A cellular automaton model, Morphozoic, is presented. Morphozoic may be used to investigate the computational power of morphogenetic fields to foster the development of structures and cell differentiation. The term morphogenetic field is used here to describe a generalized abstraction: a cell signals information about its state to its environment and is able to sense and act on signals from nested neighborhood of cells that can represent local to global morphogenetic effects. Neighborhood signals are compacted into aggregated quantities, capping the amount of information exchanged: signals from smaller, more local neighborhoods are thus more finely discriminated, while those from larger, more global neighborhoods are less so. An assembly of cells can thus cooperate to generate spatial and temporal structure. Morphozoic was found to be robust and noise tolerant. Applications of Morphozoic presented here include: 1) Conway's Game of Life, 2) Cell regeneration, 3) Evolution of a gastrulation-like sequence, 4) Neuron pathfinding, and 5) Turing's reaction-diffusion morphogenesis.
TopIntroduction
Morphogenesis is a biological process by which cells move and differentiate into organs and tissues through genetic expression and collaborative, often physical mechanisms. They become different kinds of cells, perhaps as many as 7000 kinds in our bodies (Gordon, 1999). One of the most persistent concepts of morphogenesis is the morphogenetic field (Beloussov, Opitz, & Gilbert, 1997; Alberts et al., 2002; Levin, 2011, 2012; Morozova & Shubin, 2012; Vecchi & Hernández, 2014; Beloussov, 2015) with clinical significance for human birth defects (Opitz & Neri, 2013). A morphogenetic field is a region of an embryo that has the potential to develop into a specific structure. How this happens has been subject to much investigation and debate. Some mechanisms are better understood than others. As discussed in (Tyler, 2014), which reviews the full panoply of models of morphogenetic fields, we have added the idea that a morphogenetic field is the trajectory of a two dimensional differentiation wave that triggers a step of differentiation in each cell it traverses (Gordon, 1999; Gordon & Gordon, 2016b, a).
While the mechanisms behind the transformation of an egg into an embryo (embryogenesis) and then to an adult organism have traditionally been of great interest to biologists, as a pattern formation problem it is equally intriguing to computer scientists. Part of the allure involves the spontaneous attainment of order of great complexity from geometrically simple beginnings. Even though one of the founding fathers of computer science (Leavitt, 2006; Hodges, 2014; Tyldum, 2014), Alan Turing proposed a plausible model for understanding one level of the self-organizing aspect of morphogenesis (Turing, 1952; Gordon, 2015), the process of phenotype-building has a “ghost in the machine” (Koestler, 1967), a cybernetic aspect (Gordon & Gordon, 2016a; Gordon & Stone, 2016) that has gone underappreciated.
Recent chemical experiments have revealed that while Turing’s original reaction-diffusion equations portray certain aspects of morphogenesis, they do not account for heterogeneity (Tompkins et al., 2014) or the multistep hierarchical differentiation of cells into different types (Gordon, 2015). In this study, we propose that given the right representation, simulated morphogenesis can yield solutions that are biologically plausible. Our approach, Morphozoic, models a hierarchical structure of cellular communities. Computationally, these communities are nested versions of Moore-like neighborhoods. A Moore neighborhood is the set of cells that are immediate neighbors to a cell, so in a two-dimensional square array the Moore neighborhoods contain eight cells (Weisstein, 2016b). In Morphozoic, a single higher-level cell houses an entire lower-level Moore neighborhood, down to single cells, and a set of lower-level neighborhoods compose a higher-level neighborhood (Gordon & Rangayyan, 1984a, b). While this serves as a constraint on cell-cell communication, it also serves as top-down information. This top-down information, when coupled with local, bottom-up information at different spatial scales, provides us with a mechanism for strongly emergent phenomena (Holland, 1992).