25 Reprogramming of metabolic systems by altering gene expression

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1 25 Reprogrammng of metabolc systems by alterng gene expresson E. Klpp 1, H.-G. Holzhütter 2 and R. Henrch 1 1 Insttute of Bology, Theoretcal Bophyscs, Humboldt-Unversty Berln, Invaldenstr. 42, Berln, Germany 2 Insttute of Bochemstry, Medcal Faculty, Humboldt-Unversty Berln, Monbjoustr. 2a, Berln, Germany Introducton Wth the ncreasng power of genetc analyss and DNA mcroarray technques a new knd of bologcal knowledge s avalable: t s possble to measure changes n gene expresson levels on the same tme scale as changes of metabolte concentratons. From ths knd of nvestgatons t became obvous, that the acton of the genetc apparatus can be well adjusted to the necesstes of the cell metabolsm [1,2]. In ths contrbuton a theoretcal approach s developed to explan changes n gene expresson accordng to the requrements of the metabolc apparatus. These requrements result from the fact that the metabolc system has to provde the cell wth a large number of dfferent substances as well as wth energy, for example n the form of ATP. In partcular, metabolsm has to ensure the survval of the organsm under varyng external condtons. Snce bologcal organsms are an outcome of mutaton and selecton processes, one can expect that ther functons have been optmzed durng evoluton. Hence, we try to explan certan features of these systems by applyng optmsaton prncples. In extenson to prevous nvestgatons for systems n steady state [3,4] we allow for tme dependent changes n the levels of enzyme concentratons. The model s based on the assumpton that the gene expresson apparatus can adjust to the demand of the metabolc system and can ncrease or decrease the amount of certan enzymes by a change n the expresson levels of the respectve genes. The lmted proten storng and producng capacty [4,5] s ncluded va a fxed total amount of enzyme. In partcular we treat the followng queston: whch tme-dependent dstrbuton of enzyme concentratons allows the system to fulfl a gven performance functon n an optmal 165

2 Reprogrammng of metabolc systems by alterng gene expresson way? The choce of such a performance functon s a matter of debate [3,4,6]. In the followng we consder: () unbranched metabolc chans usng as performance functon the transton tme for the converson of an ntal substrate to the end product, and () a skeleton model of the energy metabolsm n yeast usng as performance functon the tme of survval durng the dauxc shft of metabolsm. Results and dscusson Optmal expresson of enzymes n an unbranched chan under tme dependent condtons Let us consder the system wth a metabolc chan wth enzymes E whch are coded by the genes G (Scheme 1). proten synthess G 1 G 2 G genetc regulaton AA E 1 AA E 2 AA E metabolc regulaton... proten degradaton X 1 X 2 X 3 X n Fg Scheme 1 AA symbolses the amno acd pool from whch the enzymes are synthessed. We assume that the proten synthess s rather fast, such that the enzyme concentratons may be quckly adapted to the metabolc demands. In detal we consder the case where the system depcted n Fg contans only two reacton steps. Startng from X 1 (0) = C, X 2 (0) = X 3 (0) = 0 we are lookng for such a genetc programme of expressng the enzymes E 1 and E 2 that leads to a tme optmal converson of the ntal substrate nto the end product X 3. In mathematcal terms we are nterested to determne those functons E 1 (t) and E 2 (t) mnmzng the transton tme, τ, that s τ = 1 C (C X 3 )dt = 1 C 0 (X 1 + X 2 )dt mn (25.1) 0 under the constrant E 1 (t)+e 2 (t) = E tot = const. (for general defntons of transton tmes, see [7,8]). Furthermore, we assume that the rates of the two reactons may be descrbed by the blnear functons V 1 = k E 1 X 1 and V 2 = k E 2 X 2.Adetaled treatment of ths mathematcal problem leads to the nterestng result that 166

