20. Lecture WS 2005/06Bioinformatics III1 V20 The Double Description method: Theoretical framework...

20. Lecture WS 2005/06

Bioinformatics III 1

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



Review - BackgroundMetabolic pathway analysis has been recognized as a central approach for the

structural analysis of metabolic networks.

The concepts of elementary (flux) modes and extreme pathways provide two

rigorous formalisms to describe and assess pathways and have proven to be

valuable for many applications.

However, computing elementary modes is a hard computational task.

What is the basis behind these approaches?



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 conesThis geometry can be intuitively understood, noting that there are certainly

'enough' intersecting half-spaces (given by the inequalities vi ≥ 0) to have this

'pointed' effect in 0:

P contains no real line (otherwise there coexist x and -x ≠ 0 in P,

a contradiction with the constraint vi ≥ 0).

The figure suggests that a pointed polyhedral cone can be either defined in an

implicit way, by the set of constraints as we did until now, or in an explicit or

generative way, by its 'edges', the so-called extreme rays (or generating vectors)

that unambiguously define its boundaries.

In the following, we show that elementary modes always correspond to extreme

rays of a particular pointed cone as defined in (3) and that their computation

therefore matches to the so-called extreme ray enumeration problem,

i.e. the problem of enumerating all extreme rays of a pointed polyhedral cone

defined by its constraints.

!



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: P is the solution set of the linear inequalities defined by

where I is the q × q identity matrix.

Since it contains I, A is full rank and therefore P is a pointed polyhedral cone.

All v P obey S v = 0, thus the 2m first inequalities defined by A hold with

equality for all vectors in P and the inclusion condition of Definition 1 can be

restricted to the last q inequalities, i.e. the inequalities corresponding to the

reactions.

Inclusion over the zero set can be equivalently seen as containment over the set of

non-zeros in v, i.e. R(v).

Consequently, e P is an extreme ray of P if and only if:

for all e' P : R(e') R(e) e' = 0 or e' ≃ e, i.e. if and only if e is elementary.

Thus, all three conditions in (2) are fulfilled. □

TTT ISS

I

S

S

A



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 of Theorem 1We prove first that each case a) and b) defines EMs of T'.

Let e' be a flux vector defined by either case a) or b). Clearly S'e' = 0 and e' ≥ 0.

In case b) e' is not elementary only if the single forward or backward reaction

balances all internal metabolites, i.e. if the reaction does not include any internal

species. We can safely exclude this pathologic case by considering that S does not

contain a null column. Therefore, e' is elementary.

In case a), assume e' is not elementary, i.e. there exists a non-null flux vector

x' of T' not equivalent to e' such that x' ≥ 0, S'x' = 0 and R(x') R(e').

By definition of the reconfiguration, for each i Rev, at least one among

e'(i,+1) or e'(i,-1) equals zero and this holds consequently also for x'.

Thus one can define e and x, the back-configurations of e' and x'.

Now, by definition, e is an EM of T and is not equivalent to x,

Sx = 0, xi ≥ 0i for i Irrev and R(x) R(e), a contradiction.



Proof - continued

Hence each case a) and b) defines EMs of T'.

We prove now that there is no other case.

Assume there exists e' neither defined by a) nor b), such that e' ≥ 0, S'e' = 0 and

e' elementary.

For each i Rev at least one among e'(i,+1) and e'(i,-1) equals zero

(otherwise the two-cycle defined on reaction i would satisfy the constraints and

involve only a subset of the reactions of e').

Thus the back-configuration e of e' can be defined.

By definition, e is not an EM of T.

There exists x not equivalent to e such that S x = 0, xi ≥ 0i for i Irrev and

R(x) R(e).

The reconfiguration x' of x is such that

x' is not equivalent to e', x' ≥ 0, S'x' = 0 and R(x') R(e'), a contradiction. □



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.



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-

Proof Let P = P(AK+i) and let P‘ be the cone generated by the matrix R‘.

We must prove that P = P‘.

By the construction, we have rjj‘ P for all (j,j‘) J+ J- and P‘ P is clear.

Let x P. We shall show that x P‘ and hence P P‘.

Since x P, x is a nonnegative combination of rj‘s over j J, i.e. there exist

j 0 for j J such that

Jj

jjrx



Geometric version of iteration stepIf there is no positive j with j J- in the expression above then x P‘.

Suppose there is some k J- with k > 0.

Since x P, we have Aix 0. This together with (5) implies that there is a least

one h J+ with h > 0. Now by construction, hk J‘ and

rhk = (Ai rh) rk – (Airk) rh. (6)

By subtracting an appropriate positive multiple of (6) from (5), we obtain an

expression of x as a positive combination of some vectors rj (j J‘ ) with new

coefficients j where the number of positive j‘s with j J+ J- is strictly smaller

than in the first expression.

As long as there is j J- with positive j, we can apply the same transformation.

Thus we must find in a finite number of steps an expression of x without using rj

such that j J‘.

This proves x P‘, and hence P P‘. □



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



Application to central metabolism of E. coli

Redundancy removal and network compression during pre-processing results in

much smaller networks.

Using a reduction of the stochiometric matrix (entries 0 and 1) allows very fast

computation of even complex networks using a binary approach.

20. Lecture WS 2005/06Bioinformatics III1 V20 The Double Description method: Theoretical framework...

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