## Programmability of Chemical Reaction Networks

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Citations: | 8 - 2 self |

### BibTeX

@MISC{Cook_programmabilityof,

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

title = {Programmability of Chemical Reaction Networks},

year = {}

}

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### Abstract

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

### Citations

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Citation Context ...m for calculating the value. Recall that this was long before people had ever written algorithms for electronic computers, before Gödel’s famous incompleteness theorem [34] and before Turing Machines =-=[35]-=-, in short, before people had figured out any satisfactory way of standardizing or formalizing the process of mathematical calculation. Perhaps this was the way?Programmability of Chemical Reaction N... |

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Citation Context ...cs of small-scale chemical processes involving a finite number of molecules, such as occurs within cells, requires stochastic 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 expressed at fewer than a hundred copies per cell [4], averaging, for example, only 10 molecules of Lac repressor [5]. Further, obser... |

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Citation Context ...Reaction Networks are closely related to computational models such as Petri nets [9], Vector Addition Systems (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: Wha... |

453 | The Chemical Abstract Machine
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Citation Context ...f information. Communication between compartments can be achieved by diffision of specific species, or by explicit transfer reactions. This is, for example, exploited in the Chemical Abstract Machine =-=[22]-=- and in Membrane Systems [23]. Note that [24], contrary to its title, only establishes that Chemical Reaction Networks appear to be able to implement feed-forward circuits (along the lines of section ... |

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Citation Context ... exists a deterministic algorithm for calculating the value. Recall that this was long before people had ever written algorithms for electronic computers, before Gödel’s famous incompleteness theorem =-=[34]-=- and before Turing Machines [35], in short, before people had figured out any satisfactory way of standardizing or formalizing the process of mathematical calculation. Perhaps this was the way?Progra... |

350 |
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Citation Context ...n figure 1. The differences between the models are minor, amounting mostly just to different interpretations or viewpoints of the same underlying fundamental process. For example, consider Petri nets =-=[26]-=-, as shown in figure 1(b). In this model a network consists of a directed bipartite graph, having connections between places (shown as circles) and transitions (shown as black bars). The state consist... |

284 |
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Citation Context ...cs of small-scale chemical processes involving a finite number of molecules, such as occurs within cells, requires stochastic 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 expressed at fewer than a hundred copies per cell [4], averaging, for example, only 10 molecules of Lac repressor [5]. Further, obser... |

263 |
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Citation Context ...nctions were merely a strict subset of the functions that can be calculated, Herbrand in 1931 [37] and Gödel in 1934 [38] defined General Recursive Functions, which in 1936 were argued by both Church =-=[39]-=- and Turing [35] to correspond exactly to the set of all functions that can possibly be calculated in any algorithmic way. This argument was accepted by most people, and is now well known as the Churc... |

234 | Efficient exact stochastic simulation of chemical systems with many species and many channels
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Citation Context ...cs of small-scale chemical processes involving a finite number of molecules, such as occurs within cells, requires stochastic 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 expressed at fewer than a hundred copies per cell [4], averaging, for example, only 10 molecules of Lac repressor [5]. Further, obser... |

223 |
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Citation Context ...xample, only 10 molecules of Lac repressor [5]. Further, observations and computer simulations 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 Chemical Reaction Networks. Stochastic Chemical Reaction Networks ... |

186 |
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Citation Context ...xample, only 10 molecules of Lac repressor [5]. Further, observations and computer simulations 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 Chemical Reaction Networks. Stochastic Chemical Reaction Networks ... |

155 |
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Citation Context ... computational power of Stochastic Chemical Reaction Networks. Stochastic Chemical Reaction Networks are closely related to computational models such as Petri nets [9], Vector Addition Systems (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 ... |

102 |
J.Peccoud. Quantitative modeling of stochastic systems in molecular biology by using stochastic petri nets
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(Show Context)
Citation Context ...mally identical to Stochastic Chemical Reaction Networks: each place corresponds to a molecular species (the number of tokens is the number of molecules) and each transition corresponds to a reaction =-=[27]-=-. A closely related model, Vector Addition Systems (VASs), was developed and studied by Karp and Miller [10] for analyzing asynchronous parallel processes. Here, questions concern walks through an m d... |

93 |
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(Show Context)
Citation Context ...], involves showing that Register Machines (RMs) can be simulated by SCRNs for any length of time with little probability of error. Since it is known that any computer program can be compiled to a RM =-=[42, 13]-=-, we can conclude that any computer program can be effectively compiled to a SCRN. Also since there exist specific RMs known to be Turing-universal (i.e. capable of simulating any computer program), w... |

89 | Decidability issues for Petri nets – a survey
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(Show Context)
Citation Context ... Bruck In this paper, we examine the computational power of Stochastic Chemical Reaction Networks. Stochastic Chemical Reaction Networks are closely related to computational models such as Petri nets =-=[9]-=-, Vector Addition Systems (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 nondeterminis... |

