In vitro demonstration of alternate stationary states in an open enzyme system containing phosphofructokinase

The kinetic properties of an open reconstituted enzyme system are investigated with the aim of demonstrating experimentally hysteretic transitions between alternate stationary states. The approach is based on a stirred flow-through reactor containing phosphofructokinase and pyruvate kinase entrapped in polyacrylamide gel. Through the reactor is pumped a solution containing fructose 6-phosphate, ATP, and phosphoenol pyruvate as well as adenylate kinase and glucose-6-phosphate isomerase. The latter two enzymes are in excess in respect to phosphofructokinase and pyruvate kinase. According to theoretical predictions the existence of multiple stationary states could be shown experimentally within precisely definable parameters. Switches between alternate stationary states have been caused by perturbations of flow rates and of reactant concentrations.     

Estimating the distribution of dynamic invariants: illustrated with an application to human photo-plethysmographic time series

Dynamic invariants are often estimated from experimental time series with the aim of differentiating between different physical states in the underlying system. The most popular schemes for estimating dynamic invariants are capable of estimating confidence intervals, however, such confidence intervals do not reflect variability in the underlying dynamics. We propose a surrogate based method to estimate the expected distribution of values under the null hypothesis that the underlying deterministic dynamics are stationary. We demonstrate the application of this method by considering four recordings of human pulse waveforms in differing physiological states and show that correlation dimension and entropy are insufficient to differentiate between these states. In contrast, algorithmic complexity can clearly differentiate between all four rhythms.     

Parameter estimation in stochastic biochemical reactions

Gene regulatory, signal transduction and metabolic networks are major areas of interest in the newly emerging field of systems biology. In living cells, stochastic dynamics play an important role; however, the kinetic parameters of biochemical reactions necessary for modelling these processes are often not accessible directly through experiments. The problem of estimating stochastic reaction constants from molecule count data measured, with error, at discrete time points is considered. For modelling the system, a hidden Markov process is used, where the hidden states are the true molecule counts, and the transitions between those states correspond to reaction events following collisions of molecules. Two different algorithms are proposed for estimating the unknown model parameters. The first is an approximate maximum likelihood method that gives good estimates of the reaction parameters in systems with few possible reactions in each sampling interval. The second algorithm, treating the data as exact measurements, approximates the number of reactions in each sampling interval by solving a simple linear equation. Maximising the likelihood based on these approximations can provide good results, even in complex reaction systems.     

Reducing complexity in metabolic networks: making metabolic meshes manageable

The dynamics of complex systems can be effectively analyzed by judicious use of intrinsic time constants. Order of magnitude estimation based on time constants has been used successfully to examine the dynamic behavior of complicated processes. The main goal of this paper is to introduce this approach to the analysis of complex metabolic systems. Time constants and dynamic modes of motion are defined within the context of well-established linear algebra. The order of magnitude estimation is then introduced into the systemic framework. The main goals of the analysis are: to provide improved understanding of biochemical dynamics and their physiological significance, and to yield reduced dynamic models that are physiologically realistic but tractable for practical use.     

Extended Kalman Filter for Estimation of Parameters in Nonlinear State-Space Models of Biochemical Networks

It is system dynamics that determines the function of cells, tissues and organisms. To develop mathematical models and estimate their parameters are an essential issue for studying dynamic behaviors of biological systems which include metabolic networks, genetic regulatory networks and signal transduction pathways, under perturbation of external stimuli. In general, biological dynamic systems are partially observed. Therefore, a natural way to model dynamic biological systems is to employ nonlinear state-space equations. Although statistical methods for parameter estimation of linear models in biological dynamic systems have been developed intensively in the recent years, the estimation of both states and parameters of nonlinear dynamic systems remains a challenging task. In this report, we apply extended Kalman Filter (EKF) to the estimation of both states and parameters of nonlinear state-space models. To evaluate the performance of the EKF for parameter estimation, we apply the EKF to a simulation dataset and two real datasets: JAK-STAT signal transduction pathway and Ras/Raf/MEK/ERK signaling transduction pathways datasets. The preliminary results show that EKF can accurately estimate the parameters and predict states in nonlinear state-space equations for modeling dynamic biochemical networks.     

Time hierarchy in enzymatic reaction chains resulting from optimality principles

Several optimality principles reasonable for the evolution of enzyme reaction chains are formulated, considering the kinetic parameters of the enzymes to be variables. The solutions of these parametric programming problems are studied for the case of linear kinetics and turn out to be not unique in every case. The investigations are confined to stationary states. The net flux through the chain is considered a fixed parameter. The optimality criteria concern the osmotic pressure caused by the intermediates, various relaxation times, largest time scale, and controllability. They all yield distinct time hierarchies. The influence of a constraint concerning the sum of intermediate concentrations is studied. Various combinations of the single criteria are treated as multiobjective programming problems.     

