The Influence of Gene Expression Time Delays on Gierer–Meinhardt Pattern Formation Systems

Abstract

There are numerous examples of morphogen gradients controlling long range signalling in developmental and cellular systems. The prospect of two such interacting morphogens instigating long range self-organisation in biological systems via a Turing bifurcation has been explored, postulated, or implicated in the context of numerous developmental processes. However, modelling investigations of cellular systems typically neglect the influence of gene expression on such dynamics, even though transcription and translation are observed to be important in morphogenetic systems. In particular, the influence of gene expression on a large class of Turing bifurcation models, namely those with pure kinetics such as the Gierer-Meinhardt system, is unexplored. Our investigations demonstrate that the behaviour of the Gierer-Meinhardt model profoundly changes on the inclusion of gene expression dynamics and is sensitive to the sub-cellular details of gene expression. Features such as concentration blow up, morphogen oscillations and radical sensitivities to the duration of gene expression are observed and, at best, severely restrict the possible parameter spaces for feasible biological behaviour. These results also indicate that the behaviour of Turing pattern formation systems on the inclusion of gene expression time delays may provide a means of distinguishing between possible forms of interaction kinetics. Finally, this study also emphasises that sub-cellular and gene expression dynamics should not be simply neglected in models of long range biological pattern formation via morphogens.

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

“…Logistic growing domain Table 2 The Kinetics, Including Gene Expression Time Delays (Seirin-Lee et al, 2010). Models I, II, I-S and II-S are given by combining Eqs.…”

“…Similarly, whether the details of the morphogen signalling pathway impact on such patterning dynamics is unknown. In particular, while the pattern initiation that can be observed in Seirin- Lee et al (2010) for the Gierer-Meinhardt model on a stationary domain is necessary for such cascading pattern, it is by no means sufficient.…”

“…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.…”

“…Seirin- Lee et al (2010) considered gene expression time delays for both the activator and inhibitor dynamics of the pure kinetics, Gierer and Meinhardt (1972), model, on a stationary domain, plus the influence of different representations of the morphogen signalling pathways. While pattern initiation analogous to a standard Turing instability could be observed, albeit in restricted regions of parameter space and with a large patterning lag, self-organisation was also frequently accompanied by the development of oscillations and sensitivity to initial conditions.…”

Optimal barrier zones for stopping the invasion of Aedes aegypti mosquitoes via transgenic or sterile insect techniques

“…The kinetics are considered explicitly with a fully implicit treatment of diffusive transport (Morton and Mayers 1994). This numerical algorithm has been validated against the simulations of Seirin- Lee et al (2010), and it has also been confirmed that refinements in the time steps and grid size do not influence the results presented. In simulations we used 10 −3 or 10 −4 for the time step and 10 −3 for the spatial step in the simulations.…”