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Cold denaturation in the Schellman–Brandts model of globular proteins
Chemical Physics Letters 486 (2010) 65–69
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Cold denaturation in the Schellman–Brandts model of globular proteins Angelo Riccio a, Eduardo Ascolese b, Giuseppe Graziano b,* a b
Dipartimento di Scienze Applicate, Università di Napoli ‘Parthenope’, Centro Direzionale Isola C4, 80143 Napoli, Italy Dipartimento di Scienze Biologiche ed Ambientali, Università del Sannio, Via Port’Arsa 11, 82100 Benevento, Italy
a r t i c l e
i n f o
Article history: Received 26 November 2009 In final form 29 December 2009 Available online 4 January 2010
a b s t r a c t Brandts extended the zipper model devised by Schellman to treat the helix-to-coil transition by considering that residues linked by H-bonds form a spherical folded globule and that nonpolar residues preferentially cluster in the core of the globule to avoid water contact. It is shown that such a model reproduces the occurrence of two cooperative transitions, cold renaturation and hot denaturation, on increasing temperature, in agreement with experimental data. The decrease of the stabilizing contribution associated with the burial of nonpolar residues from water contact on lowering temperature is the cause of cold denaturation. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Even though it is well established that cold denaturation is a general phenomenon in the conformational stability of globular proteins [1–3], a complete molecular-level explanation of its origin and strange thermodynamic signatures has not yet been achieved [4]. A renewed interest on cold denaturation emerged in the last years from both experimental, computational and theoretical sides. In particular, (a) Temussi and colleagues characterized, by means of NMR measurements, the cold denaturation at physiological conditions, of both yeast frataxin and titin I28, without the addition of any destabilizing agent [5,6]; (b) Dias and colleagues detected, by means of molecular dynamics simulations, the cold swelling of a hydrophobic 10-monomer chain immersed in the so-called Mercedes-Benz model of water [7]; (c) we have shown the occurrence of cold denaturation in the Ikegami’s model [8], when the effect of water is properly accounted for [9,10]. Following this research line, we have tried to verify if the Schellman–Brandts model of globular proteins is able to reproduce the occurrence of cold denaturation. Schellman, in 1958, developed an elegant statistical mechanical approach (i.e., the zipper model) to treat the helix-to-coil transition of polypeptide chains [11]. Brandts, in 1969, extended the Schellman’s approach by adding two constraints: (a) the residues connected by H-bonds form a spherical folded globule; (b) nonpolar residues prefer to avoid the water contact, clustering in the core of the globule, whereas polar ones prefer to stay on the globule surface, contacting water [12]. In this manner the model should be able to describe the cooperative native globule-to-random coil transition. In fact, Brandts, by suitably selecting numerical values for the model parameters, was able to show the occurrence of a cooperative transition in such * Corresponding author. Fax: +39 0824 23013. E-mail address: [email protected] (G. Graziano). 0009-2614/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2009.12.088
a model on increasing temperature, resembling the two-state folded , unfolded transition (i.e., hot denaturation) of globular proteins [12]. We would like to confirm the reliability of the Schellman– Brandts model by showing its ability to reproduce the cold denaturation of globular proteins. Therefore, the original model has been extended in order to calculate the average excess heat capacity function that is the proper thermodynamic function to reveal the occurrence of two cooperative phase transitions on increasing temperature (i.e., cold renaturation and hot denaturation). 2. The Schellman–Brandts model According to Brandts [12], who was inspired by an original idea of Fisher [13], the folded conformation of a globular protein is considered to be a 3D spherical globule of radius Rg, made up of two different types of sites: (a) surface sites, contained in an outer spherical shell of thickness r and volume 4p[R3g (Rg r)3]/3, that are in contact with water; (b) interior sites, contained in the inner sphere whose volume is 4p(Rg r)3/3, that are out of contact with water and constitute the hydrophobic core of the structure. The total number of interior and surface sites is calculated by means of the following formulas:
8 3 < nin ¼ 4 p ðRg rÞ v res 3 :
nout ¼ 43 p
R3g ðRg rÞ3
ð1Þ
v res
where vres is the average volume occupied by a single residue, and, in line with Brandts [12], we have used vres = 125 Å3 and r = 4 Å. Fixed the number n of folded residues, the radius of the folded spherical globule Rg can be determined from:
nv res ¼
4 3 pR 3 g
ð2Þ
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A. Riccio et al. / Chemical Physics Letters 486 (2010) 65–69
