Chemical relaxation of allosteric models

CHEMICAL

RELAXATION

OF ALLOSTERIC

MODE:LS

Two modetrfor aUostencbehaviorare mnsldcredand comparedwith eachother: the mwkl ol two allosteric

irticsof the ‘pknes’ of polymerkmg titer on monomero.

I. INTRODUCTION Work on enzymes with allosteric effects is increas ing at an ewemely rapid rate. The original paper of Monad, W:/ma and Cbangeux (1965) is cited in the

Although Koshland and collaborators (Koshland, Nemchy and Filmer, 1966) (Kirtley and Koshland, 1967) also worked extensively on allosteric effects, their theory allows the presence of mixed forms of subunits within one enzyme molecule. Monad, Wy-

literature several times each week. Thii paper is not

man and Changeux (1965) allow only identical sub-

concerned with the experimental application of the

unit forms within one molecule because of its proper-

original theory (ibid.), but with its further extension. including especially the relaxation times of a slow allorteric process. With the various relations known, it becomes powible to apply chemical relaxation methods easily to allosteric enzyme systems. Certain!y, expressions for relaxation times of allosteric intelconversions have been published before

ty of functional symmetry. As this property is probably present in most enzymes with interacting subunits, only the model based on symmetry will be discussedhere.

(Kirsehner, Eigcn, Bittman and Voigbt, 1966). However, they do not consider the alternate faze of a pi lymerizing enzyme system. A comprehensive discussion of the various possabdrtresof chemical reloxafion times related to allosteric effects seems appropriate. The chemical relaxation of a polymerizing enzyme system war mentioned before quite briefly (Czerlinski, 1968).

l

Part et IMY cnleubtianr were m.dc shlk the nether UPI an svirit to the Departmentof Biochmirtry, Univwsity of Fhvide Celk~ 01 Medicine,CelnnviUe, Florida. Reprint requnb thou!4 be e/ieresscdto the aetho,, co65 Ward Bldg. 303 E.Chicw.o AYCIIUC.Chiwo. Illinois6061 1. Pat of this work w.s ruP,x.rtedby NSF-@entCB 704..

Kubin and Changeux (1966) and Changcux, Thidry, Tung and Kittel(l967) extended the interconversion model further, allowing its application to a two.dimensional matrix (a membrane). Quite :ecently the allosteric effects of polymerizing proteins were discussed (Nichol, Jackson and Winzor, 1967; Klapper and Klotz, 1968). In this latter case, the allosteric effects become dependent upon the concentration of protein present. Tbe main body of this paper will consist of four parts. First, the extent ofenzyme-saturation with linand p..I. Y,. will be derived brieflv for two alternate models, the interconverting and the polymerizing. The derivation-summary is necessary for the defhlition of the various terms r,eeded later on. The second part deals with the derivation of expressions for ths relax-

-

ation times, applicable to these models. An extensive derivation will only be given for the ~lymerizing model. In the third part, conservation of symmetry is

stant has to be multiplied by the factor i/[n-(iI)]. Here i rcprcxnts the ith step under consideration with (n > i > 1) and n representing the maximum number of ligands (Klotz, 1946). An exception is KR for the model of eq. (2) (see cq. (29)). The following detinitions hold for both models:

discussedfor the polymerizing model. In the fiial (rotosrlh)part, the various expressions for the allostefic relaxation times arc compared and discussed iarzlation to utilization in experiments.

2. IZFINITION

OF CONCEPTS

Two models arc to bc considered. The various de% nitions will be rather similar to those of Monad, Wyman and Changeux(l965). The model of ioterwnverting forms is rcprcscnted by the binding scheme:

lq.,llFl ----T---=~

[7iI

i KT

n-(1-1)

Concentrations arc indicated here by brackets in contrast to Monod et al. (1965). as extensive algebraic operations arc not initially prcscnted. The mold represcntcd by cq. (I) &tines:

(6) A8 the left side and the right ride of eq. (I) are ryotmetrical, ant may interchange them -or terchange by the ccnditions:

restrict in-

L2-19C. In contrast, the polymerizing model is represented by:

A thorough derivation (Czerlinski, 1968) leads to the extent of binding to Y,:

y, IsLca(l --- +Cu)“-~’ ca(l co)“- ’ L(l+ca)“t(lta)” .

