Assessment of the general non-linear case ..=(LK).", J'

Transcription

1 Eu. J. Bioche. 23, (993) 0 FEBS 993 Responses of etabolic systes to lage changes in enzye activities and effectos 2. The linea teatent of banched pathways and etabolite concentations. Assessent of the geneal nonlinea case J. Rankin SMALL and Henik KACSER Depatent of Genetics, Institute of Cell, Anial and Population Biology, The Univesity of Edinbugh, Scotland (Received July 2, 992) EJB We extend the analysis of unbanched chains (peceding pape) to lage paaete changes in banched systes using linea kinetic assuptions. Moe coplex elationships between flux contol coefficients and deviation indices ae established. In paticula, the deviation index in such systes depends on oe than one contol coefficient as well as on the agnitude of the enzye change. Nonadditivity of the indices is the geneal ule. Cobined changes of goups of enzyes, whethe coodinate o not, have also been foulated. Contol coefficients can be estiated fo a sall nube of independent lagechange expeients. Altenatively, the aplification factos can be calculated given the knowledge of the contol coefficients. A 'case study' using published data is pesented. The oveent of inteediate etabolites as a consequence of lage paaete changes can be dealt with in a siila anne. Expeiental ethods fo showing the adissibility of assuing the siplifying assuptions used ae suaised. Soe siulation studies show possible liits of the application of the appoach and soe aspects of the geneal, nonlinea, case ae discussed. It is concluded that, although etabolic systes ae in pinciple nonlinea, any behave, in pactice, as quasilinea systes. The elationships established between deviation indices and contol coefficients theefoe povide a pactical way of pedicting the effects of lagescale changes in paaetes fo any etabolic systes. Banched pathways ae a univesal featue of etabolis. In etabolic contol analysis, which deals with infinitesial changes, all contol coefficients ae identically defined iespective of the stuctue of the syste. Thee is additionally the banch point theoe [l 3 which elates the coefficients in banched systes to one anothe. To analyse the effects of lage changes on a banched etabolic syste we will use the concepts developed in the peceding pape in which we intoduced the definition of the 'deviation index' to aid in the analysis and pediction of the effects of lage changes. The index was defined as:..=(lk).", J' whee AJ = J' J", and AEj = EiE;, with EL =. E. Using the assuption of linea kinetics (see Eqn 3 of peceding pape), it was possible to ake a diect copaison between the deviation index of a paticula enzye change and the flux contol coefficient of that enzye. In an unbanched chain it was found that the flux contol coefficient was in Coespondence to H. Kacse, Institute of Cell, Anial and Population Biology, Cew Building, The Univesity of Edinbugh, King's Buildings, Edinbugh, Scotland EH9 3JN Abbeviation. PGI, phosphoglucose isoease. Enzye. Phosphoglucose isoease (EC ). fact nueically equal to the deviation index fo any change in enzye activity. As a consequence of this esult we wee able to intoduce a new ethod of estiating flux contol coefficients o, altenatively, a ethod to pedict the effects of lage changes in single enzyes, and in goups of enzyes, on the pathway flux as well as the effects of lage changes in extenal effectos. This has a nube of ipotant applications in the aea of DNA anipulation. We ust now enquie how fa the siple conclusions established fo an unbanched chain apply to the oe coplex banched syste. This is paticulaly elevant in biotechnological applications since, in soe of these, enzyes fo diffeent pats of the syste ay be anipulated. The second type of systeic vaiables, the concentations of inteediate etabolites, ae equally subject to changes when lage paaete vaiation is intoduced. We can theefoe define a deviation index fo etabolites, equivalent to those fo fluxes: Its elationship to the contol coefficients of the syste can be established by the sae ethodology that has been used peviously. The pediction of changes in paticula etabolites is equally ipotant fo an undestanding of biocheical systes and fo pactical puposes.

2 626 In geneal, the deviation index of enzye E, on flux J,, is given by: Fig.. Repesentation of a banched pathway. xc The use of linea kinetic assuptions, undelying the analysis pesented hee and in the pevious pape, has yielded useful elationships fo contol coefficients and deviation indices. It was shown, by intenal coheence, that the assuptions wee justified fo a nube of in vivo systes. We ust, howeve, envisage that this will not always be the case and theefoe conclude this pape by discussing the geneal nonlinea case, in pat by algebaic aguent and in pat by siulation studies. The banched pathway Fig. shows a siple banched pathway, whee each lib of the banch is catalysed by a single enzye. Each of these enzyes can also be thought of as epesenting a goup of undeteined nube whee, fo exaple: l/ea = UEa, + ll(ea2. Ka,) + ll(ea3. K,,... + ll(ea,,,. K,, a,,,). Since we ae again assuing linea kinetics and steady states, the following elationships can be applied: JA = e,. (XA SbIK,), JB = eb ' (sb xb/kb) 9 (2) Jc = e,. (sb xc/k,), whee Sb is the concentation of the banch point etabolite and of the eaining tes, e, = E,.f(k,,,, K,,,,,...) and K, ae the equilibiu constants. These thee elationships, cobined with steadystate elationship of JA = JB + Jc, (3) allows the algebaic deteination of the fou syste vaiables, JA, JB, Jc and &, in tes of the syste paaetes such as the enzye concentations [4] (and Appendix ). Following siila lines to those used in the peceding pape, elationships between deviation indices and flux contol coefficients can be deived (Appendix ). Fo the syste descibed in Fig., these elationships ae shown in Table. Note that the deviation index fo changes in enzyes with espect to that banch flux in which the enzye occus is independent of and equal to the contol coefficient and hence is exactly the sae as fo the unbanched case discussed in the peceding pape. All the consequences and applications shown in that pape apply to such a vaiation. In contast, the deviation index fo an enzye with espect to the flux though a banch othe than the one in which it occus, is dependent on the size of the enzye concentation change,, as well as on two contol coefficients. whee flux J is the flux though the banch in which enzye Ei occus. All the foulations in Table ae deivable fo this equation. Equally, if one of the banches is abolished so that the syste evets to an unbanched chain, the foulations of Table 'collapse' such that all indices ae equal to thei espective contol coefficient. Note that, as tends to unity, i.e. infinitesial changes, the function ultiplying cg in Eqn (4) tends to unity, and hence the value of the deviation index will becoe equal to the contol coefficient. This is to be expected, as the deviation index fo an infinitesial change is, of couse, the contol coefficient. If oe than one enzye occus in a banch, the denvations in Table apply to the case whee all the enzyes in a banch ae changed by the sae facto, i.e. they ae coodinate changes and the coefficients stand fo goup coefficients. Howeve, it is siple to extend this to any individual enzye (o subgoup of enzyes) within the banch. If, fo exaple, thee ae enzyes within banch B, i.e. E,,... E&,, then fo any subgoup of enzyes E,,, (including the liiting case of a single enzye), In these cases the deviation index will be with espect to a coodinated change in all enzyes of the subset and C&+ is called the goup contol coefficient of that subset. Deviation indices fo ultiplebanched systes can also be deived in a siila way, and expessions fo the deviation indices in tes of contol coefficients and can be obtained and ae of the foat given in Eqn (4). We shall now exaine the popeties and iplications of Eqns (57). Fo this pupose it is useful to ewite these equations in the fo shown in Eqns (5a), (6a) and (7a). Fo notational convenience we wite Eb,ik = Eb. We have also designated the ultiplie of a paticula contol coefficient as F, the facto which takes the banching into account, i.e. DfL = 6 a ' FA 9 Dfi E 6= a, DfL = EL ' FC. (54 (64 (74 It is instuctive to enquie whethe these factos can be > o < and what thei liits ae. Put in anothe way, we would like to know, having deteined the contol coefficients, whethe we can expect a geate o lesse effect, using a lage, fo the agnitude of the index. We shall illustate this with efeence to the following pactical question: what ae the consequences if we attepted to incease the output of JB by inceasing one, o oe, enzye in the B banch? Eqn (6a) pedicts that the value of the index with

