Canalization, bifurcation and gene circuit models

In the oldest held text, bifurcation is a description of anatomy. Bateson's 1894 catalogue of variation records ribs and costal cartilages that broaden and divide. In one entry a left fourth rib becomes very broad and bifurcates in front, leaving large spaces at the division.¹ He notes that in three of the cases the division affected the fourth rib, and in others the rib was probably the fourth or fifth.¹ Here the word names a visible split in a structure. It makes no claim about dynamics, parameters or stable states. The dynamical sense appears only in the later work described below.

Manu and colleagues (2009) analysed the gap gene system of the Drosophila blastoderm with dynamical systems theory. They represent the state of a gene circuit as a point in a concentration space, called phase space. A time-varying solution is a curve in that space, and a steady state is a single point.² Crombach and colleagues (2012) credit Reinitz and colleagues with the pioneering case study of this method on the gap gene network. They say it has not been applied more widely, partly because the quantitative data set of spatial gap protein patterns took many years to establish.⁴ Lobo and Levin (2015) add that expression-profiling methods yield models without spatial information, so they do not fit patterning data.⁵

Manu and colleagues argue that the Bcd concentration acts as a bifurcation parameter of the gap gene equations. If separate attractors were created and removed in sequence as Bcd varied, they say, noise in Bcd would cause rapid switching between them. They propose instead a one dimensional manifold that depends smoothly on maternal Hb, and they describe it as robust to such fluctuations.³ They also suggest the attractors they found may be saddle points in a higher dimensional space, acting as way points.³ Meyer and colleagues (2017) analyse a slime mold decision model and report a sub-critical pitchfork bifurcation in which three basins of attraction give way to two.⁶

Lobo, Lobikin and Levin (2017) reverse-engineered a signaling network for melanocyte conversion in Xenopus larvae. They searched in silico all combinations of up to three reagents and found one that gave an intermediate phenotype. The model showed a bifurcation producing an attractor 25% of the way between wild type and full conversion.⁷ Ramstead, Badcock and Friston (2018) frame living systems as occupying a small, bounded viability set within their phase space.⁸ Manicka and Levin (2019) found a cyclic hyperpolarized attractor in a bioelectric network model that was not required during training and emerged on its own.⁹