Sensitivity of pathway rate to activities of substrate-cycle enzymes: Application to gluconeogenesis and glycolysis

J. theor. Biol. (1984) 111, 635-658

Sensitivity of Pathway Rate to Activities of Substrate-Cycle Enzymes: Application to Gluconeogenesis and Glycolysis DAVID M. REGEN AND SIMON J. PILKIS

Department of Physiology, ; "anderbilt University School of Medicine, Nashville, Tennessee 37232, U.S.A. (Received 12 May 1983, and in final form 30 April 1984) In a study of metabolic regulation, it is frequently useful to consider the degree to which an enzyme can influence the rate of its pathway. The most productive expression of rate-controlling influence is the fractional change in pathway rate per fractional change in enzyme activity (called control strength or sensitivity coefficient). We have developed a system for considering how a substrate-cycle enzyme's control strength depends on its flux and reaction order and on related features of other enzymes of its pathway. We have applied this system to the gluconeogenic pathway of rat liver and the glycolytic pathway of bovine sperm, where enough fluxes and reaction orders have been published to allow valid estimates of several control strengths. In normal fed animals where gluconeogenesis is slow and unidirectional substrate-to-product and product-to-substrate fluxes are comparable, all substrate-cycle limbs have very high and similar control strengths regardless of their flux rates and positions in the pathway. The activity of a step affects all substrate-cycle control strengths similarly as it affects unidirectional end-to-end fluxes relative to net rate. Control strengths of non-substrate-cycte enzymes are negligible compared to those of substrate cycles. In fasting animals, on the other hand, where unidirectional Pyr ~ Glc flux is much greater than Glc--> Pyr flux, upstream enzymes (near Pyr) have a regulatory advantage over downstream enzymes (near Glc). In this circumstance, control strength of each substrate-cycle enzyme is inversely related to rate limitingness between its substrate and the pathway substrate. Because the Pyr/PEP cycle is significantly rate limiting, the control strength of the P y r ~ PEP limb is much greater than that of pyruvate kinase and all downstream enzymes. In the glycolytic pathway of bovine sperm, strong product inhibition of hexokinase detracts greatly from its rate limitingness and control strength, which are very small despite its position at the beginning of the pathway and its large free energy. Because the glucose-transport-hexokinase segment is not rate limiting, phosphofructo 1-kinase has almost as much control strength as it would have as the first enzyme of the pathway, and because the F6P/FDP cycle is only moderately rate limiting, Fru-l,6-P2ase and enzymes further downstream have substantial control strengths. When glycolysis is accelerated by stimulation of phosphofructo 1-kinase, control strength shifts from 635

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© 1984 Academic Press Inc. (London) Ltd.

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phosphofructo-l-kinase and all downstream enzymes to the transporthesokinase segment. These studies extend the sensitivity analysis of Kacser & Burns (1973) to substrate cycles and show how measured pathway properties can be used to estimate sensitivities.

Introduction

The literature is replete with references to rate-limiting, rate-determining or pacemaking enzymes. The characterization of enzymes with such phrases is usually well justified; but, as emphasized by Kacser & Burns (1973), the degree to which a given enzyme determines pathway rate is rarely expressed or even considered quantitatively. Kacser & Burns proved several theorems showing how the sensitivity of pathway rate to an enzyme's activity depends on coefficients of the pathway. The sensitivity expression which they and others have used is the fractional change of pathway rate per fractional change of enzyme activity. Kacser & Burns (1973, 1979) called this expression "sensitivity coefficient" and others (Higgins, 1965; Rapoport, Heinrich and Rapoport, 1976; Groen et al., 1982) used the more popular name "control strength". Control strength of a substrate-cycle enzyme in a reversible pathway can be an awkward quantity. For example, the fractional rate change (per fractional activity change) approaches +infinity as the net rate approach zero, owing to decreased pathway substrate, increased pathway product, increased activity of substrate-cycle enzymes promoting reverse flux, or decreased activity of substrate-cycle enzymes promoting forward flux. Under these circumstances, where existing rate is not a measure of "system size", sensitivity expressions, in which rate changes are normalized to system size rather than existing rate, may contribute to sensitivity analysis. When Kacser & Burns presented their theory in its most complete form (1973), they did not deal with substrate cycles or pathways which can operate in either direction. Several of the generalizations which they expressed were not intended for substrate-cycle enzymes or reversible pathways, with the result that some workers (Hammerstedt & Lardy, 1983) have drawn the wrong conclusions about the relation between substrate-cycle rates and the rate-determining role of substrate-cycle enzymes. Other workers have treated this topic systematically (e.g. Stein & Blum, 1978), with an emphasis quite different from that of Kacser & Burns. Our approach will be to extend the logic of Kacser & Burns to substrate cycles, to show how substrate-cycle control strengths depend on pathway properties. Two methods have been used to evaluate control strength (Groen et al., 1982). One is to observe the fractional rate change incident to a known fractional

SUBSTRATE-CYCLE

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637

decrease of enzyme activity due to a specific non-competitive inhibitor of the enzyme. The other is to calculate control strength from measurements of the determinants of control strength. We will apply the latter method to the gluconeogenic pathway of rat liver and the glycolytic pathway of bovine sperm, where enough data are available for valid estimates.

Theory DEFINITION

OF CONTROL STRENGTH

Pathway control theory is concerned primarily with the responsiveness of pathway rates to changes in catalytic abilities of its enzymes. An enzyme's catalytic ability can change owing to a change in its Vmaxor changes in its affinities for substrates and products. Like Kacser & Burns (1973), we will deal only with effects of Vmaxchanges. We will analyse and evaluate control strengths defined as follows: Control strength = (d U/U)/(de/e)

(1)

where U is net pathway rate and e is amount of an enzyme or its VmaxThere are other U's whose changes are worth considering, and there are other ways of normalizing dU/de to express an enzyme's regulatory influence. One might legitimately define other control strengths according to the rate being affected (e.g. a unidirectional end-to-end flux) or the enzyme property (e.g. affinity) whose regulatory impact is under consideration. In other words, control strength is a broad concept, one area of which will be treated here. CONTROL

STRENGTHS

OF SUBSTRATE-CYCLE ENZYMES

VS N O N - C Y C L E

ENZYMES

The following analysis is for a pathway consisting of a substrate (S) several intermediates and a product (P). The conversion of each substance to the next in series is called a "step". The forward and reverse unidirectional fluxes of a step may be catalyzed by the same enzyme (e.g. isomerase, mutase) or by separate enzymes (e.g. phosphofructo l-kinase and Fru-l,6P2ase). The metabolite on one side of a step may influence the net rate of that step by promoting unidirectional flux to the metabolite on the other side of the step or by inhibiting unidirectional flux from the other metabolite to itself or both. If the step is not a substrate cycle, the effects of adjacent metabolites on its net rate can be expressed in the following equation: U = k~bEA]~b - k~[B] ~

(2a)