3 Table. Relationship between deviation indices and flux contol coefficients in a banched pathway. 627 Monitoed flux D fo vaiation in E, Eb Table 2. Effect on F, of diffeent values of the contol coefficients espect to flux JB is the sae as the (peviously deteined) goup coefficient. Eqn (7a) pedicts the effect of this change on flux.ic. The sign and agnitude of this deviation index depends on the sign and agnitude of a and also on the facto : F =, a). Since a is negative in value (incease in Eb deceases Jc) and cs is positive, the su in backets is always a posi tive nube. The atio ( l)/ goes fo 0 to as inceases. If (a cs). ( )l is >, Fc is negative and o"& is positive. This, howeve, can only occu if, as a esult of the change, thee is a evesal of the Jc flux (i.e. the diection of flux is now fo X, to S,,). We shall concentate on the case whee thee has been no evesal of flux and theefoe, since Fc is positive, PA is negative, i.e. the sae sign as a. The question eains as to what the quantitative liits of Fc ae. Its value will depend on the absolute and elative values of the two contol coefficients. These ae deteined by elasticity coefficients and a, the faction of JA going into JB [l3, 5. Within these constaints, the coefficients can be consideed independent of one anothe and a whole atix of possible values can be obtained. We shall, howeve, discuss two inteesting cases. a) If and a ae vey sall, then, iespective of a and, F, = and PA = a. The lowe liit of F, =. b) If (e a). (l)/ is = (but < l), Fc can becoe vey lage and has an uppe liit of. In this case W& + e a Two paticula, nueical, exaples ae given in Table 2. In the fist ow of this table we have set the coefficients low such that E,, has vey little contol ove fluxes JB and Jc. The calculation shows that the deviation index is nealy equal to the espective contol coefficient. The second ow assues an equal but high contol by Eb on both fluxes. The esult is that (with an = 20) the deviation index has a agnitude appoxiately 3.45 ties geate than the agnitude of the coesponding contol coefficient. In the liit this could tend towads 20. A siila analysis of Eqn (5a) to that caied out above fo Eqn (7a) would show that the liits of FA ae to ~0 but that D2 is always positive and can take any value between 0 and. Cobined changes within banched systes It is often found in pactice that changes in single enzyes do not yield useful enhanceents of the desied flux [6,7]. The question theefoe aises whethe elevation of seveal enzyes siultaneously will poduce a geate change and what infoation is equied to pedict such a change. In the peceding pape we showed that in an unbanched pathway the deviation indices wee additive, i.e. k D& = x D$. The situation in a banched syste is, as the following exaples will show, not as siple. Thee ae two diffeent cases to be consideed. Case : Cobined goup of enzyes fo the sae banch 2= The additivity shown fo the unbanched syste applies equally to the banched syste when the enzye concentations which ae changed by the sae facto,, occu in the banch which is onitoed. Thus, in ou exaple (Fig. ), if k of the enzyes in banch B ae changed, then the deviation index fo flux JB can be expessed as k k = x G,# i= i.e. the esult can be pedicted fo a knowledge of the individual contol coefficients alone. Note also that, since the individual deviation indices ae independent of, it is not necessay fo the individual deviation indices to have been obtained using the sae value. (9)

4 628 If the flux of inteest is the flux though one of the othe banches (i.e. in the above exaple eithe flux JA o Jc) then the individual deviation indices ae no longe additive. Taking the case of the effect of siultaneous changes in two of the enzyes in banch B, Eh,l and EbJ on flux Jc, the individual deviation indices ae given as follows: 0 C = E6.t a,, * '. (G, G,,> while the cobined deviation index is not given by DSE + DS, but by *L,. (0) (a,, a,). whee av is equal to (a, + a,> and a,, is equal to (a, + Cj$,>, i.e. both ae goup coefficients. Note that both coefficients with espect to flux C, have negative values. Copaison of Eqn (0) with the equations fo the individual deviation indices indicates that the latte ae not additive. Futheoe, povided (c", a g,). <, i.e. no e vesa of flux Jc, the cobined deviation index will be geate in agnitude than the su of the two individual indices. Hence, the individual indices ae not sufficient but knowledge of the individual contol coefficients, o the goup contol coefficients, as well as, is equied to pedict the cobined change. = a,. Case 2: Cobined goup ade up of enzyes fo diffeent banches Individual deviation indices ae also not additive when the effects of a cobined change in enzye concentations fo diffeent banches ae consideed. We will fist conside when enzye concentation changes occu within the onitoed banch as well as in the coon pathway, JA. This ight be consideed as useful stategy to incease the JB flux. If the concentations of enzyes E, and E, (i.e. all enzyes in banches A and B) ae changed by the sae facto, and the esponse of flux JB is onitoed, the deviation index fo this case can be shown (see Appendix ) to be given by the following expession : By consideing the type of elationship fo the individual changes (see Table l), it is seen that the su of the individual deviation indices does not equal the cobined deviation index. We will now conside the case whee none of the enzyes whose concentations ae changed ae fo the banch flux onitoed. If enzyes E, and E, (i.e. all enzyes in banches A and C) ae changed by a facto and the esponse onitoed in flux J,, then it can be shown (Appendix ) that the deviation index fo this cobined change tuns out to be siply : Dz,, = a + a. (2) By copaing this equation with the type of elationship shown in Table, howeve, which shows the fo of the expession fo the individual changes, DS and Da, it can again be seen that the individual indices ae nonadditive. Finally, we can conside what the outcoe would be if the concentations of all these enzyes, E,, Eh E, (o the espective goups) ae changed by a facto. The esult (Appendix ) is: DS,,., = a a. (3) By the suation theoe, the ighthand side of Eqn (3) is equal to, and hence so is this deviation index. This, then, assets that the siultaneous elevation of the concentation of all these thee (goup) enzyes by a facto will esult in J' =. J" (see Eqn ). This is the deivation, using linea kinetics, of the coodinate theoe as epoted in [S], which cae to the sae conclusion but was quite geneal fo any nonlinea syste. The elationships shown fo the cobined deviation indices shown in Eqns ( 3) only apply to the situation whee thee is eithe only one enzye pe banch, o whee the concentations of all the enzyes within a banch ae odified in an identical anne. It is possible to deive elationships which expess deviation indices in tes of contol coefficients fo oe geneal situations, whee the concentations of a subset of enzyes within one banch ae changed in cobination with a subset fo anothe banch (see Appendix ). These elationships tend to be slightly oe coplex in natue than those shown befoe. They ay nevetheless be useful in pactical cicustances. In suay, we have shown that fo banched systes, deviation indices fo individual changes cannot, in ost cases, be siply added togethe to obtain deviation indices fo cobined changes. Howeve, it is possible to obtain an expession fo the cobined deviation index in tes of contol coefficients. A late section will show how individual changes can be used to deteine these coefficients, and hence useful infoation elating to cobined changes can be obtained fo individual changes. Pediction of changes in flux As was the case fo an unbanched syste, it is possible, by eaanging the appopiate equation fo the deviation index, to obtain an expession fo the aplification facto, f = JVJ", fo any flux in a banched syste. In all cases, the aplification facto can be expessed in tes of the deviation index as: f= D. Hence, the aplification facto fo flux JB due to a change in the goup of enzyes Eh,ik is given by:

