BIOPOLYMERS
VOL. 14, 19-32 (1975)
A Quantitative Model of Cooperative Effects in Aspartate Transcarbamylase and Related Hybrid Molecules MARK DIESENDORF, Department of Applied Mathematics, S.G.S., Australian National University, Canberra, A.C.T. 2600, Australia
Synopsis The hypothesis is made that cooperative (homotropic) effects in aspartate transcarbamylase (ATCase) and closely related hybrid molecules can be treated on the same basis, by expressing the effects in terms of cooperative interactions between catalytic sites. Then, provided the hybrid C,,,C,,,[R] exhibits cooperativity experimentally, it is deduced that at least two parameters are required for a consistent description of cooperative effects in ATCase. On this basis, a quantitative model is proposed that has cooperative interactions of two types: 1) a cooperative interaction of strength U between a pair of native catalytic sites situated in different catalytic trimers, and joined by a regulatory dimer; 2 ) an indirect cooperative interaction 0, between any pair of native catalytic sites in the same catalytic trimer, which is transmitted via two regulatory dimers. The model is consistent with the observation that the isolated catalytic trimer does not exhibit cooperativity, and that cooperative effects require the integrity of the regulatory dimers. In this model, the hybrid molecule C,,,C,,,[R] becomes simply a one-dimensional Ising model for three sites with pair interactions 0,and the hybrid C,,,C,,,[R] becomes an Ising model for two sites wit,h interaction U / 3 . The model of native ATCase is essentially two Ising systems (the catalytic subunits), each with three sites having intrasubunit pair interactions 0, coupled together with three intersubunit pair interactions U . The parameters 0and U can be determined by comparing the exact solutions for the saturation curves of the two Ising models with the curves obtained from cooperativity measurements on the corresponding hybrid molecules. The theory can be readily tested, when accurate experimental data become available, by substituting the values of 0 and U determined from the hybrids into the model of native ATCase, or into models of other hybrids.
INTRODUCTION The saturation curves for the binding of aspartate or carbamyl phosphate to the allosteric enzyme aspartate transcarbamylase (referred to as ATCase hereinafter) are sigmoidal in shape. This observation, taken together with other data, suggests strongly that the binding of ligand by the enzyme is cooperative.' It is generally accepted that allosteric enzymes consist of a small number of subunits, each of which contains one or more distinct binding sites, and that cooperative binding of ligand to the sites 19
@ 1975 by John Wiley & Sons, Inc.
20
DIESENDORF
results from conformational changes produced by the binding. However, beyond this, there is debate concerning tho molecular mechanisms of cooperative binding. I n the “concerted” m e ~ h a n i s m it , ~ is postulated t.hat the enzyme moleculo exists in equilibrium betwecn (at least) two conformational states R and T , which haw: diffrrcnt affinities for ligand. If, in the absence of ligand, the moleculc exists preferentially in the R stat(:, whilc it,binds ligand prcfcrcntially in thc I’ state, thrn addition of ligand will produce a coordinated transition of tho whole molc.culc to thc T statc, leading to a sigmoidal relation betwccn the fraction of sitcs occupied by ligand, and the ligand concentration. Utilizing the data available in 1967, Changeux and Rubin4 applied the concertctd modd t o ATCase. I n thc “sequential” mod(ll,5it is assumed that the subunits of the enzyme molecule change their conformational statc individually upon substrate binding. I n contrast to the concerted case, thc occupation of site i increases (dccreascs) the probability of occupation of site j ; i.c., there is a positive (negative) coopemfive interaction between sites i and j , which accounts for the sigmoidal character of the saturation curvcs. It was shown by Thompson6 tha t sigmoidal saturation curves could be derived, within the conceptual framework of thc scquent,ial model, from the one-dimensional Ising model of equilibrium statistical mechanics. Thc Ising model is well known in statistical m e c h a ~ i i c sand ~ . ~ has been utilized previously in biology by Zimm and Braggg and more rc:ccntly by ;\Iontroll and Goello*ll t o account for the melting curve of DNA. Both the concerted and sequrntial models can provide good fits to the observed saturation curvw of ATCasc upon adjustment of the parameters of the models. The concerted model requires two paramet,crs, L and c, while the sequent,ial model6 requires only one parameter, the interaction energy U bctwecn neighboring catalytic sites. (We assume that the