particular, it is shown that by combining the methods of proper lumping and empirical balanced truncation it is often possible to produce more accurate reductions than can be obtained by the use of either method in isolation

On Projection-Based Model Reduction of Biochemical Networks-- Part II: The Stochastic Case

In this paper, we consider the problem of model order reduction of stochastic biochemical networks. In particular, we reduce the order of (the number of equations in) the Linear Noise Approximation of the Chemical Master Equation, which is often used to describe biochemical networks. In contrast to other biochemical network reduction methods, the presented one is projection-based. Projection-based methods are powerful tools, but the cost of their use is the loss of physical interpretation of the nodes in the network. In order alleviate this drawback, we employ structured projectors, which means that some nodes in the network will keep their physical interpretation. For many models in engineering, finding structured projectors is not always feasible; however, in the context of biochemical networks it is much more likely as the networks are often (almost) monotonic. To summarise, the method can serve as a trade-off between approximation quality and physical interpretation, which is illustrated on numerical examples.Comment: Submitted to the 53rd CD