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International Journal of Biological Macromolecules journal homepage: www.elsevier.com/locate/ijbiomac
Determination of the glass-transition temperature of proteins from a viscometric approach
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Karol Monkos ∗ Department of Biophysics, Medical University of Silesia, H. Jordana 19, 41-808 Zabrze 8, Poland
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Article history: Received 11 March 2014 Received in revised form 29 August 2014 Accepted 27 November 2014 Available online xxx
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Keywords: Viscosity Albumin Avramov’s model Glass-transition temperature Gordon–Taylor equation
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1. Introduction
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All fully hydrated proteins undergo a distinct change in their dynamical properties at glass-transition temperature Tg . To determine indirectly this temperature for dry albumins, the viscosity measurements of aqueous solutions of human, equine, ovine, porcine and rabbit serum albumin have been conducted at a wide range of concentrations and at temperatures ranging from 278 K to 318 K. Viscosity–temperature dependence of the solutions is discussed on the basis of the three parameters equation resulting from Avramov’s model. One of the parameter in the Avramov’s equation is the glass-transition temperature. For all studied albumins, Tg of a solution monotonically increases with increasing concentration. The glasstransition temperature of a solution depends both on Tg for a dissolved dry protein Tg,p and water Tg,w . To obtain Tg,p for each studied albumin the modified Gordon–Taylor equation was applied. This equation describes the dependence of Tg of a solution on concentration, and Tg,p and a parameter depending on the strength of the protein–solvent interaction are the fitting parameters. Thus determined the glasstransition temperature for the studied dry albumins is in the range (215.4–245.5) K. © 2014 Published by Elsevier B.V.
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The temperature at which the properties of material change from liquid-like to solid-like is called glass-transition temperature Tg . In this temperature the viscosity exceeds a value of 1013 poise (or 1012 Ns/m2 ). The glass-transition behavior of synthetic polymers and molecular liquids is well known. Recently, the glass-transition behavior of biopolymers, including proteins and polysaccharides, has received increased attention [1–16]. Most of globular proteins have an ordered three-dimensional flexible structure. This structural, or conformational flexibility is a key component in proteins functions. The functions of proteins like for instance ligand binding, require some flexibility because all interactions lead to at least small rearrangements of atoms and, in consequence, to conformational changes. The energy of the protein is changed with change of its conformation, and like in glasses this can be described by the conformational energy landscape. According to the concept developed by Frauenfelder [17], a protein molecule can assume a very large number of nearly isoenergetic conformational substates, valleys in the protein energy landscape. Protein dynamics involve transitions between its conformational substates. As temperature decreases these transitions become increasingly slower, and at a
∗ Tel.: +48 32 2720142. E-mail address: [email protected]
certain temperature the protein becomes frozen in a specific substate. The freezing of transitions between different conformational substates involves a change in the thermal energy of the protein, and consequently, a change in the heat capacity. As a result, the protein undergoes a glass transition. In the vicinity of glass transition temperature the sharp change in temperature dependence of various physical properties, such as heat capacity, density, elastic modulus, etc. is observed. This allows experimental determination of Tg . The glass transition temperature in hydrated proteins has mainly been estimated by calorimetric and rheological measurements [10,15]. However, it is difficult to measure glass transition temperature experimentally for globular proteins in the dry state. In the present paper Tg for several dry mammalian serum albumins has been obtained from viscosity measurements of aqueous solutions of the albumins, the Avramov’s model and the modified Gordon–Taylor equation. Mammalian serum albumins are moderately large proteins, with nearly identical molecular mass of about 66.5 kDa [18]. Their primary structure is constituted by a single polypeptide chain of about 580 amino-acid residues. Albumins from different mammals exhibit high amino-acid sequence identity with each other [19]. However detail determination of amino-acid sequences for several of them showed some differences [19]. The differences in amino-acid sequences, in turn, cause some differences in the three-dimensional structure of the albumins and considerable differences in their physicochemical properties in solution. It was
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demonstrated by different experimental techniques such as dielectric spectroscopy [20], liquid chromatography [21], electrophoresis [22], viscometry [23–25], calorimetry and fluorescence anisotropy [26,27], circular dichroism [28] or fluorescence spectroscopy and modeling [29]. The study presents the results of viscosity measurements on aqueous solutions of human serum albumin (HSA), equine serum albumin (ESA), ovine serum albumin (OSA), porcine serum albumin (PSA) and rabbit serum albumin (RSA) at temperatures ranging from 278 K to 318 K and over a wide range of concentrations. For each albumin the viscosity-temperature dependence, for a fixed concentration, is analyzed on the basis of equation resulting from the Avramov’s model [30]. One of the parameter in this equation is the glass-transition temperature. It appears that the glass-transition temperature of a solution, for each studied albumin, increases with increasing concentration. To establish the glass-transition temperature of the studied dry albumins, in turn, a modified Gordon–Taylor formulae is applied.
