The Fusion of Control Analysis and Temporal Analysis of Metabolic Systems

J. theor. Biol. (1996) 182, 327–331

The Fusion of Control Analysis and Temporal Analysis of Metabolic Systems J S. E† Department of Biochemistry, The University, PO Box 147, Liverpool L69 3BX, U.K.

Metabolic control analysis and the study of the transient response of metabolic systems had coincident births in 1973. They developed along parallel lines until in 1989/90 their complete fusion occurred. It was evident that the control of the transient response of metabolism could be described in terms of general control properties, such as the flux and concentration control coefficients and elasticities. Consequently, it is possible to define temporal control coefficients which relate to the lifetimes of individual metabolite pools or to the total system temporal response. These control coefficients are readily expressed in terms of the flux and concentration control coefficients. Therefore, to analyse the control of metabolism is also to analyse its temporal response. 7 1996 Academic Press Limited

like a ray of light, the idea of metabolic control analysis (Kacser & Burns, 1973). Control analysis superficially appeared to ignore any specific model of metabolism and retain great generality. On reflection, control analysis itself contains a metabolic model but one which is very loosely defined in order to achieve generality. Control analysis, for the first time, indicated how one might quantify the effects of individual enzymes on metabolic control and even indicated what sort of experimentation would be necessary to achieve this. It also established the complete interdependence of behaviour of enzymes and metabolites in such a system. At about the same time Heinrich & Rapoport (1975) developed a similar but more restrictive approach applied to linear enzymatic chains and even described a control coefficient that measured the transient response of individual metabolites. Their approach was more mathematical and less accessible for the average biologist and therefore Kacser & Burns’s (1973) methodology became established in most people’s minds.

The Beginnings of Metabolic Control Analysis 1973 was an important year in the development of my approach to theoretical biology. For a long time I had thought that it should be the aim of biologists to develop general, theoretical descriptions of biological systems akin to those developed by physicists and physical chemists to describe the systems in which they were interested. Unfortunately, in biochemistry, most applications of theory to metabolic systems had involved computer simulations. These were deficient in three major respects. Firstly, the biochemical systems themselves were often ill-defined. Secondly, the parameters which were required to be known for such simulations, namely enzyme properties and kinetic mechanisms, pool sizes and localisation, metabolite and effector concentrations, etc., were not only unknown but were probably almost impossible to determine. And thirdly, if a simulation for one set of parameters was successful, it was not at all clear how generalisations could be made from this specific and unique set of conditions. Models, if they are of any use and most importantly if they are to be testable, should be predictive. This was not the case with most computer simulations. Into this muddle was injected,

The Transient Behaviour of the Linear Enzymatic Chain Also in 1973 I became interested in linear enzymatic chains for an entirely different reason. A colleague

† E-mail: jse.liv.ac.uk 0022–5193/96/190327 + 05 $25.00/0

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. . 

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had been working with a complex enzyme system the response of which he believed to be hysteretic. He explained the hysteretic behaviour in terms of conformational changes in the component proteins. It seemed to me that the hysteresis could be explained without invoking special properties for the enzymes. The system was indeed complex and was assayed using several expensive coupling enyzmes. These were administered in the minimum concentrations possible owing to cost and it occurred to me that the lag involved in product formation by the system could be the result of the slow establishment of a steady state in the intermediate pools. This turned out to be correct and I published a paper entitled ‘‘Coupled enzyme assays: a general expression for the transient’’ (Easterby, 1973). Earlier papers by Hess and co-workers had dealt with this problem but had never gone beyond systems involving three component enzymes (Barwell & Hess, 1970) and the basic approach had been established by McClure (1969). However, my analysis allowed for an unlimited number of irreversible coupled enzymes and therefore allowed all-important generalisations to be made.

