BIOPY.YSICAL CHEMISTRY 2 (1974) 377-384-o NORTH-HOLLAND PUBLISHING COhfPANk’

POSSIBLE IMPLICATIONS FOR ALLOSTERIC MODELS OF HEMOGLOBIN OF THE LAWS OF COMBINATION WITH CARBON MONOXIDE

UNDER PHOTODISSOCIATJiNG CONDITIONS

Edward WHITEHEAD Istituto di Chimica Biologicq dell’llniversit~ di Roma.

and Centro di Stblogia Uolecolare deI C.N.R.. Citta Universitarin. 00185 Rome. Italy

Received 3 hIay 1974

It is shown that the data obtained by Brunosi et al. on binding of carbon monoxide to hemoglobin in photodissociat- ing conditions is explaiied by a simple allosteric model, the tw-nfiguratian exclusive-binding induced-fit model to- gether with the simplest kinetic assumptions. Alternative models are considered. It is possible to calculate from the data of Brunori et al. a dissociation rate constant for carbon monoxide which agrees well with an independently determined rate constant

1. Introduction

Brunori et al. [l] studied carbon monoxide binding to human hemoglobin under conclitions of constant illumination where the apparent affinity of this l&and for the protein can be decreased by a factor of as much as IO3 due to z process of photodissociation. The main experimental conclusions from this study are that

(i) the shape of the saturation curves as a function of log ligand concentration is not changed by light, i.e., the curves are all parallel, co-operativity is not changed;

(ii) the affinity is a Linear function of light intensity; (iii) the Bohr effect (the effect of pH on affmity)

is invariant with li&t intensity; and (iv) in photodissociating conditions change of tem-

perature does not lead to change in the shapes of the saturation curves, as has long been known to be also the case in the dark. The effect of temperature on affinity in the presence of light is different from that in the dark, while in the dark increase in temperature leads to decrease in &and affinity, in photodissociat- ing conditions it increrrses affinity.

For the purposes of mathematical treatment the statements (i)-(iv) will be assumed to be exact laws.

Insofar as they are exact, these laws as well as the classical laws of invariance of co-operativity with pH and temperature in the dark are completely accounted

for by a simple model of allosteric equilibria - the tw_oconfiguration exclusive binding induced-fit model, plus the simplest possible kinetic assumptions_

2. The model

Consider an oligomer whose protomers can exist in only two configurations. termed T and R, and a~- sume that a l&and w can be found only to R. If there is only one site per protomer for w and if the protein does not undergo an association-dissociation reaction then at equilibrium the relation between fractional saturation a of the protein by the ligand and the concentration w of the ligand <J in solution has the form

(1)

where B is the association constant for binding of the l&and to the configuration R, q the number of proto- mers and the Lj a series of constants relating to the

Jntsraciians between protomers (see, e-g., eq. (26) of ref- [2] ; eq- (7) &reef, [3I ). This may be written

x2 = [f&?/f1 +Bw)]* (1 +Bw) ‘I m

where%? is a functional operator. It can be shown that tfie J&W- form is valici for fhe two~o~~guration ex- clusive binding hypothesis even in the presence of dis- sociation of the ofig5mer [4j. Nclw the Jon~~~~abJ~~hed invariance of co-operativity ofligand binding to hemo~obin with pW and teznperarure may be expJained {S) by the in&iced-fit hypotiesis in which it is as- sumed, in addition to all the above, fhat for aJJ measurable a, Bw 3 I. [This implies that there is never a significant proporiion of free R in the system.) Applying this st%dtion to eq. (21, we obtain

5 =%?(&J) . (3)

3. Laws (i) and (ii)

As well a5 the above a~~rnpii~~s about aGosteric equiir%ria, it is assumed that in the absence of light the reactions leading to the binding 0f CO to a sub- unit in the R form, and the dissociation of the RCO compiex may be represented

I R+co$Rca. (4’4)

No*w B till be determined by two velocity canstanb, 6’ and 6, respectively the forward and backward v&o- &y constants for the reaction:

ssiY/E,.

Tim the facts can be explained by the reasonable ~sumption &at light causes an extra ““oft? process

RCO+hu~R=+CQ, (9

and thar rhe f&d-order rare constant of this process is proportional to JJght intensity, 1, and to the absorption coefficient, e, and quantum yield, ~9, of tie chemaphore. Thus the first-order rate constant for the pkatuinduceci process (S) would be GEL Ttre total first-order rate constant for the d~co~pos~r~on of RCO directly into R + CO is the sum olF those for (4) and (51, i.e.

