Modeling and simulation of nitrogen regulation in Corynebacterium glutamicum

Modeling the dynamic behaviour of biochemical systems at a molecular level aims at understanding and predicting the interactions of macromolecules inside the cell. Models of small subsystems based on differential equations not only prepare the way for the long-term goal of understanding a whole cell, but are inherently valuable due to their ability to predict the behaviour of the subsystem for varying external conditions or parameters. Nitrogen supply is essential for prokaryotes, thus the nitrogen uptake is an interesting target for model building. The goal is to provide new information about the interactions of the relevant proteins by performing various simulations.A model based on piecewise linear differential equations is formulated for the nitrogen uptake in Corynebacterium glutamicum. We theoretically derive a model for biochemical networks and introduce a general method for the parameter estimation which is also applicable in the case of very short time series. This approach is applied to a special system concerning the nitrogen uptake using Western blot experiments. The equations are set up for the main components of this system, the optimization problem for parameter estimation is formulated and solved, and simulations for the evaluation of the model as well as for predictions are carried out.We show that model building based on differential equations can also, when only a few measurements are performed, lead to a satisfactory model which provides valuable insights into the way it’s network components function. For example, we are able to make predictions about the maximal value of the time course as well as the steady-state level of the signal transduction protein GlnK in case of restricted activity of the proteases when considering the transition of nitrogen starvation to nitrogen excess or vice versa.     

A Gaussian Mixed Model for Learning Discrete Bayesian Networks

In this paper, we address the problem of learning discrete Bayesian networks from noisy data. A graphical model based on a mixture of Gaussian distributions with categorical mixing structure coming from a discrete Bayesian network is considered. The network learning is formulated as a maximum likelihood estimation problem and performed by employing an EM algorithm. The proposed approach is relevant to a variety of statistical problems for which Bayesian network models are suitable—from simple regression analysis to learning gene/protein regulatory networks from microarray data.     

利用贝叶斯网络分析基因表达数据

利用基因表达数据提出一种新的网络模型—贝叶斯网络,发现基因的互作.一个贝叶斯网络是多变量联合概率分布的有向图模型,表示变量间的条件独立属性.首先我们阐明贝叶斯网络如何表示基因间的互作,然后介绍从基因芯片数据学习贝叶斯网络的方法.     

Buffering in cyclic gene networks

We consider cyclic chains of unidirectionally coupled delay differential–difference equations that are mathematical models of artificial oscillating gene networks. We establish that the buffering phenomenon is realized in these system for an appropriate choice of the parameters: any given finite number of stable periodic motions of a special type, the so-called traveling waves, coexist.     

Modelling non-stationary dynamic gene regulatory processes with the BGM model

Recently, a Bayesian network model for inferring non-stationary regulatory processes from gene expression time series has been proposed. The Bayesian Gaussian Mixture (BGM) Bayesian network model divides the data into disjunct compartments (data subsets) by a free allocation model, and infers network structures, which are kept fixed for all compartments. Fixing the network structure allows for some information sharing among compartments, and each compartment is modelled separately and independently with the Gaussian BGe scoring metric for Bayesian networks. The BGM model can equally be applied to both static (steady-state) and dynamic (time series) gene expression data. However, it is this flexibility that renders its application to time series data suboptimal. To improve the performance of the BGM model on time series data we propose a revised approach in which the free allocation of data points is replaced by a changepoint process so as to take the temporal structure into account. The practical inference follows the Bayesian paradigm and approximately samples the network, the number of compartments and the changepoint locations from the posterior distribution with Markov chain Monte Carlo (MCMC). Our empirical results show that the proposed modification leads to a more efficient inference tool for analysing gene expression time series.     

Polynomial representations of piecewise-linear differential equations arising from gene regulatory networks

We describe generic sliding modes of piecewise-linear systems of differential equations arising in the theory of gene regulatory networks with Boolean interactions. We do not make any a priori assumptions on regulatory functions in the network and try to understand what mathematical consequences this may have in regard to the limit dynamics of the system. Further, we provide a complete classification of such systems in terms of polynomial representations for the cases where the discontinuity set of the right-hand side of the system has a codimension 1 in the phase space. In particular, we prove that the multilinear representation of the underlying Boolean structure of a continuous-time gene regulatory network is only generic in the absence of sliding trajectories. Our results also explain why the Boolean structure of interactions is too coarse and usually gives rise to several non-equivalent models with smooth interactions.     