Identification of small scale biochemical networks based on general type system perturbations

New technologies enable acquisition of large data-sets containing genomic, proteomic and metabolic information that describe the state of a cell. These data-sets call for systematic methods enabling relevant information about the inner workings of the cell to be extracted. One important issue at hand is the understanding of the functional interactions between genes, proteins and metabolites. We here present a method for identifying the dynamic interactions between biochemical components within the cell, in the vicinity of a steady-state. Key features of the proposed method are that it can deal with data obtained under perturbations of any system parameter, not only concentrations of specific components, and that the direct effect of the perturbations does not need to be known. This is important as concentration perturbations are often difficult to perform in biochemical systems and the specific effects of general type perturbations are usually highly uncertain, or unknown. The basis of the method is a linear least-squares estimation, using time-series measurements of concentrations and expression profiles, in which system states and parameter perturbations are estimated simultaneously. An important side-effect of also employing estimation of the parameter perturbations is that knowledge of the system's steady-state concentrations, or activities, is not required and that deviations from steady-state prior to the perturbation can be dealt with. Time derivatives are computed using a zero-order hold discretization, shown to yield significant improvements over the widely used Euler approximation. We also show how network interactions with dynamics that are too fast to be captured within the available sampling time can be determined and excluded from the network identification. Known and unknown moiety conservation relationships can be processed in the same manner. The method requires that the number of samples equals at least the number of network components and, hence, is at present restricted to relatively small-scale networks. We demonstrate herein the performance of the method on two small-scale in silico genetic networks.     

Concerted simulations reveal how peroxidase compound III formation results in cellular oscillations

A major problem in mathematical modeling of the dynamics of complex biological systems is the frequent lack of knowledge of kinetic parameters. Here, we apply Brownian dynamics simulations, based on protein three-dimensional structures, to estimate a previously undetermined kinetic parameter, which is then used in biochemical network simulations. The peroxidase-oxidase reaction involves many elementary steps and displays oscillatory dynamics important for immune response. Brownian dynamics simulations were performed for three different peroxidases to estimate the rate constant for one of the elementary steps crucial for oscillations in the peroxidase-oxidase reaction, the association of superoxide with peroxidase. Computed second-order rate constants agree well with available experimental data and permit prediction of rate constants at physiological conditions. The simulations show that electrostatic interactions depress the rate of superoxide association with myeloperoxidase, bringing it into the range necessary for oscillatory behavior in activated neutrophils. Such negative electrostatic steering of enzyme-substrate association presents a novel control mechanism and lies in sharp contrast to the electrostatically-steered fast association of superoxide and Cu/Zn superoxide dismutase, which is also simulated here. The results demonstrate the potential of an integrated and concerted application of structure-based simulations and biochemical network simulations in cellular systems biology.     

Personality development in psychotherapy: a synergetic model of state-trait dynamics

Theoretical models of psychotherapy not only try to predict outcome but also intend to explain patterns of change. Studies showed that psychotherapeutic change processes are characterized by nonlinearity, complexity, and discontinuous transitions. By this, theoretical models of psychotherapy should be able to reproduce these dynamic features. Using time series derived from daily measures through internet-based real-time monitoring as empirical reference, we earlier presented a model of psychotherapy which includes five state variables and four trait variables. In mathematical terms, the traits modulate the shape of the functions which define the nonlinear interactions between the variables (states) of the model. The functions are integrated into five coupled nonlinear difference equations. In the present paper, we model how traits (dispositions or competencies of a person) can continuously be altered by new experiences and states (cognition, emotion, behavior). Adding equations that link states to traits, this model not only describes how therapeutic interventions modulate short-term change and fluctuations of psychological states, but also how these can influence traits. Speaking in terms of Synergetics (theory of self-organization in complex systems), the states correspond to the order parameters and the traits to the control parameters of the system. In terms of psychology, trait dynamics is driven by the states—i.e., by the concrete experiences of a client—and creates a process of personality development at a slower time scale than that of the state dynamics (separation of time scales between control and order parameter dynamics).     