2. Materials and methods The following products of the Sigma (USA) were used in the study: HSA at pH 7.0 (A 1653), ESA (A 9888), OSA (A 3264), PSA (A 2764) and RSA (A 0639). HSA at pH 4.7 was purchased from Polish Chemical Reagents factories. Albumins were used without further purification for all measurements. Aqueous solutions were prepared by dissolving the crystallized albumins in distilled water. To remove possible undissolved fragments the solutions were treated with filter papers. The samples were cooled in a refrigerator (up to 277 K) until just prior to viscometry measurements, when they were warmed from 278 K to 318 K. The pH values of thus prepared solutions were as follows: 7.0 or 4.7 for HSA, 7.4 for ESA, 7.05 for OSA, 6.6 for PSA and 7.0 for RSA. The isoelectric point pI of the studied albumins is: (4.7–4.95) for HSA, (4.65–4.9) for ESA, (4.6–4.9) for OSA, (4.6–4.9) for PSA and (4.6–5.3) for RSA [22]. The pH values of the solutions changed slightly in the whole range of concentrations. The above given values are the average pH. The viscosity measurements of albumins solutions were conducted by using an Ubbelohde-type capillary microviscometer with a flow time for water of 28.5 s at 298 K. It was placed in a waterbath controlled thermostatically with a precision of ±0.1 K and was mounted so that it always occupied the same position in the bath. Flow times were recorded to within 0.1 s. The microviscometer was calibrated using cooled boiled distilled water and the same microviscometer was used for all measurements. Measurements started after a few minutes delay to ensure that the system reached equilibrium. For each concentration, the solution was passed once through the microviscometer before any measurements were made. For most concentrations the viscosity measurements were taken from 278 K to 318 K mainly by steps of 5 K. At temperatures slightly higher than 318 K the thermal denaturation of the studied albumins occurs and the lower the protein concentration the higher the denaturation temperature. The viscosity of the studied albumins is discussed here in the mono-disperse range, i.e. from 8.2 kg/m3 up to 369 kg/m3 for HSA at pH 7.0, from 9.5 kg/m3 up to 328 kg/m3 for HSA at pH 4.7, from 13 kg/m3 up to 367 kg/m3 for ESA, from 36 kg/m3 up to 317 kg/m3 for OSA, from 34 kg/m3 up to 195 kg/m3 for PSA and from 14 kg/m3 up to 300 kg/m3 for RSA. For higher concentrations the aggregations of albumins occur and solutions become poly-disperse. The problem is discussed in detail elsewhere [23,24]. The above concentration ranges may be expressed by a suitable range of weight fractions as follows: from 0.0082 up to 0.34 for HSA at pH 7.0, from 0.0096 up to 0.306 for HSA at pH 4.7, from 0.0128 up to 0.333 for ESA, from 0.0359 up to 0.293 for OSA, from 0.0341 up to 0.185 for
PSA and from 0.0138 up to 0.277 for RSA. Solution densities were measured by weighing. Albumin concentrations were determined using a dry weight method in which the samples were dried at high temperature for several hours. 3. Results and discussion
(c, T ) = ∞ (c) exp
(c) T
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˛(c)
where ∞ (c), (c) and ˛(c) are concentration dependent parameters. To fit the viscosity from the Avramov’s relation to the experimental values of viscosity, the numerical values of the parameters ∞ (c), (c) and ˛(c) are needed. The calculations of these parameters were conducted by applying a non-linear regression procedure in the computational statistical program. Fig. 1 shows the results of viscosity measurements for HSA at pH 4.7, ESA and OSA aqueous solutions for high concentrations. The curve shows the fit to the experimental points according to relation (1), with the parameters obtained by the mentioned above method. As seen this function gives very good fit over the whole range of measured temperatures. For the smaller concentrations the situation is the same. This is also the case for the other albumins discussed here. In Avramov’s relation (1) the parameter (c) = Tg (c)ε1/˛(c) . The parameter Tg (c) denotes the glass transition temperature for a solution and the quantity ε means the ratio of the activation energy corresponding to its value at the maximum of the probability distribution function to a dispersity of the activation energy.
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One of the models of viscous flow for glass-forming systems is the Avramov’s model [30]. According to the model molecules in a flowing liquid jump from the holes formed by the nearest neighbors to one of the adjoining holes. During the jumps the molecule has to overcome some energy barrier which, in general, is different for different molecules. The frequency of the jumps decreases exponentially with increasing the energy barrier. The assumption that the jumps frequency follows a Poisson distribution allows calculation of the average jump frequency. It depends on a dispersity and a maximum value of the energy barrier. Moreover, in the model one assumes that viscosity of the liquid is inversely proportional to the average frequency of these jumps. As a final result the temperature dependence of liquid viscosity is obtained. For solutions, when viscosity depends both on temperature and concentration, this dependence can be written in the following way:
η [cP]
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Fig. 1. Temperature dependence of the viscosity of HSA at pH 4.7 (•), ESA () and OSA () aqueous solutions for concentrations: c = 328, 367 and 317 kg/m3 , respectively. The curves show the fit obtained by using Eq. (1) with the parameters: ∞ (c) = 2135 cP, (c) = 3058 K and ˛(c) = 9118 for HSA; ∞ (c) = 5761 cP, (c) = 3424 K and ˛(c) = 5302 for ESA; ∞ (c) = 318 cP, (c) = 3308 K and ˛(c) = 5233 for OSA.
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Tg =
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wp Tg,p + kww Tg,w wp + kww
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in which wp and ww are weight fractions of the protein and water, and Tg,p and Tg,w are glass-transition temperature of the dry protein and water, respectively. The parameter k is related to the strength of protein–water interaction, and as has been shown by Couchmann [33] it is equivalent to the ratio of the heat capacity changes at Tg of pure components. To apply the above relation for solutions some transformations are necessary. The weight fractions of the protein and water are: wp = mp /(mp + mw ) and ww = mw /(mp + mw ) where mp and mw are masses of the dissolved protein and water in a solution, respectively. Because the mole numbers of the dissolved protein and water: Np = mp /Mp and Nw = mw /Mw , where Mp and Mw denote their molecular masses then wp = Np Mp /(Np Mp + Nw Mw ) and ww = Nw Mw /(Np Mp + Nw Mw ). Taking into account that the molar fractions of the dissolved protein and water are: Xp = Np /(Np + Nw ) and Xw = Nw /(Np + Nw ) the relation (2) one can transform into the following form:
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Xp Mp Tg,p + kXw Mw Tg,w Xp Mp + kXw Mw
(3)
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As has been shown in earlier study [34], the molar fraction of the dissolved protein can express by the solution concentration in the following way:
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c w (Mp /Mw ) − c(w (Mp /Mw ) − 1)
(4)
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where w and are the water density and the effective specific volume of a protein, respectively, and c denotes the solution concentration in kg/m3 . The effective specific volume is a coefficient of proportionality between the effective molar volume and the molar mass of a macrosolute. Because the molar fraction of water Xw = 1 − Xp therefore the substitution of the above relation into Eq. (3) gives the final form of the glass-transition temperature for a solution:
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c(Tg,p − kw Tg,w ) + kw Tg,w c(1 − kw ) + kw
(5)
The effective specific volume of all studied albumins was obtained previously and it is equal to: 1.78 × 10−3 m3 /kg for HSA at pH 4.7, 2.15 × 10−3 m3 /kg for HSA at pH 7.0 [35], 2.32 × 10−3 m3 /kg for ESA, 3.84 × 10−3 m3 /kg for PSA, 1.98 × 10−3 m3 /kg for RSA [23] and 2.08 × 10−3 m3 /kg for OSA [24]. The glass-transition temperature for pure bulk water is well-known and its the most frequently cited value is Tg,w = 136 K [16,32,36–39]. It is obvious that the glasstransition temperature for a solution Tg (c) in the limit of zero concentration should be equal to the glass-transition temperature for water. It allows calculation of the parameter ε for the studied albumins. The numerical values of the parameter are presented in Table 1. In his original work [30] Avramov determined the values of ε for many inorganic liquids. He obtained values of ε in the range (25.1–35.1) with the average value (30.5 ± 2). As shown in Table 1 the values of ε obtained for the studied albumins lie in this range, or are very close to the lower limit of the values obtained by Avramov. The glass-transition temperature for a solution can be calculated by using the values of the parameters (c) and ˛(c) from Avramov’s relation and taking into account that (c) = Tg (c)ε1/˛(c) . Thus obtained values of Tg (c), for each studied albumin, are shown
Table 1 The numerical values of the parameters ε, Tg,p and k of the investigated albumins obtained from the fit of Eq. (5) to the experimental points. Albumin
ε
Tg,p [K]
HSA pH 4.7 HSA pH 7.0 ESA OSA PSA RSA
23.74 25.22 25.25 24.61 24.24 25.07
245.5 245.0 217.1 217.0 215.5 215.4
± ± ± ± ± ±
k 3.8 6.2 4.3 6.1 5.8 7.0
0.2616 1.473 1.435 0.7702 0.5707 0.6624
± ± ± ± ± ±
0.0294 0.154 0.159 0.1198 0.089 0.1102
Values of Tg,p and k are expressed as mean ± SE.
