Abstract
The effect of temperature on the extent of association of self-assembling protein polymers is expressed mathematically in terms of the van't Hoff enthalpy of polymerization, AHV.H.. This quantity has been experimentally defined in two ways-from the respective temperature derivatives of the critical polymerization concentration and of the fractional conversion of protein into polymer. These two definitions are shown not to be identical, except in certain limits. 

fined in two ways-from the respective temperature derivatives of the critical polymerization concentration and of the fractional conversion of protein into polymer. These two definitions are shown not to be identical, except in certain limits. In terms of both definitions, it is shown that AHV.H. depends not only upon the enthalpy changes but also upon the corresponding equilibrium constants for the various equilibria involved in polymer formation. This has two consequences: (i) large AHv.H. values may result from reactions having small calorimetric enthalpy changes; and (ii) AHV.H. can depend strongly on temperature. These considerations are applied to two systems for which there exist considerable experimental data-namely, hemoglobin S and tubulin. The large discrepancy between the calorimetric and van't Hoff enthalpies for the polymerization of tubulin is shown to be explicable in terms of these considerations.

Various biologically important self-assembling systems composed of protein subunits have been investigated in vitro, utilizing temperature to regulate assembly. The parameter expressing the effect of temperature on the extent of assembly is the van't Hoff enthalpy of polymerization, AHv.H.. This quantity has been reported for a number of systems (1) (2) (3) (4) (5) (6) (7) . In two of these systems, tubulin (1-3) and bacterial flagellin (4, 5) , there exists a large discrepancy between LAHV.H. and the calorimetrically measured enthalpy change, \Hcal. In the hemoglobin S system, on the other hand, A\H2aj -LHv.H.. Several difficulties in particular attend the interpretation of van't Hoff data in such complex systems as these. The size of the cooperative unit to which the "mole" refers is generally unknown in the van't Hoff procedure. Moreover, AHv.H. may be sensitive to the mechanism in multistep transitions (8 

Qi -Qo [3] and extrapolations must again be made (Fig. 2 ). An equilibrium constant is then defined by:

and, since AH = RT2 blnK/bT, we find RT2 (Af)

Proc. Natl. Acad. Sci. USA 74 (1977) Determination of the critical concentration, C*, from a plot of equilibrium polymer concentration, P (or an observable, Q oC P), versus total protein concentration, C, for an arbitrary system. At some specified degree of polymerization, the curve is extrapolated back to the C axis.

At the point (PO.5, CO.5) shown above, half of the protein is polymerized and C* = CO.5 -PO.5 (bP/?C)-.. The temperature derivative of C* gives A *HI. (see text). Inset. P versus C for an infinitely cooperative system. For C < C*, no polymer can exist and all protein is in the form of nuclei, N (N = C). As C increases beyond C*, the additional protein goes into the polymeric form and P = C -C*.