5 629 This is identical to the aplification facto fo an enzye in an unbanched chain. This expession is valid fo a change in concentation of a single enzye E$,i and fo any nube of the enzyes in banch B. The aplification facto fo flux J, due to a change in E, can also be obtained fo Eqn (4) and Table, and is given by : a* This expession is also valid fo any subset of enzyes fo banch A, with eplaced by the su of the contol coefficients with espect to the enzyes in the subset. It is also possible to obtain expessions fo aplification factos when enzyes fo diffeent banches ae changed siultaneously. Hence, by eaanging Eqn (ll), the aplification facto fo the change in flux JB due to a siultaneous change in E, and E, is given as:. [ (c2 + ( ) fk = * (7) (2% ) (a + a) ( ) Although coplex, this expession shows that it is possible to pedict the effects of a lage concentation change in two goups of enzyes fo contol coefficients, all deteined at the oiginal state, and the value of. Eqns (6) and (7) wee deived assuing that the two goups of enzyes wee changed by an equal aount. It is possible to obtain elationships between aplification factos and contol coefficients whee the enzye changes ae diffeent (see Appendix ). The esulting expessions ae only slightly oe coplex in foat, e.g. the coesponding expession fo Eqn (7) whee E, is changed by a facto, and Eb is changed by a facto, is given by: Thus a knowledge of thee contol coefficients, togethe with the two enzye factos,, and,, allows us to pedict the outcoe of such an expeient. The following section will show how the aplification factos obtained fo individual enzye changes can be used to deteine all the flux contol coefficients. Deteination of fzux contol coeficients in banched systes If it is shown that in a banched syste, the assuption of linea kinetics is justified (see Test fo nonlineaity in peceding pape and Appendix 4 in this pape), then the aplification factos can be used to deteine the flux contol coefficients of the syste. Taking Fig. as an exaple, thee ae nine flux contol coefficients to be deteined (i.e. thee is a flux contol coefficient fo each of the thee enzyes with espect to each of the thee fluxes). A single LIE, change in one of the thee enzyes and the onitoing of the effect on the fluxes allows the aplification factos of this enzye with espect to the thee fluxes to be estiated. This in tun, using the elationships between the values off and C fo a banched syste (see Table Al, Appendix 3), will give estiates of the flux contol coefficients of this enzye with espect to the thee fluxes. A futhe thee flux contol coefficients can be deteined by aking an analogous AE, change in one of the othe two enzyes. The final thee flux contol coefficients can be deteined using the suation theoe which states that, fo any flux, the su of the flux contol coefficients ust add up to unity. With two sepaate single AE opeations, theefoe, it is possible to deteine all nine flux contol coefficients (if data ae available fo changes in the thid enzye, then independent estiates fo the values of these final thee flux contol coefficients with espect to this enzye can be obtained). The values of the individual contol coefficients thus obtained can be used, if equied, to pedict the deviation index, o aplification factos, fo a cobined change by insetion into the appopiate elationship fo that paticula cobined index (see pevious sections). Fluxes to expansion It has been shown theoetically and expeientally [5, 9 that in gowing systes the steadystate flux though an appaently unbanched pathway ay diffe depending on whee in the pathway the flux is easued. This is due to a eduction of the flux at each etabolite (equal to the poduct of the gowth ate and the concentation of the etabolite) due to the expansion of the syste caused by gowth. This eduction was teed the flux to expansion o siply the expansion flux. Expansion fluxes occu in any gowing syste, egadless of pathway stuctue. Such gowing systes can be thought of as highly banched pathways, with a banch point at evey etabolite. Ou ethod of analysing and pedicting the effects of lage changes in enzyes in banched pathways can theefoe be applied to such systes, povided these lage changes do not affect the gowth ate to any significant degee. A case study of sucose and stach synthesis Kuckebeg et al. [lo] have studied the effect of deceasedactivity utants of phosphoglucose isoease (PGI) on sucose and stach synthesis in the leaves of the plant Clakia xantiana. Thee ae two fos of this enzye in this plant: one fo (PGI,) is located in the plastid copatent while the othe (PGL) is located in the cytosolic copatent. Duing photosynthesis, PGI, is involved in the synthesis of stach while PGI, is involved in sucose synthesis; the fluxes catalysed by PGI, and PGI, can be estiated by easuing the ates of stach and sucose synthesis, espectively. These fluxes ae banches off the Calvin cycle flux with tiose phosphate as the banch point etabolite; hence any analysis of lage changes in enzye activity ust use the foulations of the type shown in Table. Kuckebeg et al. [lo] obtained a educed activity utant of PGI, and two diffeent educed activity utants of PGI, and by genetic cosses wee able to obtain stains showing diffeent cobinations of the vaious activities of PGI, and the vaious activities of PGL. These stains wee copaed with the wildtype stain in thei ates of photosynthesis, stach and sucose synthesis. Using thei published data (Table 3 of [lo]) fo the conditions of satuating light inten

