Eberhard O. Voit

Recasting nonlinear differential equations as S-systems: a canonical nonlinear form

An enormous variety of nonlinear differential equations and functions have been recast exactly in the canonical form called an S-system. This is a system of nonlinear ordinary differential equations, each with the same structure: the change in a variable is equal to a difference of products of power-law functions. We review the development of S-systems, prove that the minimum for the range of equations that can be recast as S-systems consists of all equations composed of elementary functions and nested elementary functions of elementary functions, give a detailed example of the recasting process, and discuss the theoretical and practical implications. Among the latter is the ability to solve numerically nonlinear ordinary differential equations in their S-system form significantly faster than in their original form through utilization of a specially designed algorithm.

Simulation and validation of modelled sphingolipid metabolism in Saccharomyces cerevisiae

Mathematical models have become a necessary tool for organizing the rapidly increasing amounts of large-scale data on biochemical pathways and for advanced evaluation of their structure and regulation. Most of these models have addressed specific pathways using either stoichiometric or flux-balance analysis, or fully kinetic Michaelis–Menten representations, metabolic control analysis, or biochemical systems theory. So far, the predictions of kinetic models have rarely been tested using direct experimentation. Here, we validate experimentally a biochemical systems theoretical model of sphingolipid metabolism in yeast. Simulations of metabolic fluxes, enzyme deletion and the effects of inositol (a key regulator of phospholipid metabolism) led to predictions that show significant concordance with experimental results generated post hoc. The model also allowed the simulation of the effects of acute perturbations in fatty-acid precursors of sphingolipids, a situation that is not amenable to direct experimentation. The results demonstrate that modelling now allows testable predictions as well as the design and evaluation of hypothetical ‘thought experiments’ that may generate new metabolomic approaches.

Parameter estimation in biochemical systems models with alternating regression

BackgroundThe estimation of parameter values continues to be the bottleneck of the computational analysis of biological systems. It is therefore necessary to develop improved methods that are effective, fast, and scalable.ResultsWe show here that alternating regression (AR), applied to S-system models and combined with methods for decoupling systems of differential equations, provides a fast new tool for identifying parameter values from time series data. The key feature of AR is that it dissects the nonlinear inverse problem of estimating parameter values into iterative steps of linear regression. We show with several artificial examples that the method works well in many cases. In cases of no convergence, it is feasible to dedicate some computational effort to identifying suitable start values and search settings, because the method is fast in comparison to conventional methods that the search for suitable initial values is easily recouped. Because parameter estimation and the identification of system structure are closely related in S-system modeling, the AR method is beneficial for the latter as well. Specifically, we show with an example from the literature that AR is three to five orders of magnitudes faster than direct structure identifications in systems of nonlinear differential equations.ConclusionAlternating regression provides a strategy for the estimation of parameter values and the identification of structure and regulation in S-systems that is genuinely different from all existing methods. Alternating regression is usually very fast, but its convergence patterns are complex and will require further investigation. In cases where convergence is an issue, the enormous speed of the method renders it feasible to select several initial guesses and search settings as an effective countermeasure.

Mechanistic simulations of inflammation: current state and future prospects.

Inflammation is a normal, robust physiological process. It can also be viewed as a complex system that senses and attempts to resolve homeostatic perturbations initiated from within the body (for example, in autoimmune disease) or from the outside (for example, in infections). Virtually all acute and chronic diseases are either driven or modulated by inflammation. The complex interplay between beneficial and harmful arms of the inflammatory response may underlie the lack of fully effective therapies for many diseases. Mathematical modeling is emerging as a frontline tool for understanding the complexity of the inflammatory response. A series of articles in this issue highlights various modeling approaches to inflammation in the larger context of health and disease, from intracellular signaling to whole-animal physiology. Here we discuss the state of this emerging field. We note several common features of inflammation models, as well as challenges and prospects for future studies.

Biochemical systems theory and metabolic control theory: 1. fundamental similarities and differences

Abstract Biochemical Systems Theory (BST) was developed in the late 1960s to explicate the integrated behavior of intact biochemical systems—specific dynamic behavior as well as general principles of design—in relation to the properties of their underlying molecular elements. This approach was used successfully in a number of biochemical and other biological applications throughout the 1970s and 1980s. A related approach, Metabolic Control Theory (MCT), was proposed in the mid 1970s. Its developments generally have followed without reference the analogous developments in BST, and its proponents have treated the two approaches as if they were unrelated. Detailed comparison of the fundamental structures of BST and MCT shows that, although there are some superficial differences, both in fact are based upon the same underlying formalism. Molecular descriptions in MCT comprise a special case of those in BST. Systemic descriptions differ with respect to the level of aggregation assumed. The aggregation at the level of net increase or net decrease of each system constituent found in BST is shown to produce the more revealing and useful theory, and results presented elsewhere [41] suggest that this level of aggregation also provides a more accurate description of the system. At this fundamental level, MCT represents a special case of BST, for the content and range of validity of BST are more inclusive than those of MCT.

