- Email: [email protected]
Investigation of hydration effect of the proteins by phenomenological thermostatistical methods
ARTICLE IN PRESS
Physica A 361 (2006) 255–262 www.elsevier.com/locate/physa
Investigation of hydration effect of the proteins by phenomenological thermostatistical methods Gorkem Oylumluoglua,, Fevzi Bu¨yu¨kkılıc- b, Dog˘an Demirhanb a
Department of Physics, Faculty of Arts and Sciences, Mugla University, 48170-Kotekli, Mugla, Turkey b Department of Physics, Faculty of Science, Ege University, 35100-Bornova, I˙zmir, Turkey Received 21 February 2005; received in revised form 23 June 2005 Available online 11 August 2005
Abstract In this study variations with respect to temperature of entropy increment DS, arising in the dissolution of proteins in water, have been investigated by the methods of statistical thermodynamics. In this formalism, effective electric field E and total dipole moment M are taken as thermodynamical variables. The partition function given by Bakk et al. [Physica A 304 (2002) 355–361] has been used in a modified form while obtaining the free energy. In the constructed phenomenological theory, experimental data are taken from the study of Privalov [J. Chem. Thermodyn. 29 (1997) 447–474; J. Mol. Biol. 213 (1990) 385; T.E. Creighton, Protein Folding, W.H. Freeman and Company, New York, 1995, pp. 83–151] and relevant parameters have been found by fitting to experimental curves. The variations of entropy increments DS have been investigated in the temperature range 265–350 K. r 2005 Elsevier B.V. All rights reserved. Keywords: Thermodynamics; Classical statistical mechanics
1. Introduction Proteins is the common name of complex macromolecules which are formed by gathering up of a great number amino acids and which play unremitting roles in the Corresponding author.
E-mail address: [email protected] (G. Oylumluoglu). 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.07.004
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lifetimes of all the living creatures. There are approximately 20 different amino acids in the structure of the proteins in the form of chain molecules. The order or arrangement of these 20 units is sensitive and definite. This order determines the nature of the protein and as a result identifies its function [5]. Each protein molecule is formed by an original combination and consecutive placing of different numbers of amino acids. Proteins, under their own natural physiological conditions, spontaneously open from one-dimensional structures to three-dimensional structures in time intervals starting from milliseconds and reaching to minutes [6,7]. The three-dimensional structures of the proteins depend on various environmental factors in the cell. Unfolded proteins return to their natural states when the environmental factors are removed exhibiting the feature that the three-dimensional structure of a protein solely depends on its amino acid ordering information. This situation gives the hope to some research workers that protein folding could be reconstructed by computer simulations [8]. But the difficulty of these simulations even for small proteins is in view. In this study, the model of the proteins is presented in Section 2. In Section 3, the thermodynamical quantities related to the variation of the additional entropy increment DS have been given and these parameters are related to the parameters of the proteins by using statistical thermodynamics. Finally, starting from the curves of the entropy increment, relevant conclusions on the protein structures are presented. The thermodynamical variables of fluids are represented by the physical quantities U (energy), V (volume) and N (particle number), whereas their corresponding parameters in the experimental measurements are T (temperature), P (pressure) and m (chemical potential) [9]. In a similar manner, for the magnetic systems the thermodynamical variables are represented by the physical quantities U (energy), H (magnetic field) and N (particle number) and their corresponding experimental parameters on T (temperature), M (magnetization) and m (chemical potential) [9,10]. As thermodynamical quantities of the proteins, internal energy (U), electric field (E) and particle number (N) could be taken as macroscopic variables. When the system tends to thermodynamical equilibrium, to demonstrate the connection with thermodynamics, the system is controlled by internal energy parameters namely temperature (T), electric field parameter i.e., dipole moment (M) and particle number parameter i.e., chemical potential (m).