3 Klpp et al. X1 ATP 7 1 ADP X2 2 8 X4 NAD NAD NADH ATP ADP 3 4 ADP ATP NADH NAD NADH X CO2 Fg Smplfed model of the central metabolsm: reacton 1: upper part of glycolyss wth consumpton of 2 molecules ATP; reacton 2: lower part of glycolyss wth formaton of 2 molecules ATP and one molecule NADH; reacton 3: pyruvate decarboxylase/alcohol dehydrogenase 1; reacton 4: alcohol dehydrogenase 2 and subsequent reactons leadng to the formaton of Acetyl-CoA; reacton 5: lumped reactons of the TCA cycle leadng to formaton of 4 molecules NADH; reacton 6: oxdatve phosphorylaton; reacton 7: external ATP consumpton; reacton 8: external NADH consumpton. X 1 : glucose; X 2 : pool of trose phosphates; X 3 : pool of pyruvate and Acetyl-CoA; X 4 : ethanol. Systems equatons: V 1 = E 1 X 1 AT P, V 2 = E 2 X 2 NAD ADP, V 3 = E 3 X 3 NADH, V 4 = E 4 X 4 NAD, V 5 = E 5 X 3 NAD, V 6 = E 6 NADHADP, V 7 = k 7 AT P, V 8 = k 8,NADH dnadh/dt = dnad/dt = V 2 V 3 +V 4 +4V 5 V 6 V 8, dadp/dt = dat P/dt = 2V 1 2V 2 3V 6 +V 7, dx 1 /dt = V 1, dx 2 /dt = 2V 1 V 2, dx 3 /dt = V 2 V 3 + V 4 V 5, dx 4 /dt = V 3 V 4 the optmal E 1 (t) and E 2 (t) are not contnuous functons. Instead, the enzyme concentratons change n a dscontnuous manner between two constant values at a swtchng tme t = T. In partcular one gets wth 0 t<t: E 1 = E tot,e 2 = 0 (25.2) ( ) E T<t< : = = , E tot 2 ( ) E = = , (25.3) E tot 2 T = ( ) 1 2 ln k E tot 3 5 = k E tot (25.4) Ths soluton mples that n the frst phase (t < T ): X 1 + X 2 = C and n the second phase (t > T ): X 2 /X 1 = const. It s nterestng to note that durng the second phase E tot s dstrbuted among the two enzymes accordng to the Golden rato. The basc characterstcs of the optmal solutons gven n eqs were confrmed by applyng numercal methods, n partcular by usng a genetc algorthm. To ths end, the tme axs s dvded nto a fnte number of tme ntervals T. Each speces s charactersed by a randomly selected dstrbuton of enzyme concentratons along the chan wthn the dfferent tme ntervals and the optmum tme dependent profles E (T ) are determned by mutaton and selecton. 167

4 Reprogrammng of metabolc systems by alterng gene expresson 1.2 T 1 T 2 T 3 X4 Metabolte Concentraton s X1 X3 ATP X 2 NADH X 3 ATP NADH Tme Fg Tme course of the metabolte concentratons startng from a steady state at t = 0. The frst nterval lasts from zero tll t = T 1 = 60, here the enzyme concentratons E (1) are vald. The second nterval wth E (2) ends at t = T 3 = 83.02, where the ATP concentraton droped below the crtcal value. T 2 = s the survval tme n case that the system s not allowed to swtch the enzyme concentratons but has to work the whole tme wth E (1) (dashed lnes). Dotted lnes ndcate the tme courses after reachng the crtcal values (at T 3 ). Intal values and parameters: X 1 (0) = X 2 (0) = X 3 (0) = 1, X 4 (0) = 10, NADH(0) = 1 NAD(0) = 0.7, AT P(0) = 1 ADP(0) = 0.8, AT P crt = 0.7, NADH crt = 0.5, k 7 = 3, k 8 = 0.1 (tme, concentratons, and rate constants n arbtrary unts). Optmal expresson of enzymes of the central metabolsm of Saccharomyces cerevsae durng dauxc shft The presented approach s appled to a mnmal model of the central metabolsm of S. cerevsae to explan the relatve changes of the gene expresson levels durng the dauxc shft as descrbed n [1]. In these experments t has been observed that the gene expresson (RNA levels) for the enzymes of glycolyss and ethanol formaton are down regulated durng the depleton of glucose whereas the gene expresson for enzymes of the TCA cycle, and of some steps of the bosynthetc pathways (gluconeogeness, glycogen formaton) s ncreased. Smlar results have been obtaned by cultvatng yeast cells under glucose lmtaton condtons over 250 generatons [9]. To account for ths behavour as result of evolutonary optmzaton we consdered a smplfed reacton scheme depcted n Fg The consdered performance functon s the tme T the cells may survve gven a lm- 168

