J. Mol. Biol. (1976) 108, 151-178
The Functional Relationship between Polymerization and Catalytic Activity of Beef Liver Glutamate Dehydrogenase I. T h e o r y RICHARD JONATHAN COHEN AND GEORGE B. BENEDEK
Department of Physics, Center for Materials Science and Engineering and the Harvard-M.I.T. Program in Health Sciences and Technology Massachusetts Institute of Technology, Cambridge, Mass 02139, U.S.A. (Received 15 March 1976, and in revised form 12 July 1976) In our initial report on this subject (Cohen et al., 1975) we presented evidence that the reversible polymerization of beef liver glutamate dehydrogenase may play an important role in the allosteric control of this enzyme. At that time we proposed a simple quantitative model for analyzing the detailed relationship between the distribution of enzyme polymers and the catalytic activity of the enzyme solution. The model was tested by means of quasi-elastic light-scattering spectroscopic determinations of polymer distribution and biochemical determinations of enzyme activity. In this present paper we complete the mathematical development of the original model. In addition, subsequent more extensive experimental investigations have led us to modify the original model to obtain even closer agreement between theory and experiment. Hence we present, in addition to the original model, two alternative versions of the model. The original model presumed that the enzyme polymers must be homogenous, in that all of the constituents of a given polymer had to be in the same eonformational state. The two alternative models deal explicitly with the possibility of forming polymers composed of elements in different conformational states. In the final section of this paper we develop the theoretical framework necessary to relate the distribution of polymers predicted by any of the three models with the distribution of diffusion constants measured by quasi-elastic light-scattering spectroscopy. In the following paper (Cohen et al., 1976) we compare the predictions of each of the three models with a detailed set of experimental investigations. 1. I n t r o d u c t i o n Beef liver glutamate dehydrogenase is a mitochondrial enzyme which catalyzes the oxidative deamination of glutamic acid: glutamate H NAD + H O H - ~- ~-ketoglutarate H N A D H H N H +. PariUa & Goodman (1974) present evidence that in rive glutamate dehydrogenase (GDH) operates primarily in the direction of glutamate deamination. This reaction is an important step in the catabolic pathways of m a n y of the amino acids (McGilvery, 1970; Stadtman, 1966). Thus GDH~ could act physiologically as a control element Abbreviation used: GDH, glutamate dehydrogenase. 151
152
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COHEN AND G. B. B E N E D E K
regulating the rate at which proteins are used as an energy source for cellular metabolism. Indeed it is found t h a t the high energy triphosphates, G T P and ATP, act as potent allosteric inhibitors of G D H and t h a t the relatively energy-poor diphosphate A D P acts as an allosteric activator (Frieden, 1963a). Thus the accumulation of energy-rich triphosphates m a y well signal the enzyme to reduce the rate of amino acid catabolism while, conversely, increased concentrations of A D P m a y provide the signal to increase the rate of conversion of proteins into energy. I n addition the activity of G D H is allosterically modulated b y a wide assortment of other effector molecules (Eisenberg, 1971; Frieden, 19635; Sund et al., 1976). Although all the details have not been worked out, it is clear t h a t a nexus of biochemical feedback loops is involved in setting the level of G D H activityt. Thus it appears t h a t the fine control of G D H activity m a y be of central significance in the regulation of amino acid metabolism. Hence, the study of the mechanisms of the allosteric control of G D H is not only ~f importance with respect to understanding the relationship between molecular structure and function, but it is also i m p o r t a n t with regard to understanding the detailed nature of cellular control mechanisms. The early studies of bovine G D H revealed t h a t it possessed an interesting physical property which subsequently was found to be shared by a number of other regulatory enzymes (Levitzki & Koshland, 1972 ; Duncan etal., 1972). Namely, it was found t h a t G D H forms aggregates in vitro when it is present in high concentration (Olson & Anfinsen, 1952; Friedcn, 1958; K u b o etal., 1959). Significantly, it is estimated t h a t in the cell mitochondria the enzyme is present in this high concentration range, several milligrams per milliliter (Tomkins et al., 1963; Sund, 1964). The degree of aggregation was found to be a reversible function of total enzyme concentration. Also the number and size of the aggregates varied strongly and reversibly with the concentration of the effector molecules. The addition of inhibitor molecules such as G T P to a solution of the enzyme could almost completely abolish the aggregation. The subsequent addition of activator molecules such as A D P could reverse this effect and restore the aggregation (Yielding & Tomkins, 1960; Wolff, 1962; Frieden, 1968). On the basis of these findings it was first believed t h a t it was necessary for the enzyme to be in an aggregated form for it to be active (Tomkins et al., 1961). This contention was examined by Fisher et al. (1962). They measured the mean molecular weight of the aggregates by measuring the intensity of light scattered from solutions of the enzyme, and they simultaneously determined the enzyme activity. As the concentration of the enzyme was changed from 0-I mg/ml to 4 mg/ml, they found t h a t the mean molecular weight of the aggregates increased by a factor of three while the specific catalytic activity (the activity per mg of enzyme) stayed constant. Fisher et al. (1962) concluded t h a t the activity of the enzyme was therefore independent of its state of aggregation. The prevailing view subsequent to the publication of Fisher's results was t h a t the aggregation of G D H is perhaps interesting, but biologically insignificant. However, Fricden & Colman (1967) subsequently found t h a t the dependence of enzyme activity on the concentration of various purine nucleotide effector molecules t Moreover, when GDH is allosterically inhibited with reference to its ability to oxidatively glutamate, it seems to have an increased ability to function as an alanine dchydrogenase (Tomkins etal., 1965). In this paper we will refer to the activation and inhibition of GDH with reference only to its activity as a glutamate dehydrogenase. deaminate
POLYMERIZATION AND ACTIVITY OF GDH. I
153