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c [kg/m3] Fig. 2. Plot of the glass-transition temperature Tg (c) of the HSA at pH 4.7 (•), OSA () and ESA () aqueous solutions versus concentration. The curves show the fit according to relation (5) with Tg,w = 136 K and the parameters: Tg,p = 245.5 K, k = 0.2616 and = 1.78 × 10−3 m3 /kg for HSA at pH 4.7; Tg,p = 217 K, k = 0.7702 and = 2.08 × 10−3 m3 /kg for OSA; Tg,p = 217.1 K, k = 1.435 and = 2.32 × 10−3 m3 /kg for ESA.
in Figs. 2 and 3. When the effective specific volume of albumin and the glass-transition temperature for water are known, the only unknown parameters in the relation (5) are Tg,p and a parameter k. They must be taken into account as adjustable parameters. The calculations of these parameters were conducted by applying a non-linear regression procedure in the computational statistical program. Thus obtained parameters are gathered in Table 1. As seen in Figs. 2 and 3 the function from relation (5) gives then good approximation to the values of Tg (c) obtained in the above described method. Various attempts to estimate the glass transition temperature in hydrated globular proteins indicate that the glass transition does
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A classical thermodynamic treatment for the glass-transition temperature of binary mixtures was given by Gordon and Taylor [31]. The authors derived an expression for Tg of mixtures as a function of pure component properties. The formalism can also be applied to blends and solutions [32]. For the mixtures of protein and water the Gordon–Taylor expression has the form:
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c [kg/m3] Fig. 3. Plot of the glass-transition temperature Tg (c) of the PSA (•), RSA () and HSA at pH 7 () aqueous solutions versus concentration. The curves show the fit according to relation (5) with Tg,w = 136 K and the parameters: Tg,p = 215.5 K, k = 0.5707 and = 3.84 × 10−3 m3 /kg for PSA; Tg,p = 215.4 K, k = 0.6624 and = 1.98 × 10−3 m3 /kg for RSA; Tg,p = 245 K, k = 1.473 and = 2.15 × 10−3 m3 /kg for HSA at pH 7.
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not occur at a single temperature or in a narrow range of temperature but occur over a broad temperature range [38]. For instance, analysis of viscoelestic properties of lysozyme, myoglobine and albumin suggests the existence of a broad glass transition for temperatures in the range from 130 to 240 K, depending on hydration level [40]. The glass transition in hydrated hemoglobin extends from about 150 K up to denaturation temperature [38]. Molecular dynamic simulations of the small DNA oligonucleotide duplex d(CGCGCG)2 in aqueous solution give a glass transition temperature at 223–234 K [41]. The onset temperature of the glass transition in hydrated proteins is almost independent of the hydration level whereas the width of the transition decreases with increased hydration level [10]. Some authors have pointed out that difficulties of exact determination of Tg in proteins are caused by their complex secondary and tertiary structure, which generates a gradual (and not sharp) increment in heat capacity during the glass transition [5]. The glass transition temperatures for studied albumins solutions obtained in this work (Figs. 2 and 3), are above the glass transition temperature of water (136 K) up to 195 K for HSA at pH 7.0, 216 K for HSA at pH 4.7, 186 K for ESA, 179 K for OSA, 181 K for PSA and 178 K for RSA. As shown, they are comprised within the range of the glass transition temperatures for the hydrated proteins mentioned above with literature. In addition, comparison of the above values with the results given in Table 1 shows that the glass transition temperature of albumin in a dry state is always higher than the glass transition temperature of these albumin solutions. Unfortunately, as far as I know, there is no data in the literature concerning the glass transition temperature of albumin in the dry state with which you might be able to compare. It should be noted that for some proteins the values of Tg obtained by different method can be quite different. For instance, for myoglobin Tg obtained from differential scanning calorimetry method lies in the range 190 K to 210 K [10], and obtained from molecular dynamics simulation is equal to 220 K [42]. So, a confrontation and comparison of the glass transition temperature for a given protein obtained from different methods is highly desirable. As seen in Table 1, the glass transition temperature of the HSA obtained by analyzing the solutions at the isoelectric point and the neutral pH is almost exactly the same. However, the values of the parameter k – which is related to the strength of protein–water interaction – in the Gordon–Taylor equation are, in this case, quite different. In our earlier paper [43] it is shown that such quantities as the activation energy and entropy of viscous flow, the intrinsic viscosity, the effective specific volume, the self-crowding factor and the Huggins coefficient depend on the solution pH. At the same time it has been shown that the conformation and stiffness of HSA molecules in solutions at the isoelectric point and at neutral pH are the same. This suggests that protein conformation and stiffness have a great influence on its glass transition temperature. It is well known that albumins derived from different species have similar (or identical) molecular weight and exhibit high amino acid sequence identity with each other. The amino acid sequence similarity among mammalian albumin cause that they have a similar (but not identical) conformations [23,24,35]. This, in turn, leads to the differentiation of their glass transition temperature. Knowledge of the glass transition temperature of the protein is important because Tg is an indicator of its thermostability [3]. The higher the glass transition temperature of the protein, the lesser protein is susceptible to unfolding and denaturation at lower temperatures. 