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On the physiological significance of positive and negative co-operativity
J. theor. Biol. (1981) 93,395-401
On the Physiological Significance of Positive and Negative Co-operativity ROBIN
GHOSH
Abteilung
fiir Biophysikalische Chemie, Biozentrum, strasse 70, CH-4056 Basel, Switzerland
(Received
19 December
Klingelberg-
1980, and in revised form 30 April
1981)
Negative co-operativity of binding was first predicted from the allosteric model of Koshland, Nemethy & Filmer (1966) and established experimentally for the glyceraldehyde-3-phosphate dehydrogenase present in rabbit muscle (Conway & Koshland, 1968; Meunier & Dalziel, 1978). Catalytic negative co-operativity (which may not reflect binding co-operativity) has also been well established for a number of systems: purine nucleoside phosphorylase (Savage & Spencer, 1979), alkaline phosphatase (Waight, Lett & Bardsley, 1977), AMP deaminase (Raggi & RannieriRaggi, 1979), glycogen synthase (Stilling, 1979), methanol oxidase (Van Dijken, 1975), methanol dehydrogenase (Ghosh, 1980a,b), pyruvate kinase (Gregory, 1980), hexokinase (Meunier et al., 1974). Negative co-operativity of binding has also been proposed for many hormone receptors (Levitski, 1974), though this has recently been disputed (Pollet, Standert & Haase, 1980). Kinetic rather than binding negative co-operativity does not affect the conclusions to be presented below. Indeed, it could be argued that selective pressure will not distinguish between these two possibilities. Despite the evidence that kinetic and binding negative co-operativity c,an .be observed for many enzymes, a physiological rationale for this phenomenon has been difficult to find. However, several workers have attempted an explanation of the phenomena (Levitski & Koshland, 1969; L.evitski, 1974; Cornish-Bowden, 1975; Friedrich, 1979) with varying degrees of success. The present study attempts to show a fresh light on this subject by adopting the general approach given by Cornish-Bowden (‘1975), but also by using a recently introduced phenomenological analysis of co-operative systems (Ainsworth, 1977). Recently Cornish-Bowden (1975) has shown that the sensitivity of a reaction characterized by the Hill equation to changes in the concentration of a ligand A is given by: dF ----= nP(lT) (1) d In [A] 305
0022-5193/81/220395+07$02.00/0
@ 1981 Academic
Press Inc. (London)
Ltd
396
K.
GHOSH
where Y is the fractional saturation of the enzyme with the ligand and II is an index of co-operativity. Equation (1) requires no arbitrary assumptions about the association constants involved and demonstrates that at all fractional saturations a negatively co-operative enzyme (n < 1) will be less sensitive to ligand than a Michaelis-Menten or positively co-operative system. The principal difficulty with the Hill equation is that it is never an exact description of a co-operative enzyme reaction but is applicable only as an approximation at high fractional saturations. However, any realistic analysis using a rational polynomial discription would encounter once again the difficulty of assigning arbitrary values to the association constants, thereby losing its forcefulness and simplicity. Recently a convenient and realistic description of co-operative binding or steady state kinetics has been introduced by Ainsworth (1977). The phenomenological description (the exponential model) relates the fractional saturation ( F) of an enzyme to the substrate concentration (A) by means of an exponential term:
y= [Al
QA
exp
(kA
P)
1 + [A](YA exp (kA Y) ’
(2)
where (YA is the affinity of the unbound enzyme for the substrate A, and of the reaction. kA may be zero, negative, or positive (corresponding to Michaelis-Menten kinetics and negative and positively co-operative kinetics, respectively) and bears a simple relation to the Hill coefficient (n): kA is the co-operativity
l.3)