2. Model For different aspects of the folding and unfolding transitions of the proteins, various models have been proposed. A simple yet notable model among them is the zipper model which defines the helix chain [6,7]. Another model where the solvent is represented as dipoles in the unfolding of proteins was given by Warshel and Levitt [11] and the applications of the model were presented by Russell [12], Fan [13] and Avbelj [14]. A simple model of protein unfolding has been developed by Bakk [1]. Bakk started by choosing water molecules
ARTICLE IN PRESS G. Oylumluoglu et al. / Physica A 361 (2006) 255–262
257
as classical electric dipoles. He demanded that non-polar dissolution should include important physical properties in modelling of the effect on the folding of the proteins. In the model, in the medium of the formed electric field, the folding and unfolding of the proteins have been taken as the basis. Here, the electric field is not an external field but has been used in the modelling of the ice-like behavior exhibited by the water molecules around the non-polar surfaces [15]. The electric field is a result of the effective behavior of the non-polar dissolvent on the protein unfolding. The modified partition function of the protein system could be taken as [16] h m i ZðT; E; mÞ ¼ exp exp zðT; E; N ¼ 1Þ . (1) kT Here, m represents the chemical potential and the term z ¼ expðbmÞ, which corresponds to the pH of the system has been introduced to the partition function. Single-particle partition function is zðT; E; N ¼ 1Þ ¼ 4p ebbm
2 =2
1 sinh be , be
(2)
where effective energy is given by e ¼ þ bhmi [1]. On the other hand, average dipole moment has the form hmi ¼ Lðbe Þ. One of the characteristics of the folding and unfolding of the proteins is that the temperature interval of the process depends on environmental conditions and particularly on the pH values of the solution. This task is supplied by the term z ¼ expðbmÞ in the partition function. The role of the electric field here is to model a kind of stiffness (ice-like behavior) that water molecules exhibit around non-polar surfaces. Thus, it is not a real external electric field, but merely an effective behavior upon non-polar solvation of protein folding.
3. Variations of the entropy increment with respect to temperature Taking electric field E and total dipole moment M as thermodynamical variables, the first law of thermodynamics could be written for the proteins: dU ¼ TdS MdE .
(3)
Entropy increment could also be written by taking the total differentials of entropy S ¼ SðT; MÞ: qS qS dS ¼ dT þ dM . (4) qT M qM T Writing down the expressions ðqS=qTÞM ¼ DC M =T in Eq. (4) and then integrating one could get for the entropy increment: Z T Z MðTÞ qS DS ¼ C M dT þ dM . (5) qM T Tt MðT t Þ
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The free energy of the system has been obtained from the relation F ¼ kT ln Z giving F ¼
4 expðbð12 bm2 mÞÞp sinhðbe Þ b2 e
.
(6)
The thermodynamical quantities of the system has been determined using the entropy relation S ¼ ðqF =qTÞE which in turn leads to 2 expðbð12 bm2 mÞÞkp be
S¼
fð2e b coshðbe Þ þ ð4 þ bbm2 2bmÞ sinhðbe ÞÞg .
ð7Þ
The total dipole moment has been calculated from the expression M ¼ ðqF =qEÞT resulting in M¼
4 expðbð12 bm2 mÞÞp b2 2e
fe b coshðbe Þ sinhðbe Þg .
(8)
The energy of the system is given by U ¼ ð1=ZÞðqZ=qbÞ and when the partition function is substituted one obtains U¼
expðbð12 bm2 mÞÞ b2 e fð4pe b coshðbe Þ þ 2pð2 þ bbm2 2bmÞ sinhðbe ÞÞg .
ð9Þ
Additional heat capacity at effective field has been calculated by making use of the equation DC E ¼ kb2 ðqU=qbÞE which gives DC E ¼
expðbð12 bm2 mÞÞkp f8e b coshðbe Þ be ð4e bðbbm2 þ 2bmÞÞ coshðbe Þ þ ð8 þ bðb2 m4 b 8mÞÞ sinhðbe Þ b4bm2 ðbm 1Þ sinhðbe Þ þ 4b2 ð2e þ m2 Þ sinhðbe Þg .