could be substantially modified by varying the level of enzyme concentration. This phenomenon was related to the aggregation of the enzyme which increases with increasing enzyme concentration. It was suggested that this effect might play a physiological role in vivo. Frieden (1967) and Nichol et al. (1967) suggested a possible mechanism by which the polymerization process might be able to influence the dependence of enzyme activity on the concentration of various ligands. It was suggested that the polymeric forms of enzyme (vis-a-vis the monomeric form) are characterized by a different set of affinities for substrate and for each of the effector molecules, and perhaps by a different set of rate constants for the breakdown of enzyme-substrate complex. According to this scheme, at low enzyme concentration the enzyme would be predominantly in the monomeric form, and at high enzyme concentration the enzyme would be predominantly in the polymeric forms. Thus at low enzyme concentration the behavior of the enzyme solution would reflect the properties of the enzyme monomer. At higher enzyme concentrations the behavior of the enzyme solution would reflect the properties of the polymeric forms. However, this analysis did not adequately account for the results of Fisher and others (Iwatsubo & Pantaloni, 1967 ; Josephs et al., 1972). Moreover, the overall biological significance of the aggregation process remained unclear. In fact, despite the findings of Frieden & Colman (1967), the consensus in the recent literature is that the aggregation process is of little or no biological importance (Fisher, 1973; Sund et al., 1976). Our investigations, on the other hand, clearly indicate that aggregation may play a vital role in the allosteric regulation of GDH (Cohen et al., 1975). Furthermore, we believe that elucidating the precise function of aggregation in the control of this particular enzyme may significantly contribute to our general understanding of allosteric control mechanisms. Over the past fifteen years a number of quantitative theories have been proposed that analyze the mechanisms by which various ligands influence the catalytic activity of the regulatory enzymes (Monod et al., 1964; Koshland et al., 1966; Herzfeld & Stanley, 1974). The usual situation envisioned in these theories is that of an isolated enzyme molecule interacting with much smaller ligand molecules. There has been much less analysis of the interactions between enzyme molecules and the possible role played by such protein-protein interactions in the regulatory process. Moreover, these theories are usually examined experimentally under conditions of very low protein concentration. Under such conditions protein-protein interactions would be expected to be very weak. It is hot at all clear that these theories apply, in their present form, to the conditions of high protein concentration found in vivo. Furthermore, the prevalent procedure of extrapolating experimental determinations of the chemical characteristics of an enzyme made at low protein concentration in vitro, to conditions of high protein concentrations found in vivo, would appear to be open to serious question if such protein-protein interactions are important. The possibly important, but poorly understood nature of the protein-protein interaction was alluded to by Monod et al. (1962) in their original paper on the allosteric hypothesis : Thus while aUosteric agents frequently appear to affect the state of aggregation of the sensitive proteins, the activating or inhibitory effects of the same agents do not seem necessarily to depend upon the association-dissociation reaction itself. The nature of the indirect correlation which appears, nevertheless, to exist between the two classes of effects remains to be explored and interpreted.
154
R. J. COHEN AND G. B. B E N E D E K
In a preliminary report (Cohen et al., 1975) we presented a simple quantitative model for analyzing the detailed relationship between the aggregative process and the level of catalytic activity of bovine GDH. This model led to a direct and simple understanding of the results of Fisher et al. (1962) and the experimental observations of Frieden & Colman (1967). I t also accounted quantitatively for our own experimental determinations of polymer distribution and enzymatic activity. Moreover, this model elucidates in general terms the role played by the aggregation in the control of the catalytic activity of GDH. I n this model it is proposed that the enzyme monomer t can exist in one of two conformational states, an active x form and an inactive y form. The x form demonstrates a greater tendency to polymerize to form x polymers than does the y form to form y polymers. The ratio of the concentration of the free x monomer to the concentration of the free y monomer is determined solely b y the concentration of allosteric effectors present. However, the concentration of effectors does not affect the values of the two polymerization constants which characterize the tendency of each form of the enzyme to polymerize$. I n contrast to Frieden (1967) and Niehol et al. (1967) we proposed that the polymerization process p e r se does not affect any of the catalytic parameters of the enzyme monomer. In fact it is assumed that the catalytic activity of an enzyme monomer of a given type is precisely the same whether it is free in solution or polymerbound. The polymerization process, however, does affect the overall catalytic activity of the enzyme solution since it affects the balance between the total (i.e. free or polymer~bound) number of x monomers and the total number of y monomers present in solution. The dependence of enzyme activity on the concentration of effector molecules should be markedly altered at high enzyme concentration, as Frieden & Colman (1967) in fact observed experimentally, because, according to our theory, at high enzyme concentration the polymerization process can dramatically shift the balance between the total number of x and the total number of y monomers present. On the other hand, under the conditions of Fisher's experiments, essentially all of the GDH monomerie units were in the active x form at all enzyme concentrations. Under such conditions, changing the total enzyme concentration leads to a change in the distribution of polymers but no change in the fraction of the enzyme in the active form. Hence, no change in the specific catalytic activity was observed (a more detailed explanation of Fisher's experiment is found later in the text). I n our initial report on the subject we presented our first experimental tests of the model by means of quasi-elastic light-scattering spectroscopic determinations of polymer distribution and biochemical determinations of enzyme activity. These results indicated that the model was quite successful in predicting both the distribution of enzyme polymers and the biochemical activity of the enzyme solution as a function of total enzyme concentration and effector molecule concentrations. I t was shown that the net effect of the aggregation is : (a) to cause the allosteric transition What we refer to as the GDH monomer is in fact itself composed of six subunits which do not normally dissociate. Hence our GDH monomer is often referred to as the oligomer in the literao ture. This construction is similar to a qualitative scheme proposed by Bitensky r al. (1965), based on a suggestion of Frieden (1963c), in an effort to understand qualitatively why the degree of aggregation of GDH varied as a function of effector concentration. However, this s c h e m e w a s not analyzed quantitatively nor was it used to examine the possible role of aggregatior~ irL ir~. fluenclng the level of activity of the enzyme.
