Kinetics of the allosteric transition of aspartate transcarbamylase chemical quench studies

J. Mol. Riol. (1984) 176, 523-534

Kinetics of the Allosteric Transition of Aspartate Transcarbamylase Chemical Quench Studies H. KTHARA’~,

T. E. BARMAN', ’ European D-6900

2 INSERM

P. T. JONES’ AND M. F.

Moonu’

Molecular Biology Laboratory Heidelberg, West Germany

U 128, C’NRS, (Received

34033 Montpellier

Cedex, France

23 December 1983)

IJsing a chemical quench device, the rate of synthesis of carbamyl aspartate from the substrates aspartate and carbamyl phosphate was followed as a function of the time between mixing the enzyme with substrates and quenching with trichloroacetic acid. This function, which is linear at long times, shows (at 4°C) a transient lag phase of product of roughly 10ms. However, when the catalytic subunit (in which the enzymatic activity is desensitized) is used instead of the enzyme, the lag disappears. Therefore the lag seems to be associated with the control functions of the enzyme, i.e. to represent the allosteric transition involved in substrate-substrate (homotropic) co-operativity. Thus the relaxation time for the activation process is roughly 10 ma The implications of these results are examined.

1. Introduction transcarbamylase of Escherichia coli (EC 2.1.3.2) catalyses the reaction (carbamyl phosphate carbamyl aspartate and with L-aspartate giving orthophosphate) that commits aspartate to pyrimidine biosynthesis. The key position of ATCase$ is used to control this pathway (it is activated by substrates or by ATP, and is inhibited by CTP), and it is one of the most intensively studied allosteric enzymes (for reviews, see Gerhart, 1970; Hammes & Wu, 1971c; Jacobson & Stark, 1973; Kantrowitz et al., 1980a,b). The native enzyme consists of three regulatory subunits (each composed of two chains, and with two binding sites for ATP or CTP) and of two enzymatically active (but desensitized) catalytic subunits (each composed of three chains, and containing three active sites). Detailed structures are known for two forms of the enzyme without bound Aspartate

t Present address: Department of Physics, Jichi Medical School, Minamikawachi-Machi, TochigiKen, ,Japan 329-04. 1 Abbreviations used: ATCase, aspartate transcarbamylase from E. coli (EC 2.1.3.2); CTP, cytidine triphosphate; DTE, dithioerythritol; EDTA, ethylenediaminetetraacetic acid; PMSF, phenylmethane sulphonic acid; TCA, trichloroaeetie acid. 523 0(~2-2830/84/200523-12

$03.00/O

@> 1984 Academic

Press Inc. (London)

Ltd.

521

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substrate (Honzatko et nl.. 1982). The binding of substrate analogues to t,hesc causes a large conformational change, originally seen by hydrodynamic techniques (Gerhart & Schachman, 1968; nubin & Cannell, 1975). and recently studied in more structural detail by low-angle X-ray scattering (Moody et al.. 1979) and X-ray crystallography (Ladner et al., 1982; Altman et al., 1982). To determine the mechanism of the allosteric transition, we must investigate its kinetics. The first question is the approximate speed of the transition; this is relevant both to its biological control function and to the technical problems of studying it. Hammes’ group conducted an extensive series of temperature-jump experiments (e.g. see Hammes & Wu, 1971a,h), but they found a complex array of optically monitored time constants that has left uncertain the rate of the allosteric transition. Keck & Schuster (1976) and Keck (1980), who studied the binding of tetraiodofluorescein to ACTase, also found a rich variety of time constants. Furthermore, none of these studies used the natural substrates. To clarify the question, we decided to measure the rate at which substrates induce increased enzyme activity, a characteristically allosteric property. This means following the initial formation of product, an approach that has been used to investigate the initial steps of enzyme reaction pathways (Gutfreund, 1971,1972), and that depends upon two pre-conditions: the ability to sample the reaction mix and the availability of a specific and sensitive assay for at times below k,:, product. Both of these can be satisfied, the former by using the rapid flow-quench technique (Gutfreund, 1969), and the latter by the method of Prescott & Jones (1969). Having established in this way the rate of co-operative homotropy, we use (unpublished results) stopped-flow experiments to connect this rate with the time constants observed, using temperature-jump, by Hammes’ group. By monitoring the reaction in these different ways, we can make a start on the long undertaking of assigning the rates of all the different processes that occur in the enzyme during its allosteric transition.

