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.     

An efficient method for calculation of dynamic logarithmic gains in biochemical systems theory

Biochemical systems theory (BST) characterizes a given biochemical system based on the logarithmic gains, rate-constant sensitivities and kinetic-order sensitivities defined at a steady state. This paper describes an efficient method for calculation of the time courses of logarithmic gains, i.e. dynamic logarithmic gains L(Xi, Xj; t), which expresses the percentage change in the value of a dependent variable Xi at a time t in response to an infinitesimal percentage change in the value of an independent variable Xj at t=0. In this method, one first recasts the ordinary differential equations for the dependent variables into an exact canonical nonlinear representation (GMA system) through appropriate transformations of variables. Owing to the structured mathematical form of this representation, the recast system can be fully described by a set of numeric parameters, and the differential equations for the dynamic logarithmic gains can be set up automatically without resource to computer algebra. A simple general-purpose computer program can thus be written that requires only the relevant numeric parameters as input to calculate the time courses of the variables and of the dynamic logarithmic gains for both concentrations and fluxes. Unlike other methods, the proposed method does not require to derive any expression for the partial differentiation of flux expressions with respect to each independent variable. The proposed method has been applied to two kinds of reaction models to elucidate its usefulness.     

Computation of a Stabilizing Set of Feedback Matrices bf a Large-scale Nonlinear Musculoskeletal Dynamic Model

The purpose of this study is to present a general mathematical framework to compute a set of feedback matrices which stabilize an unstable nonlinear anthropomorphic musculoskeletal dynamic model. This method is activity specific and involves four fundamental stages. First, from muscle activation data (input) and motion degrees-of-freedom (output) a dynamic experimental model is obtained using system identification schemes. Second, a nonlinear musculoskeletal dynamic model which contains the same number of muscles and degrees-of-freedom and best represents the activity being considered is proposed. Third, the nonlinear musculoskeletal model (anthropomorphic model) is replaced by a family of linear systems, parameterized by the same set of input/ output data (nominal points) used in the identification of the experimental model. Finally, a set of stabilizing output feedback matrices, parameterized again by the same set of nominal points, is computed such that when combined with the anthropomorphic model, the combined system resembles the structural form of the experimental model. The method is illustrated in regard to the human squat activity.     

Computation of a stabilizing set of feedback matrices of a large-scale nonlinear musculoskeletal dynamic model

The purpose of this study is to present a general mathematical framework to compute a set of feedback matrices which stabilize an unstable nonlinear anthropomorphic musculoskeletal dynamic model. This method is activity specific and involves four fundamental stages. First, from muscle activation data (input) and motion degrees-of-freedom (output) a dynamic experimental model is obtained using system identification schemes. Second, a nonlinear musculoskeletal dynamic model which contains the same number of muscles and degrees-of-freedom and best represents the activity being considered is proposed. Third, the nonlinear musculoskeletal model (anthropomorphic model) is replaced by a family of linear systems, parameterized by the same set of input/output data (nominal points) used in the identification of the experimental model. Finally, a set of stabilizing output feedback matrices, parameterized again by the same set of nominal points, is computed such that when combined with the anthropomorphic model, the combined system resembles the structural form of the experimental model. The method is illustrated in regard to the human squat activity.     

Evolutionary optimization with data collocation for reverse engineering of biological networks

MOTIVATION: Modern experimental biology is moving away from analyses of single elements to whole-organism measurements. Such measured time-course data contain a wealth of information about the structure and dynamic of the pathway or network. The dynamic modeling of the whole systems is formulated as a reverse problem that requires a well-suited mathematical model and a very efficient computational method to identify the model structure and parameters. Numerical integration for differential equations and finding global parameter values are still two major challenges in this field of the parameter estimation of nonlinear dynamic biological systems. RESULTS: We compare three techniques of parameter estimation for nonlinear dynamic biological systems. In the proposed scheme, the modified collocation method is applied to convert the differential equations to the system of algebraic equations. The observed time-course data are then substituted into the algebraic system equations to decouple system interactions in order to obtain the approximate model profiles. Hybrid differential evolution (HDE) with population size of five is able to find a global solution. The method is not only suited for parameter estimation but also can be applied for structure identification. The solution obtained by HDE is then used as the starting point for a local search method to yield the refined estimates.     

