The effects of cell density and metabolite flux on cellular dynamics

Unbounded solutions of models for glycolysis

The Selkov oscillator, a simple description of glycolysis, is a system of two ordinary differential equations with mass action kinetics. In previous work the authors established several properties of the solutions of this system. In the present paper we extend this to prove that this system has solutions which diverge to infinity in an oscillatory manner at late times. This is done with the help of a Poincaré compactification of the system and a shooting argument. This system was originally derived from another system with Michaelis-Menten kinetics. A Poincaré compactification of the latter system is carried out and this is used to show that the Michaelis-Menten system, like that with mass action, has solutions which diverge to infinity in a monotone manner. It is also shown to admit subcritical Hopf bifurcations and thus unstable periodic solutions. We discuss to what extent the unbounded solutions cast doubt on the biological relevance of the Selkov oscillator and compare it with other models for the same biological system in the literature.

Comparison of Deterministic and Stochastic Regime in a Model for Cdc42 Oscillations in Fission Yeast

Oscillations occur in a wide variety of essential cellular processes, such as cell cycle progression, circadian clocks and calcium signaling in response to stimuli. It remains unclear how intrinsic stochasticity can influence these oscillatory systems. Here, we focus on oscillations of Cdc42 GTPase in fission yeast. We extend our previous deterministic model by Xu and Jilkine to construct a stochastic model, focusing on the fast diffusion case. We use SSA (Gillespie's algorithm) to numerically explore the low copy number regime in this model, and use analytical techniques to study the long-time behavior of the stochastic model and compare it to the equilibria of its deterministic counterpart. Numerical solutions suggest noisy limit cycles exist in the parameter regime in which the deterministic system converges to a stable limit cycle, and quasi-cycles exist in the parameter regime where the deterministic model has a damped oscillation. Near an infinite period bifurcation point, the deterministic model has a sustained oscillation, while stochastic trajectories start with an oscillatory mode and tend to approach deterministic steady states. In the low copy number regime, metastable transitions from oscillatory to steady behavior occur in the stochastic model. Our work contributes to the understanding of how stochastic chemical kinetics can affect a finite-dimensional dynamical system, and destabilize a deterministic steady state leading to oscillations.

Bogdanov-Takens bifurcations in the enzyme-catalyzed reaction comprising a branched network

There have been some results on bifurcations of codimension one (such as saddle-node, transcritical, pitchfork) and degenerate Hopf bifurcations for an enzyme-catalyzed reaction system comprising a branched network but no further discussion for bifurcations at its cusp. In this paper we give conditions for the existence of a cusp and compute the parameter curves for the Bogdanov-Takens bifurcation, which induces the appearance of homoclinic orbits and periodic orbits, indicating the tendency to steady-states or a rise of periodic oscillations for the concentrations of the substrate and the product.

A multi-time-scale analysis of chemical reaction networks: II. Stochastic systems

We consider stochastic descriptions of chemical reaction networks in which there are both fast and slow reactions, and for which the time scales are widely separated. We develop a computational algorithm that produces the generator of the full chemical master equation for arbitrary systems, and show how to obtain a reduced equation that governs the evolution on the slow time scale. This is done by applying a state space decomposition to the full equation that leads to the reduced dynamics in terms of certain projections and the invariant distributions of the fast system. The rates or propensities of the reduced system are shown to be the rates of the slow reactions conditioned on the expectations of fast steps. We also show that the generator of the reduced system is a Markov generator, and we present an efficient stochastic simulation algorithm for the slow time scale dynamics. We illustrate the numerical accuracy of the approximation by simulating several examples. Graph-theoretic techniques are used throughout to describe the structure of the reaction network and the state-space transitions accessible under the dynamics.

Insights into collective cell behaviour from populations of coupled chemical oscillators

Biological systems such as yeast show coordinated activity driven by chemical communication between cells. Experiments with coupled chemical oscillators can provide insights into the collective behaviour.

Stochastic analysis of reaction-diffusion processes

Reaction and diffusion processes are used to model chemical and biological processes over a wide range of spatial and temporal scales. Several routes to the diffusion process at various levels of description in time and space are discussed and the master equation for spatially discretized systems involving reaction and diffusion is developed. We discuss an estimator for the appropriate compartment size for simulating reaction-diffusion systems and introduce a measure of fluctuations in a discretized system. We then describe a new computational algorithm for implementing a modified Gillespie method for compartmental systems in which reactions are aggregated into equivalence classes and computational cells are searched via an optimized tree structure. Finally, we discuss several examples that illustrate the issues that have to be addressed in general systems.

