## Computation with finite stochastic chemical reaction networks (2008)

### Cached

### Download Links

Venue: | Natural Computing |

Citations: | 21 - 5 self |

### BibTeX

@ARTICLE{Soloveichik08computationwith,

author = {David Soloveichik and Matthew Cook and Erik Winfree and Jehoshua Bruck},

title = {Computation with finite stochastic chemical reaction networks},

journal = {Natural Computing},

year = {2008},

pages = {615--633}

}

### OpenURL

### Abstract

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 error-free Turing universal computation is impossible in this setting. Nevertheless, we show a design of a chemical computer that achieves fast and reliable Turing-universal 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; Turing-universal computation; probabilistic computation 1. Introduction. Many

### Citations

771 |
Exact stochastic simulation of coupled chemical reactions
- Gillespie
- 1977
(Show Context)
Citation Context ...modeling cellular processes (e.g. [25]) and is included in numerous software packages [26, 27, 28, 29]. The algorithms used for modeling stochastic kinetics are usually based on Gillespie’s algorithm =-=[30, 31, 32]-=-. Consider a solution containing p species. Its state is a vector z ∈ N p (where N = {0, 1, 2, . . . }) specifying the integral molecular counts of the species. A reaction α is a tuple 〈l,r, k〉 ∈ N p ... |

528 |
Introduction to the Theory of Computation
- Sipser
- 1997
(Show Context)
Citation Context ...if one can prove that there can be no algorithm that solves it correctly in all cases; the classic undecidable problem is the halting problem: determine whether or not a given TM will eventually halt =-=[36]-=-. Now suppose by way of contradiction that someone claims to have an errorless way of simulating any TM in a SCRN. Say it is claimed that if the TM halts then the state species corresponding to the ha... |

454 | The chemical abstract machine
- Berry, Boudol
- 1989
(Show Context)
Citation Context ...so been suggested via a Turing machine (TM) simulation on a polymer [5, 6], via cellular automaton simulation in self-assembly [7], or via compartmentalization of molecules into membrane compartments =-=[8, 9]-=-. These approaches rely on the geometrical arrangement of a fixed set of parts to encode information. This allows unbounded computation to be performed by molecular systems containing only a limited s... |

286 |
Stochastic kinetic analysis of developmental pathway bifurcation in phage lambda-infected Escherichia coli cells
- Arkin, Ross, et al.
- 1998
(Show Context)
Citation Context ...omputer simulations have confirmed that stochastic effects can be physiologically significant [22, 23, 24]. Consequently, the stochastic model is widely employed for modeling cellular processes (e.g. =-=[25]-=-) and is included in numerous software packages [26, 27, 28, 29]. The algorithms used for modeling stochastic kinetics are usually based on Gillespie’s algorithm [30, 31, 32]. Consider a solution cont... |

237 | Efficient Exact Stochastic Simulation of Chemical Systems with Many Species and Many Channels
- Gibson, Bruck
- 2000
(Show Context)
Citation Context ...modeling cellular processes (e.g. [25]) and is included in numerous software packages [26, 27, 28, 29]. The algorithms used for modeling stochastic kinetics are usually based on Gillespie’s algorithm =-=[30, 31, 32]-=-. Consider a solution containing p species. Its state is a vector z ∈ N p (where N = {0, 1, 2, . . . }) specifying the integral molecular counts of the species. A reaction α is a tuple 〈l,r, k〉 ∈ N p ... |

230 |
Stochastic mechanisms in gene expression
- McAdams, Arkin
- 1997
(Show Context)
Citation Context ...ith some key control factors present in quantities under a dozen [20, 21]. Experimental observations and computer simulations have confirmed that stochastic effects can be physiologically significant =-=[22, 23, 24]-=-. Consequently, the stochastic model is widely employed for modeling cellular processes (e.g. [25]) and is included in numerous software packages [26, 27, 28, 29]. The algorithms used for modeling sto... |

189 |
Stochastic gene expression in a single cell
- Elowitz, Levine, et al.
- 2002
(Show Context)
Citation Context ...ith some key control factors present in quantities under a dozen [20, 21]. Experimental observations and computer simulations have confirmed that stochastic effects can be physiologically significant =-=[22, 23, 24]-=-. Consequently, the stochastic model is widely employed for modeling cellular processes (e.g. [25]) and is included in numerous software packages [26, 27, 28, 29]. The algorithms used for modeling sto... |