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where U is net rate, A and B are adjacent upstream and downstream metabolites, kab and k ~ are substrate and product action coefficients which are proportional to enzyme abundance, and the exponents, ab and/3a, are fractional changes in the respective terms per fractional changes in metabolite concentrations. If the step is a substrate cycle, a second pair o f terms is needed for the reverse limb:

U = (kab[A] "b - ko= [B] a= ) - (kb.[B] b" - k,~o[A] "°)

(2b)

where kb. is the substrate-dependence coefficient of the reverse limb and k=0 is the product-action coefficient o f the reverse limb, and the exponents, ba and aft, are the fractional changes in the respective terms per fractional changes in metabolite concentrations. The difference, kab[A] ab- k~,[B] ~', is forward-limb rate. The difference, kb,,[B] b'~-k,,o[A] "~, is reverse-limb rate. Most substrate-cycle limbs are irreversible, so these rates are the forward and reverse unidirectional fluxes o f the step, U,b and Ub,. If the step under consideration is not a substrate cycle, then a fractional change in e results in the same fractional change in all step coefficients. The generalizations expressed by Kacser & Burns (1973) refer to this kind of step. If the step under consideration is a substrate cycle, then a fractional change in the forward limb's e results in the same fractional change in k,b and k ~ with no change in kb~ or k~o, and a fractional change in the reverse limb's e results in the same fractional change in kbo and k,,~ with no change in k~b or k~. We will use the following symbols for control strengths associated with these three combinations of coefficient change: Z = d In U / d In kabb~

(3a)

X = d In U / d In k~b

(3b)

Y = d In U / d In kb~

(3c)

where d In kabb, is a fractional change in all step coefficients, d In k~b is a fractional change in k,b and k ~ , and d In kb, is a fractional change in kba and k~. Z may be viewed as the sum o f two antagonistic effects and it relates to X and Y as follows Z = X+ Y

(4)

Z is the rate limitingness o f the step whether or not it is a substrate cycle. If the step is a substrate cycle, X is the control strength of its forward limb and Y is the control strength of its reverse limb. If the step is a substrate cycle and net rate is positive, then Y is negative, and X is bigger than Z. Moreover, X and Y may both be much bigger than Z, if forward and

SUBSTRATE-CYCLE

ENZYMES

639

reverse limbs exhibit high rates relative to net rate. If the step is not a substrate cycle, then Y = 0 and Z is the control strength (i.e. X = Z). One could define sensitivities of pathway rate to each step coefficient and each exponent, and one could define sensitivities of the coefficients and exponents to more independent variables such as coreactants, coproducts or regulators. Sensitivity of pathway rate to the latter would then be calculated by chain rules and summations. We will define two imaginary sensitivity expressions for later use: V~ = fractional change in pathway rate per fractional change in both k~b and k~, Wb = fractional change in pathway rate per fractional change in both kb~ and k~. DEPENDENCE

OF CONTROL

CONCENTRATIONS

AND

STRENGTHS

EXPONENTS

ON COEFFICIENTS, OF THE PATHWAY

A simplifying concept expressed by Kacser & Burns is that the relative values of control strengths can be determined in an imaginary experiment from the fractional change in one enzyme concentration which counteracts a fractional change in another so that net pathway rate does not change. This approach can apply to any coe~cient or any quantitative property of any enzyme of the pathway, but we will apply it only to e effects. Another important fact expressed by Kacser & Burns is that the sum of all Z values is unity. From equation (4) we see that the sum of all X and Y values is unity. With these relations, one can assign approximate values to control strengths, if one knows the values of enough equation (2) terms and exponents. The dependences of control strengths on pathway properties can be examined with a simple model pathway A *-~ B ~-~ C ~-~ D ~--~E

(5)

As explained by Kacser and Burns, the ratio of adjacent Z values depends on the "elasticities" of the adjacent steps with respect to the intervening metabolite pool. An elasticity is the fractional change in a step's net rate per fractional change in metabolite concentration (other things being equal). The elasticity, e, of step A / B with respect to B is abeb = d In °bU/d In B

(6a)

where the superscript identifies the step and the subscript identifies the intermediate. The elasticity of step B / C with respect to B is bCeb= d In bcU/d In B.

(6b)

According to equation (2) step elasticities relate to coefficients as follows.

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~beb = - ( bakba[ B] b~ + ~c~k~[ B]~")/ U bee b =

(7a)

( bckbc[ B ] bc + fl ykav[ B ]~V ) / U

(7b)

where fly refers to product actions of B on the C-~ B limb, if the B / C step is a cycle. Relative Z values of adjacent steps are inversely proportional to their relative elasticities with respect to their intervening metabolite (Kacser & Burns, 1973).

Zb~/ Zob = --°%,,~bomb = (bakba[B] b'~+ flaka~,[B]~'~)/(bckb~[B] b~ + fl'yk~,[B]~'r).

(8)

Applying this relation down the line, one can show how relative Z values depend on step coefficients. The essentials of these relations were derived by Kacser & Burns from the imaginary counteraction experiments. We can perform the same imaginary experiments to show how substratecycle control strengths relate to each other, depending on coefficients, concentrations and exponents. To see the relation between X and Y of a given step, we can ask what fractional kba and k ~ change would counteract a given fractional k~b and k ~ change, so that U would not change. Since the counteracting coefficient changes are within a step, [A] and [B] would not change. For step A / B , the condition is [from equation (2)]:

[A] ~b dk~b+[A] "~ d k ~ - [ B ] b~ dkb~ - [ B ] ~ d k ~ = 0

(9a)

SO

d In k ~ b l d In kb~ =

( k b ~ [ B ] b~ - k ~ [ A ] ~ ) l ( k ~ b [ A ]

ab -

k~,[B]~"). (9b)

Control strengths are inversely proportional to the fractional coefficient changes needed for a given small fractional rate effect, so

Yab/Xab

=

-(kba[B] b'~ - k~,~[A]~'a)/(k,,b[A] ab - kt3~[B]~").

(10a)

Thus, control strengths of a substrate cycle's limbs relate to each other as their fluxes relate to each other. If the substrate cycle limbs are irreversible, the relation is Y,,b/ Xob = -- Ub~/ Uab

(10b)

where Uo is unidirectional flux from metabolite i to metabolite j. To see the relation between Y of one step and X of the next step, we can ask what fractional kbc and k~ change would counteract a given fractional kba and k ~ change, so that U does not change. The intervening metabolite will change, and the counteracting coefficient changes will relate to each other according to the dependences on that metabolite. For steps

SUBSTRATE-CYCLE ENZYMES

641

A / B and B~ C, the condition is [B] bc dkbc+kbc d[B]bc-[C]Va dk,~ + ka, d[B]O" = 0

(lla)

[B] ba dkbo+kba d [ B ] b ~ - [ A ] ~ dk,,e+ka,, d[B] a~ =0.

(lib)

This reduces to d In kbo/d In kbc = Zbc(kb,[B] bc - kvo[C]Va)/Zob(kbo[B] b~ - k~[A]~a). (12) Since control strengths are inversely proportional to the fractional coefficient changes needed for a given small fractional rate effect, Xbc/Yah

= -Zbc(kbc[n]

bc -

kva[C]V~)/Z,b(kb~[n] b~ - k,,~ [ a ] ~ ) -

(13a)

If the substrate-cycle limbs are irreversible, the relation is, Xb,/ Yah = -Zb~Ub~/ ZobUb~.

