Gillespie algorithm - Revision history

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2025-06-23T22:53:13Z

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	{{Short description\|Method for stochastic equation systems}}		{{Short description\|Method for stochastic equation systems}}
	In [[probability theory]], the '''Gillespie algorithm''' (or the '''Doob–Gillespie algorithm''' or '''stochastic simulation algorithm''', the '''SSA''') generates a statistically correct trajectory (possible solution) of a [[stochastic]] equation system for which the [[reaction rates]] are known. It was created by [[Joseph L. Doob]] and others (circa 1945), presented by [[Dan Gillespie]] in 1976, and popularized in 1977 in a paper where he uses it to simulate chemical or biochemical systems of reactions efficiently and accurately using limited computational power (see [[stochastic simulation]]).<ref>{{Cite journal \|last=Gillespie \|first=Daniel T. \|date=2007-05-01 \|title=Stochastic Simulation of Chemical Kinetics \|url=https://www.annualreviews.org/doi/10.1146/annurev.physchem.58.032806.104637 \|journal=Annual Review of Physical Chemistry \|language=en \|volume=58 \|issue=1 \|pages=35–55 \|doi=10.1146/annurev.physchem.58.032806.104637 \|pmid=17037977 \|bibcode=2007ARPC...58...35G \|issn=0066-426X}}</ref> As computers have become faster, the algorithm has been used to simulate increasingly complex systems. The algorithm is particularly useful for simulating reactions within cells, where the number of [[reagent]]s is low and keeping track of every single reaction is computationally feasible. Mathematically, it is a variant of a [[dynamic Monte Carlo method]] and similar to the [[kinetic Monte Carlo]] methods. It is used heavily in [[computational systems biology]].{{citation needed\|date=June 2012}}		In [[probability theory]], the '''Gillespie algorithm''' (or the '''Doob–Gillespie algorithm''' or '''stochastic simulation algorithm''', the '''SSA''') generates a statistically correct trajectory (possible solution) of a [[stochastic]] equation system for which the [[reaction rates]] are known. It was created by [[Joseph L. Doob]] and others (circa 1945), presented by [[Dan Gillespie]] in 1976, and popularized in 1977 in a paper where he uses it to simulate chemical or biochemical systems of reactions efficiently and accurately using limited computational power (see [[stochastic simulation]]).<ref>{{Cite journal \|last=Gillespie \|first=Daniel T. \|date=2007-05-01 \|title=Stochastic Simulation of Chemical Kinetics \|url=https://www.annualreviews.org/doi/10.1146/annurev.physchem.58.032806.104637 \|journal=Annual Review of Physical Chemistry \|language=en \|volume=58 \|issue=1 \|pages=35–55 \|doi=10.1146/annurev.physchem.58.032806.104637 \|pmid=17037977 \|bibcode=2007ARPC...58...35G \|issn=0066-426X\|url-access=subscription }}</ref> As computers have become faster, the algorithm has been used to simulate increasingly complex systems. The algorithm is particularly useful for simulating reactions within cells, where the number of [[reagent]]s is low and keeping track of every single reaction is computationally feasible. Mathematically, it is a variant of a [[dynamic Monte Carlo method]] and similar to the [[kinetic Monte Carlo]] methods. It is used heavily in [[computational systems biology]].{{citation needed\|date=June 2012}}

	==History==		==History==

37.171.162.41: This section had nothing to do with the Gillespie algorithm. The content of the section just refer to a model of chemical reactions where the Gillespie algorithm was used for simulations.

2025-01-23T21:54:36Z

This section had nothing to do with the Gillespie algorithm. The content of the section just refer to a model of chemical reactions where the Gillespie algorithm was used for simulations.

← Previous revision		Revision as of 21:54, 23 January 2025
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	That was just a simple example, with two reactions. More complex systems with more reactions are handled in the same way. All reaction rates must be calculated at each time step, and one chosen with probability equal to its fractional contribution to the rate. Time is then advanced as in this example.		That was just a simple example, with two reactions. More complex systems with more reactions are handled in the same way. All reaction rates must be calculated at each time step, and one chosen with probability equal to its fractional contribution to the rate. Time is then advanced as in this example.