88 |
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Citation Context ...e able to implement a desired target function. We call this the gate implementability question. The terms gate and function will be used interchangeably here. It has been known since the time of Post =-=[30]-=- that, given a set of functions of boolean values, only a finite number of tests need to be done to know whether a particular target function can or cannot be implemented by them, if function values, ... |

75 |
Universality in elementary cellular automata
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Citation Context ...han the gold standard computational systems such as Turing Machines. On the other hand, we have become accustomed to proofs that even some of the simplest systems are capable of universal computation =-=[45, 46]-=-, meaning that they are in some senses equivalent in power to Turing Machines, and thus predicting their eventual behavior is impossible even in theory. Chemical reaction networks have been shown to b... |

74 |
Thermodynamics of computation- a review
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(Show Context)
Citation Context ...ent in power to Turing Machines, and thus predicting their eventual behavior is impossible even in theory. Chemical reaction networks have been shown to be universal when combined with polymer memory =-=[18]-=- or membraneseparated compartmental memory [22], but researchers have previously assumed that on their own, a finite number of species in a well-mixed medium can only perform bounded computations [24,... |

57 |
Zum Hilbertschen Aufbau der reellen Zahlen
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Citation Context ...ardizing or formalizing the process of mathematical calculation. Perhaps this was the way?Programmability of Chemical Reaction Networks 15 It turned out that this was not the way. In 1928, Ackermann =-=[36]-=- showed that there is a limit to how fast a Primitive Recursive Function can grow (depending on how many other functions are used to help define it), and there turn out to exist simple deterministic a... |

55 |
On the power of the splicing operation
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Citation Context ...energy per computational step can be achieved. Since then, there have been many more formal works proving that biochemical reactions that act on polymers can perform Turinguniversal computation (e.g. =-=[19, 20, 21]-=-). In all of these studies, unlike the work presented here, there are an infinite number of distinct chemical species (polymers with different lengths and different sequences) and thus, formally, an i... |

52 |
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Citation Context .... Cook et al. Specifically, given a volume V , for any state A = (q1,...,qm), the rate of reaction α in that state is ρα(A) = kαV m∏ i=1 (qi) rα,i V rα,i where q r def = q! = q(q − 1) ···(q − r + 1). =-=(1)-=- (q − r)! Since the solution is assumed to be well stirred, the time until a particular reaction α occurs in state A is an exponentially distributed random variable with the rate parameter ρα(A); i.e.... |

43 |
On undecidable propositions of formal mathematical systems
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- 1965
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Citation Context ...ynomial. Not long after Ackermann’s paper made it clear that Primitive Recursive Functions were merely a strict subset of the functions that can be calculated, Herbrand in 1931 [37] and Gödel in 1934 =-=[38]-=- defined General Recursive Functions, which in 1936 were argued by both Church [39] and Turing [35] to correspond exactly to the set of all functions that can possibly be calculated in any algorithmic... |

41 | Self-assembled circuit patterns
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(Show Context)
Citation Context ...han the gold standard computational systems such as Turing Machines. On the other hand, we have become accustomed to proofs that even some of the simplest systems are capable of universal computation =-=[45, 46]-=-, meaning that they are in some senses equivalent in power to Turing Machines, and thus predicting their eventual behavior is impossible even in theory. Chemical reaction networks have been shown to b... |

41 | Arkin A: Stochastic mechanisms in gene expression - HH - 1997 |

41 | AJ, Siggia ED, Swain PS (2002) Stochastic gene expression in a single cell. Science Signalling 297 - MB, Levine |

35 |
Chemical implementation of neural networks and turing machines
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Citation Context ... affect the speed with which the computation occurs, but do not change the eventuality. Circuits of the same general flavor as the one above can be modified to work with mass action chemical kinetics =-=[25, 24]-=-, showing that individual boolean logic gates can be constructed, and that they can be connected together into a circuit. This provides for efficient computation but is a non-uniform model: the number... |

30 | Fast computation by population protocols with a leader
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Citation Context ... [10], the corresponding questions of probability are undecidable [14, 15]. This result derives from showing that Stochastic Chemical Reaction Networks can simulate Register Machines [16] efficiently =-=[17]-=- within a known error bound that is independent of the unknown number of steps prior to halting [14]. This result – that when answers must be guaranteed to be correct, computational power is limited, ... |

25 |
An excitable gene regulatory circuit induces transient cellular differentiation
- Suel, Garcia-Ojalvo, et al.
(Show Context)
Citation Context ...xample, only 10 molecules of Lac repressor [5]. Further, observations and computer simulations 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 Chemical Reaction Networks. Stochastic Chemical Reaction Networks ... |