(13b)

Equation (8) can be derived from equations (4), (10) and (13), showing that the system is internally consistent and compatible with that of Kacser & Burns. From equation (8) and the fact that the sum of Z values in a pathway is unity, one can derive explicit expressions for the dependence of Z values on pathway coefficients. In our example pathway they are Z~b = 1/ ( 1 + Rb + RbRc + RbR~Rd)

(14a)

Zbc = R d (1 + Rb + RbR, + RbR,Rd )

(14b)

Z~d = RbRc/ ( 1 + Rb + RbR~ + RbRcRa )

(14c)

Zae = RbR~Ra / (1 + Rb + RbR~ + RbR~Rd ).

(14d)

In these expressions, R~ is the absolute value of the elasticity of the upstream step with respect to metabolite i relative to the elasticity of the downstream step with respect to metabolite/--as given in equation (8). This pattern of numerators and denominator terms could be extended to a pathway of any number of steps. Each Z value is the rate-limitingness of the step. From equations (4) and (10a), we see that the control strength of a substrate-cycle's forward limb is the step's rate limitingness (Z) multiplied by the forward limb's rate relative to net pathway rate. The control strength of a substrate cycle's reverse limb is the step's rate limitingness (Z) multiplied by the reverse limb's rate relative to net pathway rate. For step B / C : X~,c = ZbcUbc/ U

(15a)

Ybc = --Zb, U,b/U

(15b)

provided the limbs are irreversible.

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The above equations show how control strengths, however evaluated, depend on pathway properties. They allow one to predict control strengths from measured and/or assumed pathway properties. Our equations are designed for simplest analysis of substrate-cycle actions. A more general but more complicated treatment is possible. OTHER

NORMALIZATIONS

Although we will carry out our analysis in terms of control strengths as defined above, it is worth noting that other normalized d U/de expressions can be useful for particular purposes. To examine how pathway responsiveness to an enzyme's regulation depends on the enzyme's activity over a wide range, it is useful to normalize d U to a constant reference (rather than U) expressing responsiveness to a particular regulation when the enzyme in question is at a standard activity. It is also possible to renormalize the control strengths to other measures of pathway activity or regulability which do not become zero under any condition. The values of these sensitivity expressions reflect rate responses under all circumstances the way control strengths reflect rate responses in unidirectional pathways--actual responsiveness relative to potential responsiveness or relative to the fluxes whose changes constitute the response. We suggest that those interested in this problem consider one or both of the following renormalizations. One is to divide all control strengths by V~- Wp, where Vs is the fractional change in net rate per fractional change in the pathway-substrate coefficient(s) (k~b and k ~ of first step) and Wp is the fractional change in pathway rate per fractional change in the pathway-product coefficient(s) (kb~ and ka~ of last step). In sensitivity expressions renormalized this way, rate changes are normalized to standard rate changes rather than net rate. A similar renormalization is to multiply all control strengths by U/( Usp + Ups), where Usp is unidirectional S--,,P flux and Ups is unidirectional P-->S flux. This effectively changes the normalizing reference from U to the sum of end-toend unidirectional fluxes. Enzyme regulation affects rate by affecting both end-to-end fluxes, so their sum represents the substrate of enzyme regulation, just as U is the substrate of enzyme regulation in a unidirectional pathway. The resulting sensitivity expression is rate change (per fractional activity change) relative to the unidirectional fluxes available to be affected. Neither of these conversions changes the relative values of the sensitivity expressions, so the renormalized values still show how control is shared among the enzymes. One can calculate a unidirection end-to-end flux from measured and assumed step fluxes, based on the relation U~k= UuUjk/(Uji+ Ujk). It says

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643

that unidirectional flux forward across two steps is unidirectional flux forward across the first multiplied by the probability that the first-step product (j) will make the second step (to k) rather than go back (to i). To calculate unidirectional substrate-to-product flux, one calculates unidirectional flux across the first two steps, then treats these two steps as the first step (ij) and the third step as the second (jk) and repeats the calculation. The process is repeated until the last step is the jk step. This is much easier and more flexible than driving an equation for the whole series. IMPLICATIONS OF THE THEORY

Equations (14) and (15) summarize the important determinants of control strength. If the coefficients of a step cannot be changed independently by e changes, then the control strength of the enzyme is the Z value of the step, its rate-limitingness. This would be the case with most monomolecular rearangements and many irreversible, non-substrate-cycle, bimolecular reactions exhibiting product inhibition (e.g. hexokinase without phosphohexose phosphatase). Since an enzyme's elasticity is proportional to its activity, its control strength (X = Z) declines and those of other enzymes increase as its activity increases relative to others in the pathway. Increasing the activities of these enzymes transfers control strength to other enzymes. Likewise, the Z value (rate limitingness) of a substrate cycle declines as the activities of its enzymes increase, thereby enhancing the control strengths of other enzymes. However, this may increase the control strengths of the substrate cycle's enzymes. The effect seen in equation (15) of increasing cycle fluxes relative to net rate may outweigh the reduction of rate limitingness. In other words, the increase of X and Y relative to Z, may outweigh the reduction of Z. These relations will be seen in the examples which follow.

Application Evaluation of an enzyme's control strength in situ is difficult. Each of the two available methods is applicable only under special, restricted circumstances. The control strength of an enzyme is best determined from the fractional reduction of pathway rate incident to a small fractional inhibition of the enzyme by a specific non-competitive inhibitor. Specific non-competitive inhibitors are very rare. The procedure usually involves an assumption about the activity-vs-inhibitor relation (e.g. that non-specific inhibitor binding is unimportant), and it always involves an assumption that activity changes are not obscured owing to regulation of the same enzyme (or others) by signals outside the pathway.

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One can sometimes estimate control strengths from measured and/or assumed pathway properties. The significant properties are the sizes of equation 2 terms and the exponents of their metabolite dependences. The coefficients and concentrations need not be known. Whenever product inhibition is not significant (Greek-letter terms<< English-letter terms), the terms and exponents are fluxes and their reaction orders. HEPATIC

