Gene Expression Time Delays and Turing Pattern Formation Systems

Abstract

The incorporation of time delays can greatly affect the behaviour of partial differential equations and dynamical systems. In addition, there is evidence that time delays in gene expression due to transcription and translation play an important role in the dynamics of cellular systems. In this paper, we investigate the effects of incorporating gene expression time delays into a one-dimensional putative reaction diffusion pattern formation mechanism on both stationary domains and domains with spatially uniform exponential growth. While oscillatory behaviour is rare, we find that the time taken to initiate and stabilise patterns increases dramatically as the time delay is increased. In addition, we observe that on rapidly growing domains the time delay can induce a failure of the Turing instability which cannot be predicted by a naive linear analysis of the underlying equations about the homogeneous steady state. The dramatic lag in the induction of patterning, or even its complete absence on occasions, highlights the importance of considering explicit gene expression time delays in models for cellular reaction diffusion patterning.

Aberrant Behaviours of Reaction Diffusion Self-organisation Models on Growing Domains in the Presence of Gene Expression Time Delays

“…1(a1) and 1(a2), even for an unrealistically small gene expression time delay (of about 15 seconds for the parameters used). The fact the very prospect of patterning is sensitive to perturbations in the initial conditions is a level of sensitivity that has not been observed in previous studies (Gaffney and Monk, 2006). Further, even when the final pattern appears to be similar for different initial conditions, the temporal profile of the patterning can be very sensitive to the initial conditions, as illustrated by Figs.…”

“…A direct generalisation of the calculations presented by Gaffney and Monk (2006) once more reveals that the equations of a linearised stability analysis for the models summarised by Eq. (4) and Table 2 do not admit temporal oscillations at the Turing bifurcation point.…”

“…This cascade to robust self-organisation just requires an initial patterning which forms a central peak with domain growth; such features are commonly observed and virtually independent of the details of the Turing patterning kinetics in standard reaction diffusion models [ibid]. This mechanism of pattern formation has been observed in the Schnakenberg system in the presence of gene expression time delays leading to robust spatial patterning, though for restricted parameter choices (Gaffney and Monk, 2006). However, whether such cascades are inherited for other choices of interaction kinetics is unexplored once gene expression time delays are incorporated.…”

“…However, due to domain growth, these cells were not at spatial location x at time t − τ . Instead these cells were previously located at x τ ; an explicit expression can be found by backtracking along the characteristic generated by the velocity vector field starting at (x, t) (Gaffney and Monk, 2006). However, an explicit expression is not required below due to the consideration of Lagrangian coordinates.…”

“…Initial investigations of such issues have been pursued by Gaffney and Monk (2006) and Seirin-Lee et al (2010). In particular, the former explored the influence of gene expression time delays for activator production within the Schnakenberg (1979) model, an exemplar of cross kinetics.…”