number of catalytic sites per molecule can be determincd independently.) Hence prcdictions beyond the saturation curve of ATCasc are required to differentiatc between thc models. Changeux and Rubin4 claimed that a sequential model of ATCasck is ruled out because it entails a linear relation between the fraction of molecules in a given conformation and the fraction of molecules occupied by substratc, and that such a relation is contradicted by the data. Hon-evor the data (r.g., sedimentation velocity experiments) refer t o the conformation of thr: uhole molecule, while a sequential model only requires progressive conformational changes a t the subunit level. A sequential model does not necessarily make any prediction about conformational changes of t he whole molecule. Further experimental tests were suggested by t,he ability of the Ising model formulation t o provide the framework for a quantitative treatment of inhibition and activation,e while t,he equilibrium model was extended t o the time-dcpendent casc.12 However, the structural model utilized for ATCase (eight catalytic sitrs in a circlc, with a regulatory site between each pair of catalytic sitcs6) has been superseded by experiment, and so a reformulation
COOPERATIVE EFFECTS I N ATCASE
21
of the sequential model, expressed in terms of cooperative interactions between catalytic sites, is required. We utilize the structural models of ATCase proposed by Gerhartl and Cohlberg et al.,13in which the enzyme is considered t o be composed of two catalytic subunits and three regulatory subunits. A catalytic subunit is a trimer of polypeptide chains, with each chain containing a single site for binding aspartate and a single separate site for binding carbamyl phosphate. Thus there are six distinct catalytic sites for each substrate, per ATCase molecule. A regulatory subunit consists of two polypeptide chains. Each regulatory dimer links two polypeptide chains situated in different catalytic trimers. The present work, a quantitative model of cooperative effects for the equilibrium system, attempts t o provide a unified t,reatment of ATCase and intersubunit hybrid ATCase-like molecules. These “hybrids” contain catalytic subunits in which one, two, or all three of the catalytic sites for a given substrate have been inact,ivated (i.e., cannot bind a substrate molecule). The following notation is utilized. A catalytic trimer wit,h one native and two inactivated sites (for example) is denoted by C,,,7. The hybrid of two such catalytic trimers, linked together by regulatory dimers, is denoted by C,,,C,,,[R]. In particular, the hybrid C,,,C,,,[R] is native ATCase. The basic hypothesis underlying the present model is that the only important effect on the cooperativity of ATCase, of inactivating a certain catalytic site, is t.o remove the cooperative interactions in which that particular site normally participates. It is shown below that, provided the hybrid C,,,C,,,[R] can be observed t o exhibit cooperativity, then a single-parameter sequential model6 is inadequate. It is therefore deduced logically that, consistent with the basic hypotheses, there must be a t least two independent types of cooperative interaction in ATCase, and hence a t least, two paramet.ers are required for a consistent quantitative description. Each parameter is determined independently from the cooperativity of thc appropriate hybrid molecule, each of which is shown t o be a one-dimensional Ising system. The values of these parameters can be substituted in the equation derived for the saturation curve of native ATCase, which turns out to be two coupled onedimensional k i n g systems, t o test if the observed value of the Hill coefficient of ATCase is regained. The model is also applied to a third hybrid, which depends on both parameters, and an equation for its saturation curve and hence for its Hill coefficient is derived.
DERIVATION OF THE MODEL Three basic experimental results are necessary for the derivation of the model. 1) It is well known that the intact isolated catalytic trimer of native sites does not exhibit cooperative substrate binding. Further, studies of intra-
DIESENDORF
22
TABLE I Observed and Calculated Cooperativity in Native ATCase and Succinylated Hybrids Species CL C n n n C n n n [RI b CnnnCss,[R1 c CnsL'nss[Rl
d c,",c,,,lnI c Two isolated native catalytic subunits
Weight factors for One-Paramet,er Model
Observed Hill Coefficientb
I 0 1 /9
1.6 1.3 1.12 Not reported 1.0
4/9
0
Weight factor (compared with unity for native ATCase) contributed by each species to rooperativity according t o the simple one-parameter sequential model of Fig. I . Preliminary data from Pigiet,I5 as reported by Schachman,16which relate t o binding of aspartate; correction factors are required for these values.