25 |
Unpredictable iterations
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Citation Context ...cessary for a Stochastic Chemical Reaction Network to be universal.Programmability of Chemical Reaction Networks 27 Another model that turns out to be related is a lesser known model called Fractran =-=[11]-=-, shown by Conway to be Turing universal. A Fractran program consists of an ordered list of rational numbers (see figure 1(d)). Execution is deterministic: starting with a positive integer n as input,... |

21 | On the computational power of biochemistry
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(Show Context)
Citation Context ...energy per computational step can be achieved. Since then, there have been many more formal works proving that biochemical reactions that act on polymers can perform Turinguniversal computation (e.g. =-=[19, 20, 21]-=-). In all of these studies, unlike the work presented here, there are an infinite number of distinct chemical species (polymers with different lengths and different sequences) and thus, formally, an i... |

20 |
Chemical kinetics is turing universal
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Citation Context ...ents can be achieved by diffision of specific species, or by explicit transfer reactions. This is, for example, exploited in the Chemical Abstract Machine [22] and in Membrane Systems [23]. Note that =-=[24]-=-, contrary to its title, only establishes that Chemical Reaction Networks appear to be able to implement feed-forward circuits (along the lines of section 3), making them empirically at least P-hard.... |

20 |
A Curious Nim-type game
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Citation Context ...l thus move to the list of next lower index. It remains to show that these lists eventually get exhausted as the algorithm progresses deeper into the tree. For readers familiar with the game of Chomp =-=[41]-=-, this process is quite similar to Chomp on infinite boards. The Bound To each point in the search tree, with its state and its lists, we can assign a positive integer as described below. We will see ... |

19 |
Persistence of vector replacement systems is decidable
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(Show Context)
Citation Context ...of d given vectors Vα ∈ Z m , and each point in the walk must have no negative coordinates. Whether it is possible to walk from a point x to a point y (the reachability question) is in fact decidable =-=[28]-=-. It is also decidable whether it is possible for a walk to enter a linearly-defined subregion [29] – a special case is whether the i th component of the point ever becomes non-zero (the producibility... |

18 | Computation with Finite Stochastic Chemical Reaction Networks
- Soloveichik, Cook, et al.
(Show Context)
Citation Context ...works, it it particularly interesting that whereas most questions of possibility concerning the behavior of these models are decidable [10], the corresponding questions of probability are undecidable =-=[14, 15]-=-. This result derives from showing that Stochastic Chemical Reaction Networks can simulate Register Machines [16] efficiently [17] within a known error bound that is independent of the unknown number ... |

15 |
Begründung der elementaren Arithmetik durch die rekurrierende Denkweise ohne Anwendung scheinbarer Verënderlichen mit unendlichem Ausdehnungsbereich
- Skolem
- 1923
(Show Context)
Citation Context ...with the largest amount of S2 and returning the corresponding amount of S1. 5.1 Primitive Recursive Functions Primitive Recursive Functions were first investigated in the 1920’s, starting with Skolem =-=[33]-=-, who pointed out that many standard functions on nonnegative integers can be defined using just function composition and recursion, starting with just the successor function. This surprising fact is ... |

15 | Four fast universal Turing machines
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(Show Context)
Citation Context ...the original Turing Machine, but only by a linear factor—if the original machine took O(t 16 ) steps, the queue program will take O(t 17 ) steps. (Slightly more efficient implementations are possible =-=[43, 14]-=-.) With this queue architecture, the challenge of detecting a single molecule is avoided; all we need is a scheme that allows the system to be able to read the high order memory digit quickly and accu... |

13 | LM, Elowitz MB (2006) An excitable gene regulatory circuit induces transient cellular differentiation. Nature 440: 545–550 - GM, Garcia-Ojalvo, et al. |

12 |
Does replication-induced transcription regulate synthesis of the myriad low copy number proteins of Escherichia coli? BioEssays 17
- Guptasarma
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(Show Context)
Citation Context ...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 expressed at fewer than a hundred copies per cell =-=[4]-=-, averaging, for example, only 10 molecules of Lac repressor [5]. Further, observations and computer simulations have shown that stochastic effects resulting from these small numbers may be physiologi... |

12 | Biological computing
- Kurtz, Mahaney, et al.
- 1997
(Show Context)
Citation Context ...energy per computational step can be achieved. Since then, there have been many more formal works proving that biochemical reactions that act on polymers can perform Turinguniversal computation (e.g. =-=[19, 20, 21]-=-). In all of these studies, unlike the work presented here, there are an infinite number of distinct chemical species (polymers with different lengths and different sequences) and thus, formally, an i... |

12 |
A multiparameter analysis of the boundedness problem for vector addition systems
- Rosier, Yen
(Show Context)
Citation Context ...questions of possibility listed above are known to be decidable, their complexity is sometimes not so clear. The complexity of the problem for X=“grow without bound” is known to be doubly exponential =-=[32]-=-, but the complexity of the problem for X=“reach a certain state” has been an open problem for decades [9]. Even though double exponential complexity sounds quite complex, the complexity of these type... |

