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

Dynamical robustness of biological networks with hierarchical distribution of time scales

We propose the concepts of distributed robustness and r-robustness, well adapted to functional genetics. Then we discuss the robustness of the relaxation time using a chemical reaction description of genetic and signalling networks. First, we obtain the following result for linear networks: for large multiscale systems with hierarchical distribution of time scales the variance of the inverse relaxation time (as well as the variance of the stationary rate) is much lower than the variance of the separate constants. Moreover, it can tend to 0 faster than 1/n, where n is the number of reactions. We argue that similar phenomena are valid in the nonlinear case as well. As a numerical illustration we use a model of signalling network that can be applied to important transcription factors such as NFkB

Kinetic Path Summation, Multi--Sheeted Extension of Master Equation, and Evaluation of Ergodicity Coefficient

We study the Master equation with time--dependent coefficients, a linear kinetic equation for the Markov chains or for the monomolecular chemical kinetics. For the solution of this equation a path summation formula is proved. This formula represents the solution as a sum of solutions for simple kinetic schemes (kinetic paths), which are available in explicit analytical form. The relaxation rate is studied and a family of estimates for the relaxation time and the ergodicity coefficient is developed. To calculate the estimates we introduce the multi--sheeted extensions of the initial kinetics. This approach allows us to exploit the internal ("micro")structure of the extended kinetics without perturbation of the base kinetics.Comment: The final journal versio

Asymptotology of Chemical Reaction Networks

The concept of the limiting step is extended to the asymptotology of multiscale reaction networks. Complete theory for linear networks with well separated reaction rate constants is developed. We present algorithms for explicit approximations of eigenvalues and eigenvectors of kinetic matrix. Accuracy of estimates is proven. Performance of the algorithms is demonstrated on simple examples. Application of algorithms to nonlinear systems is discussed.Comment: 23 pages, 8 figures, 84 refs, Corrected Journal Versio

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

Model complexity reduction of chemical reaction networks using mixed-integer quadratic programming

The model complexity reduction problem of large chemical reaction networks under isobaric and isothermal conditions is considered. With a given detailed kinetic mechanism and measured data of the key species over a finite time horizon, the complexity reduction is formulated in the form of a mixed-integer quadratic optimization problem where the objective function is derived from the parametric sensitivity matrix. The proposed method sequentially eliminates reactions from the mechanism and simultaneously tunes the remaining parameters until the pre-specified tolerance limit in the species concentration space is reached. The computational efficiency and numerical stability of the optimization are improved by a pre-reduction step followed by suitable scaling and initial conditioning of the Hessian involved. The proposed complexity reduction method is illustrated using three well-known case studies taken from reaction kinetics literature. © 2012 Elsevier Ltd. All rights reserved

Mathematical modeling of microRNA-mediated mechanisms of translation repression

MicroRNAs can affect the protein translation using nine mechanistically different mechanisms, including repression of initiation and degradation of the transcript. There is a hot debate in the current literature about which mechanism and in which situations has a dominant role in living cells. The worst, same experimental systems dealing with the same pairs of mRNA and miRNA can provide ambiguous evidences about which is the actual mechanism of translation repression observed in the experiment. We start with reviewing the current knowledge of various mechanisms of miRNA action and suggest that mathematical modeling can help resolving some of the controversial interpretations. We describe three simple mathematical models of miRNA translation that can be used as tools in interpreting the experimental data on the dynamics of protein synthesis. The most complex model developed by us includes all known mechanisms of miRNA action. It allowed us to study possible dynamical patterns corresponding to different miRNA-mediated mechanisms of translation repression and to suggest concrete recipes on determining the dominant mechanism of miRNA action in the form of kinetic signatures. Using computational experiments and systematizing existing evidences from the literature, we justify a hypothesis about co-existence of distinct miRNA-mediated mechanisms of translation repression. The actually observed mechanism will be that acting on or changing the limiting "place" of the translation process. The limiting place can vary from one experimental setting to another. This model explains the majority of existing controversies reported.Comment: 40 pages, 9 figures, 4 tables, 91 cited reference. The analysis of kinetic signatures is updated according to the new model of coupled transcription, translation and degradation, and of miRNA-based regulation of this process published recently (arXiv:1204.5941). arXiv admin note: text overlap with arXiv:0911.179