A COMPOUND POISSON MODEL FOR LEARNING DISCRETE BAYESIAN NETWORKS

We introduce here the concept of Bayesian networks, in compound Poisson model, which provides a graphical modeling framework that encodes the joint probability distribution for a set of random variables within a directed acyclic graph. We suggest an approach proposal which offers a new mixed implicit estimator. We show that the implicit approach applied in compound Poisson model is very attractive for its ability to understand data and does not require any prior information. A comparative study between learned estimates given by implicit and by standard Bayesian approaches is established. Under some conditions and based on minimal squared error calculations, we show that the mixed implicit estimator is better than the standard Bayesian and the maximum likelihood estimators. We illustrate our approach by considering a simulation study in the context of mobile communication networks.     

Posterior consistency via precision operators for Bayesian nonparametric drift estimation in SDEs

We study a Bayesian approach to nonparametric estimation of the periodic drift function of a one-dimensional diffusion from continuous-time data. Rewriting the likelihood in terms of local time of the process, and specifying a Gaussian prior with precision operator of differential form, we show that the posterior is also Gaussian with the precision operator also of differential form. The resulting expressions are explicit and lead to algorithms which are readily implementable. Using new functional limit theorems for the local time of diffusions on the circle, we bound the rate at which the posterior contracts around the true drift function.     

利用动态贝叶斯网构建基因调控网络的研究进展

构建基因调控网络是21世纪人类科学所面临的重要挑战之一。基因调控网络是一个基因组内基因相互作用而形成的关系网络,它从全基因组水平上以系统和全局的角度来研究复杂的生命现象及其本质。本文阐述了近几年来此领域的研究进展,着重介绍利用动态贝叶斯网络重构基因调控网络的若干模型,包括加权核l1模型,正则化模型、高斯混合贝叶斯网模型和自回归时间变化模型。     

<Emphasis Type="Italic">Graph_sampler</Emphasis>: a simple tool for fully Bayesian analyses of DAG-models

Bayesian networks (BNs) are widely used graphical models usable to draw statistical inference about directed acyclic graphs. We presented here Graph_sampler a fast free C language software for structural inference on BNs. Graph_sampler uses a fully Bayesian approach in which the marginal likelihood of the data and prior information about the network structure are considered. This new software can handle both the continuous as well as discrete data and based on the data type two different models are formulated. The software also provides a wide variety of structure prior which can depict either the global or local properties of the graph structure. Now based on the type of structure prior selected, we considered a wide range of possible values for the prior making it either informative or uninformative. We proposed a new and much faster jumping kernel strategy in the Metropolis–Hastings algorithm. The source C code distributed is very compact, fast, uses low memory and disk storage. We performed out several analyses based on different simulated data sets and synthetic as well as real networks to discuss the performance of Graph_sampler.     

Periodic Plane Waves for the Oregonator

The Oregonator is a set of differential equations proposed by R. J. Field and R. M. Noyes as a model for the oscillating chemical reaction first studied by B. P. Belousov and A. M. Zhabotinskii. In this paper it is shown that the associated diffusion equations have periodic plane waves for parameter values not covered in earlier work. This amounts to studying a singularly perturbed system when nothing is known about the stability of periodic solutions for the reduced system.     

Homogenization of a three-phase flow model in porous media

We give homogenization results for an immiscible and incompressible three-phase flow model in a heterogeneous petroleum reservoir with periodic structure, including capillary effects. We consider a model which leads to a coupled system of partial differential equations which includes an elliptic equation and two nonlinear degenerate parabolic equations of convection–diffusion types. Using two-scale convergence, we get an homogenized model which governs the global behavior of the flow. The determination of effective properties require the numerical resolution of local problems in a standard cell.     

Periodic problem for an evolution equation with a quadratic and a cubic nonlinearity

We study conditions for the existence of a solution of a periodic problem for a model nonlinear equation in the spatially multidimensional case and consider various types of large time asymptotics (exponential and oscillating) for such solutions. The generalized Kolmogorov-Petrovskii-Piskunov equation, the nonlinear Schrödinger equation, and some other partial differential equations are special cases of this equation. We analyze the solution smoothing phenomenon under certain conditions on the linear part of the equation and study the case of nonsmall initial data for a nonlinearity of special form. The leading asymptotic term is presented, and the remainder in the asymptotics of the solution is estimated in a spatially uniform metric.     

