The dynamic systems approach to control and regulation of intracellular networks

Systems theory and cell biology have enjoyed a long relationship that has received renewed interest in recent years in the context of systems biology. The term 'systems' in systems biology comes from systems theory or dynamic systems theory: systems biology is defined through the application of systems- and signal-oriented approaches for an understanding of inter- and intra-cellular dynamic processes. The aim of the present text is to review the systems and control perspective of dynamic systems. The biologist's conceptual framework for representing the variables of a biochemical reaction network, and for describing their relationships, are pathway maps. A principal goal of systems biology is to turn these static maps into dynamic models, which can provide insight into the temporal evolution of biochemical reaction networks. Towards this end, we review the case for differential equation models as a 'natural' representation of causal entailment in pathways. Block-diagrams, commonly used in the engineering sciences, are introduced and compared to pathway maps. The stimulus-response representation of a molecular system is a necessary condition for an understanding of dynamic interactions among the components that make up a pathway. Using simple examples, we show how biochemical reactions are modelled in the dynamic systems framework and visualized using block-diagrams.     

Finding complex oscillatory phenomena in biochemical systems. An empirical approach

Starting with a model for a product-activated enzymatic reaction proposed for glycolytic oscillations, we show how more complex oscillatory phenomena may develop when the basic model is modified by addition of product recycling into substrate or by coupling in parallel or in series two autocatalytic enzyme reactions. Among the new modes of behavior are the coexistence between two stable types of oscillations (birhythmicity), bursting, and aperiodic oscillations (chaos). On the basis of these results, we outline an empirical method for finding complex oscillatory phenomena in autonomous biochemical systems, not subjected to forcing by a periodic input. This procedure relies on finding in parameter space two domains of instability of the steady state and bringing them close to each other until they merge. Complex phenomena occur in or near the region where the two domains overlap. The method applies to the search for birhythmicity, bursting and chaos in a model for the cAMP signalling system of Dictyostelium discoideum amoebae.     

Quasi-multiparameter sensitivity measure for robustness analysis of complex biochemical networks

Robustness is the ability to resume reliable operation in the face of different types of perturbations. Analysis of how network structure achieves robustness enables one to understand and design cellular systems. It is typically true that all parameters simultaneously differ from their nominal values in vivo, but there have been few intelligible measures to estimate the robustness of a system's function to the uncertainty of all parameters.We propose a numerical and fast measure of a robust property to the uncertainty of all kinetic parameters, named quasi-multiparameter sensitivity (QMPS), which is defined as the sum of the squared magnitudes of single-parameter sensitivities. Despite its plain idea, it has hardly been employed in analysis of biological models. While QMPS is theoretically derived as a linear model, QMPS can be consistent with the expected variance simulated by the widely used Monte Carlo method in nonlinear biological models, when relatively small perturbations are given. To demonstrate the feasibility of QMPS, it is employed for numerical comparison to analyze the mechanism of how specific regulations generate robustness in typical biological models.QMPS characterizes the robustness much faster than the Monte Carlo method, thereby enabling the extensive search of a large parameter space to perform the numerical comparison between alternative or competing models. It provides a theoretical or quantitative insight to an understanding of how specific network structures are related to robustness. In circadian oscillators, a negative feedback loop with multiple phosphorylations is demonstrated to play a critical role in generating robust cycles to the uncertainty of multiple parameters.     

Graphical approach to model reduction for nonlinear biochemical networks

Model reduction is a central challenge to the development and analysis of multiscale physiology models. Advances in model reduction are needed not only for computational feasibility but also for obtaining conceptual insights from complex systems. Here, we introduce an intuitive graphical approach to model reduction based on phase plane analysis. Timescale separation is identified by the degree of hysteresis observed in phase-loops, which guides a "concentration-clamp" procedure for estimating explicit algebraic relationships between species equilibrating on fast timescales. The primary advantages of this approach over Jacobian-based timescale decomposition are that: 1) it incorporates nonlinear system dynamics, and 2) it can be easily visualized, even directly from experimental data. We tested this graphical model reduction approach using a 25-variable model of cardiac β(1)-adrenergic signaling, obtaining 6- and 4-variable reduced models that retain good predictive capabilities even in response to new perturbations. These 6 signaling species appear to be optimal "kinetic biomarkers" of the overall β(1)-adrenergic pathway. The 6-variable reduced model is well suited for integration into multiscale models of heart function, and more generally, this graphical model reduction approach is readily applicable to a variety of other complex biological systems.     