POLYMERIZATION
AND ACTIVITY
OF GDH. I
155
of the enzyme from inactive to active form to occur at a lower level of allosterie activation, where the level of aUosteric activation is a measure of the relative concentrations of allosteric activators and inhibitors ; and (b) to m a k e this allosteric transition a more abrupt function of the level of allosteric activation. Thus this model not only provides a rational means of connecting the distribution of enzyme polymers with the level of enzymatic activity, but it also clearly indicates a potentially import a n t biological role for the polymerization process. Subsequent, more extensive, experimental measurements indicate t h a t a further refinement of the model yields predictions in better quantitative agreement with the experimental data. The modification involves relaxing the restriction t h a t excludes the formation of mixed x - y polymers. As will be seen, this modification preserves the central features of the original model. I n this present paper we complete the mathematical development of the original version of the model (Model I) t h a t was sketched in our initial report (Cohen et al., 1975). I n addition we work out mathematically the predictions of two alternative models (Models I I and I I I ) in which we relax the restriction precluding the formation of mixed x - y polymers. I n the final section of this paper we show how to calculate the diffusion constant distribution measured b y quasi-elastic light-scattering spectroscopy from the polymer distribution predicted b y a n y of the three models. I n the following paper (Cohen et al., 1976) we compare the predictions of each of the three models with a detailed set of experimental investigations.
2. Theory of Enzyme Activity and Polymerization I n this section we present and solve three versions of a theoretical model t h a t connects in detail the distribution of the enzyme polymers with the biochemical activit y of the enzyme solution. All of the models assume t h a t the enzyme monomer can exist in two conformational states, x or y. All the models also assume t h a t the mode of aggregation is one of linear polymerization. Model I is the same as the model which we proposed in outline form in our earlier report. I n Models I I and I I I , in contrast to Model I, the possibility of forming mixed x - y polymers is permitted. I n Model I I it is assumed t h a t both the x - y and y - x binding constants are each equal to the geometric mean of the x - x and y - y binding constants. I n Model I I I the x - y binding constant is assumed to be equal to the y - y binding constant, and the y - x binding constant is assumed to equal the ~-x binding constant (or vice versa). These proposed relationships correspond to possible simple physical structures of the x and y monomers. Numerical calculation of the predictions of a n y of these three models requires a knowledge of the values of the same four fundamental quantities: the x - x binding constant, the y - y binding constant, the degree of allosteric activation (which determines the ratio of the concentrations of free x monomer to free y monomer), and the total enzyme concentration. (a) Model I The scheme of Model I is presented in Figure l(a). The equilibrium constant K 0 fixes the relative proportion of free monomeric G D H in the active (x) or inactive (y) forms. The value of K 0 is in turn determined b y the concentration of effector molecules present. Here we assume t h a t the x form of the enzyme polymerizes to form rigid
156
R.J.
COHEN
AND
G. B . B E N E D E K
AIIoster=c effectors determine h"o
Eqmlibrmm
Equilibriufj
constant for y polymerization
constant for x polymerization
Ky y
Kx
polymers .= =' Y ~
x 4
:
x polymers
Ko
Equilibrium constant for the reactive active monomer
reaction
(a)
m units
n units
io iololololo1+1ololo P2
"4-
m + n units
Iolol :--Io Iolo I~ PP
---
Iolo Io Io I ol P2 §
a =xory
(b] FIG. 1. S c h e m a t i c r e p r e s e n t a t i o n Model I.
