Data2Dynamics: a modeling environment tailored to parameter estimation in dynamical systems.

UNLABELLED: Modeling of dynamical systems using ordinary differential equations is a popular approach in the field of systems biology. Two of the most critical steps in this approach are to construct dynamical models of biochemical reaction networks for large datasets and complex experimental conditions and to perform efficient and reliable parameter estimation for model fitting. We present a modeling environment for MATLAB that pioneers these challenges. The numerically expensive parts of the calculations such as the solving of the differential equations and of the associated sensitivity system are parallelized and automatically compiled into efficient C code. A variety of parameter estimation algorithms as well as frequentist and Bayesian methods for uncertainty analysis have been implemented and used on a range of applications that lead to publications. AVAILABILITY AND IMPLEMENTATION: The Data2Dynamics modeling environment is MATLAB based, open source and freely available at http://www.data2dynamics.org. CONTACT: andreas.raue@fdm.uni-freiburg.de SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Superoxide production by cytochrome bc1 complex: a mathematical model.

Reactive oxygen species (ROS) are involved in the pathophysiology of several diseases (e.g. Alzheimer or atherosclerosis) and also in the aging process. The main source of ROS in aerobic organisms is the electron transport chain (ETC) in the inner mitochondrial membrane. Superoxide is produced at complexes I and III of the ETC, starting a complex network of ROS reactions. To achieve a deeper mechanistic understanding of how ROS are generated by complex III, we developed a mathematical model that successfully describes experimental data of complex III activity in various rat tissues, the production of ROS with and without antimycin and ROS generation depending on different values of the membrane potential ∆Ψ. The model also reinforces the idea of ubiquinone acting as a redox mediator between heme bL and oxygen, as proposed earlier.

Modeling mechanistic biological networks: an advanced Boolean approach.

MOTIVATION: The understanding of the molecular sources for diseases like cancer can be significantly improved by computational models. Recently, Boolean networks have become very popular for modeling signaling and regulatory networks. However, such models rely on a set of Boolean functions that are in general not known. Unfortunately, while detailed information on the molecular interactions becomes available in large scale through electronic databases, the information on the Boolean functions does not become available simultaneously and has to be included manually into the models, if at all known. RESULTS: We propose a new Boolean approach which can directly utilize the mechanistic network information available through modern databases. The Boolean function is implicitly defined by the reaction mechanisms. Special care has been taken for the treatment of kinetic features like inhibition. The method has been applied to a signaling model combining the Wnt and MAPK pathway. AVAILABILITY: A sample C++ implementation of the proposed method is available for Linux and compatible systems through http://code.google.com/p/libscopes/wiki/Paper2011.

Selective benefits of damage partitioning in unicellular systems and its effects on aging.

Cytokinesis in unicellular organisms sometimes entails a division of labor between cells leading to lineage-specific aging. To investigate the potential benefits of asymmetrical cytokinesis, we created a mathematical model to simulate the robustness and fitness of dividing systems displaying different degrees of damage segregation and size asymmetries. The model suggests that systems dividing asymmetrically (size-wise) or displaying damage segregation can withstand higher degrees of damage before entering clonal senescence. When considering population fitness, a system producing different-sized progeny like budding yeast is predicted to benefit from damage retention only at high damage propagation rates. In contrast, the fitness of a system of equal-sized progeny is enhanced by damage segregation regardless of damage propagation rates, suggesting that damage partitioning may also provide an evolutionary advantage in systems dividing by binary fission. Indeed, by using Schizosaccharomyces pombe as a model, we experimentally demonstrate that damaged proteins are unevenly partitioned during cytokinesis and the damage-enriched sibling suffers from a prolonged generation time and accelerated aging. This damage retention in S. pombe is, like in Saccharomyces cerevisiae, Sir2p- and cytoskeleton-dependent, suggesting this to be an evolutionarily conserved mechanism. We suggest that sibling-specific aging may be a result of the strong selective advantage of damage segregation, which may be more common in nature than previously anticipated.

A model for the spatiotemporal organization of DNA replication in Saccharomyces cerevisiae.

DNA replication in eukaryotes is considered to proceed according to a precise program in which each chromosomal region is duplicated in a defined temporal order. However, recent studies reveal an intrinsic temporal disorder in the replication of yeast chromosome VI. Here we provide a model of the chromosomal duplication to study the temporal sequence of origin activation in budding yeast. The model comprises four parameters that influence the DNA replication system: the lengths of the chromosomes, the explicit chromosomal positions for all replication origins as well as their distinct initiation times and the replication fork migration rate. The designed model is able to reproduce the available experimental data in form of replication profiles. The dynamics of DNA replication was monitored during simulations of wild type and randomly perturbed replication conditions. Severe loss of origin function showed only little influence on the replication dynamics, so systematic deletions of origins (or loss of efficiency) were simulated to provide predictions to be tested experimentally. The simulations provide new insights into the complex system of DNA replication, showing that the system is robust to perturbation, and giving hints about the influence of a possible disordered firing.

Competition for enzymes in metabolic pathways: implications for optimal distributions of enzyme concentrations and for the distribution of flux control.