(7)

@q. (7) is identical with eq. (2) of Monad et ad.(1965). The model. reprcscnted by cq. (2), is now: M,kz._[Tol % [RI”

w

As the *ft si4l.cof cq. (2) is not the (functional) ntir-

(2)

rot-image of the right side, no additiooal degree of freedom irawilable. The above relation L * I *c un no longer be maintained forlhe model of cq. (2). A thorwghderivation

In both schemes.KR and KT represent the intrinsic dissociation con&ant of F from the T-form or the R. form, respectively. To obtain the dissociation constant

ofan individual

step m terms Pf the equilibrium cancentrations of thi8 step, th: intrinsic dirsociation cow

yF_

(Czriiirki,

196@lea$s to:

[R]a+nM]R]“ca(l~

[R](I

+a)t nM[R]”

A divilion o*numerator

(1

+a)”

and dcIominator

(9)

by [R] is

CHEMICAL RELAXATION OF ALLOSTERIC MODELS avoided~ineq. (9), as its denominator represents the analytical (total) concentration of enzyme in terms of the monomer, 5. If the analytical concentration of the enzyme in terms of the polymer is desired, the concentration [A] has to be replaced by [To] according to eq. (8). If n = 2 (dimerisation), one may easily solve the quadraticequation for the expression ofci:

F

tea)

YF’&.

1 +a+2M[R](l +CI# (7) of Niehol, Jackson and Winzor (I 967) converts to eq. (9)above with their M,,r =hf~pY~,p= l,q =tr. K,t [S] =aKc[S] sac, [A] = [R] and their [c) = M[R]“, in accordancs with the model of eq. (2) and their general defmitions. A comparison between eqs. (7) and (9) reveals that the two constants L and M may be brought into a for ml relation (no: representing equivalence of models): (12)

If

formally n :: 1, then L = hf. Nevertheless, the letter hi is not replaced by the le:ter L to facilitate a joining of the two models. However. an allosterx interconversion may occur in the munomer, in the polymer, or eye” in both states (expandable further to a cycle). Niihd, Jackson and Winzor (1967) do not discuss thtSe extensions. Now the ratio of intrinsic constants c is not restricted to the relation e 4 1, wiiich corresponds to the highly preferred binding of ligand F to the monomer R. A special case, c= 0, is represented by exclusive binding of F to R (which is also fomuBy identical with competition between F-biding and polymerizing). For this special case, eq. (9) simplitias to:

For equal binding to both forms, interaction between sites is no longer evident. The limit c + - corresponds to exclusive binding of the l&and F to the polymer. Only the product ~a may then be considered, reducing eq. (9) to:

1 +a+nMR”-’

(15)

The nwneratm corresponds to the denominator, differentiated with respect to ca and then multiplied by ca.

3. THE ‘ALLOSTERIC RELAXATION TIME The vertical reaction steps in the models of eqs. (I) and (2) are consideredmuch faster than the horizontal reaction steps, which are then defined as determining the ‘allosteric’ (slowest) relaxation iime. Under the condition of ‘buffering’ by ligaad F(reprosented by CE3 c$), the left ride of the models is practically isolated from the right side, leading to very simple expressions ior 7, and ,2 (Czerlinski. 1968). Expressions ior the slowest chemical relaxation time, TV,of system (I) were derived previously (Kirschner, Eigen, Bittman and Voight, 1966;Czerlinski. 1961). Two degenerate eaaasmay be considered in , theirderivation. The terminology oirect. 2 is again employed as in the recent elaborate derivation (Czorlinski, 1968). The result for the case k,,,= k2i+4 (OGiCn)

was:

One obtains for the alterrate case kti+,

y,=-A----

(14)

(11)

Eq.

L =nMIp]“-’

c * 1, there is the intermediate case, c= 1. Then eq. (9) as wtll as eq. (7) reduce to:

Y, = ti[R]“-I ca(l +~a)“I +nM[R]“-‘(1 CCC)”

For diierisation eq. (9) reduces to: Y =a+2M[R]ca(l

221

(13)

Eq. (7j above may be similarly simplified for c = 0, aa axtenaivaly rmployed in the original paper (donod, Wyman and Changeax, 1965). Aaida of preferential binding for the polymer with

(0Giin):

=kzltS (171

These equations will be compared with others, to be derived below. Expressions for the slowest chemical relaxation

time, referring to system (2). were briefly reported previously (Czerlinski, 1968). As their derivation was not given, it will be reported here, following closelj

Now the following defmitions are introduced: T,=e,+Aq,

(23)

an already publis4cd method (Czcrlinski, 1968). Because of the cyclic nature of system (2) all n+ polyme&tmn steps have to bc considered for the slowest

R=FR+AcR,

(24)

mlawatian pmccss. In the time range of r3. DE+ would

Q=e,+AcQ.

(25)

I

bavc to write ntl $

differential equations of the t@:

Ti = -kzi+,

T, + kZi+2R”-‘Qi.

(18)

In this equation Tt, R and Q represent the concenlrn~ Nuns of these components. If one would consider an

The cmceotration chrmgesare small compared !o the equilibriumconecntrations and terms containing (Ac$ can be neglected. Remembering this, one ob tatns upon substituting eq. (23), (24)and eq. (22):

(25) into

unspecified time range. the right side would have to contain four additional terms (or fwo: for iii0 and iw). These additional terms, however,vanish in the time range of 7, under the initially introduced conditions (TV * 5;). In the next algebraic step, n + 1 equations of the type of eq. (18) are added together:

-$,$ r-0

Ti= - 5 i=O

The dissociation constant KR for the model under consideration (that of eq. (2)). is not an intrinsic’ one, but simply defined by

kZ,+,Ti

e

n + c i=o

Q

k2i+2R”-iQi.