The phenomenological description of catalytic co-operativity is obtained by setting F = v/ VA where v and VA are the reaction rates obtained when A is non-saturating and saturating, respectively. It has been shown (Ainsworth, 1977; Ainsworth & Gregory, 1978) that equation (2) will fit a wide range of steady state kinetic data to within the accuracy of the fit provided by a rational polynomial, and for the present purpose, may be considered exact. The sensitivity(s) of a reaction may be described by the equation (4) S=-= dln Y dln[A]
(1-Y) l-kAv(l-Y)
(4)
which in common with equation (l), requires no arbitrary assumptions about the association constants and the double logarithmic derivative is formerly equivalent to the fractional change in saturation with a fractional
PHYSIOLOGICAL
IMPORTANCE
OF
CO-OPERATIVITY
391
increase of substrate concentration ((A F/ q)/(A[A]/[A]). This function is the usual definition of sensitivity given by workers in metabolic simulation. The sensitivity functions for equation (4) and the Hill reaction (n (1 - F)) respectively, are shown in Fig. 1. Using equation (l), Cornish-Bowden
FIG. 1. Changes in the sensitivity of an enzyme reaction with changes in the fractional saturation of the enzyme for reactions which exhibit different degrees 0-f co-operativity. The sensitivity (s) was determined from (a) the Hill equation where S = d In Y/d In [A] = n(l - Y); and (b) the exponential model (S is given by equation (10) in the text). The degree of co-operativity was given by (a) n = 2.67 (k = 2.5); (b) n = 2 (k = 2); (c) n = 1.33 (k = 1); (d) n = 1 (k =O); (e) n =0.5 (k = -8.0).
(1974) has concluded that a negatively co-operative system is less sensitive to changes in ligand concentration than a Michaelis-Menten or positively co-operative system, at all fractional saturations. Clearly, the same conclusion holds for the more exact description of co-operative kinetics given by the exponential model, except at the extremes of saturation. At P = 0 however, the sensitivity of the Hill system is n, whereas that for exponential model is always 1. That this latter value is exact can be shown by integrating
398
R.
GHOSH
equation (2) at the extremes of saturation (S is therefore constant, Crabtree & Newsholme, 1978): u =h[AIS
(51
where u = kcat p (k,,, is the catalytic constant) and A is a constant of integration. When S = 1, the velocity of the reaction will depend stoichiometrically upon the substrate concentration. This condition is always true for a Michaelis-Menten and higher order system at very low fractional saturations of an enzyme with substrate. Interestingly, at Y = 0.5, S reduces to 0.5n for any realistic reaction. The qualitative behaviour of Fig. la suggests that two distinct concepts might be useful in an appraisal of the roles of positive and negative co-operativity in metabolism. The first concept is the sensitivity of the reaction (S) (defined above) which has been much discussed in the literature (Higgins, 1965; Heinrich, Rappaport & Rappaport, 1980; Crabtree & Newsholme, 1978). The second concept is the response (R) of the sensitivity to changes in the fractional saturation of a controlling enzyme (or receptor). Differentiation of equation (4) leads to: R-
dS_ka-l-k/&2-Y) (l- kAv(ldY
F))*
*
(6)
The properties of R are trivial but interesting. When Y+ 0, then R -+ ka - 1, which is positive or negative if kA > 1 or kA < 1, respectively, when F’-+ 1, then R + - 1 for all values of kA. The most important result, however, is obtained when P = 0.5: RO.5 = -n.
(7)
Equations (5) and (7) show that the phenomenological definitions of cooperativity can be related to the parameters relevant to metabolic control, at least for simple curve shapes, where n is maximal or minimal at F = O-5. For control enzymes, this condition is frequently obtained. Inspection of the S and R functions shows that at low substrate concentrations a positively co-operative enzyme is always more sensitive to changes in flux than a Michaelis-Menten or negatively co-operative system. However, the response of this sensitivity to rapid local fluctuations is large only for k values greater than (n > 1.2). This might be of significance for enzymes that control energy generating sequences, which usually operate at fluxes far below full capacity. Thus, rapid fluctuations of substrates or cofactors will interfere only minimally with the all-important task of energy generation.
