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
S connectivity matrix– represents a network
A
v1 b1 b2
 1
S   1
 1
1
0
0
0
1
0
b1
b3
0
0 
1 
B
A
B
C
v1
b2
C
Each column represents reaction (n reactions)
Each row represents a metabolite (m metabolites)
b3
Stoicheometric Matrix
• S is a linear transformation of of the flux
vector
dx
 Sv
dt
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)
dx
 Sv  0
dt
Svs s  0
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.
•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
v s s  wi ri
Choosing a Basis for a Biological Network
Two types of reactions in open systems:
– Elementary reactions (internal) only have positive flux
v i 0
– Exchange fluxes can include diffusion and are
considered bidirectional
0  bi 0
A
b1
B
v1
b2
C
b3
B
Null Space defined by:
v3
v2
v1
r1
A
v4
r2
v5
D
v6







1
0
0
0
 v1 
 
 1 0 0  1 0   v2   0 

 
1  1 0 0 0   v3   0 
 

0 1  1 0 1  v4   0 
   
0 0 0 1  1  v5   0 
v 
 6
C
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
 1
 0 
 


 1
  1
 1
  1
  w1r1  w2r2
v  v4    v6 
 1
 0 
 0
 1 
 


 0
 1 
 


Nonnegative linear basis vectors for null space




r1, r2   





1
1
1
1
0
0
0

 1
 1

0
1 
1 
1 1


 0 1
1

1
1

1
0

0

1

0
0
   p1 , p2 
1
1 
1 
p1
B
1

1
1
 p1 , p2   
1
0

0

1

0
0

1
1 
1 
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 unique
• Number of convex basis bigger than the
dimension of the null space
• Elementary reactions positive and have
upper bound
0  vi  vi max
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
v ss  k p k
k 1
Convex basis vectors are edges of polytope
that contain steady state flux vectors
v ss  k p k
k 1
Where pk are the edges, or extreme states, and k are the
weights that are positive and bounded,
0  i ,min  i   i ,max
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
v ss  k p k
k 1
0 k   k ,max
[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 Type I
. Primary system
.
0
.
.
vn
b1  0
bnE
Type II
Futile cycle
Type III
Internal cycles
0
0
0
0
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