subunit hybrids of isolated catalyt,ic subunits demonstrate t h a t the relative cmymo mtivitivs of the species C,,,, C,,,, C,,v,v, C,,7,are in the rat,io 3 :2: 1 : O . Thus cnzymti activky of an isolated catalytic trimcr is proportional t o the number of native polypeptide chains in a catalytic subunit (see review by SchachmanI6). 2 ) The regulatory dimers are essential, not only for hetcrotropic (inhibitory or activatory) intcractions at rcgulatory sites, but also for the homotropic (coopcrativc) intcractions bctwecn catalytic sites in the intact enzyme. IGl1' 3 ) Cooperative effects arc' manifcst in the sigmoidal saturation curve or adsorption isothcrm obtaincd for thc uptake of aspartate or carbamyl phosphatc.. A moasurc of thc strcngth of coopcrativc effects in the steady state is thc Hill coc4icicmt 7 i H , which is determined by the slopc of tht: saturation curve F ( a ) a t one-half saturation1X[see Eq. (30) belo\v]. The third basic c~xpc~rimcmtal observation roquircd comprises the saturation curves (or a t [ R ]and CnSsCnSS [R]. lcast the Hill cocficicnts) of the hybrids CnnnCSsS At prescvit thwe arc no oxisting data of t'he lattor kind for which it has been demonstrated that inactivation it.sclf docs not introduce an extraneous coopc,rativity int,o tho mol(wdc. It is true that an existing method of inactivating sites on ATCasc. is t o succinylatc the polypeptide chains containing thosc. sitos.14 llcasurcmcnts of cooperativity of succinylated ATCase-lilie hybrids havc becn made by Pigiet,l5 and a preliminary report of the results has been publishcdlfi (Tablr I). However it is likely t,hat the succiriylation proci:ss, by introducing additional charges into the molecule, could modify coopcrativc irit c>ract,ionsbetn-cm sites (Schachman, private communication). For this roason \vc only quote succinylation data as an oxamplc of the kind of observations rcquirc.d, and t o suggwt a hypothesis for construct,ing the prosctnt model of ATCasc. Observation 2), talcm togdhcr with the structural models of ATCase constructc~don th(1 basis of d ( ~ c t r o nmicrographs and a variety of physical and chemical data by Gcirhart' and Cohlbcrg ct a1.,I3makc it highly plausiblc
COOPERATIVE EFFECTS IN ATCASE
a.
23
2
1
:I1 6
C.
6
&
d.
& Fig. 1. Catalytic sites and interactions in ATCase and intersubunit hybrids according to a simple one-parameter sequential model. Sites labeled 1, 2, 3 comprise one inactivated sites-0; regulacatalytic trimer, and 4, 5, 6 the other. Native sites--.; tory dimer mediating a cooperative interaction (-) between native catalytic sites; regulatory dimer not participating in a cooperative interaction (-------). a-native ATCase C,,,C,,,[R] ; b h y b r i d C,,,C,,,[R] ; c(i)-(iiitthe three possible arrangements for C,,,C,,,[R] ; d(i)-(iii)-the three arrangemenh for Cn,,sC,,,,s(R). This simple model fails qualitatively to account for the preliminary cooperativity observations (Table I).
that a t least one contribution to cooperativity in ATCase and in intersubunit hybrids comes from a direct interaction between pairs of catalytic sites, each situated in a different catalytic trimer and linked by a regulatory dimer. Indeed this is just the “basic allosteric unit” of the mechanism of Markus et al.17 Could this single type of Cooperative interaction account even qualitatively for the cooperativity observations (Table I) for succinylated hybrids? Consider a simple sequential model in which cooperative interactions, characterized by a single parameter, only exist between those pairs of catalytic sites joined directly by regulatory dimers. It is clear that both catalytic sites participating in a cooperative pair interaction must be native. Figure 1 shows schematically the simple sequential model applied to the species listed in Table I. The six catalytic sites are circles, the regulatory dimers not participating in a cooperative interaction are broken lines, and
24
DIESENDORF
t h e regulatory dimers mediating a cooperative interaction are unbroken lines. Table I also gives the relative neight that each diagram contributes t o the cooperativity, taking a weight factor unity for native ATCase. Species c has weight 1/9 because only one of the three possible diagrams, lc(iii), can contribute, and this diagram has 1/3 the number of cooperative interactions as species a. Provided tha t thc observed cooperativity of specicls b (Hill coefficient 1.3) is not all due t o the succinylation process, the one-parameter sequential model of Figure 1 fails for species 6; i.e., any observed cooperativity of species b cannot result from a direct interaction between catalytic sites situated in different catalytic trimers. Therefore, any cooperativity of species b not arising from the inactivation process must result from a separate, second kind of cooperative interaction, hich occurs betwern the sites of t h e native subunit C,,, only. Honever, by observation a), there cannot be a direct cooperative interaction between sitcs 1, 2, and 3. Hence a Cooperative interaction between sites 1 and 3 (for example) must propagate indirectly via the t n o rcgulatory dimers joining site 1 to site 4 and site 6 t o site 3. We know already t ha t there must be a relatively strong interaction between polypeptide chains \\ ithin a catalytic trimer, since the enzyme dissociates easily into catalytic trimers and regulatory dimers, rather than into mixed subunits. Direct evidence of such an intrasubunit interaction has been reported.lg Hence the strong noncooperative interaction between sites 4 and 6 servrs only t o join the cooperative links 1-4 and 6-3, which mediate an indirect cooperative interaction between native sites 1 and 3. If the hybrid molecule b has the same symmetries as ATCase, it follow tha t therc will be similar indirect cooperative interactions between sites 1 and 2, 2 and 3, and 3 and 1, in case b. Characterizing the indirect cooperative interaction between two native sites in the same catalytic trimer by a dimensionless parameter 0, wr represent our two-parameter model of the hybrid ATCase-like molecules in Figure 2. Figure 2b, for the hybrid molecule CnnnCSSS [ R ]is just a one-dimensional Ising model for three sites interacting via a nearest neighbor potential proportional t o ii. The hybrid molecule C,,,C,,,[R] of Figure 2c has only one native site in each catalytic trimer. Hence there can be no indirect interaction between catalytic sites, and for this particular case, our model reduces to the simple sequential model with interaction parameter U . Given that catalytic site 1 of one trimer is native, then there are three possible arrangements of the catalytic sites in the other trimer. However only Figure 2c(iii) can contribute to the cooperativity of this hybrid. Figure 2c(iii) represents an Ising model for two interacting sites. Because only one of t h e three diagrams contributes, the interaction parameter for species c will be U / 3 , assuming t ha t one-third of hybrid c molecules arc of type c(iii). Figure 2a represents our two-parameter model of the cooperative interaction in native ATCase. Within each catalytic trimw, the sites interact indirectly in pairs n i t h the intrraction parameter while thr trimers are
u,
COOPERATIVE EFFECTS I N ATCASE
b
25
2-9 2 ‘V’ 3
L
5
0
0
0 6
d.