PHYSIOLOGICAL
IMPORTANCE
OF
CO-OPERATIVITY
399
Clearly, the opposite situation obtains for enzymes that exhibit negative co-operativity. The flux catalyzed by these enzymes will be insensitive to large changes in the fractional saturation of substrate, but will respond quickly to rapid local fluctuations. The role of such enzymes is difficult to establish a priori but a specific consideration of examples in this new light may help to clarify this picture. A well-established example of negative co-operativity is that observed for rabbit muscle glyceraldehyde-3-phosphate dehydrogenase (G3PdH). This system has recently been examined by Friedrich (1979) but will be treated here in a somewhat different way. In muscle, G3PdH participates principally in NADH formation for lactate generation during exercise. In yeast, the enzyme is positively co-operative, and is responsible for the production of NADH necessary for the formation of ethanol, which is subsequently excreted into the medium. In the muscle system, therefore, the flux catalyzed by this enzyme is not very sensitive to the availability of NAD’, but may experience a rapid increase in sensitivity with small fluctuations of this cofactor. In yeast, however, the enzyme is very sensitive to changes in the fractional saturatiop of NAD’. These different types of behaviour might reflect different functional modes of metabolism. For instance, in muscle, G3PdH is functioning in an environment experiencing the rapid fluctuations of metabolites that occur during contraction and must respond accordingly. In yeast, the enzyme must maintain a steady flux of NADH during anaerobic fermentation despite the rapid changes in flux that are produced during the normal functioning of the cell. This picture is somewhat strengthened by a consideration of the lbehaviour of rabbit muscle pyruvate kinase. At neutral pH (muscle at rest), this enzyme is hyperbolic with respect to ADP. However, at more acidic pH’s (approx. pH 6.4) the kinetics of the enzyme become noticeably lsigmoidal (Gregory, 1980). During exercise (i.e. acidic conditions). Therefore, the enzyme becomes very sensitive to the fractional saturation of ADP, maximizing the flux so as to facilitate the production of ATP for contraction but is relatively insensitive to the rapid fluctuations of the nucleotide that might result from perturbation of the creatine kinase equilibrium. It is becoming increasingly clear that the observed co-operativity of an enzyme not an immutable property but can be decreased and even reversed by a change of assay conditions e.g. for a two-substrate enzyme, different co-operativities (k values) may be observed for one substrate at different concentrations of a second substrate. One of the best examples of this type of behaviour is given by the methanol oxidase present in the methylotrophic
400
R.
GHOSH
yeast Hansenula polymorpha (Van Dijken, 1975). At high concentrations of methanol, the kinetics of the enzymes are positively co-operative with respect to oxygen, with the inflection point occurring in the region y,,, = 0.5. (This corresponds to the physiological oxygen concentration). As the concentration of methanol is decreased, however, the kinetics of the enzyme show a decrease in co-operativity and eventually take negative values of k. This behaviour might be rationalized as follows. At high concentrations of methanol and physiological concentrations of oxygen, the flux, and the sensitivity of the enzyme to changes in the methanol concentration are maximized, thereby optimizing the flow of carbon into biosynthesis and energy generation. At low availabilities of methanol, biosynthesis is reduced to a basal level, and the carbon flow is decreased to supply only the minimal energy demands of the cell. It is probably advantageous that this minimal flux is rather insensitive to local fluctuations in oxygen consumption, so that a steady flow of carbon can be maintained. In this case the R function is superfluous since rapid oxygen fluctuations do not occur. Recently Friedrich (1979) has made the important point that enzymes should not be considered in isolation in any referral of their functional properties to a metabolic overview. However, if the above relations are realistic. in principle the kinetic parameters obtained by in vitro methods may often be assigned a meaningful physiological significance. I would like to thank Drs A. N. Lane and R. B. Gregory for many helpful and stimulating discussions. REFERENCES AINSWORTH, S. (1977). J. theor. Biol. 68, 391. AINSWORTH, S. & GREGORY, R. B. (1978). J. theor. Biol. 75,97. BELL, J. E. & DALZIEL, K. (1978). Eur. J. Biochem. 82,483. CORNISH-BOWDEN, A. (1974). J. theor. Biol. 51, 233. CRABTREE, B. & NEWSHOLME, E. A. (1978). Eur. J. Biochem. 89,19. FRIEDRICH, R. (1979). J. theor. Biol. 81, 527. GHOSH, R. (1980~). Ph.D. thesis: University of Sheffield, England. GHOSH, R. (19806). Biochem. Sot. Trans. 9, in press. GREGORY, R. B. (1980). Ph.D. thesis: University of Sheffield, England. HEINRICH, R., RAPPAPORT, S. M. & RAPPAPORT, T. A. (1980). Prog. Biophys. Mol. Biol. 32, 1. HIGGINS, J. J. (1965). In: Confrol of Energy Metabolism (B. Chance, R. N. Estabrook and S. R. Williamson eds). pp. 13-46. New York and London: Academic Press. KOSHLAND, D. E., JR., NEMETHY, G. & FILMER, D. (1966). Biochemisfry 5,365. LEVITSKI, A. (1974). J. theor. Biol. 44, 367. LEVITSKI, A. & KOSHLAND, D. E., JR. (1969). Proc. natn. Acad. Sci. U.S.A. 62, 1121. MEUNIER, J.-C. & DALZIEL, (1978). Eur. J. Biochem. 82,483. MEUNIER, J.-C., But, .I., NAVARRO, A. & RICHARD, J. (1974). Eur. J. Biochem. 49,209.