2. Materials and Methods (a) Chemicals Dilithium carbamyl phosphate was purchased from Gerva and further purified by precipitation from 50% ethanol (Gerhart & Pardee, 1962). (The carbamyl phosphate solution was freshly prepared just before each experiment to avoid hydrolysis (Allen & Jones, 1964).) Potassium aspartate was from ICN Pharmaceuticals, and EDTA, TCA, diacetylmonoxime and antipyrene( I-phenyl-2,3-dimethyl-pyrazolone) from Merck, DTE and PMSF from Sigma. N-Carbamyl-I%aspartic acid was from Sigma and CTP (disodium salt) from Pharma Waldhof. Antipyrenef 1,5-dimethyl-2-phenyl-3-pyrazolone) and diacetylmonoxime for the calorimetric enzyme assay were from Merck. (b) Enzyme preparation

and reaction

conditions

The diploid strain of E. coli was grown in a 4000 1 fermenter at the Gesellschaft fiir Biologische Forschung, Stockheim. ATCase was prepared by a modification (Foote, Leberman, Jones & Moody, unpublished results) of Gerhart & Holoubek’s (1967) method. Catalytic subunit was prepared by the procedure of Yang et al. (1978). The enzyme was buffered with 20 mM-potassium phosphate (pH 7) and 0.1 mM-EDTA, 0.1 mM-DTE and

ATCase

ALLOSTERIC

TRANSITION

KINETICS

525

0.25 mM-PMSF were often added. The enzyme was concentrated to 73 mg/ml using an Amicon model 8MC filter unit with a PM 30 membrane, and the preparation was cleared by low-speed centrifugation immediately before the experiment. (The concentration was determined by spectrophotometry, assuming an extinction coefficient at 280 nm of 0.59 cm’mg-’ for ATCase (Gerhart & Holoubek, 1967) and 0.70 cm2mg-’ (Collins & Stark, 1969) for catalytic subunit.) All the experiments were done at pH 7 since, in phosphate buffer at pH 7, ATCase is predominantly in the inactive form. (c) Assay for product Carbamyl aspartate was assayed according to Method II of Prescott & Jones (1969) with some modifications: (1) 1 ml of the colour mix was added to 0.6 ml of each sample; (2) after being left in the dark overnight, each sample was heated at 56°C for 70 min; (3) each sample was centrifuged twice, before the colour mix was added and just before its absorbance was measured. The extinction coefficient of carbamyl aspartate at 466 nm was 4.6 x lo3 optical density units/m01 on the average but, as it was sensitive to slight changes in experimental conditions, the sensitivity was calibrated each time by including standard carbamyl aspartate solutions in the same rack as a control. (d) Quenched-jlow experiments The rapid quench-flow apparatus used in this work (Barman et al., 1980) takes samples in the time range 4 ms to 280 ms and was thermostatically controlled at +4 kO.2 deg.C. In a typical experiment, the apparatus rapidly mixed l-1 ml of enzyme dissolved in buffer with an equal volume of substrate dissolved in the same buffer. The concentrations of enzyme and substrate used are shown in the Figure legends. After a suitable interval, the reaction was quenched by the injection of 2.2 ml of 0.5 M-TCA, and the mixture assayed for carbamyl aspartate. Estimates of the background absorbance in the assay were obtained by the following procedure: 1.1 ml of O-5 M-TCA was mixed manually with 1-1 ml of enzyme dissolved in buffer, and immediately 1.1 ml of 0.5 M-TCA was mixed manually with an equal volume of substrate dissolved in the same buffer. The 2 solutions were immediately mixed together manually. Because of the quenching of the enzyme, no substrate should have been formed, but any contaminating carbamyl aspartate, or other substances giving a colour reaction in the assay, would contribute to the blank value. This value was measured for each syringe filling, and was used as the absorbance value at t = 0 (D(0) in the equations given below). (e) Steady-state enzyme assays Substrates (and, if relevant, CTP), at the same concentration as in the corresponding quench experiment, were reacted in the presence of ATCase (0.1 FM in active sites) of the same lot as that used in the corresponding quench experiment. Samples were taken from 1 to 10 min after the reaction started, quenched and assayed. k,,, was estimated from a plot, of carbamyl aspartate concentration against time. (f) Data analysis of lag curries The measured absorbances, without equation:

any correction

D(t) = k,t-(k,-k,)7jI-exp

or averaging,

were used to fit the

(-t/r))+B,

where D(t) is the absorbance, z (s) is the average time constant, k, (s-l) is the steady-state rate constant (i.e. the rate constant when t is large), ki is the “initial” rate constant (i.e. when t is small), and B is the value of D(t) that would be obtained at t = 0. (B, a constant

,526

H

K I H A R A E 7’ 2.1/,

adjusted by the least-squares program, should be distinguished from the experimental values of n(t) used at t = 0, which were> measured as explained in section (d) above.) The paramrtrrs of this equation [for the interpretation and justification of which. SW IXscussion, section (b)) were fitted by a non-linear least-squares procedure. which is feasible in bhis case because the number of sample t,imes is not too large. The progra,m calculated the values of least-squares residuals for a range of z. k, and /ci values (for each set of which /I was fitted by linear least-squares). The range and number of trial values was undrr operator control, and the interactive program presents the results so that the operator can clhoose new values. The optimum set of values was considered to he rPac%hed when thr lowest residual was obtained at the bottom of a minimum showing approximately quadratic behaviour in all planes, and when this optimum set also gave a theoretical rut-v? in reasonable agreement with the experimental points.

3. Results (a) Initial

formation

of carbamyl

asparate

by the eataEytic subz~nit

The initial formation of product by the catalytic subunit is shown in Figure 1. The results fit a straight line passing through the origin. From its slope, a steadystate rate of 39 s-l is obtained, which agrees well with the manually measured steady-state rate (30 s-l). Although a very small transient phase (lag or burst) cannot be excluded, its amplitude would have to be below 0.1 M product/mol enzyme, and its duration would have to be less than 4 ms.

1 0

20

40

60 Time

SO

100

(msl

FIG. 1, Production of carbamyl aspartate by the catalytic subunit of ATCase at a concentration of 107 pm-active sites in the reaction mixture. (The buffer was 20 mM-phosphate buffer, pH 7, with 0.1 mu-EDTA and 0.25 mM-PMSF, at 4°C.) Catalytic subunit was mixed with both carbamyl phosphate (2 mM in the reaction mixture) and aspartate (10 mM in the reaction mixture). The reaction mixture was quenched with 0.5 M-TCA after the time indicated on the abscissa, and carbamyl aspartate was assayed by the modified Method II of Prescott & Jones (1969), the sensitivity of the assay being calibrated by a standard concentration of carbamyl aspartate. The “blank” value (at t = 0) was obtained by adding substrate solution to ATCase solution that had just been quenched with TCA. A least-squares straight line is shown.

