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Dynamic protein clusterization in supercritical region of the phase diagram of water–protein–salt solutions
J. of Supercritical Fluids 95 (2014) 68–74
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Dynamic protein clusterization in supercritical region of the phase diagram of water–protein–salt solutions S.P. Rozhkov ∗ , A.S. Goryunov Institute of Biology, Federal State Budget Institution, Karelian Research Centre, RAS, Pushkinskaya St., 11, Petrozavodsk, Russia
a r t i c l e
i n f o
Article history: Received 13 May 2014 Received in revised form 28 July 2014 Accepted 31 July 2014 Available online 9 August 2014 Keywords: Protein clusters Phase diagrams Thermodynamic stability (Quasi)Spinodal Critical point
a b s t r a c t A protein solution phase diagram, constructed on the basis of the thermodynamic description of protein–solvent interaction, is discussed. A thermodynamic mechanism of formation of protein clustersoligomers and mesoscopic clusters in the supercritical region of a protein solution phase diagram, is considered. Variation in the chemical potential of water, which depends on solution composition, is the parameter which describes the interaction. Equations for the spinodal and critical points of the phase diagram, in which the critical composition (protein to salt concentration ratio) of the system is related to the physico-chemical characteristics of protein (charge, number of adsorbed ions) and salt (activity), are proposed. The equations are used to construct the phase diagrams of protein solution in the water chemical potential, protein and salt concentration planes, protein solubility and salt concentration plane and to relate effective temperature and critical composition at which a critical point and a spinodal are achieved. The approach proposed provides a deeper insight into the formation of globular-reticular and cellular structures in water–protein solution, induced by phase transformation in the pre-critical and in the supercritical region of the phase diagram. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The phase diagrams (PD) of globular protein solutions [1,2] are essential for the understanding of protein crystallization, analysis of association and aggregation phenomena in physiological and biotechnological processes and structural-dynamic changes in proteins responsible for the pathology of condensation diseases. The interaction of globular proteins in concentrated solutions is often described using a model of solid spheres with a short-range attraction potential [3]. In this case, two types of phase transitions (PT): (1) liquid–solid (L–S), characterized by the co-existence of crystalline and liquid phases, and (2) liquid–liquid (L–L), in which both a diluted and a concentrated phase exist, are expected. Such an approach results in one of the versions of phase equilibrium diagrams known for systems of gas–liquid type, where first-order PT is transformed into critical PT at certain temperature and concentration. Here, protein molecules in diluted solution are regarded as gas and protein molecules in a concentrated dense phase as liquid. As protein molecules in solutions exhibit attractive interactions with a range about 1/4 of the particle diameter, L–L type PT is found to be metastable relative to L and
∗ Corresponding author. Tel.: +7 9142765264. E-mail address: [email protected] (S.P. Rozhkov). http://dx.doi.org/10.1016/j.supflu.2014.07.028 0896-8446/© 2014 Elsevier B.V. All rights reserved.
S. Such PT is revealed systematically in protein lysozyme solutions [4], some ␥-crystallines [5] and antibodies [6]. As temperature decreases, droplets of more concentrated protein solution, which is in metastable equilibrium with a less concentrated phase, are formed. Two liquid layers, differing in density, may be formed some time later or upon the centrifugation of such solutions [6,7]. Fig. 1b shows schematically a phase diagram of protein solution used most commonly to analyze its phase state [1,2] and, particularly, L–L type PT. Here, the presence of stable, metastable and unstable regions and their boundaries – binodal and spinodal lines – makes it possible to establish the one-to-one correspondence of special points on these curves with the points on the curves for the water chemical potential (1 − 1 0 ) of protein solution versus protein concentration m2 [8]. The isotherm (1 − 1 0 ), shown in Fig. 1a, is similar to the isotherm which describes first-order PT gas–liquid and a loop, analogous to van der Waals loop, is formed by the isotherm between points A and D. Within the spinodal (curve 3 in Fig. 1b, which separates the PD regions of instability and metastability) any fluctuation of concentration results in phase transition with instant gel formation. Metastable and stable PD regions are separated by binodal (curve 4). The metastable phase equilibrium L↔L may arise on the curve, if concentration fluctuations result in sufficiently high heterogeneity. Equilibrium protein concentration in solution in the presence of a crystalline phase L↔S is described by a solubility
S.P. Rozhkov, A.S. Goryunov / J. of Supercritical Fluids 95 (2014) 68–74
Fig. 1. Schematic representation of isotherms (a) and a typical phase diagram (b) of the model system water–protein in the water chemical potential, protein concentration (1 − 1 0 , m2 ) plane (a), and in the temperature, protein concentration plane (b). Figures denote: isotherm with a van der Waals loop (1), solubility line (dotted, 2), spinodal line (3), binodal line (4), region of experimentally observed oligomers and dynamic mesoscopic protein clusters (hatched, 5), gelation line (heavy dotted, 6), solidus line (7). L↔S denotes equilibrium of liquid and solid phases. L↔L denotes metastable equilibrium of diluted and dense phases. Point C is the isotherm inflection point. Combination of inflection points for a set of isotherms in the supercritical region gives a quasi-spinodal [19].