5 Klpp et al. Table 25.1 Enzyme concentratons. In the frst phase, the E (1) are fxed to the steadystate values, n the second phase the E (2) may adjust to allow maxmal survval tme. Reacton number E (1) E (2) E (2) /E (1) 1 ( upper glycolyss ) ( lower glycolyss ) ( ethanol formaton ) ( ethanol degradaton ) ( TCA cycle ) ( respratory chan ) ted ntal concentraton of glucose. The cells are consdered to be alve as long as the concentratons of ATP and NADH are above crtcal values ATP crt and NADH crt. Optmal tme courses for the enzyme concentratons E (t) of reactons 1 through 6 were determned for the crteron T max. Startng from a steady state the system s allowed to evolve accordng to the systems equatons. In a frst phase, the enzyme concentratons must be the same as n the steady state, n the second phase they can be adjusted (at fxed total enzyme concentraton) to yeld a longer survval tme. Optmal solutons are determned usng the genetc algorthm descrbed above. For a certan set of parameters the respectve enzyme concentratons are gven n Table 25.1 and the tme courses of the metabolte concentratons are depcted n Fg From Fg t becomes obvous that the cell can survve remarkably longer f t s allowed to change the concentratons of the ndvdual enzymes. In the presented case avalable enzyme s shfted from reactons 2 and 3 ( lower glycolyss and ethanol formaton ) to reactons 4 and 5 ( ethanol degradaton and TCA cycle ), whle reactons 1 and 6 are also nfluenced, but to a lower extend. The storage compound, ethanol, s used n dfferent tme regmes dependng on whether enzyme concentraton swtches are allowed or not. The tendency of changes corresponds to the expermental results descrbed n [1,9]. The presented approach s new snce t creates for the frst tme a brdge between two knds of well elaborated theores: genetc networks for the explanaton of genetc actvty patterns and metabolc modellng under evolutonary optmzaton assumptons [3,4,6]. References 1. DeRs, J.L., Iyer, V.R. and Brown P.O. (1997) Explorng the metabolc and genetc control of gene expresson on a genomc scale. Scence 278, Norbeck, J. and Blomberg, A. (1997) Metabolc and regulatory changes asso- 169

6 Reprogrammng of metabolc systems by alterng gene expresson cated wth growth of Saccharomyces cerevsae n 1.4 M NaCl - Evdence for osmotc nducton of glycerol dssmlaton va the dhydroxyacetone pathway. J. Bol. Chem. 272, Henrch, R. and Klpp, E. (1996) Control analyss of unbranched enzymatc chans n states of maxmal actvty. J. Theor. Bol. 182, Klpp, E. and Henrch, R. (1999) Competton for enzymes n metabolc pathways. BoSystems 54, Brown, G.C. (1991) Total cell proten concentraton as an evolutonary constrant on the metabolc control dstrbuton n cells. J. Theor. Bol. 153, Henrch, R., Montero, F., Klpp, E., Waddell, T.G. and Meléndez-Heva, E. (1997) Theoretcal approaches to the evolutonary optmzaton of glycolyss. Thermodynamc and knetc constrants. Eur. J. Bochem. 243, Henrch, R. and Rapoport, T.A. (1975) Mathematcal analyss of multenzyme systems. II. Steady state and transent control. BoSystems 7, Lloréns, M., Nuño, J.C., Rodríguez, Y., Meléndez-Heva, E. and Montero, F. (1999) Generalzaton of the theory of transton tmes n metabolc pathways: a geometrcal approach. Bophys. J. 77, Ferea, T.L., Botsten, D., Brown, P.O. and Rosenzweg, R.F. (1999) Systematc changes n gene expresson patterns followng adaptve evoluton n yeast. Proc. Natl. Acad. Sc. USA 96,