Dynamics of the B-A transition of DNA double helices

Although the transition from the B-DNA double helix to the A-form is essential for biological function, as shown by the existence of the A-form in many protein–DNA complexes, the dynamics of this transition has not been resolved yet. According to molecular dynamics simulations the transition is expected in the time range of a few nanoseconds. The B–A transition induced by mixing of DNA samples with ethanol in stopped flow experiments is complete within the deadtime, showing that the reaction is faster than ~0.2 ms. The reaction was resolved by an electric field jump technique with induction of the transition by a dipole stretching force driving the A- to the B-form. Poly[d(A-T)] was established as a favourable model system, because of a particularly high cooperativity of the transition and because of a spectral signature allowing separation of potential side reactions. The time constants observed in the case of poly[d(A-T)] with ~1600 bp are in the range around 10 µs. An additional process with time constants of ~100 µs is probably due to nucleation. The same time constants (within experimental accuracy ±10%) were observed for a poly[d(A-T)] sample with ~70 bp. Under low salt conditions commonly used for studies of the B–A transition, the time constants are almost independent of the ionic strength. The experimental data show that a significant activation barrier exists in the B–A transition and that the helical states are clearly separated from each other, in contrast to predictions by molecular dynamics simulations.     

Free energy and the kinetics of biochemical diagrams, including active transport.

In earlier papers on muscle contraction it was found very useful to relate the actual (not standard) free energy levels of the different states in the biochemical diagram of the myosin cross-bridge to the first-order rate constants governing transitions between these states and to the details of the conversion of ATP free energy into mechanical work. This same approach is applied here to other macromolecular biochemical systems, for example, carriers in active transport, and simple enzyme reactions. With the definition of free energy changes between states of diagram used here (and in the muscle papers), the rate constants of the diagram are firat order, the macromolecular transitions are effectively isomeric, the equilibrium constants are dimensionless, the free energy changes are directly related to first-order rate constant ratios, and the ratio of products of forward and backward rate constants around any cycle of the diagram is related to operational free energy changes (e.g. the in vivo free energy of ADP HYDROLYSIS). These general points are illustrated by means of particular arbitrary models, especially transport models. In contrast to the muscle case, the free energy conversion question in other biochemical systems can be handled at the less detailed, complete-cycle level rather than at the elementary transition level. There is a corresponding complete-cycle kinetics, with composite first-order rate constants for the different possible cycles (in both directions). An introductory stochastic treatment of cycle kinetics is included.     

Adding Detritus to a Nutrient-Phytoplankton-Zooplankton Model:A Dynamical-Systems Approach

The dynamics of two plankton population models are investigatedto examine sensitivities to model complexity and to parametervalues. The models simulate concentrations of nutrients, phytoplankton,zooplankton and detritus in the oceanic mixed layer. In Model1, zooplankton can graze only upon phytoplankton, whereas inModel 2 they can graze upon phytoplankton and detritus. Bothfeeding strategies are employed by zooplankton in the ocean,and both are features of models in the literature. Each modelhere consists of four coupled ordinary differential equations,and can exhibit unforced oscillations (limit cycles) of thefour concentrations. By constructing diagrams that show howsteady states and oscillations persist as each parameter isvaried, a general picture of the dynamics of each model is builtup. The addition of the detritus pool to an earlier nutrient–phytoplankton–zooplanktonmodel appears to have little influence on the dynamics whenthe zooplankton cannot graze upon the detritus (Model 1), butif the zooplankton can graze upon the detritus (Model 2), thenthe dynamics are affected in a significant way. These results,obtained using the theory of dynamical systems, enhance ourknowledge of the factors governing the dynamics of planktonpopulation models.     

Thermodynamically consistent Bayesian analysis of closed biochemical reaction systems

Background  

Estimating the rate constants of a biochemical reaction system with known stoichiometry from noisy time series measurements of molecular concentrations is an important step for building predictive models of cellular function. Inference techniques currently available in the literature may produce rate constant values that defy necessary constraints imposed by the fundamental laws of thermodynamics. As a result, these techniques may lead to biochemical reaction systems whose concentration dynamics could not possibly occur in nature. Therefore, development of a thermodynamically consistent approach for estimating the rate constants of a biochemical reaction system is highly desirable.     

Analysis of the quasi-steady-state approximation for an enzymatic one-substrate reaction

The validity of the quasi-steady state approximation for the calculation of the rate function of an isolated enzyme reaction is analysed by a detailed consideration of the time dependent process. For the characterization of the deviations of the real motion from the quasi-stationary state three kinds of error functions are used, the relaxation deficit, the relative relaxation time and the relaxation error. An improved approximation procedure is developed to calculate the transient states of the system. The maximum distance of the original motion from the quasi-stationary states is estimated by a general method. By consideration of different enzyme and substrate concentrations as well as different kinetic constants those parameter regions have been determined, where the errors of the quasi-steady state approximation do not exceed tolerated values. It is suggested how the methods can be applied to metabolic pathways.     

