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.     

Determination of metabolic fluxes in a non-steady-state system

Estimation of fluxes through metabolic networks from redistribution patterns of (13)C has become a well developed technique in recent years. However, the approach is currently limited to systems at metabolic steady-state; dynamic changes in metabolic fluxes cannot be assessed. This is a major impediment to understanding the behaviour of metabolic networks, because steady-state is not always experimentally achievable and a great deal of information about the control hierarchy of the network can be derived from the analysis of flux dynamics. To address this issue, we have developed a method for estimating non-steady-state fluxes based on the mass-balance of mass isotopomers. This approach allows multiple mass-balance equations to be written for the change in labelling of a given metabolite pool and thereby permits over-determination of fluxes. We demonstrate how linear regression methods can be used to estimate non-steady-state fluxes from these mass balance equations. The approach can be used to calculate fluxes from both mass isotopomer and positional isotopomer labelling information and thus has general applicability to data generated from common spectrometry- or NMR-based analytical platforms. The approach is applied to a GC-MS time-series dataset of (13)C-labelling of metabolites in a heterotrophic Arabidopsis cell suspension culture. Threonine biosynthesis is used to demonstrate that non-steady-state fluxes can be successfully estimated from such data while organic acid metabolism is used to highlight some common issues that can complicate flux estimation. These include multiple pools of the same metabolite that label at different rates and carbon skeleton rearrangements.     

Modelling and prediction for chaotic fir laser attractor using rational function neural network

Many real-world systems such as irregular ECG signal, volatility of currency exchange rate and heated fluid reaction exhibit highly complex nonlinear characteristic known as chaos. These chaotic systems cannot be retreated satisfactorily using linear system theory due to its high dimensionality and irregularity. This research focuses on prediction and modelling of chaotic FIR (Far InfraRed) laser system for which the underlying equations are not given. This paper proposed a method for prediction and modelling a chaotic FIR laser time series using rational function neural network. Three network architectures, TDNN (Time Delayed Neural Network), RBF (radial basis function) network and the RF (rational function) network, are also presented. Comparisons between these networks performance show the improvements introduced by the RF network in terms of a decrement in network complexity and better ability of predictability.     

A comparison of methods for fitting allometric equations to field metabolic rates of animals

We re-examined data for field metabolic rates of varanid lizards and marsupial mammals to illustrate how different procedures for fitting the allometric equation can lead to very different estimates for the allometric coefficient and exponent. A two-parameter power function was obtained in each case by the traditional method of back-transformation from a straight line fitted to logarithms of the data. Another two-parameter power function was then generated for each data-set by non-linear regression on values in the original arithmetic scale. Allometric equations obtained by non-linear regression described the metabolic rates of all animals in the samples. Equations estimated by back-transformation from logarithms, on the other hand, described the metabolic rates of small species but not large ones. Thus, allometric equations estimated in the traditional way for field metabolic rates of varanids and marsupials do not have general importance because they do not characterize rates for species spanning the full range in body size. Logarithmic transformation of predictor and response variables creates new distributions that may enable investigators to perform statistical analyses in compliance with assumptions underlying the tests. However, statistical models fitted to transformations should not be used to estimate parameters of equations in the arithmetic domain because such equations may be seriously biased and misleading. Allometric analyses should be performed on values expressed in the original scale, if possible, because this is the scale of interest.     

An Efficient Algorithm for Some Highly Nonlinear Fractional PDEs in Mathematical Physics

In this paper, a fractional complex transform (FCT) is used to convert the given fractional partial differential equations (FPDEs) into corresponding partial differential equations (PDEs) and subsequently Reduced Differential Transform Method (RDTM) is applied on the transformed system of linear and nonlinear time-fractional PDEs. The results so obtained are re-stated by making use of inverse transformation which yields it in terms of original variables. It is observed that the proposed algorithm is highly efficient and appropriate for fractional PDEs and hence can be extended to other complex problems of diversified nonlinear nature.     

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.     

