BFS/Model reduction - Revision history

Murray: /* Power-law formalisms */

2009-09-05T15:13:26Z

Power-law formalisms

← Older revision		Revision as of 15:13, 5 September 2009
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	where <math>X_i</math> is the concentration of species <math>i</math>, <math>\mu_{ij}</math> is the stochiometric coefficient, <math>\gamma_j</math> is the rate constant and <math>f_{jk}</math> is the kinetic order for the relevant reaction. Note that in this form it can exactly represent mass action dynamics, but it can also be used to approximate other biochemical dynamics (eg, Michaelis-Menten style enzymatic reactions). In the approximate case, there are formulas to figure out all of the coeffiecients based on derivatives of the full nonlinear functions evaluated an an equilibrium point. In this respect, BST is similar to a Taylor-series expansion, just using a different class of functions (sums of products of powers).		where <math>X_i</math> is the concentration of species <math>i</math>, <math>\mu_{ij}</math> is the stochiometric coefficient, <math>\gamma_j</math> is the rate constant and <math>f_{jk}</math> is the kinetic order for the relevant reaction. Note that in this form it can exactly represent mass action dynamics, but it can also be used to approximate other biochemical dynamics (eg, Michaelis-Menten style enzymatic reactions). In the approximate case, there are formulas to figure out all of the coeffiecients based on derivatives of the full nonlinear functions evaluated an an equilibrium point. In this respect, BST is similar to a Taylor-series expansion, just using a different class of functions (sums of products of powers).

	A simplified form of these dynamics can be obtained by approximating the sum by just two terms: ~~synthesis~~ and ~~degradation~~. In this case, the equations have the form		A simplified form of these dynamics can be obtained by approximating the sum by just two terms: production and consumption. In this case, the equations have the form
	<center><amsmath>		<center><amsmath>
	\frac{dX_i}{dt} = \alpha_i \prod_k X_k^{f_{k}} - \beta_i \prod_k X_k^{g_{k}}.		\frac{dX_i}{dt} = \alpha_i \prod_k X_k^{f_{k}} - \beta_i \prod_k X_k^{g_{k}}.

Murray: /* Power-law formalisms */

2009-09-05T01:09:49Z

Power-law formalisms

← Older revision		Revision as of 01:09, 5 September 2009
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	\frac{dX_i}{dt} = \sum_j \mu_{ij} \left( \gamma_j \prod_k X_k^{f_{jk}} \right)		\frac{dX_i}{dt} = \sum_j \mu_{ij} \left( \gamma_j \prod_k X_k^{f_{jk}} \right)
	</amsmath></center>		</amsmath></center>
	where <math>X_i</math> is the concentration of species <~~amsmath~~>i</~~amsmath~~>, <~~amsmath~~>\mu_{ij}</~~amsmath~~> is the stochiometric coefficient, <~~amsmath~~>\gamma_j</~~amsmath~~> is the rate constant and <~~amsmath~~>f_{jk}</~~amsmath~~> is the kinetic order for the relevant reaction. Note that in this form it can exactly represent mass action dynamics, but it can also be used to approximate other biochemical dynamics (eg, Michaelis-Menten style enzymatic reactions). In the approximate case, there are formulas to figure out all of the coeffiecients based on derivatives of the full nonlinear functions evaluated an an equilibrium point. In this respect, BST is similar to a Taylor-series expansion, just using a different class of functions (sums of products of powers).		where <math>X_i</math> is the concentration of species <math>i</math>, <math>\mu_{ij}</math> is the stochiometric coefficient, <math>\gamma_j</math> is the rate constant and <math>f_{jk}</math> is the kinetic order for the relevant reaction. Note that in this form it can exactly represent mass action dynamics, but it can also be used to approximate other biochemical dynamics (eg, Michaelis-Menten style enzymatic reactions). In the approximate case, there are formulas to figure out all of the coeffiecients based on derivatives of the full nonlinear functions evaluated an an equilibrium point. In this respect, BST is similar to a Taylor-series expansion, just using a different class of functions (sums of products of powers).

