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Rate-independent continuous inhibitory chemical reaction networks are Turing-universal

Kim Calabrese, David Doty

TL;DR

It is shown that inhibitory CRNs can compute any computable function $f:\mathbb{N}\to\mathbb{N}$.

Abstract

We study the model of continuous chemical reaction networks (CRNs), consisting of reactions such as $A+B \to C+D$ that can transform some continuous, nonnegative real-valued quantity (called a *concentration*) of chemical species $A$ and $B$ into equal concentrations of $C$ and $D$. Such a reaction can occur from any state in which both reactants $A$ and $B$ are present, i.e., have positive concentration. We modify the model to allow *inhibitors*, for instance, reaction $A+B \to^{I} C+D$ can occur only if the reactants $A$ and $B$ are present and the inhibitor $I$ is absent. The computational power of non-inhibitory CRNs has been studied. For instance, the reaction $X_1+X_2 \to Y$ can be thought to compute the function $f(x_1,x_2) = \min(x_1,x_2)$. Under an "adversarial" model in which reaction rates can vary arbitrarily over time, it was found that exactly the continuous, piecewise linear functions can be computed, ruling out even simple functions such as $f(x) = x^2$. In contrast, in this paper we show that inhibitory CRNs can compute any computable function $f:\mathbb{N}\to\mathbb{N}$.

Rate-independent continuous inhibitory chemical reaction networks are Turing-universal

TL;DR

It is shown that inhibitory CRNs can compute any computable function .

Abstract

We study the model of continuous chemical reaction networks (CRNs), consisting of reactions such as that can transform some continuous, nonnegative real-valued quantity (called a *concentration*) of chemical species and into equal concentrations of and . Such a reaction can occur from any state in which both reactants and are present, i.e., have positive concentration. We modify the model to allow *inhibitors*, for instance, reaction can occur only if the reactants and are present and the inhibitor is absent. The computational power of non-inhibitory CRNs has been studied. For instance, the reaction can be thought to compute the function . Under an "adversarial" model in which reaction rates can vary arbitrarily over time, it was found that exactly the continuous, piecewise linear functions can be computed, ruling out even simple functions such as . In contrast, in this paper we show that inhibitory CRNs can compute any computable function .
Paper Structure (11 sections, 2 theorems, 6 equations, 1 figure)

This paper contains 11 sections, 2 theorems, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Stable oscillation
  5. Construction of iCRN simulating a register machine
  6. Proof of correctness
  7. Example of construction of iCRN from register machine
  8. Conclusion
  9. Relaxing absolute inhibition.
  10. Characterizing real-valued functions.
  11. Acknowledgements.

Key Result

Lemma 3.3

Let $n \geq 3$ and $\mathcal{C}$ be the iCRN with species $\Lambda = \{X_0,X_1,\ldots,X_{n-1}\}$ and for each $0 \leq i < n$, reaction $X_i \xrightarrow[]{\substack{X_{i-1}\\ \bot}} X_{i+1},$ where $i - 1$ and $i + 1$ are both taken modulo $n$. If $\mathbf{i} = \{1X_0\}$ is the starting configuratio

Figures (1)

  • Figure 1: Plot of iCRN simulating "multiply-by-2" register machine, with input register r_in having initial value $3$. Note the species r_in decrements from 3 down to 0, and the species r_out increments from 0 up to 6, while other species oscillate.

Theorems & Definitions (10)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Definition 3.5
  • Theorem 3.7
  • proof