ð10Þ
Using the expression DC M DC E ¼ TðqM=qEÞT ½ðqE=qTÞM 2 , which has been obtained for additional heat capacity at constant total dipole moment, DC M ¼
expðbð12 bm2 mÞÞkp f4e bð2 þ bbm2 2bmÞ be coshðbe Þ þ ½8 þ bðb2 m4 b 8m 4bm2 ðbm 1Þ þ 4bð2e þ m2 ÞÞ sinhðbe Þ
k 2 b2 ð2be coshðbe Þ þ ð2 þ b2 2e Þ sinhðbe ÞÞ
½e bð4bbm2 2bmÞ coshðbe Þ ð4 þ bðbm2 þ 2be 2mÞÞ sinhðbe Þ2 g
ð11Þ
ARTICLE IN PRESS G. Oylumluoglu et al. / Physica A 361 (2006) 255–262
has been calculated. ðqS=qMÞT in Eq. (5) is calculated as qS ¼ kbe . qM T
259
(12)
When Eqs. (11) and (12) are substituted in Eq. (5), entropy increment reads b expðbm2 ðT t þ TÞ=2kT t TÞ ðbm2 =TÞ þ ð2m=T t Þ DS ¼ 4kp T exp e t 2e 2k 2 e e bm ðT t þ TÞ e cosh þ kT t sinh þ p exp 2kT t T kT t kT t n e 2e cosh 2kTe þ e ðbð2k 1Þm2 kT e þ 2ð2k2 T þ m 2kðT þ mÞÞÞ sinh kT þð4k2 T 2 e þ e ðb2 km4 42e þ 2bkm2 ðð2k 1ÞT 2mÞ o þ4kð2k2 T 2 þ 2e 2kTðT þ mÞ þ mðT þ mÞÞÞðln T ln T t ÞÞÞ . ð13Þ Now the numerical applications of the entropy increment will be considered. The unfolding and then folding of the protein molecules take place by a thermodynamical mechanism. In other words, the original structures of the macromolecules are determined thermodynamically. In Fig. 1, symbols represent experimental study whereas solid lines theoretical results. The physical quantities , b, m and T t , which have been obtained in this semi-phenomenological theory, are determined by fitting to the P experimental results of Privalov [2–4]. The fitting parameter b is given by b ¼ ij J ij , where J ij is the coupling constant between each pair of water molecules.
Fig. 1. The variation of the additional entropy with temperature.
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Table 1 Numerical values obtained by fitting the parameters in Eq. (13) to the experimental data Proteins
Symbols
m ðkJ=molÞ
ðkJ=molÞ
b ðkJ=molÞ
k ! R ðJ=kmolÞ
T t ðKÞ
Mb RNase
Square Circle
185 1000
4.92 6.25
26 19.8
8.314 8.314
298 350
e ¼ þ bm is the magnitude (strength) of the effective field, m being average dipole moment. Another fitting parameter is the chemical potential m which is a measure of the strength of interactions with water molecules. m changes by denaturants, pH or with salt concentrations. The last fitting parameter is the transition temperature at which Gibbs energy difference between native and denatured states is equal to zero. Let us give a brief discussion of the differences of magnitudes of the parameters between myoglobin and ribonuclease in terms of fitted parameters; T t for ribonuclease is higher than T t for myoglobin which is a consequence of the corresponding thermodynamical energy levels of the two proteins. Since b is a measure of the coupling of the water molecule pairs which causes hydration, one could say that myoglobin has stronger hydration properties compared to ribonuclease. physically is a bending distortion constant which has a value slightly bigger for ribonuclease relative to the value of myoglobin. Thus one could conclude that ribonuclease has a higher stiffness property compared to myoglobin. In our calculations pH is exponentially related to m. In order to approach the pH values of the proteins under consideration, m which is very highly dependent on T is obtained as seen in the Table 1.
4. Conclusions The three-dimensional structure of the proteins depends on different environmental factors inside the cell. Unfolded proteins return to their natural state when the environmental factors are removed, in a manner of exhibiting the fact that the three-dimensional structure of a protein solely depends on the information of the amino acid regularity. The probable configurations of this three-dimensional structure are very large in order. In the thermodynamical limit (N ! 1, E ! 1 and N=E ¼ const:), it is not possible to make calculations by taking into consideration each particle in the macroscopic system formed by the proteins. Thus a combination of thermodynamics and statistical physics is a necessity. In the investigation of these types of systems with statistical physics, the existence of a great number of systems that are identical to the system under consideration are taken into account and using the statistical methods for these systems, the most probable state in which the actual system could enter is determined. Generally water seems to be important in the understanding of protein folding [3,17–22] and furthermore, the peculiar feature of cold unfolding seems to be particularly sensitive to the surrounding water [23–25].