Generalisation of the Transient Response One important generalisation from such systems was that if the initial enzyme of the sequence was severely rate-limiting then, surprisingly, it did not contribute to the temporal response and the lag in end-product formation. Moreover, for the simple linear enzymatic chain, associated with each stem or enzyme was a transient time which was simply the reciprocal of the pseudo-first-order rate coefficient controlling the step and these transient times were additive when describing the system response. On closer inspection it was seen that these transients were not in fact associated with the enzymes. This apparent association was the fortuitious result of the irreversibility of the individual steps as defined in the model. The transients really represented the lifetimes or turnover times of the individual metabolite pools. This in turn indicated that the transient response of the system was simply determined by mass conservation considerations. A ‘‘black box’’ approach was evident. Whatever was fed into one end of a metabolic system must appear eventually at the other as end product. What did not immediately appear was being retained by the system to generate the intermediate pools and it was the time required to generate these pools which determined by system transient response or lag (Easterby, 1981). It was possible to describe a transient associated with each metabolite pool and

this was defined as: ti = [Si ]/J

(1)

where ti represents a pool lifetime, Si the ith metabolite of the system and J the steady-state flux to product. The lag time of the whole system was given by: t = Si [Si ]/J.

(2)

This description of the transient was completely general and independent of any particular model of metabolism. From this it was obvious that the temporal behaviour of a system was determined by pool sizes and flux; just the same quantities that figure in metabolic control analysis. It immediately became apparent that it should be possible to analyse the control of the time-dependent response of a metabolic system to modulations in individual enzyme activities in just the same way that it is possible to analyse their effect on flux and pool size. In fact it was clear that much of the data might already be available. At this point indolence entered the story. I was too lazy to gain a proper understanding of control analysis to apply it to temporal response and, so it transpired, Henrik Kacser had been too preoccupied with control analysis to bother understanding temporal response.

Temporal Control Analysis of the Linear Enzymatic Chain Nothing concentrates the mind quite like the scaffold, but in this case it was not the scaffold but an imminent meeting between Kacser and myself which spurred me to action. We were both to attend a meeting in Debrecen, Hungary in 1985 (Easterby, 1986). On the journey to Budapest I had to spend 7 hr in the transit lounge at Frankfurt airport. I whiled away the time by seeing if I could apply control analysis to t values for simple linear enzymatic chains such as those seen in coupled assay systems. For this specific case, in which all of the rate equations and their solutions may be explicitly stated it proved trivial to establish a summation property for control coefficients relating to t values. I was pleased about this as the flux summation property was the thing which had impressed me most about control analysis in 1973. Remembering that for the simple linear enzymatic chain, the transient times are the reciprocals of the pseudo-first-order rate coefficients, namely Ki /Vi , where Ki and Vi represent the Michaelis constant and maximum velocity respectively of the ith enzyme of the chain. The trivial solution goes as

   follows: t = S(Ki /Vi )

(3)

tration and flux control coefficients. A general summation property is immediately apparent.

Ceti = 1(S(Ki /Vi )/1Vi ).

(4)

and Si Ceti = Si − ti /Si ti = −1

n

n

n

i=1

i=1

i=1

s Cetij = s CeSij − s CeJi

(Vi /S(Ki /Vi )) = −Ki /Vi2 . Vi /S(Ki /Vi ) = −ti /Si ti

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and making use of the well-known results of metabolic control analysis:

(5)

where Ceti is the temporal control coefficient. Thus, a summation property is established in which the control coefficients sum to minus unity. This is analogous to the flux summation property but the minus sign is present because the time required to approach a steady state decreases as the enzyme concentrations are increased.

(10)

n

s CeSij = 0

(11)

i=1

n

s CeJi = 1

(12)

i=1

then: n

s Cetij = 0 − 1 = −1.