	==Stochastic self-assembly==
	{{Needs expansion\|date=April 2023}}
	The [[Gard model]] describes [[self-assembly]] of lipids into aggregates. Using [[stochastic]] simulations it shows the emergence of multiple types of aggregates and their evolution.

	==References==		==References==

73.165.53.153 at 01:43, 5 July 2024

2024-07-05T01:43:17Z

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	In a reaction chamber, there are a finite number of molecules. At each infinitesimal slice of time, a single reaction might take place. The rate is determined by the number of molecules in each chemical species.		In a reaction chamber, there are a finite number of molecules. At each infinitesimal slice of time, a single reaction might take place. The rate is determined by the number of molecules in each chemical species.

	Naively, we can simulate the trajectory of the reaction chamber by discretizing time, then simulate each time-step. However, there might be long stretches of time where no reaction occurs. The Gillespie algorithm samples a random waiting time until ''some'' reaction occurs, then take another random sample to decide ''which'' reaction has occurred. ~~Then it~~		Naively, we can simulate the trajectory of the reaction chamber by discretizing time, then simulate each time-step. However, there might be long stretches of time where no reaction occurs. The Gillespie algorithm samples a random waiting time until ''some'' reaction occurs, then take another random sample to decide ''which'' reaction has occurred.

	The key assumptions are that		The key assumptions are that

Cosmia Nebula: /* Idea behind the algorithm */ idea

2024-06-21T02:59:02Z

Idea behind the algorithm: idea

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 Later, Doob (1942, 1945) extended Feller's solutions beyond the case of pure-jump processes. The method was implemented in computers by [[David George Kendall]] (1950) using the [[Manchester Mark 1]] computer and later used by [[Maurice S. Bartlett]] (1953) in his studies of epidemics outbreaks. Gillespie (1977) obtains the algorithm in a different manner by making use of a physical argument.
-==Idea behind the algorithm==
+==Idea==
 Traditional continuous and deterministic biochemical [[rate equation]]s do not accurately predict cellular reactions since they rely on bulk reactions that require the interactions of millions of molecules. They are typically modeled as a set of coupled ordinary differential equations. In contrast, the Gillespie algorithm allows a discrete and stochastic simulation of a system with few reactants because every reaction is explicitly simulated. A trajectory corresponding to a single Gillespie simulation represents an exact sample from the probability mass function that is the solution of the [[master equation]].