(saturation) line (thin dotted line 2). Line 7 is the solid phase boundary, and heavy dotted line 6 is the arbitrary line of gel formation. Here, the common point of the binodal and spinodal determines the upper critical solution temperature (UCST) at a certain protein concentration. Above this point, the solution is expected to be macroscopically homogeneous. One of the most intriguing results from the point of view of supercritical phenomena in protein solutions, obtained in the past few years, is the data on small protein oligomers (clusters), which comprise several protein molecules [9–11], and dynamic mesoscopic protein clusters, hundreds of nanometers in diameter [12–14], in the supercritical region of the phase diagram (for systems with UCST, these are supercritical temperatures, and in Fig. 1b this region is hatched). Their emergence is related to a spinodal for the solution to crystal phase transition [1]. This supercritical spinodal (quasi-spinodal) is located near the gel formation line (dotted line 6 in Fig. 1b) and corresponds to slightly smaller protein concentrations. The structures formed here can also be closely connected to gel formation processes in protein solutions (incomplete gelation or amorphous precipitation [15]). On the other hand, the presence of mesoscopic clusters can be coupled to either the formation of crystalline phase nuclei or protein polymerization [13]. It appears that the presence of both clusters in the seemingly homogeneous region of the phase diagram above the critical point is typical of protein, amino acid and many inorganic systems [16]. In addition, some distinctive methods (small-angle neutron and X-ray scattering) and relevant experimental procedures are mainly used to observe small clusters [11], while other methods (dynamic light scattering, atomic force microscopy, brownian microscopy) are employed to examine mesoscopic clusters [1,14]. However, the physico-chemical and thermodynamic arguments, proposed in favour of the existence of such structures, are clearly insufficient [14–18]. On the other hand, it is the thermodynamic theory of continuous phase transitions which predicts [19] that the determinant and coefficients of stability for a supercritical phase take minimum values at the quasi-spinodal line, where the highest development of the nonequilibrium fluctuation embryos
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of both low-temperature boundary phases is reached under these conditions, creating a distinctive mesophase. This is because it still partly retains the properties of low temperature phases. Based upon the theory of thermodynamic stability to diffusion [20], adapted to the heterogeneous water–protein–salt system [21,22], we propose a thermodynamic analysis of phase formation phenomena near the critical point and cluster formation in the supercritical region of the phase diagram. Instead of protein–protein interaction we consider water–protein interaction and examine the free energy of mixing of the model system biopolymer (protein)-low molecular liquid (water + salt). Phase diagrams with UCST, common for model protein systems characterized by a critical point, a spinodal and a binodal, can thus be obtained. In this approach, however, the role of solvent and the properties of protein itself are considered to a much greater extent than in the analysis of mere protein-protein interaction. The quality of the solvent, responsible for its thermodynamic affinity to a biopolymer, is controlled by the addition of salt. However, not only ionic strength but also salt ion–protein interaction and the concentration of the salt used should be considered here. This is also essential from the biological point of view because physiological protein solutions contain certain evolutionally approved salts and because proteins are crystallized dominantly in the presence of salts. Such an approach is assumed to provide additional evidence for the thermodynamics of protein interaction in the critical and supercritical regions of the phase diagram that underlie cluster formation, gel formation, crystallization and polymerization. 2. Results The equations, describing the isotherm of the water chemical potential (1 − 1 0 ) = 1 of protein solution versus molar concentrations of protein m2 and salt m3 , the critical curve and the spinodal (quasi-spinodal) [21,22], are shown below in analytical form. They are presented graphically in corresponding coordinates in Figs. 2 and 3. The equations for 1 (m2 , m3 ) is of the form: 1 − 01 = −RT +
4m23 z 2 a
ln
1+ m1 z 2
2m 3 (2 + )
a + m2 − m2 a − m2
ln [4m23 (2 + ) − z 2 m22 ]
+ const,
(1)
where ɑ2 = (2 + )4 m3 2 /z2 . In this equation, protein is described by the parameters z and , where z is protein charge and is the number of salt ions adsorbed on a protein molecule in the specific sites of sorption. The variable describes the rate of change in the salt activity coefficient with the salt concentration (in this case, ionic strength ): ≈ −m3
∂ ∂m3
A 1/2 1 + r 1/2
−
˛i i
(2)
In the parentheses, under the differential sign, is Debye–Huckel equation for the activity coefficient of electrolyte extended for high salt concentrations by introducing the empirical corrections ˛i . As the concentration dependence of the salt activity coefficient commonly has a near-parabolic form, then as a derivative from a parabolic function is a negative value at small and moderate salt concentrations (one parabola branch) and a positive value at high salt concentrations (another parabola branch). The equation for the critical point is of the form: m2 1+ =2 m3
2
1 + (2 + ) 2 z −2 . (2 + )
(3)
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Fig. 2. Dependences of effective water chemical potential 1 = 1 − 1 0 on protein concentration m2 at different 1:1 electrolyte concentrations:10−5 M(1); 10−2 M(2); 0.15 M(3); 0.35 M(4); 3 M(5), calculated using Eq. (1). Parameter z = 10 and parameter varies from 10 (left) to 5 (middle) and 1 (right).
