Biomedical subjects

Marc R Roussel

Publications and source records attributed to Marc R Roussel.

11 recordsLinked to original sources

PubMed2006

Validation of an algorithm for delay stochastic simulation of transcription and translation in prokaryotic gene expression.

The quantitative modeling of gene transcription and translation requires a treatment of two key features: stochastic fluctuations due to the limited copy numbers of key molecules (genes, RNA polymerases, ribosomes), and delayed output due to the time required for biopolymer synthesis. Recently proposed algorithms allow for efficient simulations of such systems. However, it is critical to know whether the results of delay stochastic simulations agree with those from more detailed models of the transcription and translation processes. We present a generalization of previous delay stochastic simulation algorithms which allows both for multiple delays and for distributions of delay times. We show that delay stochastic simulations closely approximate simulations of a detailed transcription model except when two-body effects (e.g. collisions between polymerases on a template strand) are important. Finally, we study a delay stochastic model of prokaryotic transcription and translation which reproduces observations from a recent experimental study in which a single gene was expressed under the control of a repressed lac promoter in E. coli cells. This demonstrates our ability to quantitatively model gene expression using these new methods.

Algorithms↗

PubMed2006

The functional equation truncation method for approximating slow invariant manifolds: a rapid method for computing intrinsic low-dimensional manifolds.

A slow manifold is a low-dimensional invariant manifold to which trajectories nearby are rapidly attracted on the way to the equilibrium point. The exact computation of the slow manifold simplifies the model without sacrificing accuracy on the slow time scales of the system. The Maas-Pope intrinsic low-dimensional manifold (ILDM) [Combust. Flame 88, 239 (1992)] is frequently used as an approximation to the slow manifold. This approximation is based on a linearized analysis of the differential equations and thus neglects curvature. We present here an efficient way to calculate an approximation equivalent to the ILDM. Our method, called functional equation truncation (FET), first develops a hierarchy of functional equations involving higher derivatives which can then be truncated at second-derivative terms to explicitly neglect the curvature. We prove that the ILDM and FET-approximated (FETA) manifolds are identical for the one-dimensional slow manifold of any planar system. In higher-dimensional spaces, the ILDM and FETA manifolds agree to numerical accuracy almost everywhere. Solution of the FET equations is, however, expected to generally be faster than the ILDM method.

Algorithms↗

PubMed2006

A graph-theoretic method for detecting potential Turing bifurcations.

The conditions for diffusion-driven (Turing) instabilities in systems with two reactive species are well known. General methods for detecting potential Turing bifurcations in larger reaction schemes are, on the other hand, not well developed. We prove a theorem for a graph-theoretic condition originally given by Volpert and Ivanova [Mathematical Modeling (Nauka, Moscow, 1987) (in Russian), p. 57] for Turing instabilities in a mass-action reaction-diffusion system involving n substances. The method is based on the representation of a reaction mechanism as a bipartite graph with two types of nodes representing chemical species and reactions, respectively. The condition for diffusion-driven instability is related to the existence of a structure in the graph known as a critical fragment. The technique is illustrated using a substrate-inhibited bifunctional enzyme mechanism which involves seven chemical species.

Algorithms↗

PubMed2006

Oscillations of the internal CO(2) concentration in tobacco leaves transferred to low CO(2).

Measurement of the internal CO(2) concentration (Ci) in tobacco leaves using a fast-response CO(2) exchange system showed that in the light, switching from 350 microLL(-1) to a low CO(2) concentration of 36.5 microLL(-1) (promoting high photorespiration) resulted in the Ci oscillating near the value of CO(2) compensation point (Gamma*). The oscillations are highly irregular, the range of Ci varying by 2-4 microLL(-1) in substomatal cavities with a period of a few seconds. The statistical properties of the time series became stationary after a transient of approximately 100s following transfer to low CO(2). Attractor reconstruction shows that the observed oscillations are not chaotic but exhibit stochastic behavior. The period of oscillations is consistent with the duration of photorespiratory post-illumination burst (PIB). We suggest that the observed oscillations may be due to a similar mechanism to that which leads to PIB, and may play a role in switching mitochondrial operation between oxidation of the photorespiratory glycine and of the tricarboxylic acid cycle substrates.