At this point, two apparent rate constants are duced and Jefmed by:

119)

intro-

=a?--,

(27)

with o according to eq. (3) (here 6 and CR are used interchangably with [Q] and [RI). Substituting?* oi eq. (26) by eq. (27) gives:

+k-F;-t For [flP%. Combining eq. (1%) with the definitions of eq (20) and (21) gives:

& ,% Ti = -kad,, I=0

5’

i=o

Ti

n

- kc,, i~oR"-iQi~

it is (compare Czerlinski, 1966):

ACQ=lUW,Q.

Introducing eq. (b) to:

(29) into eq. (28) simplifier the latter

Although the denominator of eq. (9) represents c; ,

memayah write

cg ir! terms oi the mdividual com-

ponents: (i)

k,,,,

=kzir3

forO
1).

Eq. (20) immediately simplifies

tion to:

Then one obtains for sma!l changes: ”

o=hq$

+acp +n

c

under this condi-

k udd=ki.

AC,.

i=O

(37)

Each cycle io the system of eq. (2) imposes the Mlowing relation:

Substituting eq. (29) into eq. (32) and solving for ae, gives:

The letters ,? R. Q and Ti represent concentrations again. as for eq. (IS) to eq. (22). Now one may intro-

Combining eq. (30) and (33) gives:

duce the four rate constants of the two polymerisaxion steps and elnploy eqs. (3). (4). (5) and (27). This gives

upon simplification:

k*j+l ;+ I k2i+3

--=-C. kn+2 n-i

Similar to eq. (29), there are relations among consecutivc AC{that contain the ‘liganding’ coefficient, which was mentioned between eq. (2) and eq. (3): Aq = ea

--L n--&l)

Now employing the degeneracy k2,+, = kT,+3. eq. (39) simplifies to:

k2,%4 = k,,+, AC,.__, ,

(39)

k,i+4

(40)

3

(35)

I h use

for 1 G i G n. The resulting n equations permit one to solve for $c,. However, the composite factor is the wnc on both rides of eq. (34). The expression in brackets, therefore. directly represents the invcrsc of the ‘allosteric’ relaxation time:

Upon redefining in eq. (2 I) (increase by I), one may expressany individual even rate constant in termroik2.0ncobtains(nr~w
1

k2,+2=k2(;)Ei.

Inserting eq. (41) into eq. (2 I) one obtains:

(41)

and utilizing wt. (27),

Although this last <:qualion represents the general rxpre%iosion for the nlwest relaxation time, it contains two apparent rate constants. These comtants are quite com-

plex and should be simplified, ,f possible. Further limolitication of eo. iide~~~._ ..,_. __ . 1361 I~ , is .wssiblc. .

Incorpora!ion ofeqs. (37) and (42) into eq. (36) nnd

C, H. CZERLINSKI

224

use of tht! binomial theorem give%:

,il =k, 9 k=

lncorpcratiotl of eqs. (46) and (49) into eq. (36).and use of the binomial

.=

-=(I tcay . ix 94

theorem gives:

(43)

Using c s 0 for exclusive binding to the monomer, one obtain%for this important case:

. A linear relation results for dimerisation, where n=2. Although eq. (44) shows no pronounced dependence upon a, one should remember that ZR is dependent opon a. For the above ~0, the denominator of eq. (9) becomes: c;i =~jt(lta)+nlvE~.

(445)

For n=2, eq. (10) may be applied(with c= 0). For n>2, the equilibrium concentration CR may be computed from eq. (45) by a computer program.

(iii

k2i+2 = k2i+4

for 0 Ci G(n-

I).

The summation of the ai could be factored out. If exclusive binding to the monomer is also assmne~ (PO), one would obtain: (51)

This expression is quite strongly dapcndent upon o, while its indirect influence through Z~via sq. (45) is only of scandary importance. Eqs. (43) and (50) may be further rlt’ered for ea. elusive binding to the polymeric form: c - m, a + 0 and co as effective parameter. The reduction of eq. (43) for this case is quite minor, although c’q.(50) reduces quite substantially:

Eq. (21) simplifies immediately to: k even= ‘k 2’

(46)

Again eq. (39) is vaiid. simplifying with k,,+* = k,, to: i+l

_t

bit3 =n_i c kzitl .