6 630 Table 3. Estiates of flux contol coefficients fo enzye and flux data in Clakiu. Syste Flux Enzye D(s.e.) C,, Mean C,, a Stach Sucose PGI, (0.265) =D (0.92) =D =D PGI, =D =D b Sucose Stach PGI, PGI, C PGI, (0.93) (0.088) Photosynthesis PGI, sity, in cobination with the equations in Table, we have estiated the values of the deviation indices fo these lage changes in enzye activity, with espect to the flux though the banch in which the enzye occus (i.e. D$;b and with espect to the flux though the copeting banch (i.e. D2:;= and D$g:). The esults ae shown in Ta ble 3 a and Table 3 b espectively. In Table 3 a, the two deviation indices fo PGI, ae both estiates of the sae contol coefficient Gg;. The standad eos of the individual obsevations epoted in [ 0 indicate that the accuacy of deteination of the oiginal flux easueents was not vey good (as conceded by the authos of the pape). This will affect the accuacy of the two estiates. Copaison of the standad eos, howeve, indicates that the lage C estiate is oe eliable. The thee estiates of C%jy shown in Table 3a ae supisingly close consideing we ae, in this case, using sall changes in two easueents which have not been vey accuately obtained. In Table 3 b the ecipocal deviation indices and contol coefficients ae estiated, i.e. the effect of a change in plastid (cytosol) on the cytosol (plastid) flux. Table 3c epots the values of the deviation indices and the esulting contol coefficients fo the effects of these utants on the photosynthetic flux. It can be seen that the two estiates of the photosynthesis contol coefficient with espect to the plastid enzye, C&;, diffe soewhat in value. The standad eo (as deteined by the ethod shown in the peceding pape) of 09% fo the fist estiate ( = 0.75) is howeve elatively vey lage (0.9) while that fo the second ( = 0.5) is sall (0.09). Hence, the second estiate is a oe eliable one than the fiist. The esults also show that the cytosolic enzye has vey little contol ove the photosynthetic flux. All the standad eos epoted above wee calculated on the assuption that the enzye activities wee coect and eoless. Fo othe published data efeed to in (0, this appeas not to be the case. Since the estiates of D values can be quite sensitive to these eos with these paticula values, the confidence that can be placed on the is not vey high. If the diffeent estiates at diffeent values ae not attibutable to eo, then the discepancies ay thow doubt on the assuptions of lineaity. The above is a soewhat oe igoous analysis of the data of this seinal wok than that caied out by the authos. Although they coectly identified the easueents which would give the estiates of the contol coefficients fo an enzye with espect to the flux in the banch in which it occus, they also used this (incoect) foula fo the ecipocal flux coefficient. In fact, thei epoted values do not diffe geatly fo ou own. The cytosolic enzye appeas to have about equal (but low) contol ove its own sucose flux as ove the stach flux. The plastid enzye, on the othe hand, has consideable contol ove its own stach flux with vey little effect on the sucose flux. If this sae contol distibution wee to be found in the elevant tissue of an agicultually ipotant plant, then cloning of the PGI, gene and its aplification, would clealy be a stong candidate fo inceased stach poduction. It can be calculated that fo vey lage values of about 30 o oe, an incease in stach poduction of up to 45% could be expected, with a axiu eduction in sucose poduction of only 6%. Deviation indices fo etabolite concentations So fa we have been concened with the esponses of etabolic fluxes to lage changes in enzye concentations. To coplete the analysis, and because it ay be of pactical ipotance, we have investigated by how uch an intenal etabolite will change in esponse to such a lage change in an individual (o goup) of enzye concentation(s). In an analogous way, we can define a deviation index fo any intenal etabolite of the syste. If, fo exaple, the concentation of an enzye E, is changed by a facto, then the concentation of etabolite S, anywhee in the syste, will change fo its oiginal value, S: to a new value, F. The deviation index, and the coesponding concentation contol coefficient, fo the concentation of etabolite S, ae defined as :

7 ~. whee AS, = S,,, F,,,. As in the pevious cases, an assuption of linea kinetics allows a elationship between this deviation index and the contol coefficients of the syste to be deteined. This elationship is not a siple one. It is, howeve, possible to show (see Appendix 2) that fo any etabolite S,,, in an unbanched (20) ~ i.e. the foulation includes, unexpectedly, the flux contol coefficient as well as the concentation contol coefficient. In a banched syste, this elationship equies to be claified to indicate which flux contol coefficient appeas in the elationship. Taking the siple banched pathway used in the pevious section (Fig. ) as an illustation, it can be shown (Appendix 2) that, fo the banch etabolite S,: DsL = EL@* l(ac~>. (2) Note that, since E, was the enzye odified, a appeas in the denoinato of the ighthand side of this elationship. Altenative fos of Eqns (20) and (2) give the aplification facto fo S, in tes of the contol coefficients, e.g.: Jl (GG> G. G. Analogous elationships fo cobined changes in enzyes can also be deived (see Appendix 2). It is equally possible to estiate the value of the concentation contol coefficient fo S, by using the obseved changes in J and S, due to a change,, in Ei, i.e. the aplification factos : p a= (23) j!, In any cases this would be a siple pocedue than extapolation ethods based on etabolic contol analysis. The geneal nonlinea case In the peceding pape, we established a nube of tests fo the adissibility of assuing linea kinetics fo the equations of the syste. These, and othe possible tests, have been suaised in Appendix 4 of this pape. We also gave soe exaples fo invivo data whee this appeaed to be justified. We ust now, howeve, addess the question how fa the conclusions and theoes deived in the foegoing teatent ae invalidated, and by how uch, if the stict assuption of lineaity of the kinetic equations ae elaxed o shown not to apply. The ichness of the elationships uncoveed in ou analysis and thei intenal coheence suggest 63 that the behaviou of a nonlinea syste ay not diffe geatly fo that pesented hee. As a pactical guide, the conclusions and suggested expeients ay be pefeed to the pevious absence of any conceptual faewok to intepet, design and pedict opeations of the kind discussed hee. We can, howeve, conside what othe infoation is available which ight bidge this algebaically intactable gap. The fist ipotant point to conside is that thee is a geneal popety of lage changes, the coodinate theoe [8], which does not depend on linea assuptions. It states that if all the enzyes in a etabolic syste at steady state ae changed by the sae (unspecified) lage, o sall, facto, then all the fluxes will in(de)cease by that sae facto, and all the etabolite concentations will eain unchanged. This eans that the deviation index of the cobined changes of all n enzyes is equal to unity (fo fluxes) and equal to zeo (fo etabolites). This is so, no atte the degee of nonlineaity (e.g. satuation, feedbacklfonvad inteactions, coopeativity) involved in any of the steps. The eason fo the validity of this coodinate theoe can be undestood fo the condition that, since all the etabolite concentations eain unalteed (whateve the value of ), each equation can be epesented by the fo shown in Eqn (3) of the peceding pape. The value of k, and the values of S ae theefoe constant, the only change being in the values of E. In this case, and only this case, evey equation in the syste has a stictly linea fo. The coodinate theoe is analogous to (but diffeent fo) the suation theoe applying to infinitesial changes. The ipotant diffeence is that while in the suation theoes the values of the contol coefficients, if deteined individually, su to unity (fo fluxes) o zeo (fo etabolites), this is not so fo the deviation indices. The su of the individually deteined deviation indices, each using a facto, is, in geneal, not equal to the cobined deviation index when all n enzyes ae changed siultaneously by this facto. The sae diffeence applied when consideing the effects of a esticted goup of enzye changes. While n c&~ = c,, in gened g, z 2 &:. I =J This nonadditivity applies except fo the siplest case when the whole syste consists of an unbanched chain of enzyes o when the goup of enzyes and the flux of inteest ae contained within one unbanched section of the syste. In all othe cases the value can nevetheless be calculated (see Eqns 03). The coodinate theoe is, howeve, a useful liiting case when only a esticted nube of enzyes is changed. In any eal syste, although the nube of enzyes is vey lage, the effective nube is uch salle. By this we ean that if one consides all the n contol coefficients with espect to one flux, the vast ajoity will pobably have values indistinguishable fo 5 zeo. The eaining inoity of coefficients, pehaps between 0 and 50 in nube, will have nonzeo values, both positive and negative (although each ay still be sall in agnitude) whose su will theefoe tend to unity. This is the set of effective enzyes whose vaiation, singly o in cobination, will be capable of alteing the flux. Anothe way of putting this is to say that the distibution of contol coefficient values is vey skewed. That sae set will also be that whose indices we ae inteested in. If these ae siultaneously changed, the coodinate theoe pedicts that the flux will be changed by appoxiately the value of used in the expeient. It should be noted that in a banched syste,,=j

