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BIOINFORMATICS

expa: a program for calculating extreme

pathways in biochemical reaction networks

Steven L. Bell, Bernhard . Palsson

Department of Bioengineering, University of California, San Diego, La Jolla, CA92093, USA

ABSTRACT

Summary: The set of extreme pathways, a generating set

for all possible steady state flux maps in a biochemical reac-

tion network, can be computed from the stoichiometric matrix,

an incidence-like matrix reflecting the network topology. Here,

we describe the implementation of a well-known algorithm to

compute these pathways and give a summary of the features

of the available software.

Availability: The C-code, along with a Windows executa-

ble and sample network reaction files, are available at

http://systemsbiology.ucsd.edu

Contact: palsson@ucsd.edu

1 INTRODUCTION

The abundance of genomic data available today allows for

construction of genome-scale metabolic networks for many

organisms. The topology of the type of networks consideredhere is determined by an m n stoichiometric matrix, S,whose rows and columns represent the systems metabolites

and reactions, respectively. The dynamics of the system is

given by x(t) = Sv, where x is the m-dimensional vector ofmetabolite concentrations, denotes time-derivative, andv is a vector of fluxes which we assume is independent of

concentrations and time.

Under the assumption that the system is in steady-state,

we have that Sv = 0, and to obtain biologically feasiblesolutions to this equation, we also impose the condition that

v 0 (reversible reactions are split into two irreversible

reactions one for each direction). The set of solutions satis-fying these constraints is a so-called convex cone which can

be generated by a finite (and unique up to a multiple) number

of vectors, i.e., each biologically feasible flux vector (when

the system is in steady state) can be expressed as a non-

negative linear combination of these extreme pathways (5).

The extreme pathways are the edges of the convex cone, or

more precisely, they are conically independent, i.e., no such

vector can be expressed as a non-negative linear combination

of any other vectors in the cone. The properties and uses of

extreme pathways have been recently reviewed (2).

to whom correspondence should be addressed

2 DESCRIPTION OF THE ALGORITHM

Given a metabolic network, where the metabolites are repre-

sented by the nodes and the edges represent the associated

reactions, we compute the extreme pathways using an algo-rithm presented in (5) (see also (6)). The algorithm uses

matrix operations similar to those used in the well-known

Gaussian elimination algorithm.

For the sake of brevity, we here give a simplified version of

the extreme pathway algorithm. The algorithm may be des-

cribed by a sequence of tableaux T0, T1, . . . , T m, where theinitial tableau is given by T0 = [I S ], and the final tableauTm = [P 0 ]. Here, I is the n n identity matrix, primedenotes transpose, P is a matrix with n columns whose rowsare the extreme pathways, and, 0 is a matrix with m columnsand all entries equal to zero (determining the number of rows

in the final tableau is an open problem). Converting the right

hand matrix S to the zero matrix is done column by columnusing elementary row operations, each tableaux correspon-

ding to a column, as follows. For each 1 i m, thetableau Ti is obtained from Ti1 by first choosing a pivo-ting column of the right hand matrix (originating from S)

to zero out, column j, say. Suppose there are pos positive,neg negative, and z zero elements in column j. First, the zrows ofTi1 containing a zero in column j are copied to Ti.Then each of the pos rows is combined (using an elemen-

tary row operation) with each of the neg rows to produce a

zero in column j of Ti. More precisely, ifTi1s,j > 0 and

Ti1t,j < 0 for some s and t, then |Ti1t,j |T

i1s + |T

i1s,j |T

i1t

is the new row to be added to Ti. (Here, Ti1s denotes the sth

row, Ti1s,j is the (s, j)-element in the tableau Ti1, and |x| is

the absolute value ofx.) Finally, only rows which are coni-cally independent are retained in Ti: for any two rows x, y, ifA(x) A(y), then row x is deleted from the tableau, whereA(x) = {i : xi = 0}, the indices of the zero components ofx. Hence, the number of rows in Ti is at most z+ neg pos.

