Implications for Optimal Distributions of Enzyme Concentrations and for the Distribution of ﬂux Control

BioSystems 54 (1999) 1–14 www.elsevier.com/locate/biosystems

Competition for enzymes in metabolic pathways: Implications for optimal distributions of enzyme concentrations and for the distribution of ﬂux control

Edda Klipp *, Reinhart Heinrich

Humboldt Uni6ersity Berlin, Institute of Biology, Theoretical Biophysics, In6alidenstr. 42, D-10115, Berlin, Germany

Received 1 February 1999; accepted 25 April 1999

Abstract

The structures of biochemical pathways are assumed to be determined by evolutionary optimization processes. In the framework of mathematical models, these structures should be explained by the formulation of optimization principles. In the present work, the principle of minimal total enzyme concentration at fixed steady state fluxes is applied to metabolic networks. According to this principle there exists a competition of the reactions for the available amount of enzymes such that all biological functions are maintained. In states which fulfil these optimization criteria the enzyme concentrations are distributed in a non-uniform manner among the reactions. This result has consequences for the distribution of flux control. It is shown that the flux control matrix c, the elasticity matrix , and the vector e of enzyme concentrations fulfil in optimal states the relations cTe=e and Te=0. Starting from a well-balanced distribution of enzymes the minimization of total enzyme concentration leads to a lowering of the SD of the flux control coefficients. © 1999 Elsevier Science Ireland Ltd. All rights reserved.

Keywords: Metabolic control theory; Evolutionary optimization; Competition; Enzyme concentrations; Flux control coefﬁcient; Metabolic pathway; Mathematical modeling

1. Introduction optimal manner. On this basis one can draw conclusions about the actual states of metabolic The structural properties of metabolic systems systems using optimization criteria. Mathemati- can be considered as a result of a long-term cally, this means the application of extremum biological evolution. Therefore, one can assume principles to the models of metabolic networks. that these systems have developed towards states The following optimization criteria have been for- where they fulfil their biological function in an mulated among others: (a) maximization of steady state fluxes in metabolic networks (Heinrich et al., * Corresponding author. Tel.: +49-30-20938383; fax: +49- 1987; Pettersson, 1993; Heinrich and Klipp, 1996); 30-20938813. E-mail addresses: [email protected] (E. Klipp), (b) maximization of the catalytic efficiencies of [email protected] (R. Heinrich) isolated enzyme reactions (Albery and Knowles,

1976; Cornish-Bowden, 1976; Mavrovouniotis et amount of glycerol 3-phosphate dehydrogenase is al., 1990; Pettersson, 1992; Wilhelm et al., 1994; produced and also other enzymes involved in the Heinrich and Klipp, 1996; Pettersson, 1996; Bish glycerol synthesis were induced to a certain de- and Mavrovouniotis, 1998); (c) minimization of gree. Furthermore, an altered expression (decrease enzyme concentration in unbranched enzymatic or increase) of glycolytic enzymes was recorded. chains (Heinrich and Holzhu¨tter, 1985; Brown, These investigations indicate that cellular enzyme 1991); (d) minimization of intermediate concen- levels are not only optimized at an evolutionary trations and total osmolarity in biochemical net- time scale but can also be rapidly adopted within works (Schuster and Heinrich, 1991; Schuster et hours. al., 1991); (e) improvement of the ATP-yield in Though it is clear that biological systems can be ADP-ATP-converting systems (Heinrich et al., optimized with respect to different criteria in the 1997; Meleńdez-Hevia et al., 1997; Stephani and following the phrase ‘optimal state’ denotes a Heinrich, 1998); (f) economic design of pathways state in which the total amount of enzyme is to fulfil the function of metabolite conversion in a minimized under maintenance of network func- minimal number of steps (Meleńdez-Hevia and tion which is characterized by the steady state Isidoro, 1985; Meleńdez-Hevia et al., 1994). fluxes. This is achieved by proper adjustment of In this paper we analyze the effects of a mini- the individual enzyme concentrations. To assess mization of the total enzyme concentration on the the properties of optimal states one has to con- distribution of the individual enzyme concentra- sider non-optimized reference states with the same tions and the consequences for the distribution of steady state fluxes. In the following we consider as flux control. There are several reasons to assume ‘reference state’ a situation where all individual that the individual amounts of enzymes within a enzyme concentrations are equal to unity (in suit- cellular system should be regulated such that the able, but arbitrary units, e.g. mmol l−1). metabolic fluxes necessary for the maintenance of Evolutionary pressure or short-term adaptation cell functions can be achieved by low total enzyme leading to a special distribution of the available amount. Enzymes are osmotically active sub- amount of enzyme will also affect flux control in stances. One strategy to achieve osmotic balance metabolic networks. One may expect that opti- is, therefore, to hold the total amount of enzyme mization will result in characteristic patterns for constrained. Furthermore, enzyme synthesis is flux control coefficients and elasticities. Flux con- very expensive for the cell, energetically as well as trol coefficients are quantities which express for a with respect to the cost of material. It is, there- given steady state of the metabolic system the fore, reasonable to assume that various pathways effect of a small change of a certain reaction rate or even individual reactions compete in some on the steady state fluxes. Elasticity coefficients sense for the available resources. Recent investiga- reflect the immediate change in the rate of a tions show that an adjustment of cellular enzyme reaction caused by a change in the concentration pattern takes place not only on a long time scale of a metabolite. but already within several hours as a consequence In this paper the consequences of the minimiza- of environmental changes. DeRisi et al. (1997) tion of the total enzyme concentration will be recorded the changes of gene expression in terms analyzed under the assumption that the reaction of mRNA levels in Saccharomyces cere6isiae at rates are linear functions of the individual enzyme changing supply of glucose. Blomberg and concentrations. In Section 2, two structurally sim- coworkers (Blomberg, 1997; Norbeck and ple but important examples are investigated, the Blomberg, 1997) investigated the salt-instigated unbranched pathway of arbitrary length and the protein expression of S. cere6isiae during growth branched network consisting of three reactions in media of different salinity. To protect against connected via one common intermediate. These osmotic stress these cells accumulate glycerol, the examples indicate that there is a special relation production of which is accompanied by overall between enzyme concentrations and flux control metabolic changes. For example, an enhanced coefficients in optimal states. In Section 3, it will E. Klipp, R. Heinrich / BioSystems 54 (1999) 1–14 3

