24. lecture ws 2005/06bioinformatics iii1 v17 metabolic networks - introduction different levels for...

24. Lecture WS 2005/06

Bioinformatics III 1

V17 Metabolic Networks - Introduction

Different levels for describing metabolic networks by computational methods:

- classical biochemical pathways (glycolysis, TCA cycle, ...

- stoichiometric modelling (flux balance analysis): theoretical capabilities of an

integrated cellular process, feasible metabolic flux distributions

- automatic decomposition of metabolic networks

(elementary nodes, extreme pathways ...)

- kinetic modelling (E-Cell ...) problem: general lack of kinetic information

on the dynamics and regulation of cellular metabolism



EcoCyc Database

E.coli genome contains 4.7 million DNA bases.

How can we characterize the functional complement of E.coli and according to

what criteria can we compare the biochemical networks of two organisms?

EcoCyc contains the metabolic map of E.coli defined as the set of all known

pathways, reactions and enzymes of E.coli small-molecule metabolism.

Analyze

- the connectivity relationships of the metabolic network

- its partitioning into pathways

- enzyme activation and inhibition

- repetition and multiplicity of elements such as enzymes, reactions, and substrates.

Ouzonis, Karp, Genome Res. 10, 568 (2000)



EcoCyc Analysis of E.coli Metabolism

E.coli genome contains 4391 predicted genes, of which 4288 code for proteins.

676 of these genes form 607 enzymes of E.coli small-molecule metabolism.

Of those enzymes, 311 are protein complexes, 296 are monomers.

Organization of protein complexes. Distribution of subunit counts for all EcoCyc protein complexes. The predominance of monomers, dimers, and tetramers is obvious




ReactionsEcoCyc describes 905 metabolic reactions that are catalyzed by E. coli.

Of these reactions, 161 are not involved in small-molecule metabolism,

e.g. they participate in macromolecule metabolism such as DNA replication and

tRNA charging.

Of the remaining 744 reactions, 569 have been assigned to at least one pathway.

The next figures show an overview diagram of E. coli metabolism. Each node in

the diagram represents a single metabolite whose chemical class is encoded by

the shape of the node. Each blue line represents a single bioreaction. The white

lines connect multiple occurrences of the same metabolite in the diagram.




Reactions

The number of reactions (744) and the number of enzymes (607) differ ...

WHY??

(1) there is no one-to-one mapping between enzymes and reactions –

some enzymes catalyze multiple reactions, and some reactions are catalyzed

by multiple enzymes.

(2) for some reactions known to be catalyzed by E.coli, the enzyme has not yet

been identified.




Compounds

The 744 reactions of E.coli small-molecule metabolism involve a total of 791

different substrates.

On average, each reaction contains 4.0 substrates.

Number of reactions containing varying numbers of substrates (reactants plus products).





Each distinct substrate occurs in an average of 2.1 reactions.

Compounds



Pathways

EcoCyc describes 131 pathways:

energy metabolism

nucleotide and amino acid biosynthesis

secondary metabolism

Pathways vary in length from a

single reaction step to 16 steps

with an average of 5.4 steps.

Length distribution of EcoCyc pathways




Reactions Catalyzed by More Than one Enzyme

Diagram showing the number of reactions

that are catalyzed by one or more enzymes.

Most reactions are catalyzed by one enzyme,

some by two, and very few by more than two

enzymes.

For 84 reactions, the corresponding enzyme is not yet encoded in EcoCyc.

What may be the reasons for isozyme redundancy?

(2) the reaction is easily „invented“; therefore, there is more than one protein family

that is independently able to perform the catalysis (convergence).

(1) the enzymes that catalyze the same reaction are homologs and have

duplicated (or were obtained by horizontal gene transfer),

acquiring some specificity but retaining the same mechanism (divergence)




Enzymes that catalyze more than one reaction

Genome predictions usually assign a single enzymatic function.

However, E.coli is known to contain many multifunctional enzymes.

Of the 607 E.coli enzymes, 100 are multifunctional, either having the same active

site and different substrate specificities or different active sites.

Number of enzymes that catalyze one or

more reactions. Most enzymes catalyze

one reaction; some are multifunctional.

The enzymes that catalyze 7 and 9 reactions are purine nucleoside phosphorylase

and nucleoside diphosphate kinase.

Take-home message: The high proportion of multifunctional enzymes implies that

the genome projects significantly underpredict multifunctional enzymes!




Reactions participating in more than one pathway

The 99 reactions belonging to multiple

pathways appear to be the intersection

points in the complex network of chemical

processes in the cell.

E.g. the reaction present in 6 pathways corresponds to the reaction catalyzed by

malate dehydrogenase, a central enzyme in cellular metabolism.

Ouzonis, Karp,

Genome Res. 10, 568 (2000)



Connectivity distributions P(k) for substrates

a, Archaeoglobus fulgidus (archae);

b, E. coli (bacterium);

c, Caenorhabditis elegans (eukaryote),

shown on a log–log plot, counting

separately the incoming (In) and

outgoing links (Out) for each substrate.

kin (kout) corresponds to the number of

reactions in which a substrate

participates as a product (educt).

d, The connectivity distribution

averaged over 43 organisms.

Jeong et al. Nature 407, 651 (2000)



Properties of metabolic networks

a, The histogram of the biochemical pathway

lengths, l, in E. coli.

b, The average path length (diameter) for

each of the 43 organisms.

c, d, Average number of incoming links (c) or

outgoing links (d) per node for each

organism.

e, The effect of substrate removal on the

metabolic network diameter of E. coli. In the

top curve (red) the most connected substrates

are removed first. In the bottom curve (green)

nodes are removed randomly. M = 60

corresponds to 8% of the total number of

substrates in found in E. coli.

The horizontal axis in b– d denotes the

number of nodes in each organism. b–d,

Archaea (magenta), bacteria (green) and

eukaryotes (blue) are shown. Jeong et al. Nature 407, 651 (2000)

The diameter of the network does notgrow with N!Diameter of smallworld network growswith log N or evenlog log N!



Stoichiometric matrix

Stoichiometric matrix:

A matrix with reaction stochio-

metries as columns and

metabolite participations as

rows.

The stochiometric matrix is an

important part of the in silico

model.

With the matrix, the methods of

extreme pathway and

elementary mode analyses can

be used to generate a unique

set of pathways P1, P2, and P3

(see future lecture).

Papin et al. TIBS 28, 250 (2003)



Flux balancingAny chemical reaction requires mass conservation.

Therefore one may analyze metabolic systems by requiring mass conservation.

Only required: knowledge about stoichiometry of metabolic pathways and

metabolic demands

For each metabolite:

Under steady-state conditions, the mass balance constraints in a metabolic

network can be represented mathematically by the matrix equation:

S · v = 0

where the matrix S is the m n stoichiometric matrix,

m = the number of metabolites and n = the number of reactions in the network.

The vector v represents all fluxes in the metabolic network, including the internal

fluxes, transport fluxes and the growth flux.

)( dtransporteuseddegradeddsynthesizei

i VVVVdt

dXv



Flux balance analysis

Since the number of metabolites is generally smaller than the number of reactions

(m < n) the flux-balance equation is typically underdetermined.

Therefore there are generally multiple feasible flux distributions that satisfy the mass

balance constraints.

The set of solutions are confined to the nullspace of matrix S.

To find the „true“ biological flux in cells ( e.g. Heinzle, Huber, UdS) one needs

additional (experimental) information,

or one may impose constraints

on the magnitude of each individual metabolic flux.

The intersection of the nullspace and the region defined by those linear inequalities

defines a region in flux space = the feasible set of fluxes.

iii v



Feasible solution set for a metabolic reaction network

(A) The steady-state operation of the metabolic network is restricted to the region

within a cone, defined as the feasible set. The feasible set contains all flux vectors

that satisfy the physicochemical constrains. Thus, the feasible set defines the

capabilities of the metabolic network. All feasible metabolic flux distributions lie

within the feasible set, and

(B) in the limiting case, where all constraints on the metabolic network are known,

such as the enzyme kinetics and gene regulation, the feasible set may be reduced

to a single point. This single point must lie within the feasible set.

Edwards & Palsson PNAS 97, 5528 (2000)



SummaryFBA analysis constructs the optimal network utilization simply using

stoichiometry of metabolic reactions and capacity constraints.

For E.coli the in silico results are consistent with experimental data.

FBA shows that in the E.coli metabolic network there are relatively few critical

gene products in central metabolism.

However, the the ability to adjust to different environments (growth conditions) may

be dimished by gene deletions.

FBA identifies „the best“ the cell can do, not how the cell actually behaves under a

given set of conditions. Here, survival was equated with growth.

FBA does not directly consider regulation or regulatory constraints on the

metabolic network. This can be treated separately (see future lecture).

Edwards & Palsson PNAS 97, 5528 (2000)



V18 – extreme pathways

Price et al. Nature Rev Microbiol 2, 886 (2004)

Computational metabolomics: modelling constraints

Surviving (expressed) phenotypes must satisfy constraints imposed on the molecular

functions of a cell, e.g. conservation of mass and energy.

Fundamental approach to understand biological systems: identify and formulate

constraints.

Important constraints of cellular function:

(1) physico-chemical constraints

(2) Topological constraints

(3) Environmental constraints

(4) Regulatory constraints



Physico-chemical constraints


These are „hard“ constraints: Conservation of mass, energy and momentum.

Contents of a cell are densely packed viscosity can be 100 – 1000 times higher

than that of water

Therefore, diffusion rates of macromolecules in cells are slower than in water.

Many molecules are confined inside the semi-permeable membrane high

osmolarity. Need to deal with osmotic pressure (e.g. Na+K+ pumps)

Reaction rates are determined by local concentrations inside cells

Enzyme-turnover numbers are generally less than 104 s-1. Maximal rates are equal to

the turnover-number multiplied by the enzyme concentration.

Biochemical reactions are driven by negative free-energy change in forward

direction.



