Mathematics and biology: a Kantian view on the history of pattern formation theory

Driesch’s statement, made around 1900, that the physics and chemistry of his day were unable to explain self-regulation during embryogenesis was correct and could be extended until the year 1972. The emergence of theories of self-organisation required progress in several areas including chemistry, physics, computing and cybernetics. Two parallel lines of development can be distinguished which both culminated in the early 1970s. Firstly, physicochemical theories of self-organisation arose from theoretical (Lotka 1910–1920) and experimental work (Bray 1920; Belousov 1951) on chemical oscillations. However, this research area gained broader acceptance only after thermodynamics was extended to systems far from equilibrium (1922–1967) and the mechanism of the prime example for a chemical oscillator, the Belousov–Zhabotinski reaction, was deciphered in the early 1970s. Secondly, biological theories of self-organisation were rooted in the intellectual environment of artificial intelligence and cybernetics. Turing wrote his The chemical basis of morphogenesis (1952) after working on the construction of one of the first electronic computers. Likewise, Gierer and Meinhardt’s theory of local activation and lateral inhibition (1972) was influenced by ideas from cybernetics. The Gierer–Meinhardt theory provided an explanation for the first time of both spontaneous formation of spatial order and of self-regulation that proved to be extremely successful in elucidating a wide range of patterning processes. With the advent of developmental genetics in the 1980s, detailed molecular and functional data became available for complex developmental processes, allowing a new generation of data-driven theoretical approaches. Three examples of such approaches will be discussed. The successes and limitations of mathematical pattern formation theory throughout its history suggest a picture of the organism, which has structural similarity to views of the organic world held by the philosopher Immanuel Kant at the end of the eighteenth century

Dynamic maternal gradients control timing and shift-rates for drosophila gap gene expression

Pattern formation during development is a highly dynamic process. In spite of this, few experimental and modelling approaches take into account the explicit time-dependence of the rules governing regulatory systems. We address this problem by studying dynamic morphogen interpretation by the gap gene network in Drosophila melanogaster. Gap genes are involved in segment determination during early embryogenesis. They are activated by maternal morphogen gradients encoded by bicoid (bcd) and caudal (cad). These gradients decay at the same time-scale as the establishment of the antero-posterior gap gene pattern. We use a reverse-engineering approach, based on data-driven regulatory models called gene circuits, to isolate and characterise the explicitly time-dependent effects of changing morphogen concentrations on gap gene regulation. To achieve this, we simulate the system in the presence and absence of dynamic gradient decay. Comparison between these simulations reveals that maternal morphogen decay controls the timing and limits the rate of gap gene expression. In the anterior of the embyro, it affects peak expression and leads to the establishment of smooth spatial boundaries between gap domains. In the posterior of the embryo, it causes a progressive slow-down in the rate of gap domain shifts, which is necessary to correctly position domain boundaries and to stabilise the spatial gap gene expression pattern. We use a newly developed method for the analysis of transient dynamics in non-autonomous (time-variable) systems to understand the regulatory causes of these effects. By providing a rigorous mechanistic explanation for the role of maternal gradient decay in gap gene regulation, our study demonstrates that such analyses are feasible and reveal important aspects of dynamic gene regulation which would have been missed by a traditional steady-state approach. More generally, it highlights the importance of transient dynamics for understanding complex regulatory processes in development.The authors thankfully acknowledge the computer resources, technical expertise and assistance provided by the Barcelona Supercomputing Center-Centro Nacional de Supercomputación. BV was supported by a “la Caixa” fellowship, and a Writing-Up Fellowship at the KLI Klosterneuburg. The research group of JJ was supported by the MEC-EMBL agreement for the EMBL/CRG Research Unit in Systems Biology, European Commission grant FP7-KBBE-2011-5/289434 (BioPreDyn), and grants BFU2009-10184 and BFU2012-33775 from MINECO. JJ thanks the Wissenschaftskolleg zu Berlin (Wiko) for a 10-month fellowship in 2014/15 and AC, for a half-year fellowship in 2014/15. The Centre for Genomic Regulation (CRG) acknowledges support from MINECO, “Centro de Excelencia Severo Ochoa 2013-2017” SEV-2012-0208

A BMP-FGF Morphogen Toggle Switch Drives the Ultrasensitive Expression of Multiple Genes in the Developing Forebrain

Borders are important as they demarcate developing tissue into distinct functional units. A key challenge is the discovery of mechanisms that can convert morphogen gradients into tissue borders. While mechanisms that produce ultrasensitive cellular responses provide a solution, how extracellular morphogens drive such mechanisms remains poorly understood. Here, we show how Bone Morphogenetic Protein (BMP) and Fibroblast Growth Factor (FGF) pathways interact to generate ultrasensitivity and borders in the dorsal telencephalon. BMP and FGF signaling manipulations in explants produced border defects suggestive of cross inhibition within single cells, which was confirmed in dissociated cultures. Using mathematical modeling, we designed experiments that ruled out alternative cross inhibition mechanisms and identified a cross-inhibitory positive feedback (CIPF) mechanism, or “toggle switch”, which acts upstream of transcriptional targets in dorsal telencephalic cells. CIPF explained several cellular phenomena important for border formation such as threshold tuning, ultrasensitivity, and hysteresis. CIPF explicitly links graded morphogen signaling in the telencephalon to switch-like cellular responses and has the ability to form multiple borders and scale pattern to size. These benefits may apply to other developmental systems