8 632 Table 4. Siulation of an unbanched pathway of fou satuable evesible Michaelian enzyes. The values of V,, geneate five sets of enzye activities with thei associated satuation functions at steady state. All the K,,, values wee set to 00 except the K, of the final pool with espect to the fouth enzye which had a value of 0.. The equilibiu constants wee 0.95, 0.75,. and.0 fo enzyes 4, espectively. The initial pool size was fixed at a value of 000, while the final pool size was fixed at 0.. All units ae abitay but consistent. The values fo the esulting satuating function, ks, fo each set is also shown. Fo unsatuation, k, =. When the etabolite concentations ae equal to thei espective K,, k, = 3. When they ae 5 X K,, k, =, etc. The flux contol coefficients fo each paaete set, deteined by siulation, ae also given. Set the enzyes aking up the set of effective enzyes fo the flux though one paticula banch ay not necessaily be the enzyes which ake up the effective set fo the flux though anothe of the banches. This eduction of the syste to oe anageable popotions akes it possible fo soe pedictions to be ade. The pactical poble, howeve, is to know which enzyes constitute the effective size of the syste. It has been suggested that the futhe (kinetically) fo the flux o etabolite of inteest an enzye is, the less likely will it have a lage contol coefficient. The idea of kinetic distance is difficult to quantify since it involves not only the nube of steps between the enzye and the easued vaiable, but also depends on the agnitude of paaetes, enzye echaniss, degee of banching, cofacto coupling etc. The idea that each intevening step tends to buffe the tansission of a petubation is based on the popeties of Michaelian enzyes. These consideations then lead to the view that thee will be a clusteing of effective enzyes aound the vaiable investigated, while the effect on the vaiable attenuates the futhe you go. Thee will, howeve, be cases whee an aplification athe than an attenuation of a etabolic petubation ay occu (coopeative enzyes, banch point effects), hence kinetic distance in such cases is stongly echanis and stuctuedependent. Assuing we have identified a goup of enzyes constituting the effective size of the syste fo a paticula flux (o etabolite), this will contain, in geneal, both positive and negative coefficients (and indices) in view of the univesal occuence of banching. Applying the coodinate theoe to this goup, i.e. aising (o loweing) all the enzyes in the goup by a facto, will effectively esult in this paticula flux changing by that facto. While it ay be consideed a satisfactoy outcoe to have a pedictable change in one of the fluxes, the possible change in all the othe banch fluxes ay not have such a desiable consequence. If the banch flux(es) lead to ipotant aeas of etabolis the oveall effect ay not be satisfactoy. A diffeent stategy, ovecoing such undesiable consequences, will be discussed in a subsequent publication. In establishing the algebaic conditions fo using linea theoy, we stated in the peceding pape that the ate fo a Michaelian enzye epesented by: o : VJKz (Si SjIK,> 0. = + SiIK + SjIK, (S, S,KJ u, = e,. k, whee k,, the satuation function, was eithe equal to unity (i.e. an unsatuated enzye) o, if not, was effectively constant fo the lage vaiation iposed. To investigate this futhe, we have caied out a nube of copute siulation studies on an unbanched chain of fou enzyes. Each of the steps had substantial satuation (ks# ) and, as a esult of enzye concentation changes, the etabolite concentations (and hence the values of k,) did change. These studies wee caied out using the etabolic siulation package SCAMP [ll]. With each of five diffeent sets of fou paaetes (see Table 4), which had the effect of changing the distibution of contol coefficients, we changed the values of each of the individual enzyes and cobinations of these enzyes by a facto = 0 (also = 20 fo Set ). As a test fo the deviation fo linea expectations, Fig. 2 gives plots of the atio of the actual (siulated) flux aplification facto f, so that expected, f using the contol coefficients and applying Eqn (24) of the peceding pape: f = L Z G. p,= based on linea theoy. The figues give the atio R = f lf against the (su of) contol coefficients. Two lines, 0% on eithe side of the unity line (no deviation fo expected) ae also given. The fist point to note is that, as follows fo the coodinate theoe, when all fou enzyes wee siultaneously elevated by a facto 0 (i.e. ZCl = l), then, iespective of the contol coefficient distibution, the syste behaved quasilinealy and R =. Futheoe, the flux was elevated by a facto 0, while the steadystate etabolite concentations eained at the level befoe the elevation of the enzyes. In the othe cases, involving changes in individual enzyes o a subset of enzyes, it will be noted that ost of the R values lie within the 0% bounds. It is also evident that, up to a point, the highe values of the goup contol coefficient show geate deviations (both highe and lowe) fo the expected esult. As the goup contol coefficient appoaches close to unity, howeve, the pedictions once again becoe oe accuate. What the goup contol coefficients (both flux and etabolite) do not disclose ae the

9 d..2.4 W l )' 0.8. I a W N d N.. N d Csl.a y y: a N d.2.* ~ d Nl.. l % 0.6 u % 0.6: l N.. U du G..I U N d = W Y ~ N.I N d ln?? 2 NIT.. FIN * l l 0.2 (a) Set3, =lo O t " " ' 0 ~ " " ' ~ " ~ " " ~ ' " " " ' ' ~ ' ' CC, CC, 0.8 ). 0.6: l a * W N WN.I Nl.. Nl. e w. W.. N l Nl % d d.. l t.2. d. ' W N d N l. N l 0.2 (b) Setl, =20 O t " " " " """""" CCi 0 " " " ' " " " ' " " ' ~ ' " " CC, I.2 U N * *N....I Nl ul pll N % l NA l a: 0.4.I Nl 0.2 Q set 5, = 0 0' ' ' O ' " ". ' ' " ~ " ' ' ' " " " a Xi Fig. 2. Copaison of actual to expected values of the aplification factos fo a satuable syste. The five paaete sets used ae defined by the paaete values shown in Table 4. p' is the siulated value; f'" is the expected value using linea theoy. R is the atio Ylf'". ZCi details of the behaviou of the deviations, since they do not show the pecise oveents of the etabolite levels. In soe of the cobined enzye changes, these oveents will tend to 'copensate' each othe in tes of the satuation functions and R values close to unity ay occu, while in othes they ay einfoce each othe and the R values will divege. It is not possible to analyse these nonlinea esponses in any quantitative way since the net esult is a coplex cobination of assaction effects, satuation ef fects, absolute etabolite concentation changes and contol coefficients. What is evident howeve, is that even in the 'wost' cases, the deviations fo those pedicted by applying the linea theoy ae not disasteously wong. We have also enquied how fa anothe type of nonlineaity, naely allosteic (n = 4) negativefeedback inhibition, affects the adequacy of linea theoy. Fo this pupose we consideed a fiveenzye pathway with the second enzye being feedbackinhibited by the thid etabolite. Thee