Three types of residues are considered: (1) hydrophobic (labelled O) that prefer to be located in the inner sphere; (2) neutral (labelled N) with no preference for either surface or interior sites; (3) hydrophilic (labelled P) that prefer to be located in the outer spherical shell. Moreover, complete chain flexibility is assumed to hold: folded residues can occupy surface or interior sites independently of the position occupied by nearest neighbour residues; this is certainly an overestimate in view of the conformational restrictions imposed by the covalent connectivity of the polypeptide chain and the torsional potentials, but it proves extremely useful in performing calculations. Following the Schellman’s approach [11]: (a) the residues, independently of their identity, are connected to each other by means of H-bonds in the folded region of the chain, that has not to be necessarily a helix (this is a reliable assumption since available structural data indicate that the total number of H-bonds in the native structure of globular proteins corresponds, more-or-less, to the number of residues [14]); (b) the unfolded conformation containing no H-bonds is taken to be the reference state and its Gibbs energy is set to zero. Under these assumptions the conformational partition function of a polypeptide chain consisting of N-residues is given by:
3Dsres XN Zð3DÞ ¼ 1 þ exp ðN n þ 1Þ n¼4 R n3 nOin Dhres T Dsres Dg O exp tr exp RT RT nPin nNin P N Dg Dg exp tr expð tr Þ qcomb RT RT
ð3Þ
where Dhres and Dsres are the enthalpy and entropy changes associated with the breaking of a H-bond; exp(3Dsres/R) represents the entropic cost associated with forming the first H-bond for the need to fix three couples of dihedral angles; (N n + 1) is a statistical factor accounting for the fact that there are (N n + 1) ways of forming an n-residue folded unit in a chain of N residues [11]. The occupation numbers for interior (in) and surface (out) sites of each residue type are weighted by the appropriate Gibbs energy of transfer from surface-to-interior sites, and by a combinatorial factor qcomb that takes into account the statistical degeneracy due to the assumed complete flexibility in locating residues among interior and surface sites in the folded spherical globule. The qcomb factor is given by:
qcomb ¼
ðnOin þ nOout Þ! ðnPin þ nPout Þ! ðnNin þ nNout Þ! nPin !nPout ! nNin !nNout ! nOin !nOout !
ð4Þ
The conformational partition function proves to be entirely determined when the occupation numbers of each residue type in the folded spherical globule are calculated. This task is accomplished by the maximum term method [15] (i.e., by differentiating each term in the partition function and finding the set of occupation numbers which produce a maximum for each value of n). By using the Stirling approximation, one obtains the relations:
8 nO > < Dg Otr Dg Ntr ¼ RT ln nOin
out
P > : Dg P Dg N ¼ RT ln nPin tr tr n
out
nN out nN in nN out nN in
ð5Þ
nin ¼ nOin þ nPin þ nNin nout ¼
nOout
þ
nPout
þ
nNout
ð6Þ
Moreover, if precise values are fixed for the fractions of hydrophobic and hydrophilic residues (fO and fP, respectively), the total number of hydrophobic, hydrophilic and neutral residues can be calculated:
ð7Þ
Having fixed f O and f P, the following constraints must be satisfied as well by the occupation numbers:
(
nO ¼ nOin þ nOout nP ¼ nPin þ nPout
ð8Þ
Eqs. (5)–(8) allow the calculation of the occupation numbers for each given length of the folded spherical globule as a function of temperature. Following in large part the original choices by Brandts [12], we have fixed: N = 200 residues, Dhres = 3.8 kJ mol res1, Dsres = 22.1 J K1 mol res1, fO = 0.4 and fP = 0.2, so that fN = 0.4; the transfer Gibbs energies (expressed in kJ mol res1 units) from surface-to-interior sites are:
8 4 2 O > < Dg tr ¼ 42:70 0:2614T þ 3:312 10 T P Dg tr ¼ 1 > : N Dg tr ¼ 0
ð9Þ
By assuming Dg Ptr to be infinity, all the hydrophilic residues are forced to stay in contact with water in the outer spherical shell; this should not cause problems because hydrophilic residues are only 20% of the total. In addition, the assumption Dg Ptr = 1 simplifies the calculation of occupation numbers, since nPin ¼ 0 and nPout ¼ nP . The quadratic expression of Dg Otr has been obtained by fitting the experimental values of the hydration Gibbs energy for the Val side-chain reported by Privalov and Makhatadze over the 5– 125 °C temperature range [16], in the assumption that the Val side-chain should represent a good average for all the hydrophobic residues. These coefficients imply that the temperatures at which the transfer enthalpy and entropy changes vanish, are TH = 85.9 °C and TS = 121.5 °C, respectively. If the conformational partition function is written as P Zð3DÞ ¼ 1 þ Q ðnÞ, the probability of observing a chain with n folded residues (i.e., the folded spherical globule consists of n residues connected by n H-bonds) is given by:
PðnÞ ¼
1þ
QðnÞ P n Q ðnÞ
ð10Þ
Having determined P(n), any other thermodynamic property of the system can readily be calculated. In order to study the conformational transitions of the model protein, it is useful to evaluate, as a function of temperature, the average order parameter and the average excess heat capacity [8–10]. The average order parameter hXi is defined as:
hXi ¼ 2hf i 1
ð11Þ
where hfi is the average fraction of folded residues:
P hf i ¼
Eq. (5) are supplemented by the following further constraints:
(
8 O O > : N n ¼ ð1 f O f P Þn
n nPðnÞ
N
ð12Þ
It is clear that the average order parameter is comprised between 1 (all residues unfolded) and +1 (all residues folded). The average excess heat capacity function is given by:
P N n¼4 PðnÞDHðnÞ @hDHi @ hDC p i ¼ ¼ @T @T XN @PðnÞ XN @ DHðnÞ DHðnÞ PðnÞ ¼ þ n¼4 n¼4 @T @T
ð13Þ
The first term on the right-hand-side is the contribution due to the change in occupation probability of each folded globule as a function of temperature, whereas the second term on the right-
A. Riccio et al. / Chemical Physics Letters 486 (2010) 65–69
67
Fig. 1a. Probability of observing a chain with n folded residues, P(n), calculated at different temperatures across the cold renaturation; top: T = 15 °C; middle: T = 0 °C (corresponding to the first peak in the excess heat capacity profile of Fig. 2); bottom: T = 15 °C.