of the forms the GDH molecules may assume according to
linear x polymers, and that the y form polymerizes to form linear y polymers. However, no mixed x - y polymers are formed. The polymerization constant K x characterizes the tendency of the x form to polymerize and K ~ characterizes the tendency of the y form to polymerize. The values of K x and K ~ are not affected by the concentration of effector molecules. The catalytic activity of a GDH monomer of a given conformational type is the same whether it is free in solution or polymer bound. Thus the catalytic activity of an x polymer composed of m monomers is just m times the catalytic activity of a single x monomer. The y forms have no catalytic activity with respect to the oxidative deamination of glutamate (but m a y have activity with respect to the dehydrogenation of alanine and other amino acids; Tomkins et a l . , 1965). The mechanism of linear polymerization incorporated here is based on the
POLYMERIZATION
AND
ACTIVITY
OF GDH. I
157
work of Eisenberg, Tomkins, Reisler and others who characterized the structure and size distribution of aggregates of the active form of the enzyme using intensity lightscattering, equilibrium sedimentation and other techniques (Eisenberg & Tomkins, 1968; Reisler & Eisenberg, 1970; Reisler et al., 1970; Krause et al., 1970; Eisenberg & Reisler, 1971; Markau et al., 1971; Pilz & Sund, 1971; Reisler & Eisenberg, 1971; Thusius et al., 1975). Let the symbol P~a denote a polymer of type cr (cr = x or y) and length m (m = 1, 2, 3 . . . . ). Let C~ be the number concentration of such polymers in solution. The equilibrium between active and inactive free monomers can be quantitatively expressed as follows, C~ ----KoC ~,
(I)
where, as mentioned above, K o depends only on the concentration of effeetor molecules. This model does not explicitly examine the functional dependence of K o on
the concentration of the various effector molecules. Indeed, this problem is the entire subject matter of the Monod-Wyman-Changeux (1964) theory of allosteric transitions (Monod et al., 1964). Here we are in fact examining precisely how the aggregation process causes the catalytic activity to differ from that which would be simply expected from the dependence of K0 on the concentration of effector molecules. We may symbolically represent the equilibrium between linear chain polymers of different lengths as follows (see Fig. l(b)), Pm+n"
Applying the law of mass action to this chemical reaction, we obtain C~+. :
~;ra m n ~am~~a ~ 9
(2)
The equilibrium constant Kamnis here designated in such a way as to indicate that it can, in principle, depend on the lengths of each of the two interacting polymer chains. We will make the assumption that the free energy of bond formation is independent of the lengths of the polymer chains on either side of that bond (see Appendix). Therefore, Kan will be independent of m and n, and m a y be designated simply by the single symbol K a. Thus, a a Ca~+, = K a CmC,.
(3)
This implies Cg = K a C [ C [ C~ = K a C [ O ~
: (KaC[)2O~
a U : (KaO~)ra-IC~. Cam : K a CrnOZ
(4)
Equation (4) states that the concentration of polymers is a geometrically decreasing function of polymer length. Equation (4) in essence relates the distribution of polymers of either the active or inactive form to the concentration of monomers, C~ or The catalytic activity of the enzyme solution is proportional to the total concentration of active x-monomerie units present either as free monomers or polymer
158
R. J. C O H E N
AND
G. B. B E N E D E K
bound. Let C x be the total concentration of such active x-elements and C ~ the total concentration of inactive y-elements. Then
09 ~
~ mC~
(5a)
C y - - ~ mC~.
(5b)
I f the total concentration of enzyme monomeric units initially put into solution is denoted b y C, then we must require C = C ~ + C ~.
(5e)
We can analytically carry out the summations indicated in equation (5) b y using equation (4) as follows, co =
mO: =
m= l
m(I
oOVo-IOa
m =l
(6)
----- Ca/(1 - - g a c ~ ) 2 .
We m a y invert equation (6) to express Ca, the concentration of free a monomers, as a function of C a, the total concentration of a-type monomeric units whether free in solution or polymer bound, C a = {1 + [1 --.V/(1-b4gaca)J/(2KaCa)}/K a.
(7)
Collecting together the principal equations of the model, we have C~ =- KoC~ C = C~ § C~ C" = C~/(1 - - K~CI) 2 C ~' = VII(1 - - KYC~) 2.
Using these four equations we can in principle solve for the four unknown quantities C{, C[, C ~ and C ~ in terms of the four parameters K ~, K ~, C and K 0. F r o m this solution in conjunction with equation (4) we can calculate the distribution of polymers. T h a t is, we can calculate the quantity C~, which is the total concentration of polymers of length m in solution, = ( K ~ C f ) m - l C l + (K~C~)"-IC[.
(8)
Let us now define the quantity S, S = gffg. S is the fraction of all the G D H monomerie units which are in the active x form. Since we assume t h a t the catalytic activity of a monomeric unit is the same whether this unit is free in solution or polymer bound, the quantity S is directly proportional to the catalytic activity of the enzyme solution. We m a y denote S as the fractional catalyti~ activity of the e n z y m e solutiou. S approaches unity when all of the enzyme is in the x form and the catalytic activity of t h e enzyme solution is maximal. S approaches zero when all of the enzyme is in the y form and the catalytic activity of the enzyme solution is zero. I f the polymerization process did not occur, then the biochemical activity of the
P O L Y M E R I Z A T I O N AND ACTIVITY OF GDH. I
159
enzyme solution would be solely determined b y the concentration of effector molecules through the equilibrium constant K0. I t is elementary to show that in this case the fractional activity S would be given by S : Ko/(Ko ~- 1).
