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.     

On the Question of Self-Organization of Population Dynamics on Earth

Biophysics - A new mathematical model based on the predator–prey interactions has been proposed. Strictly analytical solution has been found for a system of nonlinear differential equations...     

Birfurcation theory applied to a simple model of a biochemical oscillator

Some results are presented relating to the question whether self-sustained oscillations are possible in a sequence of biochemical reactions with end- point inhibition. The model used has a single nonlinear ordinary differential equation coupled to a set of linear equations, with all coefficients in the linear terms equal. The explicit algebraic form of the Hopf-Friedrich bifurcation theory is used to show that when the number of coupled equations is large enough this model has a stable periodic solution when the equilibrium point of the equations has just become unstable.     

Analysis of the equations of renal network flows

Solution of the nonlinear differential equations for renal network flows are investigated, and conditions are given under which a unique solution to these equations exists.     

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.     

Modeling auditory system nonlinearities through Volterra series

A computational method is reviewed, in which the solution of systems of nonlinear differential equations is written in terms of a Volterra functional series. Results of implementing the aforementioned technique in a computer program (exploiting new software for symbolic manipulation) and of applying it to a nonlinear oscillator model are presented. The relevance of this approach to Auditory System modeling is discussed. Suggestions are given, regarding possible applications to Speech Recognition problems.     

Homochiral Growth Through Enantiomeric Cross-Inhibition

The stability and conservation properties of a recently proposed polymerization model are studied. The achiral (racemic) solution is linearly unstable once the relevant control parameter (here the fidelity of the catalyst) exceeds a critical value. The growth rate is calculated for different fidelity parameters and cross-inhibition rates. A chirality parameter is defined and shown to be conserved by the nonlinear terms of the model. Finally, a truncated version of the model is used to derive a set of two ordinary differential equations and it is argued that these equations are more realistic than those used in earlier models of that form.     

A kinetic model for the activated sludge process which considers diffusion and reaction in the microbial floc

A mathematical model has been developed which describes substrate removal, oxygen utilization, and biomass production in an aggregated microbial suspension containing the substrate as a soluble biodegradable material and a uniform floc size. It is applicable to both steady-state and transient conditions. The model, consisting of three partial differential equations and two ordinary differential equations, takes into account the flow pattern in the reactor, intraparticle mass transport of oxygen and substrate, and biochemical reaction by individual cells embedded in the floc. Efficient numerical solution of the coupled nonlinear equations is obtained using an implicit finite difference approach for both the reactor and floc equations. A convergent solution is realized through block interation utilizing the tridiagonal algorithm. Results indicate that a unifying theory of activated sludge dynamics will have to consider coupling between floc chemical kinetics and changes in the bulk liquid characteristics. Floc size emerges as an important influence on system performance. It appears necessary to distinguish between a system response caused by diffuslonal resistances and nutrient limitations within the floc and a response caused by physiological adaption when analyzing the transient behavior of an activated sludge process. Future research should be devoted to rigorous laboratory determinations of model parameters along with extensions to include limitations of nutrients other than orgabnic carbon and oxygen.     

A Finite Element Approach for Skeletal Muscle using a Distributed Moment Model of Contraction

Abstract

The present paper describes a geometrically and physically nonlinear continuum model to study the mechanical behaviour of passive and active skeletal muscle. The contraction is described with a Huxley type model. A Distributed Moments approach is used to convert the Huxley partial differential equation in a set of ordinary differential equations. An isoparametric brick element is developed to solve the field equations numerically. Special arrangements are made to deal with the combination of highly nonlinear effects and the nearly incompressible behaviour of the muscle. For this a Natural Penalty Method (NPM) and an Enhanced Stiffness Method (ESM) are tested. Finally an example of an analysis of a contracting tibialis anterior muscle of a rat is given. The DM-method proved to be an efficient tool in the numerical solution process. The ESM showed the best performance in describing the incompressible behaviour.     

A Finite Element Approach for Skeletal Muscle using a Distributed Moment Model of Contraction

The present paper describes a geometrically and physically nonlinear continuum model to study the mechanical behaviour of passive and active skeletal muscle. The contraction is described with a Huxley type model. A Distributed Moments approach is used to convert the Huxley partial differential equation in a set of ordinary differential equations. An isoparametric brick element is developed to solve the field equations numerically. Special arrangements are made to deal with the combination of highly nonlinear effects and the nearly incompressible behaviour of the muscle. For this a Natural Penalty Method (NPM) and an Enhanced Stiffness Method (ESM) are tested. Finally an example of an analysis of a contracting tibialis anterior muscle of a rat is given. The DM-method proved to be an efficient tool in the numerical solution process. The ESM showed the best performance in describing the incompressible behaviour.     

Efficient simulations of tubulin-driven axonal growth

This work concerns efficient and reliable numerical simulations of the dynamic behaviour of a moving-boundary model for tubulin-driven axonal growth. The model is nonlinear and consists of a coupled set of a partial differential equation (PDE) and two ordinary differential equations. The PDE is defined on a computational domain with a moving boundary, which is part of the solution. Numerical simulations based on standard explicit time-stepping methods are too time consuming due to the small time steps required for numerical stability. On the other hand standard implicit schemes are too complex due to the nonlinear equations that needs to be solved in each step. Instead, we propose to use the Peaceman–Rachford splitting scheme combined with temporal and spatial scalings of the model. Simulations based on this scheme have shown to be efficient, accurate, and reliable which makes it possible to evaluate the model, e.g. its dependency on biological and physical model parameters. These evaluations show among other things that the initial axon growth is very fast, that the active transport is the dominant reason over diffusion for the growth velocity, and that the polymerization rate in the growth cone does not affect the final axon length.     