The identification of nonlinear biological systems: LNL cascade models

Systems that can be represented by a cascade of a dynamic linear (L), a static nonlinear (N) and a dynamic linear (L) subsystem are considered. Various identification schemes that have been proposed for these LNL systems are critically reviewed with reference to the special problems that arise in the identification of nonlinear biological systems. A simulated LNL system is identified from limited duration input-output data using an iterative identification scheme.     

Dynamic Causal Modeling and subspace identification methods

The main contribution of the paper is in formulating the problem of detection of brain regions structure within the framework of dynamic system theory. The motivation is to see if the mature domain of experimental identification of dynamic systems can provide a methodology alternative to Dynamic Causal Modeling (DCM) which is currently used as an exclusive tool to estimate the structure of interconnections among a given set of brain regions using the measured data from functional magnetic resonance imaging (fMRI). The key tool proposed for modeling the structure of brain interconnections in this paper is subspace identification methods which produce linear state-space model, thus neglecting the bilinear term from DCM. The procedure is illustrated using a simple two-region model with maximally simplified linearized hemodynamics. We assume that the underlying system can be modeled by a set of linear differential equations, and identify the parameters (in terms of state space matrices), without any a priori constraints. We then transform the hidden states so that the implicit state matrix has a form or structure that is consistent with the generation of (region-specific) hemodynamic signals by coupled neuronal states.     

A general pre-steady-state solution to complex kinetic mechanisms

We have developed a general method for solving transient kinetic equations using Laplace transforms. Laplace transforms can be used to transform systems of differential equations that describe pre-steady-state kinetics to systems of linear algebraic equations. The general form of the pre-steady-state solution is (formula; see text) where I(t) is the time dependence of the physically observed property of the system, n is the number of intermediates, lambda i are the observed rate constants (reciprocals of the relaxation times), t is time, and Ii are the amplitude coefficients associated with each observed rate constant. We have written a program in compiled BASIC to run on a personal computer to evaluate Ii and lambda i. The program will evaluate the rate constants and coefficients of a mechanism with eight intermediates and seven relaxation times in 4 s on an 8-MHz PC-XT equipped with a math coprocessor. The most complex mechanism that we have solved, a mechanism containing 20 intermediates and 19 relaxation times, required approximately 5 min. We believe that this method will be useful to evaluate the differences in transient properties of complex biochemical mechanisms.     

Transport Effects on Multiple-Component Reactions in Optical Biosensors

Optical biosensors are often used to measure kinetic rate constants associated with chemical reactions. Such instruments operate in the surface–volume configuration, in which ligand molecules are convected through a fluid-filled volume over a surface to which receptors are confined. Currently, scientists are using optical biosensors to measure the kinetic rate constants associated with DNA translesion synthesis—a process critical to DNA damage repair. Biosensor experiments to study this process involve multiple interacting components on the sensor surface. This multiple-component biosensor experiment is modeled with a set of nonlinear integrodifferential equations (IDEs). It is shown that in physically relevant asymptotic limits these equations reduce to a much simpler set of ordinary differential equations (ODEs). To verify the validity of our ODE approximation, a numerical method for the IDE system is developed and studied. Results from the ODE model agree with simulations of the IDE model, rendering our ODE model useful for parameter estimation.     

Analytic procedures for large dimention nonlinear biochemical oscillators.

An analytic method is presented which can be used to determine if the following system of nonlinear differential equations has periodic solutions x1 = h(xn)-b1x1 xj = gj-1xj-1-bjxj j = 2, ... n A systematic dual input describing function procedure is given for constructing a function of the reaction constants R, where if R greater than 1 a periodic solution exists and if R smaller than 1 there is no periodic solution. The form of R constructed generalizes immediately to an arbitrarily large dimension. The method generalizes to cover systems displaying hysteresis kinetics, systems subject to chemical noise, and systems containing delay components. The method has been applied to a well known biochemical problem where h(xn)-k/(1 + alphaxnrho). For rho = 1, for all n, there are no stable limit cycles such that xj(t) greater than O, t larger than or equal to O. For rho = 2,n larger than or equal to 8 it is possible to construct a parameter set such that stable oscillations appear.     