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The unfolding of the globular proteins in water is a classical example. Change in the environmental conditions such as temperature, increase or decrease in pH, interaction of the solution dissolvers with protein molecule groups, gives rise to a change in the structure of the proteins. The entropy increment DS between the folding and unfolding states has a negative value in the cold folding region. This behavior shows that cold folding is realized by giving out heat and this leads to negative heat capacity; on the other hand, in the DS40 case the system is in the warm folding region and the heat capacity is positive [16]. The greatest contribution to the natural structure is provided by hydrophobic interactions. In an unfolding chain structure, hydrophobic amino acid side chains are unfolded by a regular water molecule lattice and the entropy increments of the system decreases. But when the chain undergoes folding and hydrophobic side chains are collected at intermolecular regions by getting away from water, the regular water molecules will be free and an increase in the total entropy increments of the system takes place. When the conditions return to their initial state these molecules return to their former structures. The folding and then unfolding of these molecules occur by a thermodynamical mechanism [26]. During the folding of the protein, entropy increment increases that is to say returning to the original structure of the protein is not preferred from the energy point of view. Therefore, there must be some factors which oppose the increase in entropy increments (i.e., folding) and encourage, on the other hand, the folding of the molecules. Since a covalent structure is conserved during unfolding, forces which oppose the unfolding of the protein could not be covalent interactions. Transformation of the non-polar molecules into water comes with a considerable decrease in entropy. The entropy increments of hydration of those non-polar groups must be temperature dependent since hydration leads to an important increase in the heat capacity [16]. The same process is expected for the enthalpy of hydration. In fact, if we look at the non-hydration effect during the transition of the non-polar components from gas phase to water we observe that entropy increments are negative and their magnitudes decrease with increasing temperature. Hence, one could consider that the hydration effect is responsible for the dissolution of the nonpolar compounds in water. References [1] [2] [3] [4] [5] [6] [7] [8]
A. Bakk, J.S. Hoye, A. Hansen, Physica A 304 (2002) 355–361. P.L. Privalov, J. Chem. Thermodyn. 29 (1997) 447–474. T.E. Creighton, Protein Folding, W.H. Freeman and Company, New York, 1995, pp. 83–151. P.L. Privalov, G.I. Makhatadze, J. Mol. Biol. 213 (1990) 385. T.E. Creighton, B.D. Homes, Protein Function, Wiley, New York, 1985. A. Hansen, H. Jensen, K. Sneppen, G. Zocchi, EPJ B6 (1998) 157–161. A. Hansen, H. Jensen, K. Sneppen, G. Zocchi, Physica A 270 (1999) 278–287. S. Abe, Y. Okamoto, Nonextensive Statistical Mechanics and Its Applications, Springer, Berlin, 2001, pp. 259–274.
ARTICLE IN PRESS 262 [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
G. Oylumluoglu et al. / Physica A 361 (2006) 255–262 H.B. Callen, Thermodynamics, second ed., Wiley, New York, 1960. A. Munster, Classical Thermodynamics, first ed., Wiley, New York, 1970. A. Warshel, M. Levitt, J. Mol. Biol. 103 (1976) 227–249. S.T. Russell, A. Warshel, J. Mol. Biol. 185 (1985) 389–404. Z.Z. Fan, J.K. Hwang, A. Warshel, Theor. Chem. Acc. 103 (1999) 77–80. F. Avbelj, J. Mol. Biol. 300 (2000) 1335–1359. H.S. Frank, M.W. Evans, J. Chem. Phys. 13 (1945) 507–532. G. Oylumluoglu, F. Buyukkilic, D. Demirhan, Phys. Lett. A, submitted for publication. A. Bakk, J.S. Hoye, A. Hansen, Biophys. J. 82 (2002) 713–719. W. Kauzmann, Protein Chem. 14 (1959) 1–63. G.I. Makhatadze, P.L. Privalov, Protein Chem. 47 (1995) 307–425. D. Eisenberg, A.D. McLachlan, Nature 319 (1986) 199–203. G.N. Phillips, B.M. Pettitt, Protein Sci. 4 (1995) 149–158. G.W. Robinson, C.H. Cho, Biophys. J. 77 (1999) 3311–3318. P.L. Privalov, Biochem. Mol. Biol. 25 (1990) 281–305. Y.V. Griko, P.L. Privalov, S.Y. Venyaminov, V.P. Kutyshenko, J. Mol. Biol. 202 (1988) 127–138. P. De Los Rios, G. Caldarelli, Phys. Rev. E. 62 (2000) 8449–8452. C.B. Anfinsen, Adv. Prot. Chem. 29 (1975) 205.