ei be shown that this relation holds true for (Ci)E =e = (6) j j e metabolic systems of arbitrary stoichiometry. tot In the following we denote the control coefficients in optimal states by lower case letters such that 2. Structurally simple systems c =(C ) . Eq. (6) indicates that minimization i i Ej=ej of the total enzyme concentration at fixed steady 2.1. Unbranched pathways state flux leads to a state where the distribution of flux control coefficients equals the distribution of Let us consider an unbranched metabolic path- the individual enzyme concentrations. way consisting of r enzymatic reactions trans- Special distributions for ei and ci result from forming an initial substrate P1 into a final product Eq. (4) by use of special rate laws. With: P2 via the intermediates S1,S2,..., Sn with n= Vi =Ei(Si−1ki −Sik−i) (7) r−1. All rates Vi of the individual reactions should be linearly dependent on the correspond- (S0 =P1 =const, Sn =P2 =const) the equation ing enzyme concentrations but may depend in a for the steady state flux reads: nonlinear way on the concentrations of the inter- n mediates, that is: P 5 q −P 1 i 2 k J= i=1 with q = i (8) V =E f (S ,...., S ) (1) n n i i i i 1 n % 1 5 k−i qm Under steady state conditions: j=1 Ejk−j m=j+1 (Heinrich and Klipp, 1996). Introducing Eq. (8) V =J (2) i into Eq. (4) leads to: for i=1,...., r the flux J through the chain is a 0 n n J % 1 5 function of the enzyme concentrations Ei (for an ei = Yi Yl, with Yj = qm (9) explicit expression, see below). The optimal en- N l=1 k−j m=j+1 opt zyme concentrations ei =E i in states of minimal where N denotes the numerator of Eq. (8).For opt 0 total amount of enzyme etot =E tot at fixed steady calculating J we consider a reference state where state flux J=J 0 can be determined by the varia- a given total concentration of enzymes is dis- tional equation: tributed uniformly such that Ei =Etot/n. In this ( ! r " way one obtains from Eq. (8) and Eq. (9): % u 0 ( Ej + (J(E1,...,Er)−J ) =0 (3) n Ei j=1 Y % Y E i j where u denotes the Lagrange multiplier. From tot j=1 ei = n (10) this equation it follows: n % Yl l=1 (J 1 =− (4) ( u It is easy to see that this equation implies etot/ Ei 5 Etot 1. Introducing Eq. (10) into Eq. (6) yields and for the control coefficients in optimal states: ( ei J 1 ei Yi =− (5) ci = (11) ( u n J Ei E =e J j j % Yj The left hand term in Eq. (5) represents the flux j=1 control coefficient Ciof reaction i over the flux J. In the reference state the control coefficients read: Taking into account that the sum of these coeffi- Y cients over all reaction equals unity (summation i Ci = n (12) theorem of metabolic control analysis) Eq. (5) % Y u j yields =−etot/J and in this way: j=1 4 E. Klipp, R. Heinrich / BioSystems 54 (1999) 1–14

To elucidate the effect of optimization on the the coefficients decrease. In contrast to that the distribution of flux control we will consider the optimal enzyme concentrations become differenti- SDs | of the flux control coefficients defined as: ated compared to the reference state. It is worth mentioning that the effect of mini- 1 n 1 n 2 | 2 % 2 % = C i − Ci (13) mization of the total enzyme concentration on the ni=1 ni=1 SD of control coefficients depends on the choice Due to the summation theorem the second term of the reference state. This can be illustrated on the right hand side reduces to −1/n 2.With considering the case of minimal SD | 2 =0 where Eqs. (10) and (11) one gets in the optimal state: all flux control coefficients are equal to 1/n. Using Á n Â Eq. (8) it is easy to see that such a situation can Ã n % Y Ã i be realized with enzyme concentrations Ei propor- opt 2 1 i=1 1 Etot (| ) = Ã −1Ã= −1 tional to Yi. Choosing this state as reference state n 2 n 2 n 2 e Ã Ã tot Ä % Y É (instead of Ei =Etot/n), minimization of the total j enzyme concentration leads again to Eq. (10) for j=1 (14) the optimal enzyme concentrations, to Eq. (11) Fig. 1 shows the SDs of the flux control coeffi- for the control coefficients in optimal states, and cients in the reference state and in the optimal to Eq. (14) for the corresponding SD. Since etot/ 5 | opt 2 ] | ref 2 state for an unbranched chain of five enzymes in Etot 1 one finds ( ) ( ) =0, that is, an the special case of identical kinetic properties of increase of the SD of the flux control coefficients the enzymes (qi =q, k−i =k−). It is seen that the compared to the reference state, in contrast to the SD in the optimal case is lower then the one in the case with equal enzyme concentrations in the ref- reference state for all q-values. erence state. One may prove that the relation | opt/| ref 51 The proportionality between enzyme concentra- holds true also for the general case of different tion and flux control coefficients expressed in Eq. kinetic and equilibrium constants (see Appendix (6) results also from another optimization princi- A). This means that optimization leads to a state ple which is, in some sense, inverse to the consid- where the distribution of control coefficients be- ered one, namely the principle of maximization of comes more uniform, i.e. the differences between the steady state flux at fixed total amount of enzymes (Heinrich and Holzhu¨tter, 1985; Heinrich and Klipp, 1996). This result was confirmed by Brown (1991). He stated that this proportionality is valid not only for unbranched chains but for metabolic pathways of any complexity. In the following sections we show that such a conclusion is not correct and that Eq. (6) is only a special case of a more general relation.