InternetArchiveBot: Rescuing 1 sources and tagging 0 as dead.) #IABot (v2.0.9.5

2024-06-04T02:54:41Z

Rescuing 1 sources and tagging 0 as dead.) #IABot (v2.0.9.5

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	* {{cite journal \|author=Doob, Jacob L. \|title=Topics in the Theory of Markoff Chains \|journal=Transactions of the American Mathematical Society \|volume=52 \|pages=37–64 \|year=1942 \|issue=1 \|jstor=1990152 \|doi=10.1090/S0002-9947-1942-0006633-7 \|doi-access=free }}		* {{cite journal \|author=Doob, Jacob L. \|title=Topics in the Theory of Markoff Chains \|journal=Transactions of the American Mathematical Society \|volume=52 \|pages=37–64 \|year=1942 \|issue=1 \|jstor=1990152 \|doi=10.1090/S0002-9947-1942-0006633-7 \|doi-access=free }}
	* {{cite journal \|author=Doob, Jacob L. \|title=Markoff chains – Denumerable case \|journal=Transactions of the American Mathematical Society \|volume=58 \|pages=455–473 \|year=1945 \|doi=10.2307/1990339 \|issue=3 \|jstor=1990339 }}		* {{cite journal \|author=Doob, Jacob L. \|title=Markoff chains – Denumerable case \|journal=Transactions of the American Mathematical Society \|volume=58 \|pages=455–473 \|year=1945 \|doi=10.2307/1990339 \|issue=3 \|jstor=1990339 }}
	* {{Cite book \|last1=Press \|first1=William H. \|last2=Teukolsky \|first2=Saul A. \|last3=Vetterling \|first3=William T. \|last4=Flannery \|first4=Brian P. \|year=2007 \|title=Numerical Recipes: The Art of Scientific Computing \|edition=3rd \|publisher=Cambridge University Press \|location=New York, NY \|isbn=978-0-521-88068-8 \|chapter=Section 17.7. Stochastic Simulation of Chemical Reaction Networks \|chapter-url=http://apps.nrbook.com/empanel/index.html#pg=946 }}		* {{Cite book \|last1=Press \|first1=William H. \|last2=Teukolsky \|first2=Saul A. \|last3=Vetterling \|first3=William T. \|last4=Flannery \|first4=Brian P. \|year=2007 \|title=Numerical Recipes: The Art of Scientific Computing \|edition=3rd \|publisher=Cambridge University Press \|location=New York, NY \|isbn=978-0-521-88068-8 \|chapter=Section 17.7. Stochastic Simulation of Chemical Reaction Networks \|chapter-url=http://apps.nrbook.com/empanel/index.html#pg=946 \|access-date=2011-08-17 \|archive-date=2011-08-11 \|archive-url=https://web.archive.org/web/20110811154417/http://apps.nrbook.com/empanel/index.html#pg=946 \|url-status=dead }}
	* {{cite journal \|author=Kolmogorov, Andrey N. \|title=Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung \|trans-title=On Analytical Methods in the Theory of Probability \|journal=Mathematische Annalen \|volume=104 \|pages=415–458 \|year=1931 \|doi=10.1007/BF01457949 \|s2cid=119439925 }}		* {{cite journal \|author=Kolmogorov, Andrey N. \|title=Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung \|trans-title=On Analytical Methods in the Theory of Probability \|journal=Mathematische Annalen \|volume=104 \|pages=415–458 \|year=1931 \|doi=10.1007/BF01457949 \|s2cid=119439925 }}
	* {{cite journal \|author=Feller, Willy \|title=On the Integro-Differential Equations of Purely Discontinuous Markoff Processes \|journal=Transactions of the American Mathematical Society \|volume= 48 \|pages=4885–15 \|year=1940 \|jstor=1970064 \|issue=3 \|doi=10.2307/1990095\|doi-access=free }}		* {{cite journal \|author=Feller, Willy \|title=On the Integro-Differential Equations of Purely Discontinuous Markoff Processes \|journal=Transactions of the American Mathematical Society \|volume= 48 \|pages=4885–15 \|year=1940 \|jstor=1970064 \|issue=3 \|doi=10.2307/1990095\|doi-access=free }}

LucasBrown: Adding short description: "Method for stochastic equation systems"

2024-05-18T10:35:52Z

Adding short description: "Method for stochastic equation systems"