Topological constraints


The crowding of molecules inside cells leads to topological (3D)-constraints that affect

both the form and the function of biological systems.

E.g. the ratio between the number of tRNAs and the number of ribosomes in an E.coli

cell is about 10. Because there are 43 different types of tRNA, there is less than

one full set of tRNAs per ribosome it may be necessary to configure the

genome so that rare codons are located close together.

E.g. at a pH of 7.6 E.coli typically contains only about 16 H+ ions.

Remember that H+ is involved in many metabolic reactions.

Therefore, during each such reaction, the pH of the cell changes!



Environmental constraints


Environmental constraints on cells are time and condition dependent:

Nutrient availability, pH, temperature, osmolarity, availability of electron acceptors.

E.g. Heliobacter pylori lives in the human stomach at pH = 1

needs to produce NH3 at a rate that will maintain ist immediate surrounding at a pH

that is sufficiently high to allow survival.

Ammonia is made from elementary nitrogen H. pylori has adapted by using amino

acids instead of carbohydrates as its primary carbon source.



Regulatory constraints


Regulatory constraints are self-imposed by the organism and are subject to

evolutionary change they are no „hard“ constraints.

Regulatory constraints allow the cell to eliminate suboptimal phenotypic states and to

confine itself to behaviors of increased fitness.

Q: classify the following constrains asphysico-chemical, regulatory and topological restraints ...Multiple-choice selection.



Mathematical formation of constraints


There are two fundamental types of constraints: balances and bounds.

Balances are constraints that are associated

with conserved quantities as energy, mass, redox potential, momentum

or with phenomena such as solvent capacity, electroneutrality and osmotic pressure.

Bounds are constraints that limit numerical ranges of individual variables and

parameters such as concentrations, fluxes or kinetic constants.

Both bound and balance constraints limit the allowable functional states of

reconstructed cellular metabolic networks.



Extreme Pathwaysintroduced into metabolic analysis by the lab of Bernard Palsson

(Dept. of Bioengineering, UC San Diego). The publications of this lab

are available at http://gcrg.ucsd.edu/publications/index.html

The extreme pathway

technique is based

on the stoichiometric

matrix representation

of metabolic networks.

All external fluxes are

defined as pointing outwards.

Schilling, Letscher, Palsson,

J. theor. Biol. 203, 229 (2000)



Extreme Pathways – theorem

Theorem. A convex flux cone has a set of systemically independent generating

vectors. Furthermore, these generating vectors (extremal rays) are unique up to

a multiplication by a positive scalar. These generating vectors will be called

„extreme pathways“.

Proof. omitted.



Extreme Pathways – algorithm - setup

The algorithm to determine the set of extreme pathways for a reaction network

follows the pinciples of algorithms for finding the extremal rays/ generating

vectors of convex polyhedral cones.

Combine n n identity matrix (I) with the transpose of the stoichiometric

matrix ST. I serves for bookkeeping.


J. theor. Biol. 203, 229 (2000)

S

I ST



separate internal and external fluxes

Examine constraints on each of the exchange fluxes as given by

j bj j

If the exchange flux is constrained to be positive do nothing.

If the exchange flux is constrained to be negative multiply the

corresponding row of the initial matrix by -1.

If the exchange flux is unconstrained move the entire row to a temporary

matrix T(E). This completes the first tableau T(0).

T(0) and T(E) for the example reaction system are shown on the previous slide.

Each element of this matrices will be designated Tij.

Starting with x = 1 and T(0) = T(x-1) the next tableau is generated in the following

way:


J. theor. Biol. 203, 229 (2000)



idea of algorithm

(1) Identify all metabolites that do not have an unconstrained exchange flux

associated with them.

The total number of such metabolites is denoted by .

For the example, this is only the case for metabolite C ( = 1).

What is the main idea?

- We want to find balanced extreme pathways

that don‘t change the concentrations of

metabolites when flux flows through

(input fluxes are channelled to products not to

accumulation of intermediates).

- The stochiometrix matrix describes the coupling of each reaction to the

concentration of metabolites X.

- Now we need to balance combinations of reactions that leave concentrations

unchanged. Pathways applied to metabolites should not change their

concentrations the matrix entries

need to be brought to 0.Schilling, Letscher, Palsson,

J. theor. Biol. 203, 229 (2000)



keep pathways that do not change concentrations of internal metabolites

(2) Begin forming the new matrix T(x) by copying

all rows from T(x – 1) which contain a zero in the

column of ST that corresponds to the first

metabolite identified in step 1, denoted by index c.

(Here 3rd column of ST.)

Schilling, Letscher, Palsson, J. theor. Biol. 203, 229 (2000)

1 -1 1 0 0 0

1 0 -1 1 0 0

1 0 1 -1 0 0

1 0 0 -1 1 0

1 0 0 1 -1 0

1 0 0 -1 0 1

1 -1 1 0 0 0

T(0) =

T(1) =

+



balance combinations of other pathways

(3) Of the remaining rows in T(x-1) add together

all possible combinations of rows which contain

values of the opposite sign in column c, such that

the addition produces a zero in this column.

Schilling, et al.

JTB 203, 229

1 -1 1 0 0 0

1 0 -1 1 0 0

1 0 1 -1 0 0

1 0 0 -1 1 0

1 0 0 1 -1 0

1 0 0 -1 0 1

T(0) =

T(1) =

1 0 0 0 0 0 -1 1 0 0 0

0 1 1 0 0 0 0 0 0 0 0

0 1 0 1 0 0 0 -1 0 1 0

0 1 0 0 0 1 0 -1 0 0 1

0 0 1 0 1 0 0 1 0 -1 0

0 0 0 1 1 0 0 0 0 0 0

0 0 0 0 1 1 0 0 0 -1 1



remove “non-orthogonal” pathways

(4) For all of the rows added to T(x) in steps 2 and 3 check to make sure that no

row exists that is a non-negative combination of any other sets of rows in T(x) .

One method used is as follows:

let A(i) = set of column indices j for with the elements of row i = 0.

For the example above Then check to determine if there exists

A(1) = {2,3,4,5,6,9,10,11} another row (h) for which A(i) is a

A(2) = {1,4,5,6,7,8,9,10,11} subset of A(h).

A(3) = {1,3,5,6,7,9,11}

A(4) = {1,3,4,5,7,9,10} If A(i) A(h), i h

A(5) = {1,2,3,6,7,8,9,10,11} where

A(6) = {1,2,3,4,7,8,9} A(i) = { j : Ti,j = 0, 1 j (n+m) }

then row i must be eliminated from T(x)

Schilling et al.

JTB 203, 229



repeat steps for all internal metabolites

(5) With the formation of T(x) complete steps 2 – 4 for all of the metabolites that do

not have an unconstrained exchange flux operating on the metabolite,

incrementing x by one up to . The final tableau will be T().

Note that the number of rows in T () will be equal to k, the number of extreme

pathways.

Schilling et al.

JTB 203, 229



balance external fluxes

(6) Next we append T(E) to the bottom of T(). (In the example here = 1.)

This results in the following tableau:

Schilling et al.

JTB 203, 229

T(1/E) =

1 -1 1 0 0 0

1 1 0 0 0 0 0

1 1 0 -1 0 1 0

1 1 0 -1 0 1 0

1 1 0 1 0 -1 0

1 1 0 0 0 0 0

1 1 0 0 0 -1 1

1 -1 0 0 0 0

1 0 -1 0 0 0

1 0 0 0 -1 0

1 0 0 0 0 -1




(7) Starting in the n+1 column (or the first non-zero column on the right side),

if Ti,(n+1) 0 then add the corresponding non-zero row from T(E) to row i so as to

produce 0 in the n+1-th column.

This is done by simply multiplying the corresponding row in T(E) by Ti,(n+1) and

adding this row to row i .

Repeat this procedure for each of the rows in the upper portion of the tableau so

as to create zeros in the entire upper portion of the (n+1) column.

When finished, remove the row in T(E) corresponding to the exchange flux for the

metabolite just balanced.

Schilling et al.

JTB 203, 229




(8) Follow the same procedure as in step (7) for each of the columns on the right

side of the tableau containing non-zero entries.

(In this example we need to perform step (7) for every column except the middle

column of the right side which correponds to metabolite C.)

The final tableau T(final) will contain the transpose of the matrix P containing the

extreme pathways in place of the original identity matrix.

Schilling et al.

JTB 203, 229



pathway matrix

T(final) =

PT =

Schilling et al.

JTB 203, 229

1 -1 1 0 0 0 0 0 0

1 1 0 0 0 0 0 0

1 1 -1 1 0 0 0 0 0 0

1 1 -1 1 0 0 0 0 0 0

1 1 1 -1 0 0 0 0 0 0

1 1 0 0 0 0 0 0

1 1 -1 1 0 0 0 0 0 0

1 0 0 0 0 0 -1 1 0 0

0 1 1 0 0 0 0 0 0 0

0 1 0 1 0 0 0 -1 1 0

0 1 0 0 0 1 0 -1 0 1

0 0 1 0 1 0 0 1 -1 0

0 0 0 1 1 0 0 0 0 0

0 0 0 0 1 1 0 0 -1 1

v1 v2 v3 v4 v5 v6 b1 b2 b3 b4

p1 p7 p3 p2 p4 p6 p5



Extreme Pathways for model system

Schilling et al.

JTB 203, 229

1 0 0 0 0 0 -1 1 0 0

0 1 1 0 0 0 0 0 0 0

0 1 0 1 0 0 0 -1 1 0

0 1 0 0 0 1 0 -1 0 1

0 0 1 0 1 0 0 1 -1 0

0 0 0 1 1 0 0 0 0 0

0 0 0 0 1 1 0 0 -1 1

v1 v2 v3 v4 v5 v6 b1 b2 b3 b4

p1 p7 p3 p2 p4 p6 p5

2 pathways p6 and p7 are not shown (right below) because all exchange fluxes with the exterior are 0.Such pathways have no net overall effect on the functional capabilities of the network.They belong to the cycling of reactions v4/v5 and v2/v3.



