Mathematical Representation of Reconstructed Networks

The Left Null space The Row and column spaces of S

Introduction

The system biology paradigm:

“components network in silico models phenotype”

System biology focuses on the nature of the links and their associated functional states (~phenotypes).

Cells must select useful ‘functional states’.

Functional states

E coli:

1. Outside the body (low O2, low temp)

2. Inside the body (high temp)

3. Stomach (Low PH)

4. Small intestine (low O2)

H pylori:

1. Human gastric (Low PH)

Reaction / links ‘key properties’

1. Stoichiometry. The stoichiometry is fixed, invariant between organisms for the same reactions and condition independent (pressure, pH, temp ..)

2. Relative rates. Fixed by basic thermodynamic properties which depend on conditions such as pressure, pH, temp ..

3. Absolute rates. In contrast to stoichiometry and thermodynamics, highly manipulated by the cell enzymes.

Reminder The dynamic mass balance equation:

Where,X is a vector of m metabolites.V represent vector of n ‘reaction rates’.S is m x n matrix of stoichiometric coefficients, rows represent metabolites and

columns represent reactions.

The right null space

1. For most practical purposes metabolism is in a steady state.2. The null space contains all the steady state flux distributions and is thus of

special importance to us.

0SV

dt

dXSV

nnsvsvsvXdt

d ..2211

Reminder

Constraint-based analysis

Linear programming

(Simplex)

• Each one of the generating vectors corresponds to an extreme pathway which the cell could theoretically control to reach every point in the flux cone.• A particular point within this flux cone corresponds to a given flux distribution which represents a particular metabolic phenotype.

• The analysis of the left null space of S allow us to define the achievable states of the cell and their physiological relevance.

• We look for ‘metabolic pools’ that have physiological meaningful interpretation.

Definition: The Left Null space of S

],1[

0,

rmi

sl ji

0..0

:

0..0

|||

..

|||

__

:

__

21

1

n

rm

sss

l

l

li

Span the left null space of

S

sj

All are in the column space.

(rank= r)

0LS

The Time invariants

A linear combination of individual metabolic concentrations that do not change over time is called a metabolic pool.

A dynamic motion along a reaction vector in the column space do not change the total mass in the pool.

00 Lxdt

d

dt

dxli

The concentrations space

a is a vector that gives the total concentrations of the pools. i.e.

is the conservation vector. The rows of L ( i.e. ) that span the left null space

define a concentration space. The time invariant metabolic pools resides in this

concentration space. defines an affine hyper plane. This plane does not go through the origin. The concentration vector x resides in this space.

aLx

aLx

ii axl

sli 'il

Classifying the pools

Co-Factors,

carriers

e.g. the carbon backbone in glycolysis

Reaction map Vs. Compound map

Groupings of chemical elements that move together.

TSS

Classifying the pools

Futile cycle

Internal cycle

Through flux pathways

Cofactor conservation

Primary& secondary moieties conservation

Primary moieties conservation

Simple reversible state

PAAP

11,1

11

lLS

PAAP

One Type A Pool:

Comment: The pool ‘AP+PA’ is constant both in SS and dynamic.

Reference states

We can choose that lie in the left null space. This reference state is orthogonal to

x is not orthogonal to the left null space, whereas

and are.

Now we can span the concentration space using the reaction vectors

aLx

0)( refxxL

dt

dx

)( refxx

is

refxx is

2/1

2/1

011

011

,2

,1

,22

,11

,2

,1

ref

ref

ref

ref

ref

ref

X

X

XX

XX

X

X)1(

)2(

Two conditions:

Bilinear association

APPA

The Pools Interpretation Type

1 .A+AP : Total cofactor : A

2 .P+AP : Total energy : B

110

101,

1

1

1

LS

Ordered by: A, P, AP

Carrier coupled reactionACPAPC

The pools:

1. C+CP : conservation of the substrate C.

2. A+AP : conservation of the cofactor A.

3. CP+AP : occupancy of P / total energy

4. C+A : vacancy of P / low energy state

1010

0101

1100

0011

,

1

1

1

1

Ls

The entries of x ordered by: (CP, C, AP, A)

Redox carrier coupled reaction

01010

10010

00110

01001

10001

00101

,

1

1

1

1

1

2

LS

HNADHRNADRH

The pools The pools interpretation Type

1. : Total R. : A

2. : Redox occupancy 1. : B

3. : Redox occupancy 2. : B

4. : Redox vacancy : B

5. : Total redox carrier : C

6. :Total redox carrier : C

HNAD

NADHNAD

RNAD

HRH

NADHRH

RRH

2

2

2

Multiple redox coupled reaction

NADHR

v

v

HNADHR

R

v

v

R

HNADHR

v

v

NADRH

2

6

5

4

3

2

1

2

''

