Modeling Gene Expression with Differential Equations (original) (raw)

2005

Gene expression is the process by which a gene makes its effect on a cell or organism. Linear differential equations have been explored as a model for gene expression. We discuss the shortcomings of this model, and we propose a system of nonlinear differential equations to mathematically model gene expression in prokaryotes, specifically bacteria. We investigate this biological system using explicit functions that describe the processes of protein synthesis which includes transcription, translation, degradation, and feedback in hope of shedding light on their associated rates. We analyze the transient and steady state solutions of the model and give a biological interpretation of these results.

Reconstruction of transcriptional dynamics from gene reporter data using differential equations

Bioinformatics, 2008

Promoter-driven reporter genes, notably luciferase and green fluorescent protein, provide a tool for the generation of a vast array of time-course data sets from living cells and organisms. The aim of this study is to introduce a modeling framework based on stochastic differential equations (SDEs) and ordinary differential equations (ODEs) that addresses the problem of reconstructing transcription time-course profiles and associated degradation rates. The dynamical model is embedded into a Bayesian framework and inference is performed using Markov chain Monte Carlo algorithms. Results: We present three case studies where the methodology is used to reconstruct unobserved transcription profiles and to estimate associated degradation rates. We discuss advantages and limits of fitting either SDEs ODEs and address the problem of parameter identifiability when model variables are unobserved. We also suggest functional forms, such as on/off switches and stimulus response functions to model transcriptional dynamics and present results of fitting these to experimental data.

Analysis of a simple gene expression model

2012

Gene expression is random owing to the low copy numbers of molecules in a living cell and the best way to study it is by use of a stochastic method, specifically the chemical master equation. The method is used here to derive analytically the invariant probability distributions, and expressions for the moments and noise strength for a simple gene model without feedback. Sensitivity analysis, emphasizing particularly the dependence of the probability distributions, the moments, and noise strength is carried out using Metabolic Control Analysis, which uses control coefficients that measure the response of observables when parameters change. Bifurcation analysis is also carried out. The results show that the number of mRNA molecules follows a hypergeometric probability distribution, and that noise decreases as the number of these molecules increases. Metabolic Control Analysis was successfully extended to genetic control mechanisms, with the obtained control coefficients satisfying a s...

DDE Model of Gene Expression: A Continuum Approach

ASME 2008 Dynamic Systems and Control Conference, Parts A and B, 2008

This paper presents an analytical study of the stability of the steady state solutions of a gene regulatory network with time delay. The system is modeled as a continuous network and takes the form of a nonlinear delay differential-integral equation coupled to an ordinary differential equation. Two examples are given in which the critical delay causing instability is computed.

Basic and simple mathematical model of coupled transcription, translation and degradation

Synthesis of proteins is one of the most fundamental biological processes, which consumes a significant amount of cellular resources. Despite many efforts to produce detailed mechanistic mathematical models of translation, no basic and simple kinetic model of mRNA lifecycle (transcription, translation and degradation) exists. We build such a model by lumping multiple states of translated mRNA into few dynamical variables and introducing a pool of translating ribosomes. The basic and simple model can be extended, if necessary, to take into account various phenomena such as the interaction between translating ribosomes or regulation of translation by microRNA. The model can be used as a building block (translation module) for more complex models of cellular processes.

A stochastic differential equation model for transcriptional regulatory networks

BMC Bioinformatics, 2007

Background: This work explores the quantitative characteristics of the local transcriptional regulatory network based on the availability of time dependent gene expression data sets. The dynamics of the gene expression level are fitted via a stochastic differential equation model, yielding a set of specific regulators and their contribution. Results: We show that a beta sigmoid function that keeps track of temporal parameters is a novel prototype of a regulatory function, with the effect of improving the performance of the profile prediction. The stochastic differential equation model follows well the dynamic of the gene expression levels. Conclusion: When adapted to biological hypotheses and combined with a promoter analysis, the method proposed here leads to improved models of the transcriptional regulatory networks.

A stochastic differential equation model for transcriptional regulatory networks-0

2011

Prediction of temporal gene expression

European Journal of Biochemistry, 2002

A computational approach is used to analyse temporal gene expression in the context of metabolic regulation. It is based on the assumption that cells developed optimal adaptation strategies to changing environmental conditions. Timedependent enzyme profiles are calculated which optimize the function of a metabolic pathway under the constraint of limited total enzyme amount. For linear model pathways it is shown that wave-like enzyme profiles are optimal for a rapid substrate turnover. For the central metabolism of yeast cells enzyme profiles are calculated which ensure long-term homeostasis of key metabolites under conditions of a diauxic shift. These enzyme profiles are in close correlation with observed gene expression data. Our results demonstrate that optimality principles help to rationalize observed gene expression profiles.

Model-based Inference of Gene Expression Dynamics from Sequence Information

2005

A dynamic model of prokaryotic gene expression is developed that makes considerable use of gene sequence information. The main contribution arises from the fact that the combined gene expression model allows us to access the impact of altering a nucleotide sequence on the dynamics of gene expression rates mechanistically. The high level of detail of the mathematical model is considered as an important step towards bringing together the tremendous amount of biological in-depth knowledge that has

Backward-stochastic-differential-equation approach to modeling of gene expression

Physical review. E, 2017

In this article, we introduce a backward method to model stochastic gene expression and protein-level dynamics. The protein amount is regarded as a diffusion process and is described by a backward stochastic differential equation (BSDE). Unlike many other SDE techniques proposed in the literature, the BSDE method is backward in time; that is, instead of initial conditions it requires the specification of end-point ("final") conditions, in addition to the model parametrization. To validate our approach we employ Gillespie's stochastic simulation algorithm (SSA) to generate (forward) benchmark data, according to predefined gene network models. Numerical simulations show that the BSDE method is able to correctly infer the protein-level distributions that preceded a known final condition, obtained originally from the forward SSA. This makes the BSDE method a powerful systems biology tool for time-reversed simulations, allowing, for example, the assessment of the biological...

Modeling Gene Expression with Differential Equations (original) (raw)

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