Generalisation of Temporal Control Analysis and the Summation Properties The next development in temporal control analysis came in 1989 and was again prompted by an impending meeting between Henrik Kacser and myself. We were both to attend a NATO Advanced Research Workshop on Control of Metabolic Processes in Il Ciocco (Lucca), Italy (Kacser et al., 1990; Easterby, 1990a). This time I had done my homework in advance and had managed to generalise temporal control analysis so that it was fully integrated with metabolic control analysis. However, lazy as ever, I had not bothered to write the work up and intended to deliver it briefly as a short conclusion to my talk (Easterby, 1990a). At the reception on the first evening of the meeting I met Kacser and immediately informed him that I had managed the integration. ‘‘So have we’’ he replied, ‘‘ How did you do it?’’. I scrawled a brief derivation on the back of an envelope and said ‘‘Is this how you did it?’’. ‘‘No’’, he replied, ‘‘Our derivation was rather different and involved elasticities’’, but he failed to say how. My own analysis went as follows. The lifetime of an individual metabolite pool is given by eqn (1), namely ti = [Si ]/J. Simple partial differentiation of this with respect to enzyme concentration (ei ) with all other independent variables maintained constant yields: Ceti = (1(Sj /J)/1ei .(ei J/Sj )

(6)

= {1/J .(1Sj /1ei ) − Sj /J 2 .(1J/1ei )}.Jei /Sj

(7)

= (1Sj /1ei ).ei /Sj − (1J/1ei ).ei /J

(8)

= CeSji − CeJi .

(9)

It therefore represents the difference between concen-

(13)

i=1

The control coefficients associated with the pool lifetimes therefore sum to minus unity, reflecting the fact that the lifetimes decrease as the enzyme concentration increase. A summation property, based on the concentration and flux summation properties of metabolic control analysis, is therefore applicable to temporal control as manifested in the pool lifetimes. What is also intriguing is that, at once having determined the control coefficients necessary for metabolic control analysis, one already has the data required for the analysis of the temporal response of the system. This approach applied to individual pools may also be applied to the summed pool lifetimes in the form of the total system transient defined in eqn (2): Ceti = (1(Sj Sj /J)1ei ).(ei J/Sj Sj )

(14)

= {1/J .(1Sj Sj /1ei ) − Sj Sj /J 2 .(1J/1ei )}.Jei /Sj Sj (15) = (1Sj Sj /1ei ).ei /Sj Sj − (1J/1ei ).ei /J

(16)

= CeSi j Sj − CeJi .

(17)

Here a new control coefficient has appeared, namely that associated with the summed metabolite pools. Once again a summation property applies: n

n

n

i=1

i=1

i=1

n

n

i=1

i=1

s Ceti = s CeSi j Sj − s CeJi

s Ceti = s CeSi j Sj − 1.

(18)

(19)

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The pool control coefficients, like the concentration control coefficients, may be shown by the method first proposed by Kacser & Burns (1973) to sum to zero. If each of the enzymes has its concentration modulated by an infinitesimal fraction a, then assuming linear dependence of flux on enzyme concentration, the pool sizes will remain unaltered while the pathway flux increases by fraction a. Consequently, the change in the summed pool sizes is given by:

This seemed to break the ice between us considerably and from then on we enjoyed good relations. To return to the matter in hand, the summation property had been predicted earlier by Heinrich & Rapoport (1975) as applied to linear enzymatic chains and Torres et al. (1989) had provided an experimental verification of the property. My role had been to extend it to metabolic systems in general (Easterby, 1990b). Mele´ndez-Hevia et al. (1990) also produced a similar analysis subsequent to the Il Ciocco meeting.

n

a s CeSi j Sj = 0.