Physica A 361 (2006) 255–262 www.elsevier.com/locate/physa
Investigation of hydration effect of the proteins by phenomenological thermostatistical methods Gorkem Oylumluoglua,, Fevzi Bu¨yu¨kkılıc- b, Dog˘an Demirhanb a
Department of Physics, Faculty of Arts and Sciences, Mugla University, 48170-Kotekli, Mugla, Turkey b Department of Physics, Faculty of Science, Ege University, 35100-Bornova, I˙zmir, Turkey Received 21 February 2005; received in revised form 23 June 2005 Available online 11 August 2005
Abstract In this study variations with respect to temperature of entropy increment DS, arising in the dissolution of proteins in water, have been investigated by the methods of statistical thermodynamics. In this formalism, effective electric field E and total dipole moment M are taken as thermodynamical variables. The partition function given by Bakk et al. [Physica A 304 (2002) 355–361] has been used in a modified form while obtaining the free energy. In the constructed phenomenological theory, experimental data are taken from the study of Privalov [J. Chem. Thermodyn. 29 (1997) 447–474; J. Mol. Biol. 213 (1990) 385; T.E. Creighton, Protein Folding, W.H. Freeman and Company, New York, 1995, pp. 83–151] and relevant parameters have been found by fitting to experimental curves. The variations of entropy increments DS have been investigated in the temperature range 265–350 K. r 2005 Elsevier B.V. All rights reserved. Keywords: Thermodynamics; Classical statistical mechanics
1. Introduction Proteins is the common name of complex macromolecules which are formed by gathering up of a great number amino acids and which play unremitting roles in the Corresponding author.
E-mail address: [email protected] (G. Oylumluoglu). 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.07.004
ARTICLE IN PRESS 256
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lifetimes of all the living creatures. There are approximately 20 different amino acids in the structure of the proteins in the form of chain molecules. The order or arrangement of these 20 units is sensitive and definite. This order determines the nature of the protein and as a result identifies its function [5]. Each protein molecule is formed by an original combination and consecutive placing of different numbers of amino acids. Proteins, under their own natural physiological conditions, spontaneously open from one-dimensional structures to three-dimensional structures in time intervals starting from milliseconds and reaching to minutes [6,7]. The three-dimensional structures of the proteins depend on various environmental factors in the cell. Unfolded proteins return to their natural states when the environmental factors are removed exhibiting the feature that the three-dimensional structure of a protein solely depends on its amino acid ordering information. This situation gives the hope to some research workers that protein folding could be reconstructed by computer simulations [8]. But the difficulty of these simulations even for small proteins is in view. In this study, the model of the proteins is presented in Section 2. In Section 3, the thermodynamical quantities related to the variation of the additional entropy increment DS have been given and these parameters are related to the parameters of the proteins by using statistical thermodynamics. Finally, starting from the curves of the entropy increment, relevant conclusions on the protein structures are presented. The thermodynamical variables of fluids are represented by the physical quantities U (energy), V (volume) and N (particle number), whereas their corresponding parameters in the experimental measurements are T (temperature), P (pressure) and m (chemical potential) [9]. In a similar manner, for the magnetic systems the thermodynamical variables are represented by the physical quantities U (energy), H (magnetic field) and N (particle number) and their corresponding experimental parameters on T (temperature), M (magnetization) and m (chemical potential) [9,10]. As thermodynamical quantities of the proteins, internal energy (U), electric field (E) and particle number (N) could be taken as macroscopic variables. When the system tends to thermodynamical equilibrium, to demonstrate the connection with thermodynamics, the system is controlled by internal energy parameters namely temperature (T), electric field parameter i.e., dipole moment (M) and particle number parameter i.e., chemical potential (m).