Biochemical and genomic regulation of the trehalose cycle in yeast: review of observations and canonical model analysis

The physiological hallmark of heat-shock response in yeast is a rapid, enormous increase in the concentration of trehalose. Normally found in growing yeast cells and other organisms only as traces, trehalose becomes a crucial protector of proteins and membranes against a variety of stresses, including heat, cold, starvation, desiccation, osmotic or oxidative stress, and exposure to toxicants. Trehalose is produced from glucose 6-phosphate and uridine diphosphate glucose in a two-step process, and recycled to glucose by trehalases. Even though the trehalose cycle consists of only a few metabolites and enzymatic steps, its regulatory structure and operation are surprisingly complex. The article begins with a review of experimental observations on the regulation of the trehalose cycle in yeast and proposes a canonical model for its analysis. The first part of this analysis demonstrates the benefits of the various regulatory features by means of controlled comparisons with models of otherwise equivalent pathways lacking these features. The second part elucidates the significance of the expression pattern of the trehalose cycle genes in response to heat shock. Interestingly, the genes contributing to trehalose formation are up-regulated to very different degrees, and even the trehalose degrading trehalases show drastically increased activity during heat-shock response. Again using the method of controlled comparisons, the model provides rationale for the observed pattern of gene expression and reveals benefits of the counterintuitive trehalase up-regulation.

Biochemical systems analysis of genome-wide expression data

MOTIVATION Modern methods of genomics have produced an unprecedented amount of raw data. The interpretation and explanation of these data constitute a major, well-recognized challenge. RESULTS Biochemical Systems Theory (BST) is the mathematical basis of a well-established methodological framework for analyzing networks of biochemical reactions. An existing BST model of yeast glycolysis is used here to explain and interpret the glycolytic gene expression pattern of heat shocked yeast. Our analysis demonstrates that the observed gene expression profile satisfies the primary goals of increased ATP, trehalose, and NADPH production, while maintaining intermediate metabolites at reasonable levels. Based on a systematic exploration of alternative, hypothetical expression profiles, we show that the observed profile outperforms other profiles. CONCLUSION BST is a useful framework for combining DNA microarray data with enzymatic process information to yield new insights into metabolic pathway regulation. AVAILABILITY All analyses were executed with the software PLAS(Copyright), which is freely available at http://correio.cc.fc.ul.pt/~aenf/plas.html for academic use. CONTACT VoitEO@MUSC.edu

Parameter optimization in S-system models.

BackgroundThe inverse problem of identifying the topology of biological networks from their time series responses is a cornerstone challenge in systems biology. We tackle this challenge here through the parameterization of S-system models. It was previously shown that parameter identification can be performed as an optimization based on the decoupling of the differential S-system equations, which results in a set of algebraic equations.ResultsA novel parameterization solution is proposed for the identification of S-system models from time series when no information about the network topology is known. The method is based on eigenvector optimization of a matrix formed from multiple regression equations of the linearized decoupled S-system. Furthermore, the algorithm is extended to the optimization of network topologies with constraints on metabolites and fluxes. These constraints rejoin the system in cases where it had been fragmented by decoupling. We demonstrate with synthetic time series why the algorithm can be expected to converge in most cases.ConclusionA procedure was developed that facilitates automated reverse engineering tasks for biological networks using S-systems. The proposed method of eigenvector optimization constitutes an advancement over S-system parameter identification from time series using a recent method called Alternating Regression. The proposed method overcomes convergence issues encountered in alternate regression by identifying nonlinear constraints that restrict the search space to computationally feasible solutions. Because the parameter identification is still performed for each metabolite separately, the modularity and linear time characteristics of the alternating regression method are preserved. Simulation studies illustrate how the proposed algorithm identifies the correct network topology out of a collection of models which all fit the dynamical time series essentially equally well.

Identification of metabolic system parameters using global optimization methods

BackgroundThe problem of estimating the parameters of dynamic models of complex biological systems from time series data is becoming increasingly important.Methods and resultsParticular consideration is given to metabolic systems that are formulated as Generalized Mass Action (GMA) models. The estimation problem is posed as a global optimization task, for which novel techniques can be applied to determine the best set of parameter values given the measured responses of the biological system. The challenge is that this task is nonconvex. Nonetheless, deterministic optimization techniques can be used to find a global solution that best reconciles the model parameters and measurements. Specifically, the paper employs branch-and-bound principles to identify the best set of model parameters from observed time course data and illustrates this method with an existing model of the fermentation pathway in Saccharomyces cerevisiae. This is a relatively simple yet representative system with five dependent states and a total of 19 unknown parameters of which the values are to be determined.ConclusionThe efficacy of the branch-and-reduce algorithm is illustrated by the S. cerevisiae example. The method described in this paper is likely to be widely applicable in the dynamic modeling of metabolic networks.

Biochemical Systems Theory: A Review

Biochemical systems theory (BST) is the foundation for a set of analytical andmodeling tools that facilitate the analysis of dynamic biological systems. This paper depicts major developments in BST up to the current state of the art in 2012. It discusses its rationale, describes the typical strategies and methods of designing, diagnosing, analyzing, and utilizing BST models, and reviews areas of application. The paper is intended as a guide for investigators entering the fascinating field of biological systems analysis and as a resource for practitioners and experts.