(20)

i=1

From which it follows that: n

s CeSi j Sj = 0

(21)

i=1

and from eqn (19): n

s Ceti = −1

(22)

i=1

so that the summation property applies not only to individual pool lifetimes but also to the transient time for the whole system. These aspects of temporal control analysis were communicated at the workshop at Il Ciocco referred to earlier, but somehow my manuscript disappeared after collection by the editor of the proceedings. By a remarkable coincidence a manuscript by a colleague which purported to contain data which did not agree with the flux summation property also disappeared. Henrik appeared rather amused by this, and while I know he had no part in the disappearances, I suspect that his over-zealous young disciples lacked his maturity and were drawn into contriving the ‘‘loss’’ of the papers. However, being a belt-and-braces man whose naivete´ is only skin deep, I not only had two spare hard copies of the manuscript, I also had three copies on disk! So all was well. Over the years that I had known Henrik, I thought I had noted a certain coolness towards me. However, this changed at Il Ciocco when I inadvertently managed to appeal to his sense of humour. After I had delivered the fateful paper we were sitting at the same dinner table at the end of the evening when most people were leaving. A rather rotund, red-faced colleague was making his way out of the room trailing a half empty wine bottle. Henrik commented ‘‘Old X is going to take that wine bottle to bed and empty it.’’, to which I replied ‘‘And then he’ll fill it up again’’.

Temporal Connectivity Relationships Henrik had intimated at Il Ciocco that he was more interested in elasticities and connectivities than in summation properties. Connectivity relationships can indeed be established for temporal control coefficients. The way in which flux and concentration control coefficients are related through the elasticities has been demonstrated by Kacser & Burns (1973) and by Westerhoff & Chen (1984) respectively. The same approach may be applied to temporal control coefficients. From eqn (18) it follows that: Si Ceti .oSi j = Si CeSSi .oSi j − Si CeJi .oSi j .

(23)

Here o represents the elasticity of the velocity of Ei with respect to Sj and SS represents the sum of the concentrations of all metabolite pools. Equation (23) leads to: i Sj

Si Ceti .oSi j = Si (1/SS) .Sk Sk CeSik .oSi j − Si CeJi .oSi j

(24)

Si C .o = Sk (1/SS) .Si Sk C .o − Si C .o .

(25)

t ei

i Sj

Sk ei

i Sj

J ei

i Sj

Applying the concentration connectivity relationship (Westerhoff & Chen, 1984) and the flux connectivity (Kacser & Burns, 1973) results in: Si Ceti .oSi j = −Sj /Sk Sk

(26)

Consideration of the ‘‘local’’ transients results in a connectivity relationship for the lifetimes of the individual matabolite pools: Si Cetik .oSi j = −1 0

j=k j $ k.

(27)

The connectivity property of eqn (26) may be used in conjunction with the summation property of eqn (22) to express the temporal control coefficients in terms of the elasticities and pool sizes. A specific example is useful. Taking the very simple system of two enzymes, E0 and E1 separated by a single intermediate S1 we have from the summation property [eqn (22)]: C0t + C1t = −1

(28)

   and from the connectivity property [eqn (26)]: t 0

0 1

t 1

1 1

C .o + C .o = −S1 /S1 = −1.

(29)

This results in: C0t = (o11 − 1)/(o10 − o11 )

(30)

C1t = (1 − o10)/(o10 − o11 ).

(31)

In this particular example, with a single intermediate, the control coefficients may be expressed as functions of the elasticities alone. In general however, with the exception of the control coefficients relating to the lifetimes of individual metabolite pools, the pool sizes also enter into the relationships. Conclusion It can be seen that analysis of the transient response of metabolic systems has been fully integrated into metabolic control analysis and occupies a subset of the system control properties. Shortly after this work was originally published I was looking forward to seeing Henrik Kacser again at a Gordon Conference in California. When I arrived, I was disappointed to discover that he was not there and was told that he had just had a pacemaker fitted. We did not meet again. Henrik apparently had an extraordinary life. I believe he began his scientific career in Hungary as a chemical engineer, and by way of Czechoslovakia and Ireland he finally arrived in the genetics department in Edinburgh. Consequently, he was a late developer as far as theoretical biology was concerned. Those of us who were privileged to witness his last 20 years know that he had more zest and enthusiasm than most young researchers beginning their careers. He

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made a major contribution to theoretical biology which has wide ramifications in the life sciences and is a particular inspiration to those of us interested in biochemistry. He is already sorely missed.

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