The quasi-homogenized Bakhvalov–Eglit model of a thermoviscoelastic material beyond the periodic setting

The one-dimensional model of dynamics of a thermoviscoelastic Kelvin–Voigt material provided with rapidly oscillating initial distributions of specific volume, velocity, and specific internal energy is considered. It is allowed that the rapidly oscillating initial distributions do not have any ordered microstructure: periodic, quasi-periodic, random homogeneous, and so on. We rigorously justify the homogenization procedure as the frequency of rapid oscillations tends to infinity. As the result, we construct a closed limit effective model of a thermoviscoelastic material motion. This model contains an additional kinetic equation that carries complete information on the evolution of the limit oscillation regimes. We show that if the initial data are periodic, then the constructed limit model can be reduced to a system of the classical quasi-homogenized Bakhvalov–Eglit equations.     

An optimization-based approach for the design of Bayesian networks

Bayesian networks model conditional dependencies among the domain variables, and provide a way to deduce their interrelationships as well as a method for the classification of new instances. One of the most challenging problems in using Bayesian networks, in the absence of a domain expert who can dictate the model, is inducing the structure of the network from a large, multivariate data set. We propose a new methodology for the design of the structure of a Bayesian network based on concepts of graph theory and nonlinear integer optimization techniques.     

On the Existence and Global Bifurcation of Periodic Solutions to Planar Differential Delay Equations

This paper is concerned with periodic solutions to one-parameter families of planar differential delay equations. The concept of slowly oscillating periodic solution is extended to this setting and we state the existence of an unbounded continuum of such solutions.     

Existence and exponential stability of periodic solution for impulsive delay differential equations and applications

In this paper, we consider a class of nonlinear impulsive delay differential equations. By establishing an exponential estimate for delay differential inequality with impulsive initial condition and employing Banach fixed point theorem, we obtain several sufficient conditions ensuring the existence, uniqueness and global exponential stability of a periodic solution for nonlinear impulsive delay differential equations. Furthermore, the criteria are applied to analyze dynamical behavior of impulsive delay Hopfield neural networks and the results show different behavior of impulsive system originating from one continuous system.     

Posterior contraction rates for the Bayesian approach to linear ill-posed inverse problems

We consider a Bayesian nonparametric approach to a family of linear inverse problems in a separable Hilbert space setting with Gaussian noise. We assume Gaussian priors, which are conjugate to the model, and present a method of identifying the posterior using its precision operator. Working with the unbounded precision operator enables us to use partial differential equations (PDE) methodology to obtain rates of contraction of the posterior distribution to a Dirac measure centered on the true solution. Our methods assume a relatively weak relation between the prior covariance, noise covariance and forward operator, allowing for a wide range of applications.     

MIP-based instantaneous control of mixed-integer PDE-constrained gas transport problems

We study the transient optimization of gas transport networks including both discrete controls due to switching of controllable elements and nonlinear fluid dynamics described by the system of isothermal Euler equations, which are partial differential equations in time and 1-dimensional space. This combination leads to mixed-integer optimization problems subject to nonlinear hyperbolic partial differential equations on a graph. We propose an instantaneous control approach in which suitable Euler discretizations yield systems of ordinary differential equations on a graph. This networked system of ordinary differential equations is shown to be well-posed and affine-linear solutions of these systems are derived analytically. As a consequence, finite-dimensional mixed-integer linear optimization problems are obtained for every time step that can be solved to global optimality using general-purpose solvers. We illustrate our approach in practice by presenting numerical results on a realistic gas transport network.     

Multiple Hopf bifurcations of symmetric BAM neural network model with delay

A symmetric BAM neural network model with delay is considered. Some results of Hopf bifurcations occurring at the zero equilibrium as the delay increases are exhibited. The existence of multiple periodic solutions is established using a symmetric Hopf bifurcation result of Wu [J. Wu, Symmetric functional differential equations and neural networks with memory, Transactions of the American Mathematical Society 350 (12) (1998) 4799–4838].