2. Model For different aspects of the folding and unfolding transitions of the proteins, various models have been proposed. A simple yet notable model among them is the zipper model which defines the helix chain [6,7]. Another model where the solvent is represented as dipoles in the unfolding of proteins was given by Warshel and Levitt [11] and the applications of the model were presented by Russell [12], Fan [13] and Avbelj [14]. A simple model of protein unfolding has been developed by Bakk [1]. Bakk started by choosing water molecules
ARTICLE IN PRESS G. Oylumluoglu et al. / Physica A 361 (2006) 255–262
257
as classical electric dipoles. He demanded that non-polar dissolution should include important physical properties in modelling of the effect on the folding of the proteins. In the model, in the medium of the formed electric field, the folding and unfolding of the proteins have been taken as the basis. Here, the electric field is not an external field but has been used in the modelling of the ice-like behavior exhibited by the water molecules around the non-polar surfaces [15]. The electric field is a result of the effective behavior of the non-polar dissolvent on the protein unfolding. The modified partition function of the protein system could be taken as [16] h m i ZðT; E; mÞ ¼ exp exp zðT; E; N ¼ 1Þ . (1) kT Here, m represents the chemical potential and the term z ¼ expðbmÞ, which corresponds to the pH of the system has been introduced to the partition function. Single-particle partition function is zðT; E; N ¼ 1Þ ¼ 4p ebbm
2 =2
1 sinh be , be
(2)
where effective energy is given by e ¼ þ bhmi [1]. On the other hand, average dipole moment has the form hmi ¼ Lðbe Þ. One of the characteristics of the folding and unfolding of the proteins is that the temperature interval of the process depends on environmental conditions and particularly on the pH values of the solution. This task is supplied by the term z ¼ expðbmÞ in the partition function. The role of the electric field here is to model a kind of stiffness (ice-like behavior) that water molecules exhibit around non-polar surfaces. Thus, it is not a real external electric field, but merely an effective behavior upon non-polar solvation of protein folding.
3. Variations of the entropy increment with respect to temperature Taking electric field E and total dipole moment M as thermodynamical variables, the first law of thermodynamics could be written for the proteins: dU ¼ TdS MdE .
(3)
Entropy increment could also be written by taking the total differentials of entropy S ¼ SðT; MÞ: qS qS dS ¼ dT þ dM . (4) qT M qM T Writing down the expressions ðqS=qTÞM ¼ DC M =T in Eq. (4) and then integrating one could get for the entropy increment: Z T Z MðTÞ qS DS ¼ C M dT þ dM . (5) qM T Tt MðT t Þ
ARTICLE IN PRESS G. Oylumluoglu et al. / Physica A 361 (2006) 255–262
258
The free energy of the system has been obtained from the relation F ¼ kT ln Z giving F ¼
4 expðbð12 bm2 mÞÞp sinhðbe Þ b2 e
.
(6)
The thermodynamical quantities of the system has been determined using the entropy relation S ¼ ðqF =qTÞE which in turn leads to 2 expðbð12 bm2 mÞÞkp be
S¼
fð2e b coshðbe Þ þ ð4 þ bbm2 2bmÞ sinhðbe ÞÞg .
ð7Þ
The total dipole moment has been calculated from the expression M ¼ ðqF =qEÞT resulting in M¼
4 expðbð12 bm2 mÞÞp b2 2e
fe b coshðbe Þ sinhðbe Þg .
(8)
The energy of the system is given by U ¼ ð1=ZÞðqZ=qbÞ and when the partition function is substituted one obtains U¼
expðbð12 bm2 mÞÞ b2 e fð4pe b coshðbe Þ þ 2pð2 þ bbm2 2bmÞ sinhðbe ÞÞg .
ð9Þ
Additional heat capacity at effective field has been calculated by making use of the equation DC E ¼ kb2 ðqU=qbÞE which gives DC E ¼
expðbð12 bm2 mÞÞkp f8e b coshðbe Þ be ð4e bðbbm2 þ 2bmÞÞ coshðbe Þ þ ð8 þ bðb2 m4 b 8mÞÞ sinhðbe Þ b4bm2 ðbm 1Þ sinhðbe Þ þ 4b2 ð2e þ m2 Þ sinhðbe Þg .