hand-side is the contribution due to the intrinsic heat capacity of each folded globule (there is a folded globule for each value of n P 4). 3. Results and discussion The idea to distinguish interior sites from surface sites and the possibility of forming a hydrophobic core are the cornerstone of the Schellman–Brandts model, and produce a cooperative behaviour of the system [12]. In fact, the population fractions evaluated
at different temperatures (see Figs. 1a and 1b), using the parameter values reported above, indicate that the polypeptide chain populates entirely unfolded conformations (i.e., practically with no Hbonds) or entirely folded conformations (i.e., practically with a number of H-bonds close to N, the total number of residues in the chain). There are two temperatures at which the distribution is bimodal, suggestive of a cooperative unfolded , folded transition at low temperature, and of a cooperative folded , unfolded transition at high temperature. The same picture emerges by plotting the average order parameter hXi as a function of temperature
Fig. 1b. Probability of observing a chain with n folded residues, P(n), calculated at different temperatures across the hot denaturation; top: T = 60 °C; middle: T = 75 °C (corresponding to the second peak in the excess heat capacity profile of Fig. 2); bottom: T = 90 °C.
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A. Riccio et al. / Chemical Physics Letters 486 (2010) 65–69
(see Fig. 2): the chain is unfolded at low temperature and hXi 1, it cooperatively folds at T = 0 °C and hXi 0.95 up to about 75 °C, and, on further increasing temperature, the chain unfolds populating conformations characterized by hXi 1. The average excess heat capacity function hDCpi, reported in Fig. 2, shows two endothermic peaks: the first, centred at T = 0 °C, corresponds to the cold renaturation of the chain, and the second, centred at T = 75 °C, corresponds to the hot denaturation of the chain (note that the calculated hDCpi function is zero for the unfolded conformation simply because the latter is assumed to be the reference state in the Schellman–Brandts model). This pattern of hDCpi, with the folded state possessing a lower heat capacity than the unfolded one, is entirely consistent with that determined for several small globular proteins by means of DSC measurements [17–19], demonstrating the reliability of the Schellman–Brandts model to describe the conformational stability of polypeptide chains in aqueous solutions as a function of temperature. In particular, such a model reproduces the occurrence of cold denaturation which is a subtle feature of globular proteins. In order to gain insight into the origin of cold denaturation within the Schellman–Brandts model, it is useful to further analyze the hDCpi function by dividing the first term on the right-hand-side of Eq. (13) in its two contributions:
XN
@PðnÞ XN @PðnÞ DHðnÞ ðn 3ÞDhres ¼ n¼4 n¼4 @T @T XN O O @PðnÞ D h n þ tr in n¼4 @T
ð14Þ
The first contribution (see the dotted curve in Fig. 3) is associated with the formation or breaking of inter-residue H-bonds as measured by Dhres, and produces a negative peak in correspondence of cold renaturation (because there is the formation of inter-residue H-bonds), and a positive peak in correspondence of hot denaturation (because there is the breaking of inter-residue H-bonds). The second contribution (see the solid thin curve in Fig. 3) is associated with the enthalpy gain or loss due to the transfer of nonpolar residues from water contact into protein interior, as O measured by Dhtr ¼ T 2 ½@ðDg Otr =TÞ=@T, and produces a large positive peak in correspondence of cold renaturation and a small negative peak in correspondence of hot denaturation. According to Fig. 3, the second contribution plays the dominant role for cold renaturation [note that the dashed line in Fig. 3 is the baseline provided by the second term on the right-hand-side of Eq. (13)].
Fig. 2. Temperature dependence of the average order parameter and of the average excess heat capacity function.
Fig. 3. Dotted line: contribution to hDCpi from the formation or breaking of interresidue H-bonds; solid thin line: contribution to hDCpi from the burial of hydrophobic residues into the folded core; dashed line: contribution due to the intrinsic heat capacity of each folded globule. The sum of the three terms gives the solid thick line, already plotted in Fig. 2. See text for further details.