PHYSIOLOGICAL
IMPORTANCE
OF
CO-OPERATIVITY
POLLET, R. J., STANDAERT, M. L. & HAASE, B. A. (1980). Proc. nom. Acad. 77,434o. RAGGI, A. & RANNIERI-RAGGI, M. (1979). Biochim. biophys. Acta 566, 353. REICH, J. G. & SEL’KOV, E. E. (1977). Biosystems 7,39. SAVAGE, B. & SPENCER, N. (1979). Biochem. J. 179,21. SBLLING, H. (1979). Eur. J. Biochem. 94,231. TIPTON, K. F. (1980). Biochem. Sot. Trans. 8, 242. VAN DIJKEN, J. (1975). Ph.D. thesis: University of Grsningen, The Netherlands. WAIGHT, R. D., LEFF, P. & BARDSLEY, W. G. (1977). Biochem. J. 167,787.
401 SC;. l/.S.A
On the Physiological Significance of Positive and Negative Co-operativity ROBIN
GHOSH
Abteilung
fiir Biophysikalische Chemie, Biozentrum, strasse 70, CH-4056 Basel, Switzerland
(Received
19 December
Klingelberg-
1980, and in revised form 30 April
1981)
Negative co-operativity of binding was first predicted from the allosteric model of Koshland, Nemethy & Filmer (1966) and established experimentally for the glyceraldehyde-3-phosphate dehydrogenase present in rabbit muscle (Conway & Koshland, 1968; Meunier & Dalziel, 1978). Catalytic negative co-operativity (which may not reflect binding co-operativity) has also been well established for a number of systems: purine nucleoside phosphorylase (Savage & Spencer, 1979), alkaline phosphatase (Waight, Lett & Bardsley, 1977), AMP deaminase (Raggi & RannieriRaggi, 1979), glycogen synthase (Stilling, 1979), methanol oxidase (Van Dijken, 1975), methanol dehydrogenase (Ghosh, 1980a,b), pyruvate kinase (Gregory, 1980), hexokinase (Meunier et al., 1974). Negative co-operativity of binding has also been proposed for many hormone receptors (Levitski, 1974), though this has recently been disputed (Pollet, Standert & Haase, 1980). Kinetic rather than binding negative co-operativity does not affect the conclusions to be presented below. Indeed, it could be argued that selective pressure will not distinguish between these two possibilities. Despite the evidence that kinetic and binding negative co-operativity c,an .be observed for many enzymes, a physiological rationale for this phenomenon has been difficult to find. However, several workers have attempted an explanation of the phenomena (Levitski & Koshland, 1969; L.evitski, 1974; Cornish-Bowden, 1975; Friedrich, 1979) with varying degrees of success. The present study attempts to show a fresh light on this subject by adopting the general approach given by Cornish-Bowden (‘1975), but also by using a recently introduced phenomenological analysis of co-operative systems (Ainsworth, 1977). Recently Cornish-Bowden (1975) has shown that the sensitivity of a reaction characterized by the Hill equation to changes in the concentration of a ligand A is given by: dF ----= nP(lT) (1) d In [A] 305
0022-5193/81/220395+07$02.00/0
@ 1981 Academic
Press Inc. (London)
Ltd
396
K.