2.2. The branched system

The branched system depicted in Scheme 1 is governed by the differential equation: dS =V +V +V (15) dt 1 2 3 Fig. 1. Standard deviations | of the flux control coefficients for an unbranched chain of five reaction as functions of the Using again linear rate laws which read for the |ref equilibrium constants with qi =q with k−i =k−. Curve : present system: reference state; curve |opt: optimal state; curve |opt/|ref: ratio of SDs. Vi =Ei(Piki −Sk−i) (16) E. Klipp, R. Heinrich / BioSystems 54 (1999) 1–14 5

Ji=

Ei[Piki(Ejk−j+Emk−m)−k−i(EjPjkj+EmPmkm)]

E1k−1+E2k−2+E3k−3 (18) with i, j, m=1, . . .,3; i"j"m. In this section the indices of ﬂuxes, enzyme concentrations, etc. run, as in Eqs. (16) and (18), from 1 to 3, if not stated Scheme 1. otherwise. We denote as reference states the situations where the three enzymes have the same

concentration Ei =Etot/3. Without loss of generality the concentration units can be choosen in such

a way that in the reference state Ei =1 and Etot = 3. The corresponding reference ﬂuxes are func-

tions of the concentrations Pi of the outer reactants. We will vary the concentrations of the outer reactants within a P-simplex deﬁned by the

constraint P1 +P2 +P3 =P. For further details of the simplex representation see Fig. 2. Choosing for example the kinetic parameters as

ki =k−i =k one obtains from Eq. (18) E k J = tot (3P −P) (19) i 9 i \ Each ﬂux Ji is positive for Pi P/3 and negative elsewhere. This divides the simplex into three

quadrangular regions Qi and three triangular regions Ti where in each case two fluxes have a different sign than the third one (Fig. 2). Fig. 2. P-simplex. The concentrations of the outer reactants are defined by the constraint P1 +P2 +P3 =P. Each corner Pi 2.2.1. States of minimal total enzyme is characterized by Pi =P and vanishing values for the other concentration outer reactant concentrations. The points Oi are defined by According to Eq. (16) one obtains at given P 0 and P P 2 for j"i. For k k k each flux J is i = j = / i = −i = i steady state fluxes J for the enzyme concentra- positive for P \P/3, zero at the line P =P/3 and negative i i i tions: elsewhere. This divides the simplex into six regions, Ti and Qi, where in each case two fluxes have a different sign than the Ji third one. The signs in the different regions indicate in the Ei = . (20) Piki −Sk−i given order the signs of the fluxes J1, J2, and J3.‘+’ means netto-flux directed from Pi towards S and ‘−’ means the States of minimal total enzyme concentrations opposite direction. may be found by variation of the concentration S such that: 3 dEtot d(E1 +E2 +E3) % Jik−i = = 2 =0 with i=1,....,3,oneobtains for the steady state dS dS i=1 (Piki −Sk−i) concentration of the internal metabolite: (21) E P k +E P k +E P k The obtained extremum is a minimum since 1 1 1 2 2 2 3 3 3 2 2 \ S= (17) d Etot/dS 0 which follows from sign(Piki − E1k−1 +E2k−2 +E3k−3 Sk−i)=sign(Ji) (see Eq. (16)). Eq. (21) consti- and for the three steady state fluxes: tutes a fourth-order equation for the 6 E. Klipp, R. Heinrich / BioSystems 54 (1999) 1–14 determination of the optimal intermediate concentration s=s(J1, J2, J3). Introducing its solution into Eq. (20) yields the enzyme concentrations ei allowing for the minimal total enzyme concentration etot. In Fig. 3a and Fig. 4a the optimal concentration of enzyme E1 and the minimal total enzyme concentration are depicted in the P-simplex representation for ki =k−i =k. Similar representations for e2 and e3 can be obtained by a cyclic interchange of the indices. Fig. 3b and Fig. 4b are the corresponding density plots. Each point in these

Fig. 4. The minimal total enzyme concentration etot is depicted in the P-simplex representation (a) and as a density plot (b) for

ki =k−i =k. Each point in these plots represents the solution of the minimization problem for the corresponding set of reference steady-state ﬂuxes.

plots represents the solution of the minimization problem for the corresponding set of reference steady-state ﬂuxes.

The heights of the surfaces for optimized ei vary $ between three levels with ei =0, ei 2/3, and $ ei 4/3. At the corner P1 as well as at the point O1, for example, one obtains: Fig. 3. Optimal concentration e1. (a) The optimal concentration of enzyme E is depicted in the P-simplex representation Á Â Á Â 1 e 1 4/3 for ki =k−i =k. (b) Corresponding density plot. Each point in Ã Ã Ã Ã e 2 = 2/3 (22a) these plots represents the solution of the minimization prob- Ä Å Ä Å lem for for the corresponding set of reference steady-state e 3 2/3

ﬂuxes. Similar representations for E2and E3 can be obtained by a cyclic interchange of the indices. etot =(8/9)Etot (22b) E. Klipp, R. Heinrich / BioSystems 54 (1999) 1–14 7