In more general terms, the destabilization of the folded conformation at low temperature is due to the peculiar temperature dependence of the Dg Otr function. The latter presents a minimum of about 9 kJ mol res1 at TS = 121.5 °C, and increases significantly on lowering temperature, amounting to about 4 kJ mol res1 at 0 °C (i.e., the burial from water contact of a single nonpolar residue leads to a Gibbs energy gain that increases with temperature). Since the Dg Otr function is a measure of the strength of the hydrophobic effect, it is the small magnitude of the hydrophobic effect at low temperature to cause the destabilization of the folded conformation of globular proteins [3,10,20]. Clearly, the temperature dependence of the Dg Otr function is imposed from the outside, as done in other approaches that successfully describe the stability of globular proteins against temperature [9,10,21–23]. In order to emphasize the role played by the hydrophobic core for the conformational stability, the hDCpi function has been calculated for different values of the outer shell thickness (note that, according to the selected parameter values, the folded globular unit with N = 200 has Rg = 18.14 Å, a reliable value for a small globular protein). On increasing the thickness of the outer shell from 3.5 Å to 4 Å and 4.5 Å, the number of interior sites decreases (more precisely it corresponds to 105, 95 and 85, respectively) and the thermal stability of the folded conformation decreases, since cold renaturation occurs at higher temperatures and hot denaturation at lower temperatures (see the hDCpi curves shown in Fig. 4). Since the size of the interior globule decreases on increasing the thickness of the outer shell, it decreases the number of interior sites available to nonpolar residues and the stabilization provided by the burial of the latter (i.e., some nonpolar residues are forced to occupy surface sites). It is worth noting that a combinatorial factor identical to that of Eq. (4), that assumes a complete flexibility in locating side-chains in the two spatial regions of the globule, is present in the heteropolymer collapse model developed by Dill and colleagues [21,22]. A final point that merits attention is the choice of the parameter values of the Schellman–Brandts model. Both structural and thermodynamic analyses have concluded that a folded globular domain should have at most about 200 residues [24,25]; thus the choice N = 200 is physically reliable. The energetics of the helixto-coil transition has been investigated in detail, and it resulted that the enthalpy change associated with the rupture of a H-bond amounts to about 4 kJ mol res1 [26]; the choice Dhres = 3.8
A. Riccio et al. / Chemical Physics Letters 486 (2010) 65–69
69
transitions in a folded globular unit on increasing temperature, by fixing reliable values for the model parameters; (2) the cold denaturation is caused by the decrease of the stabilizing contribution associated with the burial of nonpolar residues from water contact on lowering temperature (i.e., a decrease of the stabilizing contribution provided by the hydrophobic effect). References [1] [2] [3] [4] [5] [6] [7]
Fig. 4. Average excess heat capacity function calculated for different values of the outer shell thickness, r = 3.5 Å (curve a), 4.0 Å (curve b; it corresponds to that shown in Fig. 2), and 4.5 Å (curve c). All the other parameter values are not modified.
1
kJ mol res is in line with the experimental datum. For the conformational entropy change associated with the unfolding of globular proteins, there are no precise experimental determinations. In any case, by adding together the average value of the backbone contribution, estimated by Freire and colleagues [27], and the average value of the side-chain contribution, estimated by Doig and Sternberg [28], one obtains about 27 J K1 mol res1, a value that is not far from the present choice, Dsres = 22.1 J K1 mol res1. It does appear that the parameter values selected to perform calculations are reliable for a small globular protein. In any case, it is true that the Schellman–Brandts model is sensitive to the parameter choice, as can readily be verified by direct calculations. In conclusion, we have shown that: (1) the Schellman–Brandts model is able to reproduce the occurrence of two cooperative
[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
F. Franks, R.H.M. Hatley, H.L. Friedman, Biophys. Chem. 31 (1988) 307. P.L. Privalov, Crit. Rev. Biochem. Mol. Biol. 25 (1990) 281. F. Franks, Adv. Protein Chem. 46 (1995) 105. B. Hallerbach, H.J. Hinz, Biophys. Chem. 76 (1999) 219. A. Pastore, S.R. Martin, A. Politou, K.C. Kondapalli, T. Stemmler, P.A. Temussi, J. Am. Chem. Soc. 129 (2007) 5374. D. Sanfelice, T. Tancredi, A. Politou, A. Pastore, P.A. Temessi, J. Am. Chem. Soc. 131 (2009) 11662. C.L. Dias, T. Ala-Nissila, M. Karttunen, I. Vattulainen, M. Grant, Phys. Rev. Lett. 100 (2008) 118101. A. Ikegami, Adv. Chem. Phys. 46 (1981) 131. G. Graziano, F. Catanzano, A. Riccio, G. Barone, J. Biochem. 