Constraint-Based Analysis of Gene Deletion in a Metabolic Network

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1 Constraint-Basd Analysis of Gn Dltion in a Mtabolic Ntwork Abdlhalim Larhlimi and Alxandr Bockmayr DFG-Rsarch Cntr Mathon, FB Mathmatik und Informatik, Fri Univrsität Brlin, Arnimall, 3, Brlin, Grmany -mail: larhlimi@mi.fu-brlin.d,bockmayr@mi.fu-brlin.d Abstract. Constraint-basd modls of mtabolic ntworks us govrning constraints to rstrict potntial cllular bhavior. Th rang of possibl bhaviors, which is mathmatically dscribd by th stady-stat flux con, can b altrd by gn dltion. Svral optimization-basd approachs hav bn proposd to find, in th altrd ntwork, bhaviors optimizing a particular ntwork function. Hr, w prsnt a constraintbasd approach that allows analyzing th changs in th ovrall capabilitis of a mtabolic ntwork following a gn dltion. Furthrmor, w stablish a rlationship btwn th altrd flux con and th rvrsibility typ of th raction associatd with th dltd gn. 1 Introduction Constraint-basd modls hav bcom a fundamntal tool to study gnom-scal mtabolic ntworks [10]. Such modls us govrning constraints to rstrict potntial cllular bhavior. Th rang of all possibl bhaviors, which is mathmatically dscribd by th stady-stat flux con, can b altrd by gn dltion. Svral optimization-basd approachs hav bn dvlopd that allow computing, in th altrd ntwork, bhaviors optimizing a particular ntwork function [2]. Mathmatically, this rquirs dfining a hypothtical objctiv function. Howvr, although th assumption of optimality for a wild-typ biological systm is justifiabl, th sam assumption may not b valid for studying an altrd systm [15]. Furthrmor, ths approachs considr only optimal stats with rspct to th prdfind objctiv function. Ths particular stats form a rstrictd subst of all possibl bhaviors of th altrd systm. Hnc, ths optimization-basd approachs loos information about how th achivabl bhaviors of th ntwork could chang following a gn dltion. In this papr, w analyz th changs in th ovrall capabilitis of a mtabolic ntwork causd by gn dltion. In particular, w show how to obtain in a constraint-basd approach a dscription of th altrd stady-stat flux con. Th analysis is basd on a rfind classification of ractions. Th organization of this papr is as follows. In Sct. 2, w rcall som basic facts about mtabolic ntwork analysis and prsnt th notions of minimal mtabolic bhavior and rvrsibl mtabolic spac. This lads to a rfind classification of ractions. In Sct. 3 w us this for a constraint-basd analysis of gn dltion.

2 Systm boundary A 3 B Fig. 1. Ntwork ILLUSNET with th corrsponding lmntary mods. 2 Mtabolic ntwork analysis In th contxt of mtabolic ntwork analysis, mtabolic systms ar assumd to oprat at stady stat such that for all intrnal mtabolits th flux is balancd. In addition, th flux through ach irrvrsibl raction must b non-ngativ. Fluxs through rvrsibl ractions ar not rstrictd with rspct to thir sign. Th st of all possibl flux distributions ovr th ntwork at stady stat dfins th (stady-stat) flux con [10] C = {v R n Sv = 0, v i 0, for all i Irr}, (1) whr S is th m n stoichiomtric matrix of th ntwork, with m intrnal mtabolits (rows) and n ractions (columns), and th vctor v R n givs a flux distribution. Irr {1,...,n} dnots th st of irrvrsibl ractions in th ntwork, and Rv = {1,..., n} \ Irr th st of rvrsibl ractions. Exampl 1. For illustration w considr th hypothtical ntwork ILLUSNET dpictd in Fig. 1. It consists of two mtabolits (A,B), and fiv ractions {1,...,5}. Th flux con is dfind by C = {v R 5 Sv = 0,v i 0, for all i Irr}, with th st of irrvrsibl ractions Irr = {1, 3}, and th stoichiomtric matrix ( ) S = Th flux con contains th full rang of achivabl bhaviors of th mtabolic ntwork at stady stat. Hnc, it is of grat intrst to dscrib this con in a mathmatically and biologically maningful way. Thr ar two mathmatical ways for dscribing th flux con. Th first is an innr dscription basd on a st of gnrating vctors that span th con. Th scond is an outr dscription basd on sts of constraints, which givs a tst for dtrmining whthr a givn flux vctor blongs to th con [1]. Th concpt of lmntary mod (EM) [13, 14] has bn proposd to charactriz th flux con using an innr dscription. An EM corrsponds to a flux