Modeling biochemical networks: a cellular-automata approach

The potential of the cellular-automata (CA) method for modeling biological networks is demonstrated for the mitogen-activated protein kinase (MAPK) signaling cascade. The models derived reproduced the high signal amplification through the cascade and the deviation of the cascade enzymes from the Michaelis-Menten kinetics, evidencing cooperativity effects. The patterns of pathway change upon varying substrate concentrations and enzyme efficiencies were identified and used to show the ways for controlling pathway processes. Guidance in the selection of enzyme inhibition targets with minimum side effects is one outcome of the study.     

MultiSig: a new high-precision approach to the analysis of complex biomolecular systems

MultiSig is a newly developed mode of analysis of sedimentation equilibrium (SE) experiments in the analytical ultracentrifuge, having the capability of taking advantage of the remarkable precision (~0.1 % of signal) of the principal optical (fringe) system employed, thus supplanting existing methods of analysis through reducing the ‘noise’ level of certain important parameter estimates by up to orders of magnitude. Long-known limitations of the SE method, arising from lack of knowledge of the true fringe number in fringe optics and from the use of unstable numerical algorithms such as numerical differentiation, have been transcended. An approach to data analysis, akin to ‘spatial filtering’, has been developed, and shown by both simulation and practical application to be a powerful aid to the precision with which near-monodisperse systems can be analysed, potentially yielding information on protein-solvent interaction. For oligo- and poly-disperse systems the information returned includes precise average mass distributions over both cell radial and concentration ranges and mass-frequency histograms at fixed radial positions. The application of MultiSig analysis to various complex heterogenous systems and potentially multiply-interacting carbohydrate oligomers is described.     

Conservation analysis of large biochemical networks

MOTIVATION: Large biochemical networks pose a unique challenge from the point of view of evaluating conservation laws. The computational problem in most cases exceeds the capability of available software tools, often resulting in inaccurate computation of the number and form of conserved cycles. Such errors have profound effects on subsequent calculations, particularly in the evaluation of the Jacobian which is a critical quantity in many other calculations. The goal of this paper is to outline a new algorithm that is computationally efficient and robust at extracting the correct conservation laws for very large biochemical networks. RESULTS: We show that our algorithm can perform the conservation analysis of large biochemical networks, and can evaluate the correct conserved cycles when compared with other similar software tools. Biochemical simulators such as Jarnac and COPASI are successful at extracting only a subset of the conservation laws that our algorithm can. This is illustrated with examples for some large networks which show the advantages of our method.     

Statistically validated networks in bipartite complex systems

Many complex systems present an intrinsic bipartite structure where elements of one set link to elements of the second set. In these complex systems, such as the system of actors and movies, elements of one set are qualitatively different than elements of the other set. The properties of these complex systems are typically investigated by constructing and analyzing a projected network on one of the two sets (for example the actor network or the movie network). Complex systems are often very heterogeneous in the number of relationships that the elements of one set establish with the elements of the other set, and this heterogeneity makes it very difficult to discriminate links of the projected network that are just reflecting system's heterogeneity from links relevant to unveil the properties of the system. Here we introduce an unsupervised method to statistically validate each link of a projected network against a null hypothesis that takes into account system heterogeneity. We apply the method to a biological, an economic and a social complex system. The method we propose is able to detect network structures which are very informative about the organization and specialization of the investigated systems, and identifies those relationships between elements of the projected network that cannot be explained simply by system heterogeneity. We also show that our method applies to bipartite systems in which different relationships might have different qualitative nature, generating statistically validated networks in which such difference is preserved.     

Chemical dynamic approach to synergetics and instability in biological systems

Based on Haken's theory, self-organization or synergetics is discussed using chemical dynamics to represent an autocatalytic reaction. In a simple case the changes in a self-organizing system are given by a set of two rate equations for a primary and a partial system. When these systems mutually form a feedback loop under the adiabatic condition, the rate equation of self-organization is described by a generalized Gibbs' free energy change delta U (delta x) followed by the reaction. The sign of the parameter k'3 (k0-kex; k0, kex: rate constants with or without an external stimulus) determines the instability of the coupled system in quasi-equilibrium (k'3 approximately greater than 0; k0 greater than kex). When the product exceeds the threshold (k'3 less than 0; k0 less than kex), the system transfers into a new state, or a phase transition appears. Considering the Boltzmann distribution, the transition parameter k'3 is evaluated by an average distribution of the states and the instability is discussed using the reaction velocities vqe and vqe in the quasi-equilibrium state. As an example of this model membrane excitation is discussed briefly.     

