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Unformatted text preview: Programmability of Chemical Reaction Networks Matthew Cook 1 , David Soloveichik 2 , Erik Winfree 2 , and Jehoshua Bruck 2 1 Institute of Neuroinformatics, UZH, ETH Z¨urich, Switzerland [email protected] 2 California Institute of Technology, Pasadena, California, USA { dsolov,winfree,bruck } @ caltech.edu Summary. Motivated by the intriguing complexity of biochemical circuitry within individual cells we study Stochastic Chemical Reaction Networks (SCRNs), a for mal model that considers a set of chemical reactions acting on a finite number of molecules in a wellstirred solution according to standard chemical kinetics equa tions. 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 computa tion. We survey known connections and give new connections between SCRNs and Boolean Logic Circuits, Vector Addition Systems, Petri Nets, Gate Implementability, Primitive Recursive Functions, Register Machines, Fractran, and Turing Machines. A theme to these investigations is the thin line between decidable and undecidable questions about SCRN behavior. 1 Introduction Stochastic chemical reaction networks (SCRNs) are among the most funda mental models used in chemistry, biochemistry, and most recently, compu tational biology. Traditionally, analysis has focused on mass action kinetics, where reactions are assumed to involve sufficiently many molecules that the state of the system can be accurately represented by continuous molecular con centrations with the dynamics given by deterministic differential equations. However, analyzing the kinetics of smallscale chemical processes involving a finite number of molecules, such as occurs within cells, requires stochas tic dynamics that explicitly track the exact number of each molecular species [1, 2, 3]. For example, over 80% of the genes in the E. coli chromosome are ex pressed at fewer than a hundred copies per cell [4], averaging, for example, only 10 molecules of Lac repressor [5]. Further, observations and computer simu lations have shown that stochastic effects resulting from these small numbers may be physiologically significant [6, 7, 8]. 2 Matthew Cook, David Soloveichik, Erik Winfree, and Jehoshua Bruck In this paper, we examine the computational power of Stochastic Chem ical Reaction Networks. Stochastic Chemical Reaction Networks are closely related to computational models such as Petri nets [9], Vector Addition Sys tems (VASs) [10], Fractran [11, 12], and Register Machines (sometimes called Counter Machines) [13], and for many of these systems we can also consider stochastic or nondeterministic variants. Our initial route into this subject came through the analysis of a seemingly quite unrelated question: What dig ital logic circuits are constructible with a given set of gate types when it is...
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 Fall '09

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