On aspartate transcarbamylase kinetics

J. theor. Biol. (1980) 85,413-421

On Aspartate Transcarbamylase Kinetics EZIO MARCHI

Universidad

National

AND JORGE HORAS

de San Luis, 5700 San Luis, Argentina

(Received 21 August 1979, and in revised form 11 December 1979) We present direct calculations on aspartate transcarbamylase (ATCase) kinetics. New knowledge about ATCase quaternary structure are considered. “Decorated” Ising model is used for evaluating the statistical properties. Comparison with experimental data is considered. The theoretical results agree very well with experimental data. 1. Intrhduction There are several known allosteric enzymes, the mosi studied, to date, has been aspartate transcarbamylase (EC 2.1.3.2) (ATCase). Following experiments on ATCase, which is a compact and globular enzyme, Gerhart & Pardee (1963) and Gerhart & Schachman (1965) have demonstrated that there are separate different binding sites for substrate and for modifiers. Aspartate transcarbamylase catalyses the first reaction unique to pyrimidine biosynthesis. The product is carbamyl aspartate, which is converted via six steps to the pyrimidine nucleotidbs, CTP and UTP. In Escherichia coli the rate of synthesis of pyrimidine nucleotides is not independent of other chemical events in the bacterium, but is co-ordinated with them by a simple and effective mechanism. As long as the nucleoside phosphates produced via this pathway are converted into nucleic acids, ATCase continues to catalyse the formation of the first product, carbamyl aspartate. On the other hand, if CT? accumulates, ATCase action is inhibited and consequently the rate of production of pyrimidines is reduced. This means of metabolic control is called feedback inhibition. The inhibitor citosine triphosphate (CTP) controls or regulates the rate of its own production. There have been a number of suggestions about the number of binding sites for substrate (catalytic sites) and for modifiers (regulatory sites). Particularly, Weber (1968) presents new molecular weight determinations and ultracentrifugal analysis of the regulatory (R) and catalytic (C) chains. They indicate that the ATCase molecule contains six copies of each chain, that is to say R&6. Experiments made by Gerhart & Schachman (1965) report that the weight of the ATCase moledule is 310 000. The C subunit molecular weight 413

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414

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is about 96 000-100 000, and from here it turns out that each C subunit contains three C polypeptide chains. Similarly, the molecular weight of the R subunits is reported to be approximately 30 000. This fact indicates that each R subunit would contain two R chains. Accordingly the p-mercuribenzoate (PMB) cleaves the ATCase molecule yielding two C3 units and three R2 units, Weber (1968). On the other hand, from the theoretical point of view, a model for the reaction of the ATCase with aspartate as substrate, has been proposed by Monod-Wyman-Changeux (MWC) in Monod, Wyman & Changeux (1965). In this model, it is considered that the whole molecule would exist in two conformational states, namely: the relaxed (R) and the tense (T). This immediately permits the direct application of the Ising model as described in Thompson (1968) to the MWC model. Moreover, such application can be generalized to decorated Ising model since the (C) subunits bind substrate and the (R) subunits bind modifiers. A first approach of this generalization may be found in Thompson (1968) where it is assumed that only neighbouring (C) and (R) subunits interact. They are displayed in a closed onedimensional chain arranged in an alternating manner. The number of regulatory sites is considered to be four. By the evidence given in the discussion above, it appears to be more natural to consider the ATCase molecule divided into two Cs and three R2 subunits, which are displayed alternately (see Fig. 1). In this paper, we are concerned with the study of such a model using a simple generalization of the decorated Ising model, where the distribution of the C and R subunits are given accordingly with the previous observations. We also emphasize that the usual nearest neighbours interaction is considered. The decorated Ising Model is a statistical mechanics model that is very useful in describing the co-operative interaction between different types of sites. The introduction of this model for ATCase kinetic is indicated because there are two well defined types of sites (regulatory and catalytic), and because the S shape of the reaction rate curves suggest the co-operativity of this phenomenon. 2. Model Description

In this section, we are going to present a model for the reaction of ATCase with aspartate as substrate. We schematize a molecule of ATCase in subunits arranged as shown in Fig. 1.