Thermodynamics of Random Reaction Networks

Reaction networks are useful for analyzing reaction systems occurring in chemistry, systems biology, or Earth system science. Despite the importance of thermodynamic disequilibrium for many of those systems, the general thermodynamic properties of reaction networks are poorly understood. To circumvent the problem of sparse thermodynamic data, we generate artificial reaction networks and investigate their non-equilibrium steady state for various boundary fluxes. We generate linear and nonlinear networks using four different complex network models (Erdős-Rényi, Barabási-Albert, Watts-Strogatz, Pan-Sinha) and compare their topological properties with real reaction networks. For similar boundary conditions the steady state flow through the linear networks is about one order of magnitude higher than the flow through comparable nonlinear networks. In all networks, the flow decreases with the distance between the inflow and outflow boundary species, with Watts-Strogatz networks showing a significantly smaller slope compared to the three other network types. The distribution of entropy production of the individual reactions inside the network follows a power law in the intermediate region with an exponent of circa −1.5 for linear and −1.66 for nonlinear networks. An elevated entropy production rate is found in reactions associated with weakly connected species. This effect is stronger in nonlinear networks than in the linear ones. Increasing the flow through the nonlinear networks also increases the number of cycles and leads to a narrower distribution of chemical potentials. We conclude that the relation between distribution of dissipation, network topology and strength of disequilibrium is nontrivial and can be studied systematically by artificial reaction networks.     

A Theory of Fluid Flow in Compliant Tubes

Starting with the Navier-Stokes equations, a system of equations is obtained to describe quasi-one-dimensional behavior of fluid in a compliant tube. The nonlinear terms which cannot be shown to be small in the original equations are retained, and the resulting equations are nonlinear. A functional pressure-area relationship is postulated and the final set of equations are quasi-linear and hyperbolic, with two independent and two dependent variables. A method of numerical solution of the set of equations is indicated, and the application to cases of interest is discussed.     

Systems biology towards life in silico: mathematics of the control of living cells

Systems Biology is the science that aims to understand how biological function absent from macromolecules in isolation, arises when they are components of their system. Dedicated to the memory of Reinhart Heinrich, this paper discusses the origin and evolution of the new part of systems biology that relates to metabolic and signal-transduction pathways and extends mathematical biology so as to address postgenomic experimental reality. Various approaches to modeling the dynamics generated by metabolic and signal-transduction pathways are compared. The silicon cell approach aims to describe the intracellular network of interest precisely, by numerically integrating the precise rate equations that characterize the ways macromolecules’ interact with each other. The non-equilibrium thermodynamic or ‘lin–log’ approach approximates the enzyme rate equations in terms of linear functions of the logarithms of the concentrations. Biochemical Systems Analysis approximates in terms of power laws. Importantly all these approaches link system behavior to molecular interaction properties. The latter two do this less precisely but enable analytical solutions. By limiting the questions asked, to optimal flux patterns, or to control of fluxes and concentrations around the (patho)physiological state, Flux Balance Analysis and Metabolic/Hierarchical Control Analysis again enable analytical solutions. Both the silicon cell approach and Metabolic/Hierarchical Control Analysis are able to highlight where and how system function derives from molecular interactions. The latter approach has also discovered a set of fundamental principles underlying the control of biological systems. The new law that relates concentration control to control by time is illustrated for an important signal transduction pathway, i.e. nuclear hormone receptor signaling such as relevant to bone formation. It is envisaged that there is much more Mathematical Biology to be discovered in the area between molecules and Life.     

Steady-state voltages, ion fluxes, and volume regulation in syncytial tissues.

Equations are developed that describe the steady-state relationships among ion fluxes, solute fluxes, water flow, voltage, concentration of solute, and hydrostatic pressure in a spherically symmetrical syncytial tissue. Each cell of the syncytium is assumed to have membrane channels for Na, K, and Cl, a membrane pump for Na/K, and some concentration of intracellular protein of net negative charge. However, the surface cells and inner cells of the tissue are assumed to have different distributions of membrane transport properties, hence there is a radial circulation of fluxes and a radial distribution of forces. Some reasonable approximations are made that allow analytic solutions of the nonlinear differential equations. These solutions are used to analyze data from the frog lens and are shown to account for the known steady-state properties of this tissue. Moreover, these solutions are used to make predictions on other steady-state properties, which have not been directly measured, and graphical results on the circulation of water, ions and solute through the frog lens are presented.     

Hierarchically organized populations: interactions between individual,population, and ecosystem levels

We consider a population of individuals which can be distributed in different equivalence classes. These classes are gathered in groups so that intragroup transformations are much more frequent than intergroup ones. We study linear systems in general, illustrated by the example of coupled individual and population levels. Then, we study nonlinear systems with the example of coupled population and ecosystem levels. We give methods to derive the dynamical equations for the different levels and to calculate interlevel coupling terms. We compare the coupling effects in the linear and in the nonlinear case.     

