Modelling biochemical networks with intrinsic time delays: a hybrid semi-parametric approach

Background  

This paper presents a method for modelling dynamical biochemical networks with intrinsic time delays. Since the fundamental mechanisms leading to such delays are many times unknown, non conventional modelling approaches become necessary. Herein, a hybrid semi-parametric identification methodology is proposed in which discrete time series are incorporated into fundamental material balance models. This integration results in hybrid delay differential equations which can be applied to identify unknown cellular dynamics.     

Analysis of progress curves by simulations generated by numerical integration.

A highly flexible computer program written in FORTRAN is presented which fits computer-generated simulations to experimental progress-curve data by an iterative non-linear weighted least-squares procedure. This fitting procedure allows kinetic rate constants to be determined from the experimental progress curves. Although the numerical integration of the rate equations by a previously described method [Barshop, Wrenn & Frieden (1983) Anal. Biochem. 130, 134-145] is used here to generate predicted curves, any routine capable of the integration of a set of differential equations can be used. The fitting program described is designed to be widely applicable, easy to learn and convenient to use. The use, behaviour and power of the program is explored by using simulated test data.     

Investigation of the conditions for oscillations in two-variable models of biological processes

A procedure is presented for the direct determination of possible limit cycles that might occur in two-coupled differential equations that can arise in the modeling of certain biological phenomena. Mathematical expressions are given so that for a particular pair of differential equations the possible limit cycles, their parameters, and stability properties can be calculated by a purely algebraic process. Several examples are used to illustrate the procedure.     

Seed dispersal in a patchy environment with global age dependence

A model of seed population dynamics proposed by S. A. Levin, A. Hastings, and D. Cohen is presented and analyzed. With the environment considered as a mosaic of patches, patch age is used along with time as an independent variable. Local dynamics depend not only on the local state, but also on the global environment via dispersal modelled by an integral over all patch ages. Basic technical properties of the time varying solutions are examined; necessary and sufficient conditions for nontrivial steady states are given; and general sufficient conditions for global asymptotic stability of these steady states are established. Primary tools of analysis include a hybrid Picard iteration, fixed point methods, monotonicity of solution structure, and upper and lower solutions for differential equations.This work was supported in part by National Science Foundation Grants MCS-7903497 and MCS-790349701     

Steady state kinetics of consecutive enzyme catalysed reactions involving single substrates: procedures for the interpretation of coupled assays

Sets of differential rate equations are written describing a linear sequence of reactions occurring in solution each catalysed by a control enzyme or one of the Michaelis-Menten type. It is shown that the solutions of these equations may be formulated as a set of Maclaurin polynomials, expressing the concentration of each reactant and of final product as a function of time. From arrays of such polynomials, general expressions are induced for the first non-zero term of the series. These are used to formulate a procedure (illustrated with an example simulated by numerical integration) by which results of coupled enzymic assays may be analysed in terms of maximal velocities and apparent Michaelis constants: correlation is made with other established methods for conducting coupled assays. The present procedure assumes a steady state of enzyme-substrate complexes but not of intermediate reactants.     

小麦赤霉病流行动态的控制论模型

对小麦赤霉病流行资料进行分析,给出了流行状态微分方程的预测方法．根据不同的状态微分方程,以极大值原理建立小麦赤霉病流行动态的控制论模型,得到流行动态的最优控制轨线．结果表明,气象条件有利,品种感病,药剂防效低,则防治时间应提早．要求消灭病害彻底,防治时间也应提前．药剂防效高,小麦产量高,品种感病,则防治后的净收益就大．以小偃6号为例,给出了Malthus增长型下的病害最优控制轨线和最大净收益．     