122 (1997) 395. E. Ascolese, G. Graziano, Chem. Phys. Lett. 467 (2008) 150. J.A. Schellman, J. Phys. Chem. 62 (1958) 1485. J.F. Brandts, in: S.N. Timasheff, G.D. Fasman (Eds.), Structure and Stability of Biological Macromolecules, Marcel Dekker, New York, 1969, p. 213. H.F. Fisher, Proc. Natl. Acad. Sci. USA 51 (1964) 1285. D.F. Stickle, L.G. Presta, K.A. Dill, G.D. Rose, J. Mol. Biol. 226 (1992) 1143. T.H. Hill, An Introduction to Statistical Thermodynamics, Addison-Wesley, Reading, 1960. P.L. Privalov, G.I. Makhatadze, J. Mol. Biol. 232 (1993) 660. P.L. Privalov, Y.V. Griko, S.Y. Venyaminov, V.P. Kutyshenko, J. Mol. Biol. 190 (1986) 487. Y.V. Griko, V.P. Kutyshenko, Biophys. J. 67 (1994) 356. B. Ibarra-Molero, G.I. Makhatadze, J.M. Sanchez-Ruiz, Biochim. Biophys. Acta 1429 (1999) 384. B. Lee, Proc. Natl. Acad. Sci. USA 88 (1991) 5154. K.A. Dill, D.O.V. Alonso, K. Hutchinson, Biochemistry 28 (1989) 5439. J.K. Cheung, P. Shah, T.M. Truskett, Biophys. J. 91 (2006) 2427. D. Roccatano, A. Di Nola, A. Amadei, J. Phys. Chem. B 108 (2004) 5756. V.Z. Spassov, A.D. Karshikoff, R. Ladenstein, Protein Sci. 4 (1995) 1516. P.L. Privalov, Annu. Rev. Biophys. Biophys. Chem. 18 (1989) 47. J.M. Richardson, G.I. Makhatadze, J. Mol. Biol. 335 (2004) 1029. J.A. D’Aquino, J. Gomez, V.J. Hilser, K.H. Lee, L.M. Amzel, E. Freire, Proteins 25 (1996) 143. A.J. Doig, M.J.E. Sternberg, Protein Sci. 4 (1995) 2247.
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Cold denaturation in the Schellman–Brandts model of globular proteins Angelo Riccio a, Eduardo Ascolese b, Giuseppe Graziano b,* a b
Dipartimento di Scienze Applicate, Università di Napoli ‘Parthenope’, Centro Direzionale Isola C4, 80143 Napoli, Italy Dipartimento di Scienze Biologiche ed Ambientali, Università del Sannio, Via Port’Arsa 11, 82100 Benevento, Italy
a r t i c l e
i n f o
Article history: Received 26 November 2009 In final form 29 December 2009 Available online 4 January 2010
a b s t r a c t Brandts extended the zipper model devised by Schellman to treat the helix-to-coil transition by considering that residues linked by H-bonds form a spherical folded globule and that nonpolar residues preferentially cluster in the core of the globule to avoid water contact. It is shown that such a model reproduces the occurrence of two cooperative transitions, cold renaturation and hot denaturation, on increasing temperature, in agreement with experimental data. The decrease of the stabilizing contribution associated with the burial of nonpolar residues from water contact on lowering temperature is the cause of cold denaturation. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Even though it is well established that cold denaturation is a general phenomenon in the conformational stability of globular proteins [1–3], a complete molecular-level explanation of its origin and strange thermodynamic signatures has not yet been achieved [4]. A renewed interest on cold denaturation emerged in the last years from both experimental, computational and theoretical sides. In particular, (a) Temussi and colleagues characterized, by means of NMR measurements, the cold denaturation at physiological conditions, of both yeast frataxin and titin I28, without the addition of any destabilizing agent [5,6]; (b) Dias and colleagues detected, by means of molecular dynamics simulations, the cold swelling of a hydrophobic 10-monomer chain immersed in the so-called Mercedes-Benz model of water [7]; (c) we have shown the occurrence of cold denaturation in the Ikegami’s model [8], when the effect of water is properly accounted for [9,10]. Following this research line, we have tried to verify if the Schellman–Brandts model of globular proteins is able to reproduce the occurrence of cold denaturation. Schellman, in 1958, developed an elegant statistical mechanical approach (i.e., the zipper model) to treat the helix-to-coil transition of polypeptide chains [11]. Brandts, in 1969, extended the Schellman’s approach by adding two constraints: (a) the residues connected by H-bonds form a spherical folded globule; (b) nonpolar residues prefer to avoid the water contact, clustering in the core of the globule, whereas polar ones prefer to stay on the globule surface, contacting water [12]. In this manner the model should be able to describe the cooperative native globule-to-random coil transition. In fact, Brandts, by suitably selecting numerical values for the model parameters, was able to show the occurrence of a cooperative transition in such * Corresponding author. Fax: +39 0824 23013. E-mail address: [email protected] (G. Graziano). 0009-2614/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2009.12.088
a model on increasing temperature, resembling the two-state folded , unfolded transition (i.e., hot denaturation) of globular proteins [12]. We would like to confirm the reliability of the Schellman– Brandts model by showing its ability to reproduce the cold denaturation of globular proteins. Therefore, the original model has been extended in order to calculate the average excess heat capacity function that is the proper thermodynamic function to reveal the occurrence of two cooperative phase transitions on increasing temperature (i.e., cold renaturation and hot denaturation). 2. The Schellman–Brandts model According to Brandts [12], who was inspired by an original idea of Fisher [13], the folded conformation of a globular protein is considered to be a 3D spherical globule of radius Rg, made up of two different types of sites: (a) surface sites, contained in an outer spherical shell of thickness r and volume 4p[R3g (Rg r)3]/3, that are in contact with water; (b) interior sites, contained in the inner sphere whose volume is 4p(Rg r)3/3, that are out of contact with water and constitute the hydrophobic core of the structure. The total number of interior and surface sites is calculated by means of the following formulas:
