Marc R. Roussel

Geometry of the steady‐state approximation: Perturbation and accelerated convergence methods

The time evolution of two model enzyme reactions is represented in phase space Γ. The phase flow is attracted to a unique trajectory, the slow manifold M, before it reaches the point equilibrium of the system. Locating M describes the slow time evolution precisely, and allows all rate constants to be obtained from steady‐state data. The line set M is found by solution of a functional equation derived from the flow differential equations. For planar systems, the steady‐state (SSA) and equilibrium (EA) approximations bound a trapping region containing M, and direct iteration and perturbation theory are formally equivalent solutions of the functional equation. The iteration’s convergence is examined by eigenvalue methods. In many dimensions, the nullcline surfaces of the flow in Γ form a prism‐shaped region containing M, but this prism is not a simple trap for the flow. Two of its edges are EA and SSA. Perturbation expansion and direct iteration are now no longer equivalent procedures; they are compared in a...

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.

Invariant manifold methods for metabolic model reduction

After the decay of transients, the behavior of a set of differential equations modeling a chemical or biochemical system generally rests on a low-dimensional surface which is an invariant manifold of the flow. If an equation for such a manifold can be obtained, the model has effectively been reduced to a smaller system of differential equations. Using perturbation methods, we show that the distinction between rapidly decaying and long-lived (slow) modes has a rigorous basis. We show how equations for attracting invariant (slow) manifolds can be constructed by a geometric approach based on functional equations derived directly from the differential equations. We apply these methods to two simple metabolic models. (c) 2001 American Institute of Physics.

ON THE GEOMETRY OF TRANSIENT RELAXATION

Coupled chemical reactions are often described by (stiff) systems of ordinary differential equations (ODEs) with widely separated relaxation times. In the phase space Γ of species concentration variables, relaxation can be represented as a cascade through a nested hierarchy of smooth hypersurfaces (inertial manifolds) {Σ}: If d is the number of independent concentration variables, then Γ≡Γd⊇Σd−1⊇Σd−2⋅⋅⋅. The last three sets in this hierarchy have special chemical importance: Σ0 is the stagnation point of the ODEs, i.e., chemical equilibrium; M(≡Σ1) is the linelike slow manifold describing the dynamical steady state in closed systems; Σ(≡Σ2) is the two‐dimensional surface containing the slowest transient flow that reaches M. Thus M and Σ are the structures underlying most steady‐state and transient kinetics experiments. The ODEs describe the velocity field in Γ, which may be used to define functional equations for M, Σ, and other members in the hierarchy {Σ}. These functional equations can be solved to giv...

The Scale-Free Dynamics of Eukaryotic Cells

Temporal organization of biological processes requires massively parallel processing on a synchronized time-base. We analyzed time-series data obtained from the bioenergetic oscillatory outputs of Saccharomyces cerevisiae and isolated cardiomyocytes utilizing Relative Dispersional (RDA) and Power Spectral (PSA) analyses. These analyses revealed broad frequency distributions and evidence for long-term memory in the observed dynamics. Moreover RDA and PSA showed that the bioenergetic dynamics in both systems show fractal scaling over at least 3 orders of magnitude, and that this scaling obeys an inverse power law. Therefore we conclude that in S. cerevisiae and cardiomyocytes the dynamics are scale-free in vivo. Applying RDA and PSA to data generated from an in silico model of mitochondrial function indicated that in yeast and cardiomyocytes the underlying mechanisms regulating the scale-free behavior are similar. We validated this finding in vivo using single cells, and attenuating the activity of the mitochondrial inner membrane anion channel with 4-chlorodiazepam to show that the oscillation of NAD(P)H and reactive oxygen species (ROS) can be abated in these two evolutionarily distant species. Taken together these data strongly support our hypothesis that the generation of ROS, coupled to redox cycling, driven by cytoplasmic and mitochondrial processes, are at the core of the observed rhythmicity and scale-free dynamics. We argue that the operation of scale-free bioenergetic dynamics plays a fundamental role to integrate cellular function, while providing a framework for robust, yet flexible, responses to the environment.

