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
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2. Review - Background
Metabolic 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?
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3. Definition of Elementary Modes
Compared 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.
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4. 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.
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5. 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.
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6. A unified framework - Elementary modes as extreme rays in networks of irreversible reactions
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 pointed polyhedral cone. Dashed lines represent virtual cuts of unbounded areas
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7. Pointed polyhedral cones
This 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. !
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8. 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}
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9. 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.
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10. Extreme rays - definition
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.
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11. Extreme rays - properties
An 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).
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12. 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 □
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13. The general case
In 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.
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14. Notations
We 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
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15. 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:
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16. Theorem 1: EMs in original and in reconfigured networks
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.
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17. Proof of Theorem 1
We 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,
Sx = 0, xi ≥ 0i for i  Irrev and R(x)  R(e), a contradiction.
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18. 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. □
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19. 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.
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20. The Double Description Method
A 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.
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21. The Double Description Method
Theorem 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).
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22. The Double Description Method
Task: 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.
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23. Double Description Method: primitive form
Suppose 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).
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24. Geometric version of iteration step
The 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.
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25. Geometric version of iteration step
The newly introduced inequality Aix  0 partitions the space d into three parts:
Hi+ = {x  d : Aix > 0 }
Hi0 = {x  d : Aix = 0 }
Hi- = {x  d : Aix < 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.
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26. Geometric version of iteration step
The 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‘ = (Airj)rj‘– (Airj‘)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
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27. Geometric version of iteration step
If 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‘. □
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28. Finding seed DD pair
It 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 AKx  0  x = AK-1,   0.
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29. Primitive algorithm for DoubleDescriptionMethod
Hence 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.
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30. Towards the standard implementation
Proposition 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.
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31. Towards the standard implementation
Two 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).
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32. Algorithm for standard form of double description method
Hence we can write a straightforward variation of the DD method which produces a minimal generating set for P:
DDMethodStandard(A)
such that R is minimal
Lemma 8
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.
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33. 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.
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