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Computation with finite stochastic chemical reaction networks
 Natural Computing
, 2008
"... Abstract. A highly desired part of the synthetic biology toolbox is an embedded chemical microcontroller, capable of autonomously following a logic program specified by a set of instructions, and interacting with its cellular environment. Strategies for incorporating logic in aqueous chemistry have ..."
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Cited by 18 (5 self)
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Abstract. A highly desired part of the synthetic biology toolbox is an embedded chemical microcontroller, capable of autonomously following a logic program specified by a set of instructions, and interacting with its cellular environment. Strategies for incorporating logic in aqueous chemistry have focused primarily on implementing components, such as logic gates, that are composed into larger circuits, with each logic gate in the circuit corresponding to one or more molecular species. With this paradigm, designing and producing new molecular species is necessary to perform larger computations. An alternative approach begins by noticing that chemical systems on the small scale are fundamentally discrete and stochastic. In particular, the exact molecular counts of each molecular species present, is an intrinsically available form of information. This might appear to be a very weak form of information, perhaps quite difficult for computations to utilize. Indeed, it has been shown that errorfree Turing universal computation is impossible in this setting. Nevertheless, we show a design of a chemical computer that achieves fast and reliable Turinguniversal computation using molecular counts. Our scheme uses only a small number of different molecular species to do computation of arbitrary complexity. The total probability of error of the computation can be made arbitrarily small (but not zero) by adjusting the initial molecular counts of certain species. While physical implementations would be difficult, these results demonstrate that molecular counts can be a useful form of information for small molecular systems such as those operating within cellular environments. Key words. stochastic chemical kinetics; molecular counts; Turinguniversal computation; probabilistic computation 1. Introduction. Many
Programmability of Chemical Reaction Networks
"... 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 wellstirred solution according to standard c ..."
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Cited by 8 (2 self)
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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 wellstirred 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
Constrained Codes as Networks of Relations
"... Abstract — We address the wellknown problem of determining the capacity of constrained coding systems. While the onedimensional case is well understood to the extent that there are techniques for rigorously deriving the exact capacity, in contrast, computing the exact capacity of a twodimensional ..."
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Cited by 1 (1 self)
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Abstract — We address the wellknown problem of determining the capacity of constrained coding systems. While the onedimensional case is well understood to the extent that there are techniques for rigorously deriving the exact capacity, in contrast, computing the exact capacity of a twodimensional constrained coding system is still an elusive research challenge. The only known exception in the twodimensional case is an exact (however, not rigorous) solution to the (1, ∞)RLL system on the hexagonal lattice. Furthermore, only exponentialtime algorithms are known for the related problem of counting the exact number of constrained twodimensional information arrays. We present the first known rigorous technique that yields an exact capacity of a twodimensional constrained coding system. In addition, we devise an efficient (polynomial time) algorithm for counting the exact number of constrained arrays of any given size. Our approach is a composition of a number of ideas and techniques: describing the capacity problem as a solution to a counting problem in networks of relations, graphtheoretic tools originally developed in the field of statistical mechanics, techniques for efficiently simulating quantum circuits, as well as ideas from the theory related to the spectral distribution of Toeplitz matrices. Using our technique we derive a closed form solution to the capacity related to the PathCover constraint in a twodimensional triangular array (the resulting calculated capacity is 0.72399217...). PathCover is a generalization of the well known onedimensional (0, 1)RLL constraint for which the capacity is known to be 0.69424... Index Terms — capacity of constrained systems, capacity of twodimensional constrained systems, holographic reductions, networks of relations, FKT method, spectral distribution of Toeplitz matrices I.
A New Class of Holographic Algorithms: Fibonacci Gates
"... We introduce Fibonacci gates as a polynomial time computable primitive, and develop a theory of holographic algorithms based on these gates. The Fibonacci gates play the role of matchgates in Valiant’s theory [16]. We develop a signature theory and characterize all realizable signatures for Fibonacc ..."
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We introduce Fibonacci gates as a polynomial time computable primitive, and develop a theory of holographic algorithms based on these gates. The Fibonacci gates play the role of matchgates in Valiant’s theory [16]. We develop a signature theory and characterize all realizable signatures for Fibonacci gates. For bases of arbitrary dimensions we prove a universal bases collapse theorem. We apply this theory to give new polynomial time algorithms for certain counting problems. We also use this framework to prove that some slight variations of these counting problems are #Phard.
Supplementary information
"... SUPPLEMENTARY INFORMATION doi:10.1038/nature10262 Neural network computation ..."
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SUPPLEMENTARY INFORMATION doi:10.1038/nature10262 Neural network computation