Splitting the dynamics of large biochemical interaction networks

This article is inscribed in the general motivation of understanding the dynamics on biochemical networks including metabolic and genetic interactions. Our approach is continuous modeling by differential equations. We address the problem of the huge size of those systems. We present a mathematical tool for reducing the size of the model, master-slave synchronization, and fit it to the biochemical context.     

Selfish cellular networks and the evolution of complex organisms

Human gametogenesis takes years and involves many cellular divisions, particularly in males. Consequently, gametogenesis provides the opportunity to acquire multiple de novo mutations. A significant portion of these is likely to impact the cellular networks linking genes, proteins, RNA and metabolites, which constitute the functional units of cells. A wealth of literature shows that these individual cellular networks are complex, robust and evolvable. To some extent, they are able to monitor their own performance, and display sufficient autonomy to be termed "selfish". Their robustness is linked to quality control mechanisms which are embedded in and act upon the individual networks, thereby providing a basis for selection during gametogenesis. These selective processes are equally likely to affect cellular functions that are not gamete-specific, and the evolution of the most complex organisms, including man, is therefore likely to occur via two pathways: essential housekeeping functions would be regulated and evolve during gametogenesis within the parents before being transmitted to their progeny, while classical selection would operate on other traits of the organisms that shape their fitness with respect to the environment.     

Mathematical descriptions of biochemical networks: stability, stochasticity, evolution

In this paper, we review some fundamental aspects, as well as some new developments, in the emerging field of network biology. The focus of attention is placed on mathematical approaches to conceptual modeling of biomolecular networks with special emphasis on dynamic stability, stochasticity and evolution.     

Quantitative analysis of signaling networks

The response of biological cells to environmental change is coordinated by protein-based signaling networks. These networks are to be found in both prokaryotes and eukaryotes. In eukaryotes, the signaling networks can be highly complex, some networks comprising of 60 or more proteins. The fundamental motif that has been found in all signaling networks is the protein phosphorylation/dephosphorylation cycle--the cascade cycle. At this time, the computational function of many of the signaling networks is poorly understood. However, it is clear that it is possible to construct a huge variety of control and computational circuits, both analog and digital from combinations of the cascade cycle. In this review, we will summarize the great versatility of the simple cascade cycle as a computational unit and towards the end give two examples, one prokaryotic chemotaxis circuit and the other, the eukaryotic MAPK cascade.     

Propensity of a circadian clock to adjust to the 24h day-night light cycle and its sensitivity to molecular noise

The circadian clock of Drosophila melanogaster and its tendency to adjust to the day-night light cycle is simulated by deterministic and stochastic methods. The robustness of the locking to the light-cycle with respect to molecular noise is studied. It is found that within the model studied, the molecular noise in the stochastic simulation erases the finer injection-locking structures, stronger injection signals are needed and the locking has the character of prolonged locked time intervals with cycle slips in between. The simulations are compared to a simple injection-locking model with noise that seems to describe the overall behavior well.     

Design principles for robust biochemical reaction networks: what works, what cannot work, and what might almost work

We bring together recent results that connect the structure of a mass-action reaction network to its capacity for concentration robustness — that is, its capacity to keep the concentration of a critical bio-active species within narrow limits, even against large fluctuations in the overall supply of the network’s constituents.     

Control of Boolean networks: hardness results and algorithms for tree structured networks

Finding control strategies of cells is a challenging and important problem in the post-genomic era. This paper considers theoretical aspects of the control problem using the Boolean network (BN), which is a simplified model of genetic networks. It is shown that finding a control strategy leading to the desired global state is computationally intractable (NP-hard) in general. Furthermore, this hardness result is extended for BNs with considerably restricted network structures. These results justify existing exponential time algorithms for finding control strategies for probabilistic Boolean networks (PBNs). On the other hand, this paper shows that the control problem can be solved in polynomial time if the network has a tree structure. Then, this algorithm is extended for the case where the network has a few loops and the number of time steps is small. Though this paper focuses on theoretical aspects, biological implications of the theoretical results are also discussed.     

Modeling the mammalian circadian clock: sensitivity analysis and multiplicity of oscillatory mechanisms

We extend the study of a computational model recently proposed for the mammalian circadian clock (Proc. Natl Acad. Sci. USA 100 (2003) 7051). The model, based on the intertwined positive and negative regulatory loops involving the Per, Cry, Bmal1, and Clock genes, can give rise to sustained circadian oscillations in conditions of continuous darkness. These limit cycle oscillations correspond to circadian rhythms autonomously generated by suprachiasmatic nuclei and by some peripheral tissues. By using different sets of parameter values producing circadian oscillations, we compare the effect of the various parameters and show that both the occurrence and the period of the oscillations are generally most sensitive to parameters related to synthesis or degradation of Bmal1 mRNA and BMAL1 protein. The mechanism of circadian oscillations relies on the formation of an inactive complex between PER and CRY and the activators CLOCK and BMAL1 that enhance Per and Cry expression. Bifurcation diagrams and computer simulations nevertheless indicate the possible existence of a second source of oscillatory behavior. Thus, sustained oscillations might arise from the sole negative autoregulation of Bmal1 expression. This second oscillatory mechanism may not be functional in physiological conditions, and its period need not necessarily be circadian. When incorporating the light-induced expression of the Per gene, the model accounts for entrainment of the oscillations by light-dark (LD) cycles. Long-term suppression of circadian oscillations by a single light pulse can occur in the model when a stable steady state coexists with a stable limit cycle. The phase of the oscillations upon entrainment in LD critically depends on the parameters that govern the level of CRY protein. Small changes in the parameters governing CRY levels can shift the peak in Per mRNA from the L to the D phase, or can prevent entrainment. The results are discussed in relation to physiological disorders of the sleep-wake cycle linked to perturbations of the human circadian clock, such as the familial advanced sleep phase syndrome or the non-24h sleep-wake syndrome.     