Dynamic optimization of bioprocesses: efficient and robust numerical strategies

The dynamic optimization (open loop optimal control) of non-linear bioprocesses is considered in this contribution. These processes can be described by sets of non-linear differential and algebraic equations (DAEs), usually subject to constraints in the state and control variables. A review of the available solution techniques for this class of problems is presented, highlighting the numerical difficulties arising from the non-linear, constrained and often discontinuous nature of these systems. In order to surmount these difficulties, we present several alternative stochastic and hybrid techniques based on the control vector parameterization (CVP) approach. The CVP approach is a direct method which transforms the original problem into a non-linear programming (NLP) problem, which must be solved by a suitable (efficient and robust) solver. In particular, a hybrid technique uses a first global optimization phase followed by a fast second phase based on a local deterministic method, so it can handle the nonconvexity of many of these NLPs. The efficiency and robustness of these techniques is illustrated by solving several challenging case studies regarding the optimal control of fed-batch bioreactors and other bioprocesses. In order to fairly evaluate their advantages, a careful and critical comparison with several other direct approaches is provided. The results indicate that the two-phase hybrid approach presents the best compromise between robustness and efficiency.     

Functional nodes in dynamic neural networks for bioprocess modelling

This contribution presents a novel method for the direct integration of a-priori knowledge in a neural network and its application for the online determination of a secondary metabolite during industrial yeast fermentation. Hereby, existing system knowledge is integrated in an artificial neural network (ANN) by means of 'functional nodes'. A generalized backpropagation algorithm is presented. For illustration, a set of ordinary differential equations describing the diacetyl formation and degradation during the cultivation is incorporated in a functional node and integrated in a dynamic feedforward neural network in a hybrid manner. The results show that a hybrid modelling approach exploiting available a-priori knowledge and experimental data can considerably outperform a pure data-based modelling approach with respect to robustness, generalization and necessary amount of training data. The number of training sets were decreased by 50%, obtaining the same accuracy as in a conventional approach. All incorrect decisions, according to defined cost criteria obtained with the conventional ANN, were avoided.     

Mathematical modelling of glycolysis and adenine nucleotide metabolism of human erythrocytes. I. Reaction-kinetic statements, analysis of in vivo state and determination of starting conditions for in vitro experiments

A mathematical model of energy metabolism of human red cells is presented, which includes besides the glycolytic reactions the adenine nucleotide metabolism. The model is based on the network of chemical reactions, the thermodynamic equilibrium constants of fast reversible reactions and on the kinetic equations for irreversible enzyme reactions. The model consists of a system of 16 differential equations and allows the mathematical evaluation of metabolic levels in the steady state of energy metabolism corresponding to the in vivo state erythrocytes with the kinetic data for the enzymes derived from in vitro experiments. The dependence of the levels of metabolites in the steady state on the activity of some enzymes is analysed to characterize the regulatory properties of the system. The comparison of the steady state levels of the model with experimental data makes it possible to estimate values of some controversial enzyme parameters. Estimates of the kinetic parameters of the following intracellular processes are presented: 1) rate constant of AMP-phosphatase, 2) maximum rate of adenylate deaminase, 3) activity of adenine phosphoribosylpyrophosphate transferase and 4) adenosine transport through the cell membrane. The simulation of the preparatory phase before incubation of erythrocytes indicates, that the model also permits to compute the time course of changes of levels of metabolites. To solve the initial problem the stiff differential equation system is integrated numerically by an efficient program without the application of the quasi-steady-state approximation.     

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.     

Site-specific recombination in Schizosaccharomyces pombe and systematic assembly of a 400kb transgene array in mammalian cells using the integrase of Streptomyces phage phiBT1

We have established the integrase of the Streptomyces phage phiBT1 as a tool for eukaryotic genome manipulation. We show that the phiBT1 integrase promotes efficient reciprocal and conservative site-specific recombination in vertebrate cells and in Schizosaccharomyces pombe, thus establishing the utility of this protein for genome manipulation in a wide range of eukaryotes. We show that the phiBT1 integrase can be used in conjunction with Cre recombinase to promote the iterative integration of transgenic DNA. We describe five cycles of iterative integration of a candidate mouse centromeric sequence 80 kb in length into a human mini-chromosome within a human-Chinese hamster hybrid cell line. These results establish the generality of the iterative site-specific integration technique.     