GLUCONEOGENESIS

Groen et a t (1983) examined the relations between gluconeogenic rate and concentrations of intermediates between PEP and GIc-6-P, as these varied with lactate concentration (Pyr=0.1 Lact). Over a wide range, gluconeogenic rate was approximately proportional to Fru-1,6-P2 (free) and hexosemonophosphate concentrations, and these were approximately proportional to [PEP] 2. The second-order dependence on PEP is consistent with stoichiometry at the aldolase step. Pyruvate kinase rate was approximately proportional to [PEP] 3. This high-order dependence may have been due in part to activation by Fru-l,6-P2, which, as noted, increased with [PEP] 2. Normally, the gluconeogenic pathway is not very saturated by intermediates, for rates increase with substrate concentrations well beyond the physiological range (Exton & Park, 1967, 1969; Claus, Pilkis & Park, 1975). Therefore, reaction orders (other than pyruvate kinase) should reflect stoichiometries, and product inhibitions should be much less important than substrate dependences. Under normal conditions, the major glucosephosphorylating enzyme, glucokinase, has a high K,, and exhibits little product inhibition (Vinuela, Salas & Sols, 1963). That product inhibitions are less significant than substrate dependences is revealed by the fact that the effects of glucose on glucose production are independent of lactate concentration (Exton & Park, 1967; Clark et al., 1974; Claus et at., 1975). Under some conditions, glucose does not significantly affect unidirectional lactate-~ (glucose + glycogen) flux (Exton & Park, 1967). Under other conditions, glucose inhibits this flux--apparently by raising the level of Fru-1,6-P2 which increases pyruvate kinase activity (Rognstad, 1982). We conclude, therefore, that the gluconeogenic pathway is one in which the Greek-letter terms of equation (2) are insignificant and metabolite effects are predominantly substrate dependences (except for the effect of Fru-l,6-P2 on pyruvate kinase). We can calculate elasticities as products of reaction orders and fluxes relative to net rate [equation (7)]. We will estimate control strengths of the substrate-cycle limbs from published flux measurements. We will assume that phosphohexose isomerase is not significantly rate limiting, for this enzyme maintains near

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SUBSTRATE-CYCLE ENZYMES

equilibrium between GIc-6-P and Fru-6-P ( E x t o n & Park, 1969). The numerous steps between PEP and Fru-l,6-P2 might limit rate significantly and we will consider this possibility. However, the fact that hepatocytes convert more dihydroxyacetone to lactate than to glucose (Pilkis, Riou & Claus, 1976), despite the rather slow normal rate of the Pyr/PEP cycle (Hue, 1982) indicates that the PEP/Triose segment and lactate dehydrogenase step do not limit rate substantially. FED ANIMALS

Table 1 shows control strengths and other sensitivity expressions of enzymes in the hepatic gluconeogenic pathway of fed rats, based on most of the above assumptions and the cycle rates summarized by Hue (1982). With normal substrates and products, the net pathway rate may be slightly positive or slightly negative or zero. Had we assumed zero net rate, the X and Y values would be ±infinity, but all other sensitivity expressions would TABLE 1

Sensitivities o f gluconeogenic rate to substrate-cycle enzymes in fed rats, with simplest assumptions. Assumed rates (in hexose equivalents t z m o l / m i n . g ) were as follows: net glucose synthesis = 0.001, i.e. very slow; pyruvate kinase = O.2; phosphofructo 1-kinase = O. 17; glucokinase = O.9; unidirectional Fru. 1,6P2-~ PEP =900, i.e. very fast. The exponent on [PEP] in the PEP-~ Fru- 1,6-P2 flux terra was assumed to be 2; all others were assumed to be 1 Pyr/PEP Z X Y

0.59 119 - 118

PEP/FDP

FDP/F6P

G6P/GIc

7 x 10-s

0.34 59 -59

0-07 59 -59

Z / ( V ~ - Wp) X /( Vs - Wp) Y/( V~ - Wp)

3 x 1 0 -3 0.67 -0.67

4×10 -7

2 x 1 0 -3 0-33 -0.33

4 x 1 0 -4 0-33 -0.33

ZU/(Uspq-Ups )

4×10 -3

4 x l O -7

2 x l O -3

4 x l O -4

x u / ( u~ + u~,)

0.71

0.35

0.35

U,p+ Up,)

-0.71

-0.35

-0-35

YU/(

Vs is the fractional change in pathway rate per fractional change of equation (2) terms containing the pathway substrate. W, is the fractional change in pathway rate per fractional change in equation (2) terms containing the pathway product. Note that V~= 119 and Wp = -59, so V~- Wp = 178. Based on the unidirectional fluxes at each step, the end-to-end fluxes are U,p = 0-084 and Ups = 0.083, so ( U~p + Ups)~ U = 167. These are calculated by repeated application of the principle that unidirectional flux forward across two steps is unidirectional flux forward across the first step multiplied by unidirectional flux forward across the second step divided by the latter plus unidirectional flux backward across the first step.

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be as shown. To avoid infinite X and Y values, we assumed a low positive net rate, 0.001 l~mot/min.g. In Table 1, we assumed that the conversion of PEP to Fru-l,6-P2 is a second order reaction, consistent with data of Groen et al. (1983) and stoichiometries, but that all other reactions are first order. Specifically, the third-order dependence of pyruvate kinase rate on PEP, observed by Groen et al., was not assumed in Table 1. This table shows what the sensitivities would be in the absence of the high order dependence of pyruvate kinase on PEP or stimulation of this enzyme by Fru-l,6-P2. Under the conditions of this example (reverse flux similar to forward flux at each step), Z values are individually inversely related to fluxes, independent of position with respect to ends of the pathway but dependent on position with respect to the high-order reaction (PEP~ Fru-2,6-P2). X and Y values are similar to each other, much larger than Z values and individually not inversely related to respective fluxes. The high-order dependence of PEP-> Fru-l,6-P2 flux on PEP transfers control strengths (Z, X and Y) from all PEP/GIc enzymes to Pyr/PEP enzymes. Table 1 also illustrates that normalization to standard responses and normalization to end-to-end fluxes can be similar to each other. These alternate sensitivity expressions exhibit relative control strengths even if U were zero and X and Y were ±infinity. The renormalized control strengths show rate changes (per fractional activity changes) relative to system size under conditions where U represents system size and under conditions (as in Table 1) where U does not represent system size. Table 2 shows how control strengths are affected by various features of the pathway. Set A is the same as that of Table 1. Comparing set B with set A, we see that the third order dependence of pyruvate kinase on PEP (probably reflecting a first order dependence on PEP and on Fru-l,6-P2 which increases with [PEP] 2) shifts control strength from the Pyr/PEP cycle to all enzymes of the PEP/GIc segment (or FDP/GIc segment if the dependence reflects Fru-l,6-P2 activation). The Z, X and Y values are affected proportionally. The result is a more even distribution of control strengths among the substrate cycles. The distributions of rate limitingness (Z) and control strength (X, Y) among the substrate cycles as given in set B are accurate according to all information at our disposal (described in earlier paragraphs). That all substrate-cycle control strengths are high is also realistic, though "how high" depends on "how low" the net rate relative to unidirection end-to-end fluxes happens to be in a given liver. Comparing set C with set B, we see that a great reduction of activity in the PEP/Fru- 1,6P2 segment (flux = 0.5) increases rate limitingness of this segment to 0.16, and detracts modestly from all other control strengths. Comparing sets D and E with set B illustrates the same effect in reverse; increasing activity

SUBSTRATE-CYCLE ENZYMES

647

TABLE 2 Control strengths of gluconeogenic enzymes over net rate in fed rats, with various assumptions regarding reaction orders and step fluxes Pyr/PEP A