The structures of biochemical pathways are assumed to be determined by evolutionary optimization processes. In the framework of mathematical models, these structures should be explained by the formulation of optimization principles. In the present work, the principle of minimal total enzyme concentration at fixed steady state fluxes is applied to metabolic networks. According to this principle there exists a competition of the reactions for the available amount of enzymes such that all biological functions are maintained. In states which fulfil these optimization criteria the enzyme concentrations are distributed in a non-uniform manner among the reactions. This result has consequences for the distribution of flux control. It is shown that the flux control matrix c, the elasticity matrix epsilon, and the vector e of enzyme concentrations fulfil in optimal states the relations c(T)e = e and epsilon(T)e = 0. Starting from a well-balanced distribution of enzymes the minimization of total enzyme concentration leads to a lowering of the SD of the flux control coefficients.

Nested uncertainties in biochemical models.

Dynamic modelling of biochemical reaction networks has to cope with the inherent uncertainty about biological processes, concerning not only data and parameters but also kinetics and structure. These different types of uncertainty are nested within each other: uncertain network structures contain uncertain reaction kinetics, which in turn are governed by uncertain parameters. Here, the authors review some issues arising from such uncertainties and sketch methods, solutions and future directions to deal with them.

Biochemical networks with uncertain parameters.

The modelling of biochemical networks becomes delicate if kinetic parameters are varying, uncertain or unknown. Facing this situation, we quantify uncertain knowledge or beliefs about parameters by probability distributions. We show how parameter distributions can be used to infer probabilistic statements about dynamic network properties, such as steady-state fluxes and concentrations, signal characteristics or control coefficients. The parameter distributions can also serve as priors in Bayesian statistical analysis. We propose a graphical scheme, the 'dependence graph', to bring out known dependencies between parameters, for instance, due to the equilibrium constants. If a parameter distribution is narrow, the resulting distribution of the variables can be computed by expanding them around a set of mean parameter values. We compute the distributions of concentrations, fluxes and probabilities for qualitative variables such as flux directions. The probabilistic framework allows the study of metabolic correlations, and it provides simple measures of variability and stochastic sensitivity. It also shows clearly how the variability of biological systems is related to the metabolic response coefficients.

Control analysis of unbranched enzymatic chains in states of maximal activity.

It is shown that optimized states of metabolic systems are characterized by special distributions of control coefficients. Maximization of the steady-state flux through unbranched chains leads, under the constraint of fixed total amount of enzymes within the pathway, to a proportionality between control coefficients and enzyme concentrations. A detailed analysis is presented for two types of systems involving (a) reactions with linear kinetics and (b) reactions with Michaelis kinetics, respectively. In the first case one obtains for reactions with equilibrium constants larger than unity a monotonic decrease of enzyme concentrations and of control coefficients from the upper end to the lower end of the chain. In the second case optimization is performed by optimizing the intrinsic parameters (elementary rate constants) as well as the amounts of the enzymes. In contrast to systems with linear kinetics the results for reactions with Michaelis-Menten kinetics are dependent on the concentrations of the external reactants.

Evolutionary optimization of enzyme kinetic parameters; effect of constraints.

The distribution of the kinetic parameters of enzymic reactions is theoretically studied, on the assumption that, during evolution, the increase of reaction rates was an important target of natural selection. In extension of previous work on the optimization of enzyme kinetic parameters, the influence of constraints concerning upper limits of the individual rate constants is analyzed. The concept of "evolutionary effort" is applied to derive an expression for the cost function, leading to an overall upper limit for the values of the rate constants. The resulting optimization problem is solved for ordered mechanisms involving different numbers of elementary steps. It is shown that the optimum for the enzyme kinetic parameters strongly depends on the concentrations of the reactants. Low reactant concentrations lead generally to a tight binding of the reactants, while high concentrations result in a weak binding, favouring the rate constants of the other steps. In particular, states of maximum activity are not always characterized by maximal values of second-order rate constants. The results support the hypothesis that there is a mutual adaptation of Michaelis constants and reactant concentrations in an evolutionary timescale. In the limit of infinite values of the exponent of the cost function the results of the present "overall limit model" turn into the results of a model which takes into account individual upper limits for rate constants ("separate limit model"). The distributions of optimal rate constants are discussed in terms of free-energy profiles. The model is applied to the interpretation of the kinetic data of triosephosphate isomerase and inorganic pyrophosphatase.

Theoretical approaches to the evolutionary optimization of glycolysis: thermodynamic and kinetic constraints.

It is analyzed whether the structural design of contemporary glycolysis can be explained theoretically on the basis of optimization principles originating from natural selection during evolution. Particular attention is paid to the problem of how the kinetic and thermodynamic properties of the glycolytic pathway are related to its stoichiometry with respect to the number and location of ATP-coupling sites. The mathematical analysis of a minimal model of unbranched energy-converting pathways shows that the requirement of high ATP-production rate favours a structural design that includes not only ATP-producing reactions (P-sites) but also ATP-consuming reactions (C-sites). It is demonstrated that, at fixed overall thermodynamic properties of a chain, the ATP-production rate may be enhanced by kinetic optimization. The ATP-production rate is increased if the C-sites are concentrated at the beginning and all the P-sites at the end of the pathway. An optimum is attained, which is characterized by numbers of coupling sites corresponding to those found in glycolysis. Various extensions of the minimal model are considered, which allow the effects of internal feedback-regulations, variable enzyme concentrations, and the symmetric branching of glycolysis at the aldolase step to be considered.