Inspection of Fig. 3A and B together with Fig. 2 P1 =P/3, where the fluxes are J1 =0 and J2 =− shows that ei is, in general, close to the highest J3 =(1/2)k(P2 −P3). Here, the analytical solution level, if the corresponding flux Ji has the opposite of Eq. (21) yields s=(1/2)(P2 +P3) and the corre- sign as the other two fluxes. This is the case, for sponding optimal enzyme concentrations are e1 = \ example, in the region Q1 where J1 0 and 0 and e2 =e3 =1. As expected, no enzyme (e1)is B B J2, J3 0, and in the region T1 where J1 0 and necessary to maintain a zero net flux J1. Accord- \ J2, J3 0. In the interior of these two regions the ingly, the case Pi =P/3 (centre of the simplex) solution (Eq. (22a)) for e1 is also approximately where all fluxes vanish is characterised by vanish- fulfilled. The fluxes in these two regions represent ing concentrations of all three enzymes. different situations. In the first case (region Q ) 1 The optimal concentration e1 depicted in Fig. one substrate (P ) is transformed into two prod- 1 3A and B and similar results for e2 and e3 yield ucts (P2,P3), and in the second case (region T1) the minimal total enzyme concentration repre- two substrates (P2,P3) are transformed into one sented in Fig. 4A and B. Since ki =k−i =k this product (P1). In both cases that reaction which plot is fully symmetrically with respect to an has to balance the other two gets the higher exchange of the indices of the external metabolites enzyme concentration in the optimal state. Close Pi. The total enzyme concentration varies in the to the boundaries of these regions rather sharp range 2/3E 5e 58/9E except for the point changes in the optimal enzyme concentrations tot tot tot Pi =P/3 where e tot =0. It is worth mentioning occur. The concentration e1 is maximal along the that in optimal states the minimal total enzyme line P P where J (2 3)k(P P ), J J 2 = 3 1 = / 1 − 2 2 = 3 = concentration is always lower than in the refer- (1 3)k(P P ), and s (1 2)(P P ) and − / 1 − 2 = / 1 + 2 ence state, whereas the individual enzyme concen- where e is the same as given in Eq. (22a). The 1 trations may be lower or higher than in the enzyme concentration e is lowest at the line 1 reference state depending on the external metabolite concentrations. In the general case the equilibrium constants may deviate from unity. In Fig. 5 optimized con-

centrations e1 are depicted for q1 =2, q2 =q3 =1/ 2 as a density plot. This distribution of equilibrium constants favours thermodynamically

the ﬂux in the direction from P1 to P2 and P3. \ B Accordingly, the region Q1 with J1 0 and J2,3 0 in the P-simplex increases compared to the

corresponding region for qi =1 (Fig. 2). As a

consequence, the region Q1 where e1 is higher then

e2 and e3 also extends towards higher values of P2

and P3, whereas the region T1 with raised e1 at B \ J1 0 and J2,3 0 shrinks. The corresponding

regions for increased e2 and increased e3 also change their shape and shift towards the line of Fig. 5. Optimal enzyme concentration e1 in the nonsymmetri- P1 =0. cal case: representation of optimal enzyme concentrations for In Fig. 6 variations of the optimal enzyme equilibrium constants deviating from unity (q =2, q =q = 1 2 3 concentrations with changing equilibrium con- 1/2) in a density plot. The ﬂux in the direction from P1 to P2 and P3 is favoured thermodynamically. Accordingly, the re- stants are shown for the case q2 =q3 =1/q1 at the gion Q with J \0 and J B0 in the P-simplex increases 1 1 2,3 point where P1 =P and P2 =P3 =0. Hence, the compared to the corresponding region for q 1 (Fig. 2), i = second and third reaction degrade metabolite S whereas the region T shrinks. The corresponding regions T , 1 2 irreversibly. For details of these dependencies see T3, and Q2,Q3 for increased e2 and increased e3 also change their shape and shift towards the line of P1 =0. the legend to Fig. 6. Note, that the tendencies at 8 E. Klipp, R. Heinrich / BioSystems 54 (1999) 1–14

m m JiC j =JjC i (25) with i"j"m. Eq. (24) contains the normalized elasticity coefficients: S (V S m i i S = ( =− Eik−i (26) Vi S Vi m which, in the following, we will abbreviate with i. Eqs. (23)–(25) represent nine equations for the nine flux control coefficients which are the elements of the flux control matrix C. One obtains: Ám m m Â 1 J1 1 J2 1 J3 1 Ãm m m Ã C=I3 −m m m 2 J1 2 J2 2 J3 1 J1 + 2 J2 + 3 J3Äm m m Å 3 J1 3 J2 3 J3 (27) Fig. 6. The dependence of the optimal enzyme concentrations on the equilibrium constants is shown for q =1/q =1/q at 1 2 3 where I3 denotes the 3×3- identity matrix. Using the point where P P and P P 0. In the case q 1, 1 = 2 = 3 = 1 Eq. (26) we can rewrite the flux control matrix C q 1, where the fluxes from P towards P and P are 2,3 1 2 3 as: thermodynamically favoured, e1 tends to its reference value E1 =1 whereas e2 and e3 tend to zero. In the opposite case 1 (q1 1) e1 tends zero where as e2 and e3 tend to their reference C=I3 − E1k−1 +E2k−2 +E3k−3 values. The curves display maxima, for e1 at q1 =1+ 3/2, for e2 and e3 at q1 =5/2− 6, and for etot at q1 =1/4. The Á Â maximum of the latter curve is characterised by e =E , J J tot tot Ã E k 2E k 3E k Ã which means that the reference state is already the optimal 1 −1 J 1 −1 J 1 −1 one. Ã 1 1 Ã ÃJ 1 J 3 Ã E2k−2 E 2k−2 E2k−2 (28) low and high q1 resemble the situation in the ÃJ2 J2 Ã unbranched chain, where in the irreversible case ÃJ 1 J 2 Ã only the reactions at the very beginning of the E3k−3 E3k−3 E 3k−3 ÄJ3 J3 Å pathway are characterized by high enzyme concentration in optimal states. 2.2.2.1. Reference states. Eq. (28) indicates that i 2.2.2. Flux control coefficients in the reference the control coefficients C i of all reactions on their state and in the optimal state own fluxes are independent of the flux distribution i and in this way also independent of the concen- The flux control coefficients C j of reactions j over the fluxes J for the branched system depicted trations of the external metabolites. Since Ei = i E 3 it follows that C i 1 k (k in Scheme 1 can be calculated using the summa- tot/ i = − −i/ −1 + k k ). In the special case k k one finds tion theorem of metabolic control analysis: −2 + −3 −i = i i " C i =2/3. The coefficients C j for i j depend on 3 the flux ratios and may be positive or negative. In % i C j =1 (23) i j=1 particular, one finds that C j =0ifJj =0 and that they tend towards infinity for Ji 0, where the the connectivity theorem: sign depends on the signs of the fluxes such that i \ " i B 3 C j 0ifsign(Ji) sign(Jj) and C j 0if % im j C j S =0 (24) sign(Ji)=sign(Jj) (Fig. 7). j=1 At corner P1 as well as at the point O1 the and the three branch-point relationships (Fell and matrix of control coefficients reads for ki =k−i = Sauro, 1985), which read for the present system: k: E. Klipp, R. Heinrich / BioSystems 54 (1999) 1–14 9