10 ~ ~~~ 634 Table 5. Siulation of a pathway with thee diffeent levels of feedback inhibition on the second enzye. An unbanched pathway of five enzyes with linea kinetics except fo the second step which was feedbackinhibited by the thid inteediate etabolite in an allosteic anne. The eaction ate though the second step was assued to be of the fo v = [(Vu,2K,z)ikJ * (SlSzlKz), whee k, = + (S3)"K,. The values of the V,IK atios wee 500, 00, 500,750, 600 and the values fo the equilibiu constants wee 0.95, 0.75,.,.0 and.5 fo enzyes 5 espectively. The initial pool size was fixed at a value of 000, while the final pool size was fixed at 0.. The values fo thee diffeent K, fo the feedback inhibition ae shown. All units ae abitay but consistent. The values fo the esulting k,, fo each diffeent K, set, is shown in the second colun. Fo no inhibition k, =. When (S3)4 is equal to the K,, k, = 2, etc. Fo each enzye, at each K, value, the flux contol coefficient and the atio, R, of siulated to pedicted aplification facto fo a change, = 0, in that enzye concentation ae given. The pedicted value was based on JJ = li (Eqn 23 of the peceding pape). 6 X0t El Ez E E E X Et ooo Ez E E E X Et E Z E E E stengths of inhibition wee consideed: Set 6, whee the K, was vey lage copaed to the concentation of the signal etabolite (S,) such that effectively no inhibition was opeating, Set 7 with a Ki such that (S,)"/Ki was appoxiately equal to and Set 8 with a Ki uch lowe than (S,)4. As a esult of the pogessive eductions in Ki (which ay be thought of as eplacing enzye E, by vaiants with geate affinity fo S,), the distibution of contol coefficient changes, as expected, putting highe values of C: and C< on the 'deand' steps of the syste. Table 5 gives soe of the esults of ou siulation fo each Set, and the R values obtained fo the odulation of each enzye in tun, by a facto = 0. As expected, Set 6 behaves as a linea syste, with R values not significantly diffeent fo. Inspection of Sets 7 and 8 shows that linea pedictions ae good fo all steps except enzye E2, whee deviations fo pedictions of up to 30% ae obseved. Nevetheless, we can suggest that the ethods developed in the foegoing analysis give a bette pediction than was available pio to this developent. One possible altenative way of tackling this nonlinea poble is to use biocheical systes theoy, as developed by Savageou (see e.g. [2, 3]), whee evey eaction in the syste is appoxiated by a powelaw function. A ajo dawback to using it as a pactical tool, howeve, is the equieent that the values ust be Table 6. Thee types of pediction of the aplification factos. RB"d = f "'' Kcad ; R'OK = ; R"" = f""/'". All R figues ae given to two decial places only. Set AE,, C<J Actualf E El, E3, El,,, E,,,, El Ez E E E El Ez E E E known of the kinetic odes (elasticities) of evey etabolite to evey eaction it acts upon. In any cases this would equie an extensive aount of difficult expeientation, paticulaly on systes which ae not yet fully chaacteised. In addition, we still have the poble of knowing ove what ange of vaiation this appoxiation ethod will be valid. Altenative ethods to that developed hee, of pedicting the effects of lage changes fo the contol coefficients, ely on soe type of extapolation fo the values of the expeientally deteined contol coefficients diectly. Two such extapolations ae available. The fist, which ay be called the 'gadient extapolation' siply extends the locally found slope of the fludenzye elationship at E" to the value assued by setting E' = e E". (This was shown in Fig. of the peceding pape.) The second ethod of extapolation, which ay be called the 'logaithic extapolation', elies on the logaithic foulation of the contol coefficient (both ethods ae descibed in Appendix 5). We have copaed both these ethods of pedicting the 'actual' (siulated) aplification facto, f = J7Jo with ou 'lineakinetic appoxiation' ethod. We have chosen thee cases fo deonstation puposes: Set (Fig. 2a) fo the satuation siulations (choosing the 'wost' five cases) and Sets 6 and 8 fo the inhibition siulations. The copaisons ae shown in Table 6. In each case, the value of the contol coefficient is followed by the actual, siulated, aplification facto (f""). The eaining coluns give the atios of actual to pedicted f values fo the thee ethods, i.e. Rg"* = jy$'ad, R'"g = fdc'p and R"" = In all cases a value of = 0 was used. Inspection of Table6 shows that, in geneal, the wost estiates wee given by $" followed by f"p. The values given b yp ae, in ost cases, vey close to the actual values i.e. R=. A linea teatent, as developed in the foegoing analysis, theefoe appeas to have not inconsideable pedictive powe. These siple siulation studies ae of couse fa fo exhaustive. An extensive exploation of nonlinea behaviou is a study in its own ight. Thee ae clealy vey any diffeent kinds of nonlineaities to be exploed, each with a vast configuation of paaete values to be con R'OP R""

11 sideed. Fo ou vey peliinay studies we believe that the esulting ultidiensional suface of R values against paaete values, if it could be visualised, will have doains of R (when linea pedictions ae easonable) but will also show 'peaks' and 'toughs' whee the pedictions ae not satisfactoy. Fo pactical puposes, a soewhat unsatisfactoy theoy is, howeve, bette than none. 635 APPENDIX Deivation of elationships between flux aplification factos o deviation indices and contol coefficients In the following deivations it is assued that all contol coefficients ae with efeence to the initial steady state, i.e. pio to any enzye changes. Solving Eqns (2) and (3) of the ain text fo the state vaiables, we find : Ea ' IEb ' (XA XBKab) + Ec ' (XA XcKac)] JA = E,/Ka + Eb + E, (All Sauo et al. [2] have shown that: Ga = c2 Q, cg = CQ CZ. We also equie the following : Eb Cg= EJK, + Eb + E, ' (A9 ('40) E, * [E,.(XA Xc/KaJ Eb. (XclKc XB/&>] Jc = E,/K, + Eb + E, ('43) Ea. xa Eb. xb/& + E,. XJK, sb = 9 (A4 EJK, + Eb + E, whee Ka, = K,. K,, Kac = K.. K,, and the eaining tes ae as descibed in the ain text. The following additional definitions ae equied befoe poceeding : (JX)' = flux, Jx, at oiginal enzye levels. (.Ix); = flux, Jx, afte the concentation of enzye Ei changed by a facto. fi = (JX);/(Jx)". Since : l/f D= the deviation index is obtainable fo5 which we shall deive fist. Befoe consideing any specific cases we will fist deive a oe geneal elationship whee the effects of abitay changes,,,, and, in enzye concentations E,, Eb and E,, espectively, on, fo exaple, flux JB ae consideed. The atio of the esulting flux to the initial flux (Eqn A2), i.e. the aplification facto f fo siultaneous changes in all enzyes, can be expessed as: b. (EJK t Eb + EJ (Ea. Xi. a f Ec. X 2. c> fk = (,. EJK, +,. Eb +,. EJ (E,. x + E,. x2) ' 645) whee x and x2 ae goups of tes which contain the paaetes of Eqns (AlA3) but not E,, Eb o E,. In ode to expess Eqn (A5) in tes of contol coefficients we need to define a = JB/JA = JB/Jc. Diffeentiation of these atios (fo Eqns A A3) gives: ' which gives ( e) + ( eb) + ( GC) =. (A4) Inseting these elationships into Eqn (A5) esults in:,. (Ga, + CSc * a) fk = ( G) *, + ( Gb)., + ( a>., ' (A5 Siilaly, if we define the following: (a) = Jc/JA, p* = /p, a* = lla and (a)* = ll(la), then by following a siila pocedue to that fo the deivation of Eqn (A5), it can be shown that: (A7 Eqns (A5Al7) can be used to obtain the aplification factos fo oe specific exaples by eplacing the vaious values with the appopiate values. Fo exaple, if the effect of a change in only Eb on flux JB is equied then by setting, and, in Eqn (A3) equal to unity we obtain: Using Eqns (A0) and (A4) in Eqn (A8) gives: i.e. Eqn (8) of the ain text. Using the elationship between an aplification facto and a deviation index, the deviation index fo this change can be expessed as (Table, ain text) : b