← Previous revision		Revision as of 10:35, 18 May 2024
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			{{Short description\|Method for stochastic equation systems}}
	In [[probability theory]], the '''Gillespie algorithm''' (or the '''Doob–Gillespie algorithm''' or '''stochastic simulation algorithm''', the '''SSA''') generates a statistically correct trajectory (possible solution) of a [[stochastic]] equation system for which the [[reaction rates]] are known. It was created by [[Joseph L. Doob]] and others (circa 1945), presented by [[Dan Gillespie]] in 1976, and popularized in 1977 in a paper where he uses it to simulate chemical or biochemical systems of reactions efficiently and accurately using limited computational power (see [[stochastic simulation]]).<ref>{{Cite journal \|last=Gillespie \|first=Daniel T. \|date=2007-05-01 \|title=Stochastic Simulation of Chemical Kinetics \|url=https://www.annualreviews.org/doi/10.1146/annurev.physchem.58.032806.104637 \|journal=Annual Review of Physical Chemistry \|language=en \|volume=58 \|issue=1 \|pages=35–55 \|doi=10.1146/annurev.physchem.58.032806.104637 \|pmid=17037977 \|bibcode=2007ARPC...58...35G \|issn=0066-426X}}</ref> As computers have become faster, the algorithm has been used to simulate increasingly complex systems. The algorithm is particularly useful for simulating reactions within cells, where the number of [[reagent]]s is low and keeping track of every single reaction is computationally feasible. Mathematically, it is a variant of a [[dynamic Monte Carlo method]] and similar to the [[kinetic Monte Carlo]] methods. It is used heavily in [[computational systems biology]].{{citation needed\|date=June 2012}}		In [[probability theory]], the '''Gillespie algorithm''' (or the '''Doob–Gillespie algorithm''' or '''stochastic simulation algorithm''', the '''SSA''') generates a statistically correct trajectory (possible solution) of a [[stochastic]] equation system for which the [[reaction rates]] are known. It was created by [[Joseph L. Doob]] and others (circa 1945), presented by [[Dan Gillespie]] in 1976, and popularized in 1977 in a paper where he uses it to simulate chemical or biochemical systems of reactions efficiently and accurately using limited computational power (see [[stochastic simulation]]).<ref>{{Cite journal \|last=Gillespie \|first=Daniel T. \|date=2007-05-01 \|title=Stochastic Simulation of Chemical Kinetics \|url=https://www.annualreviews.org/doi/10.1146/annurev.physchem.58.032806.104637 \|journal=Annual Review of Physical Chemistry \|language=en \|volume=58 \|issue=1 \|pages=35–55 \|doi=10.1146/annurev.physchem.58.032806.104637 \|pmid=17037977 \|bibcode=2007ARPC...58...35G \|issn=0066-426X}}</ref> As computers have become faster, the algorithm has been used to simulate increasingly complex systems. The algorithm is particularly useful for simulating reactions within cells, where the number of [[reagent]]s is low and keeping track of every single reaction is computationally feasible. Mathematically, it is a variant of a [[dynamic Monte Carlo method]] and similar to the [[kinetic Monte Carlo]] methods. It is used heavily in [[computational systems biology]].{{citation needed\|date=June 2012}}

Michael Hardy: /* Algorithm */

2024-05-02T20:23:58Z

Algorithm

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	5. Record <math>(\boldsymbol{x}, t)</math> as desired. Return to step 2, or else end the simulation.		5. Record <math>(\boldsymbol{x}, t)</math> as desired. Return to step 2, or else end the simulation.

	~~Where~~ <math>\~~nu_{j}~~</math> represents adding the <math>j^\text{th}</math> component of ~~your~~ given state-change vector <math>\nu</math>. This family of algorithms is computationally expensive and thus many modifications and adaptations exist, including the next reaction method (Gibson & Bruck), [[tau-leaping]], as well as hybrid techniques where abundant reactants are modeled with deterministic behavior. Adapted techniques generally compromise the exactitude of the theory behind the algorithm as it connects to the master equation, but offer reasonable realizations for greatly improved timescales. The computational cost of exact versions of the algorithm is determined by the coupling class of the reaction network. In weakly coupled networks, the number of reactions that is influenced by any other reaction is bounded by a small constant. In strongly coupled networks, a single reaction firing can in principle affect all other reactions. An exact version of the algorithm with constant-time scaling for weakly coupled networks has been developed, enabling efficient simulation of systems with very large numbers of reaction channels (Slepoy Thompson Plimpton 2008). The generalized Gillespie algorithm that accounts for the non-Markovian properties of random biochemical events with delay has been developed by Bratsun et al. 2005 and independently Barrio et al. 2006, as well as (Cai 2007). See the articles cited below for details.		where <math>\nu_j</math> represents adding the <math>j^\text{th}</math> component of the given state-change vector <math>\nu</math>. This family of algorithms is computationally expensive and thus many modifications and adaptations exist, including the next reaction method (Gibson & Bruck), [[tau-leaping]], as well as hybrid techniques where abundant reactants are modeled with deterministic behavior. Adapted techniques generally compromise the exactitude of the theory behind the algorithm as it connects to the master equation, but offer reasonable realizations for greatly improved timescales. The computational cost of exact versions of the algorithm is determined by the coupling class of the reaction network. In weakly coupled networks, the number of reactions that is influenced by any other reaction is bounded by a small constant. In strongly coupled networks, a single reaction firing can in principle affect all other reactions. An exact version of the algorithm with constant-time scaling for weakly coupled networks has been developed, enabling efficient simulation of systems with very large numbers of reaction channels (Slepoy Thompson Plimpton 2008). The generalized Gillespie algorithm that accounts for the non-Markovian properties of random biochemical events with delay has been developed by Bratsun et al. 2005 and independently Barrio et al. 2006, as well as (Cai 2007). See the articles cited below for details.