The equation for the inflection points, describing spinodal and quasi-spinodal takes the form: m2 (2 + ) =2 m3 z 2
1±
z 2 1− (2 + )2
.
(4)
The equation for the most stable phase:
2 m2 = m3 z
2+
(5)
Thus, the equations relates the critical composition of a water–protein–salt system (the ratio of the molar concentrations of the components m2 /m3 ) and the individual parameters describing protein and electrolyte. At certain ratios of variables and parameters the system reaches the critical point (Eq. (3)) and the spinodal/quasi-spinodal (Eq. (4)), or the most stable phase under these conditions (Eq. (5)). It follows from Eq. (4) that at < 0 (the decreasing branch of the activity coefficient of electrolyte, small salt concentrations) there is only one inflection point, while at > 0 (the increasing branch of the activity coefficient of electrolyte) there are two such points. Fig. 2 shows the dependences of water chemical potential on model protein concentrations calculated in accordance with Eq. (1) for several salt concentrations m3 exemplified by 1:1 electrolyte
and several ratios of the parameters z and . All the curves are of characteristic shape with an extreme in the high protein concentration range. The arrangement of the curves and the position of the extreme on the ordinate axis depend substantially on the values of the parameters z and and salt concentration. The greater z at constant , the more negative the 1 value. An inverse relationship is valid for variations in at constant z. It should be noted that the behaviour of 1 is substantially affected by salt concentration. In the moderate salt concentration range, when the parameter still has negative values, but m3 > 0.35 М (for 1:1 electrolyte), 1 attains positive values, then passes through the extreme with a rise in salt concentration, decreases to zero and passes farther into the negative value range. However, when is positive (high salt concentrations), 1 starts from strongly negative values and then increases with the rise of m3 . Thus, the dependence on salt concentration of thermodynamic stability of protein solution and its affinity to water is substantially nonlinear. Fig. 3 shows the dependences of 1 − 1 0 on salt concentration calculated for several z/ ratios at the same protein concentration. At < 0, 1 increases in the range of m3 below ∼10−4 M and above 0.1 М. In the intermediate range from 10−3 to 0.1 М 1 decreases (which may indicate a “salting-in” effect). After m3 > 0.35 М (for 1:1 electrolyte), 1 attains positive values. As
Fig. 3. Dependences of effective water chemical potential 1 = 1 − 1 0 on 1:1 electrolyte concentration at different protein concentrations: 10−2 M(1); 10−4 M(2); 10−5 M(3), calculated using Eq. (1). Parameter z = 10 and parameter varies from 10 (left) to 5 (middle) and 1 (right).
S.P. Rozhkov, A.S. Goryunov / J. of Supercritical Fluids 95 (2014) 68–74
m3 concentration continues to grow at > 0, 1 increases from very low negative values to less negative. As the parameter depends on temperature (1/ ∼ T3/2 ) (because the coefficients of Debye–Huckel equation for electrolyte activity depend on temperature), the critical composition (m2 /m3 )cr also depends on temperature, as shown earlier [22]. Fig. 4 gives the effective temperatures of the system as a function of its critical composition (m2 /m3 )cr at critical points with regard for the position of the spinodal (quasi-spinodal) line. Line 3 is located higher than the critical point line at the compositions m2 /m3 > Q and m2 /m3 < O, i.e. the system is in the supercritical temperature range of the PD under these conditions (low salt concentrations and/or high protein concentrations). In this case, line 3 is a spinodal at Q > m2 /m3 > O, but is a quasi-spinodal outside the O–Q interval. The significance of the parameters z and for the properties of protein solutions can be shown more clearly when Eqs. (3)–(5) are converted into the form of a phase diagram in the protein solubility, electrolyte concentration (log S, m3 ) plane [22,23]. Protein solubility log S in salt solutions is often described qualitatively in the form of Debye–Huckel equation (equation under the differential sign in Eq. (2)): A 1/2
log S − log S0 =
−
1 + r 1/2
˛i i
(6)
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Fig. 4. Relationship between critical composition X = (m2 /m3 )cr and effective temperature T ∼ 1/. Curves 1 and 2 describe critical points for low ( < 0) and high ( > 0) salt concentration range, respectively. Curve 3 is a spinodal line. Spinodal enters the supercritical region of the phase diagram at (m2 /m3 )cr > Q and (m2 /m3 )cr < O, and becomes quasi-spinodal. Point P corresponds to = 0. Arrows show the trend of the composition changes corresponding to the temperature changes from T1 to T2 and back. This allows keeping the critical and supercritical phase state of solution at a new temperature [31].