ATCase

ALLOSTERIC

(b) Initial

TRANSITION

527

KINETICS

formation of product by the native enzyme

Unlike the catalytic subunit, the complete enzyme does show a clear lag phase of product formation (Fig. 2(a)), which is followed by a steady state. The kinetic curves were analyzed, by the procedure described in Materials and Methods, section (f), giving the four constants z, le,, ki and B, which are defined there. Of these constants, B is not of interest and ki could be estimated only very roughly (it was somewhere between 0 and kS, and in most cases
TABLE 1

of chemical quench data

Summary

Us-7 Enzyme 0.15 0.15 0.15 0.15 0.15 0.3 0.23

Concentrations (mMj CP Asp

5 5 5 2 2 2 2

30 30 30 10 10 10 5

CTP

‘I (ms)

Rapid-quench experiments

1.0 0.5 0.0 0.0 0.0 0.0

6.2 +4 7.5+5 3.7-6.7 6.7 + 1.4 16.8f3 16.5k3.7 19.2*4

10.0+0.5 10.0+0+ 12.8kO.4 9.6kO.3 ll.5kO.5 7.1 kO.5 6.1 kO.2

Steady-state experiments 12.1 16.2 13.9 11.0 11.0 10.9

The concentrations of the reactants (all in mM) are those in the final mixture. Enzyme concentrations refer to active sites (6 per molecule of intact ATCase. Asp, aspartate; CP, carbamyl phosphate.

0

25

50

75

100

Time (ms)

200

F

180 160 140

0 I 0

20

40

60

80

100

Time (msl FIG. 2. (a) Production of carbamyl aspartate by the complete ATCase molecule at a concentration of 300~~-active sites in the reaction mixture. Other conditions were the same as for Fig. 1. The line shows the theoretical curve calculated from the equation:

D(t) = k,T-(k,-kJr{l-exp(--t/r)}+B, where D(t) is the absorbance, ki and k, are the rate constants of the enzyme before and after its transition, respectively, z is the relaxation time of the transition, and B is the blank value of D(t) at 1 = 0. (b) Comparison of the kinetics at 2 different enzyme active site concentrations in the reaction mixture: 150 pM (lower curve) and 300 PM (upper). Other conditions were the same as for (a).

ATCase

ALLOSTERIC

TRANSITION

529

KINETICS

50 Time

(msl

I

I

1

I

0

IO

20

30

Time

J

(ms)

FIN:. 3. The reaction of ATCase at low aspartak concentration (5 mM in the reaction mixture). The ATCase concentration in the reaction mixture was 225 PM, and all other conditions were the same as for Fig. 2. (a) All the data; (b) the data for the first 30 ms, during which the lag occurs.

4. Discussion (a) Reaction kinetics at the catalytic site Transient kinetic phases can provide useful information: a lag phase indicates that a rate-limiting step precedes the formation of product, but a burst indicates the reverse (Gutfreund, 1976; Engelborghs et al., 1975). From the apparent absence of any transient phase with the catalytic subunit, we conclude that neither of these situations holds, i.e. that there is no clear rate-limiting step before

530

H

ti I H .A t< A /d 7’ .-I /.

or after formation of the produc~t carbamyl aspartate on t,he enzyme (f’or anof her tbsarnplr of this. see ‘l’ravt~rs rf (11.. 1979). Therefore. the intermediates of A’IY ‘:tw should be present at similar conc.erlt~rat,ions in t ht, steady statfb. ii Gtuat,ion found with several enzymes (e.g. see Knowles, 1976).

(b) L’urve ping When substrate is added to the enzyme, it’ generates a complicated series of processes including substrate binding, subunit interactions and c>onformational changes. After some relaxation time. the enzyme reaches a new static state, and generates product from substrates at a new rate. This time is so short. that only a negligible amount of substrate is consumed, the concentration of which can therefore be considered to remain effectively constant during the transient) processes. Consequently, these processes could be described by a large set of linear first-order differential equations, yielding a large number of coupled exponential decays. But most of these are unobservable by chemical quenching, which sees only those that are associated with substantial changes in enzymatic activity and, of them, only the decays with the longest time constants. Since the observed lag is not more than four times the dead time of the apparatus, the observable time constants cannot differ by very much. If the time constants are similar. their combination mimics the effect of a single exponential decay from one act’ivity level to another; i.e. the specific enzyme activity would approximately follow the equation:

A(t)/[ E] = /c+ (ki-k,)

exp (-t/r).