B

C

D

Assumptions of Table 1 Z 0"59 X 119 Y - 118

PEP/FDP

FDP/F6P

7 x 10-5

G6P/Glc

0-34 59 -59

0-07 59 -59

Same as A but pyruvate kinase third order in [PEP] Z 0.32 1 x 10 -4 X 65 Y -65

0.57 97 -97

0.11 97 -97

Same as B but Fru-l,6-P 2 ~ PEP flux = 0.5 Z 0.27 0.16 X 55 Y -54

0.47 81 -81

0.09 81 -81

Same as B but F r u - 6 - P ~ Fru-l,6-P2 flux doubled Z 0-45 2 x 10-4 X 91 Y -90

0-40 136 -135

0.15 135 -135

Same as B but PEP--> Pyr flux doubled Z 0.19 1 × 10-' X 78 Y -77

0.68 116 -115

0.13 115 -115

of any step reduces its rate limitingness (Z) and raises all other control strengths proportionally, including X and Y of that step. Two features of this table deserve emphasis. (a) Under normal conditions, where opposing end-to-end unidirectional fluxes are comparable, the control strengths of all substrate-cycle enzymes are very high and similar to each other, regardless of enzyme activities and cycle fluxes. (b) Under these conditions, reaction orders are the main factors influencing distribution of control strength among the substrate cycles. The apparent third-order dependence of pyruvate kinase on PEP tends to enhance the influences of the Fru-6-P/Fru-l,6-P2 and Glc/GIc-6-P cycles, whether the dependence is really third order or first order in both PEP and Fru-l,6-P2. This more than neutralizes the second-order dependence of PEP--> Fru-l,6-P2 flux on PEP. FASTING A N I M A L S

Table 3 shows a similar analysis based on cycle rates from fasting animals as summarized by Hue (1982). In the presence o f normal substrate con-

648

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A N D S. J. P I L K I S

TABLE 3

Control strengths of gluconeogenic enzymes over net rate in fasting rats, with various assumptions regarding reaction orders and step fluxes. Except as noted, sensitivities are based on the following assumed rates (in hexose equivalents tzmol/ min. g ) : net glucose synthesis = 0 . 3 ; pyruvate kinase = 0 . 0 8 ; phosphofructo 1-kinase = 0 . 0 5 ; glucokinase = 0 . 4 ; unidirectional Fru- 1,6-P2 -~ PEP flux = 4 0 0 . T h e exponent of the P E P --> Fru- 1,6-P2 flux term was assumed to be 2; all others were assumed to be 1 Pyr/PEP A

B

C

D

E

Assumptions as stated above Z 0.89 X 1.13 Y -0.24

PEP/FDP

FDP/F6P

G6P/GIc

9 x l 0 -5

0.10 0.12 -0.02

7 x l0 -s 0.02 - 9 x l0 -3

Same as A but pyruvate kinase third order in [PEP] Z 0"73 2 × 10-4 0"25 X 0'93 0'29 Y -0'20 -0'04

0"02 0'04 -0'02

Same as B but Fru-l,6-P2-* PEP flux=0.2 Z 0.72 0-17 X 0.91 Y -0.19

0.10 0.12 -0.02

0.01 0.02 -0.01

Same as B but Fru-6-P~ Fru-l,6-P 2 flux doubled Z 0-74 1 x 10-4 X 0-94 Y -0.20

0-22 0-30 -0.07

0.03 0.07 -0"04

Same as B but P E P ~ P y r flux doubled Z 0.58 3 x 10-4 X 0.88 Y -0-31

0-39 0.46 -0.07

0.03 0.07 -0.04

tn set A, V~= 1-13 and W p = - 0 - 0 1 , so V ~ - W p = 1.14. In set A, unidirectional end-to-end fluxes are U,p = 0.305 and Up, = 0-005, so (U,p + Ups)~U = 1-03. Therefore, renormalized control strengths would be only a few percent less than control strenghts.

centrations, gluconeogenic rates in livers of fasting rats are about 0.3 i~mol/min.g (Exton & Park, 1967, 1969). In fasting animals, with all substrate cycles biased in the forward direction, the pathwa2~ as a whole is almost "irreversible" (unidirectional P ~ S flux = 0-02 S--> P flux). If the data for fed and fasting animals are internally consistent, the enhancement of net rate incident to fasting is associated with enhanced forward fluxes at the Pyr/PEP and FDP/F6P cycles and reduced reverse fluxes at all cycles. The large discrepancies between forward and reverse fluxes at the Pyr/PEP

SUBSTRATE-CYCLE

ENZYMES

649

cycle and the FDP/F6P cycle are reflected in the control strengths of these steps. Y is now much less than X; and X is only a little greater than Z. With the pathway now strongly biased in the forward direction, there is a marked positional effect favoring all control strengths of upstream enzymes (Z, X and Y). Comparing set B with set A, we see the effect of high-order dependence of pyruvate kinase on PEP or control of this enzyme by Fru-l,6-P2, shifting control strength from the Pyr/PEP cycle to enzymes downstream. This does not outweigh the positional effect, and the Pyr~ PEP limb remains the dominant controller, by contrast to the fed state. Comparing set C with set B shows that rate limitingness in the PEP/FDP segment would detract substantially from control strengths of downstream reactions but not upstream reactions, when the pathway is almost unidirectional. Comparing set D with set B we see that control strengths of the G6P/Glc cycle and of phosphofructo l-kinase would have been higher, had we assumed higher FDP/F6P fluxes; but the difference is inconsequential, since these downstream control strengths are so low when the pathway is almost unidirectional and upstream reactions are rate limiting. Comparing set E with set B, we see that faster Pyr/PEP cycling would result in greater control strengths of pyruvate kinase and downstream reactions with slightly less control strength of the P y r o PEP limb. These effects on pyruvate kinase and Fru-l,6-P2ase control strengths are substantial since the control strengths of these enzymes are substantial. Nevertheless, the pathway is still most sensitive to the Pyr~ PEP limb. A general conclusion from this analysis is that control strengths of substrate-cycle enzymes are not exceptionally large when the pathway is almost unidirectional. Considering Tables 2 and 3 together, we can recognize three main factors determining substrate-cycle control strengths (X and Y). (a) Control strengths of all substrate cycles increase with overall pathway activity relative to net rate. (Overall pathway activity relative to net rate is given by Vs- Wp or by ( Lisp+ Ups)/U, and this factor is cancelled out of the renormalized control strengths.) (b) If the exponents of a metabolite's actions on adjacent upstream and downstream steps differ, control strength is shifted from all steps on one side of the metabolite (the side of greater exponent) to all steps on the other side. (c) Rate limitingness between the pathway substrate and a substrate-cycle limb's substrate detracts from the substrate-cycle limb's control strength. In the fed animal, where all fluxes are large relative to net rate, the first X (or V,) is so large relative to unity that the step-down due to upstream rate limitingness is inconsequential. In the fasting animal, where some reverse fluxes are small relative to net rate, the first X (or V~) is not large relative to unity, and the step-down due to upstream rate limitingness is significant.