Carbon Dioxide↗

PubMed2006

Stochastic kinetics description of a simple transcription model.

We study a stochastic model of transcription kinetics in order to characterize the distributions of transcriptional delay and of elongation rates. Transcriptional delay is the time which elapses between the binding of RNA polymerase to a promoter sequence and its dissociation from the DNA template strand with consequent release of the transcript. Transcription elongation is the process by which the RNA polymerase slides along the template strand. The model considers a DNA template strand with one promoter site and n nucleotide sites, and five types of reaction processes, which we think are key ones in transcription. The chemical master equation is a set of ordinary differential equations in 3(n) variables, where n is the number of bases in the template. This model is too huge to be handled if n is large. We manage to get a reduced Markov model which has only 2n independent variables and can well approximate the original dynamics. We obtain a number of analytical and numerical results for this model, including delay and transcript elongation rate distributions. Recent studies of single-RNA polymerase transcription by using optical-trapping techniques raise an issue of whether the elongation rates measured in a population are heterogeneous or not. Our model implies that in the cases studied, different RNA polymerase molecules move at different characteristic rates along the template strand. We also discuss the implications of this work for the mathematical modeling of genetic regulatory circuits.

Algorithms↗

PubMed2006

Optimal observability of sustained stochastic competitive inhibition oscillations at organellar volumes.

When molecules are present in small numbers, such as is frequently the case in cells, the usual assumptions leading to differential rate equations are invalid and it is necessary to use a stochastic description which takes into account the randomness of reactive encounters in solution. We display a very simple biochemical model, ordinary competitive inhibition with substrate inflow, which is only capable of damped oscillations in the deterministic mass-action rate equation limit, but which displays sustained oscillations in stochastic simulations. We define an observability parameter, which is essentially just the ratio of the amplitude of the oscillations to the mean value of the concentration. A maximum in the observability is seen as the volume is varied, a phenomenon we name system-size observability resonance by analogy with other types of stochastic resonance. For the parameters of this study, the maximum in the observability occurs at volumes similar to those of bacterial cells or of eukaryotic organelles.

Algorithms↗

PubMed2005

The mechanism distinguishability problem in biochemical kinetics: the single-enzyme, single-substrate reaction as a case study.

A theoretical analysis of the distinguishability problem of two rival models of the single enzyme-single substrate reaction, the Michaelis-Menten and Henri mechanisms, is presented. We also outline a general approach for analysing the structural indistinguishability between two mechanisms. The approach involves constructing, if possible, a smooth mapping between the two candidate models. Evans et al. [N.D. Evans, M.J. Chappell, M.J. Chapman, K.R. Godfrey, Structural indistinguishability between uncontrolled (autonomous) nonlinear analytic systems, Automatica 40 (2004) 1947-1953] have shown that if, in addition, either of the mechanisms satisfies a particular criterion then such a transformation always exists when the models are indistinguishable from their experimentally observable outputs. The approach is applied to the single enzyme-single substrate reaction mechanism. In principle, mechanisms can be distinguished using this analysis, but we show that our ability to distinguish mechanistic models depends both on the precise measurements made, and on our knowledge of the system prior to performing the kinetics experiments.

Enzymes↗

PubMed2004

Reducing a chemical master equation by invariant manifold methods.

We study methods for reducing chemical master equations using the Michaelis-Menten mechanism as an example. The master equation consists of a set of linear ordinary differential equations whose variables are probabilities that the realizable states exist. For a master equation with s(0) initial substrate molecules and e(0) initial enzyme molecules, the manifold can be parametrized by s(0) of the probability variables. Fraser's functional iteration method is found to be difficult to use for master equations of high dimension. Building on the insights gained from Fraser's method, techniques are developed to produce s(0)-dimensional manifolds of larger systems directly from the eigenvectors. We also develop a simple, but surprisingly effective way to generate initial conditions for the reduced models.

