Temperature convergence

Protein folding cannot occur in a random fashion but must follow distinct pathways a protein of 100 amino acids with diffusional control of encounter between different parts of the chain would take (210°/2)/lO10 s 1 = 1033 s = 5 x 1024 years, or longer than the age of the universe, to fold. Levinthal s paradoxon reaches the same conclusion a protein of 100 amino acids with side chains in three different states can feature 100 / 3 states diffusion of those states would take even longer than 1024 years. Proteins cannot be stable without hydrophobic interactions convergence temperatures for unfolding enthalpies and entropies (AHm and ASm) are around 112 °C, the presumed Tmax for proteins in aqueous solution. 

L. Fu and E. Freire, On the origin of the enthalpy and entropy convergence temperatures in protein folding, Proc. Natl. Acad. Sci. USA 1992, 89, 9335-9338. 

Reciprocal plot of temperature vs. number of chain carbons for the (a) paraffins, (b) fatty acid ethyl esters, (c) 1-alcohols, and (d) iso-alkyl acids. The intercept represents the convergence temperature. 

This value of Tt is the same as the entropic convergence temperature, Tt, observed in proteins (Baldwin, 1986 Murphy etal., 1990). This is the temperature at which the denaturational entropies of globular proteins, normalized to the molecular weight or to the number of amino acid residues, take on nearly the same value when extrapolated under the assumption of constant ACP (Privalov and Khechinashvili, 1974). The significance and interpretation of this observation are discussed in more detail below. 

Analysis of the dependence of the structural thermodynamics of globular proteins on apolar surface area provides an estimation of the role of various contributions to protein stability. However, as mentioned above, proteins also show convergence temperatures that can yield similar information, given certain assumptions. 

Subsequently it was observed that the presence of convergence temperatures is also easily seen in plots of AH° or AS0 at 25°C versus ACp. If a convergence temperature exists, then this plot will be linear. The slope is equal to (298.15 — 7h) or ln(298.15/Ts) and the intercept is A// or AS. Using such plots it was shown that T% for the transfer of apolar compounds from any phase, as well as for protein dena-turation, was a universal temperature near 112°C (Murphy et al., 1990). This observation lent further support to the view that the convergence temperatures were associated with the hydrophobic effect. It must be noted also that these plots do not require the assumption that ACp be constant with temperature. 

It is important to note, however, that although group additivity with a constant component will always lead to convergence temperatures, the presence of convergence temperatures may not require this condition. Recently Lee (1991) proposed an alternative explanation of the protein convergence temperatures. In the analysis Lee showed that if the protein data are normalized to the total amount of buried surface area, the variable fraction of the total buried surface that is apolar will lead to a convergence temperature at which the apolar and polar AH0 contributions are the same on a per square angstrom basis. 

As has been discussed recently (Murphy etal., 1992), the formalism developed by Lee (1991) predicts not only the convergence behavior discussed by the author (i.e., when the apolar and polar contributions to the enthalpy are identical). For the case in which the buried polar area per residue is constant it also predicts convergence at the point at which the apolar contribution to the enthalpy is zero. In Fig. 3 we have plotted AH0 versus ACp normalized either per residue (Fig. 3a) or per buried total surface area (Fig. 3b), in order to compare the results of the two approaches. It is clear that the linearity is better when the data are normalized to the number of residues than when they are normalized to the buried surface area. This is presumably due to variabilities in the surface area calculation. The slope of the line in Fig. 3a is —72.4, which corresponds to a convergence temperature, Th, of 97.4°C for this set of proteins. If the above analysis is correct, then this temperature corresponds to The value of AH is 1.32kcal (mol res)-1 or 33.6 cal (mol A2)-1 of polar surface area. 

The slope in Fig. 3b should correspond to the temperature at which the apolar and polar contributions to AH0 are equal per square angstrom. This temperature can be calculated to be about 140°C (assuming that Th = Tk as discussed above), as shown in the figure, and is about 40°C higher than the observed convergence temperature for the residue normalized analysis. 

Fig. 3. Linear correlation of AH0 versus ACp at 25°C for protein denaturation for the proteins listed in Table IV. (a) Normalized per number of residues (b) normalized per total buried area. The line in (a) is the linear regression fit with a slope of — 72.4 corresponding to a convergence temperature of 97.4°C. The line in (b) represents the line calculated from the parameters in Table II for convergence at the temperature at which the polar and apolar contributions to AH are equal per unit area. See text for details.

Figure 8.9 Variation of the entropy convergence temperature with increasing hard-sphere radius. The thin solid line is the convergence temperature determined under the assumption that the heat capacity is independent of temperature, and the thick solid line is the exact entropy convergence temperature for spheres smaller than R < (Tww/2 (Ashbaugh and Pratt, 2004). The dashed line smoothly interpolates between the exact and constant heat capacity curves at 1.25 A and 3.3 A, respectively. The filled circle indicates the entropy convergence temperature of a methane-sized solute (7), = 382K). The open circle indicates the entropy convergence temperature based on the information model = 420 K) (Ashbaugh and Pratt, 2004).

Figure 8.10 The distribution of the number of oxygen atoms within 5.1 A of the Kr atom in aqueous solution at an elevated temperature in the region of the entropy convergence temperature (LaViolette et al, 2003). These results were obtained to investigate the possibilities of clathrate nucleation upon quenching see Filipponi etal. (1997) and Bowron etal. (1998). Note that the coordination numbers n = 20 or n = 24, which are associated with clathrate cages, are unexceptional in this distribution for the liquid solution. The subtle structure in this distribution for n below the mode may be reflective of possibilities for alternative thermodynamic phases, e.g. the coexisting gas phase, or structures with commodious cages.

Baldwin, R. L. and Muller, N., Relation between the convergence temperatures T and T in protein unfolding. Proc. Natl. Acad. Sci. USA 89, 7110-7113 (1992). 

A very remarkable feature of A/Z F) and AS CT) is that these functions reach an upper limit above a convergence temperature T 140°C, and that their specific limiting values (A/Z ) and (A5 ) per gram or per amino acid residue are nearly the same for all typical globular proteins [41]. More recent results indicate that the convergence temperature is not the same for AN (F) (FH ) as for At(T) (Ts ) [34]. 

The coefficients in Eqs. f6-29i require knowledge of the enthalpies leaving each stage and the Q values. The enthalpy values can be calculated since all x s, y s, and temperatures are known from the component mass balances and the converged temperature loop. For ideal mixtures, the enthalpies are... 

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