Test of the Inverse Monte Carlo Method for the Calculation of Interatomic Potential Energies in Atomic Liquids

Abstract

The principle purpose of this paper is to demonstrate the use of the Inverse Monte Carlo technique for calculating pair interaction energies in monoatomic liquids from a given equilibrium property. This method is based on the mathematical relation between transition probability and pair potential given by the fundamental equation of the “importance sampling” Monte Carlo method. In order to have well defined conditions for the test of the Inverse Monte Carlo method a Metropolis Monte Carlo simulation of a Lennard Jones liquid is carried out to give the equilibrium pair correlation function determined by the assumed potential. Because an equilibrium configuration is prerequisite for an Inverse Monte Carlo simulation a model system is generated reproducing the pair correlation function, which has been calculated by the Metropolis Monte Carlo simulation and therefore representing the system in thermal equilibrium. This configuration is used to simulate virtual atom displacements. The resulting changes in atom distribution for each single simulation step are inserted in a set of non-linear equations defining the transition probability for the virtual change of configuration. The solution of the set of equations for pair interaction energies yields the Lennard Jones potential by which the equilibrium configuration has been determined.     

Dynamical analysis of cellular networks based on the Green's function matrix

The complexity of cellular networks often limits human intuition in understanding functional regulations in a cell from static network diagrams. To this end, mathematical models of ordinary differential equations (ODEs) have commonly been used to simulate dynamical behavior of cellular networks, to which a quantitative model analysis can be applied in order to gain biological insights. In this paper, we introduce a dynamical analysis based on the use of Green's function matrix (GFM) as sensitivity coefficients with respect to initial concentrations. In contrast to the classical (parametric) sensitivity analysis, the GFM analysis gives a dynamical, molecule-by-molecule insight on how system behavior is accomplished and complementarily how (impulse) signal propagates through the network. The knowledge gained will have application from model reduction and validation to drug discovery research in identifying potential drug targets, studying drug efficacy and specificity, and optimizing drug dosing and timing. The efficacy of the method is demonstrated through applications to common network motifs and a Fas-induced programmed cell death model in Jurkat T cell line.     

Sensitivity analysis of stoichiometric networks: an extension of metabolic control analysis to non-steady state trajectories

A sensitivity analysis of general stoichiometric networks is considered. The results are presented as a generalization of Metabolic Control Analysis, which has been concerned primarily with system sensitivities at steady state. An expression for time-varying sensitivity coefficients is given and the Summation and Connectivity Theorems are generalized. The results are compared to previous treatments. The analysis is accompanied by a discussion of the computation of the sensitivity coefficients and an application to a model of phototransduction.     

Assessment of effects of experimental imprecision on optimized biochemical systems.

Metabolic pathways may be optimized with S-system models that prescribe profiles of control variables leading to optimal output while keeping metabolites and enzyme activities within predefined ranges. Monte Carlo simulations show how much the yield and the corresponding metabolite concentrations would be affected by inaccuracies in the experimental implementation of the prescribed profiles. For a recent model of citric acid production in Aspergillus niger, the yield is roughly normally distributed, whereas the distributions of metabolite concentrations differ greatly in shape and statistical characteristics. Even moderate inaccuracies may lead to constraint violations, which appear to be correlated with high logarithmic gains.     

An integrative simulation model linking major biochemical reactions of actin-polymerization to structural properties of actin filaments

We report on an advanced universal Monte Carlo simulation model of actin polymerization processes offering a broad application panel. The model integrates major actin-related reactions, such as assembly of actin nuclei, association/dissociation of monomers to filament ends, ATP-hydrolysis via ADP-Pi formation and ADP-ATP exchange, filament branching, fragmentation and annealing or the effects of regulatory proteins. Importantly, these reactions are linked to information on the nucleotide state of actin subunits in filaments (ATP hydrolysis) and the distribution of actin filament lengths. The developed stochastic simulation modelling schemes were validated on: i) synthetic theoretical data generated by a deterministic model and ii) sets of our and published experimental data obtained from fluorescence pyrene-actin experiments. Build on an open-architecture principle, the designed model can be extended for predictive evaluation of the activities of other actin-interacting proteins and can be applied for the analysis of experimental pyrene actin-based or fluorescence microscopy data. We provide a user-friendly, free software package ActinSimChem that integrates the implemented simulation algorithms and that is made available to the scientific community for modelling in silico any specific actin-polymerization system.     