155 |
Parallel program schemata
- Karp, Miller
- 1969
(Show Context)
Citation Context ...unts, the system can never be sure it has detected all the molecules present, and thus must decide to produce an output at some point without being certain. Formally, a theorem due to Karp and Miller =-=[35]-=- when adapted to the SCRN context (see Section A.6) rules out the possibility of error-free Turing universal computation altogether if the state of the TM head can be determined by the presence or abs... |

104 | Algorithmic self-assembly of DNA Sierpinski triangles
- Rothemund, Papadakis, et al.
(Show Context)
Citation Context ... molecule, this makes creating large circuits onerous. Computation has also been suggested via a Turing machine (TM) simulation on a polymer [5, 6], via cellular automaton simulation in self-assembly =-=[7]-=-, or via compartmentalization of molecules into membrane compartments [8, 9]. These approaches rely on the geometrical arrangement of a fixed set of parts to encode information. This allows unbounded ... |

93 |
Recursive unsolvability of Post’s problem of “tag” and other topics in theory of Turing machines
- Minsky
- 1961
(Show Context)
Citation Context ... is a rich literature on abstract models of computation that make use of integer counts, primarily because these are among the simplest Turing-universal machines known. Minsky’s register machine (RM) =-=[33]-=- is the prototypical example. A RM is a finite state machine augmented with fixed number of registers that can each hold an arbitrary non-negative integer. An inc(i, r, j) instruction specifies that w... |

86 | A rigorous derivation of the chemical master equation
- Gillespie
- 1992
(Show Context)
Citation Context ...Stochastic Model of Chemical Kinetics. The stochastic chemical reaction network (SCRN) model of chemical kinetics describes interactions involving integer number of molecules as Markov jump processes =-=[14, 15, 16, 17]-=-. It is used in domains where the traditional model of deterministic continuous mass action kinetics is invalid due to small molecular counts. When all molecular counts are large the model scales to t... |

79 | A DNA and restriction enzyme implementation of Turing Machines. In: [17] 75-119. Reliable Encoding 8
- Wilhem, Rothemund
- 1996
(Show Context)
Citation Context ...able used during the computation requires its own signaling molecule, this makes creating large circuits onerous. Computation has also been suggested via a Turing machine (TM) simulation on a polymer =-=[5, 6]-=-, via cellular automaton simulation in self-assembly [7], or via compartmentalization of molecules into membrane compartments [8, 9]. These approaches rely on the geometrical arrangement of a fixed se... |

76 | Stochastic simulation of chemical kinetics
- Gillespie
- 2007
(Show Context)
Citation Context ...modeling cellular processes (e.g. [25]) and is included in numerous software packages [26, 27, 28, 29]. The algorithms used for modeling stochastic kinetics are usually based on Gillespie’s algorithm =-=[30, 31, 32]-=-. Consider a solution containing p species. Its state is a vector z ∈ N p (where N = {0, 1, 2, . . . }) specifying the integral molecular counts of the species. A reaction α is a tuple 〈l,r, k〉 ∈ N p ... |

74 |
Thermodynamics of computation- a review
- Bennett
(Show Context)
Citation Context ...able used during the computation requires its own signaling molecule, this makes creating large circuits onerous. Computation has also been suggested via a Turing machine (TM) simulation on a polymer =-=[5, 6]-=-, via cellular automaton simulation in self-assembly [7], or via compartmentalization of molecules into membrane compartments [8, 9]. These approaches rely on the geometrical arrangement of a fixed se... |

63 |
Enzyme-free nucleic acid logic circuits
- Seelig, Soloveichik, et al.
- 2006
(Show Context)
Citation Context ...l system have been explored in the literature, both as theoretical proposals and as practical implementations. The most common and, at present, successful attempts involve simulating Boolean circuits =-=[1, 2, 3, 4]-=-. In such cases, information is generally encoded in the high or low concentrations of various signaling molecules. Since each binary variable used during the computation requires its own signaling mo... |

63 |
Stochastic approach to chemical kinetics
- McQuarrie
- 1967
(Show Context)
Citation Context ...Stochastic Model of Chemical Kinetics. The stochastic chemical reaction network (SCRN) model of chemical kinetics describes interactions involving integer number of molecules as Markov jump processes =-=[14, 15, 16, 17]-=-. It is used in domains where the traditional model of deterministic continuous mass action kinetics is invalid due to small molecular counts. When all molecular counts are large the model scales to t... |

