Reduction methods for logical control networks

In this paper some earlier stated theorems on logical control networks are proved. It is proved that a cascade of elements is equivalent to a single positive or negative element with N, number of negative elements, respectively even or odd. A loop of elements may show a cyclic behaviour with N odd or may show two possible stationary states with N even. A grafted cascade on a loop destroys the typical behaviour of the loop if the cascade is a negative chain grafted with an AND or a positive chain grafted with an OR connection; the system then admits only one stable state. Grafting a negative chain with an OR or a positive chain with an AND connection leaves the general behaviour of the loop unaffected. The detailed behaviour of two interconnected loops is extensively described. Finally it is indicated that although a complex network can be formalized as a reduced graph, its topological properties cannot always predict the final possible behaviour.     

Robust multiple-input-multiple-output control of non-linear continuous fermentation processes

In this study, the removal of nitrate (NO₃^m) ions from aqueous streams with liquid membrane technique has been investigated. Among the other parameters (temperature, pH, acceptor phase type and medium concentration), the stirring speed was chosen as process parameter. From the experimental results, it has been determined that the reaction was diffusion controlled. The transport efficiency of nitrate ions increased with increasing stirring speed. The membrane entrance and exit rate constants (k_1d, k_2m and k_2a respectively) were linearly dependent on the stirring speed ratios of 100 to 250 rpm. Coupled transport of nitrate ions through a liquid membrane in 85% n-hexane-15% tricloromethane as diluent, containing tetraoctyl ammonium chloride (TOACl) as a carrier was examined at various stirring speeds. Membrane entrance (k_1d) and exit rates (k_2m and k_2a) increase with increasing the stirring speeds. The diffusion of the nitrate ion-carrier complex through the narrow stagnant layers was found to be rate determining step. The membrane was stable during the transport experiments. There is no leakage of carrier or nitrate ion-carrier complex to both aqueous phases and also, no supplementary water penetration into the membrane. This favours interfacial reaction of nitrate ion and carrier.     

Computational models of signalling networks for non-linear control

Artificial signalling networks (ASNs) are a computational approach inspired by the signalling processes inside cells that decode outside environmental information. Using evolutionary algorithms to induce complex behaviours, we show how chaotic dynamics in a conservative dynamical system can be controlled. Such dynamics are of particular interest as they mimic the inherent complexity of non-linear physical systems in the real world. Considering the main biological interpretations of cellular signalling, in which complex behaviours and robust cellular responses emerge from the interaction of multiple pathways, we introduce two ASN representations: a stand-alone ASN and a coupled ASN. In particular we note how sophisticated cellular communication mechanisms can lead to effective controllers, where complicated problems can be divided into smaller and independent tasks.     

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.     

Bifurcation analysis of continuous biochemical reactor models

The validity of a biochemical reactor model often is evaluated by comparing transient responses to experimental data. Dynamic simulation can be a rather inefficient and ineffective tool for analyzing bioreactor models that exhibit complex nonlinear behavior. Bifurcation analysis is a powerful tool for obtaining a more efficient and complete characterization of the model behavior. To illustrate the power of bifurcation analysis, the steady-state and transient behavior of three continuous bioreactor models consisting of a small number of ordinary differential equations are investigated. Several important features, as well as potential limitations, that are difficult to ascertain via dynamic simulation are disclosed through the bifurcation analysis. The results motivate the use of dynamic simulation and bifurcation analysis as complementary tools for analyzing the nonlinear behavior of bioreactor models.     

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     

Identification of all steady states in large networks by logical analysis

The goal of generalized logical analysis is to model complex biological systems, especially so-called regulatory systems, such as genetic networks. This theory is mainly characterized by its capacity to find all the steady states of a given system and the functional positive and negative circuits, which generate multistationarity and a cycle in the state sequence graph, respectively. So far, this has been achieved by exhaustive enumeration, which severely limits the size of the systems that can be analysed. In this paper, we introduce a mathematical function, called image function, which allows the calculation of the value of the logical parameter associated with a logical variable depending on the state of the system. Thus the state table of the system is represented analytically. We then show how all steady states can be derived as solutions to a system of steady-state equations. Constraint programming, a recent method for solving constraint satisfaction problems, is applied for that purpose. To illustrate the potential of our approach, we present results from computer experiments carried out on very large randomly-generated systems (graphs) with hundreds, or even thousands, of interacting components, and show that these systems can be solved using moderate computing time. Moreover, we illustrate the approach through two published applications, one of which concerns the computation times of all steady states for a large genetic network.     