Á2/3 1/6 1/6Â C=Ã2/3 2/3 −1/3Ã. (29) Ä2/3 −1/3 2/3Å

The control matrices for the points P2,P3 and O2, O3 follow from Eq. (29) by appropriate interchange of the indices.

2.2.2.2. Optimal states. Due to the changes of the enzyme concentrations the ﬂux control coefﬁ- i cients c j in optimal states will deviate from their reference values. For ki =k−i =k one obtains from Eqs. (22a) and (28) at the corner P1 as well as at the point O1:

Á1/2 1/4 1/4Â c=Ã1/2 3/4 −1/4Ã (30) Ä1/2 −1/4 3/4Å

Fig. 8. Flux control coefficients in the optimal state. (a) The 1 control coefficient c1 in P-simplex representation for k9i =k. Values at the corners:1/2(P1) and 3/4(P2,3). In the regions Q1 $ 1 and T1, where e1 (4/3)E1, the coefficient c1 exhibits de- 1 $ creased values compared to the reference values (c1 1/ 1 $ B 2, C1 2/3). In the other regions (T2,3,Q2,3) with e1 E1 the 1 coefficient c1 is increased. These regions regions contain the line P1 =1/3 where J1 =0. Approaching this line the concentration e vanishes, while the control coefficient tends to unity. Fig. 7. Flux control coefficients in the reference state. The 1 (b) The control coefficient c1. It has the same signs, nulls and control coefficient C1 is depicted in the P-simplex representa- 2 2 singularities in the regions of the P-simplex as in the reference tion for k =k =k. C1 is zero for J =0 and tends to infinity i −i 2 2 state. The values at the corners are 1 4(P ), 1 2(P ) and for J 0 (see Eq. (26)). In the various regions (T, Q) the sign / 1 / 2 1 1 4(P ) (compare Fig. 7 for reference values). The other of the control coefficient depends on the sign of the corre- − / 3 control coefficients in optimal states can be obtained by appro- sponding fluxes: sign (Ci)=sign (−J /J ). In the corners C1 j j i 2 priate interchange of indices. assumes the values 1/6(P1), 2/3(P2), and −1/3(P3), resp. The corresponding representations for the other control coefficients Ci, i"j can be obtained by appropriate interchange of indices. j and corresponding equations at the other points The values of the control coefficients Ci are independent of the i " reactant concentrations in the reference state (=2/3 for the Pi and Oi (i 1). Fig. 8a shows the flux control 1 choosen kinetic parameters) and, therefore, are not repre- coefficient c 1 which is obtained by introducing the sented. optimal enzyme concentrations into Eq. (28). In 10 E. Klipp, R. Heinrich / BioSystems 54 (1999) 1–14

the region Q1 as well as in the region T1 charac- For ki =k−i =k numerical calculations show $ terized by increased concentration e1 (e1 4/3) that this ratio is always smaller than or equal to 1 the coefficient c 1 is decreased compared to the 3/2. The fact that this ratio is smaller than reference value. In the other regions of the P-sim- unity gives a further support to the hypothesis plex where the optimal concentration e1 is lower that the optimization of the enzyme concentra- 1 than in the reference state the coefficient c 1 is tions leads to states where control coefficients are increased. These regions contain the line P1 =1/3 distributed more uniformly than in the reference where J1 =0. Approaching this line the concen- state. tration e1 vanishes, while the control coefficient tends to unity. 2.2.3.2. Relations between control coefficients and 1 The control coefficient c 2 has in regions of the enzyme concentrations in optimal states.Inthe P-simplex the same signs as in the reference state optimized state Eq. (26) can be re-arranged to (Fig. 8B). It becomes also zero if J2 =0, i.e. for give: P =1/3 where e =0, and it has a singularity for 2 2 m opt J =0. On the corners it assumes the values c 1 = i Ji 1 2 ei =− (33) \ 1 1 B 1 sk 1/4 C 2 =1/6(P1) and c 2 =1/2 C 2 =2/3(P2) −i 1 \ 1 and c 2 =−1/4 C 2 =−1/3(P3). The other The condition (Eq. (21)) of minimized total en- control coefficients in optimal states can be ob- zyme concentration can be rewritten under con- tained by appropriate interchange of indices. sideration of Eq. (20) to give: 3 e 2k 2.2.3. General conclusions % i −i =0 (34) i=1 Ji 6 2.2.3.1. Standard de iations. It may be questioned and with the use of Eq. (33) it follows: which general conclusions concerning the change 3 of the control coefficients due to minimization of % m opt i ei =0 (35) total enzyme concentration can be drawn. In the i=1 previous section we have considered the SD of the control coefficients in the optimal compared Hence, in optimized states we have a relation to the reference state for an unbranched chain. A between the elasticities and the enzyme concen- similar consideration can be made for the trations which not necessarily holds in non-opti- branched system. Using for example the control mized states. Eq. (35) may be used to derive a simple rela- matrices given in Eqs. (29) and (30) one obtains j for the ratio of the SDs of all control coefficients tion between the control coefficients c iand the for the corners P and the points O : optimal enzyme concentrations which is indepen- i i m opt dent of the elasticties. Introducing i into Eq. | opt 3 (27) yields c j. Taking into consideration Eqs. (28) = B1 (31) i | ref 2 and (35), one derives: A general equation for the ratio of the SDs of all 3 % k control coefficients of the branched system reads: c i ek =ei (36) k=1 | opt (k +k +k ) = −1 −2 −3 The validity of this relation can be easily checked | ref (e1k−1 +e2k−2 +e3k−3) for special cases, for example, by introducing the enzyme concentrations from Eq. (22a) and 2 2 2 the control coefficients from Eq. (30). Using in- D e1k−1 e2k−2 e3k−3 + + . (32) stead of e the concentrations E =1 and for the J J J i i 1 2 3 control coefficients the values from Eq. (29) one k 2 k 2 k 2 −1 + −2 + −3 verifies that Eq. (36) is not valid for the reference J1 J2 J3 state. E. Klipp, R. Heinrich / BioSystems 54 (1999) 1–14 11

