Internalization and end flux in morphogen gradient formation

AbstractTwo simple reaction–diffusion systems of partial differential equations and auxiliary conditions governing the activities of diffusible ligands such as Dpp in anterior–posterior axis of Drosophila wing imaginal discs were previously formulated and investigated by numerical simulations in [Developmental Cell 2 (2002) 785–796]. System B focuses on diffusion, reversible binding with receptors and ligand-mediated degradation for a fixed receptor concentration uniform in time and space. System C extended this basic but meaningful model to allow for endocytosis, exocytosis and receptor synthesis and degradation. The present paper provides a mathematical underpinning for the computational studies of these two systems and some insight gained from our analysis. We will see for instance that the two boundary value problems governing the steady state for the two systems are identical in form. This result will enable us to avoid dealing with internalization explicitly when we investigate other complex morphogen activities such as the effects of (1) feedback and (2) diffusible and non-diffusible molecules competing for ligands and receptors to inhibit cell signaling and pattern formation. The principal contribution of the present work pertains to the extension of System C to allow for a ligand flux at the source end. The more general model has many significant consequences including the removal of a limitation of previous models on ligand synthesis rate for the existence of steady state behavior. Linear stability of the corresponding steady state behavior is established. While the actual decay rate of transients is less accessible in this new model, it is possible to obtain tight upper and lower bounds for the decay rate in terms of the (effective) degradation rate of the receptors and that of the ligand-receptor complexes

Selective pressures for and against genetic instability in cancer: a time-dependent problem

Genetic instability in cancer is a two-edge sword. It can both increase the rate of cancer progression (by increasing the probability of cancerous mutations) and decrease the rate of cancer growth (by imposing a large death toll on dividing cells). Two of the many selective pressures acting upon a tumour, the need for variability and the need to minimize deleterious mutations, affect the tumour's ‘choice’ of a stable or unstable ‘strategy’. As cancer progresses, the balance of the two pressures will change. In this paper, we examine how the optimal strategy of cancerous cells is shaped by the changing selective pressures. We consider the two most common patterns in multistage carcinogenesis: the activation of an oncogene (a one-step process) and an inactivation of a tumour-suppressor gene (a two-step process). For these, we formulate an optimal control problem for the mutation rate in cancer cells. We then develop a method to find optimal time-dependent strategies. It turns out that for a wide range of parameters, the most successful strategy is to start with a high rate of mutations and then switch to stability. This agrees with the growing biological evidence that genetic instability, prevalent in early cancers, turns into stability later on in the progression. We also identify parameter regimes where it is advantageous to keep stable (or unstable) constantly throughout the growth

An asymptotic description of the elastic instability of twisted thin elastic plates

The work presented here reconsiders the classical stability problem for the deformation experienced by a stretched elastic strip when its ends are subjected to small twisting moments. Singular perturbation methods enable us to describe analytically the wrinkling instability that occurs when the strip is very thin. In this case the localised structure of the instability pattern is controlled by the solution of a second-order boundary value problem with variable coefficients. The theoretical results obtained are confirmed by direct numerical simulations of the full problem