3 distribution v C \ {0} involving a minimum st of ractions, i.., th st S c (v) = {i Rv Irr v i 0} is minimal. Evry possibl flux distribution is thn a non-ngativ combination of lmntary mods. In contrast to this approach, w propos in [7, 9] an outr dscription of th flux con, basd on sts of non-ngativity constraints. This approach dfins a mtabolic bhavior as a st of irrvrsibl ractions D Irr,D, such that thr xists a flux distribution v C with D = {i Irr v i 0}. A mtabolic bhavior D is minimal (MMB), if thr is no mtabolic bhavior D D strictly containd in D. Th st of flux distributions involving only rvrsibl ractions dfins th rvrsibl mtabolic spac (RMS), RMS = {v C v i = 0, for all i Irr}, (2) which corrsponds to th linality spac of th flux con C [12]. Th dimnsion t of th rvrsibl mtabolic spac, by dfinition, is qual to th dimnsion of th linality spac of C, which is a linar subspac of R n. Th minimal mtabolic bhaviors (MMBs) ar closly rlatd to th minimal propr facs of th flux con C, i.., th facs of dimnsion t + 1 [12]. According to [7, 9], ach minimal propr fac is dscribd by its charactristic st D = {j Irr v j > 0, for som v G}. Indd, G is givn by G = {v C v i = 0, for all i Irr \ D}. (3) Th charactristic st D is uniquly dtrmind by G and all ractions from D ar proportional to ach othr. Th nxt thorm shows that th MMBs ar in a 1-1 corrspondnc with (th charactristic sts of) th minimal propr facs of th flux con C. Thorm 1 ([9]). Lt D Irr b a st of irrvrsibl ractions. Thn, th following ar quivalnt: D is a minimal mtabolic bhavior. D is th charactristic st of a minimal propr fac of th flux con. If G 1,...,G s ar th minimal propr facs of th flux con C, th corrsponding MMBs D 1,...,D s togthr with th RMS compltly dfin C, s [7, 9] for additional dtails. Exampl 2. In th mtabolic ntwork from Fig. 1, thr ar six lmntary mods { 1,..., 6 }. Th MMBs, th corrsponding minimal propr facs and th RMS ar th following: D 1 = {1}, D 2 = {3}, G 1 = {v C v 1 0, v 3 = 0}, G 2 = {v C v 3 0, v 1 = 0}, RMS = {v C v 1 = 0 and v 3 = 0} Fig. 2(a) shows a 3D illustration of th flux con C. Th RMS is a lin gnratd by th flux distribution v = (0,0,0,1, 1), i.., RMS = {λ(0,0,0,1, 1) λ R}. Th minimal propr facs G 1 and G 2 ar two half-plans.

4 Not that minimal mtabolic bhaviors satisfy a simplicity condition similar to th on that holds for lmntary mods. Furthrmor, for ach MMB D, thr xists at last on EM involving xactly th irrvrsibl ractions from D, i.., D = {i Irr i 0}. Th numbr of MMBs is typically much smallr than th numbr of EMs. For instanc, th cntral carbon mtabolism of E. coli contains mor than lmntary mods [5], but only 3560 minimal mtabolic bhaviors. Th RMS for this mtabolism is rducd to th origin, i.., RMS = {0}. Th MMBs and th RMS of a mtabolic ntwork can b dtrmind from a st of gnrators of th flux con C. Softwar packags for polyhdral computations, such as cdd [3], allow for computing ths gnrators. Th numbr of gnrators of C may b xponntial in th siz of th inquality dscription in Equation (1). For mor about complxity issus, w rfr to [4]. Basd on th concpts of MMBs and th RMS, a rfind classification of ractions has bn proposd [7]. A rvrsibl raction j Rv is calld psudoirrvrsibl if v j = 0, for all v RMS. A rvrsibl raction that is not psudoirrvrsibl is calld fully rvrsibl. Insid ach minimal propr fac, th irrvrsibl and th psudo-irrvrsibl ractions tak a uniqu dirction. Mor prcisly, w hav th following proprtis. Thorm 2 ([7]). Lt G b a minimal propr fac of th flux con C and lt j {1,...,n} b a raction. If j Irr is irrvrsibl, thn v j > 0, for all v G \ RMS, or v j = 0, for all v G. Furthrmor, v j = 0, for all v RMS. If j Rv is psudo-irrvrsibl, thn th flux v j through j has a uniqu sign in G \ RMS, i.., ithr v j > 0, for all v G \ RMS, or v j = 0, for all v G \ RMS, or v j < 0, for all v G \ RMS. For all v RMS, w hav again v j = 0. If j Rv is fully rvrsibl, thr xists v RMS such that v j 0. W can thn find pathways v +,v,v 0 G \ RMS with v + j > 0, v j < 0 and v0 j = 0. Exampl 3. In th ILLUSNET ntwork, raction 2 is psudo-irrvrsibl, whil ractions 4 and 5 ar fully rvrsibl. In th contxt of th minimal propr fac G 1, th psudo-irrvrsibl raction 2 oprats only in th forward dirction, i.., v 2 > 0 for all v G 1 \ RMS, whil it oprats in th backward dirction in th contxt of th fac G 2. In th following, rmoving a raction mans that th flux through this raction is constraind to zro. Basd on th rfind raction classification, w studid in [8] th consquncs of rmoving a raction in trms of th capabilitis of th rmaining ractions in th ntwork. A zro flux through som raction may imply a zro flux through many othr ractions. It has bn shown that th rvrsibility proprty is an important ky to lucidat intractions btwn ractions. For instanc, th rmoval of a (psudo-)irrvrsibl raction has no ffct on th fully rvrsibl ractions, and all ractions in an nzym subst [11] must hav th sam rvrsibility typ (irrvrsibl, psudo-irrvrsibl, or fully rvrsibl).