	A simplified form of these dynamics can be obtained by approximating the sum by just two terms: synthesis and degradation. In this case, the equations have the form		A simplified form of these dynamics can be obtained by approximating the sum by just two terms: synthesis and degradation. In this case, the equations have the form

Murray: /* Power-law formalisms */

2009-09-05T00:08:07Z

Power-law formalisms

← Older revision		Revision as of 00:08, 5 September 2009
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	\frac{dX_i}{dt} = \sum_j \mu_{ij} \left( \gamma_j \prod_k X_k^{f_{jk}} \right)		\frac{dX_i}{dt} = \sum_j \mu_{ij} \left( \gamma_j \prod_k X_k^{f_{jk}} \right)
	</amsmath></center>		</amsmath></center>
	where <~~amsmath~~>X_i</~~amsmath~~> is the concentration of species <amsmath>i</amsmath>, <amsmath>\mu_{ij}</amsmath> is the stochiometric coefficient, <amsmath>\gamma_j</amsmath> is the rate constant and <amsmath>f_{jk}</amsmath> is the kinetic order for the relevant reaction. Note that in this form it can exactly represent mass action dynamics, but it can also be used to approximate other biochemical dynamics (eg, Michaelis-Menten style enzymatic reactions). In the approximate case, there are formulas to figure out all of the coeffiecients based on derivatives of the full nonlinear functions evaluated an an equilibrium point. In this respect, BST is similar to a Taylor-series expansion, just using a different class of functions (sums of products of powers).		where <math>X_i</math> is the concentration of species <amsmath>i</amsmath>, <amsmath>\mu_{ij}</amsmath> is the stochiometric coefficient, <amsmath>\gamma_j</amsmath> is the rate constant and <amsmath>f_{jk}</amsmath> is the kinetic order for the relevant reaction. Note that in this form it can exactly represent mass action dynamics, but it can also be used to approximate other biochemical dynamics (eg, Michaelis-Menten style enzymatic reactions). In the approximate case, there are formulas to figure out all of the coeffiecients based on derivatives of the full nonlinear functions evaluated an an equilibrium point. In this respect, BST is similar to a Taylor-series expansion, just using a different class of functions (sums of products of powers).

	A simplified form of these dynamics can be obtained by approximating the sum by just two terms: synthesis and degradation. In this case, the equations have the form		A simplified form of these dynamics can be obtained by approximating the sum by just two terms: synthesis and degradation. In this case, the equations have the form

Murray at 23:33, 4 September 2009

2009-09-04T23:33:13Z

← Older revision		Revision as of 23:33, 4 September 2009
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	\frac{dX}{dt} = S v(X)		\frac{dX}{dt} = S v(X)
	</amsmath></center>		</amsmath></center>
	where <amsmath>X \in R^m</amsmath> is the vector of species concentrations, <amsmath>S \in R^N</amsmath> is the stoichiometry matrix and <amsmath>v:R^m \to R^N</amsmath> is the nonlinear reaction rate vector.		where <amsmath>X \in {\mathbb R}^m</amsmath> is the vector of species concentrations, <amsmath>S \in {\mathbb R}^N</amsmath> is the stoichiometry matrix and <amsmath>v:{\mathbb R}^m \to {\mathbb R}^N</amsmath> is the nonlinear reaction rate vector.

	Once you have the equation in this form, there are a couple of standard modeling techniques that are used:		Once you have the equation in this form, there are a couple of standard modeling techniques that are used:

Murray at 22:30, 4 September 2009

2009-09-04T22:30:47Z

← Older revision		Revision as of 22:30, 4 September 2009
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	This page has some notes on modeling and modeling reduction for biomolecular feedback systems.		This page has some notes on modeling and modeling reduction for biomolecular feedback systems.

	=== Modeling and Analysis ===		=== Deterministic Modeling and Analysis ===

	There's a quite large literature on both stochastic and deterministic modeling of biomolecular systems. This is something that I will write about in detail in BFS, but here are some of the basic techniques.		There's a quite large literature on both stochastic and deterministic modeling of biomolecular systems. This is something that I will write about in detail in BFS, but here are some of the basic techniques.
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	From my perspecitive, MCA/MCT is just a special case of classical concepts of sensitivity in control theory (as pointed out by Ingalls) and that seems like a good way to approach it. BST is an approximation that agrees with MCA/MCT, but really goes beyond that in terms of a (approximate) modeling methodology. In writing anything up, it would be good not to get pulled into the apparent battle between the two camps (though maybe this has finally died down).		From my perspecitive, MCA/MCT is just a special case of classical concepts of sensitivity in control theory (as pointed out by Ingalls) and that seems like a good way to approach it. BST is an approximation that agrees with MCA/MCT, but really goes beyond that in terms of a (approximate) modeling methodology. In writing anything up, it would be good not to get pulled into the apparent battle between the two camps (though maybe this has finally died down).