In fact S always approaches this value when the enzyme concentration is sufficiently reduced so that KxC
The Functional Relationship between Polymerization and Catalytic Activity of Beef Liver Glutamate Dehydrogenase I. T h e o r y RICHARD JONATHAN COHEN AND GEORGE B. BENEDEK
Department of Physics, Center for Materials Science and Engineering and the Harvard-M.I.T. Program in Health Sciences and Technology Massachusetts Institute of Technology, Cambridge, Mass 02139, U.S.A. (Received 15 March 1976, and in revised form 12 July 1976) In our initial report on this subject (Cohen et al., 1975) we presented evidence that the reversible polymerization of beef liver glutamate dehydrogenase may play an important role in the allosteric control of this enzyme. At that time we proposed a simple quantitative model for analyzing the detailed relationship between the distribution of enzyme polymers and the catalytic activity of the enzyme solution. The model was tested by means of quasi-elastic light-scattering spectroscopic determinations of polymer distribution and biochemical determinations of enzyme activity. In this present paper we complete the mathematical development of the original model. In addition, subsequent more extensive experimental investigations have led us to modify the original model to obtain even closer agreement between theory and experiment. Hence we present, in addition to the original model, two alternative versions of the model. The original model presumed that the enzyme polymers must be homogenous, in that all of the constituents of a given polymer had to be in the same eonformational state. The two alternative models deal explicitly with the possibility of forming polymers composed of elements in different conformational states. In the final section of this paper we develop the theoretical framework necessary to relate the distribution of polymers predicted by any of the three models with the distribution of diffusion constants measured by quasi-elastic light-scattering spectroscopy. In the following paper (Cohen et al., 1976) we compare the predictions of each of the three models with a detailed set of experimental investigations. 1. I n t r o d u c t i o n Beef liver glutamate dehydrogenase is a mitochondrial enzyme which catalyzes the oxidative deamination of glutamic acid: glutamate H NAD + H O H - ~- ~-ketoglutarate H N A D H H N H +. PariUa & Goodman (1974) present evidence that in rive glutamate dehydrogenase (GDH) operates primarily in the direction of glutamate deamination. This reaction is an important step in the catabolic pathways of m a n y of the amino acids (McGilvery, 1970; Stadtman, 1966). Thus GDH~ could act physiologically as a control element Abbreviation used: GDH, glutamate dehydrogenase. 151
152
R.J.
COHEN AND G. B. B E N E D E K
regulating the rate at which proteins are used as an energy source for cellular metabolism. Indeed it is found t h a t the high energy triphosphates, G T P and ATP, act as potent allosteric inhibitors of G D H and t h a t the relatively energy-poor diphosphate A D P acts as an allosteric activator (Frieden, 1963a). Thus the accumulation of energy-rich triphosphates m a y well signal the enzyme to reduce the rate of amino acid catabolism while, conversely, increased concentrations of A D P m a y provide the signal to increase the rate of conversion of proteins into energy. I n addition the activity of G D H is allosterically modulated b y a wide assortment of other effector molecules (Eisenberg, 1971; Frieden, 19635; Sund et al., 1976). Although all the details have not been worked out, it is clear t h a t a nexus of biochemical feedback loops is involved in setting the level of G D H activityt. Thus it appears t h a t the fine control of G D H activity m a y be of central significance in the regulation of amino acid metabolism. Hence, the study of the mechanisms of the allosteric control of G D H is not only ~f importance with respect to understanding the relationship between molecular structure and function, but it is also i m p o r t a n t with regard to understanding the detailed nature of cellular control mechanisms. The early studies of bovine G D H revealed t h a t it possessed an interesting physical property which subsequently was found to be shared by a number of other regulatory enzymes (Levitzki & Koshland, 1972 ; Duncan etal., 1972). Namely, it was found t h a t G D H forms aggregates in vitro when it is present in high concentration (Olson & Anfinsen, 1952; Friedcn, 1958; K u b o etal., 1959). Significantly, it is estimated t h a t in the cell mitochondria the enzyme is present in this high concentration range, several milligrams per milliliter (Tomkins et al., 1963; Sund, 1964). The degree of aggregation was found to be a reversible function of total enzyme concentration. Also the number and size of the aggregates varied strongly and reversibly with the concentration of the effector molecules. The addition of inhibitor molecules such as G T P to a solution of the enzyme could almost completely abolish the aggregation. The subsequent addition of activator molecules such as A D P could reverse this effect and restore the aggregation (Yielding & Tomkins, 1960; Wolff, 1962; Frieden, 1968). On the basis of these findings it was first believed t h a t it was necessary for the enzyme to be in an aggregated form for it to be active (Tomkins et al., 1961). This contention was examined by Fisher et al. (1962). They measured the mean molecular weight of the aggregates by measuring the intensity of light scattered from solutions of the enzyme, and they simultaneously determined the enzyme activity. As the concentration of the enzyme was changed from 0-I mg/ml to 4 mg/ml, they found t h a t the mean molecular weight of the aggregates increased by a factor of three while the specific catalytic activity (the activity per mg of enzyme) stayed constant. Fisher et al. (1962) concluded t h a t the activity of the enzyme was therefore independent of its state of aggregation. The prevailing view subsequent to the publication of Fisher's results was t h a t the aggregation of G D H is perhaps interesting, but biologically insignificant. However, Fricden & Colman (1967) subsequently found t h a t the dependence of enzyme activity on the concentration of various purine nucleotide effector molecules t Moreover, when GDH is allosterically inhibited with reference to its ability to oxidatively glutamate, it seems to have an increased ability to function as an alanine dchydrogenase (Tomkins etal., 1965). In this paper we will refer to the activation and inhibition of GDH with reference only to its activity as a glutamate dehydrogenase. deaminate
POLYMERIZATION AND ACTIVITY OF GDH. I
153