0
r‘
L
Fig. 2. Cooperative interactions in native ATCase and three intersubunit hybrids according t o the present two-parameter model. a-native ATCase; b-hybrid C,,nDC588[R]; c(i)-(iii)-the three possible diagrams for C,,,C,,,[R] ; d(i)-(iii)-the three possible diagrams for C.,,C,,,[R]. Native sites-.; inactivated-0. The cooperative interaction U between a native catalytic site (labeled 1, 2, or 3) in one catalytic trimer and a native catalytic site (labeled 4, 5, or 6) in the other trimer is shown (-). The indirect cooperative interaction 0 between neighboring native catalytic sites in the same trimer is shown (------). Only cooperative interactions are shown in Fig. 2.
coupled together by the three direct cooperative interactions with interaction parameter U . The saturation curves for a one-dimensional Ising system can be obtained exactly from the partition function.6 By matching experimental and calculated values of nR for hybrids b and c, the values of 0and U , respectively, can be obtained. By utilizing these values in the theoretical saturation curve for native ATCase (two coupled king models), derived in the next section, one can test if the model gives the observed value of nH for native ATCase. Figure 2d represents our model of the hybrid CnnSCnnS [ R ] which , requires both parameters U and 0. We derive below the saturation curve for this molecule, and show how t o obtain the value of nH,given U and 0.
DERIVATION OF SATURATION CURVES FROM THE MODEL Saturation curves for native ATCase and for the hybrids C,,,C,,,[Rl and C,,,C,,,[R], are first derived in terms of the two parameters 0and U .
DIESENDORF
26
Consider a molecule containing N distinct catalytic sites for a particular substrate. We define a variable pLiwhich takes the value 1when site i is occupied by a substrate molecule, and - 1 when site i is vacant. A given distribution of substrate molecules ‘on the enzyme (a “state” or “configuration”) is then specified by the set of numbers
+
I PI
. .,PN).
(1) If all N sites are native, then there are 2N possible states of the enzyme molecule. In a given state, the number of occupied sites N + , expressed as a function of the variable p f , is = (PI,PZ,.
However, saturation curve measurements are not performed on a single enzyme molecule, but on a whole set of molecules in different states (i.e., an “ensemble” in the language of statistical mechanics), and hence entail an intrinsic averaging process. Let the probability of a given state { p } be P ( p ) . Then the average number of occupied sites is
m+
c c
w+=
)+=fl
. . . p NC= f l N+IPPIPl
p,=fl
EN+IPlP(PCrf
=
{ rl
We now consider native ATCase represented according to the model in Figure 2a. The probability of the state { p } for the arrangement of sites and cooperative interactions shown in the figure is
P,(P} =
+
exp [ O ( f i c L I ~ PZP3
+
+ O ( P ~+P ~ + 1 + + II exp (JrJ
P3Pd
P5P6
P6Pd
N
.exp
[U(PlP4
P2P5
P3Pd1
i= 1
(4)
where the quantity 2, is known as the partition function and is defined in such a way that the total probability of all possible states is unity, i.e.
Hence
Z, =
C exp lrl
+
[ ~ ( P M
+
+ .exp [ u ( P ~ P + ~
P2P3
+ +
~ 3 ~ 1 ) D(114~5
~ 1 2 ~ 5
P5P6
+
P3Pd1.