How reactions appear in pathway matrix

In the matrix P of extreme pathways, each column is an EP and each row

corresponds to a reaction in the network.

The numerical value of the i,j-th element corresponds to the relative flux level

through the i-th reaction in the j-th EP.

Papin, Price, Palsson,

Genome Res. 12, 1889 (2002)

PPP TLM



Papin, Price, Palsson, Genome Res. 12, 1889 (2002)

A symmetric Pathway Length Matrix PLM can be calculated:

where the values along the diagonal correspond to the length of the EPs.

PPP TLM

Properties of pathway matrix

The off-diagonal terms of PLM are the number of reactions that a pair of extreme

pathways have in common.



Papin, Price, Palsson, Genome Res. 12, 1889 (2002)

One can also compute a reaction participation matrix PPM from P:

where the diagonal correspond to the number of pathways in which the given

reaction participates.

TPM PPP

Properties of pathway matrix



Application of elementary modesMetabolic network structure of E.coli determines

key aspects of functionality and regulation

Compute EFMs for central

metabolism of E.coli.

Catabolic part: substrate uptake

reactions, glycolysis, pentose

phosphate pathway, TCA cycle,

excretion of by-products (acetate,

formate, lactate, ethanol)

Anabolic part: conversions of

precursors into building blocks like

amino acids, to macromolecules,

and to biomass.

Stelling et al. Nature 420, 190 (2002)

Elementary modes will be covered in V19.

The concept is closely related to extreme

pathways. In this example , we will simply

ignore the small difference.



Metabolic network topology and phenotypeThe total number of EFMs for given

conditions is used as quantitative

measure of metabolic flexibility.

a, Relative number of EFMs N enabling

deletion mutants in gene i ( i) of E. coli

to grow (abbreviated by µ) for 90 different

combinations of mutation and carbon

source. The solid line separates

experimentally determined mutant

phenotypes, namely inviability (1–40)

from viability (41–90).


The # of EFMs for mutant strain

allows correct prediction of

growth phenotype in more than 90%

of the cases.



Robustness analysis

The # of EFMs qualitatively indicates whether a mutant is viable or not, but does

not describe quantitatively how well a mutant grows.

Define maximal biomass yield Ymass as the optimum of:

ei is the single reaction rate (growth and substrate uptake) in EFM i selected for

utilization of substrate Sk.


ki Si

iSXi e

eY

/,



Compute control-effective fluxes for each reaction l by determining the efficiency of any EFM

ei by relating the system‘s output to the substrate uptake and to the sum of all absolute

fluxes.

With flux modes normalized to the total substrate uptake, efficiencies i(Sk, ) for

the targets for optimization -growth and ATP generation, are defined as:

Can regulation be predicted by EFM analysis?


l

li

ATPi

ki

l

li

iki

e

eATPS

e

eS ,and,

Control-effective fluxes vl(Sk) are obtained by averaged weighting of the product of reaction-

specific fluxes and mode-specific efficiencies over all EFMs using the substrate under

consideration:

lki

i

liki

SAl

ki

i

liki

SXkl ATPS

eATPS

YS

eS

YSv

kk,

,1

,

,1

max/

max/

YmaxX/Si and Ymax

A/Si are optimal yields of biomass production and of ATP synthesis.

Control-effective fluxes represent the importance of each reaction for efficient and flexible

operation of the entire network.



Prediction of gene expression patterns

As cellular control on longer timescales

is predominantly achieved by genetic

regulation, the control-effective fluxes

should correlate with messenger RNA

levels.

Compute theoretical transcript ratios

(S1,S2) for growth on two alternative

substrates S1 and S2 as ratios of

control-effective fluxes.

Compare to exp. DNA-microarray data

for E.coli growin on glucose, glycerol,

and acetate.

Excellent correlation!Stelling et al. Nature 420, 190 (2002)

Calculated ratios between gene expression levels

during exponential growth on acetate and

exponential growth on glucose (filled circles

indicate outliers) based on all elementary modes

versus experimentally determined transcript

ratios19. Lines indicate 95% confidence intervals

for experimental data (horizontal lines), linear

regression (solid line), perfect match (dashed

line) and two-fold deviation (dotted line).



Summary (extreme pathways)

Price et al. Biophys J 84, 794 (2003)

Extreme pathway analysis provides a mathematically rigorous way to dissect

complex biochemical networks.

The matrix products PT P and PT P are useful ways to interpret pathway

lengths and reaction participation.

However, the number of computed vectors may range in the 1000sands.

Therefore, meta-methods (e.g. singular value decomposition) are required that

reduce the dimensionality to a useful number that can be inspected by humans.

Single value decomposition may be one useful method ... and there are more to

come.



V19 Metabolic Pathway Analysis (MPA)Metabolic Pathway Analysis searches for meaningful structural and functional units

in metabolic networks.

Today‘s most powerful methods are based on convex analysis.

Two such approaches are the elementary flux modes (Schuster et al. 1999, 2000)

and extreme pathways (Schilling et al. 2000).

Both sets span the space of feasible steady-state flux distributions by

non-decomposable routes, i.e. no subset of reactions involved in an EFM

or EP can hold the network balanced using non-trivial fluxes.

MPA can be used to study e.g.

- routing + flexibility/redundancy of networks

- functionality of networks

- idenfication of futile cycles

- gives all (sub)optimal pathways with respect to product/biomass yield

- can be useful for calculability studies in MFA

Klamt et al. Bioinformatics 19, 261 (2003)



Metabolic Pathway Analysis: Elementary Flux ModesThe technique of Elementary Flux Modes (EFM) was developed prior to extreme

pathways (EP) by Stephan Schuster, Thomas Dandekar and co-workers:

Pfeiffer et al. Bioinformatics, 15, 251 (1999)

Schuster et al. Nature Biotech. 18, 326 (2000)

The method is very similar to the „extreme pathway“ method to construct a basis

for metabolic flux states based on methods from convex algebra.

Extreme pathways are a subset of elementary modes, and for many systems, both

methods coincide.

Are the subtle differences important?



Elementary Flux ModesStart from list of reaction equations and a declaration of reversible and irreversible

reactions and of internal and external metabolites.

E.g. reaction scheme of monosaccharide Fig.1

metabolism. It includes 15 internal

metabolites, and 19 reactions.

S has dimension 15 19.

It is convenient to reduce this matrix

by lumping those reactions that

necessarily operate together.

{Gap,Pgk,Gpm,Eno,Pyk},

{Zwf,Pgl,Gnd}

Such groups of enzymes can be detected automatically.

This reveals another two sequences {Fba,TpiA} and {2 Rpe,TktI,Tal,TktII}.

Schuster et al. Nature Biotech 18, 326 (2000)



Elementary Flux ModesLumping the reactions in any one sequence gives the following reduced system:

Construct initial tableau by combining

S with identity matrix:

1 0 ... 0 0 0 1 0 0

0 1 ... 0 0 -1 0 2 0

0 0 ... 0 -1 0 0 0 1

0 0 ... 0 -2 0 2 1 -1

0 0 ... 0 0 0 0 -1 0

0 0 ... 0 1 0 0 0 0

0 0 ... 0 0 1 -1 0 0

0 0 ... 0 0 -1 1 0 0

0 0 ... 1 0 0 0 0 -1

Pgi{Fba,TpiA}Rpi reversible{2Rpe,TktI,Tal,TktII}{Gap,Pgk,Gpm,Eno,Pyk}{Zwf,Pgl,Gnd}Pfk irreversibleFbpPrs_DeoB


Ru5

P

FP

2

F6P

GA

P

R5P

T(0)=

Comment: in order not to confusethe students, only the extremepathway algorithmis necessary forthe exam.



Elementary Flux ModesAim again: bring all entries

of right part of matrix to 0.E.g. 2*row3 - row4 gives

„reversible“ row with 0 in column

10

New „irreversible“ rows with 0 entry

in column 10 by row3 + row6 and

by row4 + row7.

In general, linear combinations

of 2 rows corresponding

to the same type of directio-

nality go into the part of

the respective type in the

tableau. Combinations by

different types go into the

„irreversible“ tableau

because at least 1 reaction is

irreversible. Irreversible reactions

can only combined using positive

coefficients.Schuster et al. Nature Biotech 18, 326 (2000)

1 0 0 1 0 0

1 0 -1 0 2 0

1 -1 0 0 0 1

1 -2 0 2 1 -1

1 0 0 0 -1 0

1 1 0 0 0 0

1 0 1 -1 0 0

1 0 -1 1 0 0

1 0 0 0 0 -1

1 0 0 1 0 0

1 0 -1 0 2 0

2 -1 0 0 -2 -1 3

1 0 0 0 -1 0

1 0 1 -1 0 0

1 0 -1 1 0 0

1 0 0 0 0 -1

1 1 0 0 0 0 1

1 2 0 0 2 1 -1

T(1)=

T(0)=



Elementary Flux ModesAim: zero column 11.Include all possible (direction-wise

allowed) linear combinations of

rows.

continue with columns 12-

14. Schuster et al. Nature Biotech 18, 326 (2000)

1 0 0 1 0 0

1 0 -1 0 2 0

2 -1 0 0 -2 -1 3

1 0 0 0 -1 0

1 0 1 -1 0 0

1 0 -1 1 0 0

1 0 0 0 0 -1

1 1 0 0 0 0 1

1 2 0 0 2 1 -1

1 0 0 1 0 0

2 -1 0 0 -2 -1 3

1 0 0 0 -1 0

1 0 0 0 0 -1

1 1 0 0 0 0 1

1 2 0 0 2 1 -1

1 1 0 0 -1 2 0

-1 1 0 0 1 -2 0

1 1 0 0 0 0 0

T(2)=

T(1)=



Elementary Flux ModesIn the course of the algorithm, one must avoid

- calculation of nonelementary modes (rows that contain fewer zeros than the row

already present)

- duplicate modes (a pair of rows is only combined if it fulfills the condition

S(mi(j)) S(mk

(j)) S(ml(j+1)) where S(ml

(j+1)) is the set of positions of 0 in this row.