3. The general case are obtained from Eq. (41) and can be expressed in matrix form as follows: We consider a metabolic system consisting of r (V (V−1 reactions and n metabolites. Again we assume C=I −(dgJ)−1 N N(dgJ) (43) r (S (S that each reaction rate Vi is linearly related to the enzyme concentration Ei (and only to Ei) as given (dgJ) denotes a square matrix containing in the in Eq. (1). No restrictions are made with respect main diagonal the elements of the vector J. [For to the topology of the network except that, for more details of the derivation of control coeffi- sake of simplicity, conservation relations for cients see Heinrich and Schuster (1996)]. In the metabolites are excluded (for extensions to sys- following we have to consider the transpose CTof tems with conservation relations see below). The the flux control matrix C which reads: dynamics of the system is described by (V−1nT(VT dS CT =I −(dgJ)NT N (dgJ)−1 =NV (37) r (S (S dt (44) T where V=(V1,V2,...,Vr) denotes the vector of T As in the previous sections we compare systems reaction rates, S=(S1,S2,...,Sn) the vector of which are characterized by the same steady state metabolite concentrations, and N the stoichiomet- fluxes, but have different distributions of enzyme ric matrix. Under steady-state conditions the concentrations. In particular we are interested in equation system: states were the fluxes may be produced by a NV(S,E)=0 (38) minimal amount of total enzyme concentration. Let us compare first all states which are character- describes in an implicit manner the dependence ized by the same reaction rates S=S(E) of the metabolite concentrations on the T V =V 0 (45) enzyme concentrations E=(E1,E2,...,Er) . Us- i i ing these functions the steady state fluxes J= for i=1,...,r. We assume that the vector V 0 (J ,J ,...,J )T may be expressed as 1 2 r fulfils the steady state relation NV 0 =0 such that J=V(S(E),E). V 0 =J. Fixation of the fluxes leads by virtue of The control coefficients may be obtained by Eq. (1) to a relation between the enzyme concen- implicit differentiation of Eq. (38) with respect to trations and the metabolite concentrations such the enzyme concentrations: that: ( ( ( V S V 0 N +N =0 (39) V i (S (E (E Ei =Ei(S1,S2,...,Sn)= (46) fi By taking into account: Combining this equation with the principle of (J (V (V (S minimal total enzyme concentration: ( =( + ( ( (40) E E S E r % Etot = Ei min (47) one obtains: i=1 (J (V (V−1 (V leads to the variational equation: = I − N N (41) (E r (S (S (E (E r V 0 (f tot =− % i i =0 (48) ( 2 ( Ir denotes the r×r identity matrix. The scaled Sj i=1 f i Sj flux control coefficients: which determines the metabolite concentrations sj ( ( 0 i Jj Ji/ Ej in the optimal state. Since fi(s1, s2,...,sn)=V i /ei C j = ( ( (42) Ji Vj/ Ej (compare Eq. (46)) it follows: 12 E. Klipp, R. Heinrich / BioSystems 54 (1999) 1–14 r e (V ) % i i =0 (49) the functional requirements of the network. This 0( i=1 V i Sj S =s j j differentiation of enzyme concentrations can be This relation can be rewritten in matrix form as: viewed as a competition for the limited resources (VT for the synthesis of enzymes within cells. For (dgJ)−1e=0 (50) example, in the case of an unbranched enzymatic (S chain those enzymes which have high control where the derivatives have to be taken at the coefficients in the reference state should be ex- concentrations in the optimal state. Postmultipli- pressed in higher concentrations in the optimal cation of Eq. (44) with the vector e containing the state, while the concentrations of the enzymes optimized enzyme concentrations leads by virtue with lower control coefficients will decrease. A of Eq. (50) to: similar result is obtained for the branched pathway. cTe=e (51) Resulting from changes in the enzyme concen- Eq. (51) expresses the functional relation between trations the distribution of flux control coeffi- enzyme concentrations and flux control coefficients in states of minimal total enzyme cients in states of minimal total enzyme concen- concentration will deviate from the reference dis- tration. It represents the general form of Eq. (36) tribution. The flux control coefficients of those for enzymatic networks. enzymes with increased concentrations will de- Premultiplication of Eq. (50) with the diagonal crease during optimization. In the case of an matrix containing the optimal intermediate con- unbranched chain one arrives eventually at a state centrations, (dgs), yields: where the enzyme concentrations and the flux control coefficients show the same distribution. At Te=0 (52) first sight it may be surprising that enzymes with −1 ( ( where =(dgJ) ( V/ S) S=s(dgs) denotes the high concentrations, which implies a high rate of matrix of elasticity coefficients in optimal states. their reactions, exert high flux control, although It is easy to see that Eqs. (51) and (52) hold fast steps are usually considered to have low flux true also for systems which contain conservation control. Our analysis demonstrates once more relations. In this case Eq. (44) for the transpose of that high flux control is not only related to the the matrix of control coefficients is modified by a rate of the enzyme but also to its location in the link matrix L such that the term ((V/(S)T(dgJ)−1 chain. remains unaffected (Reder, 1988). Since the valid- For branched networks as studied in Section ity of Eq. (50) does not depend on the existence of 2.2 there is in optimal states no longer a propor- conservation relations Eqs. (51) and (52) remain tionality of concentrations of certain enzyme and valid. its flux control coefficients. Instead, one arrives at the more complex but also linear Eq. (36) which relate all control coefficients of a certain enzyme