4. Conclusions The viscosity of mammalian serum albumins aqueous solutions at temperatures from 278 K to 318 K and in a wide range of concentrations may be quantitatively described by the three parameters
equation obtained on the basis of Avramov’s model. One of these parameters is the glass-transition temperature Tg . These quantity for a solution increases nonlinearly with increasing concentration of dissolved protein. The analytical description of this dependence can be presented on the basis of a modified Gordon–Taylor relation. The glass transition temperature of dry dissolved protein can be determined from this relation. The obtained values of Tg for the studied dry albumins lie in the range (215.4–245.5) K. For each albumin glass transition temperature in the dry state is higher than the glass transition temperature of the albumin solutions. The above proposed method can be used to indirect determination of the glass-transition temperature of dry proteins, and this is very convenient and simple one. Acknowledgment This work was supported by the project of MUS: KNW-1- Q3 028/K/3/0. References [1] N. Grasmeijer, M. Stankovic, H. de Waard, H.W. Frijlink, W.L.J. Hinrichs, Biochim. Biophys. Acta 1834 (2013) 763–769. [2] D. Marsh, R. Bartucci, R. Guzzi, L. Sportelli, M. Esmann, Biochim. Biophys. Acta 1834 (2013) 1591–1595. [3] B.S. Khatkar, S. Barak, D. Mudgil, Int. J. Biol. Macromol. 53 (2013) 38–41. [4] B.C. Roughton, E.M. Topp, K.V. Camarda, Comput. Chem. Eng. 36 (2012) 208–216. [5] L. García, A. Cova, A.J. Sandoval, A.J. Müller, L.M. Carrasquel, Carbohydr. Polym. 87 (2012) 1375–1382. [6] A. Panagopoulou, A. Kyritsis, R. Sabater i Serra, J.L. Gómez Ribelles, N. Shinyashiki, P. Pissis, Biochim. Biophys. Acta 1814 (2011) 1984–1996. [7] L.T. Rodríguez Furlán, J. Lecot, A. Pérez Padilla, M.E. Campderrós, N. Zaritzky, J. Food Eng. 106 (2011) 74–79. [8] H.G. Hernández, S. Livings, J.M. Aguilera, A. Chiralt, Food Hydrocoll. 25 (2011) 1311–1318. [9] S. Khodadadi, A. Malkovskiy, A. Kisliuk, A.P. Sokolov, Biochim. Biophys. Acta 1804 (2010) 15–19. [10] H. Jansson, J. Swenson, Biochim. Biophys. Acta 1804 (2010) 20–26. [11] W. Doster, Eur. Biophys. J. 37 (2008) 591–602. [12] K. Kawai, T. Suzuki, M. Oguni, Biophys. J. 90 (2006) 1–7. [13] R. Parker, T.R. Noel, G.J. Brownsey, K. Laos, S.G. Ring, Biophys. J. 89 (2005) 1227–1236. [14] P.W. Fenimore, H. Frauenfelder, B.H. McMahon, R.D. Young, Proc. Natl. Acad. Sci. U.S.A. 101 (2004) 14408–14413. [15] G.J. Brownsey, T.R. Noel, R. Parker, S.G. Ring, Biophys. J. 85 (2003) 3943–3950. [16] M.M. Teeter, A. Yamano, B. Stec, U. Mohanty, Proc. Natl. Acad. Sci. U.S.A. 98 (2001) 11242–11247. [17] H. Frauenfelder, S.G. Sligar, P.G. Wolynes, Science 254 (1991) 1598–1603. [18] M. Dokal, D.C. Carter, F. Rüker, J. Biol. Chem. 274 (1999) 29303–29310. [19] J.X. Ho, E.W. Holowachuk, E.J. Norton, P.D. Twigg, D.C. Carter, Eur. J. Biochem. 215 (1993) 205–212. [20] P. Moser, P.G. Squire, C.T. O’Konski, J. Phys. Chem. 70 (1966) 744–756. ˇ B. Sebille, Chirality 9 (1997) 373–379. [21] L. Soltés, [22] I. Miller, M. Gemeiner, Electrophoresis 19 (1998) 2506–2514. [23] K. Monkos, Biochim. Biophys. Acta 1748 (2005) 100–109. [24] K. Monkos, J. Biol. Phys. 31 (2005) 219–232. [25] K. Monkos, Gen. Physiol. Biophys. 30 (2011) 121–129. [26] M.N. Dimitrova, H. Matsumura, A. Dimitrova, V.Z. Neitchev, Int. J. Biol. Macromol. 27 (2000) 187–194. [27] S. Ercelen, A.S. Klymchenko, Y. Mély, A.P. Demchenko, Int. J. Biol. Macromol. 35 (2005) 231–242. [28] R.H. Khan, M.S. Shabnum, Biochemistry (Moscow) 66 (2001) 1280–1285. [29] E.L. Gelamo, C.H.T.P. Silva, H. Imasato, M. Tabak, Biochim. Biophys. Acta 1594 (2002) 84–99. [30] I. Avramov, J. Non-Cryst. Solids 238 (1998) 6–10. [31] M. Gordon, J.S. Taylor, J. Appl. Chem. 2 (1952) 493–499. [32] I.I. Katkov, F. Levine, Cryobiology 49 (2004) 62–82. [33] P.R. Couchmann, Macromolecules 11 (1978) 117–119. [34] K. Monkos, Int. J. Biol. Macromol. 18 (1996) 61–68. [35] K. Monkos, Biochim. Biophys. Acta 1700 (2004) 27–34. [36] G.P. Johari, A. Hallbrucker, E. Mayer, Nature 330 (1987) 552–553. [37] A. Hallbrucker, E. Mayer, G.P. Johari, Philos. Mag. 60B (1989) 179–187. [38] G. Sartor, E. Mayer, G.P. Johari, Biophys. J. 66 (1994) 249–258. [39] G. Sartor, A. Hallbrucker, E. Mayer, Biophys. J. 69 (1995) 2679–2694. [40] V.N. Morozov, S.G. Gevorkian, Biopolymers 24 (1985) 1785–1799. [41] D. Ringe, G.A. Petsko, Biophys. Chem. 105 (2003) 667–680. [42] P.J. Steinbach, B.R. Brooks, Proc. Natl. Acad. Sci. U.S.A. 90 (1993) 9135–9139. [43] K. Monkos, Gen. Physiol. Biophys. 32 (2013) 67–78.