(we shall adopt the ccmventiurx ofwriting the effective

value of my quantity in the presence of light intensity I with a subscript I, t2u.1~ quantities with subscript 0

represent those quantities in the dark). Therefore

From (3) constant G corresponds to constant &.L+ therefore for any given constant saturation

curves [Jaw (i)f since we see in the above equation that at any tlxed light j~ten~ity w/@o is ind~~ndent of saturation, and also the linearity of the dependence of affmity on intensity [law (ii)] I as can be seen re- writing (66) in the form

or

w = w&r -WEll’bo) f (7)

The exgeriments permit an estimation of the value of&. From (7) this is equal to the horizontal Enter- cept muttip!icd by - #E in a pIot of w agaSnst inten- sity at any constant 5 (for instiince Sz = )). Thus if the absoiute intensities in the experiments of Brunori et al. were known an absofute value of ba could have heen obtained. Unfortunate& the inten- sities could not be measured absofrstel~, but only by their effect on the affmity of rnyoglobin. This means that in the first place a value of b. relative to a velocity constant of dissociation from myoglobin is what is obtaiasd dire&y from the experiments- Thus we WR write

bWXQ ~~~~b~~~~b) +kb &.sC@tb~ -= %Mb> &fbehlbrw-NKhSb, = ohibfw=Q(aj ’

since the integrated extinction coefficients of h&no- globin and myoglobin are near@ equal. Using 0.37 as the w&e E61 of&&&t,, ~~~}~~~~fb} was estimated, using the results f I] of Bruno~J et al. as being cJose to I. Now the kinetics of myo@obi’s reactions with carbon monoxide can be adequately represented [7J by the acherue:

l’he v&city constant of CO dissociatiu~ from myo-

E. Wtitehead. AlZosterfc nta.ii.k uttd pkotodissoctikobn 379

globin in the above scheme has been.estimated 181 as 0.02 s-t, so using the above ratio of the dissociation constants for hemoglobin and myoglobin, bow,, is predicted to have roughly this vaiue of 0.02 s-1 too. The dissociation veIocity canstant of carbon monoxide from fully saturated human hemoglobin - “14” in hemoglobinolagists’ terminology - has also been (and is the only li to have been) directly measured I91 as 0.01-0_015 s- 1. But this velocity constant is ex- pected from our model to be equal to b,(~b) (the relation of kinetic dissociation constants is discussed below in more detail). It can be seen that the model gives good agreement between the predicted and ob- served values of 14, when it is considered that the predicting computation uses data from five different publications.

4. Effect of pH - ?.aw (iii)

Bmnoci et al. [ 11 found that the effect of pH on affiiity was the same at any fight intensity as in the dark. The quantum yield of carbonmonoxyhemooobin is known [61 to be independent ofpH, and so therefore will be that of process (5) in our model. If then the pH variation of apparent affinity is independent af the relative contributions of the two processes - pracess (5) and the “off’ process of scheme (4) - whose sum determines this affinity and one of which is pH independent, we must conclude that the rate constant bg is also independent of pH. This is what we should expect if binding is truly to a single canfig uration, and if we think of the Bohr protons as affect- ing the equilibrium between configurations rather than interacting in a direct manner with bound ligand, fn agreement v&h &is, and with the iden- tity postulated above between b. and l,, the lat- ter is relatively independent 191 of pH.

5. The effect of temperature - law (iv)

Another constant of the model obtainable from the data of Brunori et al. 113 is the enthalpr of activa- tion of the dissociation reaction in (4). Assumtig that the process (5) and the quantum yield and extinction coefficient of the carbonmonoxyhemaglobin are not temperature-dependent, and simpWyiig eq. (7) slightly

by taking the case where 0 * “0 (a condition satisfied in many of the experiments) we see that this quantity is given by

MS =-dInboid(ifRT)=din(wllrwg)ld(KiKT),

which is available experimentally. it is stating exactly the same ttisg in other terms to say that this enthalpy of activation is equal to the difference in the enthalpies of liganding of the hemoglobin at zero and at high light intensities.