	=== Model reduction ===		=== Model reduction ===

Murray: /* Stochastic approaches */

2009-09-04T22:30:17Z

Stochastic approaches

← Older revision		Revision as of 22:30, 4 September 2009
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	=== Stochastic approaches ===		=== Stochastic approaches ===

	An area that I haven't ~~looked into~~ yet is model reduction of Markov processes, along the lines of things that Martha Grover has done, and also Mustafa Khammash and Brian Munsky.		In addition to all of the deterministic modeling work described above, it is also possible to formulate everything in terms of stochastic reactions, as detailed in Dan Gillespie's work. What's missing in this approach is much work on the ''analysis'' of the resulting stochastic dynamics (versus just simulations).

			One paper that I found is work by Kim and Sauro extending some of the ideas from metabolic control analysis (MCA/MCT) to the stochastic setting:

			* K. H. Kim and H. M. Sauro, Stochastic control analysis for biochemical reaction systems. arXiv:0904.3124v3 [q-bio.QM], 21 Aug 2009.

			Along the lines described in the previous section, the main idea here is to look at the sensitivity of the species concentrations to various (stochastic) parameters. In the context of biochemical systems, one motivation would be to look at how stochastic fluctuations in enzyme concentrations affect metabolic networks. This type of analysis should apply to a variety of biological circuits.

			An area that I haven't summarized yet is model reduction of Markov processes, along the lines of things that Martha Grover has done, and also Mustafa Khammash and Brian Munsky. Coming soon...

Murray: /* Stochastic approaches */

2009-09-04T22:29:21Z

Stochastic approaches

← Older revision		Revision as of 22:29, 4 September 2009
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	From my perspecitive, MCA/MCT is just a special case of classical concepts of sensitivity in control theory (as pointed out by Ingalls) and that seems like a good way to approach it. BST is an approximation that agrees with MCA/MCT, but really goes beyond that in terms of a (approximate) modeling methodology. In writing anything up, it would be good not to get pulled into the apparent battle between the two camps (though maybe this has finally died down).		From my perspecitive, MCA/MCT is just a special case of classical concepts of sensitivity in control theory (as pointed out by Ingalls) and that seems like a good way to approach it. BST is an approximation that agrees with MCA/MCT, but really goes beyond that in terms of a (approximate) modeling methodology. In writing anything up, it would be good not to get pulled into the apparent battle between the two camps (though maybe this has finally died down).

	~~==== Stochastic approaches ====~~

	In addition to all of the deterministic modeling work described above, it is also possible to formulate everything in terms of stochastic reactions, as detailed in Dan Gillespie's work. What's missing in this approach is much work on the ''analysis'' of the resulting stochastic dynamics (versus just simulations).

	~~One paper that I found is work by Kim and Sauro extending some of the ideas from metabolic control analysis (MCA/MCT) to the stochastic setting:~~

	* K. H. Kim and H. M. Sauro, Stochastic control analysis for biochemical reaction systems. arXiv:0904.3124v3 [q-bio.QM], 21 Aug 2009.

	Along the lines described in the previous section, the main idea here is to look at the sensitivity of the species concentrations to various (stochastic) parameters. In the context of biochemical systems, one motivation would be to look at how stochastic fluctuations in enzyme concentrations affect metabolic networks. This type of analysis should apply to a variety of biological circuits.

	=== Model reduction ===		=== Model reduction ===

Murray: /* Stochastic approaches */

2009-09-04T22:29:01Z

Stochastic approaches

← Older revision		Revision as of 22:29, 4 September 2009
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	==== Stochastic approaches ====		==== Stochastic approaches ====

			In addition to all of the deterministic modeling work described above, it is also possible to formulate everything in terms of stochastic reactions, as detailed in Dan Gillespie's work. What's missing in this approach is much work on the ''analysis'' of the resulting stochastic dynamics (versus just simulations).