Theoretical analysis of the relation between the cell state and pathways of carbon isotope fractionation in metabolic processes

On a dynamic model of fractionation of carbon isotopes in the living cell there are considered relationships between the distribution of carbon isotopes in the structures approximating basic biochemical fractions, their isotopic composition and parameters characterizing the dynamics of carbon metabolism, i.e. efficient carbon isotope separation factor in pyruvate enzymic decarboxylation, degree of its transformation at primary and secondary decarboxylation and ratios between the currents of carbon substrates. A wide range of variations of cell isotope parameters resulting from the change of its functional states was revealed. Possible applications of the relationships observed for studying biological systems are shown.     

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.     

Ultrasensitive investigations of biological systems by fluorescence correlation spectroscopy

Fluorescence correlation spectroscopy (FCS) extracts information about molecular dynamics from the tiny fluctuations that can be observed in the emission of small ensembles of fluorescent molecules in thermodynamic equilibrium. Employing a confocal setup in conjunction with highly dilute samples, the average number of fluorescent particles simultaneously within the measurement volume (approximately 1 fl) is minimized. Among the multitude of chemical and physical parameters accessible by FCS are local concentrations, mobility coefficients, rate constants for association and dissociation processes, and even enzyme kinetics. As any reaction causing an alteration of the primary measurement parameters such as fluorescence brightness or mobility can be monitored, the application of this noninvasive method to unravel processes in living cells is straightforward. Due to the high spatial resolution of less than 0.5 microm, selective measurements in cellular compartments, e.g., to probe receptor-ligand interactions on cell membranes, are feasible. Moreover, the observation of local molecular dynamics provides access to environmental parameters such as local oxygen concentrations, pH, or viscosity. Thus, this versatile technique is of particular attractiveness for researchers striving for quantitative assessment of interactions and dynamics of small molecular quantities in biologically relevant systems.     

Full-scale model of glycolysis in Saccharomyces cerevisiae.

We present a powerful, general method of fitting a model of a biochemical pathway to experimental substrate concentrations and dynamical properties measured at a stationary state, when the mechanism is largely known but kinetic parameters are lacking. Rate constants and maximum velocities are calculated from the experimental data by simple algebra without integration of kinetic equations. Using this direct approach, we fit a comprehensive model of glycolysis and glycolytic oscillations in intact yeast cells to data measured on a suspension of living cells of Saccharomyces cerevisiae near a Hopf bifurcation, and to a large set of stationary concentrations and other data estimated from comparable batch experiments. The resulting model agrees with almost all experimentally known stationary concentrations and metabolic fluxes, with the frequency of oscillation and with the majority of other experimentally known kinetic and dynamical variables. The functional forms of the rate equations have not been optimized.     

Effects of host social hierarchy on disease persistence

The effects of social hierarchy on population dynamics and epidemiology are examined through a model which contains a number of fundamental features of hierarchical systems, but is simple enough to allow analytical insight. In order to allow for differences in birth rates, contact rates and movement rates among different sets of individuals the population is first divided into subgroups representing levels in the hierarchy. Movement, representing dominance challenges, is allowed between any two levels, giving a completely connected network. The model includes hierarchical effects by introducing a set of dominance parameters which affect birth rates in each social level and movement rates between social levels, dependent upon their rank. Although natural hierarchies vary greatly in form, the skewing of contact patterns, introduced here through non-uniform dominance parameters, has marked effects on the spread of disease. A simple homogeneous mixing differential equation model of a disease with SI dynamics in a population subject to simple birth and death process is presented and it is shown that the hierarchical model tends to this as certain parameter regions are approached. Outside of these parameter regions correlations within the system give rise to deviations from the simple theory. A Gaussian moment closure scheme is developed which extends the homogeneous model in order to take account of correlations arising from the hierarchical structure, and it is shown that the results are in reasonable agreement with simulations across a range of parameters. This approach helps to elucidate the origin of hierarchical effects and shows that it may be straightforward to relate the correlations in the model to measurable quantities which could be used to determine the importance of hierarchical corrections. Overall, hierarchical effects decrease the levels of disease present in a given population compared to a homogeneous unstructured model, but show higher levels of disease than structured models with no hierarchy. The separation between these three models is greatest when the rate of dominance challenges is low, reducing mixing, and when the disease prevalence is low. This suggests that these effects will often need to be considered in models being used to examine the impact of control strategies where the low disease prevalence behaviour of a model is critical.