ð10Þ
Using the expression DC M DC E ¼ TðqM=qEÞT ½ðqE=qTÞM 2 , which has been obtained for additional heat capacity at constant total dipole moment, DC M ¼
expðbð12 bm2 mÞÞkp f4e bð2 þ bbm2 2bmÞ be coshðbe Þ þ ½8 þ bðb2 m4 b 8m 4bm2 ðbm 1Þ þ 4bð2e þ m2 ÞÞ sinhðbe Þ
k 2 b2 ð2be coshðbe Þ þ ð2 þ b2 2e Þ sinhðbe ÞÞ
½e bð4bbm2 2bmÞ coshðbe Þ ð4 þ bðbm2 þ 2be 2mÞÞ sinhðbe Þ2 g
ð11Þ
ARTICLE IN PRESS G. Oylumluoglu et al. / Physica A 361 (2006) 255–262
has been calculated. ðqS=qMÞT in Eq. (5) is calculated as qS ¼ kbe . qM T
259
(12)
When Eqs. (11) and (12) are substituted in Eq. (5), entropy increment reads b expðbm2 ðT t þ TÞ=2kT t TÞ ðbm2 =TÞ þ ð2m=T t Þ DS ¼ 4kp T exp e t 2e 2k 2 e e bm ðT t þ TÞ e cosh þ kT t sinh þ p exp 2kT t T kT t kT t n e 2e cosh 2kTe þ e ðbð2k 1Þm2 kT e þ 2ð2k2 T þ m 2kðT þ mÞÞÞ sinh kT þð4k2 T 2 e þ e ðb2 km4 42e þ 2bkm2 ðð2k 1ÞT 2mÞ o þ4kð2k2 T 2 þ 2e 2kTðT þ mÞ þ mðT þ mÞÞÞðln T ln T t ÞÞÞ . ð13Þ Now the numerical applications of the entropy increment will be considered. The unfolding and then folding of the protein molecules take place by a thermodynamical mechanism. In other words, the original structures of the macromolecules are determined thermodynamically. In Fig. 1, symbols represent experimental study whereas solid lines theoretical results. The physical quantities , b, m and T t , which have been obtained in this semi-phenomenological theory, are determined by fitting to the P experimental results of Privalov [2–4]. The fitting parameter b is given by b ¼ ij J ij , where J ij is the coupling constant between each pair of water molecules.
Fig. 1. The variation of the additional entropy with temperature.
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260
Table 1 Numerical values obtained by fitting the parameters in Eq. (13) to the experimental data Proteins
Symbols
m ðkJ=molÞ
ðkJ=molÞ
b ðkJ=molÞ
k ! R ðJ=kmolÞ
T t ðKÞ
Mb RNase
Square Circle
185 1000
4.92 6.25
26 19.8
8.314 8.314
298 350
e ¼ þ bm is the magnitude (strength) of the effective field, m being average dipole moment. Another fitting parameter is the chemical potential m which is a measure of the strength of interactions with water molecules. m changes by denaturants, pH or with salt concentrations. The last fitting parameter is the transition temperature at which Gibbs energy difference between native and denatured states is equal to zero. Let us give a brief discussion of the differences of magnitudes of the parameters between myoglobin and ribonuclease in terms of fitted parameters; T t for ribonuclease is higher than T t for myoglobin which is a consequence of the corresponding thermodynamical energy levels of the two proteins. Since b is a measure of the coupling of the water molecule pairs which causes hydration, one could say that myoglobin has stronger hydration properties compared to ribonuclease. physically is a bending distortion constant which has a value slightly bigger for ribonuclease relative to the value of myoglobin. Thus one could conclude that ribonuclease has a higher stiffness property compared to myoglobin. In our calculations pH is exponentially related to m. In order to approach the pH values of the proteins under consideration, m which is very highly dependent on T is obtained as seen in the Table 1.
4. Conclusions The three-dimensional structure of the proteins depends on different environmental factors inside the cell. Unfolded proteins return to their natural state when the environmental factors are removed, in a manner of exhibiting the fact that the three-dimensional structure of a protein solely depends on the information of the amino acid regularity. The probable configurations of this three-dimensional structure are very large in order. In the thermodynamical limit (N ! 1, E ! 1 and N=E ¼ const:), it is not possible to make calculations by taking into consideration each particle in the macroscopic system formed by the proteins. Thus a combination of thermodynamics and statistical physics is a necessity. In the investigation of these types of systems with statistical physics, the existence of a great number of systems that are identical to the system under consideration are taken into account and using the statistical methods for these systems, the most probable state in which the actual system could enter is determined. Generally water seems to be important in the understanding of protein folding [3,17–22] and furthermore, the peculiar feature of cold unfolding seems to be particularly sensitive to the surrounding water [23–25].