Here A(t) is the activity and [E] the concentration of enzyme, z (s) is the average time constant, /c, (s-l) is the steady-state rate constant (i.e. the rate constant when t is small). (This initial rate is really the rate constant after completion of all processes faster than the dead time of the machine, but before the observed relaxation process(es) has started.) The yield of carbamyl aspartate [P(t)] would then be the integral of A(t) with respect to time, i.e.:

IP(t)]/[E]

= k,t+(~i-k,)z{l-exp(-t/z)},

(1)

where the integration constant has been adjusted to make [P(O)] = 0. This simple equation was used to fit the experimental results (see Materials and Methods, section (f)). It incorporates the assumption that all the relaxation processes observable by chemical quenching have indistinguishable time constants. The validity of this assumption is shown by the accuracy with which the equation fits the data (Fig. 2(a)). It can also be shown by considering the asymptotic behaviour of the reaction. When t >> r:

Tf this asymptotic by:

IP(t)J/lE]

= k,t+(k,-k&z.

form is subtracted

from equation

# = (k,-ki)t

exp (-t/z),

(l), the difference

r$ is given

ATCase

ALLOSTERIC

TRANSITION

KINETICS

531

Time (ms)

PIG. 4. From the curve of Fig. 3(b), the asymptotic linear part was subtracted and the logarithm of this difference plotted against time. The plot is linear to at least 70% completion. The linear regression line indicates a time constant of roughly 14 ms (compared with 19.2 ms from the lag: Fig. 3(b) and Table 1, line 7).

and therefore log, t#l = log, {z(k,-k,)}

-t/z.

(2)

Figure 4 shows an experimental plot of log, 4 versus t; it fits a straight line satisfactorily, showing that the transition is adequately approximated by one time constant, to within the accuracy of our data. (c) Why is the time constant independent

of substrate concentration?

The lag showed little change when the aspartate concentration was reduced (Table 1), and we must interpret this result. The many time constants that would he given by a complete and realistic scheme will depend in different ways upon the substrate concentration. Some time constants will relate to processes involving substrate binding. The binding reactions themselves will be diffusion controlled 1976). At the concentrations and rapidt (10’ to lo* Me1 s- ‘., Gutfreund, of substrate we used, these processes are much too rapid to be followed by the quench-flow method. Other time constants will relate to the transitions of intermediate species, whose concentrations will in some cases depend upon the substrate concentration. In that case the time constants will themselves depend upon the substrate concentration. At high substrate concentrations, however, the t This assumes that no significant delay is imposed by the need for substrates to pass through a 12 L% diameter channel to gain access to the active site (Honzatko et al., 1982). But this seems no more restrictive than the entrance t,o a typical active site cleft.

H. KIHARA

532

.!C’T 3L

relevant equilibria will be displaced in the direction of the substrate-bound forms. and the time constants will then show little dependence upon substrate concentration. This is presumably the reason for our finding that the lag shows no noticeable change when the substrate concentration is halved. (The lack of any influence by CTP on the lag (Table 1) similarly implies that CTP is not affecting any relevant equilibria at the substrate concent,rations employed.) (d) Association