P6P4)I N
IT exp ( J P A (6)
i=l
The first exponent in Eqs. (4) and (6) represents the intrasite interaction energies (in dimensionless units) of the two catalytic trimers, while the second exponent describes the interactions between the two trimers. The
COOPERATIVE EFFECTS IN ATCASE
remaining exponents,
27
J p f , can be thought of as the energy of the ini
dividual sites in an external field. When U or 7 i> 0 (
VOL. 14, 19-32 (1975)
A Quantitative Model of Cooperative Effects in Aspartate Transcarbamylase and Related Hybrid Molecules MARK DIESENDORF, Department of Applied Mathematics, S.G.S., Australian National University, Canberra, A.C.T. 2600, Australia
Synopsis The hypothesis is made that cooperative (homotropic) effects in aspartate transcarbamylase (ATCase) and closely related hybrid molecules can be treated on the same basis, by expressing the effects in terms of cooperative interactions between catalytic sites. Then, provided the hybrid C,,,C,,,[R] exhibits cooperativity experimentally, it is deduced that at least two parameters are required for a consistent description of cooperative effects in ATCase. On this basis, a quantitative model is proposed that has cooperative interactions of two types: 1) a cooperative interaction of strength U between a pair of native catalytic sites situated in different catalytic trimers, and joined by a regulatory dimer; 2 ) an indirect cooperative interaction 0, between any pair of native catalytic sites in the same catalytic trimer, which is transmitted via two regulatory dimers. The model is consistent with the observation that the isolated catalytic trimer does not exhibit cooperativity, and that cooperative effects require the integrity of the regulatory dimers. In this model, the hybrid molecule C,,,C,,,[R] becomes simply a one-dimensional Ising model for three sites with pair interactions 0,and the hybrid C,,,C,,,[R] becomes an Ising model for two sites wit,h interaction U / 3 . The model of native ATCase is essentially two Ising systems (the catalytic subunits), each with three sites having intrasubunit pair interactions 0, coupled together with three intersubunit pair interactions U . The parameters 0and U can be determined by comparing the exact solutions for the saturation curves of the two Ising models with the curves obtained from cooperativity measurements on the corresponding hybrid molecules. The theory can be readily tested, when accurate experimental data become available, by substituting the values of 0 and U determined from the hybrids into the model of native ATCase, or into models of other hybrids.
INTRODUCTION The saturation curves for the binding of aspartate or carbamyl phosphate to the allosteric enzyme aspartate transcarbamylase (referred to as ATCase hereinafter) are sigmoidal in shape. This observation, taken together with other data, suggests strongly that the binding of ligand by the enzyme is cooperative.' It is generally accepted that allosteric enzymes consist of a small number of subunits, each of which contains one or more distinct binding sites, and that cooperative binding of ligand to the sites 19
@ 1975 by John Wiley & Sons, Inc.
20
DIESENDORF
results from conformational changes produced by the binding. However, beyond this, there is debate concerning tho molecular mechanisms of cooperative binding. I n the “concerted” m e ~ h a n i s m it , ~ is postulated t.hat the enzyme moleculo exists in equilibrium betwecn (at least) two conformational states R and T , which haw: diffrrcnt affinities for ligand. If, in the absence of ligand, the moleculc exists preferentially in the R stat(:, whilc it,binds ligand prcfcrcntially in thc I’ state, thrn addition of ligand will produce a coordinated transition of tho whole molc.culc to thc T statc, leading to a sigmoidal relation betwccn the fraction of sitcs occupied by ligand, and the ligand concentration. Utilizing the data available in 1967, Changeux and Rubin4 applied the concertctd modd t o ATCase. I n thc “sequential” mod(ll,5it is assumed that the subunits of the enzyme molecule change their conformational statc individually upon substrate binding. I n contrast to the concerted case, thc occupation of site i increases (dccreascs) the probability of occupation of site j ; i.c., there is a positive (negative) coopemfive interaction between sites i and j , which accounts for the sigmoidal character of the saturation curvcs. It was shown by Thompson6 tha t sigmoidal saturation curves could be derived, within the conceptual framework of thc scquent,ial model, from the one-dimensional Ising model of equilibrium statistical mechanics. Thc Ising model is well known in statistical m e c h a ~ i i c sand ~ . ~ has been utilized previously in biology by Zimm and Braggg and more rc:ccntly by ;\Iontroll and Goello*ll t o account for the melting curve of DNA. Both the concerted and sequrntial models can provide good fits to the observed saturation curvw of ATCasc upon adjustment of the parameters of the