650

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GLYCOLYSIS

S. J. P I L K I S

IN SPERM

Hammerstedt & Lardy (1983) studied glycolytic and substrate-cycle fluxes in beef sperm in artificial medium (Table 4). Respiratory poisons (antimycin A and rotenone) increased glucose uptake and glycolysis 14-fold (Table 5), apparently by virtue of phosphofructo 1-kinase activation resulting in a 70-90% fall in hexose monophosphates (product inhibitors of hexokinase) and a 10-fold rise in Fru-1,6-P2 (substrate of downstream steps). The authors made several observations of rate-metabolite relations which are germane. Net rate was approximately proportional to Fru-l,6-P2 concentration over a substantial range. We will, therefore, assume first-order dependence of Fru-1,6-P2 ~ Pyr flux on Fru-1,6-P2. Stimulation (of phosphofructo 1-kinase) did not reduce Fru-6-P more than GIc-6-P, so we will assume that isomerase is not rate limiting and will treat the hexose monophosphates as a single pool. (~arbohydrate uptake as a function of hexose monophosphate followed a relationship U=2.4-2.28([HMP]/[HMPo]) I, where [HMP] is Glc-6-P or Fru-6-P concentration and the subscript o means that which actually prevails. This empirical equation accounts for changes in glucose/HMP and fructose/HMP traffic, and suffices for a calculation of C H 2 0 / H M P elasticity and rate limitingness (Z). To calculate X and Y, we assume a high order of Fru-6-P hydrolysis, ba = 3, with an hydrolysis rate (kba[B] ha) of 0.13. The high reaction order is to account for the virtual cessation of fructose formation when HMP fell by half. To agree with the empirical relation, the exponent of hexokinase product inhibition (fla) was 0.89 and the inhibition term (k~ [B] t3~) was 2.15. The increase of Fru- 1,6-P2 -~ Fru-6-P flux incident to the rise in Fru-l,6-P2 concentration and fall in Fru-6-P concentration was less than proportional to the rise in Fru- 1,6-P2 concentration. The system therefore behaves as if the reaction order of Fru-l,6-P2 hydrolysis is less than 1 and as if this reaction is not product inhibited. Phosphofructo 1-kinase is not product inhibited in systems where this has been examined. We will, therefore, assume that the Greek-letter terms of equation (2) for the Fru-6-P/Fru-l,6-P2 step are negligible. Table 4 shows control strengths of the glycolytic pathway with two reaction orders of Fru-l,6-P2 hydrolysis and with two Fru-6-P/Fru-l,6-P2 cycling rates. The analysis reveals several interesting points which are similar regardless of these assumptions. The C H 2 0 / H M P step is not very rate limiting (Z = 0.05 to 0.06), owing largely to strong product inhibition of hexokinase. For the same reason, hexokinase is not very rate determining (X =0.1 to 0.14), despite its position at the first of the pathway and its large free energy. An increase of hexokinase results in a similar increase of positive and negative equation (2) terms of the C H 2 0 / H M P step. The great

SUBSTRATE-CYCLE

65I

ENZYMES

TABLE 4 Sensitivities of beef-sperm glycolysis to glycolytic enzymes under "basal" conditions. Except as noted, sensitivities were based on the following assumed rates (in tzmot/ h. 108 cells): glycolytic rate = O.12 ; Glc ~ Glc-6-P flux =0.25; Glu-6-P ~ Glc flux = 0 ; Fru.6.P-+ Fru flux =0.13; Fru6 - P ~ Fru-I,6-P2 flux=0.24; Fru-l,6-P2~glycolyticproduct flux = O. 12. As explained in text, Fru-6P hydrolysis was assumed to be third order, and Glc Glc-6-P flux was given by the relation, 2-42.15([ H M P ] / [ HMPo]) °'89. Except as noted, other reactions were assumed to be first order CH20/HMP A

B

C

D

Based on above assumptions Z 0"05 X 0"10 Y -0.05

F6P/FDP

FDP/Pyr

0"48 0"95 -0.48

0.48

Same as A but Fru-l,6-P2~ Fru-6-P flux halved Z 0.05 0.63 X 0. I0 0.95 Y -0.05 -0.32 Same as A but Fru-l,6-P2 hydrolysis 0-5 order Z 0.06 0.62 X 0.14 1.25 Y -0.07 -0-62 Same as C but Fru-l,6-P2~Fru-6-P flux halved Z 0-06 0"75 X 0-12 1-13 Y -0-06 -0-38

0-32

0.31

0" 19

elasticity hence low rate limitingness of the C H 2 0 / H M P step passes control strength to subsequent steps. The X (0.95 to 1.25) of the F6P/FDP step is control strength of phosphofructo 1-kinase and the Y (-0.32 to -0.62) is control strength of Fru-l,6-P2ase. Phosphofructo l-kinase is the main beneficiary of the high hexokinase elasticity and low C H : O / H M P rate limitingness and is the enzyme to which the pathway is most sensitive. A slower cycling at this step increases its rate limitingness and detracts from control strength of Fru-l,6-P2ase and downstream enzymes, but does not affect the control strength of phosphofructo 1-kinase. A lower reaction order of Fru-1,6-P2 hydrolysis raises control strengths upstream of Fru-1,6-P2 and detracts from downstream control strengths. The lower reaction order is

652

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rather like inhibiting the enzyme when enhanced phosphofructo l-kinase raises Fru-l,6-P2 level. Indeed, the apparent low order o f Fru-l,6-P2 hydrolysis could have been due to simultaneous inhibition of Fru-l,6-P2ase by the signals which stimulated phosphofructo l-kinase. Thus, the pathway is very sensitive to phosphofructo l-kinase owing to the potent product inhibition o f hexokinase, and this sensitivity is enhanced further by the low substrate dependence of Fru-l,6-P2ase but is independent o f F 6 P / F D P cycling. As glycolysis accelerates by virtue of phosphofructo 1-kinase stimulation and reduced product inhibition of hexokinase, the latter enzyme becomes more rate limiting. This is illustrated in Table 5 (vs Table 4), where p h o s p h o fructo l-kinase control strength is reduced and hexokinase control strength is increased, the transport-hexokinase segment having about three times the TABLE 5

Sensitivities of beef-sperm glycolysis to glycolytic enzymes in presence of respiratory poisons. Except as noted, sensitivities were based on the following assumed rates (in txmol/ h. 10 s cell): glycolytic rate = 1.75; Glc-6-P-> Glc f l u x = 0 ; Fru-6-P ~ Fru flux=O; Fru-6-P ~ Fru-l,6-P2 f l u x = 2 . 0 3 ; Fru-l,6-P:~Fru-6-P flux=0.28; Fru-l,6P2 -> glycolytic-product flux = 1"75. GIc -> GIc-6-Pflux was given by the relation, 2.4-0.65([ H M P ]/[ HMPo]) w.Except as noted, all reactions were assumed to be first order CH20/HMP A Based on above assumptions Z 0.73 X 0.73 Y -0 B