We use this approach to explicitly present a protein solubility function by transforming Eqs. (3)–(5) [22,23]. For example, Eq. (2) can be presented formally as: ∂ = −m3 log S ∂m3
or log S = −
dm3 . m3
(7)
Eqs. (3)–(5) after transformation relative to the parameter and corresponding integration in new coordinates attain the form, respectively: log S = ln m23 −
2m3 2z arth + const; zm2
m23 − m2 vm3 +
log S = ln −
2 z2 v2
arctan −1
log S = ln
m23
−
m22 z 2
4
4 2m3 − m2 v
m2 v
z 2 m22
(8)
z2
v2
+ const;
(9)
−1
+ const.
(10)
These dependences, calculated for some values of the parameters z and , are shown in Fig. 5. Eq. (8) describes a bell-shaped
curve of critical points with an extreme at m3 = z 2 m2 /4v
1 + 4v2 /z 2 − 1 , which corresponds to the condition
= 0. The position of the curve along the ordinate axis depends dominantly on the value of . The greater the z2 value (and the smaller the ratio 2 /z2 ), the higher the curve. Eq. (10) describes the phase diagram region which is consistent with the most stable phase, for which ∂1 /∂m2 → ∞. In Fig. 2, this is a region of the steep growth of 1 at > 0. It is assumed to correspond to the state of disordered gel. Eq. (9) describes the spinodal line, i.e. the curve, corresponding to line 3, Fig. 4 given by Eq. (3). Eq. (9) allows to present spinodal in (log S, m3 ) plane (lines 7–9 in Fig. 5) in the supercritical region (at m2 /m3 > Q and also at 2 < z2 < 22 ), where it can be considered as quasi-spinodal. It is almost the linearly decreasing function of the electrolyte concentration. Higher protein charge z2 and the number of adsorbed ions 2 result in the increase of the line slope and protein solubility in the low salt concentration range. At the same
Fig. 5. Characteristic lines calculated using Eq. (8) (critical point curves 1,2,3); Eq. (10) (curves 4–6); Eq. (9) (segments 7–9) and plotted in the protein solubility, electrolyte concentration (log S, m3 ) plane. Curves and segments are calculated at different /z ratio: /z = 4/6 (dotted); /z = 10/11 (solid); /z = 10/15 (dashed).
time, high 2 values (but smaller 2 /z2 values) result in low solubility at high salt concentrations, the critical point curve lying higher on the solubility axis. At some salt concentration quasi-spinodals reach the gel formation curve (lines 4–6 in Fig. 5). 3. Discussion The phase diagrams in the water chemical potential and protein concentration (1 , C2 ) plane have been applied earlier to a water (1)–protein (2)–salt (3) system when developing a preferential hydration theory [24]. However, they have always been constructed to a quadratic approximation 1 ∼ f(C + C2 2 ), so that a van der Waals type isotherm cannot be obtained. A cubic approximation permits plotting isothermal lines for pseudo binary system (water + salt). They are presented schematically in Fig. 6 and formed by two different coordinate planes: water chemical potential, protein concentration (1 , m2 ) plane and protein chemical potential,
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Fig. 6. Isothermal dependence of the water chemical potential 1 (T1 − T4 , left) and protein chemical potential 2 (T5 , T6 , right) versus protein concentration at constant salt concentration. ˛→ˇ is a node indicating a metastable equilibrium of two solution phases. Segment MN corresponds to unstable states of solution. Combination of MN points for a set of isotherms constitutes a spinodal line (thick arc). The top of the arc is the critical point. Thick dotted line at the top of the arc for isotherms T3 , T2 is a quasi-spinodal line. The inset shows relationship between the average free energy of mixing gM and concentration. Points ␣ and  correspond to the concentrations of metastable phases. Here, points M and N are inflection points that correspond to the points M and N on the isotherm 1 .