Algorithms↗

PubMed2004

Pattern formation in excitable media with concentration-dependent diffusivities.

We study a model of pattern formation in an excitable medium with concentration-dependent diffusivities. The reaction terms correspond to a two-variable Gray-Scott model in which the system has only one stable steady state. The diffusion coefficients of the two species are assumed to have a functional relationship with the concentration of the autocatalyst. A transition from self-replicating behavior to stationary spots is observed as the influence of the local autocatalyst concentration on the diffusion process increases. Notably, the transition occurs even though there is no change in the relative diffusivities of the activator and inhibitor. The observed time-independent patterns exhibit an unusual dependence on the size and geometry of an initial perturbation. Initial perturbations with a large spatial size, for example, sometimes revert to the homogeneous equilibrium state, whereas perturbations of smaller spatial extent develop into stable spots at the same parameter values.

Journal Article↗

PubMed2004

Reaction-diffusion models of development with state-dependent chemical diffusion coefficients.

Reaction-diffusion models are widely used to model developmental processes. The great majority of current models invoke constant diffusion coefficients. However, the diffusion of metabolites or signals through tissues is frequently such that this assumption may reasonably be questioned. We consider several different physical mechanisms leading to effective diffusion coefficients in biological tissues which vary with the local conditions, including models in which juxtacrine signaling results in the diffusion of a signal in the absence of material transport. We develop a mathematical formalism for transforming local transport laws into diffusive terms. This procedure is appropriate when the typical length scale over which the concentrations change significantly is much greater than the dimensions of a cell. We review previous developmental models which considered the possibility of state-dependent diffusion coefficients. We also provide a few new motivating examples.

Animals↗

PubMed2002

A model for the enhancement of fitness in cyanobacteria based on resonance of a circadian oscillator with the external light-dark cycle.

Fitness enhancement based on resonating circadian clocks has recently been demonstrated in cyanobacteria [Ouyang et al. (1998). Proc. Natl Acad. Sci. U.S.A.95, 8660-8664]. Thus, the competition between two cyanobacterial strains differing by the free-running period (FRP) of their circadian oscillations leads to the dominance of one or the other of the two strains, depending on the period of the external light-dark (LD) cycle. The successful strain is generally that which has an FRP closest to the period of the LD cycle. Of key importance for the resonance phenomenon are observations which indicate that the phase angle between the circadian oscillator and the LD cycle depends both on the latter cycle's length and on the FRP. We account for these experimental observations by means of a theoretical model which takes into account (i) cell growth, (ii) secretion of a putative cell growth inhibitor, and (iii) the existence of a cellular, light-sensitive circadian oscillator controlling growth as well as inhibitor secretion. Building on a previous analysis in which the phase angle was considered as a freely adjustable parameter [Roussel et al. (2000). J. theor. Biol.205, 321-340], we incorporate into the model a light-sensitive version of the van der Pol oscillator to represent explicitly the cellular circadian oscillator. In this way, the model automatically generates a phase angle between the circadian oscillator and the LD cycle which depends on the characteristic FRP of the strain and varies continuously with the period of the LD cycle. The model provides an explanation for the results of competition experiments between strains of different FRPs subjected to entrainment by LD cycles of different periods. The model further shows how the dominance of one strain over another in LD cycles can be reconciled with the observation that two strains characterized by different FRPs nevertheless display the same growth kinetics in continuous light or in LD cycles when present alone in the medium. Theoretical predictions are made as to how the outcome of competition depends on the initial proportions and on the FRPs of the different strains. We also determine the effect of the photoperiod and extend the analysis to the case of a competition between three cyanobacterial strains.

Circadian Rhythm↗