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The (Right) Null Space of S Systems Biology by Bernhard O. Polson Chapter9. Deborah Sills Walker Lab Group meeting April 12, 2007. Overview. Stoichiometric matrix Null Space of S Linear and Convex basis vectors for the null space Extreme Pathway Analysis Practical applications. A. b 1.

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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 s systems biology by bernhard o polson chapter9

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

Deborah Sills

Walker Lab Group meeting

April 12, 2007


Overview

Overview

  • Stoichiometric matrix

  • Null Space of S

  • Linear and Convex basis vectors for the null space

  • Extreme Pathway Analysis

  • Practical applications


S connectivity matrix represents a network

A

b1

B

v1

b3

b2

C

S connectivity matrix– represents a network

Each column represents reaction (n reactions)

Each row represents a metabolite (m metabolites)


Stoicheometric matrix

Stoicheometric Matrix

  • 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)


Four fundamental subspaces

Four fundamental subspaces

  • 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.


Null space of s

Null Space of S

  • Steady state flux distributions (no change in time)


Basis of the null space

Basis of the Null Space

  • 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


Basis of null space contin

Basis of Null Space contin.

  • 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


Choosing a basis for a biological network

A

b1

B

v1

b3

b2

C

Choosing a Basis for a Biological Network

Two types of reactions in open systems:

  • Elementary reactions (internal) only have positive flux

  • Exchange fluxes can include diffusion and are considered bidirectional


The right null space of s systems biology by bernhard o polson chapter9

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


Nonnegative linear basis vectors for null space

Nonnegative linear basis vectors for null space


Biologically irrelevant since no carrier or cofactor such as atp exchanges

p1

B

v3

v2

A

v4

v1

D

v5

v6

C

p2

Biologically irrelevant since no carrier or cofactor (such as ATP) exchanges


Convex bases

Convex Bases

  • 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


The right null space of s systems biology by bernhard o polson chapter9

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


The right null space of s systems biology by bernhard o polson chapter9

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


Simple 3 d example

Simple 3-D example

  • 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 links biology and math

Null space links biology and math

  • 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


Constraints in biological systems

Constraints in Biological Systems

  • Thermodynamic – reversibility

  • Mass balance

  • Maximum enzyme capacity

  • Energy balance

  • Cell volume

  • Kinetics

  • Transcriptional regulatory constraints


Constraints in biological systems contin

Constraints in Biological Systems contin.

  • 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


Biochemical reaction network and its convex steady state solution cone

Biochemical Reaction Network and its convex, steady-state solution cone

[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.]


The right null space of s systems biology by bernhard o polson chapter9

Classification of Extreme Pathways

p1……………………………………………pk

v1

.

Internalfluxes

.

.

.

vn

Exchangefluxes

b1

bnE


Extreme pathways

Extreme Pathways

  • 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


Extreme pathways1

Extreme Pathways


Extreme skeleton metabolic pathways

Extreme Skeleton Metabolic Pathways

  • 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


Basis vectors for biological system

Basis Vectors for Biological System

  • 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


Practical applications

Practical Applications

  • 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


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