			One paper that I found is work by Kim and Sauro extending some of the ideas from metabolic control analysis (MCA/MCT) to the stochastic setting:

			* K. H. Kim and H. M. Sauro, Stochastic control analysis for biochemical reaction systems. arXiv:0904.3124v3 [q-bio.QM], 21 Aug 2009.

			Along the lines described in the previous section, the main idea here is to look at the sensitivity of the species concentrations to various (stochastic) parameters. In the context of biochemical systems, one motivation would be to look at how stochastic fluctuations in enzyme concentrations affect metabolic networks. This type of analysis should apply to a variety of biological circuits.

	=== Model reduction ===		=== Model reduction ===

Murray: /* Metabolic reaction networks */

2009-09-04T22:14:51Z

Metabolic reaction networks

← Older revision		Revision as of 22:14, 4 September 2009
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	\frac{dX}{dt} = S v(X)		\frac{dX}{dt} = S v(X)
	</amsmath></center>		</amsmath></center>
	where <amsmath>X \in ~~\reals~~^m</amsmath> is the vector of species concentrations, <amsmath>S \in ~~\reals~~^N</amsmath> is the stoichiometry matrix and <amsmath>v:~~\reals~~^m \to ~~\reals~~^N</amsmath> is the nonlinear reaction rate vector.		where <amsmath>X \in R^m</amsmath> is the vector of species concentrations, <amsmath>S \in R^N</amsmath> is the stoichiometry matrix and <amsmath>v:R^m \to R^N</amsmath> is the nonlinear reaction rate vector.

	Once you have the equation in this form, there are a couple of standard modeling techniques that are used:		Once you have the equation in this form, there are a couple of standard modeling techniques that are used:

Murray: /* Metabolic reaction networks */

2009-09-04T22:12:37Z

Metabolic reaction networks

← Older revision		Revision as of 22:12, 4 September 2009
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	==== Metabolic reaction networks ====		==== Metabolic reaction networks ====
			The basic starting point for much of the work on modeling of reaction networks is the kinetic mass action model, which can be written in the form
			<center><amsmath>
			\frac{dX}{dt} = S v(X)
			</amsmath></center>
			where <amsmath>X \in \reals^m</amsmath> is the vector of species concentrations, <amsmath>S \in \reals^N</amsmath> is the stoichiometry matrix and <amsmath>v:\reals^m \to \reals^N</amsmath> is the nonlinear reaction rate vector.

			Once you have the equation in this form, there are a couple of standard modeling techniques that are used:

			''Flux Balance Analysis'': the main idea here is to analyze the set of reaction rates <amsmath>v</amsmath> that lie in the null space of <amsmath>S</amsmath> (and hence correspond to steady state conditions). We don't worried about the species concentrations themselves; just the reactions. In the case where there are many reactions and fewer species, we will get lots of possible reactions and we can explore what the space of all possible reactions tells us. An example of this sort of analysis is done by Schilling et al, who look at what you can say about pathways using this formalism:

			* C. H. Schilling, J. S. Edwards, D. Letscher and B. O. Pallson, Combining pathway analysis with flux balance analysis for comprehensive study of metabolic systems. ''Biotechnology and bioengineering'', 71(4):286-306, 2001.

			''Chemical Reaction Network (CRN) theory'': CRN theory focus on the graph structure that is encoded in the stoichiometry matrix as well as the specific type of nonlinearities that arise from mass action kinetics. The rough idea here is to "lift" the dynamics to a higher dimensional space of complexes by considering every product that occurs in the rate laws as a separate element. This then gives a set of linear dynamics in the lifted space (but with nonlinear constraints between the "states") and some analysis can be done. In particular, you can find conditions on the reaction graph under which you can have single or multiple equilibria. A nice overview of this approach that has an appealing mathematical flavor is given in the notes by Gunawardena:

			* J. Gunawardena, Chemical reaction theory for ''in-silico'' biologists. Unpublished notes, 2003.

			There is also some recent MATLAB software for doing CRN analysis called [http://people.sissa.it/~altafini/papers/SoAl09/ ERNEST]. This software can read SBML files and do network analysis, which seems pretty handy.

	==== Power-law formalisms ====		==== Power-law formalisms ====