could be substantially modified by varying the level of enzyme concentration. This phenomenon was related to the aggregation of the enzyme which increases with increasing enzyme concentration. It was suggested that this effect might play a physiological role in vivo. Frieden (1967) and Nichol et al. (1967) suggested a possible mechanism by which the polymerization process might be able to influence the dependence of enzyme activity on the concentration of various ligands. It was suggested that the polymeric forms of enzyme (vis-a-vis the monomeric form) are characterized by a different set of affinities for substrate and for each of the effector molecules, and perhaps by a different set of rate constants for the breakdown of enzyme-substrate complex. According to this scheme, at low enzyme concentration the enzyme would be predominantly in the monomeric form, and at high enzyme concentration the enzyme would be predominantly in the polymeric forms. Thus at low enzyme concentration the behavior of the enzyme solution would reflect the properties of the enzyme monomer. At higher enzyme concentrations the behavior of the enzyme solution would reflect the properties of the polymeric forms. However, this analysis did not adequately account for the results of Fisher and others (Iwatsubo & Pantaloni, 1967 ; Josephs et al., 1972). Moreover, the overall biological significance of the aggregation process remained unclear. In fact, despite the findings of Frieden & Colman (1967), the consensus in the recent literature is that the aggregation process is of little or no biological importance (Fisher, 1973; Sund et al., 1976). Our investigations, on the other hand, clearly indicate that aggregation may play a vital role in the allosteric regulation of GDH (Cohen et al., 1975). Furthermore, we believe that elucidating the precise function of aggregation in the control of this particular enzyme may significantly contribute to our general understanding of allosteric control mechanisms. Over the past fifteen years a number of quantitative theories have been proposed that analyze the mechanisms by which various ligands influence the catalytic activity of the regulatory enzymes (Monod et al., 1964; Koshland et al., 1966; Herzfeld & Stanley, 1974). The usual situation envisioned in these theories is that of an isolated enzyme molecule interacting with much smaller ligand molecules. There has been much less analysis of the interactions between enzyme molecules and the possible role played by such protein-protein interactions in the regulatory process. Moreover, these theories are usually examined experimentally under conditions of very low protein concentration. Under such conditions protein-protein interactions would be expected to be very weak. It is hot at all clear that these theories apply, in their present form, to the conditions of high protein concentration found in vivo. Furthermore, the prevalent procedure of extrapolating experimental determinations of the chemical characteristics of an enzyme made at low protein concentration in vitro, to conditions of high protein concentrations found in vivo, would appear to be open to serious question if such protein-protein interactions are important. The possibly important, but poorly understood nature of the protein-protein interaction was alluded to by Monod et al. (1962) in their original paper on the allosteric hypothesis : Thus while aUosteric agents frequently appear to affect the state of aggregation of the sensitive proteins, the activating or inhibitory effects of the same agents do not seem necessarily to depend upon the association-dissociation reaction itself. The nature of the indirect correlation which appears, nevertheless, to exist between the two classes of effects remains to be explored and interpreted.
154
R. J. COHEN AND G. B. B E N E D E K
In a preliminary report (Cohen et al., 1975) we presented a simple quantitative model for analyzing the detailed relationship between the aggregative process and the level of catalytic activity of bovine GDH. This model led to a direct and simple understanding of the results of Fisher et al. (1962) and the experimental observations of Frieden & Colman (1967). I t also accounted quantitatively for our own experimental determinations of polymer distribution and enzymatic activity. Moreover, this model elucidates in general terms the role played by the aggregation in the control of the catalytic activity of GDH. I n this model it is proposed that the enzyme monomer t can exist in one of two conformational states, an active x form and an inactive y form. The x form demonstrates a greater tendency to polymerize to form x polymers than does the y form to form y polymers. The ratio of the concentration of the free x monomer to the concentration of the free y monomer is determined solely b y the concentration of allosteric effectors present. However, the concentration of effectors does not affect the values of the two polymerization constants which characterize the tendency of each form of the enzyme to polymerize$. I n contrast to Frieden (1967) and Niehol et al. (1967) we proposed that the polymerization process p e r se does not affect any of the catalytic parameters of the enzyme monomer. In fact it is assumed that the catalytic activity of an enzyme monomer of a given type is precisely the same whether it is free in solution or polymerbound. The polymerization process, however, does affect the overall catalytic activity of the enzyme solution since it affects the balance between the total (i.e. free or polymer~bound) number of x monomers and the total number of y monomers present in solution. The dependence of enzyme activity on the concentration of effector molecules should be markedly altered at high enzyme concentration, as Frieden & Colman (1967) in fact observed experimentally, because, according to our theory, at high enzyme concentration the polymerization process can dramatically shift the balance between the total number of x and the total number of y monomers present. On the other hand, under the conditions of Fisher's experiments, essentially all of the GDH monomerie units were in the active x form at all enzyme concentrations. Under such conditions, changing the total enzyme concentration leads to a change in the distribution of polymers but no change in the fraction of the enzyme in the active form. Hence, no change in the specific catalytic activity was observed (a more detailed explanation of Fisher's experiment is found later in the text). I n our initial report on the subject we presented our first experimental tests of the model by means of quasi-elastic light-scattering spectroscopic determinations of polymer distribution and biochemical determinations of enzyme activity. These results indicated that the model was quite successful in predicting both the distribution of enzyme polymers and the biochemical activity of the enzyme solution as a function of total enzyme concentration and effector molecule concentrations. I t was shown that the net effect of the aggregation is : (a) to cause the allosteric transition What we refer to as the GDH monomer is in fact itself composed of six subunits which do not normally dissociate. Hence our GDH monomer is often referred to as the oligomer in the literao ture. This construction is similar to a qualitative scheme proposed by Bitensky r al. (1965), based on a suggestion of Frieden (1963c), in an effort to understand qualitatively why the degree of aggregation of GDH varied as a function of effector concentration. However, this s c h e m e w a s not analyzed quantitatively nor was it used to examine the possible role of aggregatior~ irL ir~. fluenclng the level of activity of the enzyme.
