Elementary Mode Analysis - Amazon S3
Transcript of Elementary Mode Analysis - Amazon S3
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Orhan Karsligil MIT, 2006
Elementary Mode AnalysisA reviewMay 2006, MIT
Orhan Karsligil
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Orhan Karsligil MIT, 2006
Major Approaches to Metabolism Modeling
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Orhan Karsligil MIT, 2006
Steady State Flux Analysis
Biochemical Reactions
Set of Reactions
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Orhan Karsligil MIT, 2006
Flux vector and Mass Balance Equation
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Orhan Karsligil MIT, 2006
Steady State Assumption
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Orhan Karsligil MIT, 2006
Constrained Flux Analysis: The Goal
Bernhard Ø. Palsson et al. BIOTECHNOLOGY AND BIOENGINEERING, VOL. 86, NO. 3, MAY 5, 2004
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Orhan Karsligil MIT, 2006
Definitions:
Metabolic networks composed of q reactions and m metabolites:
Stoichiometric matrix: Nmxq
Flux distribution: e={e1,e2,...,eq}each element describes the net rate of the corresponding reaction
The pathway: P(e) where ei≠0identified by the utilized reactions
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Orhan Karsligil MIT, 2006
Conditions behind EFM & EP
[1] Pseudo steady-state: Ne=0. (metabolite balancing equation).
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[2] Feasibility: rate ei≥0 if reaction i is irreversible.
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[3] Non-decomposability: there is no vector v (unequal to the zero vector and to e) fulfilling [1] and [2] and that P(v) is a proper subset of P(e).
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Orhan Karsligil MIT, 2006
Additional Conditions behind EP
[4] Network reconfiguration: •Each reaction must be classified either as exchange flux or as internal reaction.
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All reversible internal reactions must be split up into two separate, irreversible reactions
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No internal reaction can have a negative flux–Exchange fluxes can be reversible,but each metabolite can participate in only one exchange flux.
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[5] Systemic independence: •The set of EPs in a network (configured properly by [4]) is the minimal set of EFMs that can describe all feasible steady-state flux distributions.
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The EPs represent a convex basis in this network.–The reconfiguration [4] ensures that the set of EPs is unique.–
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Orhan Karsligil MIT, 2006
Relevant objective functions
Minimize:ATP production–nutrient uptake–redox production–metabolite production–
Maximize:biomass production (i.e. growth)–the Euclidean norm of the flux vector–
Types of objective functionsFor basic exploration and probing of solution space–To represent likely physiological objectives–To represent bioengineering design objectives–
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Orhan Karsligil MIT, 2006
α- Spectrum
Given measured fluxes calculate the minimum and maximum flux rates for each flux rate
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Orhan Karsligil MIT, 2006
α- Spectrum
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Orhan Karsligil MIT, 2006
The Strategy
Include as much data as possible•Known Fluxes–Capacities–Objective Functions (optimization)–Possible min/max ranges–
Reduce the feasibility space•
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Orhan Karsligil MIT, 2006
EFM vs EP (example)
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Orhan Karsligil MIT, 2006
EFM vs EP
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Orhan Karsligil MIT, 2006
EFM vs EP
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Orhan Karsligil MIT, 2006
What is a Null Space?
Ax=bwhere A is a mxn matrixx is a nx1 vector and b is a mx1 vector
If m=n and det|A|≠0 then there will a unique solution
If m>n then it is an overdetermined system. Projection methods are used (least squares).
If m<n then it is an underdetermined system.
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Orhan Karsligil MIT, 2006
What is Null Space? Underdetermined Systems
If Ax = b is consistent and A has full column rank then Ax = b has a unique solution
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If Ax = b is consistent and A does not have full column rank then Ax = b has infinitely many solutions.
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If Ax = b is consistent then there is exactly one solution in the row space of A and it is the solution with smallest norm. This solution is the projection onto row(A) of any solution. To find it solve AAT y = b and set x = ATy
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Orhan Karsligil MIT, 2006
What is Null Space?
For an underdetermined system:
Ax=bx=AT(AAT)-1bbut alsoAr=0 so the full solution is:x=AT(AAT)-1b+rz where r is the Null Space of matrix A
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Orhan Karsligil MIT, 2006
What is Null Space?
Multiplication with a matrix is a transformation.
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If this transformation is from a higher dimension to a lower one (n to m) then some vectors in n dimensions will be transformed to null. The space spanned by these vectors is called the Null Space.
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