8 3 < nin ¼ 4 p ðRg rÞ v res 3 :
nout ¼ 43 p
R3g ðRg rÞ3
ð1Þ
v res
where vres is the average volume occupied by a single residue, and, in line with Brandts [12], we have used vres = 125 Å3 and r = 4 Å. Fixed the number n of folded residues, the radius of the folded spherical globule Rg can be determined from:
nv res ¼
4 3 pR 3 g
ð2Þ
66
A. Riccio et al. / Chemical Physics Letters 486 (2010) 65–69
Three types of residues are considered: (1) hydrophobic (labelled O) that prefer to be located in the inner sphere; (2) neutral (labelled N) with no preference for either surface or interior sites; (3) hydrophilic (labelled P) that prefer to be located in the outer spherical shell. Moreover, complete chain flexibility is assumed to hold: folded residues can occupy surface or interior sites independently of the position occupied by nearest neighbour residues; this is certainly an overestimate in view of the conformational restrictions imposed by the covalent connectivity of the polypeptide chain and the torsional potentials, but it proves extremely useful in performing calculations. Following the Schellman’s approach [11]: (a) the residues, independently of their identity, are connected to each other by means of H-bonds in the folded region of the chain, that has not to be necessarily a helix (this is a reliable assumption since available structural data indicate that the total number of H-bonds in the native structure of globular proteins corresponds, more-or-less, to the number of residues [14]); (b) the unfolded conformation containing no H-bonds is taken to be the reference state and its Gibbs energy is set to zero. Under these assumptions the conformational partition function of a polypeptide chain consisting of N-residues is given by:
3Dsres XN Zð3DÞ ¼ 1 þ exp ðN n þ 1Þ n¼4 R n3 nOin Dhres T Dsres Dg O exp tr exp RT RT nPin nNin P N Dg Dg exp tr expð tr Þ qcomb RT RT
ð3Þ
where Dhres and Dsres are the enthalpy and entropy changes associated with the breaking of a H-bond; exp(3Dsres/R) represents the entropic cost associated with forming the first H-bond for the need to fix three couples of dihedral angles; (N n + 1) is a statistical factor accounting for the fact that there are (N n + 1) ways of forming an n-residue folded unit in a chain of N residues [11]. The occupation numbers for interior (in) and surface (out) sites of each residue type are weighted by the appropriate Gibbs energy of transfer from surface-to-interior sites, and by a combinatorial factor qcomb that takes into account the statistical degeneracy due to the assumed complete flexibility in locating residues among interior and surface sites in the folded spherical globule. The qcomb factor is given by:
qcomb ¼
ðnOin þ nOout Þ! ðnPin þ nPout Þ! ðnNin þ nNout Þ! nPin !nPout ! nNin !nNout ! nOin !nOout !
ð4Þ
The conformational partition function proves to be entirely determined when the occupation numbers of each residue type in the folded spherical globule are calculated. This task is accomplished by the maximum term method [15] (i.e., by differentiating each term in the partition function and finding the set of occupation numbers which produce a maximum for each value of n). By using the Stirling approximation, one obtains the relations:
8 nO > < Dg Otr Dg Ntr ¼ RT ln nOin
out
P > : Dg P Dg N ¼ RT ln nPin tr tr n
out
nN out nN in nN out nN in
ð5Þ
nin ¼ nOin þ nPin þ nNin nout ¼
nOout
þ
nPout
þ
nNout
ð6Þ
Moreover, if precise values are fixed for the fractions of hydrophobic and hydrophilic residues (fO and fP, respectively), the total number of hydrophobic, hydrophilic and neutral residues can be calculated:
ð7Þ
Having fixed f O and f P, the following constraints must be satisfied as well by the occupation numbers:
(
nO ¼ nOin þ nOout nP ¼ nPin þ nPout
ð8Þ
Eqs. (5)–(8) allow the calculation of the occupation numbers for each given length of the folded spherical globule as a function of temperature. Following in large part the original choices by Brandts [12], we have fixed: N = 200 residues, Dhres = 3.8 kJ mol res1, Dsres = 22.1 J K1 mol res1, fO = 0.4 and fP = 0.2, so that fN = 0.4; the transfer Gibbs energies (expressed in kJ mol res1 units) from surface-to-interior sites are:
8 4 2 O > < Dg tr ¼ 42:70 0:2614T þ 3:312 10 T P Dg tr ¼ 1 > : N Dg tr ¼ 0
ð9Þ
By assuming Dg Ptr to be infinity, all the hydrophilic residues are forced to stay in contact with water in the outer spherical shell; this should not cause problems because hydrophilic residues are only 20% of the total. In addition, the assumption Dg Ptr = 1 simplifies the calculation of occupation numbers, since nPin ¼ 0 and nPout ¼ nP . The quadratic expression of Dg Otr has been obtained by fitting the experimental values of the hydration Gibbs energy for the Val side-chain reported by Privalov and Makhatadze over the 5– 125 °C temperature range [16], in the assumption that the Val side-chain should represent a good average for all the hydrophobic residues. These coefficients imply that the temperatures at which the transfer enthalpy and entropy changes vanish, are TH = 85.9 °C and TS = 121.5 °C, respectively. If the conformational partition function is written as P Zð3DÞ ¼ 1 þ Q ðnÞ, the probability of observing a chain with n folded residues (i.e., the folded spherical globule consists of n residues connected by n H-bonds) is given by:
PðnÞ ¼
1þ
QðnÞ P n Q ðnÞ
ð10Þ
Having determined P(n), any other thermodynamic property of the system can readily be calculated. In order to study the conformational transitions of the model protein, it is useful to evaluate, as a function of temperature, the average order parameter and the average excess heat capacity [8–10]. The average order parameter hXi is defined as:
hXi ¼ 2hf i 1
ð11Þ
where hfi is the average fraction of folded residues:
P hf i ¼
Eq. (5) are supplemented by the following further constraints:
(
8 O O >
n nPðnÞ
N
ð12Þ
It is clear that the average order parameter is comprised between 1 (all residues unfolded) and +1 (all residues folded). The average excess heat capacity function is given by:
P N n¼4 PðnÞDHðnÞ @hDHi @ hDC p i ¼ ¼ @T @T XN @PðnÞ XN @ DHðnÞ DHðnÞ PðnÞ ¼ þ n¼4 n¼4 @T @T
ð13Þ
The first term on the right-hand-side is the contribution due to the change in occupation probability of each folded globule as a function of temperature, whereas the second term on the right-
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Fig. 1a. Probability of observing a chain with n folded residues, P(n), calculated at different temperatures across the cold renaturation; top: T = 15 °C; middle: T = 0 °C (corresponding to the first peak in the excess heat capacity profile of Fig. 2); bottom: T = 15 °C.
hand-side is the contribution due to the intrinsic heat capacity of each folded globule (there is a folded globule for each value of n P 4). 3. Results and discussion The idea to distinguish interior sites from surface sites and the possibility of forming a hydrophobic core are the cornerstone of the Schellman–Brandts model, and produce a cooperative behaviour of the system [12]. In fact, the population fractions evaluated
at different temperatures (see Figs. 1a and 1b), using the parameter values reported above, indicate that the polypeptide chain populates entirely unfolded conformations (i.e., practically with no Hbonds) or entirely folded conformations (i.e., practically with a number of H-bonds close to N, the total number of residues in the chain). There are two temperatures at which the distribution is bimodal, suggestive of a cooperative unfolded , folded transition at low temperature, and of a cooperative folded , unfolded transition at high temperature. The same picture emerges by plotting the average order parameter hXi as a function of temperature
Fig. 1b. Probability of observing a chain with n folded residues, P(n), calculated at different temperatures across the hot denaturation; top: T = 60 °C; middle: T = 75 °C (corresponding to the second peak in the excess heat capacity profile of Fig. 2); bottom: T = 90 °C.
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(see Fig. 2): the chain is unfolded at low temperature and hXi 1, it cooperatively folds at T = 0 °C and hXi 0.95 up to about 75 °C, and, on further increasing temperature, the chain unfolds populating conformations characterized by hXi 1. The average excess heat capacity function hDCpi, reported in Fig. 2, shows two endothermic peaks: the first, centred at T = 0 °C, corresponds to the cold renaturation of the chain, and the second, centred at T = 75 °C, corresponds to the hot denaturation of the chain (note that the calculated hDCpi function is zero for the unfolded conformation simply because the latter is assumed to be the reference state in the Schellman–Brandts model). This pattern of hDCpi, with the folded state possessing a lower heat capacity than the unfolded one, is entirely consistent with that determined for several small globular proteins by means of DSC measurements [17–19], demonstrating the reliability of the Schellman–Brandts model to describe the conformational stability of polypeptide chains in aqueous solutions as a function of temperature. In particular, such a model reproduces the occurrence of cold denaturation which is a subtle feature of globular proteins. In order to gain insight into the origin of cold denaturation within the Schellman–Brandts model, it is useful to further analyze the hDCpi function by dividing the first term on the right-hand-side of Eq. (13) in its two contributions:
XN
@PðnÞ XN @PðnÞ DHðnÞ ðn 3ÞDhres ¼ n¼4 n¼4 @T @T XN O O @PðnÞ D h n þ tr in n¼4 @T
ð14Þ
The first contribution (see the dotted curve in Fig. 3) is associated with the formation or breaking of inter-residue H-bonds as measured by Dhres, and produces a negative peak in correspondence of cold renaturation (because there is the formation of inter-residue H-bonds), and a positive peak in correspondence of hot denaturation (because there is the breaking of inter-residue H-bonds). The second contribution (see the solid thin curve in Fig. 3) is associated with the enthalpy gain or loss due to the transfer of nonpolar residues from water contact into protein interior, as O measured by Dhtr ¼ T 2 ½@ðDg Otr =TÞ=@T, and produces a large positive peak in correspondence of cold renaturation and a small negative peak in correspondence of hot denaturation. According to Fig. 3, the second contribution plays the dominant role for cold renaturation [note that the dashed line in Fig. 3 is the baseline provided by the second term on the right-hand-side of Eq. (13)].