POLYMERIZATION
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of the enzyme from inactive to active form to occur at a lower level of allosterie activation, where the level of aUosteric activation is a measure of the relative concentrations of allosteric activators and inhibitors ; and (b) to m a k e this allosteric transition a more abrupt function of the level of allosteric activation. Thus this model not only provides a rational means of connecting the distribution of enzyme polymers with the level of enzymatic activity, but it also clearly indicates a potentially import a n t biological role for the polymerization process. Subsequent, more extensive, experimental measurements indicate t h a t a further refinement of the model yields predictions in better quantitative agreement with the experimental data. The modification involves relaxing the restriction t h a t excludes the formation of mixed x - y polymers. As will be seen, this modification preserves the central features of the original model. I n this present paper we complete the mathematical development of the original version of the model (Model I) t h a t was sketched in our initial report (Cohen et al., 1975). I n addition we work out mathematically the predictions of two alternative models (Models I I and I I I ) in which we relax the restriction precluding the formation of mixed x - y polymers. I n the final section of this paper we show how to calculate the diffusion constant distribution measured b y quasi-elastic light-scattering spectroscopy from the polymer distribution predicted b y a n y of the three models. I n the following paper (Cohen et al., 1976) we compare the predictions of each of the three models with a detailed set of experimental investigations.
2. Theory of Enzyme Activity and Polymerization I n this section we present and solve three versions of a theoretical model t h a t connects in detail the distribution of the enzyme polymers with the biochemical activit y of the enzyme solution. All of the models assume t h a t the enzyme monomer can exist in two conformational states, x or y. All the models also assume t h a t the mode of aggregation is one of linear polymerization. Model I is the same as the model which we proposed in outline form in our earlier report. I n Models I I and I I I , in contrast to Model I, the possibility of forming mixed x - y polymers is permitted. I n Model I I it is assumed t h a t both the x - y and y - x binding constants are each equal to the geometric mean of the x - x and y - y binding constants. I n Model I I I the x - y binding constant is assumed to be equal to the y - y binding constant, and the y - x binding constant is assumed to equal the ~-x binding constant (or vice versa). These proposed relationships correspond to possible simple physical structures of the x and y monomers. Numerical calculation of the predictions of a n y of these three models requires a knowledge of the values of the same four fundamental quantities: the x - x binding constant, the y - y binding constant, the degree of allosteric activation (which determines the ratio of the concentrations of free x monomer to free y monomer), and the total enzyme concentration. (a) Model I The scheme of Model I is presented in Figure l(a). The equilibrium constant K 0 fixes the relative proportion of free monomeric G D H in the active (x) or inactive (y) forms. The value of K 0 is in turn determined b y the concentration of effector molecules present. Here we assume t h a t the x form of the enzyme polymerizes to form rigid
156
R.J.
COHEN
AND
G. B . B E N E D E K
AIIoster=c effectors determine h"o
Eqmlibrmm
Equilibriufj
constant for y polymerization
constant for x polymerization
Ky y
Kx
polymers .= =' Y ~
x 4
:
x polymers
Ko
Equilibrium constant for the reactive active monomer
reaction
(a)
m units
n units
io iololololo1+1ololo P2
"4-
m + n units
Iolol :--Io Iolo I~ PP
---
Iolo Io Io I ol P2 §
a =xory
(b] FIG. 1. S c h e m a t i c r e p r e s e n t a t i o n Model I.