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The unfolding of the globular proteins in water is a classical example. Change in the environmental conditions such as temperature, increase or decrease in pH, interaction of the solution dissolvers with protein molecule groups, gives rise to a change in the structure of the proteins. The entropy increment DS between the folding and unfolding states has a negative value in the cold folding region. This behavior shows that cold folding is realized by giving out heat and this leads to negative heat capacity; on the other hand, in the DS40 case the system is in the warm folding region and the heat capacity is positive [16]. The greatest contribution to the natural structure is provided by hydrophobic interactions. In an unfolding chain structure, hydrophobic amino acid side chains are unfolded by a regular water molecule lattice and the entropy increments of the system decreases. But when the chain undergoes folding and hydrophobic side chains are collected at intermolecular regions by getting away from water, the regular water molecules will be free and an increase in the total entropy increments of the system takes place. When the conditions return to their initial state these molecules return to their former structures. The folding and then unfolding of these molecules occur by a thermodynamical mechanism [26]. During the folding of the protein, entropy increment increases that is to say returning to the original structure of the protein is not preferred from the energy point of view. Therefore, there must be some factors which oppose the increase in entropy increments (i.e., folding) and encourage, on the other hand, the folding of the molecules. Since a covalent structure is conserved during unfolding, forces which oppose the unfolding of the protein could not be covalent interactions. Transformation of the non-polar molecules into water comes with a considerable decrease in entropy. The entropy increments of hydration of those non-polar groups must be temperature dependent since hydration leads to an important increase in the heat capacity [16]. The same process is expected for the enthalpy of hydration. In fact, if we look at the non-hydration effect during the transition of the non-polar components from gas phase to water we observe that entropy increments are negative and their magnitudes decrease with increasing temperature. Hence, one could consider that the hydration effect is responsible for the dissolution of the nonpolar compounds in water. References [1] [2] [3] [4] [5] [6] [7] [8]
A. Bakk, J.S. Hoye, A. Hansen, Physica A 304 (2002) 355–361. P.L. Privalov, J. Chem. Thermodyn. 29 (1997) 447–474. T.E. Creighton, Protein Folding, W.H. Freeman and Company, New York, 1995, pp. 83–151. P.L. Privalov, G.I. Makhatadze, J. Mol. Biol. 213 (1990) 385. T.E. Creighton, B.D. Homes, Protein Function, Wiley, New York, 1985. A. Hansen, H. Jensen, K. Sneppen, G. Zocchi, EPJ B6 (1998) 157–161. A. Hansen, H. Jensen, K. Sneppen, G. Zocchi, Physica A 270 (1999) 278–287. S. Abe, Y. Okamoto, Nonextensive Statistical Mechanics and Its Applications, Springer, Berlin, 2001, pp. 259–274.
ARTICLE IN PRESS 262 [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
G. Oylumluoglu et al. / Physica A 361 (2006) 255–262 H.B. Callen, Thermodynamics, second ed., Wiley, New York, 1960. A. Munster, Classical Thermodynamics, first ed., Wiley, New York, 1970. A. Warshel, M. Levitt, J. Mol. Biol. 103 (1976) 227–249. S.T. Russell, A. Warshel, J. Mol. Biol. 185 (1985) 389–404. Z.Z. Fan, J.K. Hwang, A. Warshel, Theor. Chem. Acc. 103 (1999) 77–80. F. Avbelj, J. Mol. Biol. 300 (2000) 1335–1359. H.S. Frank, M.W. Evans, J. Chem. Phys. 13 (1945) 507–532. G. Oylumluoglu, F. Buyukkilic, D. Demirhan, Phys. Lett. A, submitted for publication. A. Bakk, J.S. Hoye, A. Hansen, Biophys. J. 82 (2002) 713–719. W. Kauzmann, Protein Chem. 14 (1959) 1–63. G.I. Makhatadze, P.L. Privalov, Protein Chem. 47 (1995) 307–425. D. Eisenberg, A.D. McLachlan, Nature 319 (1986) 199–203. G.N. Phillips, B.M. Pettitt, Protein Sci. 4 (1995) 149–158. G.W. Robinson, C.H. Cho, Biophys. J. 77 (1999) 3311–3318. P.L. Privalov, Biochem. Mol. Biol. 25 (1990) 281–305. Y.V. Griko, P.L. Privalov, S.Y. Venyaminov, V.P. Kutyshenko, J. Mol. Biol. 202 (1988) 127–138. P. De Los Rios, G. Caldarelli, Phys. Rev. E. 62 (2000) 8449–8452. C.B. Anfinsen, Adv. Prot. Chem. 29 (1975) 205.