58 |
The relationship between stochastic and deterministic models for chemical reactions
- Kurtz
- 1972
(Show Context)
Citation Context ...ains where the traditional model of deterministic continuous mass action kinetics is invalid due to small molecular counts. When all molecular counts are large the model scales to the mass action law =-=[18, 19]-=-. Small molecular counts are prevalent in biology: for example, over 80% of the genes in the E. coli chromosome are expressed at fewer than a hundred copies per cell, with some key control factors pre... |

31 |
Biochemical Network Stochastic Simulator (BioNetS): software for stochastic modeling of biochemical networks
- Adalsteinsson
- 2004
(Show Context)
Citation Context ...ffects can be physiologically significant [22, 23, 24]. Consequently, the stochastic model is widely employed for modeling cellular processes (e.g. [25]) and is included in numerous software packages =-=[26, 27, 28, 29]-=-. The algorithms used for modeling stochastic kinetics are usually based on Gillespie’s algorithm [30, 31, 32]. Consider a solution containing p species. Its state is a vector z ∈ N p (where N = {0, 1... |

30 | Fast computation by population protocols with a leader
- Angluin, Aspnes, et al.
- 2008
(Show Context)
Citation Context ...cies. We view stochastic chemical kinetics as a model of computation in which information is stored and processed in the integer counts of molecules in a well-mixed solution, as discussed in [10] and =-=[11]-=- (see Sec. 5 for a comparison with our results). This type of information storage is effectively unary and thus it may seem inappropriate for fast computation. It is thus surprising that our construct... |

28 | A probabilistic-time hierarchy theorem for “Slightly Non-uniform” algorithms
- Barak
- 2002
(Show Context)
Citation Context ...annot be effectively sped up (it is impossible to build a TM that has the same input/output relationship but computes much faster). This is believed to be true even allowing some probability of error =-=[37]-=-. Since a TM can be encoded in an SCRN, if the behavior of the SCRN could be simulated very quickly, then the behavior of the TM would also be determined quickly, which would raise a contradiction. Ou... |

26 |
STOCKS: STOChastic Kinetic Simulations of biochemical systems with Gillespie algorithm
- Kierzek
- 2002
(Show Context)
Citation Context ...ffects can be physiologically significant [22, 23, 24]. Consequently, the stochastic model is widely employed for modeling cellular processes (e.g. [25]) and is included in numerous software packages =-=[26, 27, 28, 29]-=-. The algorithms used for modeling stochastic kinetics are usually based on Gillespie’s algorithm [30, 31, 32]. Consider a solution containing p species. Its state is a vector z ∈ N p (where N = {0, 1... |

25 |
An excitable gene regulatory circuit induces transient cellular differentiation
- Suel, Garcia-Ojalvo, et al.
(Show Context)
Citation Context ...ith some key control factors present in quantities under a dozen [20, 21]. Experimental observations and computer simulations have confirmed that stochastic effects can be physiologically significant =-=[22, 23, 24]-=-. Consequently, the stochastic model is widely employed for modeling cellular processes (e.g. [25]) and is included in numerous software packages [26, 27, 28, 29]. The algorithms used for modeling sto... |

22 |
Chemical kinetics is Turing universal
- Magnasco
- 1997
(Show Context)
Citation Context ...versality with chemical kinetics. However, most proposed schemes require increasing the variety of molecular species (rather than only increasing molecular counts) to perform larger computation (e.g. =-=[38]-=- which shows finite circuit computation and not Turing universal computation despite its title). Liekens and Fernando [10] have considered computation in stochastic chemistry in which computation is p... |

18 |
Adaptive stochastic-deterministic chemical kinetic simulations
- Vasudeva, Bhalla
(Show Context)
Citation Context ...ffects can be physiologically significant [22, 23, 24]. Consequently, the stochastic model is widely employed for modeling cellular processes (e.g. [25]) and is included in numerous software packages =-=[26, 27, 28, 29]-=-. The algorithms used for modeling stochastic kinetics are usually based on Gillespie’s algorithm [30, 31, 32]. Consider a solution containing p species. Its state is a vector z ∈ N p (where N = {0, 1... |