(47)

Upon redelining i tbr use in eq. (i0) (increase by 1). one may express any individual odd rate constant in terms of /cl (now I 6 i < n): kw

ktc-t =ci

(48)

Inserting eq. (48) into eq. (20) and utilizing eq. (S), one obtains:

The

aewndary effects of (ca) upon ZR OR also substantial, as the denominator of eq. (9) for exclusive binding to the polymer becomes: c~=gtn‘E~(rtcu)Y

(53)

The rxxtcentrat;lon of free R decreases strikingly as co approaches (and exceeds) unity. Another rather special case prevails if no l&and F is added: both Q and C(Lvanish. In addition, eqs. (SO) and (43) (as well as their simplified cases) become identical: r:’ = k, tnZk&-’

(54)

For the further specialiaation of dimerizingunits, one obtains an expression. which is identical with eq. (3.90) of Czerlinski (1966).

4. ON THE F’OLYMERIZINC

MECHANISM be formed ind&end;ntty, or de&ode& &on each other. If kev and lock for two oaks are 180” aoart.

.

The kinetics of chemical relaxation was treated above, as if n protomers may come together to hnl the oligomer, with ~1being any number. Mechanisti-

The two components of a polymerizing site may

cally, one would expect to encounter only bimolecular reactions. restricting the above treatment to n=2. There is one exception, however, in which poly merisation may be trealted as an n- molecular reactiun -whenever the concentration of in~crmcdiary polymers is negligibly small compared to the preferred oligomer (containing n protomers). In this case, the measured polymoleculsr rate constant is actually a composite of a bimolecular rate constant and a set of (bimolecular) dissociation constants. The measured monomolecular rate constant is then mainly governed by the rate wth which the oligomer ‘blows apart’ into fractions, which may not ne~ssarily be n monomer and a quickly disxlciatmg oligomer containing n -

protomor~. Therequirement

I

of minutely small concentrations

nut necessarily consist of 2 key and a lock. A group of. iiphatic ride chains may associate with a like grU,’ to form a hydrophobic bonding system. No longer needing a pair, protoners withsinglo polymcriz#g sites would then dimerire. A protomer may also have a complementary pair (key and lock) end a pair of groups of aliphatic aide chains, allowing two types of polymerisations to occur. To exert s!ereospecific effects, polymerizing sites hhw to contain at least three bonds (forming a plane oriented to the axis of a key or lock). One may then differentlate between heterologous and isologous bonds. Heterologous bonds consist of e donor, or P labilized hydrogen, end e distinctly different acceptor, a minus-charge or an unpaired electron, respectively. lsologour bonds c&mist of pairs of like residues,

of intermediary polymers is equivalent to the requirement of Monad. Wyman and Chunaeux (196% that the subunits in an oligomer are either all in the T-state or ali in the R-stats. To visualize these equivalent rcquirements more clearly, an extension of the analoguc inodels of Monad et al. (ibid.) will now be given. To stress the very high specificity of protomers for each

which simply associate, e.g. hydrocarbon, hlltidyl-, phenylalanyl-, tyroryl-, tryptophanyl-, carbanyl-, carbamide- and similar residues, all of them in the uncharged state. Thcx definitions are different from those of Monad et al. (1965). For a complementary pair to act as key and lock, their polymerizing site has to contain at least one heterologous bond.

other, the concepts of lock and key are wmetimes introduced. These concepts arc vczy helpful in visualizing the occunence of very specific polymerizing numbers n. The polymerizing number n is directly giww by the angle between the axis of the lock and

It 1sclear from the type of residues that aggre@ion could be critically affected by a number of agents: urcz could break hydrogen bonds, high ionic strength separnte opposite charges, soaps (or lipids) disengage hydrophobic bonds, and various pli-ranges

the axis of lhe key. If the at@ is 0”. only dmters are formed. If the angle is 120”. trimers rapresent the only possiile oligomers. Tetramers are formed by an angle of 90” and bexamers bv an an& of 60’. Linear ar. rays of theoretically unlimited length are formed for an angle of 180”. Fig. demonstrates the various polymeriwtions just meniloned. CerGdy. the mentioned angles do not nerd to be exactly fulfilled. This non~fullXment offers the possibility of a conformational change from free proto.a;er lo its polymerized form. Monomer and polymer may then in turn have highly different binding constants for l&and. The above *key’ and ‘lock’ may more generally be termed a pair of polymerizing sites. Certainly, protomers may contain more than one pair. If the pairs are

could generate charges (with histidme probably being the most sensitive element, although a terminal R. amino group might also be involved in a hydrogn bond). The polymer-monomer-equilibrium could therefore become dependent upon the concentration of these menticncd agents. While Monod et al. (1965) considered the breaking and reforming of crucial in the alloster~c transi-

_

_

.