Forced‐convergence iterative schemes for the approximation of invariant manifolds

In many dynamical systems, an invariant manifold attracts the phase‐space flow. These manifolds can be approximated by an iterative method based on a functional equation treatment. However, a convergent mapping is not automatically generated from the functional equation. Nevertheless, it is possible to construct a convergent mapping by a simple modification of the original functional equation. As an example, a convergent sequence of approximations to the slow manifold of the Michaelis–Menten system is constructed.

Observation of a chaotic multioscillatory metabolic attractor by real-time monitoring of a yeast continuous culture

We monitored a continuous culture of the yeast Saccharomyces cerevisiae by membrane‐inlet mass spectrometry. This technique allows very rapid simultaneous measurements (one point every 12 s) of several dissolved gases. During our experiment, the culture exhibited a multioscillatory mode in which the dissolved oxygen and carbon dioxide records displayed periodicities of 13 h, 36 min and 4 min. The 36‐ and 4‐min modes were not visible at all times, but returned at regular intervals during the 13‐h cycle. The 4‐min mode, which has not previously been described in continuous culture, can also be seen when the culture displays simpler oscillatory behavior. The data can be used to visualize a metabolic attractor of this system, i.e. the set of dissolved gas concentrations which are consistent with the multioscillatory state. Computation of the leading Lyapunov exponent reveals the dynamics on this attractor to be chaotic.

Dynamics and mechanisms of oscillatory photosynthesis

We classify mathematical models that can be used to describe photosynthetic oscillations using ideas from nonlinear dynamics, and discuss potential mechanisms for photosynthetic oscillations in the context of this classification. We then turn our attention to recent experiments with leaves transferred to a low CO₂ atmosphere which revealed stochastic oscillations with a period of a few seconds. Rubisco is the enzyme that takes both CO₂ and O₂ as substrates correspondingly for photosynthetic assimilation and for photorespiration. Photosynthesis depletes CO₂ and produces O₂ while respiration and photorespiration work in the opposite direction, so the product of one process becomes the reactant of the other coupled process. We examine the possibility of oscillations of CO₂ and O₂ in the leaf in relation to photorespiration. We suggest that in the cell, oscillations with a period of a few seconds, corresponding to the time between photosynthetic CO₂ fixation and photorespiratory CO₂ release, underlie the dynamics of metabolism in C₃ plants.

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.

Feedforward non-Michaelis–Menten mechanism for CO2 uptake by Rubisco: Contribution of carbonic anhydrases and photorespiration to optimization of photosynthetic carbon assimilation

Rubisco, the most abundant protein serving as the primary engine generating organic biomass on Earth, is characterized by a low catalytic constant (in higher plants approx. 3s(-1)) and low specificity for CO(2) leading to photorespiration. We analyze here why this enzyme evolved as the main carbon fixation engine. The high concentration of Rubisco exceeding the concentration of its substrate CO(2) by 2-3 orders of magnitude makes application of Michaelis-Menten kinetics invalid and requires alternative kinetic approaches to describe photosynthetic CO(2) assimilation. Efficient operation of Rubisco is supported by a strong flux of CO(2) to the chloroplast stroma provided by fast equilibration of bicarbonate and CO(2) and forwarding the latter to Rubisco reaction centers. The main part of this feedforward mechanism is a thylakoidal carbonic anhydrase associated with photosystem II and pumping CO(2) from the thylakoid lumen in coordination with the rate of electron transport, water splitting and proton gradient across the thylakoid membrane. This steady flux of CO(2) limits photosynthesis at saturating CO(2) concentrations. At low ambient CO(2) and correspondingly limited capacity of the bicarbonate pool in the stroma, its depletion at the sites of Rubisco is relieved by utilizing O(2) instead of CO(2), i.e. by photorespiration, a process which supplies CO(2) back to Rubisco and buffers the redox state and energy level in the chloroplast. Thus, the regulation of Rubisco function aims to keep steady non-equilibrium levels of CO(2), NADPH/NADP and ATP/ADP in the chloroplast stroma and to optimize the condition of homeostatic photosynthetic flux of matter and energy.