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International Journal of Biological Macromolecules xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
International Journal of Biological Macromolecules journal homepage: www.elsevier.com/locate/ijbiomac
Determination of the glass-transition temperature of proteins from a viscometric approach
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Karol Monkos ∗ Department of Biophysics, Medical University of Silesia, H. Jordana 19, 41-808 Zabrze 8, Poland
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Article history: Received 11 March 2014 Received in revised form 29 August 2014 Accepted 27 November 2014 Available online xxx
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Keywords: Viscosity Albumin Avramov’s model Glass-transition temperature Gordon–Taylor equation
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1. Introduction
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All fully hydrated proteins undergo a distinct change in their dynamical properties at glass-transition temperature Tg . To determine indirectly this temperature for dry albumins, the viscosity measurements of aqueous solutions of human, equine, ovine, porcine and rabbit serum albumin have been conducted at a wide range of concentrations and at temperatures ranging from 278 K to 318 K. Viscosity–temperature dependence of the solutions is discussed on the basis of the three parameters equation resulting from Avramov’s model. One of the parameter in the Avramov’s equation is the glass-transition temperature. For all studied albumins, Tg of a solution monotonically increases with increasing concentration. The glasstransition temperature of a solution depends both on Tg for a dissolved dry protein Tg,p and water Tg,w . To obtain Tg,p for each studied albumin the modified Gordon–Taylor equation was applied. This equation describes the dependence of Tg of a solution on concentration, and Tg,p and a parameter depending on the strength of the protein–solvent interaction are the fitting parameters. Thus determined the glasstransition temperature for the studied dry albumins is in the range (215.4–245.5) K. © 2014 Published by Elsevier B.V.
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The temperature at which the properties of material change from liquid-like to solid-like is called glass-transition temperature Tg . In this temperature the viscosity exceeds a value of 1013 poise (or 1012 Ns/m2 ). The glass-transition behavior of synthetic polymers and molecular liquids is well known. Recently, the glass-transition behavior of biopolymers, including proteins and polysaccharides, has received increased attention [1–16]. Most of globular proteins have an ordered three-dimensional flexible structure. This structural, or conformational flexibility is a key component in proteins functions. The functions of proteins like for instance ligand binding, require some flexibility because all interactions lead to at least small rearrangements of atoms and, in consequence, to conformational changes. The energy of the protein is changed with change of its conformation, and like in glasses this can be described by the conformational energy landscape. According to the concept developed by Frauenfelder [17], a protein molecule can assume a very large number of nearly isoenergetic conformational substates, valleys in the protein energy landscape. Protein dynamics involve transitions between its conformational substates. As temperature decreases these transitions become increasingly slower, and at a
∗ Tel.: +48 32 2720142. E-mail address: [email protected]
certain temperature the protein becomes frozen in a specific substate. The freezing of transitions between different conformational substates involves a change in the thermal energy of the protein, and consequently, a change in the heat capacity. As a result, the protein undergoes a glass transition. In the vicinity of glass transition temperature the sharp change in temperature dependence of various physical properties, such as heat capacity, density, elastic modulus, etc. is observed. This allows experimental determination of Tg . The glass transition temperature in hydrated proteins has mainly been estimated by calorimetric and rheological measurements [10,15]. However, it is difficult to measure glass transition temperature experimentally for globular proteins in the dry state. In the present paper Tg for several dry mammalian serum albumins has been obtained from viscosity measurements of aqueous solutions of the albumins, the Avramov’s model and the modified Gordon–Taylor equation. Mammalian serum albumins are moderately large proteins, with nearly identical molecular mass of about 66.5 kDa [18]. Their primary structure is constituted by a single polypeptide chain of about 580 amino-acid residues. Albumins from different mammals exhibit high amino-acid sequence identity with each other [19]. However detail determination of amino-acid sequences for several of them showed some differences [19]. The differences in amino-acid sequences, in turn, cause some differences in the three-dimensional structure of the albumins and considerable differences in their physicochemical properties in solution. It was
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demonstrated by different experimental techniques such as dielectric spectroscopy [20], liquid chromatography [21], electrophoresis [22], viscometry [23–25], calorimetry and fluorescence anisotropy [26,27], circular dichroism [28] or fluorescence spectroscopy and modeling [29]. The study presents the results of viscosity measurements on aqueous solutions of human serum albumin (HSA), equine serum albumin (ESA), ovine serum albumin (OSA), porcine serum albumin (PSA) and rabbit serum albumin (RSA) at temperatures ranging from 278 K to 318 K and over a wide range of concentrations. For each albumin the viscosity-temperature dependence, for a fixed concentration, is analyzed on the basis of equation resulting from the Avramov’s model [30]. One of the parameter in this equation is the glass-transition temperature. It appears that the glass-transition temperature of a solution, for each studied albumin, increases with increasing concentration. To establish the glass-transition temperature of the studied dry albumins, in turn, a modified Gordon–Taylor formulae is applied.
