Protein convergence 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. 

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. 

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. 

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. 

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.

The general thermodynamic properties of proteins reported above give rise to several questions What do the asymptotic (at Tx) values of the denaturation enthalpy and entropy mean and why are they apparently universal for very different proteins Why should the denaturation enthalpy and entropy depend so much on temperature and consequently have negative values at low temperature In other words, why is the denaturation increment of the protein heat capacity so large, with a value such that the specific enthalpies and entropies of various proteins converge to the same values at high temperature ... 

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]. 

A typical stability curve and the fractional population sizes are shown in Figure 2. It is worth noting that according to equation 16 the fractions of both the native and denatured proteins converge to zero, but never reach it. Therefore the statement at physiological temperatures the protein is in the native state must be interpreted more precisely as meaning that at physiological temperatures the native state is predominant . 

Force field calculations often truncate the non bonded potential energy of a molecular system at some finite distance. Truncation (nonbonded cutoff) saves computing resources. Also, periodic boxes and boundary conditions require it. However, this approximation is too crude for some calculations. For example, a molecular dynamic simulation with an abruptly truncated potential produces anomalous and nonphysical behavior. One symptom is that the solute (for example, a protein) cools and the solvent (water) heats rapidly. The temperatures of system components then slowly converge until the system appears to be in equilibrium, but it is not. 

Privalov et al. (1989) also reported the temperature dependence of the ellipticity at 222 nm for the proteins studied at various pH values (Fig. 28). At the highest temperature studied (80°C), the 222 nm ellipticity value for the thermally unfolded, acid-unfolded, and Gdm-HCl-unfolded proteins appear to be converging, but show a range of 2000 deg cm2/dmol out of a total of 5000 deg cm2/dmol. (ApoMb is an exception in that, as noted before, the thermally denatured protein is apparently an associated /1-sheet. However, the acid- and Gdm HC1-unfolded forms of apoMb have similar [0] 222 values at 80°C.)... 

The simplicity and accuracy of such models for the hydration of small molecule solutes has been surprising, as well as extensively scrutinized (Pratt, 2002). In the context of biophysical applications, these models can be viewed as providing a basis for considering specific physical mechanisms that contribute to hydrophobicity in more complex systems. For example, a natural explanation of entropy convergence in the temperature dependence of hydrophobic hydration and the heat denaturation of proteins emerges from this model (Garde et al., 1996), as well as a mechanistic description of the pressure dependence of hydrophobic... 

Figure 2. Energies (upper panel) and temperatures (lower panel) of the 30 replica modified parallel tempering simulation of the trp-cage protein reported in the text. The dotted line in the upper panel corresponds to the estimate of the global optimum of the free energy (obtained independently). The lower panel demonstrates a rapid equilibration of the temperatures during the simulation. The upper panel demonstrates the convergence of the energy and the rapid exchange of information between the different replicas as discussed in the text.

One of the key observations resulting from calorimetric studies of the denaturation of globular proteins is that both AH0 and AS0 of denaturation, when normalized to the number of amino acid residues in the protein (or the molecular weight), converge to common values at specific temperatures when extrapolated under the assumption of constant ACp (Privalov and Khechinashvili, 1974 Privalov, 1979 Pri-... 

In contrast to the relatively constant number of hydrogen bonds per residue, a set of proteins must bury variable amounts of apolar surface area in order to show convergence (Murphy and Gill, 1991). At the temperature at which the apolar contribution to AH° is zero, no variation would be observed in AH° per residue and the constant polar contribution is all that should be observed. The breakdown into polar and apolar interactions can also be viewed in terms of buried surface area. Proteins bury an increasing amount of surface area per residue with increasing size, but the increase is due to increased burial of apolar surface, whereas the polar surface buried remains constant. This is illustrated in Fig. 2 for 12 globular proteins that show convergence of AH°. These proteins bury a constant 39 2 A2 of polar... 

Comparison of results on thermodynamic studies of protein denaturation and hydrocarbon dissolution in water shows a number of surprising similarities and differences between these two processes. The most surprising result is the close correspondence of the temperature of convergence of the enthalpy and entropy functions for the denaturation of proteins, Tx, and the temperature 7s for the dissolution of hydrocarbons in water. 

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