Strong Turing Completeness of Continuous Chemical Reaction Networks and Compilation of Mixed Analog-Digital Programs

Abstract

Abstract. When seeking to understand how computation is carried out in the cell to maintain itself in its environment, process signals and make decisions, the continuous nature of protein interaction processes forces us to consider also analog computation models and mixed analog-digital computation programs. However, recent results in the theory of analog computability and complexity establish fundamental links with classical programming. In this paper, we derive from these results the strong (uniform computability) Turing completeness of chemical reaction networks over a finite set of molecular species under the differential semantics, solving a long standing open problem. Furthermore we derive from the proof a compiler of mathematical functions into elementary chemical reactions. We illustrate the reaction code generated by our compiler on trigonometric functions, and on various sigmoid functions which can serve as markers of presence or absence for implementing program control instructions in the cell and imperative programs. Then we start comparing our compiler-generated circuits to the natural circuit of the MAPK signaling network, which plays the role of an analog-digital converter in the cell with a Hill type sigmoid input/output functions.

CRN++: Molecular programming language

“…Previous research demonstrated techniques of achieving complex behaviors in mass-action chemistry, such as computing algebraic functions and polynomials [4,19,18], logarithms [7], implementing logic gates and finite state machines [12,16,10], and neural networks [11,18]. Moreover, the Turing completeness of chemistry has been proven, using the strategy of implementing polynomial ODEs (which have been previously shown to be Turing universal) in mass-action chemical kinetics [9]. Even though Turing complete, this translation to chemistry can result in infeasibly complex chemical reaction networks, which motivates other, more direct methods.…”

Computing with chemical reaction networks: a tutorial

“…We remark that, since states change deterministically in the continuous CRN model, this model is often called the "deterministic" CRN model -however, we do not use this terminology here to avoid possible confusion with the computational CRN models of Section 3 that also have various deterministic aspects. We now briefly sketch the computational model of continuous CRNs from [30], see that reference for the (involved) formal definition. Roughly speaking, a function f : R ≥0 → R ≥0 is called chemically-computable if there exists a continuous CRN N and a Λ-indexed vector q(x), where each entry of q(x) is a polynomial in variable x with coefficients from R ≥0 , such that, for all z ∈ R ≥0 , starting in state q(z), the state c of the CRN evolves in such a way that c(S), for some distinguished species S, approaches the value f (z) as t → ∞.…”

“…It is then shown in [30] that chemically-computable functions are exactly the functions computable by so-called General Purpose Analog Computers as defined in [11] (which is somewhat different from the original definition in [59]). In turn, General Purpose Analog Computers (as defined in [11]) are computationally equivalent to Turing machines.…”

Composable Rate-Independent Computation in Continuous Chemical Reaction Networks

“…The study of information processing within biological CRNs, as well the engineering of CRN functionality in artificial systems, motivates the exploration of the computational power of CRNs. In general, CRNs are capable of Turing universal computation [8]; however, we are often interested in restricted classes of CRNs which may have certain desired properties. Previous work distinguished two programmable features of CRNs: the stoichiometry of the reactions and the rate laws governing the reaction speeds [4].…”