A multi-time-scale analysis of chemical reaction networks: I. Deterministic systems

We consider deterministic descriptions of reaction networks in which different reactions occur on at least two distinct time scales. We show that when a certain Jacobian is nonsingular there is a coordinate system in which the evolution equations for slow and fast variables are separated, and we obtain the appropriate initial conditions for the transformed system. We also discuss topological properties which guarantee that the nonsingularity condition is satisfied, and show that in the new coordinate frame the evolution of the slow variables on the slow time scale is independent of the fast variables to lowest order in a small parameter. Several examples that illustrate the numerical accuracy of the reduction are presented, and an extension of the reduction method to three or more time scale networks is discussed.

The Intersection of Theory and Application in Elucidating Pattern Formation in Developmental Biology

We discuss theoretical and experimental approaches to three distinct developmental systems that illustrate how theory can influence experimental work and vice-versa. The chosen systems - Drosophila melanogaster, bacterial pattern formation, and pigmentation patterns - illustrate the fundamental physical processes of signaling, growth and cell division, and cell movement involved in pattern formation and development. These systems exemplify the current state of theoretical and experimental understanding of how these processes produce the observed patterns, and illustrate how theoretical and experimental approaches can interact to lead to a better understanding of development. As John Bonner said long ago'We have arrived at the stage where models are useful to suggest experiments, and the facts of the experiments in turn lead to new and improved models that suggest new experiments. By this rocking back and forth between the reality of experimental facts and the dream world of hypotheses, we can move slowly toward a satisfactory solution of the major problems of developmental biology.'

Dynamical quorum sensing: Population density encoded in cellular dynamics

Mutual synchronization by exchange of chemicals is a mechanism for the emergence of collective dynamics in cellular populations. General theories exist on the transition to coherence, but no quantitative, experimental demonstration has been given. Here, we present a modeling and experimental analysis of cell-density-dependent glycolytic oscillations in yeast. We study the disappearance of oscillations at low cell density and show that this phenomenon occurs synchronously in all cells and not by desynchronization, as previously expected. This study identifies a general scenario for the emergence of collective cellular oscillations and suggests a quorum-sensing mechanism by which the cell density information is encoded in the intracellular dynamical state.

Dynamics of two-component biochemical systems in interacting cells; synchronization and desynchronization of oscillations and multiple steady states

Systems of interacting cells containing a metabolic pathway with an autocatalytic reaction are investigated. The individual cells are considered to be identical and are described by differential equations proposed for the description of glycolytic oscillations. The coupling is realized by exchange of metabolites across the cell membranes. No constraints are introduced concerning the number of interacting systems, that is, the analysis applies also to populations with a high number of cells. Two versions of the model are considered where either the product or the substrate of the autocatalytic reaction represents the coupling metabolite (Model I and II, respectively). Model I exhibits a unique steady state while model II shows multistationary behaviour where the number of steady states increases strongly with the number of cells. The characteristic polynomials used for a local stability analysis are factorized into polynomials of lower degrees. From the various factors different Hopf bifurcations may result in leading for model I, either to asynchronous oscillations with regular phase shifts or to synchronous oscillations of the cells depending on the strength of the coupling and on the cell density. The multitude of steady states obtained for model II may be grouped into one class of states which are always unstable and another class of states which may undergo bifurcations leading to synchronous oscillations within subgroups of cells. From these bifurcations numerous different oscillatory regimes may emerge. Leaving the near neighbourhood of the boundary of stability, secondary bifurcations of the limit cycles occur in both models. By symmetry breaking the resulting oscillations for the individual cells lose their regular phase shifts. These complex dynamic phenomena are studied in more detail for a low number of interacting cells. The theoretical results are discussed in the light of recent experimental data on the synchronization of oscillations in populations of yeast cells.

Regulation of differentiation in a population of cells interacting through a common pool

We consider a model of a suspension of a cell population in a well-mixed medium. There are two chemical substances, say A and H, reacting in each cell of the population and the substance H can only diffuse from the inside of cell to the medium or vice versa across the cell membrane. The medium is well mixed that the concentration of H is kept uniform over the medium. Cells interact indirectly with each other through the medium. The differential equations governing the dynamics of the suspension are analyzed using standard techniques for differential equations. It is shown that the cell population divides into several groups in respect of the chemical concentrations as time elapses. It is also shown how the fraction of the number of cells belonging to each subgroup to the total number of cells is regulated. The results may be used to explain the mechanism for differentiation of multi-cellular organisms.

Oscillating spatial structures of a tissue

We present a simple mathematical model for the self-controlled growth of a tissue giving rise to an oscillating tissue size under certain conditions. The control is brought about by two substances (two inhibitors or one inhibitor and one nutrient) which influence the cell kinetics locally. The inhibitors are produced by the tissue itself (whereas the nutrient comes from outside but is consumed by the tissue which produces the same effect). Both diffuse freely throughout the tissue, and thus realize a communication between different parts of the tissue. In any case the tissue approaches a self-maintaining space-time structure with properties depending on the parameters of proliferation, death and inhibiting control. We discuss the conditions for this structure not to be time-independent but oscillating.