protein concentration (2 , m2 ) plane (Fig. 6). According to the Gibbs–Duhem equation, variations in the chemical potential of the components are mutually dependent, i.e. 1 is related to variations in the 2 of protein [25]. Therefore, a similar type of isotherms in the (2 , m2 ) plane (Fig. 6, right), which describe the metastable equilibrium of highly-ordered polyelectrolyte gel (protein crystal [26]) and a swollen disordered gel with an equal chemical potential of protein molecules in both phases, is assumed to exist near the critical point. The inset to Fig. 6 shows the concentration dependence of average free mixing energy gМ = x1 1 + x2 2 . Here, mole fraction х2 → 1 is arbitrarily consistent with the highly-ordered polyelectrolyte gel of protein and х1 = 1 is consistent with solvent. If a binary system is unstable in a certain concentration range and is decomposed into phases, then a convex region with a maximum is formed on the concave curve gМ which describes such a behaviour [27,28]. In this case, each metastable state of a multicomponent homogeneous system (region ␣ of the isotherm 1 ) corresponds to the equilibrium of a diluted phase with a dense phase droplet of certain size and concentration [29]. For a set of isotherms, points ˛ and ˇ will constitute a binodal line and the inflection points of the curves (M and N) a spinodal line. This is how a phase diagram, consistent with the generally accepted diagram for protein solutions [1,2], can be obtained. The function, which describes the segment ˛ˇ on the inset to Fig. 6 and relates the concentration of the system to some physico-chemical characteristics of the solution components is presented by Eq. (1). It is known that the value and sign of 1 can describe both the thermodynamic affinity of the components and thermodynamic stability of a system [20,28]. If the parameter < 0, which is valid to relatively moderate salt concentrations, then the dependence 1 (m2 ) has one inflection point in accordance with Eq. (4). The convex section of the curve 1 , which is in the negative range of 1 values (Fig. 2), indicates that in this composition range a homogeneous system is metastable and a phase transition of L–L type, together with the formation of dense phase droplets, is possible
here. The subsequent decrease of 1 without an inflection point shows that the thermodynamic stability of dense liquid increases with the growth of protein concentration. The convex section of the curve 1 in the positive value range when m3 > 0.35 M for 1:1 electrolyte indicates the complete loss of the stability of the system in this composition range and the high probability of phase transition to a new stable phase which can be formed as protein gel [2]. The subsequent sharp decrease of 1 with the growth of protein concentration and transition to the negative range shows that the newly-formed phase (gel) attains a high thermodynamic stability. The intersection of the ordinate axis by curve 1 (m2 ) when (1 − 1 0 ) = 0 is expected to correspond to the equilibrium of the possible existing phases of the system solution ↔ gel ↔ crystal. At higher salt concentrations ( > 0) the curve 1 (m2 ) is located deep in the negative range, but at certain protein concentrations and the ratio of the parameters z and it goes upwards rapidly, so that ∂1 /∂m2 → ∞ and ∂2 /∂m2 → ∞. This shows that the most stable phase may exist under these conditions. The right part of Fig. 6 suggests that this phase is a swollen disordered protein gel, rather than a crystal. In accordance with Eq. (4), the curve 1 (m2 ) has two inflection points under the condition > 0. Such dependence with two inflections is characteristic of solutions capable of separating into two metastable phases [29]. This suggests that the droplets of diluted solutions (swollen gel) may arise in the metastable ordered polyelectrolyte gel, and if the viscosity of the system formed is high enough, cellular condensation structures may be formed. On the other hand, the globular particles of a concentrated phase may arise in diluted metastable protein solutions and globular-reticular condensation structures may be formed [29]. Being macroscopically homogeneous in the quasi-spinodal range (high salt concentrations and low salt concentrations, Fig. 4), the system is expected to contain the nanoscopic non-equilibrium elements of low temperature phases. These elements could be represented by the embryos of cellular and globular structure.
S.P. Rozhkov, A.S. Goryunov / J. of Supercritical Fluids 95 (2014) 68–74
Fig. 7. Schematic presentation of a hypothetic dependence of water 1 and protein 2 chemical potentials on protein concentration in the vicinity of the critical and supercritical points: solid curve is the isotherm for the critical state; dashed arc is the distinctive feature of supercritical isotherm 1 (m2 ) revealed by the analysis of the system behaviour at the inflection point (Fig. 6) (see text). Arrows AM and BN indicate quasi-equilibrium of clusters-oligomers with diluted solution and mesoscopic clusters with ordered polyelectrolyte gel of protein, respectively.