1

_

,_

bonds

tiun, such strong changesare not really necessary. A change in the geometry of the oolvmerizina site (without fintcrmed~alel chinge in &tiber of bonds) &wld suffice to cause complete allosteric conversion. Fig. 2 demonstrates this for the involvement of heterologous bonds per rite. Then a and c may represent the site for the lock and b and d the site for the key. An incrensa

G. H. CZERLINSKI

CHEMKAL

RELAXATION

&stance OF a and c (or b and d) is associated with the generation of a binding site for the ligand F. The change of this distance in one protomer initiates the change in all protomers of the oligomer. This

in the

01: ALLOSTEI
(58)

change in distance does not need to be large, if two stable isomers have crucial sections, which change their structure only minutely. Fig. 2 could only show a one-dimensional interface betweeo the protomer:. In natute the interface is twc-dhnensional, generally a shaped surface. allowing many different chang:s i mong

227

MODELS

well as bimolecular. steps may ihen be diffusion. limited. C ne obtains for any i# n the relation:

quaternary bonds. If there are. however, only two

If the energy-gain in the last step is associated with some electronic rearrangement, 7’e should have an absorption spectrum, which is different From the monomer and all other polymers. Chemical relaxation would therefore be observed via To, The mono-

stable structures of the protomers, a change in one

molecular rate constant k, from the original scheme

protomer causes all others to change through coupling at the polymerizing sites. Then concentration of intermediate states is always minutely small. That intermediary polymers are practically nonexistent should have been made highly probable by

is then given by the conversion rate constant from To toR,. The bimolecular rate constant kz of that scheme is governed by the associaiinn rate constant From R,_ t to R,, but corrected by a ‘delayfactor’, because of the coupling-in of the other rate constants.

the earlier discussion on the polymerizing

If detection occurs via To only, the relaxation time

mechanism.

Treatment like an n-molecular polymerlsation is then justified, even though it is composed of many bimolecular steps mechanistically. As the concentration of monomer will always far exceed those of intermediary polymers, one may consider that each association step involves a monomer. Then in general one may write: nR.-(n-2)R+RZ-(n-3)R+R3.-.

_-

(n-l~R+Rj-...--R+R,_,-Rn-To

(55)

Although one could assume R,

= To. this is specifically

for pure enzyme at high concentrations (so kt can be neglected compared to the kz-term) is r:ot identical with T* oFSchwarz (1968). Although it is possible to combine allosrcric inrerconversion with polymcriutiun. thi$ complex treatment is avoided here because ofcvaluation

problems.

The combined theory requires the simultaneous determination of Four rate constants. In P narrow concentration range and with the available experimental error. many sets OF four rate constants could be made to fit the data. The total concentration-range should be large enough so that at one end a ‘pxe’ system can

avoided in order to include a special ‘locking-in’ step at the very end. The ith dissociation constant is defined

be considered, resultmg in a pair of reasonably ‘secure’ rate co~ants. They could then be employed in the

by Ibrackets omitted):

more comprehensive theory, to solve for the remaining pair of rate constants. As such a comprehe~~eive theory would reveal nothing new, it is not derived at this point. It may also be qutc difficult to cover the rather broad range in enzyme concentration required for 3 precise determination of Four rate eomtants.

R -=4

Ri

(56)

Tr-t

witi? i > 2 and RI= R. Employing the polymerizing

coxtant

the definition

of

M, one obtains (cf. eq. (58))

F: M=

-

(57)

,GKl

5. DISCUSSION The equations of se:tionf 2 and 3 will now be cxtcnsivcly d&cussed, and specifically regarding their

The real gain ht energy is obtained in the final, stabilhiog step. This gain would correspond to a relatively small mottomolecular rate constant for this 1851step Vi;?m ‘I’,, to R,). The rematig monomolecular. as

application to the invextigation of the allosteric betraviorr of enzyme systems. As systems .with two aId steric Forms have already been extensively discussed in the literature (Monad. Wyman and Changcux l9,bS;

G. H.CZERLINSKt

228

Kirschner, Eigen, Bittman and Voigt, 1966; Czcrlinski, 1968). emphasis will be placed here on the polymerti ing sy!.tem, and especially on the comparison of the two schemes(l) and (2). Fic>t, the change of the saturation-function

YF

On the other hand, ‘very high’ concentration gives:

if

[R]-m’

SM.

then YF-?_o.

ofR

(se)

the

with eithtr relaiivc ligand-concenrration a, Or enzyme concentration will be considered. Certainly, eq. (‘1) does not reveal any dependence upon enzyme cc.xentration. Thus, in a plot of Y, versw total en7.yme concentraticn cg (for c$ * cg. ar always a* sumed), one would obtain a horizontal line for any fixed value of a. This independence of cz is quite in contrast to the corresponding behavior for the polymerizinl( model, given by eq. (9). But it is alao apparent from fig. 3 that there are ranger of a, where dependence of YF upon a is not wry pmnoupced. The concenrrution of free monomer [RI in eq. ~9). is computed from the total concentration of monomer. cl;c= denominator

of cq. (9) by a computer pro&%vo

similar to Chapter 4 of Czerlinski (1968). The lower and the upper limits of Y, in fig. 3 are eaiily obtained from eq. (9). One obtains for ‘very low’ concentrations of R:

These indicated limits are easily seen in tig. 3. While fig. 3 with c a O.! represents preferenlial binding to the monomer, the opposite is obtained by c = 10: preferential binding to the polymer. Tbc resulting curves OS YF wxaus cz are shown in fig. 4. The

229

CUIY~S in fig. 3 started with a high YP. decreased to a lower value and approached this value considerably more smoothly than they ‘loft’ the upper lit. The CUIW in fig. 4 proceed just oppositely: they start

large ci: the exponent II of eq. (9) is then no longer effective. As eq. (59) is valid at very low ci, the Momber n could only show up in the shape of the cuvc ai

from a lower limit.

intermrdiate

abruptly

break off this lower limit

and approach

the upper limit

rather

rather smoothly.

values of e”x, Any such changes in shape

The differences in the upper and lower limits in the curves of tig. 3 and 4 certainly decreases as c approaches 1.0: the difference vanishes ate = 1.0. If thL curve for a = 1.0 is picked from fig. 3 as ref-

are not substantial under the conditions,chosen in fig. 6. Kc;; is kept constant at I00 yM,a substantial chaoge in the shape of the cmve of YF wsus a with c as parameter can easily be seen in fig. 7. However,

c~mce. one can easily see the inllucnce of changes of the other parameters in Iii. 5. lncreasc inn, for in-

the cb:hangeis unusual insofar, as one obtains with decreasing c initially a large shift in the point of Y, =

atlimce, lea& to a ‘steepening’ of the inflection. A chnngc in c causes some ‘steepening’, but mainly a lowering in the asymptote toward high c& es one should expect from cg. (60). Thus far. CC was the indeoendent variable. The “cxt fnu t&t% show Y, W&US a according to eq. (9). Fig. 6 shows ths effects of the parameter ci; with all other parameters kept constant: One obtains mainly a

0.5 with only slight decrease in e, one obtains a small shift in the point of YF = 0.5 and a more substantial ch@ge in slope. And something else is very clearly visible: the lower and thdupper branches of the curve become progressively more onsymmetric to each other. as E decreases. This unsymmctry was never “bserved in the model of Monad et al. (1965); it is, how-

‘sideways’ shift of the sigmoid curye (for logarithmic tiiog along absciw) with changmg 4. interesting to note that very little further ‘shift to the left’ “c-

ever. quite understandable from the structure of eq. (9). If c$ is large, then [R] is also large. But a small c in the numerator may make the second term ~fftciently small so that (at very low a), the linear first

Clf for any c$< I PM. If one increaser C$ to loo0 #M, the resulting curve shifts only sli&ttly to the right

term largely dctetmines the numerator; the denominator largely provider for the delay in the rise of the

of the ctttve for ci = 100. However, some general &anger in the shape of the sigmoid curve have to “ccur, as the very simple eq. (60) is obtained at very

curve for Y,c in fig. ‘I(compCed to c=O.I). And when the second term in the numerator finally SW passer the first term, it increases very ragidly with ~1,

It ia

230

G. H. CZERL.,NSK,

Fig. 9. Gmpnicalrcprcuntation of eq. (9) for the dashedmd the dotted curveswith n = 4,M+ = 1 ,,M, co = 5.6 PM. E - 0.01 ioz the @hod curve.The full CYNCIC%err 10 cq. (7, for wmparison and with L 5 lb0 and c = 0.1, n = 4.

producing the highly unsymmctric curve for c = 0.001. for Q = 1W pM and E is also transferred

Thecw-~e

lo fig. 8. The othcr three eurve~ of fig..8 refer to MOnod’s model of tw allosteric forms, eq. (7), It is quite evident that these iatter cwvo~ arc much atoeper than the one for the polymerizing model. Thus, in order to b:ing the slopes into agreement, the ratio c for the pclymeridng model would have to be decreased. Of particular interest is the point of half-saturation, YF = 0.5, as this is a suitable roferoncepoint for them. ical relaxaUon. For reasons of connection to previous work, the ewe L = 100 is chosen for reference, with n ~4 and c= 0.1, it is Y,= 0.5 at a = 10. These ame coordinates are then sot for the polymerizing model and cz is decreased, until it coincides with thir‘halfsaturation-value’ within 0.4%. The resulting curve is shown in fig. 9. Its slope is much too ‘flat’. If the parametcr c is now clanged from its original value ofO.1 to 0.01, the resulting dashed cuwe approacher the full cuw reasonably well, especially in the upper put, where the exponential term in the numerator dominates Y,. Any further decrease in c has comparatively little effect. The EUN~S of YF versusa acoording to eq. (7) and (9) could certainly be discussedfurther under oth1.r points of view. However. equations for relaxation times

I

,

/

CHEMKAL

RELAXAT,ON

OF ALLOSTCRK

231

VODELS

Il.

have also been developed.