Fluxes,first passage times,and the reduction of Hill diagrams

The steady-state flux resulting from the coupling of two multistate systems is considered. The dynamics of these systems are described (a) as diffusion along a continuous one-dimensional free-energy profile specified by a conformational coordinate or (b) in terms of transitions between a discrete but arbitrary number of substates. If these multistate systems are connected in a simple way, it is shown that the steady-state flux can be obtained analytically. For both the continuous and discrete cases, the exact flux is shown to be identical to that calculated from a simple kinetic scheme involving only four states, if the effective rate constants of this reduced scheme are appropriately defined in terms of the mean first passage times for moving between various points along the multistate cycles. These results clarify and quantify the manner in which the internal conformational dynamics of two multistate systems influences the steady-state flux.     

Macroscopic Equations Governing Noisy Spiking Neuronal Populations with Linear Synapses

Deriving tractable reduced equations of biological neural networks capturing the macroscopic dynamics of sub-populations of neurons has been a longstanding problem in computational neuroscience. In this paper, we propose a reduction of large-scale multi-population stochastic networks based on the mean-field theory. We derive, for a wide class of spiking neuron models, a system of differential equations of the type of the usual Wilson-Cowan systems describing the macroscopic activity of populations, under the assumption that synaptic integration is linear with random coefficients. Our reduction involves one unknown function, the effective non-linearity of the network of populations, which can be analytically determined in simple cases, and numerically computed in general. This function depends on the underlying properties of the cells, and in particular the noise level. Appropriate parameters and functions involved in the reduction are given for different models of neurons: McKean, Fitzhugh-Nagumo and Hodgkin-Huxley models. Simulations of the reduced model show a precise agreement with the macroscopic dynamics of the networks for the first two models.     

Application of a new method of nonlinear dynamical system identification to biochemical problems

The system identification method for a variety of nonlinear dynamic models is elaborated. The problem of identification of an original nonlinear model presented as a system of ordinary differential equations in the Cauchy explicit form with a polynomial right part reduces to the solution of the system of linear equations for the constants of the dynamical model. In other words, to construct an integral model of the complex system (phenomenon), it is enough to collect some data array characterizing the time-course of dynamical parameters of the system. Collection of such a data array has always been a problem. However difficulties emerging are, as a rule, not principal and may be overcome almost without exception. The potentialities of the method under discussion are demonstrated by the example of the test problem of multiparametric nonlinear oscillator identification. The identification method proposed may be applied to the study of different biological systems and in particular the enzyme kinetics of complex biochemical reactions.     

Longitudinal analysis of the dynamics and risk of coronary heart disease in the Framingham Study.

Statistical methods designed specifically for the analysis of chronic disease incidence and progression in longitudinal studies are presented. These method model the risk of acute phases of chronic disease separately from the temporal change in risk variables. This could be accomplished because, under a specific biological model of the disease mechanism, the problems of estimating the risk of an acute event and of predicting the change in risk variables are independent. Specifically, a quadratic equation relating risk variable values to chronic disease risk and a system of linear equations predicting future risk variable values from present values may beestimated separately. Taken together, they utilize the full information available in a longitudinal study on the temporal dimension of chronic disease progression. In addition, the model is found to possess a number of attractive statistical and theoretical properties. These methods are applied to longitudinal data from the Framingham Study on coronary heart disease (CHD) in males. A quadratic function relating the risk of a CHD event to selected risk variables (age, and the natural logarithms of serum cholesterol, uric acid, diastolic blood pressure and pulse pressure) was estimated from measurements made at four points equally spaced in time (two years) with a further morbidity follow-up at a fifth point. The risk function was found to predict CHD risk accurately. It showed that, apart from the linear effects of the risk variables, cohort effects, quadratic effects and interaction effects were important predictors of CHD risk. The linear regression equations used to predict future risk variable values showed that there was an intricate network of cross-temporal associations. Study of the two types of equations jointly show that putative risk variables could affect the risk of CHD incidence both directly, by being associated with higher levels of risk, and indirectly, by causing other risk variable values to change with time. The results led us to identify several different roles that risk variables might play in CHD incidence.     

Nonlinear modulation of an extraordinary wave under the conditions of parametric decay

A self-consistent set of Hamilton equations describing nonlinear saturation of the amplitude of oscillations excited under the conditions of parametric decay of an elliptically polarized extraordinary wave in cold plasma is solved analytically and numerically. It is shown that the exponential increase in the amplitude of the secondary wave excited at the half-frequency of the primary wave changes into a reverse process in which energy is returned to the primary wave and nonlinear oscillations propagating across the external magnetic field are generated. The system of ??slow?? equations for the amplitudes, obtained by averaging the initial equations over the high-frequency period, is used to describe steady-state nonlinear oscillations in plasma.