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.
Highlights
Natural biological and biochemical systems are able to construct order dynamically through constrained interactions between their constituents based on a wide range of possible inputs and environmental conditions
What role does symmetry in biological systems play in this capacity to self-organize, and, in another direction, could similar mechanisms be exploited in computation driven by interaction? By examining natural symmetry structures that arise from the algebraic analysis of discrete models of biological and related biochemical systems, we aim to begin to answer these questions
Our main goals are twofold: (i) to identify and understand mathematically the natural subsystems and hierarchical relations inside discrete models of natural systems and (ii) to use insights we find there to define a new model of computation driven by interactions that is useful for both biology and computation
Summary
Natural biological and biochemical systems are able to construct order dynamically through constrained interactions between their constituents based on a wide range of possible inputs and environmental conditions. Both consist of cyclic group C6 of order 6 acting on subsets of the state space: there is a natural subsystem (X, C6) with X consisting of the nine states Op, L, L ZYA, L Op, L Op ZYA, L A, L A ZYA, L A Op and L A Op ZYA. This can be seen from the fact that elements of the former fix the state Op while the latter do not As it is defined by restriction, the permutator group may be realized by many different subgroups of the semigroup of the automaton, as is the case for the other examples in this paper (see [31]). This specificity is not entirely absolute (there are enzymes which will catalyse the hydrolysis of an ester R–COO–R + H2O R–COOH + R –OH without much regard for R, R if there is an ester linkage between them) but for the majority of enzymes absolute specificity is the rule
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