	Partial-propensity formulations, as developed independently by both Ramaswamy et al. (2009, 2010) and Indurkhya and Beal (2010), are available to construct a family of exact versions of the algorithm whose computational cost is proportional to the number of chemical species in the network, rather than the (larger) number of reactions. These formulations can reduce the computational cost to constant-time scaling for weakly coupled networks and to scale at most linearly with the number of species for strongly coupled networks. A partial-propensity variant of the generalized Gillespie algorithm for reactions with delays has also been proposed (Ramaswamy Sbalzarini 2011). The use of partial-propensity methods is limited to elementary chemical reactions, i.e., reactions with at most two different reactants. Every non-elementary chemical reaction can be equivalently decomposed into a set of elementary ones, at the expense of a linear (in the order of the reaction) increase in network size.		Partial-propensity formulations, as developed independently by both Ramaswamy et al. (2009, 2010) and Indurkhya and Beal (2010), are available to construct a family of exact versions of the algorithm whose computational cost is proportional to the number of chemical species in the network, rather than the (larger) number of reactions. These formulations can reduce the computational cost to constant-time scaling for weakly coupled networks and to scale at most linearly with the number of species for strongly coupled networks. A partial-propensity variant of the generalized Gillespie algorithm for reactions with delays has also been proposed (Ramaswamy Sbalzarini 2011). The use of partial-propensity methods is limited to elementary chemical reactions, i.e., reactions with at most two different reactants. Every non-elementary chemical reaction can be equivalently decomposed into a set of elementary ones, at the expense of a linear (in the order of the reaction) increase in network size.

Michael Hardy: /* Algorithm */

2024-05-02T19:39:26Z

Algorithm

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	5. Record <math>(\boldsymbol{x}, t)</math> as desired. Return to step 2, or else end the simulation.		5. Record <math>(\boldsymbol{x}, t)</math> as desired. Return to step 2, or else end the simulation.

	Where <math>\nu_{j}</math> represents adding the <math>j^{th}</math> component of your given state-change vector <math>\nu</math>. This family of algorithms is computationally expensive and thus many modifications and adaptations exist, including the next reaction method (Gibson & Bruck), [[tau-leaping]], as well as hybrid techniques where abundant reactants are modeled with deterministic behavior. Adapted techniques generally compromise the exactitude of the theory behind the algorithm as it connects to the master equation, but offer reasonable realizations for greatly improved timescales. The computational cost of exact versions of the algorithm is determined by the coupling class of the reaction network. In weakly coupled networks, the number of reactions that is influenced by any other reaction is bounded by a small constant. In strongly coupled networks, a single reaction firing can in principle affect all other reactions. An exact version of the algorithm with constant-time scaling for weakly coupled networks has been developed, enabling efficient simulation of systems with very large numbers of reaction channels (Slepoy Thompson Plimpton 2008). The generalized Gillespie algorithm that accounts for the non-Markovian properties of random biochemical events with delay has been developed by Bratsun et al. 2005 and independently Barrio et al. 2006, as well as (Cai 2007). See the articles cited below for details.		Where <math>\nu_{j}</math> represents adding the <math>j^\text{th}</math> component of your given state-change vector <math>\nu</math>. This family of algorithms is computationally expensive and thus many modifications and adaptations exist, including the next reaction method (Gibson & Bruck), [[tau-leaping]], as well as hybrid techniques where abundant reactants are modeled with deterministic behavior. Adapted techniques generally compromise the exactitude of the theory behind the algorithm as it connects to the master equation, but offer reasonable realizations for greatly improved timescales. The computational cost of exact versions of the algorithm is determined by the coupling class of the reaction network. In weakly coupled networks, the number of reactions that is influenced by any other reaction is bounded by a small constant. In strongly coupled networks, a single reaction firing can in principle affect all other reactions. An exact version of the algorithm with constant-time scaling for weakly coupled networks has been developed, enabling efficient simulation of systems with very large numbers of reaction channels (Slepoy Thompson Plimpton 2008). The generalized Gillespie algorithm that accounts for the non-Markovian properties of random biochemical events with delay has been developed by Bratsun et al. 2005 and independently Barrio et al. 2006, as well as (Cai 2007). See the articles cited below for details.

