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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 conse- quences for the distribution of flux control. It is shown that the flux control matrix c, the elasticity matrix o, and the vector e of enzyme concentrations fulfil in optimal states the relations cTe=e and o 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, 0303-2647/99/$ - see front matter © 1999 Elsevier Science Ireland Ltd. All rights reserved. PII: S0303-2647(99)00059-3 2 E. Klipp, R. Heinrich / BioSystems 54 (1999) 1–14 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; Mele´ndez-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 (Mele´ndez-Hevia and tion which is characterized by the steady state Isidoro, 1985; Mele´ndez-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): % l 0 ( Ej + (J(E1,...,Er)−J ) =0 (3) n Ei j=1 Y % Y E i j where l denotes the Lagrange multiplier. From tot j=1 ei = n (10) this equation it follows: n % Yl l=1 (J 1 =− (4) ( l 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) ( l n J Ei E =e J j j % Yj The left hand term in Eq.
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