F6P/FDP

FDP/Pyr

0.23 0.27 -0-04

0.04

Same as A but Fru-l,6-P2~ Fru-6-P flux halved Z 0.73 0.25 X 0.73 0.27 Y -0 -0.02

Same as A but Fru-l,6-P2 hydrolysis 0-5 order A 0-74 0.24 X 0.74 0.28 Y -0 -0.04 D Same as C but Fru-l,6-P2-~ Fru-6-P flux halved Z 0.74 0.25 X 0.74 0"25 Y -0 -0.02

0.02

C

0.02

0.01

SU BSTRATE-CYCLE

653

ENZYMES

control strength of phosphofructo l-kinase. This relation is indifferent to F6P/FDP cycling rate and Fru-l,6-P2ase reaction order. Hammerstedt & Lardy (1983) expressed the view that phosphofructo 1-kinase is more "rate controlling" when its cycling rate is slower relative to net rate. They also expressed the view that phosphofructo 1-kinase is more "rate controlling" in the stimulated cells than in the control cells. Both of these views are opposite to our conclusions. Discussion CONTROL

STRENGTH

AND

OTHER

SENSITIVITY

EXPRESSIONS

Traditionally, sensitivity expressions are fractional changes of dependent variables per fractional changes of independent variables. Control strength as usually defined is the fractional change of net pathway rate per fractional change of an enzyme's concentration, Vmaxor other activity-proportional coefficient. We have carried out our analyses in terms of control strength defined this way. It should be recognized that control strength does not necessarily reflect the responsiveness to a change in an enzyme's affinity for substrate or product. High saturation of an enzyme with its substrate tends to reduce elasticity with respect to the substrate. This tends to increase its control strength (responsiveness to a Vm,xchange) but to reduce responsiveness to an affinity change. For example, the 02 reactive enzymes of respiration are highly saturated with 02 under normal conditions, and a change in their 02 affinity would hardly affect respiratory rates. Since most enzymes are not very saturated by their substrates, their activity coefficients such as kb~, are approximately proportional to Vmax/Km and increase comparably with a given fractional increase of Vm~xor fractional decrease of Kin. Under these circumstances, control strength reflects responsiveness to affinity changes. If one examines the control strength of a substrate-cycle enzyme and a non-substrate cycle enzyme of the same pathway as the substrate-cycle activity is varied over a wide range, one sees that the control strength values per se do not always reflect the regulatory influences of the enzymes and the significance of e changes in relation to the "size" of the system. Sometimes the regulatory influence and economic significance of regulation are better seen in unnormalized, denormalized, renormalized or differentlynormalized d U / d e expressions. A large positive control strength may not mean that a small increase in the enzyme would enhance net S -> P conversion substantially relative to system size, and a large negative control strength may not mean that a small increase in the enzyme would reduce net S ~ P

654

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S. J , P I L K I S

conversion substantially relative to system size. The X and Y values in Tables 1 and 2 are large not because the response is large but because the normalizing reference, U, is small relative to system size. If all pathway coefficients were to remain constant except Fru-l,6-P2ase activity and this were reduced slightly from that of Tables 1 and 2, its control strength could be +infinity; and if its activity were reduced slightly more, its control strength would become -1000 or other large negative value. Naively interpreted, the latter might suggest that a small increase of Fru-l,6-P2ase would reduce gluconeogenesis substantially. To appreciate what the control strength value means, one must take the normalizing reference (U) into account. By contrast, the renormalized control strengths in Table 1, would exhibit similar values under these three conditions, continuing to reflect responsiveness relative to the activities and concentrations with which the pathway operates (system size). The very small renormalized Z values under this circumstance, correctly reflect the insignificance of simultaneous increases in all coefficients of a step when P ~ S flux is comparable to S ~ P flux. In fact, the responses to substrate-cycle regulation in Tables 2 and 3 are similar, as the renormalized control strengths of Table 1 are similar to the control strengths (renormalized or not) of Table 3. While the renormalizations which we have suggested may not be ideal, they illustrate a valid p o i n t - when net rate does not reflect system size, it may be informative to normalize d U not only to U but also to a reference which reflects system size. CALCULATIONS

OF CONTROL

STRENGTH

FROM

PATHWAY

PROPERTIES

Given the dearth of specific noncompetitive inhibitors in existence and the fact that their use is not without uncertainties and assumptions, we would like to suggest that calculations based on observed and assumed pathway properties have a rightful place in sensitivity analysis. For this purpose, the significant properties are the values of the equation (2) terms for various steps or segments of the pathway and the exponents of the metabolite effects on those terms. By varying the values of the terms and exponents, one can observe the importance of errors in the measurements or assumptions. Doing this in Tables 2-5, we saw that the conclusions would be valid despite substantial errors in the measurements or assumptions. LESSONS

FROM

THE

ANALYSES

We have calculated control strengths of the gluconeogeneic pathway when its net rate is small relative to end-to-end fluxes (fed state) and when its reverse flux is insignificant relative to its net rate (fasting state). In this

SUBSTRATE-CYCLE

ENZYMES

655

pathway it was reasonable to assume that equation (2) terms of the substrate cycle steps were the corresponding fluxes and that the equation (2) exponents were the reaction orders of those fluxes. This assumption might lead to a modest underestimate of step elasticities. However, this error would be of no consequence to the calculations for fed animals, since the important determinants of control strengths were the similarities of unidirectional end-to-end fluxes and the reaction orders. The only non-unity reaction orders are those of P E P o Fru-l,6-P2 (or P E P ~ Glc) flux and PEP--> Pyr flux. These are both greater than unity and tend to cancel each other, so that control strengths of all substrate-cycle limbs are high and comparable to each other. Therefore, as the animal begins to fast, all changes of substrate-cycle-enzyme activities contribute to the increase of gluconeogenesis in proportion to the changes. Judging from the fluxes in fasting as compared to fed animals, five of the six substrate-cycle limbs may undergo cooperative activity changes. Unless we underestimated elasticities of the Pyr/PEP step or F D P / F 6 P step grossly, errors in the calculation for fasting animals would also be modest. In this case positional effects are dominant, because the first X is not high relative to unity and this is because negative equation (2) terms of some steps are small relative to positive terms. Since the Pyr/PEP step is significantly rate limiting, control strengths of enzymes acting on all intermediates (PEP to glucose) are significantly less than that of the Pyr~ PEP limb. Since the F D P / F 6 P cycle is also significantly rate limiting, control strengths of phosphofructo 1-kinase, Glc-6-Pase and hexokinase are very low. Therefore, as fasting proceeds, enzymes of the Pyr~ PEP limb become increasingly rate determining, so that small fractional increases in their activities become as important as larger fractional decreases in pyruvate kinase and larger fractional increases in Fru-l,6-P2ase. Judging from the flux rates in fed animals ( P y r ~ P E P flux=0.2, Usp =0.084), P y r ~ PEP activities would have to increase to achieve gluconeogenic rates of 0-3 in fasting animals. In fact, Pyr-~ PEP flux increases approximately to 0.38, and this is probably due in large part to the increase of PEP-carboxykinase in fasting (Shrago et al., 1963). Rognstadt & Katz (1977) obtained evidence for acute stimulation of P y r ~ P E P flux by gluconeogenic hormones. However, Groen et al. (1983) found inhibition of pyruvate kinase to be the most prominent acute action of glucagon in livers from fasting rats. The glycolytic pathway in most cells of the body is essentially irreversible, so upstream position provides a regulatory advantage in this pathway. In many ceils, glucose transport is the limiting step and has virtually all the control strength under basal conditions, where cellular glucose concentration and glucose efflux are very low. Stimulation of glucose transport tends