GHOSH
where Y is the fractional saturation of the enzyme with the ligand and II is an index of co-operativity. Equation (1) requires no arbitrary assumptions about the association constants involved and demonstrates that at all fractional saturations a negatively co-operative enzyme (n < 1) will be less sensitive to ligand than a Michaelis-Menten or positively co-operative system. The principal difficulty with the Hill equation is that it is never an exact description of a co-operative enzyme reaction but is applicable only as an approximation at high fractional saturations. However, any realistic analysis using a rational polynomial discription would encounter once again the difficulty of assigning arbitrary values to the association constants, thereby losing its forcefulness and simplicity. Recently a convenient and realistic description of co-operative binding or steady state kinetics has been introduced by Ainsworth (1977). The phenomenological description (the exponential model) relates the fractional saturation ( F) of an enzyme to the substrate concentration (A) by means of an exponential term:
y= [Al
QA
exp
(kA
P)
1 + [A](YA exp (kA Y) ’
(2)
where (YA is the affinity of the unbound enzyme for the substrate A, and of the reaction. kA may be zero, negative, or positive (corresponding to Michaelis-Menten kinetics and negative and positively co-operative kinetics, respectively) and bears a simple relation to the Hill coefficient (n): kA is the co-operativity
l.3)
The phenomenological description of catalytic co-operativity is obtained by setting F = v/ VA where v and VA are the reaction rates obtained when A is non-saturating and saturating, respectively. It has been shown (Ainsworth, 1977; Ainsworth & Gregory, 1978) that equation (2) will fit a wide range of steady state kinetic data to within the accuracy of the fit provided by a rational polynomial, and for the present purpose, may be considered exact. The sensitivity(s) of a reaction may be described by the equation (4) S=-= dln Y dln[A]
(1-Y) l-kAv(l-Y)
(4)
which in common with equation (l), requires no arbitrary assumptions about the association constants and the double logarithmic derivative is formerly equivalent to the fractional change in saturation with a fractional
PHYSIOLOGICAL
IMPORTANCE
OF
CO-OPERATIVITY
391
increase of substrate concentration ((A F/ q)/(A[A]/[A]). This function is the usual definition of sensitivity given by workers in metabolic simulation. The sensitivity functions for equation (4) and the Hill reaction (n (1 - F)) respectively, are shown in Fig. 1. Using equation (l), Cornish-Bowden
FIG. 1. Changes in the sensitivity of an enzyme reaction with changes in the fractional saturation of the enzyme for reactions which exhibit different degrees 0-f co-operativity. The sensitivity (s) was determined from (a) the Hill equation where S = d In Y/d In [A] = n(l - Y); and (b) the exponential model (S is given by equation (10) in the text). The degree of co-operativity was given by (a) n = 2.67 (k = 2.5); (b) n = 2 (k = 2); (c) n = 1.33 (k = 1); (d) n = 1 (k =O); (e) n =0.5 (k = -8.0).
(1974) has concluded that a negatively co-operative system is less sensitive to changes in ligand concentration than a Michaelis-Menten or positively co-operative system, at all fractional saturations. Clearly, the same conclusion holds for the more exact description of co-operative kinetics given by the exponential model, except at the extremes of saturation. At P = 0 however, the sensitivity of the Hill system is n, whereas that for exponential model is always 1. That this latter value is exact can be shown by integrating
398
R.