CADLIVE toolbox for MATLAB: automatic dynamic modeling of biochemical networks with comprehensive system analysis

Mathematical modeling has become a standard technique to understand the dynamics of complex biochemical systems. To promote the modeling, we had developed the CADLIVE dynamic simulator that automatically converted a biochemical map into its associated mathematical model, simulated its dynamic behaviors and analyzed its robustness. To enhance the feasibility by CADLIVE and extend its functions, we propose the CADLIVE toolbox available for MATLAB, which implements not only the existing functions of the CADLIVE dynamic simulator, but also the latest tools including global parameter search methods with robustness analysis. The seamless, bottom-up processes consisting of biochemical network construction, automatic construction of its dynamic model, simulation, optimization, and S-system analysis greatly facilitate dynamic modeling, contributing to the research of systems biology and synthetic biology. This application can be freely downloaded from http://www.cadlive.jp/CADLIVE_MATLAB/ together with an instruction.     

A complex systems approach to Arabidopsis root stem-cell niche developmental mechanisms: from molecules, to networks, to morphogenesis

Recent reports have shown that the molecular mechanisms involved in root stem-cell niche development in Arabidopsis thaliana are complex and contain several feedback loops and non-additive interactions that need to be analyzed using computational and formal approaches. Complex systems cannot be understood in terms of the behavior of their isolated components, but they emerge as a consequence of largely non-linear interactions among their components. The study of complex systems has provided a useful approach for the exploration of system-level characteristics and behaviors of the molecular networks involved in cell differentiation and morphogenesis during development. We analyzed the complex molecular networks underlying stem-cell niche patterning in the A. thaliana root in terms of some of the key dynamic traits of complex systems: self-organization, modularity and structural properties. We use these analyses to integrate the available root stem-cell niche molecular mechanisms data and postulate novel hypotheses, missing components and interactions and explain apparent contradictions in the literature.     

Investigating the dynamic behavior of biochemical networks using model families

MOTIVATION: Supporting the evolutionary modeling process of dynamic biochemical networks based on sampled in vivo data requires more than just simulation. In the course of the modeling process, the modeler is typically concerned not only with a single model but also with sequences, alternatives and structural variants of models. Powerful automatic methods are then required to assist the modeler in the organization and the evaluation of alternative models. Moreover, the structure and peculiarities of the data require dedicated tool support. SUMMARY: To support all stages of an evolutionary modeling process, a new general formalism for the combinatorial specification of large model families is introduced. It allows for automatic navigation in the space of models and excludes biologically meaningless models on the basis of elementary flux mode analysis. An incremental usage of the measured data is supported by using splined data instead of state variables. With MMT2, a versatile tool has been developed as a computational engine intended to be built into a tool chain. Using automatic code generation, automatic differentiation for sensitivity analysis and grid computing technology, a high performance computing environment is achieved. MMT2 supplies XML model specification and several software interfaces. The performance of MMT2 is illustrated by several examples from ongoing research projects. AVAILABILITY: http://www.simtec.mb.uni-siegen.de/ CONTACT: wiechert@simtec.mb.uni-siegen.de.     

Timescale analysis of rule-based biochemical reaction networks

The flow of information within a cell is governed by a series of protein–protein interactions that can be described as a reaction network. Mathematical models of biochemical reaction networks can be constructed by repetitively applying specific rules that define how reactants interact and what new species are formed on reaction. To aid in understanding the underlying biochemistry, timescale analysis is one method developed to prune the size of the reaction network. In this work, we extend the methods associated with timescale analysis to reaction rules instead of the species contained within the network. To illustrate this approach, we applied timescale analysis to a simple receptor–ligand binding model and a rule‐based model of interleukin‐12 (IL‐12) signaling in naïve CD4+ T cells. The IL‐12 signaling pathway includes multiple protein–protein interactions that collectively transmit information; however, the level of mechanistic detail sufficient to capture the observed dynamics has not been justified based on the available data. The analysis correctly predicted that reactions associated with Janus Kinase 2 and Tyrosine Kinase 2 binding to their corresponding receptor exist at a pseudo‐equilibrium. By contrast, reactions associated with ligand binding and receptor turnover regulate cellular response to IL‐12. An empirical Bayesian approach was used to estimate the uncertainty in the timescales. This approach complements existing rank‐ and flux‐based methods that can be used to interrogate complex reaction networks. Ultimately, timescale analysis of rule‐based models is a computational tool that can be used to reveal the biochemical steps that regulate signaling dynamics. © 2011 American Institute of Chemical Engineers Biotechnol. Prog., 2012     