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resulting formation of a macrophase of clusters), the number of protein bound electrolyte ions can be estimated. A critical state differs from a supercritical one both in composition m2 /m3 and in the dependence of composition m2 /m3 on temperature and the parameters z and (Eqs. (3) and (4)). Critical point temperature rises with increasing salt concentration at constant protein concentration (curve 1 in Fig. 4). At the same time, the temperature of the corresponding point at the quasispinodal decreases with increasing salt concentration (curve 3 in Fig. 4). A similar effect is also observed upon variation in protein concentration at constant salt concentration. Such a self-conjugate heterophase system can provide the basis for their self-regulation through the redistribution of components in response to variations in the temperature and/or salinity of the medium [31] (shown schematically by arrows in Fig. 4). 4. Conclusions
Using available experimental evidence and terminology for protein systems, clusters-oligomers (globular structures) and mesoscopic clusters (cellular structures) can be presented as structures that feature short-range and long-range order in macromolecular interaction, respectively. It is natural to suggest a region of reorganization of solution dynamic structure in protein solution PD, a transient region where the elements of the long-range organization, characteristic of crystals, arise and break up along with short range elements. Let us consider hypothetic isotherms which describe variations in the chemical potentials of water and protein in the supercritical range, near the critical point. As the system here is in the twophase equilibrium solution ↔ crystal, 1 = 2 and PD may be presented in the (i , m2 ) plane (Fig. 7). The chemical potential isotherm has the concave shape with a planar segment for a critical state [27] as a thermodynamically stable state. At the same time, the relative thermodynamic stability of mother liquor and the presence of embryos in the supercritical region suggests the concave shape of the supercritical isotherm with a convex segment in the range of negative values of free mixing energy, as in Fig. 7. Such a kind of isotherms can be obtained by the transformation of 1 (m2 ) (Fig. 6) using Tailor series at the inflection points. Here, the arrow AМ shows the metastable equilibrium of globular embryos with a diluted phase (1 / = 1 // ), and the arrow ВN shows the metastable equilibrium of cellular embryos with a crystalline phase (2 / = 2 // ). The former embryos are unstable relative to a crystalline phase and the latter to a diluted phase. As they arise at the different – increasing and decreasing sections of the isotherm, they cannot be in equilibrium and are separated by a potential barrier, even though they are energetically equal 1 / ≈ 2 / . However, they can appear (interconvert) one by one while overcoming the energy barrier produced by a structural factor. Basing on the wide range of experimental data on the salting out of protein lysozyme the Eq. (9) was verified. To this end quasi-spinodals as lines of the lowest thermodynamic stability of homogeneous solution in the (log S, m3 ) plane (Fig. 5) were presented in the form of Setchenov equation for the salting-out of protein by electrolyte [22,30]. Our results appear to be in good agreement with the independent experimental data both qualitatively and quantitatively (type of the dependence of solubility on pH, salt concentration, the number of ions adsorbed). The Setchenov equation coefficients were thus derived in analytical form with no correction factors used. When the charge of the protein, the number of bound salt anions, salt concentration, and the concentration of the protein salting out is known, the concentration of the protein in the cluster can be estimated [30]. And vice versa, when the concentration of the protein in the cluster is known (it can sometimes be determined after solution centrifugation and
Constructing phase diagrams of globular protein solutions, based on analysis of their interaction potential, has become common in the last few years, while a thermodynamic approach, in which the contribution of water to the phase properties of protein solutions is considered, is not sufficiently advanced. In the present study, we proceed from the qualitative understanding of water–salt–protein solution as the pseudo-binary system: low molecular liquid (water + salt) and biopolymer, while a crystal is understood as highly-ordered polyelectrolyte protein gel. The thermodynamic instability with respect to diffusion processes (the emergence of concentration heterogeneity) of a ternary water–protein–salt system upon variations in the concentration of the components is then studied quantitatively. Such an approach provides additional evidence for the thermodynamic pattern of protein interaction in the pre-critical, critical and supercritical regions of a phase diagram which results in cluster formation, gel formation and crystallization. The results obtained are largely in agreement with some concepts of the multi-step nucleation theory which has lately attracted increased attention [14]. Transition to protein–solvent interaction provides a better opportunity to represent phase diagrams in various coordinate planes and to obtain useful information on the role of some system parameters that describe the properties of protein and salts in the phase and structural properties of the system studied. This study provides arguments in favour of the existence of clusters-oligomers and mesoscopic protein clusters as thermodynamic, while shortliving, units in the supercritical region of a phase diagram, both cluster types showing evidence of phantoms of low temperature dense liquid phase and swollen gel phase. Acknowledgement This study was funded by the Russian Foundation for Basic Research (Grant 13-03-00422). References [1] P.G. Vekilov, Phase diagrams and kinetics of phase transitions in protein solutions, J. Physics – Condensed Matter 24 (2012) 193101. [2] A.C. Dumetz, A.M. Chockla, E.W. Kaler, A.M. Lenhoff, Protein phase behavior in aqueous solutions: crystallization, liquid–liquid phase separation, gels, and aggregates, Biophysical J. 94 (2008) 570–583. [3] A. Lomakin, N. Asherie, G.B. Benedek, Liquid–solid transition in nuclei of protein crystals, Proceedings of the National Academy of Sciences of the United States of America 100 (2003) 10254–10257. [4] O. Galkin, P.G. Vekilov, Control of protein crystal nucleation around the metastable liquid-liquid phase boundary, Proceedings of the National Academy of Sciences of the United States of America 97 (2000) 6277–6281. [5] M. Malfois, F. Bonnete, L. Belloni, A. Tardieu, A model of attractive interactions to account for fluid–fluid phase separation on protein solutions, J. Chemical Physics 105 (1996) 3290–3300.