2. Materials and methods The following products of the Sigma (USA) were used in the study: HSA at pH 7.0 (A 1653), ESA (A 9888), OSA (A 3264), PSA (A 2764) and RSA (A 0639). HSA at pH 4.7 was purchased from Polish Chemical Reagents factories. Albumins were used without further purification for all measurements. Aqueous solutions were prepared by dissolving the crystallized albumins in distilled water. To remove possible undissolved fragments the solutions were treated with filter papers. The samples were cooled in a refrigerator (up to 277 K) until just prior to viscometry measurements, when they were warmed from 278 K to 318 K. The pH values of thus prepared solutions were as follows: 7.0 or 4.7 for HSA, 7.4 for ESA, 7.05 for OSA, 6.6 for PSA and 7.0 for RSA. The isoelectric point pI of the studied albumins is: (4.7–4.95) for HSA, (4.65–4.9) for ESA, (4.6–4.9) for OSA, (4.6–4.9) for PSA and (4.6–5.3) for RSA [22]. The pH values of the solutions changed slightly in the whole range of concentrations. The above given values are the average pH. The viscosity measurements of albumins solutions were conducted by using an Ubbelohde-type capillary microviscometer with a flow time for water of 28.5 s at 298 K. It was placed in a waterbath controlled thermostatically with a precision of ±0.1 K and was mounted so that it always occupied the same position in the bath. Flow times were recorded to within 0.1 s. The microviscometer was calibrated using cooled boiled distilled water and the same microviscometer was used for all measurements. Measurements started after a few minutes delay to ensure that the system reached equilibrium. For each concentration, the solution was passed once through the microviscometer before any measurements were made. For most concentrations the viscosity measurements were taken from 278 K to 318 K mainly by steps of 5 K. At temperatures slightly higher than 318 K the thermal denaturation of the studied albumins occurs and the lower the protein concentration the higher the denaturation temperature. The viscosity of the studied albumins is discussed here in the mono-disperse range, i.e. from 8.2 kg/m3 up to 369 kg/m3 for HSA at pH 7.0, from 9.5 kg/m3 up to 328 kg/m3 for HSA at pH 4.7, from 13 kg/m3 up to 367 kg/m3 for ESA, from 36 kg/m3 up to 317 kg/m3 for OSA, from 34 kg/m3 up to 195 kg/m3 for PSA and from 14 kg/m3 up to 300 kg/m3 for RSA. For higher concentrations the aggregations of albumins occur and solutions become poly-disperse. The problem is discussed in detail elsewhere [23,24]. The above concentration ranges may be expressed by a suitable range of weight fractions as follows: from 0.0082 up to 0.34 for HSA at pH 7.0, from 0.0096 up to 0.306 for HSA at pH 4.7, from 0.0128 up to 0.333 for ESA, from 0.0359 up to 0.293 for OSA, from 0.0341 up to 0.185 for
PSA and from 0.0138 up to 0.277 for RSA. Solution densities were measured by weighing. Albumin concentrations were determined using a dry weight method in which the samples were dried at high temperature for several hours. 3. Results and discussion
(c, T ) = ∞ (c) exp
(c) T
(1)
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˛(c)
where ∞ (c), (c) and ˛(c) are concentration dependent parameters. To fit the viscosity from the Avramov’s relation to the experimental values of viscosity, the numerical values of the parameters ∞ (c), (c) and ˛(c) are needed. The calculations of these parameters were conducted by applying a non-linear regression procedure in the computational statistical program. Fig. 1 shows the results of viscosity measurements for HSA at pH 4.7, ESA and OSA aqueous solutions for high concentrations. The curve shows the fit to the experimental points according to relation (1), with the parameters obtained by the mentioned above method. As seen this function gives very good fit over the whole range of measured temperatures. For the smaller concentrations the situation is the same. This is also the case for the other albumins discussed here. In Avramov’s relation (1) the parameter (c) = Tg (c)ε1/˛(c) . The parameter Tg (c) denotes the glass transition temperature for a solution and the quantity ε means the ratio of the activation energy corresponding to its value at the maximum of the probability distribution function to a dispersity of the activation energy.
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One of the models of viscous flow for glass-forming systems is the Avramov’s model [30]. According to the model molecules in a flowing liquid jump from the holes formed by the nearest neighbors to one of the adjoining holes. During the jumps the molecule has to overcome some energy barrier which, in general, is different for different molecules. The frequency of the jumps decreases exponentially with increasing the energy barrier. The assumption that the jumps frequency follows a Poisson distribution allows calculation of the average jump frequency. It depends on a dispersity and a maximum value of the energy barrier. Moreover, in the model one assumes that viscosity of the liquid is inversely proportional to the average frequency of these jumps. As a final result the temperature dependence of liquid viscosity is obtained. For solutions, when viscosity depends both on temperature and concentration, this dependence can be written in the following way:
η [cP]
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t [C]
Fig. 1. Temperature dependence of the viscosity of HSA at pH 4.7 (•), ESA () and OSA () aqueous solutions for concentrations: c = 328, 367 and 317 kg/m3 , respectively. The curves show the fit obtained by using Eq. (1) with the parameters: ∞ (c) = 2135 cP, (c) = 3058 K and ˛(c) = 9118 for HSA; ∞ (c) = 5761 cP, (c) = 3424 K and ˛(c) = 5302 for ESA; ∞ (c) = 318 cP, (c) = 3308 K and ˛(c) = 5233 for OSA.
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Tg =
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wp Tg,p + kww Tg,w wp + kww
(2)
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in which wp and ww are weight fractions of the protein and water, and Tg,p and Tg,w are glass-transition temperature of the dry protein and water, respectively. The parameter k is related to the strength of protein–water interaction, and as has been shown by Couchmann [33] it is equivalent to the ratio of the heat capacity changes at Tg of pure components. To apply the above relation for solutions some transformations are necessary. The weight fractions of the protein and water are: wp = mp /(mp + mw ) and ww = mw /(mp + mw ) where mp and mw are masses of the dissolved protein and water in a solution, respectively. Because the mole numbers of the dissolved protein and water: Np = mp /Mp and Nw = mw /Mw , where Mp and Mw denote their molecular masses then wp = Np Mp /(Np Mp + Nw Mw ) and ww = Nw Mw /(Np Mp + Nw Mw ). Taking into account that the molar fractions of the dissolved protein and water are: Xp = Np /(Np + Nw ) and Xw = Nw /(Np + Nw ) the relation (2) one can transform into the following form:
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Xp Mp Tg,p + kXw Mw Tg,w Xp Mp + kXw Mw
(3)
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As has been shown in earlier study [34], the molar fraction of the dissolved protein can express by the solution concentration in the following way:
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c w (Mp /Mw ) − c(w (Mp /Mw ) − 1)
(4)
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where w and are the water density and the effective specific volume of a protein, respectively, and c denotes the solution concentration in kg/m3 . The effective specific volume is a coefficient of proportionality between the effective molar volume and the molar mass of a macrosolute. Because the molar fraction of water Xw = 1 − Xp therefore the substitution of the above relation into Eq. (3) gives the final form of the glass-transition temperature for a solution:
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c(Tg,p − kw Tg,w ) + kw Tg,w c(1 − kw ) + kw
(5)
The effective specific volume of all studied albumins was obtained previously and it is equal to: 1.78 × 10−3 m3 /kg for HSA at pH 4.7, 2.15 × 10−3 m3 /kg for HSA at pH 7.0 [35], 2.32 × 10−3 m3 /kg for ESA, 3.84 × 10−3 m3 /kg for PSA, 1.98 × 10−3 m3 /kg for RSA [23] and 2.08 × 10−3 m3 /kg for OSA [24]. The glass-transition temperature for pure bulk water is well-known and its the most frequently cited value is Tg,w = 136 K [16,32,36–39]. It is obvious that the glasstransition temperature for a solution Tg (c) in the limit of zero concentration should be equal to the glass-transition temperature for water. It allows calculation of the parameter ε for the studied albumins. The numerical values of the parameter are presented in Table 1. In his original work [30] Avramov determined the values of ε for many inorganic liquids. He obtained values of ε in the range (25.1–35.1) with the average value (30.5 ± 2). As shown in Table 1 the values of ε obtained for the studied albumins lie in this range, or are very close to the lower limit of the values obtained by Avramov. The glass-transition temperature for a solution can be calculated by using the values of the parameters (c) and ˛(c) from Avramov’s relation and taking into account that (c) = Tg (c)ε1/˛(c) . Thus obtained values of Tg (c), for each studied albumin, are shown
Table 1 The numerical values of the parameters ε, Tg,p and k of the investigated albumins obtained from the fit of Eq. (5) to the experimental points. Albumin
ε
Tg,p [K]
HSA pH 4.7 HSA pH 7.0 ESA OSA PSA RSA
23.74 25.22 25.25 24.61 24.24 25.07
245.5 245.0 217.1 217.0 215.5 215.4
± ± ± ± ± ±
k 3.8 6.2 4.3 6.1 5.8 7.0
0.2616 1.473 1.435 0.7702 0.5707 0.6624
± ± ± ± ± ±
0.0294 0.154 0.159 0.1198 0.089 0.1102
Values of Tg,p and k are expressed as mean ± SE.
