Stability analysis of a state-dependent delay differential equation for cell maturation: analytical and numerical methods

Abstract

We consider a mathematical model describing the maturation process of stem cells up to fully mature cells. The model is formulated as a differential equation with state-dependent delay, where maturity is described as a continuous variable. The maturation rate of cells may be regulated by the amount of mature cells and, moreover, it may depend on cell maturity: we investigate how the stability of equilibria is affected by the choice of the maturation rate. We show that the principle of linearised stability holds for this model, and develop some analytical methods for the investigation of characteristic equations for fixed delays. For a general maturation rate we resort to numerical methods and we extend the pseudospectral discretisation technique to approximate the state-dependent delay equation with a system of ordinary differential equations. This is the first application of the technique to nonlinear state-dependent delay equations, and currently the only method available for studying the stability of equilibria by means of established software packages for bifurcation analysis. The numerical method is validated on some cases when the maturation rate is independent of maturity and the model can be reformulated as a fixed-delay equation via a suitable time transformation. We exploit the analytical and numerical methods to investigate the stability boundary in parameter planes. Our study shows some drastic qualitative changes in the stability boundary under assumptions on the model parameters, which may have important biological implications.

Pseudospectral discretization of delay differential equations in <i>sun-star</i> formulation: Results and conjectures

“…where r M is defined in (32). We have that e M is a solution of (34) if and only if e M = Vz M with z M ∈ Y solution of the equation…”

“…Our aim is to survey what is known and to indicate possible future research directions, to obtain bounds in supremum norm in order to better understand to which extent the finite-dimensional ODE "mimics" the dynamics of the infinite-dimensional system described by the DDE. For some applications of the technique we refer to [2,8,9,32,37].…”

Effects of time delays and viscoelastic parameters in oscillatory response of cell monolayers

“…Some recent works have analysed the dynamics of the chemo-mechanical coupling through diffusion reaction equations, with applications in apical constriction in epithelia [11], monolayers on substrates [18,21] or wound healing [22]. In order to analyse the stability of such systems, we here focus on the delay and model it explicitly by resorting to a delayed cell rheology and the resulting delay differential equations, in similar manner to the stress analysis in yeast [13], neural axons conduction [3], transport in respiratory systems [5], or in cell maturation [10].…”

Long-time behavior of a size-structured population model with diffusion and delayed birth process

“…The main focus of population dynamics has been a characterization of alterations in the numbers, sizes and age distribution of individuals and of potential internal or external causes provoking these changes. In the studies of structured population equations, linear semigroup methods were successfully developed to investigate the linear stability regularity and bifurcation phenomena of solutions for linearized systems, see [16,26,31,37]. The last years have witnessed an invigorated interest in age/size-structured population dynamics due to the wide applications.…”