Improved pseudoanalytical solution for steady-state biofilm kinetics

Simple algebraic expressions for the flux of substrate into a steady-state biofilm are developed. This pseudoanalytical solution, which eliminates the need for repetitiously solving numerically a set of nonlinear differential equations, is based on an analysis of the numerical results from the numerical solution of the differential equations. The critical advantage of this new pseudoanalytical solution is that it is highly accurate for the entire range of substrate concentrations and kinetic parameters. The article also illustrates that previous pseudoanalytical solutions for steady-state biofilm kinetics are seriously inaccurate for certain ranges of substrate concentration and kinetic parameters.     

广义Lotke-Volterra生态模型的非线性奇摄动近似解

非线性奇摄动问题在国际学术界中是一个重要的研究对象。它涉及到许多学科。在一些生态现象中,原始的研究方法只是采取某些简单观察和统计数据来得到结论。但是它对生态现象的实质的研究达不到效果。近来在国际上提出了研究生态学的动力学方法,即人们首先把它归化为代表它的现象本质的微分方程的模型,然后用数学方法来求解对应的方程,最后研究关于生物和数学理论的动力学方面的规律。目前,非线性摄动问题已经被广泛地研究。许多学者已经研究了一些近似理论。近似求解方法已被发展,包括平均法,边界层法,匹配渐近展开和多尺度法等等。研究非线性广义Lotke-Volterra捕食-被捕食生态模型,一个简单而有效的摄动方法被应用到捕食-被捕食生态模型。提出了捕食-被捕食的一个模型,它是一个微分方程系统,并用小的正参数按幂级数展开未知函数,然后得到关于幂级数的系数的方程,并求出它们的解。于是利用摄动方法得到了原问题解的渐近展开式。得到了它是原模型解是一个好的近似的结论,它是一个解析展开式并且能保持其解析运算。最后,给出了一个对应的例子,它说明得到的解具有很好的精度。     

Morphologically accurate reduced order modeling of spiking neurons

Accurately simulating neurons with realistic morphological structure and synaptic inputs requires the solution of large systems of nonlinear ordinary differential equations. We apply model reduction techniques to recover the complete nonlinear voltage dynamics of a neuron using a system of much lower dimension. Using a proper orthogonal decomposition, we build a reduced-order system from salient snapshots of the full system output, thus reducing the number of state variables. A discrete empirical interpolation method is then used to reduce the complexity of the nonlinear term to be proportional to the number of reduced variables. Together these two techniques allow for up to two orders of magnitude dimension reduction without sacrificing the spatially-distributed input structure, with an associated order of magnitude speed-up in simulation time. We demonstrate that both nonlinear spiking behavior and subthreshold response of realistic cells are accurately captured by these low-dimensional models.     

非线性高阶阻尼时滞微分方程的振动定理

研究具有阻尼项的非线性时滞微分方程,给出了使得方程的一切解振动的两个充分条件.一些实例说明,本文的结果在判定非线性阻尼方程的振动性时较文献中的结果更为有效.     

A new method of identifying a systems based on the minimal quadratic discrepancy criterion for biophysics problems

A wide range of biophysical systems are described by nonlinear dynamic models mathematically presented as a set of ordinary differential equations in the Cauchy explicit form: [formula: see text] Fij(X1(t),..,XN(t),t), (i = 1,...,N, j = 1,...,M), where Fij (X1(t), ..., XN(t), t) is a set of basis functions satisfying the Lipschitz condition. We investigate the problem of evaluation of model constants aij (the system identification) using experimental data about the time dependence of the dynamic parameters of the system Xi(t). A new method of system identification for the class of similar nonlinear dynamic models is proposed. It is shown that the problem of identifying an initial nonlinear model can be reduced to the solution of a system of linear equations for the matrix of the dynamic model constants [aj]i. It is proposed to determine the set of dynamic model constants aij using the criterion of minimal quadratic discrepancy for the time dependence of the set of dynamic parameters Xi(t). An important special case of the nonlinear model, the quadratic model, is considered. Test problems of identification using this method are presented for two nonlinear systems: the Van der Pol type multiparametric nonlinear oscillator and the strange attractor of Ressler, a widely known example of dynamic systems showing the stochastic behavior.     

Transient behavior of population density with competition for resources

In this paper we derive a perturbation expansion of some nonlinear diffusion equations proposed by J. G. Skellam and M. Kimura. The solution is derived by converting the differential equations into integral equations by means of the Green's function for the diffusion equation. This research was supported in part by the Office of Naval Research under Contract Nonr-595(17).     

Accurate computation of the stable solitary wave for the FitzHugh-Nagumo equations

Comparisons are made between three different methods for computing the stable solitary wave solution for the FitzHugh-Nagumo equations which consist of a nonlinear diffusion equation coupled to an ordinary differential equation in time. They model the Hodgkin-Huxley equations which describe the propagation of the nerve impulse down the axon. Two of the methods involve the travelling wave equations. Previous accurate numerical computations of these equations as an initial-value problem using a shooting method lead to inaccurate values for the wave speed; however, nonlinear corrections to the initial values are shown to yield accurate values. A boundary-value method applies asymptotic boundary conditions and uses a spline-collocation code called COLSYS for numerical solution of boundary-value problems which leads to accurate wave profiles and speeds. The third method is to solve an initial-boundary-value problem with an adaptive outgoing wave condition for the partial differential equations where the solitary wave emerges as the stable long time solution. The concept of a wave integral is introduced and they are derived to determine the wave speed used in the adaptive boundary condition and to measure the closeness of the computed solutions to the exact solitary wave solution.This work was supported in part by the Natural Sciences and Engineering Research Council Canada under Grant A4559 and by the John Simon Guggenheim Memorial Foundation