Uses and misuses of progress curve analysis in enzyme kinetics

Progress curve analysis is a convenient tool for the characterization of enzyme action: a single reaction mixture provides multiple experimental measured points for continuously varying amounts of substrates and products with exactly the same enzyme and modulator concentrations. The determination of kinetic parameters from the progress curves, however, requires complex mathematical evaluation of the time-course data. Some freely available programs (e.g. FITSIM, DYNAFIT) are widely applied to fit kinetic parameters to user-defined enzymatic mechanisms, but users often overlook the stringent requirements of the analytic procedures for appropriate design of the input experiments. Flaws in the experimental setup result in unreliable parameters with consequent misinterpretation of the biological phenomenon under study. The present commentary suggests some helpful mathematical tools to improve the analytic procedure in order to diagnose major errors in concept and design of kinetic experiments.     

Exclusion probabilities for single-locus paternity analysis when related males compete for matings

The statistical power of single-locus paternity analyses has previously been assessed by calculating an expected exclusion probability ( E ), the probability of excluding a randomly chosen nonfather. This E -statistic assumes that putative sires are a random selection of individuals from a panmictic study population. In species that display male natal philopatry, closely related individuals may be the principal competitors for paternity. In such structured populations, the E statistic will overestimate exclusion probability because males competing for paternity are more closely related than males chosen randomly from the population. A suite of loci thought to be sufficient for a panmictic population may frequently incorrectly assign paternity to close relatives of true sires. This study provides equations for calculating the expected probability of excluding a close male relative of the genetic sire ( Erel ) for any genotyping system that uses codominant markers. We also describe the use of Monte Carlo modelling to estimate exclusion probabilities when multiple male relatives compete for paternity. We show that the utility of a set of codominant markers will depend on the breeding behaviour and social system of the species in question.     

Efficiency considerations in the analysis of inter-observer agreement

The reliability of binary assessments is often measured by the proportion of agreement above chance, as estimated by the kappa statistic. In this paper, we develop a model to estimate inter-rater and intra-rater reliability when each of the two observers has the opportunity to obtain a pair of replicate measurements on each subject. The model is analogous to the nested beta-binomial model proposed by Rosner (1989, 1992). We show that the gain in precision obtained from increasing the number of measurements per rater from one to two may allow fewer subjects to be included in the study with no net loss in efficiency for estimating the inter-rater reliability.     

Dynamic modelling and analysis of biochemical networks: mechanism-based models and model-based experiments

Systems biology applies quantitative, mechanistic modelling to study genetic networks, signal transduction pathways and metabolic networks. Mathematical models of biochemical networks can look very different. An important reason is that the purpose and application of a model are essential for the selection of the best mathematical framework. Fundamental aspects of selecting an appropriate modelling framework and a strategy for model building are discussed. Concepts and methods from system and control theory provide a sound basis for the further development of improved and dedicated computational tools for systems biology. Identification of the network components and rate constants that are most critical to the output behaviour of the system is one of the major problems raised in systems biology. Current approaches and methods of parameter sensitivity analysis and parameter estimation are reviewed. It is shown how these methods can be applied in the design of model-based experiments which iteratively yield models that are decreasingly wrong and increasingly gain predictive power.     

Dynamic Bayesian sensitivity analysis of a myocardial metabolic model

Dynamic compartmentalized metabolic models are identified by a large number of parameters, several of which are either non-physical or extremely difficult to measure. Typically, the available data and prior information is insufficient to fully identify the system. Since the models are used to predict the behavior of unobserved quantities, it is important to understand how sensitive the output of the system is to perturbations in the poorly identifiable parameters. Classically, it is the goal of sensitivity analysis to asses how much the output changes as a function of the parameters. In the case of dynamic models, the output is a function of time and therefore its sensitivity is a time dependent function. If the output is a differentiable function of the parameters, the sensitivity at one time instance can be computed from its partial derivatives with respect to the parameters. The time course of these partial derivatives describes how the sensitivity varies in time.When the model is not uniquely identifiable, or if the solution of the parameter identification problem is known only approximately, we may have not one, but a distribution of possible parameter values. This is always the case when the parameter identification problem is solved in a statistical framework. In that setting, the proper way to perform sensitivity analysis is to not rely on the values of the sensitivity functions corresponding to a single model, but to consider the distributed nature of the sensitivity functions, inherited from the distribution of the vector of the model parameters.In this paper we propose a methodology for analyzing the sensitivity of dynamic metabolic models which takes into account the variability of the sensitivity over time and across a sample. More specifically, we draw a representative sample from the posterior density of the vector of model parameters, viewed as a random variable. To interpret the output of this doubly varying sensitivity analysis, we propose visualization modalities particularly effective at displaying simultaneously variations over time and across a sample. We perform an analysis of the sensitivity of the concentrations of lactate and glycogen in cytosol, and of ATP, ADP, NAD⁺ and NADH in cytosol and mitochondria, to the parameters identifying a three compartment model for myocardial metabolism during ischemia.