4. Discussion Ei to the concentrations of all enzymes. Rewriting Eq. (36) in matrix form yields Eq. (51). This In the present paper we have shown that mini- equation is shown to be of general validity for all mization of the total enzyme concentration in metabolic networks provided that the enzyme metabolic networks at fixed steady state fluxes concentrations enter the rate equations in a linear lead generally to states where the enzyme concen- manner. The proportionality of enzyme concentrations differ from those in a reference state. trations and flux control coefficients found for the Starting from an even distribution some enzyme unbranched chain is a special case of this general concentrations increase and others decrease dur- relation. According to Eq. (51) the vector of the ing optimization whereas the total enzyme con- optimal enzyme concentrations can be viewed as centration decreases. The optimal individual an eigenvector of the transpose of the flux control concentrations are, therefore, better adapted to matrix to the eigenvalue 1. In that respect this E. Klipp, R. Heinrich / BioSystems 54 (1999) 1–14 13 relation resembles the summation theorem Ck=k, zyme concentrations may then be obtained by where k denotes any vector of the nullspace of the vanishing values of the individual enzyme concen- stoichiometric matrix (Reder, 1988; Heinrich and trations and the flux constraints are fulfilled by Schuster, 1996). The latter equation indicates that k infinite values of the metabolite concentrations. is an eigenvector of C to the eigenvalue 1. Unique solutions could be found by introducing Minimization of total enzyme concentration may upper limits for the metabolite concentrations, for contribute to a more uniform distribution of the example due to osmotic constraints. flux control among the enzymes. For the two structurally simple systems considered this ten- dency is expressed by a lowering of the SDs of these Acknowledgements coefficients compared to a reference state with non-differentiated values of enzyme concentra- We thank Holger Nathansen for ideas at the tions. This effect can be a part of an answer to the proof given in the appendix and H.-G. Holzhu¨tter question raised by Mazat et al. (1996): ‘Why are for stimulating discussions. most flux control coefficients so small?’, at least for unbranched chains where all flux control coeffi- Appendix A. Proof for (¬ 2)ref ](s2)opt for an cients are positive and their sum is constrained to unbranched chain unity. Whether or not the SD is decreased during optimization depends on the choice of the reference With the notations x = Y assertion (| 2)ref ] state as demonstrated in Section 2.1. i i (| 2)opt holds true if: As shown in Section 3 minimization of the total 3 2 enzyme concentration leads not only to a special % 2 5 % 4 % x i x i xj (A1) relation between control coefficients and enzyme i i j ] concentrations (Eq. (51)) but also to a special for xi 0. The validity Eq. (A1) may be proved as relation between elasticity coefficients and enzyme follows. One derives easily: concentrations (Eq. (52)). It is worth mentioning 3 % 2 % 6 % 4 2 % 2 2 2 that there is a direct relation between these two A= x i = x i +3 x i x j + x i x j x k, i i i"j i"j"k equations. Eq. (52) results from Eq. (51) if the latter (A2) equation is premultiplied by the transpose of the elasticity matrix and by taking into account the and: T T 2 connectivity theorem in the form C =0 which is % 4 % % 6 % 4 2 % 5 B= x i xj = x i + x i x j +2 x i xj fulfilled also for the optimal case. i j i i"j i"j In the case that the system does not contain % 4 conservation relations the optimal enzyme concen- + x i xjxk, (A3) i"j"k trations for a system with r reactions and n metabo- and in this way: lites are determined by n steady state conditions for the metabolite concentrations, r−n constraints for % 5 4 2 B−A=2 (x i xj −x i x j ) the independent steady state fluxes and n conditions i"j for the elasticities given Eq. (52). The latter equa- ¿¹¹¹¹Ë¹¹¹¹À T 1 tion results from the minimization of the total % 4 2 2 2 enzyme concentration by variation of the metabo- + (x i xjxk −x i x j x k). (A4) lite concentrations which are the only quantities i"j"k ¿¹¹¹¹¹¹Ë¹¹¹¹¹¹À which may be changed at fixed fluxes and unknown T 2 enzyme concentrations. This procedure for deter- For the first sum T to be nonnegative it is sufficient mining e may fail in the case that the system 1 i to show that the individual terms: contains irreversible reactions. The reason is that 5 4 2 5 4 2 some metabolite concentrations drop out from Eq. T1(i, j)=x i xj −x i x j +x j xi −x j x i (A5) (48) and remain undetermined. Minimal total en- are nonnegative. Since: 14 E. Klipp, R. Heinrich / BioSystems 54 (1999) 1–14