- flux modes violating the sign restriction for the irreversible reactions.

Final tableau

T(5) =

This shows that the number of rows may decrease or increase in the course of the

algorithm. All constructed elementary modes are irreversible.


1 1 0 0 2 0 1 0 0 0 ... ... 0

-2 0 1 1 1 3 0 0 0 ... ...

0 2 1 1 5 3 2 0 0

0 0 1 0 0 1 0 0 1

5 1 4 -2 0 0 1 0 6

-5 -1 2 2 0 6 0 1 0 ... ...

0 0 0 0 0 0 1 1 0 0 ... ... 0



Two approaches for Metabolic Pathway Analysis?A pathway P(v) is an elementary flux mode if it fulfills conditions C1 – C3.

(C1) Pseudo steady-state. S e = 0. This ensures that none of the

metabolites is consumed or produced in the overall stoichiometry.

(C2) Feasibility: rate ei 0 if reaction is irreversible. This demands that only

thermodynamically realizable fluxes are contained in e.

(C3) Non-decomposability: there is no vector v (unequal to the zero vector

and to e) fulfilling C1 and C2 and that P(v) is a proper subset of P(e). This is

the core characteristics for EFMs and EPs and supplies the decomposition

of the network into smallest units (able to hold the network in steady state).

C3 is often called „genetic independence“ because it implies that the

enzymes in one EFM or EP are not a subset of the enzymes from another

EFM or EP.

Klamt & Stelling Trends Biotech 21, 64 (2003)



Two approaches for Metabolic Pathway Analysis?The pathway P(e) is an extreme pathway if it fulfills conditions C1 – C3 AND

conditions C4 – C5.

(C4) Network reconfiguration: Each reaction must be classified either as

exchange flux or as internal reaction. All reversible internal reactions must

be split up into two separate, irreversible reactions (forward and backward

reaction).

(C5) Systemic independence: the set of EPs in a network is the minimal set

of EFMs that can describe all feasible steady-state flux distributions.




Two approaches for Metabolic Pathway Analysis?


A C P

B

D

A(ext) B(ext) C(ext)R1 R2 R3

R5

R4 R8

R9

R6

R7



Reconfigured Network


A C P

B

D


R5

R4 R8

R9

R6

R7bR7f

3 EFMs are not systemically independent:EFM1 = EP4 + EP5EFM2 = EP3 + EP5EFM4 = EP2 + EP3



Property 1 of EFMs


The only difference in the set of EFMs emerging upon reconfiguration consists in

the two-cycles that result from splitting up reversible reactions. However, two-

cycles are not considered as meaningful pathways.

Valid for any network: Property 1

Reconfiguring a network by splitting up reversible reactions leads to the same set of

meaningful EFMs.



Software: FluxAnalyzerWhat is the consequence of when all exchange fluxes (and hence all

reactions in the network) are irreversible?


Then EFMs and EPs always co-incide!



Property 2 of EFMs


Property 2

If all exchange reactions in a network are irreversible then the sets of

meaningful EFMs (both in the original and in the reconfigured network) and

EPs coincide.



Reconfigured Network


A C P

B

D


R5

R4 R8

R9

R6

R7bR7f

3 EFMs are not systemically independent:EFM1 = EP4 + EP5EFM2 = EP3 + EP5EFM4 = EP2 + EP3



Comparison of EFMs and EPs


Problem EFM (network N1) EP (network N2)

Recognition of 4 genetically indepen- Set of EPs does not contain

operational modes: dent routes all genetically independent

routes for converting (EFM1-EFM4) routes. Searching for EPs

exclusively A to P. leading from A to P via B,

no pathway would be found.





Problem EFM (network N1) EP (network N2)

Finding all the EFM1 and EFM2 are One would only find the

optimal routes: optimal because they suboptimal EP1, not the

optimal pathways for yield one mole P per optimal routes EFM1 and

synthesizing P during mole substrate A EFM2.

growth on A alone. (i.e. R3/R1 = 1),

whereas EFM3 and

EFM4 are only sub-

optimal (R3/R1 = 0.5).





EFM (network N1)

4 pathways convert A

to P (EFM1-EFM4),

whereas for B only one

route (EFM8) exists.

When one of the

internal reactions (R4-

R9) fails, for production

of P from A 2 pathways

will always „survive“. By

contrast, removing

reaction R8 already

stops the production of

P from B alone.

EP (network N2)

Only 1 EP exists for

producing P by substrate A

alone, and 1 EP for

synthesizing P by (only)

substrate B. One might

suggest that both

substrates possess the

same redundancy of

pathways, but as shown by

EFM analysis, growth on

substrate A is much more

flexible than on B.

Problem

Analysis of network

flexibility (structural

robustness,

redundancy):

relative robustness of

exclusive growth on

A or B.





EFM (network N1)

R8 is essential for

producing P by substrate

B, whereas for A there is

no structurally „favored“

reaction (R4-R9 all occur

twice in EFM1-EFM4).

However, considering the

optimal modes EFM1,

EFM2, one recognizes the

importance of R8 also for

growth on A.

EP (network N2)

Consider again biosynthesis

of P from substrate A (EP1

only). Because R8 is not

involved in EP1 one might

think that this reaction is not

important for synthesizing P

from A. However, without this

reaction, it is impossible to

obtain optimal yields (1 P per

A; EFM1 and EFM2).

Problem

Relative importance

of single reactions:

relative importance of

reaction R8.





EFM (network N1)

R6 and R9 are an enzyme

subset. By contrast, R6

and R9 never occur

together with R8 in an

EFM. Thus (R6,R8) and

(R8,R9) are excluding

reaction pairs.(In an arbitrary composable

steady-state flux distribution they

might occur together.)

EP (network N2)

The EPs pretend R4 and R8

to be an excluding reaction

pair – but they are not

(EFM2). The enzyme

subsets would be correctly

identified. However, one can construct simple

examples where the EPs would also

pretend wrong enzyme subsets (not

shown).

Problem

Enzyme subsets

and excluding

reaction pairs:

suggest regulatory

structures or rules.





EFM (network N1)

The shortest pathway

from A to P needs 2

internal reactions (EFM2),

the longest 4 (EFM4).

EP (network N2)

Both the shortest (EFM2)

and the longest (EFM4)

pathway from A to P are not

contained in the set of EPs.

Problem

Pathway length:

shortest/longest

pathway for

production of P from

A.





EFM (network N1)

All EFMs not involving the

specific reactions build up

the complete set of EFMs

in the new (smaller) sub-

network. If R7 is deleted,

EFMs 2,3,6,8 „survive“.

Hence the mutant is

viable.

EP (network N2)

Analyzing a subnetwork

implies that the EPs must be

newly computed. E.g. when

deleting R2, EFM2 would

become an EP. For this

reason, mutation studies

cannot be performed easily.

Problem

Removing a

reaction and

mutation studies:

effect of deleting R7.





EFM (network N1)

For the case of R7, all

EFMs but EFM1 and

EFM7 „survive“ because

the latter ones utilize R7

with negative rate.

EP (network N2)

In general, the set of EPs

must be recalculated:

compare the EPs in network

N2 (R2 reversible) and N4

(R2 irreversible).

Problem

Constraining

reaction

reversibility:

effect of R7 limited to

B C.

Q: Discuss the relevance of the elementary modes 1 to Xfor the following properties ...



Minimal cut sets in biochemical reaction networksConcept of minimal cut sets (MCSs): smallest „failure modes“ in the network

that render the correct functioning of a cellular reaction impossible.

Klamt & Gilles, Bioinformatics 20, 226 (2004)

Right: fictitious reaction network NetEx.

The only reversible reaction is R4.

We are particularly interested in the flux

obR exporting synthesized metabolite X.

Characterize solution space by

computing elementary modes.



Elementary modes of NetEx


One finds 4 elementary modes for NetEx.

3 of them (shaded) allow the production of metabolite X.



Cut set


Now we want to prevent the production of metabolite X.

demand that there is no balanced flux distribution possible which involves obR.

Definition. We call a set of reactions a cut set (with respect to a defined

objective reaction) if after the removal of these reactions from the network no

feasible balanced flux distribution involves the objective reaction.

A trivial cut set if the reaction itself: C0 = {obR}. Why should we be interested in

other solutions as well?

- From an engineering point of view, it might be desirable to cut reactions at the

beginning of a pathway.

- The production of biomass is usually not coupled to a single gene or enzyme, and

can therefore not be directly inactivated.



Cut set


Another extreme case is the removal of all reactions except obR .. not efficient!

E.g. C1 = {R5,R8} is a cut set already

sufficient for preventing the production of X.

Removing R5 or R8 alone is not sufficient.

C1 is a minimal cut set

Definition. A cut set C (related to a

defined objective reaction) is a

minimal cut set (MCS) if no proper

subset of C is a cut set.

Q: discuss the following minimal cut sets MCS1, MCS2.



Remarks


(1) An MCS always guarantees dysfunction as long as the assumed network

structure is currect. However, additional regulatory circuits or capacity restrictions

may allow that even a proper subset of a MCS is a cut set.

The MCS analysis should always be seen from a purely structural point of view.

(2) After removing a complete MCS from the network, other pathways producing

other metabolites may still be active.

(3) MCS4 = {R5,R8} clearly stops production of X.

What about MCS6 = {R3,R4,R6}?

Cannot X be still be produced via R1, R2, and R5?

However, this would lead to accumulation of B and is therefore physiologically

impossible.



Algorithm for computing MCSs


The MCSs for a given network and objective reaction are members of the power

set of the set of reaction indices and are uniquely determined.

A systematic computation must ensure that the calculated MCSs are:

(1) cut sets („destroying“ all possible balanced flux distributions involving the

objective reaction), and

(2) that the MCSs are really minimal, and

(3) that all MCSs are found.