which determines AKHEH at any chosen temperature or degree of conversion. For example, at the point at which one-half of the protein is converted to polymer, AZHf.H = 4RT2.5 (.11 This procedure determines AHf.H from of/laT at any given value of C; thus, one expects that AHfH will also generally depend on C, in contrast with AHC*. We turn now to the models. MODELS
Isodesmic case
This case involves sequential monomeric addition, the equilibrium constant, K, being the same for each step:
Isodesmic case
The total protein concentration, C, is a constant, independent of temperature, T, and is related to the concentration of free monomer, A, by the relationship: [7] Defining all aggregates of size >n to be polymerst, we find the polymer concentration, P(A), to be:
Isodesmic case
The concentration of nuclei is N(A) = C -P(A).
Isodesmic case
The fractional conversion of protein into polymer is then [10] We may invert the above equation to obtain
Isodesmic case
[11]
Isodesmic case
Substituting Eq. 11 into Eq. 9 gives f = f(e). [15] lymerization (9, 11) , according to which subunits associate to form unstable oligomers or "nuclei" that are themselves capable of undergoing further association to form stable polymers only if they exceed a certain critical size, n. The critical protein concentration therefore simply reflects the minimal amount of total protein required by mass action to force the aggregation through a series of n thermodynamically unfavorable nucleation steps. This process occurs during the kinetic lag period. For an idealized system displaying infinite cooperativity, a plot of polymer and nuclei concentrations (P and N, respectively) at various concentrations of total protein, C, is given in Fig. 1 inset. It may be seen that, as C increases, all of the protein initially goes into the form of nuclei. Once C exceeds C*, polymer increases linearly with C, while the nuclei concentration remains at the critical value. The model discussed here approaches this behavior to any arbitrarily chosen degree, for appropriate values of the parameters, which are: a, the equilibrium constant for nucleation; K, the equilibrium constant for propagation (K [16] * As in the isodesmic case, the appropriate summations can be done to express N, P, C, and f as functions of concentration of 1171 free monomer, [A] . In terms of the previously defined reduced variables, we find: [18] [19] It can be seen from [21] [22]
Isodesmic case
[23]
Isodesmic case
6O(A) = iV(A) + P(A), P(A) [24] [25]
Isodesmic case
A plot of f versus @ for various values of n is shown in Fig.  3 . The critical concentration is evaluated as follows. We note from Eq. 23 that P(A) possesses a singularity at A = 1. Thus, the maximal value NA(A) can take (@*) is just [26] or C* = K-1(1 -/K)-2 that, once C > C*, P = C -C*; hence f = 1-c*/C [note that the concentration at which f becomes 0 is therefore C* (Fig. 3) However, the fact that intermediates have not been detected by ultracentrifugation (12) suggests that a/K may indeed be << 1.
Tubulin polymerization
The value for AHC* for this system has been reported to be 21 kcal/mol (1). Alternatively, a value of about 25 kcal/mol, depending on the concentration, may be calculated for AHf.H.
Tubulin polymerization
from the data of ref. 3 . Both values are much larger than the recently reported (13) calorimetric value for LHK of 0 ± 1 kcal/mol. Tubulin appears to be a more complicated system than hemoglobin S for several reasons: (i) there exists for tubulin a variety of microtubule-associated proteins whose function has not yet been completely established (14-16); (ii) stable oligomers of tubulin which may be reaction intermediates are known to exist (17, 18) ; (iii) the role of nucleotides and possibly of nucleotide hydrolysis is still uncertain (13, 19) . Even if the answers to these questions were known, realistic models would involve too many adjustable parameters to permit unambiguous interpretations of the available thermodynamic data. A few conclusions can be drawn, however. 1. Even neglecting complicated reaction mechanisms, there need be no discrepancy between the experimental values for AHV.H. and AHcai. To illustrate this point, we consider model 2 and take AHK = 0 and a/K = 0.8. From Eq. 28 we find AHV.H. 20 kcal = -zH,(2u/K)/(1 -u/K) which gives ZHq -2.5 kcal/mol. Considering that the protein solutions are very dilute on a molar basis, because of the high molecular weight of the tubulin dimer, such a small value for 'H, would have been difficult to detect by scanning calorimetry (13) . A similar conclusion follows from models based on thermally induced conformational changes responsible for "activation" of monomers or intermediates (9).
Tubulin polymerization
2. There is no thermodynamic necessity for polymerization-linked GTP hydrolysis, because nucleotides could facilitate assembly allosterically (19) . Indeed, if the nucleotides regulated assembly and possessed sizeable binding enthalpies, this alone would be sufficient to account for the effect of temperature on assembly.
Tubulin polymerization
3. Because AHK 0, the chain-propagating step (which presumably accounts for the bulk of polymer formation) is entropically driven. Thus, we see that comparatively small enthalpy changes, because they may be magnified by potentially large equilibrium constants, can be sufficient to account for the strong temperature dependence of this system. It is conceivable that, when the factors governing the in vivo regulation of assembly are known, a similar situation may perCell Biology: Sutherland Proc. Natl. Acad. Sci. USA 74 (1977) tain-i.e., the effect of a regulator on f or C* may be large not because of a large difference in chemical potential between bound and unbound regulator per se, but because such differences in chemical potential as do exist are magnified by the pertinent equilibrium constants.
DISCUSSION
Although the preceding considerations clearly establish the need for taking the mechanism and type of experiment into consideration in interpreting van't Hoff measurements, there are problems in applying such analyses to real polymerizing systems. Some systems exhibit hysteresis (1, 4) ; accordingly, the transitions may not be thermally reversible and may involve states of metastable equilibrium. Calorimeters detect only overall enthalpy changes. Apart from the uncertainty resulting from the measurement of these usually small quantities, it may not be possible to measure enthalpy changes for separate steps of the overall reaction. These treatments assume solution ideality. This approximation may be acceptable for tubulin and flagellin, wherein protein concentrations are typically <1% by weight, but its consequence with concentrated solutions is un- As noted with tubulin, realistic models generally involve too many parameters to permit their specification uniquely. This is particularly true of systems characterized by length-regulating (20) and other factors (9) .
DISCUSSION
The preceding considerations were applied only to tubulin and to hemoglobin S. In spite of the availability of data for bacterial flagellin polymerization, this system was not discussed for several reasons: (i) the van't Hoff studies (21) were carried out at a temperature (400) at which partial denaturation may have occurred (22) ; (ii) the two calorimetric studies disagree (4, 5) , although this could be due to species differences; and (iii) the flagellar subunits may exist in a kinetically determined metastable state (4) .
DISCUSSION
The formal extension of the preceding analyses to volume rather than enthalpy changes is straightforward and will not be discussed here. Interpretative difficulties must once again be anticipated, however, unless both the mechanism and the AVs for the individual reaction steps can be determined. We have likewise omitted from consideration the multitude of in vivo microtubule studies (23) (24) (25) (26) 

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