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

Chapter9

Deborah SillsWalker 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

S connectivity matrix– represents a network

A

C

Bv1

b2

b3

b1

1001

0101

0011

S

3211 bbbv

C

B

A

Each column represents reaction (n reactions)

Each row represents a metabolite (m metabolites)

Stoicheometric Matrix

• S is a linear transformation of of the flux vector

Svx

dt

d

x = metabolite concentrations (m, 1)

S = stoicheometric matrix (m,n)

v = flux vector (n,1)

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

• Steady state flux distributions (no change in time)

0ssSv

0Svx

dt

d

Basis of the Null Space

• Basis spans space of matrix

• Null space spanned by (n-r) basis vectorsWhere 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.•Null space orthogonal to row space of S

•Basis vectors form columns of matrix R

ii rwssv

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

Two types of reactions in open systems:

– Elementary reactions (internal) only have positive flux

– Exchange fluxes can include diffusion and are considered bidirectional

A

C

Bv1

b2

b3

b1

0iv

00 ib

C

A

B

D

v1

v3v2

v5

v4

v6

0

0

0

0

110000

101100

000110

010011

6

5

4

3

2

1

v

v

v

v

v

v

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

r2

r1

21 rrv 2164

1

1

0

1

1

0

0

0

1

1

1

1

wwvv

Nonnegative linear basis vectors for null space

10

10

01

11

11

01

, 2r1r

10

11 21,

10

10

11

01

01

11

pp

C

A

B

D

v1

v3v2

v5

v4

v6

10

10

11

01

01

11

, 21 pp

p1

p2

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

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

max0 ii vv

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

1k

kkss pv

1k

kkss pv

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

max,min,0 iii

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

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

•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 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

• Thermodynamic – reversibility

• Mass balance

• Maximum enzyme capacity

• Energy balance

• Cell volume

• Kinetics

• Transcriptional regulatory constraints

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 i,max

• Effects of gene deletion can also be examined

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.]

1k

kkss pv max,0 kk

Type I

Primary system

Type II

Futile cycle

Type III

Internal cycles

0 0 0

0 0 0

p1……………………………………………pk

b1

vn

.

.

.

.

v1

bnE

Classification of Extreme Pathways

Internal fluxes

Exchange fluxes

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 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• 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

• 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