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.
Highlights
The research of PhG was supported by the Deutsche Forschungsgemeinschaft (DFG), Project Number 214819831, and by the ERC (Gergely Röst’s starting Grant EPIDELAY 658, No 259559)
The first approach is by means of a partial differential equation (PDE) of transport type for the progenitor cells, coupled with two ordinary differential equations (ODE) for, respectively, stem and mature cells
Pseudospectral techniques are particular kinds of spectral methods where the approximating space is chosen as the space of polynomials of a fixed degree M ∈ N, and the projection is done through collocation: a finite number of conditions is imposed on a set of points, called collocation nodes
Summary
We consider a model for cell maturation studied in PDE formulation by Doumic et al (2011) and in DDE formulation by Getto and Waurick (2016). These ingredients were derived by Marciniak-Czochra et al (2009) for a multicompartment model describing hematopoietic stem cells producing leukocytes, and later considered by Getto and Marciniak-Czochra (2015) for the multi-compartment model as well as for the present model. They are based on the assumption that the individual stem cell division rate (dw(v)) and the fraction of self-renewal (s(v)) are regulated by a single external feedback mechanism through some signalling molecules.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