Algorithm given in lecture is omitted.



Applications of MCSs


Target identification and repressing cellular functions

A screening of all MCSs allows for the identification of the best suitable

manipulation. For practical reasons, the following conditions should be fulfilled:

- usually, a small number of interventions is desirable (small size of MCS)

- other pathways in the network should only be weakly affected

- some of the cellular functions might be difficult to run off genetically or by

inhibition, e.g. if many isozymes exist for a reaction.



Applications of MCSs


Structural fragility and robustness

If we assume that each reaction in a metabolic network has the same probability to

fail, small MCSs are most probable to be responsible for a failing objective function.

Define a fragility coefficient Fi as the

reciprocal of the average size of all

MCSs in which reaction i participates.

Besides the essential reaction R1, reactionR5 is most crucial for the objective reaction.

Q: give the definition of the fragility coefficientand discuss its meaning.



Conclusion


An MCS is a irreducible combination of network elements whose simultaneous

inactivation leads to a guaranteed dysfunction of certain cellular reactions or

processes.

MCSs are inherent and uniquely determined structural features of metabolic

networks similar to EMs.

The computation of MCSs and EMs becomes challenging in large networks.

Analyzing the MCSs gives deeper insights in the structural fragility of a given

metabolic network and is useful for identifying target sets for an intended

repression of network functions.



V20 The Double Description method:Theoretical framework behind EFM and EP

in „Combinatorics and Computer Science Vol. 1120“ edited by Deza, Euler, Manoussakis, Springer, 1996:91



Definition of Elementary ModesCompared to gene regulatory processes, metabolism involves fast reactions and

high turnover of substances.

it is often assumed that metabolite concentrations and reaction rates are

equilibrated (constant) on the timescale of study.

The metabolic system is then considered to be in quasi steady state implying

S v = 0 , S: stoichiometric matrix, v: feasible flux distributions.

Thermodynamics imposes the rate of each irreversible reaction to be nonnegative.

Consequently the set of feasible flux vectors is restricted to

P = {v q : S v = 0 and vi ≥ 0, i Irrev} (1)

P is a set of q-vectors that obey a finite set of homogeneous linear equalities and

inequalities, namely

- the |Irrev| inequalities defined by vi ≥ 0, i Irrev and

- the m equalities defined by S v = 0.



Elementary Flux Modes

Metabolic pathway analysis serves to describe the infinite set P of feasible states by

providing a finite set of vectors that allow the generation of any vectors of P and are

of fundamental importance for the overall capabilities of the metabolic system.

One of these sets is the so-called set of elementary (flux) modes (EMs).

For a given flux vector v, we note R(v) = {i : vi ≠ 0} the set of indices of the reactions

participating in v.

R(v) can be seen as the underlying pathway of v.



Elementary Flux Modes

By definition, a flux vector e is an elementary mode (EM) if and only if it fulfills the

following three conditions:

(2)

In other words, e is an EM if and only if

- it works at quasi steady state,

- is thermodynamically feasible and

- there is no other non-null flux vector (up to a scaling) that both satisfies these

constraints and involves a proper subset of its participating reactions.

With this convention, reversible modes are here considered as two vectors of

opposite directions.

tyelementari :'or 'or 0'':' allfor

yfeasibilit ynamical thermod:,0e

statesteady quasi :.. ,

i

eeeeeeee

0eSe

RRP

IrrevieiP



In the particular case of a metabolic system with only irreversible reactions,

the set of admissible reactions reads:

P = {v q : S v = 0 and vi 0, i Irrev } (3)

P is in this case a pointed polyhedral cone.

The set of feasible metabolic fluxes described before is therefore – by definition –

a convex polyhedral cone.

A unified framework - Elementary modes as extreme rays in networks of irreversible reactions

A pointed polyhedral cone. Dashed lines represent virtual cuts of unbounded areas



Pointed polyhedral cone – more precise

Definition P is a pointed polyhedral cone of d if and only if P is defined by a

full rank h × d matrix A (rank(A) = d) such that,

Insert: the rank of a matrix is the dimension of the range of the matrix,

corresponding to the number of linearly independent rows or columns of the matrix.

The h rows of the matrix A represent h linear inequalities, whereas the full rank

mention imposes the "pointed" effect in 0. Note that a pointed polyhedral cone is,

in general, not restricted to be located completely in the positive orphant as in (3).

For example, the cone considered in extreme-pathway analysis may have

negative parts (namely for exchange reactions), however, by using a particular

configuration it is ensured that the spanned cone is pointed.

P = P(A) = {x d : A x 0}



Extreme rays

A vector r is said to be a ray of P(A) if r ≠ 0 and for all α > 0, α · r P(A).

We identify two rays r and r' if there is some α > 0 such that r = α · r' and we

denote r ≃ r', analogous as introduced above for flux vectors.

For any vector x in P(A), the zero set or active set Z(x) is the set of inequality

indices satisfied by x with equality.

Noting Ai• the ith row of A, Z(x) = {i : Ai•x = 0}.

Zero sets can be used to characterize extreme rays.



Extreme rays - definition

Definition 1

Let r be a ray of the pointed polyhedral cone P(A).

The following statements are equivalent:

(a) r is an extreme ray of P(A)

(b) if r' is a ray of P(A) with Z(r) Z(r') then r' ≃ r

Since A is full rank, 0 is the unique vector that solves all constraints with equality.

The extreme rays are those rays of P(A) that solve a maximum but not all

constraints with equalities. This is expressed in (b) by requiring that no other ray in P(A) solves the same constraints plus additional ones with equalities.

Note that in (b) Z(r) = Z(r') consequently holds.



Extreme rays - propertiesAn important property of the extreme rays is that they form a finite set of

generating vectors of the pointed cone:

any vector of P(A) can be expressed as a non-negative linear combination of

extreme rays,

The converse is also true:

all non-negative combinations of extreme rays lie in P(A).

The set of extreme rays is the unique minimal set of generating vectors of

a pointed cone P(A) (up to positive scalings).



Elementary modes

Lemma 1: EMs in networks of irreversible reactions

In a metabolic system where all reactions are irreversible, the EMs are exactly the

extreme rays of P = {v q : S v = 0 and v ≥ 0}.

Proof: omitted. □



The general caseIn the general case, some reactions of the metabolic system can be reversible.

Consequently, A does not contain the identity matrix and P (as given in (1))

is not ensured to be a pointed polyhedral cone anymore.

Because they contain a linear subspace, non-pointed polyhedral cones cannot be

represented properly by a unique set of generating vectors composed of extreme

rays.

One way to obtain a pointed polyhedral cone from (1) is to split up each reversible

reaction into two opposite irreversible ones.

(This is routinely done for the construction of extreme pathways.

Therefore, EPs directly correspond to a flux cone).

This virtual split essentially does not change the outcome:

the EMs in the reconfigured network are practically equivalent

to the EMs from the original network.



NotationsWe denote the original reaction network by T and the reconfigured network

(with all reversible reactions split up) by T'.

The reactions of T are indexed from 1 to q.

Irrev denotes the set of irreversible reaction indices and Rev the reversible ones.

An irreversible reaction indexed i gives rise to a reaction of T' indexed i.

A reversible reaction indexed i gives rise to two opposite reactions of T' indexed by

the pairs (i,+1) and (i,-1) for the forward and the backward respectively.

The reconfiguration of a flux vector v q of T is a flux vector

v' Irrev Rev × {-1;+1} of T' such that



Notations

Let S' be the stoichiometry matrix of T'. S' can be written as S' = [S –SRev] where

SRev consists of all columns of S corresponding to reversible reactions.

Note that if v is a flux vector of T and v' is its reconfiguration then S v = S' v'.

If possible, i.e. if v' Irrev Rev × {-1;+1} is such that for any reversible reaction index

i Rev at least one of the two coefficients v'(i,+1) or v'(i,-1) equals zero,

then we define the reverse operation, called back-configuration that maps v' back to

a flux vector v such that:



Theorem 1: EMs in original and in reconfigured networks

Theorem 1

Let T be a metabolic system and T' its reconfiguration by splitting up

reversible reactions.

Then the set of EMs of T' is the union of

a) the set of reconfigured EMs of T

b) the set of two-cycles made of a forward and a backward reaction

of T' derived from the same reversible reaction of T.

Proof. omitted.



Double Description Method (1953)

All known algorithms for computing EMs are variants of the

Double Description Method.

- derive simple & efficient algorithm for extreme ray enumeration, the so-called

Double Description Method.

- show that it serves as a framework to the popular EM computation methods.



The Double Description MethodA pair (A,R) of real matrices A and R is said to be a double description pair or

simply a DD pair if the relationship

A x 0 if and only if x = R for some 0

holds. Clearly, for a pair (A,R) to be a DD pair, the column size of A has to

equal the row size of R, say d.

For such a pair,

the set P(A) represented by A as

is simultaneously represented by R as

A subset P of d is called polyhedral cone if P = P(A) for some matrix A,

and A is called a representation matrix of the polyhedral cone P(A).

Then, we say R is a generating matrix for P. Clearly, each column vector of a

generating matrix R lies in the cone P and every vector in P is a nonnegative

combination of some columns of R.

0: AxxA dP

0 somefor : λRλxx d



The Double Description MethodTheorem 1 (Minkowski‘s Theorem for Polyhedral Cones)

For any m n real matrix A, there exists some d m real matrix R such that

(A,R) is a DD pair, or in other words, the cone P(A) is generated by R.

The theorem states that every polyhedral cone admits a generating matrix.

The nontriviality comes from the fact that the row size of R is finite.

If we allow an infinite size, there is a trivial generating matrix consisting of all

vectors in the cone.

Also the converse is true:

Theorem 2 (Weyl‘s Theorem for Polyhedral Cones)

For any d n real matrix R, there exists some m d real matrix A such that (A,R)

is a DD pair, or in other words, the set generated by R is the cone P(A).



The Double Description MethodTask: how does one construct a matrix R from a given matrix A, and the converse?