	Partial-propensity formulations, as developed independently by both Ramaswamy et al. (2009, 2010) and Indurkhya and Beal (2010), are available to construct a family of exact versions of the algorithm whose computational cost is proportional to the number of chemical species in the network, rather than the (larger) number of reactions. These formulations can reduce the computational cost to constant-time scaling for weakly coupled networks and to scale at most linearly with the number of species for strongly coupled networks. A partial-propensity variant of the generalized Gillespie algorithm for reactions with delays has also been proposed (Ramaswamy Sbalzarini 2011). The use of partial-propensity methods is limited to elementary chemical reactions, i.e., reactions with at most two different reactants. Every non-elementary chemical reaction can be equivalently decomposed into a set of elementary ones, at the expense of a linear (in the order of the reaction) increase in network size.		Partial-propensity formulations, as developed independently by both Ramaswamy et al. (2009, 2010) and Indurkhya and Beal (2010), are available to construct a family of exact versions of the algorithm whose computational cost is proportional to the number of chemical species in the network, rather than the (larger) number of reactions. These formulations can reduce the computational cost to constant-time scaling for weakly coupled networks and to scale at most linearly with the number of species for strongly coupled networks. A partial-propensity variant of the generalized Gillespie algorithm for reactions with delays has also been proposed (Ramaswamy Sbalzarini 2011). The use of partial-propensity methods is limited to elementary chemical reactions, i.e., reactions with at most two different reactants. Every non-elementary chemical reaction can be equivalently decomposed into a set of elementary ones, at the expense of a linear (in the order of the reaction) increase in network size.

Michael Hardy: /* Algorithm */

2024-05-02T19:38:33Z

Algorithm

← Previous revision		Revision as of 19:38, 2 May 2024
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	:<math>\tau = \frac{1}{\sum_{j}a_{j}(\boldsymbol{x})}\log\left(\frac{1}{r_{1}}\right),</math>		:<math>\tau = \frac{1}{\sum_{j}a_{j}(\boldsymbol{x})}\log\left(\frac{1}{r_{1}}\right),</math>
	and		and
	:<math>j =</math> the smallest integer satisfying <math>\sum_{j'=1}^j a_{j'}(\boldsymbol{x}) > r_2 \sum_j a_j (\boldsymbol{x}).</math>		:<math>j ={}</math>the smallest integer satisfying <math>\sum_{j'=1}^j a_{j'}(\boldsymbol{x}) > r_2 \sum_j a_j (\boldsymbol{x}).</math>

	Utilizing this generating method for the sojourn time and next reaction, the direct method algorithm is stated by Gillespie as		Utilizing this generating method for the sojourn time and next reaction, the direct method algorithm is stated by Gillespie as

Michael Hardy: /* Reversible binding of A and B to form AB dimers */

2024-05-02T19:35:24Z

Reversible binding of A and B to form AB dimers

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	The total reaction rate, <math>R_\mathrm{TOT}</math>, at time ''t'' is then given by		The total reaction rate, <math>R_\mathrm{TOT}</math>, at time ''t'' is then given by

	<math>R_\mathrm{TOT}=k_\mathrm{D}n_\mathrm{A}n_\mathrm{B}+k_\mathrm{B}n_\mathrm{AB}</math>		: <math>R_\mathrm{TOT}=k_\mathrm{D}n_\mathrm{A}n_\mathrm{B}+k_\mathrm{B}n_\mathrm{AB}</math>

	So, we have now described a simple model with two reactions. This definition is independent of the Gillespie algorithm. We will now describe how to apply the Gillespie algorithm to this system.		So, we have now described a simple model with two reactions. This definition is independent of the Gillespie algorithm. We will now describe how to apply the Gillespie algorithm to this system.