qf the lag

with the allosteric transition

The lag phase of product formation we have found with native ATCase could be caused by delays either at the active site (accumulation of intermediates), or in the coupling between subunits. The isolated catalytic subunit has far fewer possibilities for such coupling, and shows none of the allosteric properties associated with this. Since it shows no lag, we conclude that the lag is dependent upon subunit-subunit coupling. Now an increase in enzyme activity brought about by such coupling in the presence of high substrate concentrations is called homotropic co-operativity, and is one of the characteristic properties of allosteric enzymes. Thus the lag gives us the rate of this aspect of the allosteric control process. (e) Comparison with other results Our experiments give the first direct, unambiguous observation of the lag due to the allosteric transition of ATCase. The estimate we obtain, about 10 ms at 4”C, lies between two other well-established time constants for allosteric transitions from the inactive to the active form: 0.4 ms for human haemoglobin at 22°C (Ferrone & Hopfield, 1976); and 0.2 to 5 s for yeast glyceraldehyde phosphate dehydrogenase at 40°C (Kirschner, 1968). This estimate is difficult to compare with other kinetic data concerning ATCase. The extensive series of temperature-jump studies reported by Hammes’ group (e.g. see Hammes & Wu, 1971a,b,c) never used the natural substrates, so the rate of the allosteric transition might be quite different. Keck & Schuster (1976) and Keck (1980) studied the binding of tetraiodofluorescein; but this compound does not by itself cause the major conformational change produced by substrate analogues (Moody et al., 1979). It would be most interesting to know whether the activation has the same speed as the quaternary structure change that is usually supposed to cause it. To test this, we need a method for specifically monitoring the gross conformational change. The only rapid method available appears to be X-ray scattering (Moody et al., 1979), so we have been constructing an apparatus for fast kinetic X-ray scattering experiments (Moody et al., 1980; Fowler et al., 1983). The results presented here show that, even near the lowest temperature attainable in ordinary buffers, the reaction is still 10 to 100 times too fast for measurement with existing synchrotron radiation X-ray sources. It will therefore be necessary to slow the transition further, e.g. by the use of low temperatures reached with the aid of cryoprotectants, before the associated conformational change can be followed by X-ray scattering.

ATCase

ALLOSTERIC

TRANSITION

KINETICS

533

Our kinetic results can be compared, however, with other rate processes of the enzyme. At least at 4”C, the lag for homotropic co-operativity is about an order of magnitude less than the time taken to react two substrate molecules. Thus it. seems unlikely that a special physiological role is played by the rate at which the enzyme activity is increased when substrates are bound. However, that rate is coupled, Ga an equilibrium constant, to the reverse rate, i.e. the rate at which enzyme activity is reduced when substrates (or products) dissociate; and it would be desirable for this rate also to be reasonably fast, even when the equilibrium lies strongly on the side of the more active form (as is the case at higher substrate concentrations). (f) Mechanism

of the allosteric

transition

Activation of ATCase by substrate analogues (and presumably also by substrates) is accompanied by a large quaternary structure change, in which the catalytic trimers (100,000 M, each) move apart by 11 to 12 A (Ladner et al., 1982). Supposing that this change causes the activation, could the size of the movement account for the duration of the lag? If the movement were obstructed by no entropy or activation energy barrier, its rate could be estimated along the lines of McCammon & Karplus’s (1977) study of hinge-bending in IgG antibodies. Since the molecular sizes are greater, but the movements are smaller, that paper gives us an order of magnitude estimate of 10-s s for the change. This is smaller than our experimental value by a factor of the order of 106. But the large quaternary structure change must alter so many bonding patterns that it would be expected to have some activation energy. If this entirely accounts for the factor of order 106, then the free energy of activation must be roughly 7 to 8 kcal/mol, i.e. roughly 1 to 2 kcal/mol per asymmetric unit. The value of 1 to 2 kcal/mol is much less than the energy required to break a hydrogen bond (3 kcal/mol) or a salt linkage (5 kcal/mol), and yet the increased separation of the catalytic trimers after activation very likely breaks the polar interactions (three per asymmetric unit) between residues in the Rl-C4-Cl interface (Honzatko et al., 1982). However, the surprisingly small free energy of activation per asymmetric unit might be accounted for in two possible ways. Since the process is slow, the transition state might be stabilized by solvation or by processes that increase the entropy. Alternatively (or in addition), the units could undergo their changes sequentially rather than in parallel, so the activation energy for the entire conformational change need be no greater than the activation energy per asymmetric unit. The advantage of this sort of mechanism is seen in the process of removing a push-on lid, which is much more difficult by a pathway that preserves the circular symmetry of the system. The extensive rearrangements involved in phage sheath contraction do not all occur simultjaneously, but generate a contraction wave that breaks the translational symmetry of the sheath (Moody, 1973). Such a symmetry-breaking transition would also be consistent with other evidence that the point group symmetry of ATCase is not always strictly preserved (for reviews, see Jacobson & Stark, 1973; Dembo & Rubinow, 1977; Kantrowitz et al., 198Oa,b).

