Rate-independent Computation in Continuous Chemical Reaction Networks

Understanding the algorithmic behaviors that are in principle realizable a chemical system is necessary for rigorous understanding of design principles biological regulatory networks. Further, advances synthetic biology herald time when we will be able to rationally engineer complex systems and idealized formal models become blueprints engineering. Coupled interactions well-mixed solution commonly formalized as reaction networks (CRNs). However, despite widespread use CRNs natural sciences, range computational exhibited by not well understood. Here, study following problem: What functions f : ℝ k → can computed CRN, which CRN eventually produces correct amount “output” molecule, no matter rate at reactions proceed? This captures previously unexplored but very class computations: For example, X 1 + 2 Y thought compute function y = min ( x , ). Such robust sense it whether its evolution governed standard model mass-action kinetics, alternatives such Hill-function or Michaelis-Menten other arbitrary chemistry respect (fundamentally digital) stoichiometric constraints (what reactants products?). We develop reachability relation based on broad notion “what could happen” if rates vary arbitrarily over time. Using reachability, define stable computation analogously probability distributed computing connect with seemingly stronger rate-independent convergence limit t ∞ under wide generalized laws. Besides direct mapping concentration nonnegative analog value, also consider “dual-rail representation” represent negative values difference two concentrations allows composition modules. prove rate-independently computable only piecewise linear (with rational coefficients) continuous (dual-rail representation), non-negative discontinuities occurring some inputs switch from zero positive (direct representation). The many contexts where powerful targets implementation, combined systematic construction these functions, demonstrate potential computation.