AH* =AK~-LW-. (8s)

From Brunori et d_‘s values of +7 and - 11.5 kcallmole tespectivety for the quantities on the right-hand side, Af@ is approximately +18.5 kcdjmole. This may be compared with the value of 24 kcaI/mole estimated (91 for directly measured I,. The discrepancy between the two is equivalent to a better accord between the estimates of 6, and 1, at some temperatures than others.

6. Discussion

In this paper certain laws true to within a certain error of human hemogIobin, have been derived from a model, so supporting the model as an account of the real situation in this hemoglobin, However. the field af dlosteric models is strewn with unwarranted in- ferences from experimental results springing from failure to recognise clearly which predictions are due ‘to which features of the madels. For this reason. and ti VWJ of the Wure usefulness of the photodissocia- tion method for other systems, it wiCL be important and illuminating to examine this relation closely in the present case, and to consider alternative models.

Concerning the laws (i) and (ii), the essential features of the made\ respansible for these are: the exclusive bittding hypothesis. i.e., the assumption that CO is bound to one configuration only of the subunits, and the assumption that the processes of dissociation and photodissociation from this c~nfigu-

ration can be adequately represented by scheme (4) and (5). The assumptions of only two configurations and of induced-f&wise binding are not necessary to these predictions, but were made above for convenience of exposition in the context of a discussion of human hemoglobin. Laws that are explained by the last two

380 E. Whitehead, AIIosteric models and photodissocidttin

assumptions are tte invariance of co-operativity with pH and temperature in the dark and the Iinear relation between ligand saturation and configuration-dependent parameters. These last laws are not v&d for hernogio- bins in genera& but only for some better-known mam- malian (notably human and sheep) hemoglobins. The co-operativity of I&and binding by many fish hemo- gIobins, and among mammalian forms, cat and some abnormal human hzmoglobins is on the other hand pH-dependent [IO-I 21.

To see that the first-mentioned assumptions above are adequate to predict laws (i) and (ii) consider eq. (Z)_ An equation of this form represents the case where binding of the Iigand is to only one configuration however many non-binding configurations there may be. This equation can be written in the form:

S = f(L?w) _ (9)

Light affects B in the way already specified and the rest of the argument is exactly the same whether we use eq. (3) or (9)_

The .Enetic assumptions implicit in schemes (4) and (5) are essential to the explanations we have given.

A significant point about the assumption concerning photodissociation required by the explanation is that

processes such as E!Ji +hv + E!CZo +iw, are excluded, or in other terms the quantum yield is independent of the degree of Iiganding (which has recently been confiimed experimentally by Brunori - personal communication) - or again that the photodissociation terms enters into the binding equation with the same statistica weights as does b,. This need not exclude transfer of the photoexcitation between subunits, but does mean that such transfer cannot be specifically towards the liganded subunits.

With regard to the dark dissociation process, the pathway RCO + TCO + T + CO must be assumed to contribute negligibly to it. A significant role for this pathway would allow the dissociaiion rate constants for CO to vary with the degree of liganding, but would

not be compatible with the photodissociation experi- ments, whilst on the other hand in the model being proposed these constrants are independent of the

degree of liganding. There is a general belief, based on kinetic and equiIibrium studies and conflicting with the proposed model (or any model having its essential characteristics, as already emphasized by

Brunori et al.) that these constants do vary with degree of Iiganding. Neither the author, not his colleagues at Rome have been able to devise a model satisfying both this belief and the photodissociatiort laws that is satisfactory on other grounds. It can onIy be noted that the belief in question is not based on any direct

measurements of different CO dissociation rate con- stants (ii has not been measured for i = 1,3,3). The values of the equilibrium constants for CO binding to sheep hemoglobin [ I31 on which the belief is mainly based are also in conflict with the theory [I41 of the relation of CO and 0, binding that gives a good ac- count of the overall equilibrium behaviour of these ligands with hemoglobin. On the other hazd the photo- dissociation experiments are the ones which most directly reflect effects of all the dissociation rate con- stants. It is quite reasonable to suspect current beliefs concerning CO dissociation kinetics may be mistaken; perhaps the experimental equilibrium constants for CO at extreme low and high saturations are affected by some extra factor that is not very important to the overal pattern of binding of CO_ On the other hand the oxygen dissociation rate constants certainly do vary with degree of liganding. It is therefore expected that the photodissociation behaviour of oxyhemoglobin, which Professor Brunori is now preparing to study, will be quite different from that of carbonmonoxy- hemogIobin*. Co-operativeIy should not be invariant with light, nor, under photodissociating conditions, with pH or temperature. Binding curv-z~ should be- come asymmetrIc_ The equations ha= not suggested any particular simple positive features.