models. The concerted model requires two paramet,crs, L and c, while the sequent,ial model6 requires only one parameter, the interaction energy U bctwecn neighboring catalytic sites. (We assume that the number of catalytic sites per molecule can be determincd independently.) Hence prcdictions beyond the saturation curve of ATCasc are required to differentiatc between thc models. Changeux and Rubin4 claimed that a sequential model of ATCasck is ruled out because it entails a linear relation between the fraction of molecules in a given conformation and the fraction of molecules occupied by substratc, and that such a relation is contradicted by the data. Hon-evor the data (r.g., sedimentation velocity experiments) refer t o the conformation of thr: uhole molecule, while a sequential model only requires progressive conformational changes a t the subunit level. A sequential model does not necessarily make any prediction about conformational changes of t he whole molecule. Further experimental tests were suggested by t,he ability of the Ising model formulation t o provide the framework for a quantitative treatment of inhibition and activation,e while t,he equilibrium model was extended t o the time-dcpendent casc.12 However, the structural model utilized for ATCase (eight catalytic sitrs in a circlc, with a regulatory site between each pair of catalytic sitcs6) has been superseded by experiment, and so a reformulation
COOPERATIVE EFFECTS I N ATCASE
21
of the sequential model, expressed in terms of cooperative interactions between catalytic sites, is required. We utilize the structural models of ATCase proposed by Gerhartl and Cohlberg et al.,13in which the enzyme is considered t o be composed of two catalytic subunits and three regulatory subunits. A catalytic subunit is a trimer of polypeptide chains, with each chain containing a single site for binding aspartate and a single separate site for binding carbamyl phosphate. Thus there are six distinct catalytic sites for each substrate, per ATCase molecule. A regulatory subunit consists of two polypeptide chains. Each regulatory dimer links two polypeptide chains situated in different catalytic trimers. The present work, a quantitative model of cooperative effects for the equilibrium system, attempts t o provide a unified t,reatment of ATCase and intersubunit hybrid ATCase-like molecules. These “hybrids” contain catalytic subunits in which one, two, or all three of the catalytic sites for a given substrate have been inact,ivated (i.e., cannot bind a substrate molecule). The following notation is utilized. A catalytic trimer wit,h one native and two inactivated sites (for example) is denoted by C,,,7. The hybrid of two such catalytic trimers, linked together by regulatory dimers, is denoted by C,,,C,,,[R]. In particular, the hybrid C,,,C,,,[R] is native ATCase. The basic hypothesis underlying the present model is that the only important effect on the cooperativity of ATCase, of inactivating a certain catalytic site, is t.o remove the cooperative interactions in which that particular site normally participates. It is shown below that, provided the hybrid C,,,C,,,[R] can be observed t o exhibit cooperativity, then a single-parameter sequential model6 is inadequate. It is therefore deduced logically that, consistent with the basic hypotheses, there must be a t least two independent types of cooperative interaction in ATCase, and hence a t least, two paramet.ers are required for a consistent quantitative description. Each parameter is determined independently from the cooperativity of thc appropriate hybrid molecule, each of which is shown t o be a one-dimensional Ising system. The values of these parameters can be substituted in the equation derived for the saturation curve of native ATCase, which turns out to be two coupled onedimensional k i n g systems, t o test if the observed value of the Hill coefficient of ATCase is regained. The model is also applied to a third hybrid, which depends on both parameters, and an equation for its saturation curve and hence for its Hill coefficient is derived.
DERIVATION OF THE MODEL Three basic experimental results are necessary for the derivation of the model. 1) It is well known that the intact isolated catalytic trimer of native sites does not exhibit cooperative substrate binding. Further, studies of intra-
DIESENDORF
22
TABLE I Observed and Calculated Cooperativity in Native ATCase and Succinylated Hybrids Species CL C n n n C n n n [RI b CnnnCss,[R1 c CnsL'nss[Rl
d c,",c,,,lnI c Two isolated native catalytic subunits
Weight factors for One-Paramet,er Model
Observed Hill Coefficientb
I 0 1 /9
1.6 1.3 1.12 Not reported 1.0
4/9
0
Weight factor (compared with unity for native ATCase) contributed by each species to rooperativity according t o the simple one-parameter sequential model of Fig. I . Preliminary data from Pigiet,I5 as reported by Schachman,16which relate t o binding of aspartate; correction factors are required for these values.