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The Journal of Supercritical Fluids journal homepage: www.elsevier.com/locate/supflu
Dynamic protein clusterization in supercritical region of the phase diagram of water–protein–salt solutions S.P. Rozhkov ∗ , A.S. Goryunov Institute of Biology, Federal State Budget Institution, Karelian Research Centre, RAS, Pushkinskaya St., 11, Petrozavodsk, Russia
a r t i c l e
i n f o
Article history: Received 13 May 2014 Received in revised form 28 July 2014 Accepted 31 July 2014 Available online 9 August 2014 Keywords: Protein clusters Phase diagrams Thermodynamic stability (Quasi)Spinodal Critical point
a b s t r a c t A protein solution phase diagram, constructed on the basis of the thermodynamic description of protein–solvent interaction, is discussed. A thermodynamic mechanism of formation of protein clustersoligomers and mesoscopic clusters in the supercritical region of a protein solution phase diagram, is considered. Variation in the chemical potential of water, which depends on solution composition, is the parameter which describes the interaction. Equations for the spinodal and critical points of the phase diagram, in which the critical composition (protein to salt concentration ratio) of the system is related to the physico-chemical characteristics of protein (charge, number of adsorbed ions) and salt (activity), are proposed. The equations are used to construct the phase diagrams of protein solution in the water chemical potential, protein and salt concentration planes, protein solubility and salt concentration plane and to relate effective temperature and critical composition at which a critical point and a spinodal are achieved. The approach proposed provides a deeper insight into the formation of globular-reticular and cellular structures in water–protein solution, induced by phase transformation in the pre-critical and in the supercritical region of the phase diagram. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The phase diagrams (PD) of globular protein solutions [1,2] are essential for the understanding of protein crystallization, analysis of association and aggregation phenomena in physiological and biotechnological processes and structural-dynamic changes in proteins responsible for the pathology of condensation diseases. The interaction of globular proteins in concentrated solutions is often described using a model of solid spheres with a short-range attraction potential [3]. In this case, two types of phase transitions (PT): (1) liquid–solid (L–S), characterized by the co-existence of crystalline and liquid phases, and (2) liquid–liquid (L–L), in which both a diluted and a concentrated phase exist, are expected. Such an approach results in one of the versions of phase equilibrium diagrams known for systems of gas–liquid type, where first-order PT is transformed into critical PT at certain temperature and concentration. Here, protein molecules in diluted solution are regarded as gas and protein molecules in a concentrated dense phase as liquid. As protein molecules in solutions exhibit attractive interactions with a range about 1/4 of the particle diameter, L–L type PT is found to be metastable relative to L and
∗ Corresponding author. Tel.: +7 9142765264. E-mail address: [email protected] (S.P. Rozhkov). http://dx.doi.org/10.1016/j.supflu.2014.07.028 0896-8446/© 2014 Elsevier B.V. All rights reserved.
S. Such PT is revealed systematically in protein lysozyme solutions [4], some ␥-crystallines [5] and antibodies [6]. As temperature decreases, droplets of more concentrated protein solution, which is in metastable equilibrium with a less concentrated phase, are formed. Two liquid layers, differing in density, may be formed some time later or upon the centrifugation of such solutions [6,7]. Fig. 1b shows schematically a phase diagram of protein solution used most commonly to analyze its phase state [1,2] and, particularly, L–L type PT. Here, the presence of stable, metastable and unstable regions and their boundaries – binodal and spinodal lines – makes it possible to establish the one-to-one correspondence of special points on these curves with the points on the curves for the water chemical potential (1 − 1 0 ) of protein solution versus protein concentration m2 [8]. The isotherm (1 − 1 0 ), shown in Fig. 1a, is similar to the isotherm which describes first-order PT gas–liquid and a loop, analogous to van der Waals loop, is formed by the isotherm between points A and D. Within the spinodal (curve 3 in Fig. 1b, which separates the PD regions of instability and metastability) any fluctuation of concentration results in phase transition with instant gel formation. Metastable and stable PD regions are separated by binodal (curve 4). The metastable phase equilibrium L↔L may arise on the curve, if concentration fluctuations result in sufficiently high heterogeneity. Equilibrium protein concentration in solution in the presence of a crystalline phase L↔S is described by a solubility
S.P. Rozhkov, A.S. Goryunov / J. of Supercritical Fluids 95 (2014) 68–74
Fig. 1. Schematic representation of isotherms (a) and a typical phase diagram (b) of the model system water–protein in the water chemical potential, protein concentration (1 − 1 0 , m2 ) plane (a), and in the temperature, protein concentration plane (b). Figures denote: isotherm with a van der Waals loop (1), solubility line (dotted, 2), spinodal line (3), binodal line (4), region of experimentally observed oligomers and dynamic mesoscopic protein clusters (hatched, 5), gelation line (heavy dotted, 6), solidus line (7). L↔S denotes equilibrium of liquid and solid phases. L↔L denotes metastable equilibrium of diluted and dense phases. Point C is the isotherm inflection point. Combination of inflection points for a set of isotherms in the supercritical region gives a quasi-spinodal [19].