Evaluation of four methods for computing parameter sensitivities in dynamic system models

Computation of state sensitivities with respect to parameters can be a difficult and costly numerical problem when the number of states and parameters is large, or when sensitivities must be computed repeatedly, as with many optimization algorithms. Four methods are evaluated in terms of solution accuracy, and computer-time and storage requirements: direct numerical integration of the complete sensitivity-system differential equations, a reduced-order method based on the controllable states of the sensitivity system, a numerical-quadratures technique applied directly to the analytic solution of the original system, and an approach based on the solution of the transition matrix. Three linear system models, with four different types of inputs, were used as test cases, the largest having 6 states and 12 parameters. The reduced-order method was the most time-efficient in a majority of cases, but it was prone to numerical instability problems in certain situations which may be encountered in applications. It also had the largest storage requirements. For the highest-order system, only direct numerical integration and the transition-matrix method produced sufficiently accurate results for most applications, because of matrix-inversion problems with the other methods. For impulse inputs, the transition-matrix and the numerical-quadratures methods overall were the most computationally efficient, but the transition-matrix approach required much more memory storage.     

A random-walk model of human mortality and aging

Consideration is made of the roles of certain types of state space and time scales for a random-walk model of individual physiological status change and death. Because the actual measurement of physiological variables omits many variables relevant to survival, we are forced to view this model as operating in a stochastic state space for a population of individuals where only the frequency distributions are deterministic. In this stochastic state space, under the assumption that the “history” of prior movement contains no additional information, the forward partial differential equation is obtained for the distribution of a population whose movement in the selected space is determined by the randomwalk equations. If the initial distribution of the population in the state space is normal, then certain assumptions about movement and mortality will operate to preserve normality thereafter. Under the assumption of normality, simultaneous ordinary differential equations can be derived from the forward partial differential equation defining the distribution function. Examination of the ordinary simultaneous differential equations shows how parameters for certain models of aging and mortality can be obtained.     

A hybrid representation approach for modelling complex dynamic bioprocesses

This paper considers the use of hybrid models to represent the dynamic behaviour of biotechnological processes. Each hybrid model consists of a set of non linear differential equations and a neural model. The set of differential equations attempts to describe as much as possible the phenomenology of the process whereas neural networks model predict some key parameters that are an essential part of the phenomenological model. The neural model is obtained indirectly, that is, using the prediction errors of one or more state variables to adjust its weights instead of successive presentations of input-output data of the neural network. This approach allows to use actual measurements to derive a suitable neural model that not only represents the variation of some key parameters but it is also able to partly include dynamic behaviour unaccounted for by the phenomenological model. The approach is described in detail using three test cases: (1) the fermentation of glucose to gluconic acid by the micro-organism Pseudomonas ovalis, (2) the growth of filamentous fungi in a solid state fermenter, and (3) the propagation of filamentous fungi growing on a 2-D solid substrate. Results for the three applications clearly demon- strate that using a hybrid model is a viable alternative for modelling complex biotechnological bioprocesses.     

Simulation of biochemical systems

A computer program has been developed which solves sets of first-order non-linear differential equations which are typical of those encountered in biochemistry. The program includes its own algebraic compiler, which writes the subroutines needed by the integration routine, which is a fourth-order Runge-Kutta process incorporating variable step-length. The routine is simple to use and very fast in operation.     

Three steady state situation in an open chemical reaction system. I

In an isothermal continuously stirred tank reactor (open chemical reaction system) fed by sulphuric acid solutions of bromate, bromide and cerium)(III) bistability (three steady state situation) is experimentally observed. This remarkable behavior, based on the instability of one steady state, has important consequences for the understanding of excitability and biochemical control mechanisms. The mass-balance equations for the reactor and the chemical mechanism of the reaction are combined into a simple mathematical model. The behavior of the resulting nonlinear differential equations is examined analytically and by a graphical integration procedure (method of isoclines). Using realistic kinetic data, the model shows the same behavior as observed in the experiment.     

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.