And the conditions of fig.

wS quite small,in fig. Comparatively hi& values were em&ved in ties. 7 and 8. The wne hieh values w are empioy;d in Ii&-l 2. The non-polymerizirg model

9 seem quite suitable for their consideration, all three

is again urcd as reference. Its upper value {at low a)

curves come together a’r Y.c = 0.5. If even*subxrip!ed

know

ate ccmstantsafe equal. eq. (16)and (50) apply. rewectivelv. Pie. 10 shows the restdtin~curves with a ai independent variable. Although the dashed curve canes from about the same value at low a. they deviala “ely much at high a. Tbhir rise can be qui:c rut!-

.

I

srantial, and very large time-differences are obtained br the limits. as one may easily derive from the mentioned equations by proper substitution. If edd-subscripttd rate constants are now equal, eq. (I 7) and (43) at~ply, respectively. Fig. 1 I shows the rercllting curves for the recitwxcal relaxation time canstont venus a. The full curve reaches the value 2 sac-t at (r = 10. At this pein!, bath terms of the equation contribute to r-t in equal amount (as kl = I :ecee-‘).The CWVEB for the polymerizing model change here in a much less pronounced manner than the curve for two allwtefic conformers. It is also quite well vis.ible from the dotted cwve that the upper asymptote is reached rather smoothly compared to the approach of the lower asymptote (with 7-t

= 1

see- t). This

seems lo be generally characteristic of the polymerizing model. The anatytical (= total) concentrat: n of monomer

r; ,)

below that of the polymerizing modei, which

r “.

---\

‘\

\

\

‘...

‘(.,

\

‘.., ._ .,.,

\

\

‘

\

‘.

‘\

\

‘,i--

,.I

+.*cj;

,,.

‘i,& .... .. 1.4

-------.F-7

= 3.L III.,.” - 4.0 = 0.1 inall faS$,

C. tt.CZERLtNSKt

232

... ‘\

“r.,,.

I

II

\

‘...,

‘,

Y,.*Y3-

Ifc; Pc’l and(ktc; + k2) P(kj for the slow relaxation time (K2,,

‘...,\

II

k3

r-‘=kg

‘_ \

$1

\

u

‘,...\

‘4,

I,

Pip.14.

Eq. (43) and

c)J= 3.1

plotted versusCT

= kz/kl):

+k,

(62)

+c’i

, Thisis the ‘primitive’

ib

scheme,

basic to the curves of fig. 10. On the other hand, one may also consider the alternate reaction scheme:

kt

4 Y4 -q--

If againc’j % c’: is in contrast to these limits in fig. Il. The dotted

(51)

+ k4), oneobtains

The slope of the sigmoid cuwe is positive, if r-’

NM in attcases. Futthemorc,

4. e = 0.0, for tl;e f”U curve: n = 8, c = 0.1 for*he dnshCd cun’oand n = 4, c = 0.1 for the dotted CWVC.OtherwiseM-‘b = IllH aIWayI,81abwC:k,=, xc-l. ” =

4

‘Y3.

Y

-Y3

and(k,caz+kz)*(k~+k4),

(63)

one ob-

tains for the slow relaxation time:

cuwe finally approaches the limit of the full curve. But its slope is still not as steep as that bf the referencectuve. And there is very little increase in slop with further decrease in c. I~I fig. 10, the total change in r-1 was only about one order of magnitude. This would correspond to the alloacric model with I, = 10. Fia. 13 shows this allostetic model. But aside of coop&ivity with n = 4, ttottcooperativity with n = I is also demonstrated. The dopes of the dashed and the dotted curves are somewhat different, but not very strikingly, although c = 0.1 for all cases.One should keep in miad that there the ordinate is linearly divided in contrast to all other sauhs with 7-t. The full wwe refers to the polymer&S model with C: = 3.1 uM. With the ch&n &utteier-values it is rather simit: in shape to the dashed

curve. The same cttrve with cg = 3.1 JIM is shown again as full curve in fia. 14. referrina a&e nenerallv to this tola1 monometeOncenttation.-Tie effects ofehanges in c sod n are also sbowtt. If n = 1 for the allortetic conversion model, one obtains a simple cycle with two (dnw) monomolecular step6 and two (fast) bimolccukr ones. However, this is not yet the simplest scheme for obtaining sigmoid ctttvcs for r-t verstu ligandconcentration. One such scheme is represented by

