Robust biomolecular finite automata

Abstract

We present a uniform method for translating an arbitrary nondeterministic finite automaton (NFA) into a deterministic mass action input/output chemical reaction network (I/O CRN ) that simulates it. The I/O CRN receives its input as a continuous time signal consisting of concentrations of chemical species that vary to represent the NFA's input string in a natural way. The I/O CRN exploits the inherent parallelism of chemical kinetics to simulate the NFA in real time with a number of chemical species that is linear in the size of the NFA. We prove that the simulation is correct and that it is robust with respect to perturbations of the input signal, the initial concentrations of species, the output (decision), and the rate constants of the reactions of the I/O CRN.are still nearly synonymous), but the field has made progress in the present century at a rate whose increase is reminiscent of Moore's law. The achievements of molecular programming are far too numerous to survey here, but they include the self-assembly of virtually any two-or three-dimensional nanoscale structure that one wants to prescribe [19, 25, 27, 35, 42], DNA strand displacement networks that simulate logic circuits and neural networks [31][32][33], and molecular robots that perform various functions while either walking on nanoscale tracks or floating free in solution [13,15,18, 37, 41, 45, 46]. All this has been achieved in real laboratory experiments, and applications to synthetic biology, medicine, and computer electronics are envisioned. Theoretical progress includes demonstrations that various molecular programming paradigms are, in principle, Turing universal [3,14,17, 22, 38, 43, 44], thereby indicating that the full generality and creativity of algorithmic computation may be deployed in molecular and biological arenas.Our objective in this paper is to begin mitigating the "in principle" of the preceding sentence. This is important for two reasons. First, although such theoretical results are steps in the right direction, processes that require unrealistically precise control of unrealistically large numbers of molecules simply cannot be implemented. Second, processes that can be implemented, but only with inordinately precise control of parameters are inherently unreliable and hence inherently unsafe in many envisioned applications. Our objective here is thus to identify a class of computations that can be implemented robustly in the molecular world, i.e., implemented in such a way that they will provably perform correctly, even when crucial parameters are perturbed by small amounts. Future research can then strive to enhance this robustness and to extend the class of computations that enjoy it.In this paper we give a uniform method for translating nondeterministic finite automata to chemical reaction networks that implement them robustly. Nondeterministic finite automata (NFAs) are over half a century old [34] and far from Turing universal, but they have many applications and remain an active research topic [8,9, 26]. Applications of...

Robust finite automata in stochastic chemical reaction networks

“…Perhaps the closest related work to ours is that of Klinge et al [ 18 ] on implementations of robust FSAs in deterministic CRNs. That work uses an ODE model as opposed to our stochastic one and proves theoretical results on robustness, whereas we opt instead for empirical characterization of behaviour under various sets of reaction rate parameters.…”

“…This means that there may be more than one possible next state given the current state and the supplied input symbol. There are several interpretations of this, including that all possible next states are somehow tried in parallel (as explored by Klinge et al [ 18 ]), as is commonly assumed in theoretical computer science studies of non-deterministic automata. An alternative interpretation would be for the system to commit completely to one of the possible transitions.…”

Robust Real-time Computing with Chemical Reaction Networks

“…First, note that N is uniformly bounded on all of its valid inputs. Moreover, if we apply Theorem 4.8 to the CRN described above, we obtain a Turing machine which not only behaves identically but can be transformed back into the same CRN [14]. As such, these are truly inverse statements.…”

“…Notice, however, that Theorem 4.8 says nothing about the existence of a robust CRN capable of simulating a NFA. For that, we turn to [14] for a CRN with a more restrictive notion of robustness which nonetheless satisfies the definitions we derived here and Theorem 4.8. We briefly summarize this below as an illustrative example.…”

“…In order to prove this theorem, we first prove a supporting lemma. In [14], Klinge, Lathrop, and Lutz provide a general CRN construction for nondeterministic finite automata (NFAs). These NFAs utilize a dual rail system for each state Z with zptq « 1 indicating that the NFA is in state Z at time t and zptq « 0 indicating the NFA is not in state Z while the complementary species Z has the opposite meaning.…”