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c [kg/m3] Fig. 2. Plot of the glass-transition temperature Tg (c) of the HSA at pH 4.7 (•), OSA () and ESA () aqueous solutions versus concentration. The curves show the fit according to relation (5) with Tg,w = 136 K and the parameters: Tg,p = 245.5 K, k = 0.2616 and = 1.78 × 10−3 m3 /kg for HSA at pH 4.7; Tg,p = 217 K, k = 0.7702 and = 2.08 × 10−3 m3 /kg for OSA; Tg,p = 217.1 K, k = 1.435 and = 2.32 × 10−3 m3 /kg for ESA.
in Figs. 2 and 3. When the effective specific volume of albumin and the glass-transition temperature for water are known, the only unknown parameters in the relation (5) are Tg,p and a parameter k. They must be taken into account as adjustable parameters. The calculations of these parameters were conducted by applying a non-linear regression procedure in the computational statistical program. Thus obtained parameters are gathered in Table 1. As seen in Figs. 2 and 3 the function from relation (5) gives then good approximation to the values of Tg (c) obtained in the above described method. Various attempts to estimate the glass transition temperature in hydrated globular proteins indicate that the glass transition does
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A classical thermodynamic treatment for the glass-transition temperature of binary mixtures was given by Gordon and Taylor [31]. The authors derived an expression for Tg of mixtures as a function of pure component properties. The formalism can also be applied to blends and solutions [32]. For the mixtures of protein and water the Gordon–Taylor expression has the form:
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c [kg/m3] Fig. 3. Plot of the glass-transition temperature Tg (c) of the PSA (•), RSA () and HSA at pH 7 () aqueous solutions versus concentration. The curves show the fit according to relation (5) with Tg,w = 136 K and the parameters: Tg,p = 215.5 K, k = 0.5707 and = 3.84 × 10−3 m3 /kg for PSA; Tg,p = 215.4 K, k = 0.6624 and = 1.98 × 10−3 m3 /kg for RSA; Tg,p = 245 K, k = 1.473 and = 2.15 × 10−3 m3 /kg for HSA at pH 7.
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not occur at a single temperature or in a narrow range of temperature but occur over a broad temperature range [38]. For instance, analysis of viscoelestic properties of lysozyme, myoglobine and albumin suggests the existence of a broad glass transition for temperatures in the range from 130 to 240 K, depending on hydration level [40]. The glass transition in hydrated hemoglobin extends from about 150 K up to denaturation temperature [38]. Molecular dynamic simulations of the small DNA oligonucleotide duplex d(CGCGCG)2 in aqueous solution give a glass transition temperature at 223–234 K [41]. The onset temperature of the glass transition in hydrated proteins is almost independent of the hydration level whereas the width of the transition decreases with increased hydration level [10]. Some authors have pointed out that difficulties of exact determination of Tg in proteins are caused by their complex secondary and tertiary structure, which generates a gradual (and not sharp) increment in heat capacity during the glass transition [5]. The glass transition temperatures for studied albumins solutions obtained in this work (Figs. 2 and 3), are above the glass transition temperature of water (136 K) up to 195 K for HSA at pH 7.0, 216 K for HSA at pH 4.7, 186 K for ESA, 179 K for OSA, 181 K for PSA and 178 K for RSA. As shown, they are comprised within the range of the glass transition temperatures for the hydrated proteins mentioned above with literature. In addition, comparison of the above values with the results given in Table 1 shows that the glass transition temperature of albumin in a dry state is always higher than the glass transition temperature of these albumin solutions. Unfortunately, as far as I know, there is no data in the literature concerning the glass transition temperature of albumin in the dry state with which you might be able to compare. It should be noted that for some proteins the values of Tg obtained by different method can be quite different. For instance, for myoglobin Tg obtained from differential scanning calorimetry method lies in the range 190 K to 210 K [10], and obtained from molecular dynamics simulation is equal to 220 K [42]. So, a confrontation and comparison of the glass transition temperature for a given protein obtained from different methods is highly desirable. As seen in Table 1, the glass transition temperature of the HSA obtained by analyzing the solutions at the isoelectric point and the neutral pH is almost exactly the same. However, the values of the parameter k – which is related to the strength of protein–water interaction – in the Gordon–Taylor equation are, in this case, quite different. In our earlier paper [43] it is shown that such quantities as the activation energy and entropy of viscous flow, the intrinsic viscosity, the effective specific volume, the self-crowding factor and the Huggins coefficient depend on the solution pH. At the same time it has been shown that the conformation and stiffness of HSA molecules in solutions at the isoelectric point and at neutral pH are the same. This suggests that protein conformation and stiffness have a great influence on its glass transition temperature. It is well known that albumins derived from different species have similar (or identical) molecular weight and exhibit high amino acid sequence identity with each other. The amino acid sequence similarity among mammalian albumin cause that they have a similar (but not identical) conformations [23,24,35]. This, in turn, leads to the differentiation of their glass transition temperature. Knowledge of the glass transition temperature of the protein is important because Tg is an indicator of its thermostability [3]. The higher the glass transition temperature of the protein, the lesser protein is susceptible to unfolding and denaturation at lower temperatures. 