Simultaneous inference for ratios of linear combinations of general linear model parameters

Consider a general linear model with p -dimensional parameter vector beta and i.i.d. normal errors. Let K(1), ..., K(k ), and L be linearly independent vectors of constants such that L(T)beta not equal 0. We describe exact simultaneous tests for hypotheses that Ki(T)beta/L(T)beta equal specified constants using one-sided and two-sided alternatives, and describe exact simultaneous confidence intervals for these ratios. In the case where the confidence set is a single bounded contiguous set, we describe what we claim are the best possible conservative simultaneous confidence intervals for these ratios - best in that they form the minimum k -dimensional hypercube enclosing the exact simultaneous confidence set. We show that in the case of k = 2, this "box" is defined by the minimum and maximum values for the two ratios in the simultaneous confidence set and that these values are obtained via one of two sources: either from the solutions to each of four systems of equations or at points along the boundary of the simultaneous confidence set where the correlation between two t variables is zero. We then verify that these intervals are narrower than those previously presented in the literature.     

Ring1b-mediated H2A ubiquitination associates with inactive X chromosomes and is involved in initiation of X inactivation

Histone modifications are thought to serve as epigenetic markers that mediate dynamic changes in chromatin structure and regulation of gene expression. As a model system for understanding epigenetic silencing, X chromosome inactivation has been previously linked to a number of histone modifications including methylation and hypoacetylation. In this study, we provide evidence that supports H2A ubiquitination as a novel epigenetic marker for the inactive X chromosome (Xi) and links H2A ubiquitination to initiation of X inactivation. We found that the H2A-K119 ubiquitin E3 ligase Ring1b, a Polycomb group protein, is enriched on Xi in female trophoblast stem (TS) cells as well as differentiating embryonic stem (ES) cells. Consistent with Ring1b mediating H2A ubiquitination, ubiquitinated H2A (ubH2A) is also enriched on the Xi of both TS and ES cells. We demonstrate that the enrichment of Ring1b and ubH2A on Xi is transient during TS and ES cell differentiation, suggesting that the Ring1b and ubH2A are involved in the initiation of both imprinted and random X inactivation. Furthermore, we showed that the association of Ring1b and ubH2A with Xi is mitotically stable in non-differentiated TS cells.     

Biochemical switching device: monocyclic enzyme system

The switching characteristics of a monocyclic enzyme system, in which two enzymes share substrates or co-factors in a cyclic manner, such as, --> X(1) + B + E(1) right arrow over left arrow A + E(1) + X(2) -->, --> X(3) + A + E(2) right arrow over left arrow B + E(2) + X(4) --> (E(1), E(2) are enzymes, X(1), X(3) are substrates, X(2), X(4) are products, A, B are cofactors), were demonstrated using computer simulations. The detailed mathematical models of biochemically possible cyclic enzyme systems were built up and the effects of rate constants and the effects of initial concentrations of enzymes and cofactors on switching characteristics were discussed. The cyclic enzyme system could function as a switching circuit when the initial concentrations of enzymes or cofactors are over a certain threshold value. Based on the present results, we further discuss the dynamic characteristics of a biochemical reactor system (bioreactor) involving this cyclic enzyme system as a switching controller.     

Transient behavior of a chemotaxis system modelling certain types of tissue inflammation

A spatially-distributed mathematical model for the inflammatory response to bacterial invasion of tissue is proposed which includes leukocyte motility and chemotaxis behavior and chemical mediator properties explicitly. This system involves three coupled nonlinear partial differential equations and so is not amenable to analysis. Using scaling arguments and singular perturbation techniques, an approximating system of two coupled nonlinear ordinary differential equations is developed. This system now permits analysis by phase plane methods. Using the approximating model, the dependence of the dynamic behavior of the inflammatory response upon key process parameters, including leukocyte chemotaxis, is studied.This work has been supported by the Deutsche Forschungsgemeinschaft     

植被系统中植物与环境因子相互作用的动态模拟

本文在综合考虑现有植被发展动态模型的基础上提出包含植被,环境因子,人类或其它外力干扰三者间相互作用情况下的植被系统一般动态模型,并从控制论的角度阐述了植被与环境因子间的耦合,反馈作用。最后以线性逼近和微分拟合的方法求取模型的局部解,进而作出外延预测。实例分析表明,模型预测对实际观测有较好的跟踪性能。     

NMR spin exchange kinetics at equilibrium in membrane transport and enzyme systems

A set of differential equations is formulated to describe the rapid exchange (time scale, approximately 0.01 to approximately 10 s) of a labelled solute across the membranes of cells in suspension. The labelling is achieved with nuclear magnetic resonance by exposure of the system to a high intensity radio-frequency pulse, and the excited nuclei relax to the equilibrium state with a short half life. An analytical expression for the decay of the magnetic resonance signal is presented; the solution involves the determination of eigenvalues, of an array of Laplace-Carson transformed differential equations, by use of the general solution of a quartic polynomial. Simulations of the behaviour of the exchange system using various conditions of cell number, rate constants and nuclear magnetic relaxation times are presented. The marked concentration dependence of the extent of reaction at a given time has not previously been reported for nuclear magnetic resonance exchange systems and is a feature anticipated from the known saturability of several membrane transport systems including glucose transport into human erythrocytes. The theory is readily generalized to other model systems by appropriate reinterpretation of the physical meaning of various parameters; the general form of the solution holds in many biological contexts other than membrane transport and includes equilibrium enzyme kinetics.     