5 In th following, w ar intrstd in gn dltion. Mor prcisly, w study how th flux con is changd if a raction associatd with th dltd gn is rmovd. This raction will b calld th targt raction. Basd on th mathmatical rsults abov, th following sction stablishs a rlationship btwn th changs in th flux con and th rvrsibility typ of th targt raction. 3 Constraint-basd analysis of gn dltion Lt τ {1,...,n} b th targt raction associatd with th dltd gn. To simulat gn dltion, w constrain th flux through raction τ to zro. This lads to th altrd flux con C = {v R n Sv = 0, v τ = 0, v i 0, for all i Irr} (4) which contains all possibl stady-stat flux distributions ovr th altrd ntwork. Th altrd flux con C can b dscribd using an xisting dscription of th flux con C. To do this, th lmntary mod approach taks advantag of th consrvation proprty of EMs: if th flux through a raction is constraind to zro, th st of EMs of th altrd ntwork is th st of all EMs which do not involv this raction [6, 14]. Exampl 4. In th ILLUSNET ntwork, if w constrain th flux through raction 4 to zro, th lmntary mods of th altrd flux con C ar 1, 5 and 6, which do not involv raction 4. To dscrib th mtabolic ntwork aftr a gn dltion, th EM approach dos not xplicitly tak into account th rvrsibility typ of th targt raction. Howvr, this bcoms possibl by using minimal mtabolic bhaviors and th rvrsibl mtabolic spac. Mathmatically, th con C dfind in Equation (4) is also givn by C = C {v R n v τ = 0}. Thrfor, w may dduc an outr dscription of th altrd con C from an outr dscription of th original con C. This dduction dpnds on th rvrsibility typ of th targt raction τ. 3.1 Rmoving an irrvrsibl raction In analogy with EMs, thr is th following consrvation proprty for MMBs: if th flux through an irrvrsibl raction is constraind to zro, th st of MMBs of th nw ntwork is th st of all MMBs which do not involv this raction. Hnc, if raction τ is irrvrsibl, this consrvation proprty guarants that th MMBs of C ar xactly th MMBs D of C for which j D. Th rvrsibl mtabolic spac dos not chang, i.., RMS = RMS.