In this use, the sigmoid cttrve for I- 1 versus c!j demOttStmteSa negative slope, similar to the cttrves of fig. I I and the ones thereafter. These Ian two ‘txitttitive’ expressions for relaxation times should ce&dy considered even &fore the cyclic reaction with

be

fmt

n = 1 (ttonzooperativity). The above systems of eq. (61) and (63) were originally and rather extensively treated by this author (Czerlinski, 1964). The resulting eqr. (62) and (64) for r-1 were later compared with equations, derived from somewhat more extended reaction systems (Czerlinski, 1967). These systems ittchtded one which corresponds to system (1) at the very beghttting of this paper with n = 1. Although chemical relaxation experiments ma) easily be possible near hzlf-saturation. such experi. ments may be rather difficult to conduct at very low and at very hi& saturation. If kz/kl B 1, the relaxation curves.,from which the characteristic time coortantr are evaluated, will vanish in the noise at low a. If inversely, k2,,+l/kZn+Z 2+ I, the same difficulty itt eValtt.atios 0CC”tSat 1al8Ca. whrt ,a”@ Of Q aCttt-

ally leads to detectable reluation times, depends I!?O” the model, the size of the individual parameters it-d the relevant signal-to-noise-ratio of the detecting instrument. As Kirschner (1966) demonstrated, an .&qua@ range of a con be covered, also allowing employment of chemical relaxation methods to the elucidation of systems with allosteric behavior. 111summary, a distinction between the alternate models, represented by eqs. (I) and (Z), is best accnmpliled by observing Y, (or rTt lnot sbhownl) as a function of c$, keeping all other parameters constant. The model of allosteric interconversion (eq. (1)) would show ni: dependence upon ci The polymerizing model (= eq. (2)) would show the dependences. which are demonstrated in figs. 3 and 4. However, such dependence could only show up, if a is neither very large, nor very small (compared to unity, see the mentioned figures; always e + I .O). In addition, the extreme conditionsof eqs. (59) and (60) should not be fulfiied. Positively expressed: tbr concentration of free monomer. [R 1. should be of the general order of magnitude ofM- ‘,‘tntt) (and the same condition should roughly be fulfilled for cl). It is apparent from eq. (8) that only M -*!(n*t) can be linearly re. lated to concentration and therefore orly the con~tancy ofM-t/w’) bhould be fulfilled (and nor of M-J.One may then also equate for on:‘,,: W-‘/t”+‘) = I.O#M. And thisspecitic value ofM-t tml) was chosen, to allow directly the normalisa ion of the abrcbsa: if 4 is given in PM, the numbers at the abscissa of figs. 3 and 4 are dimensionless and given in c;/Jf-l/w).

6. SUMMARY Two models for allostoric behavior are considered and compared with each other: The model of two al. lustedc formsaccording to Monad, Wyman and Chsngeux (1965) - the polymerizing model contrining

monomer yd highest polymeranly. Saturation functions I’, ard brietlv derived and extensivelv discussed. Ccnerai e~ression~ for the relaxation times of both models are derived (hamotropic interaction only) and their dependence upon individual parameters is dis- \ cussed. Similarities and differences betwcan the vuious resulting curves are shown, to facilitate in the differentiation among the models. The postulate .,f exclusive presence of monomer and highest palyme~ implies structural requirements, which .w discussed in some detail. These requirements are reduced to angular characteristics of the ‘plants’ of polymcrizing sites on monomers.

Chsngeux. J.-P.. J. Tbicry. Y. Tung and C. Kittei. ,967. Prac. Natl. Acad. Se,. U.S. 57. 335. C~erlinslri.C...t9h4. J. Thcor. Viol. 7.463. t~rLxb”rkl. G.. 1966. CbemiCLt Rolzwatlon. An l”l‘Od”Ctiori (0 IbC Theory an* Application of Stepwirc Pcrtwbatio” Wilrcul Dckker. New Yark,Chap,. 3. Cxrlinrki. C.. 1967. J. Thrur. Siol. 17, 343. Cecrlinrki, G.. 19611, Chemical Rclaxation hlalhodr, in: Thew elical and Rxpcrm~cntal Hiophysics, vol. 2, cd. A. Cole wucc1 DekkLY. NW York, rbwter 5. Kknchnur. K.. M. Eigcn. R. Rittman and B. Vast. ,966. Proc. Natl. Acad. Sei. U.S. 56, 1661. Kutloy. M. E. and D. E. Kcebland Jr.. 1967.1. Mot. Cbem. 242,4192.

Klappcr,M. H.nnd I. M. Ktotz, 1968. Biocbemirt~y 7.223. Kbatr. 1. M.. 1946. Arch. Siochem. 9. 109. Korhland. 0. E.. Jr..C. Nemethyand D. Filmer. 1966. Biuchemistry 5, 365. Nicbol. I.. ti.. W. 1. H. Jacksonmd D. J. Winror, ,967. Bmcbemiwy 6. 2449. Rubin. \f. M. ilnd J..P,Cbangcux, 1966.J. Mol. Uwl. 21, 265. Scbuau, 6., ,968, Rev. Modern Phyr. 40.206.