16 |
Reconstruction of genetic circuits
- Sprinzak, Elowitz
- 2005
(Show Context)
Citation Context ...l system have been explored in the literature, both as theoretical proposals and as practical implementations. The most common and, at present, successful attempts involve simulating Boolean circuits =-=[1, 2, 3, 4]-=-. In such cases, information is generally encoded in the high or low concentrations of various signaling molecules. Since each binary variable used during the computation requires its own signaling mo... |

15 | Four fast universal Turing machines
- Neary, Woods
- 2009
(Show Context)
Citation Context ...3]. 3 Thus, the first question is whether we can simulate a TM instead of the much slower RM? We achieve this in our next construction where we simulate an abstract machine called a clockwise TM (CTM)=-=[34]-=- which is only quadratically slower than a regular TM (Lemma A.9). Our second question is whether it is possible to speed up computation by increasing the molecular counts of some species. After all, ... |

12 |
Elowitz and Stanislas Leibler. A synthetic oscillatory network of transcriptional regulators
- Michael
(Show Context)
Citation Context ...chemical systems suggests that reaction networks on the scale of our construction may soon become feasible. The state of the art in synthetic biology progressed from the coupling of 2-3 genes in 2000 =-=[12]-=-, to the implementation of over 100 deoxyribonuclease logic gates in vitro in 2006 [13]. Our construction is sufficiently simple that significant aspects of it may be implemented with the technology o... |

12 |
Does replication-induced transcription regulate synthesis of the myriad low copy number proteins of Escherichia coli? BioEssays 17
- Guptasarma
- 1995
(Show Context)
Citation Context ...alent in biology: for example, over 80% of the genes in the E. coli chromosome are expressed at fewer than a hundred copies per cell, with some key control factors present in quantities under a dozen =-=[20, 21]-=-. Experimental observations and computer simulations have confirmed that stochastic effects can be physiologically significant [22, 23, 24]. Consequently, the stochastic model is widely employed for m... |

10 |
Medium scale integration of molecular logic gates in an automaton
- Macdonald, Li, et al.
(Show Context)
Citation Context ...oon become feasible. The state of the art in synthetic biology progressed from the coupling of 2-3 genes in 2000 [12], to the implementation of over 100 deoxyribonuclease logic gates in vitro in 2006 =-=[13]-=-. Our construction is sufficiently simple that significant aspects of it may be implemented with the technology of synthetic biology of the near future.sCOMPUTATION WITH FINITE STOCHASTIC CHEMICAL REA... |

8 |
Darko Stefanovic. Deoxyribozyme-based ligase logic gates and their initial circuits
- Stojanovic, Semova, et al.
(Show Context)
Citation Context ...l system have been explored in the literature, both as theoretical proposals and as practical implementations. The most common and, at present, successful attempts involve simulating Boolean circuits =-=[1, 2, 3, 4]-=-. In such cases, information is generally encoded in the high or low concentrations of various signaling molecules. Since each binary variable used during the computation requires its own signaling mo... |

6 | Turing complete catalytic particle computers
- Liekens, Fernando
- 2006
(Show Context)
Citation Context ...racy” species. We view stochastic chemical kinetics as a model of computation in which information is stored and processed in the integer counts of molecules in a well-mixed solution, as discussed in =-=[10]-=- and [11] (see Sec. 5 for a comparison with our results). This type of information storage is effectively unary and thus it may seem inappropriate for fast computation. It is thus surprising that our ... |

6 |
Molecular-scale logic gates
- Silva, McClenaghan
(Show Context)
Citation Context |

5 | Networks of Relations
- Cook
- 2005
(Show Context)
Citation Context ... that the following reachability question is decidable: given points x and y, is there a walk that reaches some point y ′ ≥ y from x [35]. The correspondence between VASs and SCRNs is straightforward =-=[39]-=-. First consider chemical reactions in which no species occurs both as a reactant and as a product (i.e. reactions that have no catalysts). When such a reaction α = 〈l,r, k〉 occurs, the state of the S... |

4 |
Genes VII
- Levin
- 1999
(Show Context)
Citation Context ...alent in biology: for example, over 80% of the genes in the E. coli chromosome are expressed at fewer than a hundred copies per cell, with some key control factors present in quantities under a dozen =-=[20, 21]-=-. Experimental observations and computer simulations have confirmed that stochastic effects can be physiologically significant [22, 23, 24]. Consequently, the stochastic model is widely employed for m... |

1 |
Prasanna de Silva. Molecular-scale logic gates
- A
(Show Context)
Citation Context |