GHOSH
equation (2) at the extremes of saturation (S is therefore constant, Crabtree & Newsholme, 1978): u =h[AIS
(51
where u = kcat p (k,,, is the catalytic constant) and A is a constant of integration. When S = 1, the velocity of the reaction will depend stoichiometrically upon the substrate concentration. This condition is always true for a Michaelis-Menten and higher order system at very low fractional saturations of an enzyme with substrate. Interestingly, at Y = 0.5, S reduces to 0.5n for any realistic reaction. The qualitative behaviour of Fig. la suggests that two distinct concepts might be useful in an appraisal of the roles of positive and negative co-operativity in metabolism. The first concept is the sensitivity of the reaction (S) (defined above) which has been much discussed in the literature (Higgins, 1965; Heinrich, Rappaport & Rappaport, 1980; Crabtree & Newsholme, 1978). The second concept is the response (R) of the sensitivity to changes in the fractional saturation of a controlling enzyme (or receptor). Differentiation of equation (4) leads to: R-
dS_ka-l-k/&2-Y) (l- kAv(ldY
F))*
*
(6)
The properties of R are trivial but interesting. When Y+ 0, then R -+ ka - 1, which is positive or negative if kA > 1 or kA < 1, respectively, when F’-+ 1, then R + - 1 for all values of kA. The most important result, however, is obtained when P = 0.5: RO.5 = -n.
(7)
Equations (5) and (7) show that the phenomenological definitions of cooperativity can be related to the parameters relevant to metabolic control, at least for simple curve shapes, where n is maximal or minimal at F = O-5. For control enzymes, this condition is frequently obtained. Inspection of the S and R functions shows that at low substrate concentrations a positively co-operative enzyme is always more sensitive to changes in flux than a Michaelis-Menten or negatively co-operative system. However, the response of this sensitivity to rapid local fluctuations is large only for k values greater than (n > 1.2). This might be of significance for enzymes that control energy generating sequences, which usually operate at fluxes far below full capacity. Thus, rapid fluctuations of substrates or cofactors will interfere only minimally with the all-important task of energy generation.
PHYSIOLOGICAL
IMPORTANCE
OF
CO-OPERATIVITY
399
Clearly, the opposite situation obtains for enzymes that exhibit negative co-operativity. The flux catalyzed by these enzymes will be insensitive to large changes in the fractional saturation of substrate, but will respond quickly to rapid local fluctuations. The role of such enzymes is difficult to establish a priori but a specific consideration of examples in this new light may help to clarify this picture. A well-established example of negative co-operativity is that observed for rabbit muscle glyceraldehyde-3-phosphate dehydrogenase (G3PdH). This system has recently been examined by Friedrich (1979) but will be treated here in a somewhat different way. In muscle, G3PdH participates principally in NADH formation for lactate generation during exercise. In yeast, the enzyme is positively co-operative, and is responsible for the production of NADH necessary for the formation of ethanol, which is subsequently excreted into the medium. In the muscle system, therefore, the flux catalyzed by this enzyme is not very sensitive to the availability of NAD’, but may experience a rapid increase in sensitivity with small fluctuations of this cofactor. In yeast, however, the enzyme is very sensitive to changes in the fractional saturatiop of NAD’. These different types of behaviour might reflect different functional modes of metabolism. For instance, in muscle, G3PdH is functioning in an environment experiencing the rapid fluctuations of metabolites that occur during contraction and must respond accordingly. In yeast, the enzyme must maintain a steady flux of NADH during anaerobic fermentation despite the rapid changes in flux that are produced during the normal functioning of the cell. This picture is somewhat strengthened by a consideration of the lbehaviour of rabbit muscle pyruvate kinase. At neutral pH (muscle at rest), this enzyme is hyperbolic with respect to ADP. However, at more acidic pH’s (approx. pH 6.4) the kinetics of the enzyme become noticeably lsigmoidal (Gregory, 1980). During exercise (i.e. acidic conditions). Therefore, the enzyme becomes very sensitive to the fractional saturation of ADP, maximizing the flux so as to facilitate the production of ATP for contraction but is relatively insensitive to the rapid fluctuations of the nucleotide that might result from perturbation of the creatine kinase equilibrium. It is becoming increasingly clear that the observed co-operativity of an enzyme not an immutable property but can be decreased and even reversed by a change of assay conditions e.g. for a two-substrate enzyme, different co-operativities (k values) may be observed for one substrate at different concentrations of a second substrate. One of the best examples of this type of behaviour is given by the methanol oxidase present in the methylotrophic
400
R.