subunit hybrids of isolated catalyt,ic subunits demonstrate t h a t the relative cmymo mtivitivs of the species C,,,, C,,,, C,,v,v, C,,7,are in the rat,io 3 :2: 1 : O . Thus cnzymti activky of an isolated catalytic trimcr is proportional t o the number of native polypeptide chains in a catalytic subunit (see review by SchachmanI6). 2 ) The regulatory dimers are essential, not only for hetcrotropic (inhibitory or activatory) intcractions at rcgulatory sites, but also for the homotropic (coopcrativc) intcractions bctwecn catalytic sites in the intact enzyme. IGl1' 3 ) Cooperative effects arc' manifcst in the sigmoidal saturation curve or adsorption isothcrm obtaincd for thc uptake of aspartate or carbamyl phosphatc.. A moasurc of thc strcngth of coopcrativc effects in the steady state is thc Hill coc4icicmt 7 i H , which is determined by the slopc of tht: saturation curve F ( a ) a t one-half saturation1X[see Eq. (30) belo\v]. The third basic c~xpc~rimcmtal observation roquircd comprises the saturation curves (or a t [ R ]and CnSsCnSS [R]. lcast the Hill cocficicnts) of the hybrids CnnnCSsS At prescvit thwe arc no oxisting data of t'he lattor kind for which it has been demonstrated that inactivation it.sclf docs not introduce an extraneous coopc,rativity int,o tho mol(wdc. It is true that an existing method of inactivating sites on ATCasc. is t o succinylatc the polypeptide chains containing thosc. sitos.14 llcasurcmcnts of cooperativity of succinylated ATCase-lilie hybrids havc becn made by Pigiet,l5 and a preliminary report of the results has been publishcdlfi (Tablr I). However it is likely t,hat the succiriylation proci:ss, by introducing additional charges into the molecule, could modify coopcrativc irit c>ract,ionsbetn-cm sites (Schachman, private communication). For this roason \vc only quote succinylation data as an oxamplc of the kind of observations rcquirc.d, and t o suggwt a hypothesis for construct,ing the prosctnt model of ATCasc. Observation 2), talcm togdhcr with the structural models of ATCase constructc~don th(1 basis of d ( ~ c t r o nmicrographs and a variety of physical and chemical data by Gcirhart' and Cohlbcrg ct a1.,I3makc it highly plausiblc
COOPERATIVE EFFECTS IN ATCASE
a.
23
2
1
:I1 6
C.
6
&
d.
& Fig. 1. Catalytic sites and interactions in ATCase and intersubunit hybrids according to a simple one-parameter sequential model. Sites labeled 1, 2, 3 comprise one inactivated sites-0; regulacatalytic trimer, and 4, 5, 6 the other. Native sites--.; tory dimer mediating a cooperative interaction (-) between native catalytic sites; regulatory dimer not participating in a cooperative interaction (-------). a-native ATCase C,,,C,,,[R] ; b h y b r i d C,,,C,,,[R] ; c(i)-(iiitthe three possible arrangements for C,,,C,,,[R] ; d(i)-(iii)-the three arrangemenh for Cn,,sC,,,,s(R). This simple model fails qualitatively to account for the preliminary cooperativity observations (Table I).
that a t least one contribution to cooperativity in ATCase and in intersubunit hybrids comes from a direct interaction between pairs of catalytic sites, each situated in a different catalytic trimer and linked by a regulatory dimer. Indeed this is just the “basic allosteric unit” of the mechanism of Markus et al.17 Could this single type of Cooperative interaction account even qualitatively for the cooperativity observations (Table I) for succinylated hybrids? Consider a simple sequential model in which cooperative interactions, characterized by a single parameter, only exist between those pairs of catalytic sites joined directly by regulatory dimers. It is clear that both catalytic sites participating in a cooperative pair interaction must be native. Figure 1 shows schematically the simple sequential model applied to the species listed in Table I. The six catalytic sites are circles, the regulatory dimers not participating in a cooperative interaction are broken lines, and
24
DIESENDORF
t h e regulatory dimers mediating a cooperative interaction are unbroken lines. Table I also gives the relative neight that each diagram contributes t o the cooperativity, taking a weight factor unity for native ATCase. Species c has weight 1/9 because only one of the three possible diagrams, lc(iii), can contribute, and this diagram has 1/3 the number of cooperative interactions as species a. Provided tha t thc observed cooperativity of specicls b (Hill coefficient 1.3) is not all due t o the succinylation process, the one-parameter sequential model of Figure 1 fails for species 6; i.e., any observed cooperativity of species b cannot result from a direct interaction between catalytic sites situated in different catalytic trimers. Therefore, any cooperativity of species b not arising from the inactivation process must result from a separate, second kind of cooperative interaction, hich occurs betwern the sites of t h e native subunit C,,, only. Honever, by observation a), there cannot be a direct cooperative interaction between sitcs 1, 2, and 3. Hence a Cooperative interaction between sites 1 and 3 (for example) must propagate indirectly via the t n o rcgulatory dimers joining site 1 to site 4 and site 6 t o site 3. We know already t ha t there must be a relatively strong interaction between polypeptide chains \\ ithin a catalytic trimer, since the enzyme dissociates easily into catalytic trimers and regulatory dimers, rather than into mixed subunits. Direct evidence of such an intrasubunit interaction has been reported.lg Hence the strong noncooperative interaction between sites 4 and 6 servrs only t o join the cooperative links 1-4 and 6-3, which mediate an indirect cooperative interaction between native sites 1 and 3. If the hybrid molecule b has the same symmetries as ATCase, it follow tha t therc will be similar indirect cooperative interactions between sites 1 and 2, 2 and 3, and 3 and 1, in case b. Characterizing the indirect cooperative interaction between two native sites in the same catalytic trimer by a dimensionless parameter 0, wr represent our two-parameter model of the hybrid ATCase-like molecules in Figure 2. Figure 2b, for the hybrid molecule CnnnCSSS [ R ]is just a one-dimensional Ising model for three sites interacting via a nearest neighbor potential proportional t o ii. The hybrid molecule C,,,C,,,[R] of Figure 2c has only one native site in each catalytic trimer. Hence there can be no indirect interaction between catalytic sites, and for this particular case, our model reduces to the simple sequential model with interaction parameter U . Given that catalytic site 1 of one trimer is native, then there are three possible arrangements of the catalytic sites in the other trimer. However only Figure 2c(iii) can contribute to the cooperativity of this hybrid. Figure 2c(iii) represents an Ising model for two interacting sites. Because only one of t h e three diagrams contributes, the interaction parameter for species c will be U / 3 , assuming t ha t one-third of hybrid c molecules arc of type c(iii). Figure 2a represents our two-parameter model of the cooperative interaction in native ATCase. Within each catalytic trimw, the sites interact indirectly in pairs n i t h the intrraction parameter while thr trimers are
u,
COOPERATIVE EFFECTS I N ATCASE
b
25
2-9 2 ‘V’ 3
L
5
0
0
0 6
d.