ASPARTATE

TRANSCARBAMYLASE

KINETICS

0

FIG.

1. Schematics

of the ATCase

0

415

0.

molecule.

The full circles represent R chains and the open circles are the C chains. In the figure we also show the different interactions among sites, which are given by various letters. The nearest neighbours interaction energies are given by f, u and a. Second nearest neighbours interaction energies are represented by c, d and e. Finally, the third nearest neighbours interaction energy is given by 6, which correspond to the interaction between two consecutive C chains. We note that in this way we generalize the models already presented in the bibliography for the decorated Ising model, Thompson (1968). All other energies are non-zero except for f. This is due to the experimental fact that ATCase in the presence of CTP only, fits the classical MichaelisHenri equation, i.e. there are no apparent interactions among regulatory sites. To introduce the CTP in the model shown in Fig. 1, we take a set of parameters Vi, i = 1,2, . . . , 3n with values +l if the ith regulatory site is occupied by CTP and -1 if the ith regulatory site is unoccupied by CTP. On the other hand the parameters bi: i = 1,2, . . . ,3n specify the states of the catalytic sites in a similar way. Therefore the corresponding partition function in terms of the Ising model can be written as follows:

2n+2 wa-24% i=l

xie

lb i=l

*(l+vL)

ie2

-(l+v,i-1)(uusi-3+drrgi-~+e~3i-2)

e-

(l+u,,)(uaoi-2+d~3,,1+CCLOI-3)

e-(l+v,)(uwI+dfi,)

e-(l+qMer,)

i=2 x e-(1+~2,+2kws,)

-(l+u*,+,)(ur~.+du3,-1) (1)

Here n is the number of CJ subunits, I and Y represent the stacking energies of the regulatory and catalytic sites respectively. Finally the last

416

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four terms in the above expression take into account the respective end effects. The sum is taken over all the possible configurations and

t/J) = (Ph. * . ,IU~

(VI = (VI,. . . , 4.

Performing the sum over the states (v), and transforming tials into sum of hyperbolic functions, we will obtain: Z=K

c ii (rr) i=l

(l+wypJh*

all the exponen-

(l+W,CL31-2~3r-l)(l+Wa~3i-lILL3i)

n-1 x iFl

(1 +W*P3iP3i+l)

I!

C1 + W&3i-2p31)

i=l

x (I+

WdP3i-1)(1

fi

[l

+ Bl(l

+ WdJ-3i-2)1

i=2

+ wecL3i-3)

X ifi [l +Bi(l x[l

+B,(l

x[l

+B211+W,CL3n)l

+Wu/d3i-3)U

+w,/JLI)(l

+0.kP3i-2)1C1

+W&3i-4)[(1

-eWdru2)1[1

+B3(1

+O,P3n)(l

+B2(1

+UePl)l

+wfP3n-l)l

(2)

where the corresponding d, e, v,

parameters are related by wi = tanh (i)i : y, a, 6, c,

B1 = exp (21) cash (v) cash (d) cash (e),

B2 = exp (21) cash (e)

B3 = exp (21) cash (v) cash (d)

and finally K = [cash (y)]3n[cosh (a)l”[cosh

@)I”-‘[cash

(c)3”.

In order to solve explicitly by the recursive method described in Marchi & Vila (1980), we have to sum first over (pi) obtaining:

x (1 +wwd[l

+&Cl

+4~3)U

+wcsz)(l

+w41

3n xin3(1+o,wi)

ii

(l+

~&3i-2~3i--l)(l+~w,cL3i-IcL3i~

i=2 n-1

x i!l

Cl+

WbLL3iP3i+l)

,Ij2

t1 + ~cPSi-2EL3i)

ii i=2

II1 +Bl(l

+ %P3i-2)l

ASPARTATE

X (1+0&3i-4)(1 x [l

+Bz(l+

TRANSCARBAMYLASE

+W&Si-2)[1

+B3(1

417

KINETICS

+%P3n)C1

+WdCL3n-1)1

(3)

Wc13")l

where the coefficients are given by WY @a

1

WC

%wz

%WC@o,

with matrix multiplication.