H. KIH-1Ki I

,531

I P7’.,lL 1

Vl’r thank Dr (:. Hervb for sending fresh cultures of the diploid strain of E. coli and Dr W. Wania (Gesellschaft fur Biologische Forschung, Stockheim) and A. M. Foote (EMBL) for growing it in a large fermmt,ation. H.K. thanks the EMBL for a fellowship.

REFEREXCES Allen. C. M. ,Jr & Jones, M. E. (1964). Biochemistry, 3, 12381247. Altman, R. B., Ladner, .J. E. & I,ipscomb, W. X. (1982). Biochem.

Biophys.

Res. (lommun.

108, 592-595. Barman, T. E., Brun. A. & Travers, F. (1980). Eur. J. Biochem. 110. 397-403. Collins, K. D. & Stark, G. R. (1969). .I. Biol. Chem. 244. 1869-1877. Dembo, M. & Rubinow, S. 1. (1977). Biophys. J. 18, 245-267. Dubin, S. B. & Cannell. I). 8. (1975). Biochemistry, 14, 1922195. Engelborghs, Y., Marsh, A. & Gutfreund, H. (1975). Hiochem. J. 151, 47-50. Ferrone, F. ,4. & Hopfield, J. J. (1976). Proc. Nat. Acad. Sci., U.S.A. 73, 4497-4501. Fowler, A. G., Foote, A. M., Moody, M. F.. Vachette, P., Provencher, S. W., Gabriel, A., Bordas, J. & Koch, M. H. J. (1983). J. Biophys. Biochem. Methods, 7, 317-329. (ierhart. ,J. C. (1970). Curr. Top. Cell. Regul. 2, 275-325. Gerhart, ,J. C. & Holoubek, H. (1967). J. Riot Chem. 242, 288662892. Gerhart, J. C. & Pardee, A. B. (1962). J. Biol. Ghem. 237, 891-896. Gerhart, J. C. & Schachman, H. K. (1968). Biochemistry, 7, 538-552. 16, 229-249. Gutfreund, H. (1969). Methods Enzymol. Gutfreund, H. (1971). Annu. Rev. Biochem. 40, 315-344. Gutfreund, H. (1972). Enzymes: Physical Principles, Wiley, Interscience, London. Gutfreund, H. (1976). Prog. Biophys. Mol. Biol. 29, 161-165. 10, 1051-1057. Hammes, G. G. & Wu, C. (197la). Biochemistry, Hammes, G. G. & Wu. C. (19716). Biochemistry, 10, 2150-2156. Hammes, G. 0. & Wu, C. (197lc). Science, 172, 1205~1211. Honzatko, R. B., Crawford, J. L., Monaco, H. L., Ladner. J. E., Edwards, B. F. P., Evans, D. R., Warren, S. G., Wiley. D. C.. Ladner. R. C. 8: Lipscomb, W. N. (1982). J. Mol.

Biol. 160, 219-263. Jacobson, G. R. & Stark, G. R. (1973). In The Enzymes (Boyer, P. D., ed.), vol. 9. pp. 225.-308, Academic Press, Xew York and London. Kantrowitz, E. R., Pastra-Landis. S. C. & Lipscomb, W. K’. (1980a). Trends Biochem. Sci.

5, 1244128. Kantrowitz, E. R., Pastra-Landis, S. (1. & Lipscomb, W’. PJ. (19086). Trends Biochem. Sci. 5, 150-153. Keck. I’. C. (1980). Ph.D. thesis, University of Connecticut. Keck, P. C. & Schuster, T. M. (1976). Biophys. J. 16, 203a. Kirschner, K. (1968). In Regulation of enzyme activity and allosteric interactions (Kvamme, E. & Phil. A., eds), pp. 39958, Universitetsforlaget, Oslo. Knowles,
by G. A. Gilbert