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The (Right) Null Space of S Systems Biology by Bernhard O. Polson Chapter9

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The (Right) Null Space of SSystems Biology by Bernhard O. PolsonChapter9

Deborah Sills

Walker Lab Group meeting

April 12, 2007

- Stoichiometric matrix
- Null Space of S
- Linear and Convex basis vectors for the null space
- Extreme Pathway Analysis
- Practical applications

A

b1

B

v1

b3

b2

C

Each column represents reaction (n reactions)

Each row represents a metabolite (m metabolites)

- S is a linear transformation of of the flux vector

x = metabolite concentrations (m, 1)

S = stoicheometric matrix (m,n)

v = flux vector (n,1)

- Column space, left null space, row space, and (right) null space

Iman Famili and Bernhard O. Palsson, Biophys J. 2003 July; 85(1): 16–26.

- Steady state flux distributions (no change in time)

- Basis spans space of matrix
- Null space spanned by (n-r) basis vectors
Where n = number of metabolites

r = rank of S (number of linearly independent rows and columns)

- Exampes of bases: linear basis, orthonormal basis, convex basis

- Null space orthogonal to row space of S

- Basis vectors form columns of matrix R

ri i[1, n-r]

SR= 0

- Every point in vector space, uniquely defined by set linearly independent basis vectors

A

b1

B

v1

b3

b2

C

Two types of reactions in open systems:

- Elementary reactions (internal) only have positive flux
- Exchange fluxes can include diffusion and are considered bidirectional

r1

B

r2

v3

v2

A

v4

v1

D

v5

v6

C

Null Space defined by:

Matrix full rank, and dimension of null space r-n = 6 - 4 = 2

Column 4, 6 don’t contain pivot, so solve null space in terms of v4 and v6

p1

B

v3

v2

A

v4

v1

D

v5

v6

C

p2

- Convex bases unique
- Number of convex basis bigger than the dimension of the null space
- Elementary reactions positive and have upper bound

Allowable fluxes are in a hyperbox bounded by planes parallel to each axis as defined by vi,max

Convex Basis

- Hyperbox contains all fluxes (steady state and dynamic)

- Sv=0 is hyperplane that intersects hyperbox forming finite segment of hyperplane

- Intersection is polytope in which all steady state fluxes lie

- Polytope spanned by convex basis vectors, which are edges of polytope, with restricted ranges on weights

Convex basis vectors are edges of polytope that contain steady state flux vectors

Where pk are the edges, or extreme states, and ak are the weights that are positive and bounded,

- Aside(from Schilling et al., BiotechBioeng.2000.
- Convex polyhedron is a region in Rn determined by linear equalities and inequalities
- Polytope is bounded polyhedron
- Polyhedral cone if every ray through the origin and any point in the polyhedron are completely contained in polyhedron

- Node with three reactions forms simple flux split

- Min and max constraints form box that is intercepted by plain formed by flux balance

0=v3 – v1- v2= [(-1, -1, 1)•(v1, v2 , v3)]

- 2D polytope spanned by two convex basis vectors

b1= (v1, 0, v3)

b2= (0, v2, v3)

- Null space represents all functional, phenotypic states of network
- Each point in polytope represent one network function or one phenotypic state
- Edges of flux cone are unique extreme pathways
- Extreme pathways describe range of fluxes that are permitted

- Thermodynamic – reversibility
- Mass balance
- Maximum enzyme capacity
- Energy balance
- Cell volume
- Kinetics
- Transcriptional regulatory constraints

- Constraints can help to determine effect of various parameters on achievable states of network
- Examples: enzymopathies can reduce certain maximum fluxes, reducing ai,max
- Effects of gene deletion can also be examined

[Nathan D. Price, Jennifer L. Reed, Jason A. Papin, Iman Famili and Bernhard O. Palsson , Analysis of Metabolic Capabilities Using Singular Value Decomposition of Extreme Pathway Matrices. The Biophysical Society, 2003.]

Classification of Extreme Pathways

p1……………………………………………pk

v1

.

Internalfluxes

.

.

.

vn

Exchangefluxes

b1

bnE

- Type I involve conversion of primary inputs to primary outputs
- Type II involve internal exchange carrier metabolites such as ATP and NADH
- Type III are internal cycles with no flux across system boundaries

- Glycolysis has five extreme pathways
- Two type I represent in secretion of two metabolites
- One type II represent dissipation of phosphate bond – futile pathway
- Two type III that will have no net flux

- ri makes the nodes in the flux map “link neutral”, because it is orthogonal to all the rows of S simultaneously
- Network-based pathway definition, and use basis vectors (pi) that represent these pathways

- Convex bases offers a mathematical analysis of the null space that makes biochemical sense
- Flux-balance analysis mostly used so far to analyze single species
- Analysis of complex communities challenge, but possible to limit study to core model
- Stolyar et al., 2007 used multispecies stoicheometric metabolic model to predict mutualistic interactions between sulfate reducing bacteria and methanogen