These two problems are computationally equivalent.

Farkas‘ Lemma shows that (A,R) is a DD pair if and only if (RT,AT) is a DD pair.

A more appropriate formulation of the problem is to require the minimality of R:

find a matrix R such that no proper submatrix is generating P(A).

A minimal set of generators is unique up to positive scaling when we assume the

regularity condition that the cone is pointed, i.e. the origin is an extreme point of

P(A).

Geometrically, the columns of a minimal generating matrix are in 1-to-1

correspondence with the extreme rays of P.

Thus the problem is also known as the extreme ray enumeration problem.

No efficient (polynomial) algorithm is known for the general problem.



Double Description Method: primitive formSuppose that the m d matrix A is given and let(This is equivalent to the situation at the beginning of constructing EPs or EFMs: we only know S.)

The DD method is an incremental algorithm to construct a d m matrix R

such that (A,R) is a DD pair.

Let us assume for simplicity that the cone P(A) is pointed.

Let K be a subset of the row indices {1,2,...,m} of A and let AK denote the

submatrix of A consisting of rows indexed by K.

Suppose we already found a generating matrix R for AK, or equivalently,

(AK,R) is a DD pair. If A = AK ,we are done.

Otherwise we select any row index i not in K and try to construct a DD pair

(AK+i, R‘) using the information of the DD pair (AK,R).

Once this basic procedure is described, we have an algorithm to construct a

generating matrix R for P(A).

0 AxxA :P



Geometric version of iteration stepThe procedure can be easily understood geometrically by looking at the

cut-section C of the cone P(AK) with some appropriate hyperplane h in d

which intersects with every extreme ray of P(AK) at a single point.

Let us assume that the cone is pointed and

thus C is bounded.

Having a generating matrix R means that all

extreme rays (i.e. extreme points of the

cut-section) of the cone are represented

by columns of R.

Such a cutsection is illustrated in the Fig.

Here, C is the cube abcdefgh.



Geometric version of iteration stepThe newly introduced inequality Aix 0 partitions the space d into three parts:

Hi+ = {x d : Aix > 0 }

Hi0 = {x d : Aix = 0 }

Hi- = {x d : Aix < 0 }

The intersection of Hi0 with P and the new extreme points i and j in the cut-section

C are shown in bold in the Fig.

Let J be the set of column indices of R. The rays rj (j J ) are then partitioned into

three parts accordingly:

J+ = {j J : rj Hi+ }

J0 = {j J : rj Hi0 }

J- = {j J : rj Hi- }

We call the rays indexed by J+, J0, J- the positive, zero, negative rays with

respect to i, respectively.

To construct a matrix R‘ from R, we generate new | J+| | J-| rays lying on

the ith hyperplane Hi0 by taking an appropriate positive combination of each

positive ray rj and each negative ray rj‘ and by discarding all negative rays.

Comment: remember that all raysinside the cone and on its surface and are valid solutions.



Geometric version of iteration stepThe following lemma ensures that we have a DD pair (AK+i ,R‘), and provides the

key procedure for the most primitive version of the DD method.

Lemma 3 Let (AK,R) be a DD pair and let i be a row index of A not in K.

Then the pair (AK+i ,R‘) is a DD pair, where R‘ is the d |J‘ | matrix with column

vectors rj (j J‘) defined by

J‘ = J+ J0 (J+ J-), and

rjj‘ = (Airj)rj‘– (Airj‘)rj for each (j,j‘) J+ J-



Finding seed DD pairIt is quite simple to find a DD pair (AK,R) when |K| = 1, which can serve as the

initial DD pair.

Another simple (and perhaps the most efficient) way to obtain an initial DD form of

P is by selecting a maximal submatrix AK of A consisting of linearly independent

rows of A.

The vectors rj‘s are obtained by solving the system of equations

AK R = I

where I is the identity matrix of size |K|, R is a matrix of unknown column vectors

rj, j J.

As we have assumed rank(A) = d, i.e. R = AK-1 , the pair (AK,R) is clearly a DD

pair, since AKx 0 x = AK-1, 0.



Primitive algorithm for DoubleDescriptionMethodHence we write the DD method in procedural form:

The method given here is very primitive, and the straightforward implementation

will be quite useless, because the size of J increases very fast and goes beyond

any tractable limit.

This is because many vectors rjj‘ the algorithm generates (defined in Lemma 3)

are unnessary. We need to avoid generating redundant vectors.



Towards the standard implementationProposition 4. Let r be a ray of P, G := { x : AZ(r) x = 0}, F := G P and

rank(AZ(r) ) = d – k. Then

(a) rank(A Z(r){i} ) = d – k + 1 for all i Z(r),

(b) F contains k linearly independent rays,

(c) if k 2 then r is a nonnegative combination of two distinct rays

r1 and r2 with rank(AZ(ri)) > d – k, i = 1,2.

A ray r is said to be extreme if it is not a nonnegative combination of two rays of P

distinct from r.

Proposition 5. Let r be a ray of P. Then

(a) r is an extreme ray of P if and only if the rank of the matrix AZ(r) is d – 1,

(b) r is a nonnegative combination of extreme rays of P.

Corollary 6. Let R be a minimal generating matrix of P.

Then R is the set of extreme rays of P.



Towards the standard implementationTwo distinct extreme rays r and r‘ of P are adjacent if the minimal face of P

containing both contains no other extreme rays.

Proposition 7. Let r and r‘ be distinct rays of P.

Then the following statements are equivalent

(a) r and r‘ are adjacent extreme rays,

(b) r and r‘ are extreme rays and the rank of the matrix AZ(r) Z(r‘) is d – 2,

(c) if r‘‘ is a ray with Z(r‘‘) Z(r) Z(r‘) then either r‘‘ ≃ r or r‘‘ ≃ r.

Lemma 8. Let (AK,R) be a DD pair such than rank(AK) = d and let i be a row index

of A not in K. Then the pair (AK+i , R‘) is a DD pair, where R‘ is the d | J‘| matrix

with column vectors rj (j J‘) defined by

J‘ = J+ J0 Adj

Adj = {(j,j‘) J+ J- : rj and rj‘ are adjacent in P(AK)} and

r = (Ai rj ) rj‘ – (Airj ) rj for each (j,j‘) Adj.

Furthermore, if R is a minimal generating matrix for P(AK) then R‘ is a minimal

generating matrix for P(AK+i).



Algorithm for standard form of double description methodHence we can write a straightforward variation of the DD method which produces

a minimal generating set for P:

To implement DDMethodStandard, we must check for each pair of extreme rays

r and r‘ of P(AK) with Ai r > 0 and Ai r‘ < 0 whether they are adjacent in P(AK).

As stated in Proposition 7, there are two ways to check adjacency, the

combinatiorial and the algebraic way. While it cannot be rigorously shown which

method is more efficient, in practice, the combinatorial method is always faster.

DDMethodStandard(A)

such that R is minimal

Lemma 8

Q: Describe briefly the strategy of the double decomposition method.What is the connection to the algorithm to compute extreme pathways?



V21 Current metabolomicsReview:

(1) recent work on metabolic networks required revising the picture of separate

biochemical pathways into a densely-woven metabolic network

(2) The connectivity of substrates in this network follows a power-law.

(3) Constraint-based modeling approaches (FBA) were successful in analyzing the

capabilities of cellular metabolism including

- its capacity to predict deletion phenotypes

- the ability to calculate the relative flux values of metabolic reactions, and

- the capability to identify properties of alternate optimal growth states

in a wide range of simulated environmental conditions

Open questions

- what parts of metabolism are involved in adaptation to environmental conditions?

- is there a central essential metabolic core?

- what role does transcriptional regulation play?



Distribution of fluxes in E.coli

Stoichiometric matrix for E.coli strain MG1655 containing 537 metabolites and

739 reactions taken from Palsson et al.

Apply flux balance analysis to characterize solution space

(all possible flux states under a given condition).

Nature 427, 839 (2004)

Aim: understand principles that govern

the use of individual reactions under

different growth conditions.

j

jiji vSAdt

d0

vj is the flux of reaction j and Sij is the stoichiometric coefficient of reaction j.



Optimal states

Using linear programming and adapting constraints for each reaction flux vi of the

form imin ≤ vi ≤ i

max, the flux states were calculated that optimize cell growth on

various substrates.

Plot the flux distribution for active (non-zero flux) reactions of E.coli grown in a

glutamate- or succinate-rich substrate.

Denote the mass carried by reaction j producing (consuming) metabolite i by

Fluxes vary widely: e.g. dimensionless flux of succinyl coenzyme A synthetase

reaction is 0.185, whereas the flux of the aspartate oxidase reaction is 10.000

times smaller, 2.2 10-5.

jijij vSv ˆ



Overall flux organization of E.coli metabolic network

a, Flux distribution for optimized biomass production

on succinate (black) and glutamate (red) substrates.

The solid line corresponds to the power-law fit

that a reaction has flux v

P(v) (v + v0)- , with v0 = 0.0003 and = 1.5.

d, The distribution of experimentally determined fluxes

from the central metabolism of E. coli shows

power-law behaviour as well, with a best fit to

P(v) v- with = 1.

Both computed and experimental flux distribution

show wide spectrum of fluxes.

Almaar et al., Nature 427, 839 (2004)



Use scaling behavior to determine local connectivity

The observed flux distribution is compatible with two different potential local

flux structures:

(a) a homogenous local organization would imply that all reactions producing

(consuming) a given metabolite have comparable fluxes

(b) a more delocalized „high-flux backbone (HFB)“ is expected if the local flux

organisation is heterogenous such that each metabolite has a dominant

source (consuming) reaction.

Schematic illustration of the hypothetical scenario in which

(a) all fluxes have comparable activity, in which case we expect kY(k) 1 and

(b) the majority of the flux is carried by a single incoming or outgoing reaction,

for which we should have kY(k) k . Almaar et al., Nature 427, 839 (2004)



Measuring the importance of individual reactions

To distinguish between these 2 schemes for each metabolite i produced

(consumed) by k reactions, define


2

11ˆ

ˆ,

k

jk

l ilv

ijv

ikY

where vij is the mass carried by reaction j which produces (consumes) metabolite i.

