Attractor-Based Obstructions to Growth in Homogeneous Cyclic Boolean Automata

We consider a synchronous Boolean organism consisting of N cells arranged in a circle, where each cell initially takes on an independently chosen Boolean value. During the lifetime of the organism, each cell updates its own value by responding to the presence (or absence) of diversity amongst its two neighbours’ values. We show that if all cells eventually take a value of 0 (irrespective of their initial values) then the organism necessarily has a cell count that is a power of 2. In addition, the converse is also proved: if the number of cells in the organism is a proper power of 2, then no matter what the initial values of the cells are, eventually all cells take on a value of 0 and then cease to change further. We argue that such an absence of structure in the dynamical properties of the organism implies a lack of adaptiveness, and so is evolutionarily disadvantageous. It follows that as the organism doubles in size (say from m to 2m) it will necessarily encounter an intermediate size that is a proper power of 2, and suffers from low adaptiveness. Finally we show, through computational experiments, that one way an organism can grow to more than twice its size and still avoid passing through intermediate sizes that lack structural dynamics, is for the organism to depart from assumptions of homogeneity at the cellular level

Attractor-Based Obstructions to Growth in Homogeneous Cyclic Boolean Automata

We consider a synchronous Boolean organism consisting of N cells arranged in a circle, where each cell initially takes on an independently chosen Boolean value. During the lifetime of the organism, each cell updates its own value by responding to the presence (or absence) of diversity amongst its two neighbours’ values. We show that if all cells eventually take a value of 0 (irrespective of their initial values) then the organism necessarily has a cell count that is a power of 2. In addition, the converse is also proved: if the number of cells in the organism is a proper power of 2, then no matter what the initial values of the cells are, eventually all cells take on a value of 0 and then cease to change further. We argue that such an absence of structure in the dynamical properties of the organism implies a lack of adaptiveness, and so is evolutionarily disadvantageous. It follows that as the organism doubles in size (say from m to 2m) it will necessarily encounter an intermediate size that is a proper power of 2, and suffers from low adaptiveness. Finally we show, through computational experiments, that one way an organism can grow to more than twice its size and still avoid passing through intermediate sizes that lack structural dynamics, is for the organism to depart from assumptions of homogeneity at the cellular level

Morphogenesis and Growth Driven by Selection of Dynamical Properties

Organisms are understood to be complex adaptive systems that evolved to thrive in hostile environments. Though widely studied, the phenomena of organism development and growth, and their relationship to organism dynamics is not well understood. Indeed, the large number of components, their interconnectivity, and complex system interactions all obscure our ability to see, describe, and understand the functioning of biological organisms. Here we take a synthetic and computational approach to the problem, abstracting the organism as a cellular automaton. Such systems are discrete digital models of real-world environments, making them more accessible and easier to study then their physical world counterparts. In such simplified synthetic models, we find that the structure of the cellular network greatly impacts the dynamics of the organism as a whole. In the physical world, for example, the network property wherein some cells depend on phosphorus produces the cyclical boom-bust dynamics of algae on the surface of a pond. Using techniques of synthetic biology and cellular automata, such local properties can be abstractly specified, and the long-term, system-wide, and dynamical consequences of localized assumptions can be carefully explored. This thesis explores the potential impacts of Darwinian selection of dynamical properties on long term cellular differentiation and organism growth. The focus here is on the relationship between organism homogeneity (or heterogeneity) and the dynamical properties of robustness, adaptivity, and chromatic symmetry. This dissertation applies an experimental approach to test the following three hypotheses: (1) cellular differentiation increases the expected robustness in an organism’s dynamics, (2) cellular differentiation leads to more uniform adaptivity as the organism grows, and (3) for organisms with symmetry, growth by segment elongation is more likely than growth by segment reduplication. To explore these hypotheses, we address several obstacles in the experimental study of dynamical systems, including computational time limits and big data

Heterogeneity and its Impact on Thermal Robustness and Attractor Density

There is considerable research relating the structure of Boolean networks to their state space dynamics. In this paper, we extend the standard model to include the effects of thermal noise, which has the potential to deflect the trajectory of a dynamical system within its state space, sending it from one stable attractor to another. We introduce a new “thermal robustness” measure, which quantifies a Boolean network’s resilience to such deflections. In particular, we investigate the impact of structural homogeneity on two dynamical properties: thermal robustness and attractor density. Through computational experiments on cyclic Boolean networks, we ascertain that as a homogeneous Boolean network grows in size, it tends to underperform most of its heterogeneous counterparts with respect to at least one of these two dynamical properties. These results strongly suggest that during an organism’s growth and morphogenesis, cellular differentiation is required if the organism seeks to exhibit both an increasing number of attractors and resilience to thermal noise