Origination of Pareto distribution in complex dynamic systems

The Pareto distribution, whose probability density function can be approximated at sufficiently great chi as rho(chi) - chi(-alpha), where alpha > or = 2, is of crucial importance from both the theoretical and practical point of view. The main reason is its qualitative distinction from the normal (Gaussian) distribution. Namely, the probability of high deviations appears to be significantly higher. The conception of the universal applicability of the Gauss law remains to be widely distributed despite the lack of objective confirmation of this notion in a variety of application areas. The origin of the Pareto distribution in dynamic systems located in the gaussian noise field is considered. A simple one-dimensional model is discussed where the system response in a rather wide interval of the variable can be quite precisely approximated by this distribution.     

LucidDraw: Efficiently visualizing complex biochemical networks within MATLAB

Background  

Biochemical networks play an essential role in systems biology. Rapidly growing network data and versatile research activities call for convenient visualization tools to aid intuitively perceiving abstract structures of networks and gaining insights into the functional implications of networks. There are various kinds of network visualization software, but they are usually not adequate for visual analysis of complex biological networks mainly because of the two reasons: 1) most existing drawing methods suitable for biochemical networks have high computation loads and can hardly achieve near real-time visualization; 2) available network visualization tools are designed for working in certain network modeling platforms, so they are not convenient for general analyses due to lack of broader range of readily accessible numerical utilities.     

A systems approach to unravel complex water management institutions

The study unravels the complexity of water management institutions by analysing the interactive nature of actors and rules to a particular water-related problem, using a systems approach in a hamlet in the Indian Himalayas. The approach builds on the strengths of institutional analysis development framework, but makes amendments to suit complex and adaptive water management institutions. It applies multiple research methods to collect both qualitative and quantitative information at different contextual levels. The information collected is applied in Bayesian belief network model to identify differential rules in influencing water management. Systems perspective in a problem context helped to comprehensively understand the socio-political process of water management by identifying broad array of actors and rules constraining water management, and at the same time identify actors and rules facilitating agents and their agency for a change in the water management process. In this socio-political process, the study reveals human entities – stakeholders, actors and agents – occupy different positions, which they actively shift in a problem context and when agents pursue ‘projects’ by integrating diverse rules and resources to remain adaptive. It is this adaptive and dynamic behaviour that contemporary programmes and policies fail to acknowledge. In this dynamic behaviour of the transformative capacity or power is everywhere, but they are displayed, maintained and upheld, only when agents pursue their ‘project’ by negotiating with other agents. The paper highlights the importance of comprehensive approach, in contrast to simplistic, linear and single package reforms to manage water. Such approach calls for conscious designing of rules and, at the same time, enabling actors to design rules. A conscious designing of rules is required to regulate water distribution, to build the capabilities of the poor, and to be adaptive to institutional and bio-physical crises. It calls for the development of infrastructures to further actors and agent's capabilities to design rules for informed water-related decisions. Such an approach will contribute towards sustainable water resource management.     

Thermodynamics of stoichiometric biochemical networks in living systems far from equilibrium

The principles of thermodynamics apply to both equilibrium and nonequilibrium biochemical systems. The mathematical machinery of the classic thermodynamics, however, mainly applies to systems in equilibrium. We introduce a thermodynamic formalism for the study of metabolic biochemical reaction (open, nonlinear) networks in both time-dependent and time-independent nonequilibrium states. Classical concepts in equilibrium thermodynamics-enthalpy, entropy, and Gibbs free energy of biochemical reaction systems-are generalized to nonequilibrium settings. Chemical motive force, heat dissipation rate, and entropy production (creation) rate, key concepts in nonequilibrium systems, are introduced. Dynamic equations for the thermodynamic quantities are presented in terms of the key observables of a biochemical network: stoichiometric matrix Q, reaction fluxes J, and chemical potentials of species mu without evoking empirical rate laws. Energy conservation and the Second Law are established for steady-state and dynamic biochemical networks. The theory provides the physiochemical basis for analyzing large-scale metabolic networks in living organisms.