If all reactions producing (consuming) metabolite i have comparable vij

values, Y(k,i) scales as 1/k.

If, however, the activity of a single reaction dominates we expect

Y(k,i) 1 (independent of k).



Characterizing the local inhomogeneity of the flux net

a, Measured kY(k) shown as a function of k for

incoming and outgoing reactions, averaged over

all metabolites, indicates that Y(k) k-0.27.

Inset shows non-zero mass flows, v^ij, producing

(consuming) FAD on a glutamate-rich substrate.

an intermediate behavior is found between the

two extreme cases.

the large-scale inhomogeneity observed in the

overall flux distribution is also increasingly valid at

the level of the individual metabolites.

The more reactions that consume (produce) a

given metabolite, the more likely it is that a single

reaction carries most of the flux, see FAD.




Clean up metabolic network

Simple algorithm removes for each metabolite systematically all reactions

but the one providing the largest incoming (outgoing) flux distribution.

The algorithm uncovers the „high-flux-backbone“ of the metabolism,

a distinct structure of linked reactions that form a giant component

with a star-like topology.




FBA-optimized network on glutamate-rich substrateHigh-flux backbone for FBA-optimized metabolic

network of E. coli on a glutamate-rich substrate.

Metabolites (vertices) coloured blue have at least one

neighbour in common in glutamate- and succinate-rich

substrates, and those coloured red have none.

Reactions (lines) are coloured blue if they are identical

in glutamate- and succinate-rich substrates, green if a

different reaction connects the same neighbour pair,

and red if this is a new neighbour pair. Black dotted

lines indicate where the disconnected pathways, for

example, folate biosynthesis, would connect to the

cluster through a link that is not part of the HFB. Thus,

the red nodes and links highlight the predicted changes

in the HFB when shifting E. coli from glutamate- to

succinate-rich media. Dashed lines indicate links to the

biomass growth reaction.


(1) Pentose Phospate (11) Respiration

(2) Purine Biosynthesis (12) Glutamate Biosynthesis (20) Histidine Biosynthesis

(3) Aromatic Amino Acids (13) NAD Biosynthesis (21) Pyrimidine Biosynthesis

(4) Folate Biosynthesis (14) Threonine, Lysine and Methionine Biosynthesis

(5) Serine Biosynthesis (15) Branched Chain Amino Acid Biosynthesis

(6) Cysteine Biosynthesis (16) Spermidine Biosynthesis (22) Membrane Lipid Biosynthesis

(7) Riboflavin Biosynthesis (17) Salvage Pathways (23) Arginine Biosynthesis

(8) Vitamin B6 Biosynthesis (18) Murein Biosynthesis (24) Pyruvate Metabolism

(9) Coenzyme A Biosynthesis (19) Cell Envelope Biosynthesis (25) Glycolysis

(10) TCA Cycle



Interpretation

Only a few pathways appear disconnected indicating that although these pathways

are part of the HFB, their end product is only the second-most important source for

another HFB metabolite.

Groups of individual HFB reactions largely overlap with traditional biochemical

partitioning of cellular metabolism.




How sensitive is the HFB to changes in the environment?


b, Fluxes of individual

reactions for glutamate-rich

and succinate-rich conditions.

Reactions with negligible flux

changes follow the diagonal

(solid line).

Some reactions are turned off

in only one of the conditions

(shown close to the

coordinate axes). Reactions

belonging to the HFB are

indicated by black squares,

the rest are indicated by blue

dots. Reactions in which the

direction of the flux is

reversed are coloured green.

Only the reactions in the high-flux territory

undergo noticeable differences!

Type I: reactions turned on in one conditions and

off in the other (symbols).

Type II: reactions remain active but show an

orders-in-magnitude shift in flux under the two

different growth conditions.

Q: dicuss how E.coli respondsto a change of substrate fromglutamate to succinate anddraw the corres-ponding pointsinto the chart.



Flux distributions for individual reactions

Shown is the flux distribution for four selected

E. coli reactions in a 50% random environment.

a Triosphosphate isomerase;

b carbon dioxide transport;

c NAD kinase;

d guanosine kinase.

Reactions on the v curve (small fluxes)

have unimodal/gaussian distributions (a and

c). Shifts in growth-conditions only lead to small

changes of their flux values.

Reactions off this curve have multimodal

distributions (b and d), showing several

discrete flux values under diverse conditions.

Under different growth conditions they show

several discrete and distinct flux values.




Summary Metabolic network use is highly uneven (power-law distribution) at the global level

and at the level of the individual metabolites.

Whereas most metabolic reactions have low fluxes, the overall activity of the

metabolism is dominated by several reactions with very high fluxes.

E. coli responds to changes in growth conditions by reorganizing the rates of

selected fluxes predominantly within this high-flux backbone.

Apart from minor changes, the use of the other pathways remains unaltered.

These reorganizations result in large, discrete changes in the fluxes of the HFB

reactions.



The same authors as before used Flux Balance Analysis to examine utilization

and relative flux rate of each metabolite in a wide range of simulated

environmental conditions for E.coli, H. pylori and S. cerevisae:

consider in each case 30.000 randomly chosen combinations where each uptake

reaction is a assigned a random value between 0 and 20 mmol/g/h.

adaptation to different conditions occurs by 2 mechanisms:

(a) flux plasticity: changes in the fluxes of already active reactions.

E.g. changing from glucose- to succinate-rich conditions alters the flux of 264

E.coli reactions by more than 20%

(b) less commonly, adaptation includes structural plasticity, turning on

previously zero-flux reactions or switching off active pathways.



The two adaptation method mechanisms allow for the possibility of a group of

reactions not subject to structural plasticity being active under all environmental

conditions.

Assume that active reactions were randomly distributed.

If typically a q fraction of the metabolic reactions are active under a specific

growth condition,

we expect for n distinct conditions an overlap of at least qn reactions.

This converges quickly to 0.

Emergence of the Metabolic Core



However, as the number of conditions increases, the curve converges to a

constant enoted by the dashed line, identifying the metabolic core of an organism.

Red line : number of reactions that are always active if activity is randomly

distributed in the metabolic network. The fact that it converges to zero indicates

that the real core represents a collective network effect, forcing a group of

reactions to be active in all conditions.

Emergence of the Metabolic Core(A–C) The average relative size of the number of reactions that are always active as a function of the number of sampled conditions (blackline) for (A) H. pylori, (B) E. coli, and (C) S. cerevisiae.(D and E) The number of metabolic reactions (D) and the number of metabolic core reactions (E) in the three studied organisms.



The constantly active reactions form a tightly connected cluster!

All reactions that are found to be active in each of the 30,000 investigated external conditions are shown. Metabolites that contribute directly tobiomass formation are colored blue, while core reactions (links) catalyzed by essential (or nonessential) enzymes are colored red (or green).(Black-colored links denote enzymes with unknown deletion phenotype.) Blue dashed lines indicate multiple appearances of a metabolite, while links with arrows denote unidirectional reactions. Note that 20 out of the 51 metabolites necessary for biomass synthesis are not present in the core, indicating that they are produced (or consumed) in a growth-condition-specific manner. Blue and brown shading: folate and peptidoglycan biosynthesis pathways White numbered arrows denote current antibiotic targets inhibited by: (1) sulfonamides, (2) trimethoprim, (3) cycloserine, and (4) fosfomycin. A few reactions appear disconnected since we have omitted the drawing of cofactors.

Metabolic Core of E.coli



The metabolic cores contain 2 types of reactions:

(a) reactions that are essential for biomass production under all environment

conditions (81 of 90 in E.coli)

(b) reactions that assure optimal metabolic performance.

Metabolic Core Reactions



(A) The number of overlapping metabolic reactions in the

metabolic core of H. pylori, E. coli, and S. cerevisiae.

The metabolic cores of simple organisms (H. pylori and

E.coli) overlap to a large extent.

The largest organism (S.cerevisae) has a much larger

reaction network that allows more flexbility the relative

size of the metabolic core is much lower.

(B) The fraction of metabolic reactions catalyzed by

essential enzymes in the cores (black) and outside the

core in E. coli and S. cerevisiae.

Reactions of the metabolic core are mostly

essential ones.

(C) One could assume that the core represents a subset

of high-flux reactions. This is apparently not the case.

The distributions of average metabolic fluxes for the

core and the noncore reactions in E. coli are very

similar.

Characterizing the Metabolic Cores



- Adaptation to environmental conditions occurs via structural plasticity and/or flux

plasticity.

Here: identification of a surprisingly stable metabolic core of reactions that are

tightly connected to eachother.

- the reactions belonging to this core represent potential targets for antimicrobial

intervention.

Summary



Integrated Analysis of Metabolic and Regulatory Networksis omitted ...



V22 Modelling Dynamic Cellular Processes

John Tyson Bela Novak

Mathematical description of signalling

pathways helps answering questions like:

(1) How do the magnitudes of signal output

and signal duration depend on the kinetic

properties of pathway components?

(2) Can high signal amplification be coupled

with fast signaling?

(3) How are signaling pathways designed to

ensure that they are safely off in the absence

of stimulation, yet display high signal

amplification following receptor activation?

(4) How can different agonists stimulate the

same pathway in distinct ways to elicit a

sustained or a transient response, which can

have dramatically different consequences?

Heinrich et al. Mol. Cell. 9, 957 (2002)



Protein synthesis and degradation: linear response

RkSkkdt

dR210

S = signal strength (e.g. concentration of mRNA)

R = response magnitude (e.g. concentration of protein)

Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003)

A steady-state solution of a differential equation, dR/dt = f(R) is a constant Rss

that satisfies the algebraic equation f(Rss) = 0. In this case

2

10

k

SkkR

ss



Protein synthesis and degradation

RkSkkdt

dR210


Solid curve: rate of removal of the

response component R.