12 636 D3b E6 (A20 If the esponse of JB to a change in E, is equied then by setting, and, in Eqn (AlS) equal to unity we obtain Eqn (3) of the ain text gives the deviation index of flux JB with espect to a cobined changed in all enzyes, with, =, =, =. In this case Eqn (AlS) educes to: Using the elationships shown in Eqns (A7) and (A8), this can be eexpessed as + CEa. (, l)/a (A23 e. (a l)/a ' Cobining this esult with Eqn (A6) esults in Eqn (9) of the ain text, i.e. Again, the deviation index as expessed in Table, follows fo this elationship, i.e. Following a siila pocedue, all the eaining expessions in Table can also be deived. Likewise, the effects of cobined changes can be consideed along the sae lines. By setting, = in Eqn (A5) we can obtain the aplification facto fo the cobined change of E, and E, on flux JB. If both these enzyes ae changed by the sae facto (i.e., =, = ) then the aplification facto can be expessed as follows : The coesponding deviation index (Eqn of the ain text) is theefoe: + (4 + GW ((24 + G). (A3 Hence, the deviation index fo this cobined change is given by DZ,b,c = (A32 =4+4+@. (A33 A siila analysis can be caied out to deive the aplification factos and deviation indices with espect to the othe fluxes in the syste, with equivalent esults. In all the above deivations, it was assued that the thee enzyes (Ea, E, and Ec) epesented goups of enzyes within a banch, and that the concentation of all enzyes within a paticula goup wee changed by an identical facto,. If soe, o all, of the enzyes within the goup ae changed by diffeent values then the above elationships ust be odified. Fo illustation puposes we will assue that & in Fig., and the above deivations, epesents two enzyes E,, and b2 and the elationship between the 'effective' value of Eb and the two enzye concentations is: o I/& = /Eh + /(&2. &I) (A34 Eb = Ebl Ebl ' EhZ. Kbl + Eb2 ' Kbl What is the effect on the syste if Ebl is changed by a facto,, in cobination with a change in Eb2 by a facto h2? Since we know what the effect of any change in Eb on the syste will be, we can appoach this poble by consideing fist the esulting change in the effective value of E,. Using the elationship (A3S) we find that, as a esult of the cobined change in Ebl and Eb2 the level of Eb changes by a facto, h,eff given by: _ El % b.efl The coesponding aplification facto, whee, #, can be expessed as follows:,.,. [I ((24 + 4). (, l)/a] f2 = (, +, ). (, I) c&. (, ) Eqn (2) of the ain text gives the deviation index of flux JB with espect to a cobined changed in E, and E,, with, =, =. This can also be deived fo Eqn (A5) by setting, equal to unity. Afte eaangeent and soe substitution, this esults in: Using elationship (A37), it can be shown that, fo any syste vaiable V (flux o etabolite) the atio of the contol coefficient with espect to Ebl and the contol coefficient with espect to Eb is given by: and siilaly : The elationship fo the coesponding deviation index is theefoe : Cobining Eqns (A37), (A38) and (A39) gives: o";,. = 4 + a. (A29 Hence, in any of the aplification facto elationships

13 shown in the ain text o in this Appendix, substituting b with b,eff will give the aplification facto fo any changes in Ebl and Eb2. Fo exaple, if Rbl is changed by a facto bl while Eb2 eains constant, then the effective change in Eb is given by: o,,eff = bl * Gb + c&l ' bl 'gbl bl ('44 ) b,eff 'EbI bl ~ (A43. ff" = ~ b,eff Gb bl * WI l) + x2ae2. 2) xn2/(en. n) The aplification facto fo a change in flux JA due to a ~iii(ei. i) + ~2/(E2. 2) +.. Xni/(En. n) change in Eb is given by:. (A whee the x tes contain paaetes only, but not any of the E, tes, and will be diffeent fo diffeent etabolites. If all enzyes in the syste ae changed, and j is the facto by which E, is changed in each of the, then the esulting concentation of S,,, will be : The vaious, values ae not necessaily equal in agnitude. The atio, f" = S/%, of the new concentation to the old, is given by: l/el + xje xn2/en XiiIEi + ~2/E2 + '.' xni/en The flux though this pathway can be expessed in a siila anne by: To obtain the aplification facto fo a change in Ebl only, we can eplace in this elationship with (bl,eff l)/b,er, i.e. using the esult of Eqn (A42): fja b,eff. bl whee Xo and X, ae extenal pools. We define the atio, 7,. as the atio between flux J and etabolite S, i.e. Note, howeve, that : J q =. S, Gbl. ~ b,eff bl Although we consideed only two enzyes within the banch, it is tivial to genealise this to any nube of enzyes, and to conside the effects of changes to any subgoup of these enzyes. bl ". The aplification facto, as expessed in Eqn (A45), fo this set of changes can be expessed in tes of coefficients as: Cz/l + C Cg/n f?" =. (A49) Gl/l + cj,/2.. ' + C&/,, This elationship is geneal in that all enzyes ae assued to have been changed. By setting j fo those enzyes which do not change to, howeve, we can obtain the elationships fo oe specific cases. If, fo exaple, only Ei is changed then : Using Eqn (A45) and noting that suation theoes fo these coefficients ae : APPENDIX 2 Deivation of elationships between etabolite aplification factos o deviation indices and contol coefficients In an unbanched pathway showing linea kinetics an exact algebaic epesentation of any etabolite concentation in tes of the syste paaetes can be obtained. Since we ae concened with the effects of changes in enzye activities on the etabolite concentations, it is convenient to expess the elationship between any etabolite concentation S, and the enzye activities as: and we obtain: xcg= (45) i= i=l ('453)

