Gene network models robust to spatial scaling and noisy input

Abstract

Many biological systems are inherently noisy, yet demonstrate robustness to perturbations and changes in external influences. Such is the case in the Bicoid–Hunchback (Bcd–Hb) system, which is critical to axis specification in the developing Drosophila embryo. We use this system as motivation to explore the larger problem of how precise patterning can be achieved under imprecise conditions. While evidence suggests Bicoid gradients are uncorrelated with respect to embryo length, downstream genes, such as Hb, are expressed in a precise manner with regard to position along the anterior–posterior (AP)-axis. In addition to precision under variability of embryo length, Hb also exhibits robustness to perturbations to the regulatory network, gene dosage, and temperature. Understanding the reduced variability of patterns in this system is of interest to both experimentalists and theoreticians, lending itself well to the field of mathematical modeling. In this paper, a class of reaction–diffusion models is presented, which produce precise patterns, despite receiving noisy input and other perturbations to the system. An essential property of the network includes the existence of a strong inhibitor for the Hb representative, where the strength of the inhibition is directly related to the amount of variation that can be tolerated. With a higher inhibitory effect, larger perturbations of Bcd can be made with relatively small changes to the location of the Hb boundary. Network topology and interaction strength are the essential properties of the minimal model giving rise to the robust features, and possible interpretations are made with regard to the Bcd–Hb system.