The appropriate model for oxygen reaction, then, wiIl probably be different from that for carbon mono.uide. Behaviour under photodissociating condi- tions thus reflects both aIIosteric equilibria and kinetics. The preceding discussion because of the simpIicity of the mode1 used has not yet brought out the essential character of binding under photodissociat ing conditions. A system under such conditions is not of course an equilibrium, but a steady-state, fully analogous to a steady-state of an enzyme during catalysis, and can be treated by the same mathemati-

* A condition under which the Same laws ar for CO would obtain is that the reaction of oxygen with the R form be even faster than tit with dithionite, and that the R +T transformation be much slower than the ROz -+ TO, tWlSf0m thL

E_ Whitehead. AUusmic models andphofodismciafrim 381

cal methods, notably the topological method of King and Altuiari [16]. Such steady-states in the general case (even in single-site proteins) allow kinetic contri- butions to co-operativity (see ref. [2] for review and references). These contributions are always dependent on the circulation of material around cIosed pathways, e.g., carbon monoxide can be in the sum result taken on by one configuration and discharged by another, a thing that does not happen at equilibrium. This circu- lation will not be significant when certain pathways are overwhelmingly favoured over others. Only in these and other special circumstances will the equation of binding in photodissociating conditions talcs a simple form similar to a possible equilibrium with only some of the constants changed by the photodissociation process, as happens in our proposed model. In such circumstances some, but not all species are present in their equilibrium ratios. As an example, if in a protein with an arbitrary number of configurations, liganded subunits may not undergo configurational transitions, light only changes the values of the B‘s without other- wise changing the binding equation from that at equilibrium-

A circumstance under which a pmtid simplification of the general steady-state binding equations results is when the configurational transitions are rapid in comparison with the binding and dissociation reactions. The binding of a single Iigand can then be represented by an equation of Adair form, of which eq. (10) is an example.

Considering now alternative models, nonexclusive binding models, i.e., ones with binding of ligand to more than one conformation have been proposed for hemoglobin ]17-211. These models do not in general predict the laws (i)-(iv), and in this respect the ob- servstions of Brunori et al. give them no support. To see this, note that in such models equilibrium binding is determined by more than one association constant (S). While we have seen that the change of behaviour under photodissociation is not in general like that due to a change of B’s in an equilibrium scheme, neverthe- less, used with care, and when something can be as- sumed about the constants, such a way of thinking is often useful for the rapid prediction of qualitative effects. Thus Iight might change one of the B’s much more than another in such a way that the system might come to resemble an excIusive binding equili- brium scheme, with consequent change in co-opera-

tivity. This is an extreme case, but light will in general change co-operativity in a nonexclusive scheme.

Still, it could be that there exists a departure from laws (i) and (ii) which is within the errors of the ex- periments, while the nonexclusive binding models may predict such smnff systematic effects which are not as yet observed experimentally in normal human hemoglobin. The same, it has been argued [S] , is true of their predictions for the effects of pH and temperature in the dark. This corresponds to the fact that what is often being proposed in these modeIs is something but slightly different from exciusive binding (and there would seem little enough justifica- tion for insisting on these minor modifications). The extent to which such models are falsified by the experiments is a matter for detailed calculation; but the photodissociation experiments certainly impose further constraint on nonexclusive models as accounts of this system.

Nonexclusive binding models with a kinetic basis and therefore susceptible to confrontation with photo- dissociation data have been proposed by Hopfield et al. [21] and Szabo and Karplus [22] _

The attempt at a computation for the former model revealed some defects in it which will be detailed_

The value of the “nonexclusivity constant”, c, proposed by Hopfield et al. 1211 is 800 for CO. They do not appear to have noticed that in order to account for the correct value of the co-operativity (II z Z-8) this value of c demands a revision of their initially assumed value of the “allosteric constant”, L. The revised value then alters the assignment of states of the protein which was the basis for their calculation of c and its four component velocity constants, such as to require an even more extreme value of c_ More- over, the equilibrium binding curves predicted by the revised values of the constants become noticeably asymmetric, in contradiction to the facts. However for illustrative purposes we shall investigate the effect of photodissociation using Hopfield et al.,% kinetic constants and the eq.(lO) corresponding to the symmetric transition model when all configurational changes are assumed rapid.