2 2 2 Heinrich, R., Klipp, E., 1996. Control analysis of unbranched T1(i, j)=xixj(x i +xixj +x j )(xi −xj) (A6) enzymatic chains in states of maximal activity. J. Theor. ] ] one obtains T1(i, j) 0 and, therefore, T1 0.For Biol. 182, 243–252. Heinrich, R., Schuster, S., 1996, The regulation of cellular the sum T2 to be nonnegative it is sufficient to show that the individual terms: systems. Chapman and Hall, New York. Heinrich, R., Montero, F., Klipp, E., Waddell, T.G., Meleń- 4 4 4 2 2 2 dez-Hevia, E., 1997. Theoretical approaches to the evolu- T2(i, j,k)=x i xjxk +x j xixk +x kxixj −3x i x j x k (A7) tionary optimization of glycolysis. Thermodynamic and kinetic constraints. Eur. J. Biochem. 243, 191–201. are nonnegative. One obtains: Mavrovouniotis, M.L., Stephanopoulus, G., Stephanopoulus, G., 1990. Estimation of upper bounds for the rates of 3 3 3 T2(i, j,k)=xixjxk(x i +x j +x k −3xixjxk) (A8) enzymatic reactions. Chem. Eng. Commun. 93, 211–236. Mazat, J.-P., Reder, C., Letellier, T., 1996. Why are most flux Without loss of generality one may assume that control coefficients so small? J. Theor. Biol. 182, 253–258. 5 ] xi xj,xk such that xj −xi =d1 0 and xk −xi = Meleńdez-Hevia, E., Isidoro, A., 1985. The game of the pen- ] tose phosphate cycle. J. Theor. Biol. 117, 251–263. d2 0. In this way one obtains: Meleńdez-Hevia, E., Waddell, T.G., Montero, F., 1994. Opti- 3 3 2 2 T2(i, j, k)=xixjxk[d 1 +d 2 +3xi(d 1 +d 2 −d1d2)] mization of metabolism: The evolution of metabolic pathways toward simplicity through the game of the pentose ] 3 3 2 xixjxk(d 1 +d 2 +3xi(d1 −d2) ) (A9) phosphate cycle. J. Theor. Biol. 166, 201–220. ] ] Meleńdez-Hevia, E., Waddell, T.G., Heinrich, R., Montero, such that T2(i, j,k) 0 and, therefore, T2 0. F., 1997. Theoretical approaches to the evolutionary optimization of glycolysis. 2. Chemical analysis. Eur. J. Biochem. 244, 527–543. References Norbeck, J., Blomberg, A., 1997. Metabolic and regulatory changes associated with growth of Saccharomyces cere- 6isiae in 1.4 M NaCl — Evidence for osmotic induction of Albery, W.J., Knowles, J.R., 1976. Evolution of enzyme func- glycerol dissimilation via the dihydroxyacetone pathway. J. tion and the development of catalytic efficiency. Biochem- Biol. Chem. 272, 5544–5554. istry 15, 5631–5640. Pettersson, G., 1992. Evolutionary optimization of the cata- Bish, D.R., Mavrovouniotis, M.L., 1998. Enzymatic-reaction lytic efficiency of enzymes. Eur. J. Biochem. 206, 289–295. rate limits with constraints on equilibrium-constants and Pettersson, G., 1993. Optimal kinetic design of enzymes in a experimental parameters. BioSystems 47, 37–60. Blomberg, A., 1997. Osmoresponsive proteins and functional linear metabolic pathway. Biochim. Biophys. Acta 1164, assessment strategies in Saccharomyces cere6isiae. Elec- 1–7. trophoresis 18, 1429–1440. Pettersson, G., 1996. A new approach for determination of the Brown, G.C., 1991. Total cell protein concentration as an selectivity favoured kinetic design of enzyme reactions. J. evolutionary constraint on the metabolic control distribu- Theor. Biol. 183, 179–183. tion in cells. J. Theor. Biol. 153, 195–203. Reder, C., 1988. Metabolic control theory: a structural ap- Cornish-Bowden, A., 1976. The effect of natural selection on proach. J. Theor. Biol. 135, 175–201. enzyme catalysis. J. Mol. Biol. 101, 1–9. Schuster, S., Heinrich, R., 1991. Minimization of intermediate DeRisi, J.L., Iyer, V.R., Brown, P.O., 1997. Exploring the concentrations as a suggested optimality principle for bio- metabolic and genetic control of gene expression on a chemical networks. I. Theoretical analysis. J. Math. Biol. genomic scale. Science 278, 680–686. 29, 425–442. Fell, D.A., Sauro, H.M, 1985. Metabolic control and its Schuster, S., Schuster, R., Heinrich, R., 1991. Minimization of analysis. Additional relationships between elasticities and intermediate concentrations as a suggested optimality prin- control coefficients. Eur. J. Biochem. 148, 555–561. ciple for biochemical networks. II. Time hierarchy, enzy- Heinrich, R., Holzhu¨tter, H.-G., 1985. Efficiency and design of matic rate laws, and erythrocyte metabolism. J. Math. simple metabolic systems. Biomed. Biochim. Acta 44, 959– Biol. 29, 425–442. 969. Stephani, A., Heinrich, R., 1998. Kinetic and thermodynamic Heinrich, R., Holzhu¨tter, H.-G., Schuster, S., 1987. A theoret- principles determining the structural design of ATP-pro- ical approach to the evolution and structural design of ducing systems. Bull. Math. Biol. 60, 505–543. enzymatic networks; linear enzymatic chains, branched Wilhelm, T., Hoffmann-Klipp, E., Heinrich, R., 1994. An pathways and glycolysis of erythrocytes. Bull. Math. Biol. evolutionary approach to enzyme kinetics: optimization of 49, 539–595. ordered mechanisms. Bull. Math. Biol. 56, 65–106.