Controllability of non-linear biochemical systems

Mathematical methods of biochemical pathway analysis are rapidly maturing to a point where it is possible to provide objective rationale for the natural design of metabolic systems and where it is becoming feasible to manipulate these systems based on model predictions, for instance, with the goal of optimizing the yield of a desired microbial product. So far, theory-based metabolic optimization techniques have mostly been applied to steady-state conditions or the minimization of transition time, using either linear stoichiometric models or fully kinetic models within biochemical systems theory (BST). This article addresses the related problem of controllability, where the task is to steer a non-linear biochemical system, within a given time period, from an initial state to some target state, which may or may not be a steady state. For this purpose, BST models in S-system form are transformed into affine non-linear control systems, which are subjected to an exact feedback linearization that permits controllability through independent variables. The method is exemplified with a small glycolytic-glycogenolytic pathway that had been analyzed previously by several other authors in different contexts.     

The stochastic nature of biochemical networks

Cell behaviour and the cellular environment are stochastic. Phenotypes vary across isogenic populations and in individual cells over time. Here we will argue that to understand the abilities of cells we need to understand their stochastic nature. New experimental techniques allow gene expression to be followed in single cells over time and reveal stochastic bursts of both mRNA and protein synthesis in many different types of organisms. Stochasticity has been shown to be exploited by bacteria and viruses to decide between different behaviours. In fluctuating environments, cells that respond stochastically can out-compete those that sense environmental changes, and stochasticity may even have contributed to chromosomal gene order. We will focus on advances in modelling stochasticity, in understanding its effects on evolution and cellular design, and on means by which it may be exploited in biotechnology and medicine.     

Combining genomics, metabolome analysis, and biochemical modelling to understand metabolic networks

Now that complete genome sequences are available for a variety of organisms, the elucidation of gene functions involved in metabolism necessarily includes a better understanding of cellular responses upon mutations on all levels of gene products, mRNA, proteins, and metabolites. Such progress is essential since the observable properties of organisms - the phenotypes - are produced by the genotype in juxtaposition with the environment. Whereas much has been done to make mRNA and protein profiling possible, considerably less effort has been put into profiling the end products of gene expression, metabolites. To date, analytical approaches have been aimed primarily at the accurate quantification of a number of pre-defined target metabolites, or at producing fingerprints of metabolic changes without individually determining metabolite identities. Neither of these approaches allows the formation of an in-depth understanding of the biochemical behaviour within metabolic networks. Yet, by carefully choosing protocols for sample preparation and analytical techniques, a number of chemically different classes of compounds can be quantified simultaneously to enable such understanding. In this review, the terms describing various metabolite-oriented approaches are given, and the differences among these approaches are outlined. Metabolite target analysis, metabolite profiling, metabolomics, and metabolic fingerprinting are considered. For each approach, a number of examples are given, and potential applications are discussed.     

Top-down analysis of temporal hierarchy in biochemical reaction networks

The study of dynamic functions of large-scale biological networks has intensified in recent years. A critical component in developing an understanding of such dynamics involves the study of their hierarchical organization. We investigate the temporal hierarchy in biochemical reaction networks focusing on: (1) the elucidation of the existence of "pools" (i.e., aggregate variables) formed from component concentrations and (2) the determination of their composition and interactions over different time scales. To date the identification of such pools without prior knowledge of their composition has been a challenge. A new approach is developed for the algorithmic identification of pool formation using correlations between elements of the modal matrix that correspond to a pair of concentrations and how such correlations form over the hierarchy of time scales. The analysis elucidates a temporal hierarchy of events that range from chemical equilibration events to the formation of physiologically meaningful pools, culminating in a network-scale (dynamic) structure-(physiological) function relationship. This method is validated on a model of human red blood cell metabolism and further applied to kinetic models of yeast glycolysis and human folate metabolism, enabling the simplification of these models. The understanding of temporal hierarchy and the formation of dynamic aggregates on different time scales is foundational to the study of network dynamics and has relevance in multiple areas ranging from bacterial strain design and metabolic engineering to the understanding of disease processes in humans.     

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.     

Synchronization of non-linear biochemical oscillators coupled by diffusion

The behaviour of similar coupled non-linear oscillators of the type \(\dot x\) =f(x, y, µ \(\dot y\) =g(x, y, µ is to be investigated. The oscillators are assumed to be coupled by diffusion gradients. If some conditions on the magnitude of the diffusion coefficients are satisfied, it is proved that: 1) if the oscillators have the same period (identical value of the parameter μ) and different phases before coupling, after coupling they tend to synchronize the phases; 2) if the periods of the oscillators are not too different (in terms of the values of the parameter μ) before coupling, after coupling they tend to oscillate with the same period. It is suggested the possible role of diffusion as a synchronizing mechanism in some biological phenomena.     