Wdf+

W&o,

0. WY w4da WC

aA I[1 ai

a2

1

= A(a

(4)

ai

The vector a’ has co-ordinates

a: =‘(

1 + B2)B3W"

+ (1 + B3)B2+

(3 ai = (1 + B2)Bwd + B~Bsodw~~ ai = We

mention

at this

(1

+ B~)B~w~w, + BzB~wLw~.

point

that

(p3)

in the

sum

of (3) means

(CL39 CL49 - * * , P3n).

For simplicity, we abbreviate all the factors from the first appearing product in the expression (3) by A(cL~)B(cL~)C(CC~)~(~~)~(~~)~(~~) Gb3n-1,

~3").

Three further steps with the recursive method, give rise to the following expression:

Z=ZZ4k C. C (rr”) (1”s)

(a~+a:Cc5+a~CC6+a3C15CL6)(l+OaCL5CL6)

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E. MARCH1

AND

J. HORAS

where the coefficients are now obtained as: a: 5 al

as=

a:

= Da’=

A(a)I?(d,

u)C(b)B(e,

2

u)a

II a: 1

= W&Y I

WC wawcwy

l+Bx ’

WY

&WcWy WC I

wcwy

1

B lW,Wu

Bw,

Blw, Bo,w,

(1 +B)

Bw,

Bwd

Bwwe

Bwuwi

Bww&i

WyWb

Bwewd

wb

: &o&j

Blw,

Bw,,wd

l+B ’

1 W&c

wa

W&Y WLlWC

WC2

wowy

WY

Bw, Bo, i Bwow,

Bwi

Blwd BWdw, B&wi BW&&

II

B 1% (1 +B)

BlWdW,

Bwzou

%4Whi

Bw,

WyWb

wy

BWci

=

?’

w,j

Wb

1

WyWb

a

wb Wy@b WY

wy

2

1

(7)

Bud& I

where we emphasize that D =A(a)I?(d, u)C(b)B(e, v)A(a). One now is allowed to perform the sum as far as 3k + 2 obtaining, thus, a similar expression as (6) with the only change 1 + k. In other words a3k+1= Dka’. As a particular case for k = n - 1, we have a3n-2 = D”-‘a’. Performing a further sum, it appears that 2 =23”-2 X (I+

C (a:“-’ C (w,,) (rsn-l) wy/-h,-d1+

x (1 +wy~3nNl

+a:“-1p3n-l+a~n-1~3n

%/.Q-IcL~~)[~ +Bz(l

+ B(l

+ w&3n)tl+

+ai”-1p3n-lp3n) ad!&-1)

(8)

+we~3n)l

where the coefficients are now obtained as a 3”-1 = C(a)D”-‘a

la

Summing over (p3,,-i) and then over (p3n) it is easily seen that the partition function has the following final shape: 2 = 23”KE&(e,

v)C(a)D”-‘a’,

(9)

where B2(e, v) is the matrix obtained from B(e, U) by deleting the last two

ASPARTATE

TRANSCARBAMYLASE

419

KINETICS

rows and replacing Bi by BJ, last matrix is given by E = [Cl

+ J32)

+ Bzwewo,,

Cl+

Bzb,

+ B2d1.