Dashed lines: rates of production of R

for various values of signal strength.

Filled circles: steady-state solutions

where rate of production and rate of

removal are identical.

Wiring diagram rate curve signal-response curve

Steady-state response R as a

function of signal strength S.

2

10

k

SkkR

ss

Parameters:k0 = 0.01k1 = 1k2 = 5



phosphorylation/dephosphorylation: hyperbolic response


TP

PPTP

PPPT

SRkSkkR

SRkSRkSRkRk

RkRkRRSk

112

1112

2210

A steady-state solution:

Skk

SRR T

ssP

12

,

PPT

P RkRRSkdt

dR21

RP: phosphorylated form of the response element

RP= [RP]

Pi: inorganic phosphate

RT = R + RP total concentration of R

Parameters:k1 = k2 = 1RT = 1

Q: the time-dependenceof RP is given.Derive the steady-stateconcentration RP,ss.



phosphorylation/dephosphorylation: sigmoidal response


Modification of (b) where the phosphorylation and

dephosphorylation are governed by Michaelis-

Menten kinetics:

Pm

P

PTm

PTP

RK

Rk

RRK

RRSk

dt

dR

2

2

1

1

PmPTPTmP

PTm

PT

Pm

P

Pm

P

PTm

PT

RKRRSkRRKRk

RRK

RRSk

RK

Rk

RK

Rk

RRK

RRSk

2112

1

1

2

2

2

2

1

10

steady-state solution of RP The biophysically acceptable solutions

must be in the range 0 < RP < RT:

T

m

T

m

T

ssP

R

K

R

KkSkG

R

R21

21

, ,,,

with the „Goldbeter-Koshland“ function G:

uKuvuKvJuvuKvJuv

uK

KJvuG

4

2

),,,(

2



phosphorylation/dephosphorylation: sigmoidal response

This mechanism creates a switch-like signal-response curve which is

called zero-order ultrasensitivity.

(a), (b), and (c) give „graded“ and reversible behavior of R and S.

„graded“: R increases continuously with S

reversible: if S is change from Sinitial to Sfinal , the response at Sfinal is the same

whether the signal is being increased (Sinitial < Sfinal) or decreased (Sinitial > Sfinal).

Element behaves like a buzzer: to activate the response, one must push hard

enough on the button.Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003)

Parameters:k1 = k2 = 1RT = 1Km1 = Km2 = 0.05



perfect adaptation: sniffer


Here, the simple linear response element is supplemented

with a second signaling pathway through species X.

XkSkdt

dX

RXkSkdt

dR

43

21

32

41

2

1

4

3

kk

kk

Xk

SkR

k

SkX

ss

ss

The response mechanism exhibits perfect adaptation to the signal:

although the signaling pathway exhibits a transient response to changes in signal

strength, its steady-state response Rss is independent of S.

Such behavior is typical of chemotactic systems, which respond to an abrupt

change in attractants or repellents, but then adapt to a constant level of the signal.

Our own sense of smell operates this way we call this element „sniffer“.



perfect adaptation

Right panel: transient response (R, black) as a function of stepwise increases

in signal strength S (red) with concomitant changes in the indirect signaling

pathway X (green).

The signal influences the response via two parallel pathways that push the

response in opposite directions. This is an example of feed-forward control.

Alternatively, some component of a response pathway may feed back on the

signal (positive, negative, or mixed).

Tyson et al., Curr.Opin.Cell.Biol. 15, 221 (2003)

Parameters:k1 = k2 = 2k3 = k4 = 1



Positive feedback: Mutual activation


E: a protein involved with R

EP: phosphorylated form of E

Here, R activates E by phosphorylation,

and EP enhances the synthesis of R.

4343

210

,,, JJkRkGRE

RkSkREkdt

dR

P

P



mutual activation: one-way switch

As S increases, the response is low until S exceeds a critical value Scrit at which

point the response increases abruptly to a high value.

Then, if S decreases, the response stays high.

between 0 and Scrit, the control system is „bistable“ – it has two stable

steady-state response values (on the upper and lower branches, the solid lines)

separated by an unstable steady state (on the intermediate branch, the dashed

line).

This is called a one-parameter bifurcation. Tyson et al., Curr.Opin.Cell.Biol. 15, 221 (2003)



Negative feedback: homeostasis


In negative feedback, the response counteracts the effect of the stimulus.

Here, the response element R inhibits the enzyme E catalyzing its synthesis.

Therefore, the rate of production of R is a sigmoidal decreasing function of R.

4343

20

,,, JJRkkGRE

RSkREkdt

dR

Negative feedback in a two-component

system X R | X can also exhibit

damped oscillations to a stable steady state

but not sustained oscillations.



Negative feedback: oscillatory response


There are two ways to close the negative feedback loop:

(1) RP inhibits the synthesis of X

(2) RP activates the degradation of X.

Sustained oscillations require at least 3 components:

X Y R |X

Left: example for a negative-feedback control loop.

Pm

P

PTm

PTPP

Pm

P

PTm

PTP

P

RK

Rk

RRK

RRYk

dt

dR

YK

Yk

YYK

YYXk

dt

dY

XRkXkSkkdt

dX

6

6

5

5

4

4

3

3

'

2210



Negative feedback: oscillatory response

Feedback loop leads to

oscillations of X (black),

YP (red), and RP (blue).


Within the range Scrit1 < S

< Scrit2, the steady-state

response RP,ss is unstable.

Within this range, RP(t)

oscillates between RPmin

and RPmax.

Again, Scrit1 and Scrit2 are bifurcation points.

The oscillations arise by a generic mechanism

called „Hopf bifurcation“.

Negative feedback has ben proposed as a basis for

oscillations in protein synthesis, MPF activity, MAPK

signaling pathways, and circadian rhythms.

Q: set up the 3 ODEs to describe the time-dependentbehavior of X, YP, and RP.How does the response Rfollow the signal S?



Positive and negative feedback: Activator-inhibitor oscillations

R is created in an autocatalytic

process, and then promotes the

production of an inhibitor X,

which speeds up R removal.


4343

65

'

2210

,,, JJkRkGRE

XkRkdt

dX

RXkRkSkREkdt

dR

P

P

The classic example of such a system is cyclic AMP production in the slime mold. External cAMP binds to a surface receptor, which stimulates adenylate cyclase to produce and excrete more cAMP. At the same time, cAMP-bindingpushes the receptor into an inactive form. After cAMP falls off, the inactive form slowly recovers its ability tobind cAMP and stimulate adenylate cyclase again.



V23 Stochastic simulations of cellular signalling

Traditional computational approach to chemical/biochemical kinetics:

(a) start with a set of coupled ODEs (reaction rate equations) that describe the

time-dependent concentration of chemical species,

(b) use some integrator to calculate the concentrations as a function of time given

the rate constants and a set of initial concentrations.

Successful applications : studies of yeast cell cycle, metabolic engineering,

whole-cell scale models of metabolic pathways (E-cell), ...

Major problem: cellular processes occur in very small volumes and

frequently involve very small number of molecules.

E.g. in gene expression processes a few TF molecules may interact with a

single gene regulatory region.

E.coli cells contain on average only 10 molecules of Lac repressor.



Include stochastic effects

(Consequence1) modeling of reactions as continuous fluxes of matter is no

longer correct.

(Consequence2) Significant stochastic fluctuations occur.

To study the stochastic effects in biochemical reactions stochastic formulations of

chemical kinetics and Monte Carlo computer simulations have been used.

Daniel Gillespie (J Comput Phys 22, 403 (1976); J Chem Phys 81, 2340 (1977))

introduced the exact Dynamic Monte Carlo (DMC) method that connects the

traditional chemical kinetics and stochastic approaches.

Assuming that the system is well mixed, the rate constants appearing in these two

methods are related.



Dynamic Monte Carlo

In the usual implementation of DMC for kinetic simulations, each reaction is

considered as an event and each event has an associated probability of occurring.

The probability P(Ei) that a certain chemical reaction Ei takes place in a given time

interval t is proportional to an effective rate constant k and to the number of

chemical species that can take part in that event.

E.g. the probability of the first-order reaction

X Y + Z

would be k1Nx with Nx :number of species X, and

k1 : rate constant of the reaction

Similarly, the probability of the reverse second-order reaction

Y + Z X

would be k2NYNZ.

Resat et al., J.Phys.Chem. B 105, 11026 (2001)



Dynamic Monte Carlo

As the method is a probabilistic approach based on „events“, „reactions“ included in

the DMC simulations do not have to be solely chemical reactions.

Any process that can be associated with a probability can be included as an event

in the DMC simulations.

E.g. a substrate attaching to a solid surface can initiate a series of chemical

reactions.

One can split the modelling into the physical events of substrate arrival, of

attaching the substrate, followed by the chemical reaction steps.




Basic outline of the direct method of Gillespie

(Step i) generate a list of the components/species and define the initial distribution

at time t = 0.

(Step ii) generate a list of possible events Ei (chemical reactions as well as

physical processes).

(Step iii) using the current component/species distribution, prepare a probability

table P(Ei) of all the events that can take place.

Compute the total probability

P(Ei) : probability of event Ei .

(Step iv) Pick two random numbers r1 and r2 [0...1] to decide which event E will

occur next and the amount of time by which E occurs later since the most recent

event.


)(itotEPP



Basic outline of the direct method of Gillespie

Using the random number r1 and the probability table,

the event E is determined by finding the event that satisfies the relation


1

1

11

ii

itoti

EPPrEP

The second random number r2 is used to obtain the amount of time between the

reactions

2ln

1r

Ptot

As the total probability of the events changes in time, the time step between

occurring steps varies.

Steps (iii) and (iv) are repeated at each step of the simulation.

The necessary number of runs depends on the inherent noise of the system and on

the desired statistical accuracy.

weighted sampling (PW-DMC) is omitted.

24. lecture ws 2005/06bioinformatics iii1 v17 metabolic networks - introduction different levels for...

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