GHOSH
yeast Hansenula polymorpha (Van Dijken, 1975). At high concentrations of methanol, the kinetics of the enzymes are positively co-operative with respect to oxygen, with the inflection point occurring in the region y,,, = 0.5. (This corresponds to the physiological oxygen concentration). As the concentration of methanol is decreased, however, the kinetics of the enzyme show a decrease in co-operativity and eventually take negative values of k. This behaviour might be rationalized as follows. At high concentrations of methanol and physiological concentrations of oxygen, the flux, and the sensitivity of the enzyme to changes in the methanol concentration are maximized, thereby optimizing the flow of carbon into biosynthesis and energy generation. At low availabilities of methanol, biosynthesis is reduced to a basal level, and the carbon flow is decreased to supply only the minimal energy demands of the cell. It is probably advantageous that this minimal flux is rather insensitive to local fluctuations in oxygen consumption, so that a steady flow of carbon can be maintained. In this case the R function is superfluous since rapid oxygen fluctuations do not occur. Recently Friedrich (1979) has made the important point that enzymes should not be considered in isolation in any referral of their functional properties to a metabolic overview. However, if the above relations are realistic. in principle the kinetic parameters obtained by in vitro methods may often be assigned a meaningful physiological significance. I would like to thank Drs A. N. Lane and R. B. Gregory for many helpful and stimulating discussions. REFERENCES AINSWORTH, S. (1977). J. theor. Biol. 68, 391. AINSWORTH, S. & GREGORY, R. B. (1978). J. theor. Biol. 75,97. BELL, J. E. & DALZIEL, K. (1978). Eur. J. Biochem. 82,483. CORNISH-BOWDEN, A. (1974). J. theor. Biol. 51, 233. CRABTREE, B. & NEWSHOLME, E. A. (1978). Eur. J. Biochem. 89,19. FRIEDRICH, R. (1979). J. theor. Biol. 81, 527. GHOSH, R. (1980~). Ph.D. thesis: University of Sheffield, England. GHOSH, R. (19806). Biochem. Sot. Trans. 9, in press. GREGORY, R. B. (1980). Ph.D. thesis: University of Sheffield, England. HEINRICH, R., RAPPAPORT, S. M. & RAPPAPORT, T. A. (1980). Prog. Biophys. Mol. Biol. 32, 1. HIGGINS, J. J. (1965). In: Confrol of Energy Metabolism (B. Chance, R. N. Estabrook and S. R. Williamson eds). pp. 13-46. New York and London: Academic Press. KOSHLAND, D. E., JR., NEMETHY, G. & FILMER, D. (1966). Biochemisfry 5,365. LEVITSKI, A. (1974). J. theor. Biol. 44, 367. LEVITSKI, A. & KOSHLAND, D. E., JR. (1969). Proc. natn. Acad. Sci. U.S.A. 62, 1121. MEUNIER, J.-C. & DALZIEL, (1978). Eur. J. Biochem. 82,483. MEUNIER, J.-C., But, .I., NAVARRO, A. & RICHARD, J. (1974). Eur. J. Biochem. 49,209.
PHYSIOLOGICAL
IMPORTANCE
OF
CO-OPERATIVITY
POLLET, R. J., STANDAERT, M. L. & HAASE, B. A. (1980). Proc. nom. Acad. 77,434o. RAGGI, A. & RANNIERI-RAGGI, M. (1979). Biochim. biophys. Acta 566, 353. REICH, J. G. & SEL’KOV, E. E. (1977). Biosystems 7,39. SAVAGE, B. & SPENCER, N. (1979). Biochem. J. 179,21. SBLLING, H. (1979). Eur. J. Biochem. 94,231. TIPTON, K. F. (1980). Biochem. Sot. Trans. 8, 242. VAN DIJKEN, J. (1975). Ph.D. thesis: University of Grsningen, The Netherlands. WAIGHT, R. D., LEFF, P. & BARDSLEY, W. G. (1977). Biochem. J. 167,787.
401 SC;. l/.S.A