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.     

New Karplus equations for 2JHH, 3JHH, 2JCH, 3JCH, 3JCOCH, 3JCSCH, and 3JCCCH in some aldohexopyranoside derivatives as determined using NMR spectroscopy and density functional theory calculations

The (1)H-(13)C coupling constants of methyl alpha- and beta-pyranosides of D-glucose and D-galactose have been measured by one-dimensional and two-dimensional (1)H-(13)C heteronuclear zero and double quantum, phase sensitive J-HMBC spectra to determine a complete set of coupling constants ((1)J(CH), (2)J(CH), (3)J(CH), (2)J(HH), and (3)J(HH)) within the exocyclic hydroxymethyl group (CH(2)OH) for each compound. In parallel with these experimental studies, structure, energy, and potential energy surfaces of the hydroxymethyl group for these compounds were determined employing quantum mechanical calculations at the B3LYP level using the 6-311++G( * *) basis set. Values of the vicinal coupling constants involving (1)H and (13)C in the C5-C6 (omega) and C6-O6 (theta) torsion angles in the aldohexopyranoside model compounds were calculated with water as the solvent using the PCM method. To test the relationship between (3)J(CXCH) (X=C, O, S) and torsion angle C1-X (phi) around the anomeric center, the conformations of 24 derivatives of glucose and galactose, which represent sequences of atoms at the anomeric center of C-glycosides (C-C bond), O-glycosides (C-O bond), thioglycosides (C-S bond), glycosylamines (C-N bond), and glycosyl halides (C-halogen (F/Cl) bond) have been calculated. Nonlinear regression analysis of the coupling constants (1)J(C1,H1), (2)J(C5,H6R), (2)J(C5,H6S), (2)J(C6,H5), (3)J(C4,H6R), (3)J(C4,H6S), (2)J(H6R,H5), and (3)J(H5,H6R) as well as (3)J(CXCH) (X=C, O, S) on the dihedral angles omega, theta, and phi have yielded new Karplus equations. Good agreement between calculated and experimentally measured coupling constants revealed that the DFT method was able to accurately predict J-couplings in aqueous solutions.     

Energy cost of treadmill and floor walking at self-selected paces

Oxygen uptake-velocity regression equations were developed for floor and level treadmill walking by having two groups of men, aged 19-29 years (n = 20) and 55-66 years (n = 22), walk at four self-selected paces, from "rather slowly" to "as fast as possible". A two-variable quadratic model relating VO2 (ml X kg-1 X min-1) to velocity (m X s-1) was adopted for prediction purposes. However, age and fatness significantly (P less than 0.05) interacted with treadmill walking speed, while age alone significantly interacted with floor speed. In addition, a significant difference was found between the energy cost of floor and treadmill walking. For example at the normal walking speed of 1.33 m X s-1, the energy cost for the treadmill (age 55-66 years) was 10.58 ml X kg-1 X min-1 and for the floor, 11.04 ml X kg-1 X min-1 (P less than 0.05). Four quadratic equations are therefore presented, one each for floor and treadmill in each of the two age-groups. The percent variance explained was between 87 and 95% for each of these equations.     

Dynamic output feedback stabilization for nonlinear systems based on standard neural network models

A neural-model-based control design for some nonlinear systems is addressed. The design approach is to approximate the nonlinear systems with neural networks of which the activation functions satisfy the sector conditions. A novel neural network model termed standard neural network model (SNNM) is advanced for describing this class of approximating neural networks. Full-order dynamic output feedback control laws are then designed for the SNNMs with inputs and outputs to stabilize the closed-loop systems. The control design equations are shown to be a set of linear matrix inequalities (LMIs) which can be easily solved by various convex optimization algorithms to determine the control signals. It is shown that most neural-network-based nonlinear systems can be transformed into input-output SNNMs to be stabilization synthesized in a unified way. Finally, some application examples are presented to illustrate the control design procedures.