Instability of the steady state solution in cell cycle population structure models with feedback.

We show that when cell-cell feedback is added to a model of the cell cycle for a large population of cells, then instability of the steady state solution occurs in many cases. We show this in the context of a generic agent-based ODE model. If the feedback is positive, then instability of the steady state solution is proved for all parameter values except for a small set on the boundary of parameter space. For negative feedback we prove instability for half the parameter space. We also show by example that instability in the other half may be proved on a case by case basis.

Noise-induced dispersion and breakup of clusters in cell cycle dynamics.

We study the effects of random perturbations on collective dynamics of a large ensemble of interacting cells in a model of the cell division cycle. We consider a parameter region for which the unperturbed model possesses asymptotically stable two-cluster periodic solutions. Two biologically motivated forms of random perturbations are considered: bounded variations in growth rate and asymmetric division. We compare the effects of these two dispersive mechanisms with additive Gaussian white noise perturbations. We observe three distinct phases of the response to noise in the model. First, for weak noise there is a linear relationship between the applied noise strength and the dispersion of the clusters. Second, for moderate noise strengths the clusters begin to mix, i.e. individual cells move between clusters, yet the population distribution clearly continues to maintain a two-cluster structure. Third, for strong noise the clusters are destroyed and the population is characterized by a uniform distribution. The second and third phases are separated by an order-disorder phase transition that has the characteristics of a Hopf bifurcation. Furthermore, we show that for the cell cycle model studied, the effects of bounded random perturbations are virtually indistinguishable from those induced by additive Gaussian noise, after appropriate scaling of the variance of noise strength. We then use the model to predict the strength of coupling among the cells from experimental data. In particular, we show that coupling must be rather strong to account for the observed clustering of cells given experimentally estimated noise variance.

Universality of stable multi-cluster periodic solutions in a population model of the cell cycle with negative feedback.

We study a population model where cells in one part of the cell cycle may affect the progress of cells in another part. If the influence, or feedback, from one part to another is negative, simulations of the model almost always result in multiple temporal clusters formed by groups of cells. We study regions in parameter space where periodic 'k-cyclic' solutions are stable. The regions of stability coincide with sub-triangles on which certain events occur in a fixed order. For boundary sub-triangles with order 'rs1', we prove that the k-cyclic periodic solution is asymptotically stable if the index of the sub-triangle is relatively prime with respect to the number of clusters k and neutrally stable otherwise. For negative linear feedback, we prove that the interior of the parameter set is covered by stable sub-triangles, i.e. a stable k-cyclic solution always exists for some k. We observe numerically that the result also holds for many forms of nonlinear feedback, but may break down in extreme cases.

Cell cycle dynamics in a response/signalling feedback system with a gap.

We consider a dynamical model of cell cycles of n cells in a culture in which cells in one specific phase (S for signalling) of the cell cycle produce chemical agents that influence the growth/cell cycle progression of cells in another phase (R for responsive). In the case that the feedback is negative, it is known that subpopulations of cells tend to become clustered in the cell cycle; while for a positive feedback, all the cells tend to become synchronized. In this paper, we suppose that there is a gap between the two phases. The gap can be thought of as modelling the physical reality of a time delay in the production and action of the signalling agents. We completely analyse the dynamics of this system when the cells are arranged into two cell cycle clusters. We also consider the stability of certain important periodic solutions in which clusters of cells have a cyclic arrangement and there are just enough clusters to allow interactions between them. We find that the inclusion of a small gap does not greatly alter the global dynamics of the system; there are still large open sets of parameters for which clustered solutions are stable. Thus, we add to the evidence that clustering can be a robust phenomenon in biological systems. However, the gap does effect the system by enhancing the stability of the stable clustered solutions. We explain this phenomenon in terms of contraction rates (Floquet exponents) in various invariant subspaces of the system. We conclude that in systems for which these models are reasonable, a delay in signalling is advantageous to the emergence of clustering.