0
r‘
L
Fig. 2. Cooperative interactions in native ATCase and three intersubunit hybrids according t o the present two-parameter model. a-native ATCase; b-hybrid C,,nDC588[R]; c(i)-(iii)-the three possible diagrams for C,,,C,,,[R] ; d(i)-(iii)-the three possible diagrams for C.,,C,,,[R]. Native sites-.; inactivated-0. The cooperative interaction U between a native catalytic site (labeled 1, 2, or 3) in one catalytic trimer and a native catalytic site (labeled 4, 5, or 6) in the other trimer is shown (-). The indirect cooperative interaction 0 between neighboring native catalytic sites in the same trimer is shown (------). Only cooperative interactions are shown in Fig. 2.
coupled together by the three direct cooperative interactions with interaction parameter U . The saturation curves for a one-dimensional Ising system can be obtained exactly from the partition function.6 By matching experimental and calculated values of nR for hybrids b and c, the values of 0and U , respectively, can be obtained. By utilizing these values in the theoretical saturation curve for native ATCase (two coupled king models), derived in the next section, one can test if the model gives the observed value of nH for native ATCase. Figure 2d represents our model of the hybrid CnnSCnnS [ R ] which , requires both parameters U and 0. We derive below the saturation curve for this molecule, and show how t o obtain the value of nH,given U and 0.
DERIVATION OF SATURATION CURVES FROM THE MODEL Saturation curves for native ATCase and for the hybrids C,,,C,,,[Rl and C,,,C,,,[R], are first derived in terms of the two parameters 0and U .
DIESENDORF
26
Consider a molecule containing N distinct catalytic sites for a particular substrate. We define a variable pLiwhich takes the value 1when site i is occupied by a substrate molecule, and - 1 when site i is vacant. A given distribution of substrate molecules ‘on the enzyme (a “state” or “configuration”) is then specified by the set of numbers
+
I PI
. .,PN).
(1) If all N sites are native, then there are 2N possible states of the enzyme molecule. In a given state, the number of occupied sites N + , expressed as a function of the variable p f , is = (PI,PZ,.
However, saturation curve measurements are not performed on a single enzyme molecule, but on a whole set of molecules in different states (i.e., an “ensemble” in the language of statistical mechanics), and hence entail an intrinsic averaging process. Let the probability of a given state { p } be P ( p ) . Then the average number of occupied sites is
m+
c c
w+=
)+=fl
. . . p NC= f l N+IPPIPl
p,=fl
EN+IPlP(PCrf
=
{ rl
We now consider native ATCase represented according to the model in Figure 2a. The probability of the state { p } for the arrangement of sites and cooperative interactions shown in the figure is
P,(P} =
+
exp [ O ( f i c L I ~ PZP3
+
+ O ( P ~+P ~ + 1 + + II exp (JrJ
P3Pd
P5P6
P6Pd
N
.exp
[U(PlP4
P2P5
P3Pd1
i= 1
(4)
where the quantity 2, is known as the partition function and is defined in such a way that the total probability of all possible states is unity, i.e.
Hence
Z, =
C exp lrl
+
[ ~ ( P M
+
+ .exp [ u ( P ~ P + ~
P2P3
+ +
~ 3 ~ 1 ) D(114~5
~ 1 2 ~ 5
P5P6
+
P3Pd1.
P6P4)I N
IT exp ( J P A (6)
i=l
The first exponent in Eqs. (4) and (6) represents the intrasite interaction energies (in dimensionless units) of the two catalytic trimers, while the second exponent describes the interactions between the two trimers. The
COOPERATIVE EFFECTS IN ATCASE
remaining exponents,
27
J p f , can be thought of as the energy of the ini
dividual sites in an external field. When U or 7 i> 0 (