4. Conclusions The viscosity of mammalian serum albumins aqueous solutions at temperatures from 278 K to 318 K and in a wide range of concentrations may be quantitatively described by the three parameters
equation obtained on the basis of Avramov’s model. One of these parameters is the glass-transition temperature Tg . These quantity for a solution increases nonlinearly with increasing concentration of dissolved protein. The analytical description of this dependence can be presented on the basis of a modified Gordon–Taylor relation. The glass transition temperature of dry dissolved protein can be determined from this relation. The obtained values of Tg for the studied dry albumins lie in the range (215.4–245.5) K. For each albumin glass transition temperature in the dry state is higher than the glass transition temperature of the albumin solutions. The above proposed method can be used to indirect determination of the glass-transition temperature of dry proteins, and this is very convenient and simple one. Acknowledgment This work was supported by the project of MUS: KNW-1- Q3 028/K/3/0. References [1] N. Grasmeijer, M. Stankovic, H. de Waard, H.W. Frijlink, W.L.J. Hinrichs, Biochim. Biophys. Acta 1834 (2013) 763–769. [2] D. Marsh, R. Bartucci, R. Guzzi, L. Sportelli, M. Esmann, Biochim. Biophys. Acta 1834 (2013) 1591–1595. [3] B.S. Khatkar, S. Barak, D. Mudgil, Int. J. Biol. Macromol. 53 (2013) 38–41. [4] B.C. Roughton, E.M. Topp, K.V. Camarda, Comput. Chem. Eng. 36 (2012) 208–216. [5] L. García, A. Cova, A.J. Sandoval, A.J. Müller, L.M. Carrasquel, Carbohydr. Polym. 87 (2012) 1375–1382. [6] A. Panagopoulou, A. Kyritsis, R. Sabater i Serra, J.L. Gómez Ribelles, N. Shinyashiki, P. Pissis, Biochim. Biophys. Acta 1814 (2011) 1984–1996. [7] L.T. Rodríguez Furlán, J. Lecot, A. Pérez Padilla, M.E. Campderrós, N. Zaritzky, J. Food Eng. 106 (2011) 74–79. [8] H.G. Hernández, S. Livings, J.M. Aguilera, A. Chiralt, Food Hydrocoll. 25 (2011) 1311–1318. [9] S. Khodadadi, A. Malkovskiy, A. Kisliuk, A.P. Sokolov, Biochim. Biophys. Acta 1804 (2010) 15–19. [10] H. Jansson, J. Swenson, Biochim. Biophys. Acta 1804 (2010) 20–26. [11] W. Doster, Eur. Biophys. J. 37 (2008) 591–602. [12] K. Kawai, T. Suzuki, M. Oguni, Biophys. J. 90 (2006) 1–7. [13] R. Parker, T.R. Noel, G.J. Brownsey, K. Laos, S.G. Ring, Biophys. J. 89 (2005) 1227–1236. [14] P.W. Fenimore, H. Frauenfelder, B.H. McMahon, R.D. Young, Proc. Natl. Acad. Sci. U.S.A. 101 (2004) 14408–14413. [15] G.J. Brownsey, T.R. Noel, R. Parker, S.G. Ring, Biophys. J. 85 (2003) 3943–3950. [16] M.M. Teeter, A. Yamano, B. Stec, U. Mohanty, Proc. Natl. Acad. Sci. U.S.A. 98 (2001) 11242–11247. [17] H. Frauenfelder, S.G. Sligar, P.G. Wolynes, Science 254 (1991) 1598–1603. [18] M. Dokal, D.C. Carter, F. Rüker, J. Biol. Chem. 274 (1999) 29303–29310. [19] J.X. Ho, E.W. Holowachuk, E.J. Norton, P.D. Twigg, D.C. Carter, Eur. J. Biochem. 215 (1993) 205–212. [20] P. Moser, P.G. Squire, C.T. O’Konski, J. Phys. Chem. 70 (1966) 744–756. ˇ B. Sebille, Chirality 9 (1997) 373–379. [21] L. Soltés, [22] I. Miller, M. Gemeiner, Electrophoresis 19 (1998) 2506–2514. [23] K. Monkos, Biochim. Biophys. Acta 1748 (2005) 100–109. [24] K. Monkos, J. Biol. Phys. 31 (2005) 219–232. [25] K. Monkos, Gen. Physiol. Biophys. 30 (2011) 121–129. [26] M.N. Dimitrova, H. Matsumura, A. Dimitrova, V.Z. Neitchev, Int. J. Biol. Macromol. 27 (2000) 187–194. [27] S. Ercelen, A.S. Klymchenko, Y. Mély, A.P. Demchenko, Int. J. Biol. Macromol. 35 (2005) 231–242. [28] R.H. Khan, M.S. Shabnum, Biochemistry (Moscow) 66 (2001) 1280–1285. [29] E.L. Gelamo, C.H.T.P. Silva, H. Imasato, M. Tabak, Biochim. Biophys. Acta 1594 (2002) 84–99. [30] I. Avramov, J. Non-Cryst. Solids 238 (1998) 6–10. [31] M. Gordon, J.S. Taylor, J. Appl. Chem. 2 (1952) 493–499. [32] I.I. Katkov, F. Levine, Cryobiology 49 (2004) 62–82. [33] P.R. Couchmann, Macromolecules 11 (1978) 117–119. [34] K. Monkos, Int. J. Biol. Macromol. 18 (1996) 61–68. [35] K. Monkos, Biochim. Biophys. Acta 1700 (2004) 27–34. [36] G.P. Johari, A. Hallbrucker, E. Mayer, Nature 330 (1987) 552–553. [37] A. Hallbrucker, E. Mayer, G.P. Johari, Philos. Mag. 60B (1989) 179–187. [38] G. Sartor, E. Mayer, G.P. Johari, Biophys. J. 66 (1994) 249–258. [39] G. Sartor, A. Hallbrucker, E. Mayer, Biophys. J. 69 (1995) 2679–2694. [40] V.N. Morozov, S.G. Gevorkian, Biopolymers 24 (1985) 1785–1799. [41] D. Ringe, G.A. Petsko, Biophys. Chem. 105 (2003) 667–680. [42] P.J. Steinbach, B.R. Brooks, Proc. Natl. Acad. Sci. U.S.A. 90 (1993) 9135–9139. [43] K. Monkos, Gen. Physiol. Biophys. 32 (2013) 67–78.
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