'

The pools The pools interpretation Type

1. : Total R. : A

2. : Redox occupancy 1. : B

3. : Redox occupancy 2. : B

4. : Redox vacancy : B

5. : Total redox carrier 1 : C

6. :Total redox carrier 2 : C

HNAD

NADHNAD

RRNAD

HRHRH

HRNADHRH

HRRRRH

'

'

'

''

22

22

22

(1)

(2)

(3)

Glycolysis

PCPCPC

APCPCPCC

PAPPCPCPCPCPC

APPCPCPCPCPCC

3323

23366

332313266

3323132666

22

22

22432Type B pools:

High energy

Conservation of P

Low energy

Stand alone inorganic P

TCA cycle

NHN

NC

NHHCCHCH

HCCHC

CHCCHCH

56222

5624

56222

22

22 • Exchanging carbon group.

• Recycled C4 moiety which ‘carries’ the two carbon group that is oxidized.

• H group that contains the redox inventory in the system.

• Redox vacancy.

• Total cofactor pool.

Summary: Left Null space of S

Contains dynamic invariants. A convex basis for this space is biological meaningful and can be

found. Three basic types of convex basis vectors can be defined. The metabolic pools can be displayed on the compound map –

similar to pathways in a flux map. Integration of time derivatives leads to bounded affine space of

concentrations.

The affine space of concentrations1. All the concentrations states, dynamic and steady, lie in this space.2. A suitable reference state can be defined (parallel to the left null

space and orthogonal to the column space). The shifted concentration space is spanned by the Si’s.

The column space

Contain the time derivatives of the concentrations. Spanned by the reaction vectors. Change in the flux levels determine the location of

in the column space. Fast reactions that quickly come to SS reduce the

column space dimension on slower time scales. Reduction in the columns space dimension leads to

effective additional dimension in the left null space. Constraints on the fluxes induce constraints on the

‘s. Hence the column space is a closed space.

nnsvsvsvxdt

d ..2211

dt

dx

dt

dx

Example 1 OHOOH 2222 22

22OH 2O OH 2

O 2 2 1

H 2 0 2

1

222

2222

2222

2

2

22

2

1

0

0

9

22

22

22

202

122

2121

v

H

O

R

dt

d

OHOH

OHOOH

OHOOH

dt

d

OH

O

OH

dt

dx

dt

d

l

l

sT

The left null space will be spanned by the elemental matrix

LS = ES = 0

R is a group of concentrations changing over time

Example 2

3

2

1

21 X

v

v

XX

21

2

1

32

31

321

0

0

4

VV

l

l

R

dt

d

XX

XX

XXX

dt

d

110

101

111

2

1

1

l

l

sT

Example 3

AMPATPADP

ATPADP

2

(Ignore the P for a moment)

A

R

R

dt

d

AMPADPATP

AMPADPATP

ADPATP

AMP

ADP

ATP

dt

d2

1

2

111

121

011

Note: If one reaction is fast compare to the other, we get ‘L’ shape

The row space

The row space contain all the thermodynamic driving forces (i.e. fluxes).

The individual reaction fluxes form an orthogonal basis for the raw space.

Each reaction has a natural thermodynamic basis vector. Since the fluxes are constrained, All the fluxes are in a rectangle

in the positive orthant. The null space lies within the rectangle and its orthogonal

complement is the row space.

Constraints on the flux values The magnitude of the individual fluxes is constrained. These constraints are derived from:1. The limitation on the concentration.2. Upper limit on the kinetic constants. The turnover rate of an enzyme complex X:

Where the total amount of enzyme ( ) present is limited to X + e. Bilinear association of substrate to an enzyme:The rate is:

Where is the size of the most limiting conservation pool of which Xi is a member.

The total amount of enzyme (alone) limits the flux through enzymatic pathway.

The release step of a product from enzyme is often the rate limiting step in enzyme catalysis.

totalekeXkXkV 111 )(

totale

totalibib eakeXkV

ia

Thermodynamic driving forces If the fluxes are imbalanced, there will be a net generation or

elimination of compounds in the network.

Since the r’s are fixed and the V’s are bounded, the inner product is also bounded.

)cos(, iiii VrVr

dt

dX

The column space is naturally spanned by the reaction vectors.

The row space can be represented by an orthogonal basis formed by the individual fluxes with values only in the positive orthant.

The magnitude of the individual fluxes is limited by kinetics and caps on concentration values.

This limitation also limits the possible value of the time derivatives and thus the column space.

The column and row spaces are closed.

Summary: The row and column spaces of S

Thanks