Reduction, integration and emergence in biochemical networks

Most studies of molecular cell biology are based upon a process of decomposition of complex biological systems into their components, followed by the study of these components. The aim of the present paper is to discuss, on a physical basis, the internal logic of this process of reduction. The analysis is performed on simple biological systems, namely protein and metabolic networks. A multi-sited protein that binds two ligands x and y can be considered the simplest possible biochemical network. The organization of this network can be described through a comparison of three systems, i.e. XY, X and Y. X and Y are component sub-systems that collect states x(i) and y(j), respectively, i.e. protein states that have bound either i molecules of x (whether or not these states have also bound y), or j molecules of y (whether or not these states have bound x). XY is a system made up of the specific association of X and Y that collects states x(i)y(j). One can define mean self-informations per node of the network, , and . Reduction of the system XY into its components is possible if, and only if, ,is equal to the sum of and . If is smaller than the sum of and , the system is integrated, for it has less self-information than the set of its components X and Y. It can also occur that , be larger than the sum of and . Hence, the system XY displays negative integration and emergence of self-information relative to its components X and Y. Such a system is defined as complex. Positive or negative integration of the system implies it cannot be reduced to its components. The degree of integration can be measured by a function , called mutual information of integration. In the case of enzyme networks, emergence of self-information is associated with emergence of catalytic activity. Moreover, if the enzyme reaction is part of a metabolic sequence, its mutual information of integration can be increased by an effect of context of this sequence.     

Genetic control of flower morphogenesis in Arabidopsis thaliana: a logical analysis.

MOTIVATION: A large number of molecular mechanisms at the basis of gene regulation have been described during the last few decades. It is now becoming possible to address questions dealing with both the structure and the dynamics of genetic regulatory networks, at least in the case of some of the best-characterized organisms. Most recent attempts to address these questions deal with microbial or animal model systems. In contrast, we analyze here a gene network involved in the control of the morphogenesis of flowers in a model plant, Arabidopsis thaliana. RESULTS: The genetic control of flower morphogenesis in Arabidopsis involves a large number of genes, of which 10 are considered here. The network topology has been derived from published genetic and molecular data, mainly relying on mRNA expression patterns under wild-type and mutant backgrounds. Using a 'generalized logical formalism', we provide a qualitative model and derive the parameter constraints accounting for the different patterns of gene expression found in the four floral organs of Arabidopsis (sepals, petals, stamens and carpels), plus a 'non-floral' state. This model also allows the simulation or the prediction of various mutant phenotypes. On the basis of our model analysis, we predict the existence of a sixth stable pattern of gene expression, yet to be characterized experimentally. Moreover, our dynamical analysis leads to the prediction of at least one more regulator of the gene LFY, likely to be involved in the transition from the non-flowering state to the flowering pathways. Finally, this work, together with other theoretical and experimental considerations, leads us to propose some general conclusions about the structure of gene networks controlling development.     

Periodic metabolic systems: Oscillations in multiple-loop negative feedback biochemical control networks

Summary For a general multiple loop feedback inhibition system in which the end product can inhibit any or all of the intermediate reactions it is shown that biologically significant behaviour is always confined to a bounded region of reaction space containing a unique equilibrium. By explicit construction of a Liapunov function for the general n dimensional differential equation it is shown that some values of reaction parameters cause the concentration vector to approach the equilibrium asymptotically for all physically realizable initial conditions. As the parameter values change, periodic solutions can appear within the bounded region. Some information about these periodic solutions can be obtained from the Hopf bifurcation theorem. Alternatively, if specific parameter values are known a numerical method can be used to find periodic solutions and determine their stability by locating a zero of the displacement map. The single loop Goodwin oscillator is analysed in detail. The methods are then used to treat an oscillator with two feedback loops and it is found that oscillations are possible even if both Hill coefficients are equal to one.     

Instantaneous flux control analysis for biochemical systems.

Linear sensitivity analysis of steady-state control of enzymic systems has been extended to non-steady states yielding sensitivity coefficients which provide non-intuitive insights into the behavior of the system and the sites of metabolic control, and which are quantitative counterparts to traditional qualitative concepts. Because this information is provided in a readily understood format, these coefficients serve as convenient indices of metabolic control. This treatment was applied to a simple test system, consisting of two enzymes and one non-enzymatic reaction, which exhibits oscillatory behavior. The results indicate that oscillations in the concentrations of the intermediate metabolites are regulated almost exclusively by the second enzyme. Control of the flux through the pathway is apportioned equally among the three reactions during periods of low net flux, but it is due almost exclusively to the second enzyme during periods of high net flux.