Symmetry structure in discrete models of biochemical systems: natural subsystems and the weak control hierarchy in a new model of computation driven by interactions

Abstract

Interaction computing is inspired by the observation that cell metabolic/regulatory systems construct order dynamically, through constrained interactions between their components and based on a wide range of possible inputs and environmental conditions. The goals of this work are to (i) identify and understand mathematically the natural subsystems and hierarchical relations in natural systems enabling this and (ii) use the resulting insights to define a new model of computation based on interactions that is useful for both biology and computation. The dynamical characteristics of the cellular pathways studied in systems biology relate, mathematically, to the computational characteristics of automata derived from them, and their internal symmetry structures to computational power. Finite discrete automata models of biological systems such as the lac operon, the Krebs cycle and p53-mdm2 genetic regulation constructed from systems biology models have canonically associated algebraic structures (their transformation semigroups). These contain permutation groups (local substructures exhibiting symmetry) that correspond to 'pools of reversibility'. These natural subsystems are related to one another in a hierarchical manner by the notion of 'weak control'. We present natural subsystems arising from several biological examples and their weak control hierarchies in detail. Finite simple non-Abelian groups are found in biological examples and can be harnessed to realize finitary universal computation. This allows ensembles of cells to achieve any desired finitary computational transformation, depending on external inputs, via suitably constrained interactions. Based on this, interaction machines that grow and change their structure recursively are introduced and applied, providing a natural model of computation driven by interactions.In the 21st century, algebra will be as important for biology as chemistry was in the 20th century.-Günter P. Wagner

Heterotic computing: exploiting hybrid computational devices

“…Building on many years of prior work, Nehaniv et al [6] present a mathematical model based on finite automata for diverse biological systems, that can be extended hierarchically to any level of aggregation of constituent parts. They identify the symmetries and the substructures that lead to interesting behaviour and interpret how control and dependencies flow through the network.…”

“…Among the recurring features of our examples, we note the importance of including transduction between these substrates in any analysis, because extra computation can be hiding in the transduction process. Indeed, in a paper in this issue, Nehaniv et al [6] explicitly model transduction as 'transformers' connecting automata into networks, and their 'transformers' are themselves a special type of automaton. Another aspect common to many of our examples is the incomplete understanding of such systems due to their hybrid nature being little studied as such.…”

Combining Graph Transformations and Semigroups for Isotopic Labeling Design

“…While the modeling approach in Egri-Nagy and Nehaniv (2008) is applied on a different level of abstraction (the transformations of a semigroup map molecules onto each other, in contrast to our approach where atoms are mapped), there is an interesting similarity. In Nehaniv et al (2015), local substructures that exhibit symmetry on a network level are found to be permutation groups. Such permutation groups are considered as natural and important subsystems (a ''pool of local reversibility'').…”

Co-Designing the Computational Model and the Computing Substrate

“…Such a dynamical system has a higher level meta-dynamics. Approaches such as BIOMICS [8,28] and MGS [12,33] provide suggestive starting points for such languages.…”