6 3.2 Rmoving a psudo-irrvrsibl raction In this cas, only MMBs in which raction τ is not involvd ar kpt, and nw ons ar gnratd. Th gnration of nw MMBs rlis on th adjacncy proprty of th minimal propr facs of th con C [3]. Indd, considr th hyprplan H = {v R n v τ = 0}. Lt H + = {v R n v τ > 0} (rsp. H = {v R n v τ < 0}) b th positiv (rsp. ngativ) half-spac supportd by th hyprplan H. Thn H partitions th st of minimal propr facs of C into thr parts: th st J + of positiv minimal propr facs G for which G \RMS H +, th st J of ngativ minimal propr facs G for which G \ RMS H, and th st J 0 of zro minimal propr facs containd in H. Th nw minimal propr facs of C ar obtaind by combining a positiv and a ngativ minimal propr fac that ar containd in a (t + 2) dimnsional fac of th flux con C [3]. Again, t dnots th dimnsion of th RMS. Sinc only positiv combinations ar prformd, all irrvrsibl ractions dfining a minimal propr fac G 1 will dfin a nw fac G if th lattr is obtaind by combining G 1 with anothr minimal propr fac G 2. Th MMB D associatd with G is thn th union of th MMBs D 1 and D 2 associatd with G 1 and G 2, rspctivly. Thrfor, th nw MMBs of th con C can b computd by (a) idntifying positiv and ngativ MMBs associatd with th positiv and ngativ minimal propr facs of C, (b) computing all possibl unions btwn positiv and ngativ MMBs of C, and (c) kping only thos which ar minimal. Finally, th rvrsibl mtabolic spac dos not chang, i.., RMS = RMS. 3.3 Rmoving a fully rvrsibl raction Th uniqu ffct of rmoving a fully rvrsibl raction is th rduction of th dimnsion of th rvrsibl mtabolic spac, i.., dim(rms ) = dim(rms) 1. Th MMBs of C and C ar th sam. G 1 2 G RMS 1 G RMS G 1 G 2 RMS (a) (b) (c) Fig. 2. Hr, 2(a) givs a 3D illustration of th flux con C, 2(b) shows th altrd flux con C aftr rmoving th irrvrsibl raction 3. Finally, 2(c) shows th altrd con C aftr rmoving th fully rvrsibl raction 4.

7 Exampl 5. Fig. 2 shows th flux con C and th altrd con C dpnding on th rvrsibility typ of th targt raction. Fig. 2(b) shows th con C aftr th rmoval of th irrvrsibl raction 3. In this cas, th rvrsibl mtabolic spac dos not chang and only th MMB D 1, which dos not involv raction 3, is still an MMB for th con C. Fig. 2(c) shows th con C aftr th rmoval of th fully rvrsibl raction 4. In this cas, th flux con bcoms pointd, i.., th rvrsibl mtabolic spac is rducd to th origin {0}, and th MMBs of C and C ar th sam. Th rsults abov can b xtndd to prdict th ffct on th flux con whn constraining th rvrsibility of som raction. If a rvrsibl raction ι is constraind to oprat in th positiv (rsp. ngativ) dirction only, th rsulting flux con will b C = C {v R n v ι 0} (rsp. C = C {v R n v ι 0}). Again, th dscription of C can b dducd from that of th flux con C dpnding on th rvrsibility typ of raction ι. Indd, if ι is psudoirrvrsibl, th MMBs of C ar th MMBs of th altrd con C, dfind in Equation (4), togthr with th MMBs corrsponding to th positiv (rsp. ngativ) minimal propr facs of C. Th rvrsibl mtabolic spac dos not chang, i.., RMS = RMS. On th othr hand, if ι is fully rvrsibl, th MMBs of C ar th MMBs of C, togthr with a nw MMB D = {ι} and dim(rms ) = dim(rms) 1. 4 Conclusion In this papr, w hav shown that th outcom of dlting a gn or constraining th rvrsibility of som raction mainly dpnds on th rvrsibility typ of th targt raction. Possibl ffcts includ not only changs in th stady-stat flux con, but also changs in th rvrsibility of ractions. Indd, som rvrsibl ractions bcom unabl to oprat in th forward and backward dirction, whil othrs bcom unabl of carrying any flux undr stady-stat conditions. Th importanc of a targt raction can thn b assssd by th amount of all ths changs. Rfrncs 1. B. L. Clark. Stability of complx raction ntworks. In I. Prigogin and S.A. Ric, ditors, Advancs in Chmical Physics, volum 43, pags John Wily & Sons, J. S. Edwards and B. O. Palsson. Mtabolic flux balanc analysis and th in silico analysis of schrichia coli k-12 gn dltions. BMC bioinformatics, 1:1, K. Fukuda and A. Prodon. Doubl dscription mthod rvisitd. In Combinatorics and Computr Scinc, pags Springr, LNCS 1120, L. Khachiyan, E. Boros, K. Borys, K. Elbassioni, and V. Gurvich. Gnrating all vrtics of a polyhdron is hard. In SODA 06: 7th Annual ACM-SIAM Symposium on Discrt Algorithms, pags , Nw York, 2006.

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