of the forms the GDH molecules may assume according to
linear x polymers, and that the y form polymerizes to form linear y polymers. However, no mixed x - y polymers are formed. The polymerization constant K x characterizes the tendency of the x form to polymerize and K ~ characterizes the tendency of the y form to polymerize. The values of K x and K ~ are not affected by the concentration of effector molecules. The catalytic activity of a GDH monomer of a given conformational type is the same whether it is free in solution or polymer bound. Thus the catalytic activity of an x polymer composed of m monomers is just m times the catalytic activity of a single x monomer. The y forms have no catalytic activity with respect to the oxidative deamination of glutamate (but m a y have activity with respect to the dehydrogenation of alanine and other amino acids; Tomkins et a l . , 1965). The mechanism of linear polymerization incorporated here is based on the
POLYMERIZATION
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ACTIVITY
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157
work of Eisenberg, Tomkins, Reisler and others who characterized the structure and size distribution of aggregates of the active form of the enzyme using intensity lightscattering, equilibrium sedimentation and other techniques (Eisenberg & Tomkins, 1968; Reisler & Eisenberg, 1970; Reisler et al., 1970; Krause et al., 1970; Eisenberg & Reisler, 1971; Markau et al., 1971; Pilz & Sund, 1971; Reisler & Eisenberg, 1971; Thusius et al., 1975). Let the symbol P~a denote a polymer of type cr (cr = x or y) and length m (m = 1, 2, 3 . . . . ). Let C~ be the number concentration of such polymers in solution. The equilibrium between active and inactive free monomers can be quantitatively expressed as follows, C~ ----KoC ~,
(I)
where, as mentioned above, K o depends only on the concentration of effeetor molecules. This model does not explicitly examine the functional dependence of K o on
the concentration of the various effector molecules. Indeed, this problem is the entire subject matter of the Monod-Wyman-Changeux (1964) theory of allosteric transitions (Monod et al., 1964). Here we are in fact examining precisely how the aggregation process causes the catalytic activity to differ from that which would be simply expected from the dependence of K0 on the concentration of effector molecules. We may symbolically represent the equilibrium between linear chain polymers of different lengths as follows (see Fig. l(b)), Pm+n"
Applying the law of mass action to this chemical reaction, we obtain C~+. :
~;ra m n ~am~~a ~ 9
(2)
The equilibrium constant Kamnis here designated in such a way as to indicate that it can, in principle, depend on the lengths of each of the two interacting polymer chains. We will make the assumption that the free energy of bond formation is independent of the lengths of the polymer chains on either side of that bond (see Appendix). Therefore, Kan will be independent of m and n, and m a y be designated simply by the single symbol K a. Thus, a a Ca~+, = K a CmC,.
(3)
This implies Cg = K a C [ C [ C~ = K a C [ O ~
: (KaC[)2O~
a U : (KaO~)ra-IC~. Cam : K a CrnOZ
(4)
Equation (4) states that the concentration of polymers is a geometrically decreasing function of polymer length. Equation (4) in essence relates the distribution of polymers of either the active or inactive form to the concentration of monomers, C~ or The catalytic activity of the enzyme solution is proportional to the total concentration of active x-monomerie units present either as free monomers or polymer
158
R. J. C O H E N
AND
G. B. B E N E D E K
bound. Let C x be the total concentration of such active x-elements and C ~ the total concentration of inactive y-elements. Then
09 ~
~ mC~
(5a)
C y - - ~ mC~.
(5b)
I f the total concentration of enzyme monomeric units initially put into solution is denoted b y C, then we must require C = C ~ + C ~.
(5e)
We can analytically carry out the summations indicated in equation (5) b y using equation (4) as follows, co =
mO: =
m= l
m(I
oOVo-IOa
m =l
(6)
----- Ca/(1 - - g a c ~ ) 2 .
We m a y invert equation (6) to express Ca, the concentration of free a monomers, as a function of C a, the total concentration of a-type monomeric units whether free in solution or polymer bound, C a = {1 + [1 --.V/(1-b4gaca)J/(2KaCa)}/K a.
(7)
Collecting together the principal equations of the model, we have C~ =- KoC~ C = C~ § C~ C" = C~/(1 - - K~CI) 2 C ~' = VII(1 - - KYC~) 2.
Using these four equations we can in principle solve for the four unknown quantities C{, C[, C ~ and C ~ in terms of the four parameters K ~, K ~, C and K 0. F r o m this solution in conjunction with equation (4) we can calculate the distribution of polymers. T h a t is, we can calculate the quantity C~, which is the total concentration of polymers of length m in solution, = ( K ~ C f ) m - l C l + (K~C~)"-IC[.
(8)
Let us now define the quantity S, S = gffg. S is the fraction of all the G D H monomerie units which are in the active x form. Since we assume t h a t the catalytic activity of a monomeric unit is the same whether this unit is free in solution or polymer bound, the quantity S is directly proportional to the catalytic activity of the enzyme solution. We m a y denote S as the fractional catalyti~ activity of the e n z y m e solutiou. S approaches unity when all of the enzyme is in the x form and the catalytic activity of t h e enzyme solution is maximal. S approaches zero when all of the enzyme is in the y form and the catalytic activity of the enzyme solution is zero. I f the polymerization process did not occur, then the biochemical activity of the
P O L Y M E R I Z A T I O N AND ACTIVITY OF GDH. I
159
enzyme solution would be solely determined b y the concentration of effector molecules through the equilibrium constant K0. I t is elementary to show that in this case the fractional activity S would be given by S : Ko/(Ko ~- 1).
In fact S always approaches this value when the enzyme concentration is sufficiently reduced so that KxC