656

D. M. REGEN

AND

S, J . P I L K I S

to increase cellular glucose (hence, glycolytic rate), glucose ettlux relative to influx and transport elasticity--transferring control strength downstream. In other tissues, e.g. brain, glucose transport is not rate limiting and glucose efflux (brain to capillary) is at least half of glucose influx (Growdon et al., 1971), so transport is rather elastic under basal conditions and subsequent steps can exert control. We do not know whether transport in sperm is significantly rate limiting. The high Glc-6-P levels under basal conditions suggest that it is not, and the fall in Glc-6-P associated with rate enhancement would suggest that transport stimulation is not responsible for the rate enhancement. The results are consistent with the view that transport and hexokinase are not rate limiting under basal conditions, that transport exhibits great elasticity owing to efflux comparable to influx (due to high transport activity relative to net rate) and that hexokinase exhibits great elasticity owing to strong product inhibition (again, due to high transport and hexokinase activities relative to net rate). As phosphofructo l-kinase activation reduces GIc-6-P level and product inhibition, the greater glucose phosphorylation would reduce cellular glucose and efflux. As this happens, both steps of the transporthexokinase segment become more rate limiting (Z) and rate determining (X) and control strength of phosphofructo l-kinase (X) decreases (Table 5). Lacking information on cell glucose levels or efflux rates, we do not know the distribution of rate limitingness and control strength between transport and hexokinase. There is a link between the range of rate increase achievable by stimulation of phosphofructo 1-kinase and control strength of this enzyme. The highest rate (2.4 by extrapolation) is the positive equation 2 term of the C H 2 0 / H M P step or segment and the difference between this rate and basal rate is the sum of negative terms (2.28) under basal conditions. Given the large size of the negative terms relative to net rate, elasticity of this step would be very large even if the exponent of hexokinase inhibition were substantially less than 0.89. The elasticity of this step and that of hexose phosphate isomerase means that these early steps are not rate limiting and other steps will be. The control strength of phosphofructo 1-kinase is, therefore, almost as great as it would be if it were the first enzyme of the pathway. Since it is in a substrate cycle, its control strength is its X (not Z), and this is almost independent of F6P/FDP cycling rate. USEFULNESS OF SENSITIVITY ANALYSIS Sensitivity analysis is a systematic examination of the factors determining responsiveness of a system to changes in quantitative features of its com-

SUBSTRATE-CYCLE ENZYMES

657

ponents. The theoretical development identifies those determining factors. Attempts to apply the theory illustrate the operation of those factors. Assigning Z, X and Y values to a step formalizes and crystallizes ones information a b o u t its rate limitingness and the rate determiningness o f its enzymes whether or not that information is exact. It is very difficult to appreciate the implications o f the theory without examples. The applications illustrate that differences between exponents o f metabolite effects on adjacent steps shift control strength from all enzymes on one side o f the metabolite (the side o f greater exponent) to all enzymes on the other side. They illustrate the regulatory advantage o f upstream position, owing to the detracting effect o f upstream rate limitingness on d o w n s t r e a m control strength. Despite this advantage, upstream enzymes were not the most rate determining in two o f our four examples. In one case, high reverse fluxes relative to net rate resulted in high and c o m p a r a b l e control strengths o f all substrate-cycle enzymes, so high that the detracting effect o f upstream rate limitingness was insignificant. In another, strong product inhibition o f the first-step's forward limb rendered the first step non-rate-limiting and nonrate-controlling. Without sensitivity analysis, these relations could not be expressed with conviction and their quantitative meaning would remain vague. We are not the first to apply sensitivity analysis to substrate cycles (cf. Stein & Blum, 1978; G r o e n et al., 1982, 1983), though we may be the first to formalize the extension of Kacser and Burns' elegant logic to substrate cycles [equation (15)]. Despite the simplicity of that logic, its mathematical expression is rather involved, and several of its implications were revealed by application to particular pathways. This work was supported by NIH Grants AM 18270 and AM 18733. REFERENCES CLARK, M. C., KNEER, N. M, BOSCH,A. L. & LARDY, H. A. (1974). J. Biol. Chem. 249, 5695.

CLAUS, T. H., PILKIS,S. J. • PARK, C. R. (1975). Biochim. Biophys. Acta 404, 110. EXTON, J. H. & PARK, C. R. (1967). Z Biol. Chem. 242, 2622. EXTON, J. H. & PARK, C. R. (1969). J. Biol. Chem. 244, 1424.. GROEN, A. K., VAN DER MEER, R., WESTERHOFF, H. V., WANDERS, R. J. A., AKERBOOM, T. P. M. & "lAGER,J. M. (1982). In: Metabolic Compartmentation (Sies, H. ed), pp. 9-37. London: Academic Press. GROEN, A. K., VERVOORN,R. C., VAN DER MEER, R. & TAGER,J. M. (1983). J. Biol. Chem. 258, 14346. GROWDON, W. A., BRA'I"FON,T. S., HOUSTON, M. C., TARPLEY, H. L. & REGEN, D. M. (1971). Am. £ Physiol. 221, 1738. HAMMERSTEDT, R. H. & LARDY, H. A. (1983). J. Biol. Chem. 258, 8759. HIGGINS,J. (1965). Control of Energy Metabolism. (Chance, B., Estabrook, R. K. & Williamson, J. R. eds), pp. 13-46. New York: Academic Press.

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HUE, L. (1982). In: Metabolic Compartmentation (Sies, H. ed), pp. 71-97. London: Academic Press. KACSER, H. & BURNS, J. A. (1973), Symp. Soc. Exp. Biol. 32, 65. KACSER, H. & BURNS, J. A. (1979). Biochemical Society Transactions 7, 1149. PILKIS, S. J., RIOU, J. P. & CLAUS, T. H. (1976). J. Biol. Chem. 251, 7841. RAPOPORT, T. A., HEINRICH, R. & RAPOPORT, S. M. (t976). Biochem. J. 154, 449. ROGNSTAD, R. (1982). Arch. Biochem. Biophys. 217, 498. ROGNSTAD, R. & KATZ, J. (1977). J. Biol. Chem. 252, 1831. SHgAGO, E., LARDY, J. A., NORDLIE, R. C. & FOSTER, D. O. (1963). Z Biol. Chem. 238, 3188. STEIN, R. B. & BLUM, J. J. (1978). J. Theor. Biol. 72, 487. VINUELA, E., SALAS, M. & SOLS, A. (1963). k Biol. Chem. 238, 1175.