Thus, we have gotten an explicit form for the partition function of the system. In our case n = 2, and the partition function in the case of the molecule of ATCase has been in this way obtained. 3. Reaction Rate Computation In this paragraph, we are going to compute the reaction rate function for the reaction cited. This can be performed as follows: it is clear that the reaction rate is proportional to the average number of occupied sites N which is evaluated as follows: (10) Therefore

the real reaction rate is given by the function

(11) where the parameter a = exp 2y is the ratio of the probability that a site is occupied to the probability that the site is unoccupied, so it seems natural to interpret a as a measure of the concentration of substrate. In order to obtain an explicit expression of equation (11) we let II = 2 and we perform the derivative of equation (9) with respect to y. In doing this we make use of the matrix product derivation rule, and take into account that the derivatives of B (e, V) and a ’ with respect to y are zero; the reaction rate is then given by u)C(a)Du’+E~2(e, +E&(c,

u)C(fz)~n’ I/ E&e,

u)C(a)Du’

u)rdWDa, Y cash’ y +f tanh y.

Y

(12)

We are now interested in fitting the main result of our model, equation (12), with a typical set of experimental data. In the chosen experiment, Gerhart & Pardee (1963) the experimental conditions are: the reaction mixture contained 3.6 x 10m3 M carbamyl

420

E.

MARCH1

AND

J. HORAS

I I.0 -

.F F(a)

0.5

.

/

FIG. 2. Reaction rate determined by our theoretical model (solid line). The experimental points are shown by dots, the parameters take the values: B = 0.001; a = 0.91; b = 0.83; c = 0.75; d = -0,013; e = -0.01; cl= -0.015.

phosphate, aspartate varied as indicated in the figures, pH 7.0, O-04 M potassium phosphate buffer and 9-O x lo-* g of enzyme protein per ml. Figure 2, no CTP (control curve), fig. 3 the same conditions but in presence of 2 x lop4 M CTP. It was observed in several experiments (Gerhart & Pardee, 1961,1963; and Gerhart & Schachman, 1965) that the presence of carbamyl phosphate, another substrate, did not influence the inhibition by CTP. I I.0

-

/

FM

0.5 /

i

4

a

FIG. 3. Reaction rate in presence of 2 x lo-* M CTP, determined by our theoretical model (solid line). The experimental points are shown by dots, here the values of the parameters are: B = 0.986; a = 0.97 b = 0.93; c = 0.89; d = -0.28; e = -0.19; v = -0.3.

ASPARTATE

TRANSCARBAMYLASE

421

KINETICS

The computated control curves are shown in Fig. 2. Here the interaction parameters have been adjusted in order to obtain a good fit with the experimental data. On the other hand, with +2x 10m4 M CTP present in the reaction, the reaction rate curve becomes a little bit flatter in the entire zone. This curve is displayed in Fig. 3, which is again compared with the experimental values obtained by Gerhart & Pardee (1963). The large difference between the corresponding values of parameter B in the situations described by both figures is due to the fact that B takes into account the presence of CTP. Therefore in Fig. 2, which describes control situation, the corresponding value has to be small. On the other hand in Fig. 3 the value of B has to be larger due to the presence of CTP. All other parameters also increased when CTP concentration increases. Following Gerhart & Pardee (1963) this may be interpreted as that when the interactions becomes stronger, the active sites becomes slightly more distorted, and as a result substrate affinity is reduced. The minus signs in the parameters d, e and D are related with the fact that they represent interaction energies between an inhibitor site and a catalytic site (repulsive interaction). Finally we would like to remark that the good fit obtained in our model is based in a detailed treatment of intersite interactions. This work waspartially supported by a grant of SUCYT at UNSL, Argentina. REFERENCES GERHART, J. C. & PARDEE, A. B. (1962). J. biol. Chem., 237,891. GERHART, J. C. & PARDEE, A. B. (1963). Cold Spring Harbor Symp. Quant. GERHART, J. C. & SCHACHMAN, H. K. (1965). Biochemistry, 4,1054. MARCHI,

E. & VILA,

J. (1980).

Biol.

28,491.

J. Phys. A (in press).

MONOD, J., WYMAN, J. & CHANGEUX. J. P. (1965). J. mol. Biol. 12,88. THOMPSON, C. J. (1968). In Mathematical Statistical Mechanics, p. 187. Macmillan. WEBER,K.(~~~~). Nature 218,116.