382 E. Whtiehead, Ah’osteric models and pho~odissoclrtim

where

and where c = Boo/f3~ _ The values ofL consistent with c = 800 and nllt =

2.8 at equilibrium are 2.6 X IOe3 and 9 X IO-1a. With the second value fight reduces n, ,* to 1.9. This value would presumably not have been chosen because with oxygen (c = 280) it gives n, ,z = 1.5. With the other vaIue ofL saturating light would reduce nllS to 2.3. A change of this order would have been detected in the experiments_ There is also an obvious deviation from law (ii)_ Thzs if this model were con- sidered broadly appIicabIe to the material of the photodissociation experiments, human hemogI0bi.n at high dilutions, it would be eliminated. If alternatively the hypothesis is made that liganded subunit cannot undergo configurational transitions, in which case the binding equation is of eqitilibrium form, the deviations with L = 2.6 X 1W3 are Iess and might escape detec- tion. This hardly rescues nonexclusivity since with these values of the constants, at 90% saturation the low-affinity conformation is only saturated to the extent of I.570 at most in the dark_ This is equivaIent to saying that the exckive-binding model is a good appro.ximation. The difference between the two com- putations illustrates the influence of Icizetics on the steady state photodissociation behaviour.

The constants proposed by Eopfield et al. are de- rived from measured equilibrium constants about which reservations were expressed above.

Szabo and KarpIus [22] in their account assume implicitly that the dissociation rate constants for the two conformations are equai *_ This is consistent with the view of the overall CO dissociation kinetics taken here, though not witi the usual view- It is then true that this model describes the photodissociation laws exactly as does the excIusive binding model. But so does a model with thirty-six different cortfigura- tions, all having the same “off’ constants (at all

T Strictly that the e of each of the two or more different spzcies is proportional to their dark dissociation velocity constznts. If the latter are different the author knows of no physical justification or precedent for such a proportion- ality.

temperatures!). If it be objected that a model with thirty-six binding conformations is more complicated than one with two, then so is the latter more compli- cated than a model with one. In other words, rather than supporting any particular nonexclusive model, the tendancy of these authors’ arguments is towards the position that the photodissociation observations, ideaIIy described by laws (i)-(iv), contain no infor- mation about allosteric equilibria (though they do contain the same value of bg calculated above). Since this is obviously relevant to the degree of confidence to be felt with any model, it wiIl be further commented on in the conclusion.

Returning to the analysis of the reIation of our own hypotheses and predictions, the exclusive binding hypothesis and the already mentioned kinetic assump- tions are again by themselves adequate to predict invariance of the Bohr effect with respect to light [law (iii)] _ By the concept of configuration i&If one expects this to hold in an exclusive binding model. It is an indication of the participation of only one dissociation rate constant and one intrinsic equilibrium binding constant over a range of conditions where if there were more than one binding configuratitin, more than one constant would in general be expected to influence the observations. In other words it makes the conditions nonexclusive modeIs have to satisfy yet more stringent than do observations at a single pH, while the exclusive binding hypothesis has no difficul- ty in meeting it.

Where the induced fit hypothesis comes in is in expIaining an important feature of the Bohr effect namely invariance of co-operativity in the dark (and thereby automatically through the laws already con- sidered in the light). The Iongknown pH and tempera- ture invariance of the co-operativity ofligand binding to hemoglobin are highly relevant to the choice be- tween induced-fit and nonexclusive models. The argument eventuaIly becomes numerical: the former model predicts pH-invariance while the latter models not, but these may perhaps do so to within some ap- proximation. The spirit of the rival models in this re- spect was commented above. The point could be phrased saying that it is the typical predictions of the induced-fit hypothesis that show up strongly with nor- mal human hemoglobin, while nothing corresponding to tkte typical predictions of the nonexclusive models is seen.