10 |
Rekursive funktionen. Akadémiai Kiadó
- Péter
- 1953
(Show Context)
Citation Context ...hether a given algorithm halts.16 Matthew Cook, David Soloveichik, Erik Winfree, and Jehoshua Bruck While most people turned their attention at this point to General Recursive Functions, Rózsa Péter =-=[40]-=- continued to develop the theory of Primitive Recursive Functions, treating them not as a historical mistake, but as an opportunity for study. Her work makes it clear that the following definition is ... |

9 |
The decidability of the reachability problem for vector addition systems (preliminary version
- Sacerdote, Tenney
- 1977
(Show Context)
Citation Context ... it is possible to walk from a point x to a point y (the reachability question) is in fact decidable [28]. It is also decidable whether it is possible for a walk to enter a linearly-defined subregion =-=[29]-=- – a special case is whether the i th component of the point ever becomes non-zero (the producibility question). The correspondence between Vector Addition Systems, Stochastic Chemical Reaction Networ... |

8 |
Fractran: A simple universal programming language for arithmetic
- Conway
- 1987
(Show Context)
Citation Context ...power of Stochastic Chemical Reaction Networks. Stochastic Chemical Reaction Networks are closely related to computational models such as Petri nets [9], Vector Addition Systems (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 t... |

7 |
Sur la non-contradiction de l’arithmétique, Jounal für die reine und angewandteMathematik 166: 1–8. English translation in [Herbrand
- Herbrand
- 1971
(Show Context)
Citation Context ... all mathematical calculations. Not long after Ackermann’s paper made it clear that Primitive Recursive Functions were merely a strict subset of the functions that can be calculated, Herbrand in 1931 =-=[37]-=- and Gödel in 1934 [38] defined General Recursive Functions, which in 1936 were argued by both Church [39] and Turing [35] to correspond exactly to the set of all functions that can possibly be calcul... |

6 | Termination problems in chemical kinetics
- Zavattaro, Cardelli
- 2008
(Show Context)
Citation Context ...works, it it particularly interesting that whereas most questions of possibility concerning the behavior of these models are decidable [10], the corresponding questions of probability are undecidable =-=[14, 15]-=-. This result derives from showing that Stochastic Chemical Reaction Networks can simulate Register Machines [16] efficiently [17] within a known error bound that is independent of the unknown number ... |

6 | Turing complete catalytic particle computers
- Liekens, Fernando
- 2006
(Show Context)
Citation Context ...els are decidable [10], the corresponding questions of probability are undecidable [14, 15]. This result derives from showing that Stochastic Chemical Reaction Networks can simulate Register Machines =-=[16]-=- efficiently [17] within a known error bound that is independent of the unknown number of steps prior to halting [14]. This result – that when answers must be guaranteed to be correct, computational p... |

6 |
Peccoud J: Quantitative modeling of stochastic systems in molecular biology by using stochastic Petri nets
- PJE
- 1998
(Show Context)
Citation Context ...mally identical to Stochastic Chemical Reaction Networks: each place corresponds to a molecular species (the number of tokens is the number of molecules) and each transition corresponds to a reaction =-=[27]-=-. A closely related model, Vector Addition Systems (VASs), was developed and studied by Karp and Miller [10] for analyzing asynchronous parallel processes. Here, questions concern walks through an m d... |

5 | Networks of Relations
- Cook
- 2005
(Show Context)
Citation Context ...or circuits built of relations, rather than functions, then the ability to reuse values makes this question become decidable, whereas it is undecidable if values, once produced, can only be used once =-=[31]-=-. It is natural to wonder, if fan-out is not available, might the gate implementability question become undecidable, as it did for relations? First of all, we have to be clear about what we mean by “c... |

4 |
Genes VII
- Levin
- 1999
(Show Context)
Citation Context ...r species [1, 2, 3]. For example, over 80% of the genes in the E. coli chromosome are expressed at fewer than a hundred copies per cell [4], averaging, for example, only 10 molecules of Lac repressor =-=[5]-=-. Further, observations and computer simulations have shown that stochastic effects resulting from these small numbers may be physiologically significant [6, 7, 8].2 Matthew Cook, David Soloveichik, ... |

4 |
Kommunikation mit automaten
- CA
- 1962
(Show Context)
Citation Context ... minor, amounting mostlyProgrammability of Chemical Reaction Networks 551 just to different interpretations or viewpoints of the same underlying fundamental process. For example, consider Petri nets =-=[26]-=-, as shown in Fig. 1(b). In this model, a network consists of a directed bipartite graph, having connections between places (shown as circles) and transitions (shown as black bars). The state consists... |