Andrea Pagnani pagnani@isi.it ISI Foundation Turin Outlook - - PowerPoint PPT Presentation

Andrea Pagnani

The space of solutions of metabolic systems

ISI Foundation Turin

pagnani@isi.it

Outlook

• Metabolic modeling
• Inferring the space of solution by message-passing
• Biological applications
• Conclusions and perspectives

Metabolic Network

A C B

ν1 ν2 ν3 b1 b2 b3

A,B,C: components/metabolites

ν1, ν2, ν3 : Internal fluxes b1, b2, b3 : External Fluxes

uptake, secretion

Balance Equations dA dt dB dt = dA dt

−1 −1 1 1 1 −1 1 −1 −1

ν1 ν2 ν3 b1 b2 b3

.

Matrix Formalism

d X dt = ˆ S · ν − b

cell membrane

dA dt = −ν1 − ν2 + b1 dB dt = ν1 + ν3 − b2 dC dt = ν2 − ν3 − b3

Steady state metabolism

Time constants describing metabolic transients are fast: ~ order msec. to sec. Time constants associated to cell growth: ~ order of hours to days. Under these hypothesis the cell is in a (quasi)-steady state

ˆ S · ν = b d X dt = 0

Constraining the space of fluxes

Fluxes must be positive and cannot exceed νMAX:

The steady state mass balance equation together the inequalities

• n fluxes define a convex polytope

if N < M (as in metabolism)

0 ≤ νi ≤ νMAX

i

i ∈ (1, · · · , N)

Each equation defines a plane All equations define the space of all feasible fluxes.

The space of feasible solutions is a high-dimensional polytope To measure the volume of the polytope we will:

• Discretize the space (Riemann integration)
• Transform the linear system into a constraint satisfaction

problem (CSP) defined on a sparse topology.

• Approximate the volume of the polytope with the number of

solutions of the associated CSP .

• Set up a message-passing strategy for solving the CSP
• As a byproduct we can measure the single flux distribution

functions for all fluxes

Outline of the computational strategy

V Π

Discretizing the problem

Integration a la Riemann

Λǫ

The volume is proportional to the number of cube intesecting Π ˆ S · ν = b

0 ≤ νi ≤ νMAX

i

i ∈ (1, · · · , N)

{

notation) by the same Eqs. ariables νi ∈ {0, 1, ..., qmax

i

}, part of qmax × νmax, where

• r qmax

i

equal where the ∈ {

i

• f qmax × νmax

i

, ularity of the appro

= but now and qmax is the granularity of the approximation.

121 21

ν1 ν8 ν32 ν5 ν3 b2

2ν3 + ν5 − ν1 − 2ν8 − ν32 = b2

δ(2ν3 + ν5 − ν1 − 2ν8 − ν32; b2)

The function node imposes a hard constraint M functional nodes N variable nodes (fluxes)

The constraint satisfaction problem

!( ) ({ } ) ( ) ν ν =

∈ ∈ − ∈

∏ ∏

P P

a l l a a A i i d i I

i

ν ν

1

Metabolites Fluxes Bethe approximation (exact on trees and large locally tree-like structures)

• !i " a(#): the probability that flux i takes value # in the

absence of reaction a.

• ma " i(#): the non-normalized probability that the bal-

ance in reaction a is fulfilled given that flux i takes value #.

m s b u

a i i a l l a l a l a l l a i i

l l a i

→ ∈ → ∈ →

=        

∑ ∑ ∏

∈

( ) ; ( )

, { } \

\

ν δ ν ν µ

ν a i i a b i i b i a

C m ( ) ( )

\

ν ν =

→ → ∈

∏

Belief propagation

a1 a2 a3 a4 a5 a6 a7 a8 x1 x2 x3 x4 x5 x6 x7 x8

m1→8 ( x8 )

µ

6 → 1

( x

6

)

µ7→1(x6) µ5→1(x6)

Check of the performance on artificial data

1000 realizations x point Fixed α = M/N = 1/3 N=12 is the maximal value for lrs

LRS package [http://cgm.cs.mcgill.ca/~avis

. Avis D, Fukuda K: A pivoting algorithm for con- vex hulls and vertex enumeration of arrange- ments and polyhedra. Discrete Comput. Geom. 1992, 8(3):295–313.

0.001 0.01 0.1 1 10 100 1000 10000 100000 1 10 100 1000 τ[sec] N

lrs BP

Prop to N

Wibak, et al.

• J. Theoretical
• Biol. 2004.

Human Red Blood Cell

Network: N=46 M=34 Method: Montecarlo Sampling

Computation time: minutes

BP marginals

Computation time: ~3 sec! (on a comparable PC)

Impact of gene knock out in E-Coli central metabolism

Most of the high impact fluxes are involved in the glycolytic pathway

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 h 2

• h

H E X 1 P G K G A P D G L C P G 1 P P g l y c

• g

e n P G M E N O P D H A C O N T S U C D 1 i C S f a d h 2 f a d F U M c

• 2

M D H T P I S0 - SKO

0.1 0.2 0.3 0.4 0.5 10 20 30 40 50 60 70 80 90 100 S0 - SKO Flux knock-out percentage h2o h HEX1 PGK GAPD GLCP G1PP glycogen PGM ENO PDH ACONT SUCD1i CS fadh2 fad FUM co2 MDH TPI

Simulating enzimopaties

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 S0 - SKO <ν> HEX1 GLCP G1PP glycogen PDH CS 100% knock-out 75% knock-out

Relevance of fluxes as a function of their mean-velocity

G L Y C O L Y S I S K R E B S C Y C L E

E.Coli Central Metabolism

HEX1 G1PP GLCP CS PDH

0.1 1 10 0.01 0.1 1 < ν > A(ν - ν0)-γ

Pfit(ν) = A (ν − ν0)γ

γ = 1.48(5) ν0 = 0.00020(8)

E.Coli organism wide metabolism

N = #of reactions = 1025 M = #of metabolites = 626 40 min. on a standard laptop

Conclusions and Perspectives

• Cell metabolism is a very well defined biological description of
• rganisms (universal, stoich. parameters are integer, etc.)
• One needs fast algorithms for analyzing metabolism:
• 1. Global characterization of the space of solution
• 2. Fast in-silico flux-KO of organism-wide metabolic systems.

Work in progress:

• Conserved groups in collaboration with A. De Martino

Work in collaboration with

• Alfredo Braunstein (Politecnico, Torino, Italy)
• Roberto Mulet (University of Havana, Cuba)

Extended collaborators: Martin Weigt (ISI), Hamed Mahamoudi (ISI), Riccardo Zecchina (Politecninco Torino & ISI), Enzo Marinari (Univ. Roma 1), Andrea De Martino (University of Roma 1), Ginestra Bianconi (ICTP)

Estimating the size of the solution space of metabolic networks Braunstein A, Mulet R, Pagnani A BMC Bioinformatics 2008, 9:240 (19 May 2008) The space of feasible solutions in metabolic networks A Braunstein, R Mulet and A Pagnani

• J. Phys.: Conf. Ser. 95 (2008) 012017