Approximate Bisimulations for Sodium Channel Dynamics

Abstract. This paper shows that, in the context of the Iyer et al. 67-variable cardiac myocycte model (IMW), it is possible to replace the detailed 13-state continuous-time MDP model of the sodium-channel dy-namics, with a much simpler Hodgkin-Huxley (HH)-like two-state sodium-channel model, while only incurring a bounded approximation error. The technical basis for this result is the construction of an approximate bisim-ulation between the HH and IMW channel models, both of which are input-controlled (voltage in this case) continuous-time Markov chains. The construction of the appropriate approximate bisimulation, as well as the overall result regarding the behavior of this modified IMW model, in-volves: (1) The identification of the voltage-dependent parameters of the m and h gates in the HH-type channel, based on the observations of the IMW channel. (2) Proving that the distance between observations of the two channels never exceeds a given error. (3) Exploring the sensitivity of the overall IMW model to the HH-type sodium-channel approximation. Our extensive simulation results experimentally validate our findings, for varying IMW-type input stimuli

Comprehensive review of models and methods for inferences in bio-chemical reaction networks

The key processes in biological and chemical systems are described by networks of chemical reactions. From molecular biology to biotechnology applications, computational models of reaction networks are used extensively to elucidate their non-linear dynamics. The model dynamics are crucially dependent on the parameter values which are often estimated from observations. Over the past decade, the interest in parameter and state estimation in models of (bio-) chemical reaction networks (BRNs) grew considerably. The related inference problems are also encountered in many other tasks including model calibration, discrimination, identifiability, and checking, and optimum experiment design, sensitivity analysis, and bifurcation analysis. The aim of this review paper is to examine the developments in literature to understand what BRN models are commonly used, and for what inference tasks and inference methods. The initial collection of about 700 documents concerning estimation problems in BRNs excluding books and textbooks in computational biology and chemistry were screened to select over 270 research papers and 20 graduate research theses. The paper selection was facilitated by text mining scripts to automate the search for relevant keywords and terms. The outcomes are presented in tables revealing the levels of interest in different inference tasks and methods for given models in the literature as well as the research trends are uncovered. Our findings indicate that many combinations of models, tasks and methods are still relatively unexplored, and there are many new research opportunities to explore combinations that have not been considered—perhaps for good reasons. The most common models of BRNs in literature involve differential equations, Markov processes, mass action kinetics, and state space representations whereas the most common tasks are the parameter inference and model identification. The most common methods in literature are Bayesian analysis, Monte Carlo sampling strategies, and model fitting to data using evolutionary algorithms. The new research problems which cannot be directly deduced from the text mining data are also discussed

Dynamic and static limitation in multiscale reaction networks, revisited

International audienceThe concept of limiting step gives the limit simplification: the whole network behaves as a single step. This is the most popular approach for model simplification in chemical kinetics. However, in its simplest form this idea is applicable only to the simplest linear cycles in steady states. For such the simplest cycles the nonstationary behaviour is also limited by a single step, but not the same step that limits the stationary rate. In this paper, we develop a general theory of static and dynamic limitation for all linear multiscale networks. Our main mathematical tools are auxiliary discrete dynamical systems on finite sets and specially developed algorithms of ``cycles surgery" for reaction graphs. New estimates of eigenvectors for diagonally dominant matrices are used. Multiscale ensembles of reaction networks with well separated constants are introduced and typical properties of such systems are studied. For any given ordering of reaction rate constants the explicit approximation of steady state, relaxation spectrum and related eigenvectors (``modes") is presented. In particular, we proved that for systems with well separated constants eigenvalues are real (damped oscillations are improbable). For systems with modular structure, we propose to select such modules that it is possible to solve the kinetic equation for every module in the explicit form. All such ``solvable" networks are described. The obtained multiscale approximations that we call ``dominant systems" are computationally cheap and robust. These dominant systems can be used for direct computation of steady states and relaxation dynamics, especially when kinetic information is incomplete, for design of experiments and mining of experimental data, and could serve as a robust first approximation in perturbation theory or for preconditioning