Fig. 2. Temperature dependence of the average order parameter and of the average excess heat capacity function.
Fig. 3. Dotted line: contribution to hDCpi from the formation or breaking of interresidue H-bonds; solid thin line: contribution to hDCpi from the burial of hydrophobic residues into the folded core; dashed line: contribution due to the intrinsic heat capacity of each folded globule. The sum of the three terms gives the solid thick line, already plotted in Fig. 2. See text for further details.
In more general terms, the destabilization of the folded conformation at low temperature is due to the peculiar temperature dependence of the Dg Otr function. The latter presents a minimum of about 9 kJ mol res1 at TS = 121.5 °C, and increases significantly on lowering temperature, amounting to about 4 kJ mol res1 at 0 °C (i.e., the burial from water contact of a single nonpolar residue leads to a Gibbs energy gain that increases with temperature). Since the Dg Otr function is a measure of the strength of the hydrophobic effect, it is the small magnitude of the hydrophobic effect at low temperature to cause the destabilization of the folded conformation of globular proteins [3,10,20]. Clearly, the temperature dependence of the Dg Otr function is imposed from the outside, as done in other approaches that successfully describe the stability of globular proteins against temperature [9,10,21–23]. In order to emphasize the role played by the hydrophobic core for the conformational stability, the hDCpi function has been calculated for different values of the outer shell thickness (note that, according to the selected parameter values, the folded globular unit with N = 200 has Rg = 18.14 Å, a reliable value for a small globular protein). On increasing the thickness of the outer shell from 3.5 Å to 4 Å and 4.5 Å, the number of interior sites decreases (more precisely it corresponds to 105, 95 and 85, respectively) and the thermal stability of the folded conformation decreases, since cold renaturation occurs at higher temperatures and hot denaturation at lower temperatures (see the hDCpi curves shown in Fig. 4). Since the size of the interior globule decreases on increasing the thickness of the outer shell, it decreases the number of interior sites available to nonpolar residues and the stabilization provided by the burial of the latter (i.e., some nonpolar residues are forced to occupy surface sites). It is worth noting that a combinatorial factor identical to that of Eq. (4), that assumes a complete flexibility in locating side-chains in the two spatial regions of the globule, is present in the heteropolymer collapse model developed by Dill and colleagues [21,22]. A final point that merits attention is the choice of the parameter values of the Schellman–Brandts model. Both structural and thermodynamic analyses have concluded that a folded globular domain should have at most about 200 residues [24,25]; thus the choice N = 200 is physically reliable. The energetics of the helixto-coil transition has been investigated in detail, and it resulted that the enthalpy change associated with the rupture of a H-bond amounts to about 4 kJ mol res1 [26]; the choice Dhres = 3.8
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transitions in a folded globular unit on increasing temperature, by fixing reliable values for the model parameters; (2) the cold denaturation is caused by the decrease of the stabilizing contribution associated with the burial of nonpolar residues from water contact on lowering temperature (i.e., a decrease of the stabilizing contribution provided by the hydrophobic effect). References [1] [2] [3] [4] [5] [6] [7]
Fig. 4. Average excess heat capacity function calculated for different values of the outer shell thickness, r = 3.5 Å (curve a), 4.0 Å (curve b; it corresponds to that shown in Fig. 2), and 4.5 Å (curve c). All the other parameter values are not modified.
1
kJ mol res is in line with the experimental datum. For the conformational entropy change associated with the unfolding of globular proteins, there are no precise experimental determinations. In any case, by adding together the average value of the backbone contribution, estimated by Freire and colleagues [27], and the average value of the side-chain contribution, estimated by Doig and Sternberg [28], one obtains about 27 J K1 mol res1, a value that is not far from the present choice, Dsres = 22.1 J K1 mol res1. It does appear that the parameter values selected to perform calculations are reliable for a small globular protein. In any case, it is true that the Schellman–Brandts model is sensitive to the parameter choice, as can readily be verified by direct calculations. In conclusion, we have shown that: (1) the Schellman–Brandts model is able to reproduce the occurrence of two cooperative
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