Summary. Motivated by the intriguing complexity of biochemical circuitry within individual cells we study Stochastic Chemical Reaction Networks (SCRNs), a formal model that considers a set of chemical reactions acting on a finite number of molecules in a well-stirred solution according to standard chemical kinetics equations. SCRNs have been widely used for describing naturally occurring (bio)chemical systems, and with the advent of synthetic biology they become a promising language for the design of artificial biochemical circuits. Our interest here is the computational power of SCRNs and how they relate to more conventional models of computation. We survey known connections and give new connections between SCRNs and

Abstract. Bennett’s proposed chemical Turing machine is one of the most important thought experiments in the study of the thermodynamics of computation. Yet the sophistication of molecular engineering required to physically construct Bennett’s hypothetical polymer substrate and enzyme has deterred experimental implementations. Here we propose a chemical implementation of stack machines — a Turing-universal model of computation similar to Turing machines — using strand displacement cascades as the underlying chemical primitive. More specifically, the mechanism described herein is the addition and removal of monomers from the end of a polymer, controlled by strand displacement logic. We capture the motivating feature of Bennett’s scheme — that physical reversibility corresponds to logically reversible computation, and arbitrarily little energy per computation step is required. Further, as a method of embedding logic control into chemical and biological systems, polymer-based chemical computation is significantly more efficient than geometry-free chemical reaction networks. 1

Abstract. We consider nondeterministic and probabilistic termination problems in a process algebra that is equivalent to basic chemistry. We show that the existence of a terminating computation is decidable, but that termination with any probability strictly greater than zero is undecidable. Moreover, we show that the fairness intrinsic in stochastic computations implies that termination of all computation paths is undecidable, while it is decidable in a nondeterministic framework. 1

Designing a programmable molecular machine is a fundamental problem in nano-technology. DNA is a good candidate for building these machines due to its small size and combinatorial nature. One experimentally promising direction towards obtaining such machines is a DNA walker, which is a DNA structure that moves along some pre-assembled substrate. A DNA walker can also be a useful tool for controllable molecular transportation. In this paper, we propose and analyze DNA walkers that can simulate arbitrary Turing machines within expected linear (in the execution length of the Turing Machine) time. We also develop a formal model and proof techniques for analyzing DNA walkers. Specifically, we define two classes of “safety properties”, and prove that a walker will operate correctly and efficiently, even in the presence of adversarial operations which disrupt the system, as long as it satisfies these properties, which our walkers do. Our walker consumes much less energy than previous proposed designs (linear as opposed to quadratic in the running time of the Turing machine being simulated), and is the first that can directly simulate arbitrary Turing Machines and the first that satisfies the safety properties.

Abstract—Many natural systems, including chemical and biological systems, can be modeled using stochastic switching circuits. These circuits consist of stochastic switches, called pswitches, which operate with a fixed probability of being open or closed. We study the effect caused by introducing an error of size ǫ to each pswitch in a stochastic circuit. We analyze two constructions—simple series-parallel and general series-parallel circuits—and prove that simple series-parallel circuits are robust to small error perturbations, while general series-parallel circuits are not. Specifically, the total error introduced by perturbations of size less than ǫ is bounded by a constant multiple of ǫ in a simple series-parallel circuit, independent of the size of the circuit. However, the same result does not hold in the case of more general series-parallel circuits. In the case of a general stochastic circuit, we prove that the overall error probability is bounded by a linear function of the number of pswitches. I.

The behavior of some stochastic chemical reaction networks is largely unaffected by slight inaccuracies in reaction rates. We formalize the robustness of state probabilities to reaction rate deviations, and describe a formal connection between robustness and efficiency of simulation. Without robustness guarantees, stochastic simulation seems to require computational time proportional to the total number of reaction events. Even if the concentration (molecular count per volume) stays bounded, the number of reaction events can be linear in the duration of simulated time and total molecular count. We show that the behavior of robust systems can be predicted such that the computational work scales linearly with the duration of simulated time and concentration, and only polylogarithmically in the total molecular count. Thus our asymptotic analysis captures the dramatic speedup when molecular counts are large, and shows that for bounded concentrations the computation time is essentially invariant with molecular count. Finally, by noticing that even robust stochastic chemical reaction networks are capable of embedding complex computational problems, we argue that the linear dependence on simulated time and concentration is likely optimal. 1

This paper explores the use of negative (i.e., repulsive) interactions in the abstract Tile Assembly Model defined by Winfree. Winfree in his Ph.D. thesis postulated negative interactions to be physically plausible, and Reif, Sahu, and Yin studied them in the context of reversible attachment operations. We investigate the power of negative interactions with irreversible attachments, and we achieve two main results. Our first result is an impossibility theorem: after t steps of assembly, Ω(t) tiles will be forever bound to an assembly, unable to detach. Thus negative glue strengths do not afford unlimited power to reuse tiles. Our second result is a positive one: we construct a set of tiles that can simulate an s-space-bounded, t-time-bounded Turing machine, while ensuring that no intermediate assembly grows larger than O(s), rather than O(s · t) as required by the standard Turing machine simulation with tiles. 1

We present a process algebra for DNA computing, discussing compilation of other formal systems into the algebra, and compilation of the algebra into DNA structures. 1

This paper presents a collection of computational modules implemented with chemical reactions: an inverter, an incrementer, a decrementer, a copier, a comparator, and a multiplier. Unlike previous schemes for chemical computation, ours produces designs that are dependent only on coarse rate categories for the reactions (“fast ” vs. “slow”). Given such categories, the computation is exact and independent of the specific reaction rates. We validate our designs through stochastic simulations of the chemical kinetics. Although conceptual for the time being, our methodology has potential applications in domains of synthetic biology such as biochemical sensing and drug delivery. We are exploring DNA-based computation via strand displacement as a possible experimental chassis.

We explore the expressive power of languages that naturally model biochemical interactions with relative to languages that naturally model only basic chemical reactions, identifying molecular association as the basic mechanism that distinguishes the former from the latter. We use a process algebra, the Biochemical Ground Form (BGF), which extends with primitives for molecular association CGF, a process algebra proved to be equivalent to the traditional notations for describing basic chemical reactions. We first observe that, differently from CGF, BGF is Turing universal as it supports a finite precise encoding of Random Access Machines, a well-known Turing powerful formalism. Then we prove that the Turing universality of BGF derives from the interplay between the molecular primitives of association and dissociation. In fact, the elimination from BGF of the primitives already present in CGF does not reduce the computational strength of the process algebra, while if either association or dissociation is removed then BGF is no longer Turing complete. 1.