← Previous revision		Revision as of 02:54, 4 June 2024
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	* {{cite journal \|author=Doob, Jacob L. \|title=Topics in the Theory of Markoff Chains \|journal=Transactions of the American Mathematical Society \|volume=52 \|pages=37–64 \|year=1942 \|issue=1 \|jstor=1990152 \|doi=10.1090/S0002-9947-1942-0006633-7 \|doi-access=free }}		* {{cite journal \|author=Doob, Jacob L. \|title=Topics in the Theory of Markoff Chains \|journal=Transactions of the American Mathematical Society \|volume=52 \|pages=37–64 \|year=1942 \|issue=1 \|jstor=1990152 \|doi=10.1090/S0002-9947-1942-0006633-7 \|doi-access=free }}
	* {{cite journal \|author=Doob, Jacob L. \|title=Markoff chains – Denumerable case \|journal=Transactions of the American Mathematical Society \|volume=58 \|pages=455–473 \|year=1945 \|doi=10.2307/1990339 \|issue=3 \|jstor=1990339 }}		* {{cite journal \|author=Doob, Jacob L. \|title=Markoff chains – Denumerable case \|journal=Transactions of the American Mathematical Society \|volume=58 \|pages=455–473 \|year=1945 \|doi=10.2307/1990339 \|issue=3 \|jstor=1990339 }}
	* {{Cite book \|last1=Press \|first1=William H. \|last2=Teukolsky \|first2=Saul A. \|last3=Vetterling \|first3=William T. \|last4=Flannery \|first4=Brian P. \|year=2007 \|title=Numerical Recipes: The Art of Scientific Computing \|edition=3rd \|publisher=Cambridge University Press \|location=New York, NY \|isbn=978-0-521-88068-8 \|chapter=Section 17.7. Stochastic Simulation of Chemical Reaction Networks \|chapter-url=http://apps.nrbook.com/empanel/index.html#pg=946 }}		* {{Cite book \|last1=Press \|first1=William H. \|last2=Teukolsky \|first2=Saul A. \|last3=Vetterling \|first3=William T. \|last4=Flannery \|first4=Brian P. \|year=2007 \|title=Numerical Recipes: The Art of Scientific Computing \|edition=3rd \|publisher=Cambridge University Press \|location=New York, NY \|isbn=978-0-521-88068-8 \|chapter=Section 17.7. Stochastic Simulation of Chemical Reaction Networks \|chapter-url=http://apps.nrbook.com/empanel/index.html#pg=946 \|access-date=2011-08-17 \|archive-date=2011-08-11 \|archive-url=https://web.archive.org/web/20110811154417/http://apps.nrbook.com/empanel/index.html#pg=946 \|url-status=dead }}
	* {{cite journal \|author=Kolmogorov, Andrey N. \|title=Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung \|trans-title=On Analytical Methods in the Theory of Probability \|journal=Mathematische Annalen \|volume=104 \|pages=415–458 \|year=1931 \|doi=10.1007/BF01457949 \|s2cid=119439925 }}		* {{cite journal \|author=Kolmogorov, Andrey N. \|title=Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung \|trans-title=On Analytical Methods in the Theory of Probability \|journal=Mathematische Annalen \|volume=104 \|pages=415–458 \|year=1931 \|doi=10.1007/BF01457949 \|s2cid=119439925 }}
	* {{cite journal \|author=Feller, Willy \|title=On the Integro-Differential Equations of Purely Discontinuous Markoff Processes \|journal=Transactions of the American Mathematical Society \|volume= 48 \|pages=4885–15 \|year=1940 \|jstor=1970064 \|issue=3 \|doi=10.2307/1990095\|doi-access=free }}		* {{cite journal \|author=Feller, Willy \|title=On the Integro-Differential Equations of Purely Discontinuous Markoff Processes \|journal=Transactions of the American Mathematical Society \|volume= 48 \|pages=4885–15 \|year=1940 \|jstor=1970064 \|issue=3 \|doi=10.2307/1990095\|doi-access=free }}