14 , 638 The esulting expession fo the coesponding deviation index (Eqn 2 of the ain text) follows fo Eqn (A54) i.e. Table Al. Estiation of flux contol Coefficients in a banched syste fo aplification factos., Othe oe coplex changes, involving diffeent subsets of enzye changes, can be obtained in a siila anne. Fo banched systes, the deivation is slightly oe coplex but follows a siila line of aguent. Taking the syste shown in Fig. as an exaple, if the atio of any flux J, to the banch point etabolite S, is defined as: f2 h X B. 7; = J, /&, (A56 then, by using Eqn (Al)(A4) and following an analogous pocedue to that fo the deivation of Eqn (A3), it can be shown that the aplification facto of etabolite &, due to changes in concentations of all thee enzyes (by diffeing aounts) is given by: ( c$)., + ( ~ $., + ( ~ $., E& = ( Ga>., + (I Qb)., + (I Q), ' (A57 Fo this equation, the aplification facto fo all possible cobinations of enzye changes can be obtained. Fo exaple, if only Eb is changed (i.e., =, = l), then the aplification facto is given by: A?= Futheoe, since ( c$) + ( + C$ + ( Gb)., + ( Gc) ( G) + ( cz) ( cg) + ( cg) + ( CZ) =, ( G) + ( Gb) + ( c;) =, then, noting that Cz! = Gz e, cqf Eb.(,) +, A?= Cg. ( b) +,. (A58) f$ b f2, h f$ f? p and hence the coesponding deviation index is AL The above pocedue can be caied out to deive elationships fo all the othe cobinations of enzye changes and etabolites onitoed. Siilaly, if the banched schee of Fig. 3. is futhe coplicated by the addition of oe banches and oe enzyes within the banches then analogous elationships ae obtained. In fact, the elationships shown in Eqns (A55) and (A6) ae geneal fo any single enzye change, with espect to any etabolite in the syste, no atte what the coplexity of banching. APPENDIX 3 Estiation of flux contol coefficients fo lage enzye concentation changes in banched systes Thee ae nine flux contol coefficients in the banched syste shown in Fig.. Table A shows the individual changes in enzye concentations (a,, and J, and the esulting aplification factos equied to estiate each of the contol coefficients. As stated in the ain text, one paticula enzye change and the onitoing of the effects on the thee fluxes will give the flux contol coefficients of that enzye with espect to all thee fluxes (any colun in Table Al). A change in a second enzye will give a futhe thee contol coefficients (a second colun of Table A). Using the suation theoe, the final thee flux contol coefficients can be obtained. Altenatively, o in addition, a lage change in the final enzye concentation will give second estiates of the final thee coefficients, which can be used as a test of the adissibility of the linea kinetic assuptions. APPENDIX 4 Tests fo nonlineaity In this pape, and the pevious pape in this seies, we have established a nube of tests fo showing the adissibility of the assuption of kinetic lineaity. These tests ae suaised in this appendix. Fist, in an unbanched syste, the flux deviation index fo a change in one (o oe) enzye concentation is found

15 639 to be nueically equal to the flux contol coefficient with espect to the enzye(s) changed i.e. D = C. Thus, the value of this deviation index is independent of the size of. Hence, if two changes with diffeent values ae ade fo the sae enzye and the esulting easued deviation indices have the sae value, then the assuptions of kinetic lineaity can be assued to be valid fo these changes. If significantly diffeent values ae obtained, then soe fo of nonlineaity ust be exeting soe influence on the flux. This paticula test is also valid fo banched systes when the flux onitoed and the enzye(s) changed ae fo the sae banch of the syste. Second, it was shown in the peceding pape that, if the initial pool X, of an unbanched chain is changed by a facto, then the esulting value of the deviation index will be independent of and equal to the flux esponse coefficient R of the pool i.e. *D = R, whee *D is defined as in the peceding pape as: Note that, in contast to the 'noal' deviation index, D, the scaling facto in this D is the fludpool atio easued at the oiginal point. The equivalence of this sot of deviation index and the esponse coefficient is also valid fo any intenal etabolite in the unbanched chain and also fo any flux o etabolite in a banched syste. Since the value of the deviation index is independent of, then, again, two diffeent changes in the initial pool should give the sae value if the assuption of linea kinetics is valid. Monitoing the esponse of oe than one vaiable will povide exta checks on the validity of the assuptions. Thid, it should be eebeed that the second test will fail if the change in extenal pool has not only a diect, assaction, effect on the syste, but has othe, nonlinea, inteactions such as inductiodepession echaniss. If it is found that two diffeent changes in extenal pool give diffeent values fo the esulting deviation indices, then this ay be due to a nonlinea inteaction of the initial pool with the fist enzye. Thus the test fo nonlineaity would fail even though all inteediate etabolites show linea kinetics with espect to the enzyes and the elationships deived fo enzye changes would still be valid. One coon, nonlinea, elationship between the initial pool and the ate of the fist eaction occus if the fist enzye is potentially satuable with espect to the initial pool, and the ate equation is of a MichaelisMenten type, i.e. Vax,l.(X, WG) u, = L. + Xo In this case, if thee diffeent changes in the extenal pool ae ade and the values of the aplification factos and deviation indices ae obtained, then the following elationship between the values is tue, povided all the intenal etabolites show linea kinetics with espect to the enzyes: *D2*D, *D,*D, fi f2 h.l whee *Di andj ae the deviation index and aplification facto fo the ith change. Fouth, if the elationships shown in these two papes ae used to estiate contol coefficients and the values obtained ae consistent with those obtained fo anothe ethod, then this suggests that the assuption of linea kinetics was valid. If oe than thee diffeent changes can be ade fo a paticula enzye, then a diect fludenzye pofile can be obtained and the 'goodness of fit' between the data and a ectangula hypebolic function can be used as a diect test of the suitability of the linea kinetic assuption (see case studies in the peceding pape). The tests suaised above ae not testing whethe the assuption of linea kinetics is valid unde all conditions and fo all possible changes, but tests whethe the esults obtained fo the paticula lage changes ade can be analysed in tes of the elationships deived in ou two papes. APPENDIX 5 Estiation of flux aplification factos by extapolation In the ain text we use two diffeent extapolations fo the values of the contol coefficients to obtain estiates of the esulting flux J' (via the aplification facto f) due to a change in enzye concentation Ei by a facto. The fist, which we call the 'gadient extapolation', is deived as follows. The definition of a flux contol coefficient is : aj" EP ce= ~. ~. aep J" Using a value of C$, but extapolating the slope at the initial point to E = Ep, we obtain: AJ ~ &.. AEi J" Ep Assuing that the value of the contol coefficient does not change, by expanding and eaanging this equation we can obtain: (E". _ J' Ep) =C$ > J" Ep ($')gad = +. ( ). The second extapolation, which we call the 'logaithic extapolation', esults fo the logaithic foulation of the contol coefficient, i.e. : a In Jo ce=. a In E: If this elationship is extapolated to finite changes, then we obtain: AlnJ= CGdlnE,. Expanding this elationship, and eaanging, esults in : In J' In J" = c", (In E: In EP) = ci (In Ep + In In Ep)