3 IMPLEMENTATION

The tableaux are implemented as matrices (two-dimensional

arrays) using pointers to pointers as described in Appendix B

Bioinfor matics Oxford University Press 2004; all rights reserved.

Bioinformatics Advance Access published December 21, 2004

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Bell and Palsson

of (4). This means that the rows of the matrices are not neces-

sarily stored in contiguous locations in memory (since rows

are added and deleted each iteration, it would be inefficient

to store the matrices in continuous chunks of memory). For

each iteration i = 1, 2, . . . ,m, two matrices are used, onecontaining the tableau from the previous iteration, Ti1, andthe other for constructing the next tableau, Ti. These matri-ces are in sparse form, i.e., only the non-zero elements are

stored in memory. The column indices corresponding to the

non-zero elements in each row of the matrices are accoun-

ted for by bit map representations of the matrices (similiar to

the ones used in (3)). These bitmaps are also implemented

as matrices of the type described above, but here each row

consists ofw words, where w = (m + n)/sizeof(word),i.e., each row is represented by a pointer which points to a

location in memory of size w words.

For the following, assume (for simplicity) that the expres-sion inside the ceiling operator of the definition ofw is aninteger. The conical independence check is done with AND

logical bit operations using bit representations of the rows

(ifx and y are bit rows and ( x[i]) & y[i] is nonzero, forsome 0 i w 1 then A(x) A(y), where deno-tes bit negation). Each of the candidate pos neg rows (if and are rows of opposite signs, then a new row, , isconstructed by doing = | , where | denotes bitwiseOR and the operation is performed word by word) is checked

against the existing rows of the next bit matrix, and conditio-

nal on the outcome of the test, its bit representation is added

to the bit array and a corresponding sparse row is added to

the next sparse matrix (at the start of the iteration the nextmatrix consist of the z zero rows determined by the current

column). The pivoting column chosen in each iteration is one

whose quantity pos neg is a minimum (of the columns notyet processed). This choice minimizes the amount of work

done when constructing the next tableau and may be thought

of as a local (or greedy) optimization strategy (an intere-

sting open problem is how to choose pivoting columns based

on some global optimization criterion see the poster from

RECOMB04 on our web page for some numerical compari-

sons of different schemes for picking columns). The conical

independence check described above is by far the most com-

putationally intensive part of the algorithm. To descrease thenumber of checks necessary, it may be possible to partition

the rows into equivalence classes based on the zero index sets

A() so that only a subset of the neg pos rows need bechecked for a particular candidate row (3). Such a partition

is, in general, network dependendent, and the number of clas-

ses must be large enough to outweigh the additional expense

of implementing the partitioning scheme. We have, as of yet,

not found such a scheme.

There have been other implementations of the extreme

pathway algorithm described above (see for example (1) and

(3), but to our knowledge none where the source-code has

been freely available.

4 FEATURES

The open-source software comes with a command line inter-

face, where the user is provided with input options and helpmenu by typing expa with no arguments or expa -h,

respectively. To calculate the extreme pathways, the user has

the option of specifying the network topology in the form

of the stoichiometric matrix or the corresponding metabolic

reactions.

The matrix file is an ascii file where each row of the matrix

constitutes a line (ended by a new line character) and each

matrix entry is separated by white space. In addition, the user

must supply the dimensions of the matrix and the number and

types of the so-called exchange fluxes (see (5)). Being able to

use the stoichimetric matrix as input is useful if preprocessing

of the matrix is desired (for example, permuting or removingcolumns).

The reaction file is also an ascii file and contains all the

reactions in the metabolic network, one on each line of the

file. The exact form of the entries is described in the README

file on our website, and several sample files are provided

there as well.

The extreme pathways are output to a file named

Paths.txt in matrix form, where each row is a pathway.

ACKNOWLEDGEMENT

Financial support for this work was provided by a grant from

the National Institute of Health (GM68837).

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