E. Whirehtwd. Allosteric models and photodissxhttin 383

But cat hemoglobin does show a marked effect typical of nonexclusive binding. While affinity for oxygen increases with pH, so does cooperativity [ 113 . On an exclusive-binding non-induced-fit model with positive co-operativity the opposite effect is expected. Possible simple explanations for the Bohr effect in cat hemoglobin are therefore: a nonexclusive binding model; an exclusive binding model with more than two configurations; and an exclusive binding model where B varies with pH (however as explained below the third of these is practically a special case of the second). Here is an example where the photadissocia- tion method should be useful for distinguishing be- tween models since there is a better chance than in human hemoglobin that the effects of nonexclusivity show up as deviations from the laws that apply to the latter.

Let us consider the possibility of explaining the Bohr effect in human hemoglobin without invoking induced-fit by postulating that variation of pH changes only one constant, B. The argument on photo- dissociation was based on change (by light) of B. With or without induced-fit this leaves co-operativity unchanged. Clearly in the same way, change of B by any agent is a possible formal explanation of its changing affinity without change of co-operativity. Applied to the effect of pH in the dark it seems at frost somewhat contrary to notions of configuration_ This formal explanation could however coincide with a corlfigurational one if we assume, in addition to a PH.-independent co-operative configurational change, other pH dependent non-co-operative ones. Wyman [12] has proposed just this in a “nonexclusive” model (one with two such effective pH dependent B’s). Ac- cording to his special assumption which cause both B’s vary in the same way with pH, exact invariance of co-aperativity with pH in the dark results. L.&e other nonexclusive models it needs further special assumptions

to survive the photodissociation experiments. The idea can be put in an “exclusive” form (binding only to R which is a set of configurations related to each other throqh non-co-operative transitions)‘and then in general it satisfies laws (i) and (ii) but not (iii). The case where (iii) is satisfied without any special kinetic or other assumptions is when the binding is to one of these sub-configurations only -just a special case of exclusive binding. It must be added that in the type of model we are now considering, if it is not to reduce

to exchrsive binding, and’induced fit, the observables - the m, m and the other configuration-dependent parameters which vary linearly with saturation [7] are all correlated with non-co-operative transitions, and none as yet with the cooperative transition which must still be invoked. Law (iv) does not automatically fol- low either from exclusive binding or induced-fit. It requires the configurational equilibria to be tempera- ture independent, i.e., the free energy differences between R and T to be entirely entropic. The induced- fit hypothesis is much more permissive - only the in- teractive parts of these free energy differences and not the much larger non-interactive parts need be purely entropic. In general (and with our usual kinetic assump- tions) in the hypothesis of exclusive binding without induced-fit the effect of temperature on co-operariviry would be the same in the light as in the dark (this is just saying that law (i) holds at different temperatures) but the effect on affniQ would be quantitatively different, and eq. (8) would always apply, where the AJY on the right-hand side are obtainable from

M=d Inw,/d(l/RT) ,

where w, is the median l&and concentration as defined by Wyman [14].

The proposals of Szabo and Karplus 1221 show that if one is prepared to accept very special kinetic assumptions infinite latitude is allowed the description of allosteric equilibria so far as the restrictions posed by the photodissociation experiments are concerned. The nearer the various “off’ constants are to one another, the less constraint these experiments pose on schemes of equilibria. On the other hand the model proposed here allows considerable though not infinite latitude with respect to the kinetic constants (it suf- fices that b;-fLi, all r‘, is not comparable with or greater than 6k) by making more special assumptions with respect to equihbria. These are however naturally suggested by the experimental equilibrium behaviour. It is not pretended that the induced fit model accurately accounts for all the facts taken into account here. Discrepancies of note, though perhaps not extremely grave, are the slight pH dependence of the directly measured E4; and the difference in the estimated AEFr of 1, and bO. These might be weaknesses through which the way could be forced towards a better des- cription of the system. But at this moment it seems fair to believe that an improved model for hemoglobin

384 E. Whirehead. Albsteric models and plrotodi.smciLltion

equilibria and kinetics together would be situated somewhere within a small volume of the relevant parameter space which also contains the present model

Acknowledgement

I am greatIy indebted to Professors E. Antonini, M. Brunori and 1. Wyman for introducing me to the subject of photodissociation and for many heipful discussions and stimulating criticism. Computer cal- culntions used in this work were done at the Medical Biological Laboratory of the TNO, Rijswijk, Holland and the University of Rome. I thank Drs. Wijnans and Salerno for assistance with computations. This work has been financed by the Consiglio Nazionale delle Ricerche and by Euratom- This is publication No. 1000 of the

Euratom Biology Department of which the author is a staff member.

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