(saturation) line (thin dotted line 2). Line 7 is the solid phase boundary, and heavy dotted line 6 is the arbitrary line of gel formation. Here, the common point of the binodal and spinodal determines the upper critical solution temperature (UCST) at a certain protein concentration. Above this point, the solution is expected to be macroscopically homogeneous. One of the most intriguing results from the point of view of supercritical phenomena in protein solutions, obtained in the past few years, is the data on small protein oligomers (clusters), which comprise several protein molecules [9–11], and dynamic mesoscopic protein clusters, hundreds of nanometers in diameter [12–14], in the supercritical region of the phase diagram (for systems with UCST, these are supercritical temperatures, and in Fig. 1b this region is hatched). Their emergence is related to a spinodal for the solution to crystal phase transition [1]. This supercritical spinodal (quasi-spinodal) is located near the gel formation line (dotted line 6 in Fig. 1b) and corresponds to slightly smaller protein concentrations. The structures formed here can also be closely connected to gel formation processes in protein solutions (incomplete gelation or amorphous precipitation [15]). On the other hand, the presence of mesoscopic clusters can be coupled to either the formation of crystalline phase nuclei or protein polymerization [13]. It appears that the presence of both clusters in the seemingly homogeneous region of the phase diagram above the critical point is typical of protein, amino acid and many inorganic systems [16]. In addition, some distinctive methods (small-angle neutron and X-ray scattering) and relevant experimental procedures are mainly used to observe small clusters [11], while other methods (dynamic light scattering, atomic force microscopy, brownian microscopy) are employed to examine mesoscopic clusters [1,14]. However, the physico-chemical and thermodynamic arguments, proposed in favour of the existence of such structures, are clearly insufficient [14–18]. On the other hand, it is the thermodynamic theory of continuous phase transitions which predicts [19] that the determinant and coefficients of stability for a supercritical phase take minimum values at the quasi-spinodal line, where the highest development of the nonequilibrium fluctuation embryos
69
of both low-temperature boundary phases is reached under these conditions, creating a distinctive mesophase. This is because it still partly retains the properties of low temperature phases. Based upon the theory of thermodynamic stability to diffusion [20], adapted to the heterogeneous water–protein–salt system [21,22], we propose a thermodynamic analysis of phase formation phenomena near the critical point and cluster formation in the supercritical region of the phase diagram. Instead of protein–protein interaction we consider water–protein interaction and examine the free energy of mixing of the model system biopolymer (protein)-low molecular liquid (water + salt). Phase diagrams with UCST, common for model protein systems characterized by a critical point, a spinodal and a binodal, can thus be obtained. In this approach, however, the role of solvent and the properties of protein itself are considered to a much greater extent than in the analysis of mere protein-protein interaction. The quality of the solvent, responsible for its thermodynamic affinity to a biopolymer, is controlled by the addition of salt. However, not only ionic strength but also salt ion–protein interaction and the concentration of the salt used should be considered here. This is also essential from the biological point of view because physiological protein solutions contain certain evolutionally approved salts and because proteins are crystallized dominantly in the presence of salts. Such an approach is assumed to provide additional evidence for the thermodynamics of protein interaction in the critical and supercritical regions of the phase diagram that underlie cluster formation, gel formation, crystallization and polymerization. 2. Results The equations, describing the isotherm of the water chemical potential (1 − 1 0 ) = 1 of protein solution versus molar concentrations of protein m2 and salt m3 , the critical curve and the spinodal (quasi-spinodal) [21,22], are shown below in analytical form. They are presented graphically in corresponding coordinates in Figs. 2 and 3. The equations for 1 (m2 , m3 ) is of the form: 1 − 01 = −RT +
4m23 z 2 a
ln
1+ m1 z 2
2m 3 (2 + )
a + m2 − m2 a − m2
ln [4m23 (2 + ) − z 2 m22 ]
+ const,
(1)
where ɑ2 = (2 + )4 m3 2 /z2 . In this equation, protein is described by the parameters z and , where z is protein charge and is the number of salt ions adsorbed on a protein molecule in the specific sites of sorption. The variable describes the rate of change in the salt activity coefficient with the salt concentration (in this case, ionic strength ): ≈ −m3
∂ ∂m3
A 1/2 1 + r 1/2
−
˛i i
(2)
In the parentheses, under the differential sign, is Debye–Huckel equation for the activity coefficient of electrolyte extended for high salt concentrations by introducing the empirical corrections ˛i . As the concentration dependence of the salt activity coefficient commonly has a near-parabolic form, then as a derivative from a parabolic function is a negative value at small and moderate salt concentrations (one parabola branch) and a positive value at high salt concentrations (another parabola branch). The equation for the critical point is of the form: m2 1+ =2 m3
2
1 + (2 + ) 2 z −2 . (2 + )
(3)
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Fig. 2. Dependences of effective water chemical potential 1 = 1 − 1 0 on protein concentration m2 at different 1:1 electrolyte concentrations:10−5 M(1); 10−2 M(2); 0.15 M(3); 0.35 M(4); 3 M(5), calculated using Eq. (1). Parameter z = 10 and parameter varies from 10 (left) to 5 (middle) and 1 (right).
The equation for the inflection points, describing spinodal and quasi-spinodal takes the form: m2 (2 + ) =2 m3 z 2
1±
z 2 1− (2 + )2
.
(4)
The equation for the most stable phase:
2 m2 = m3 z
2+
(5)
Thus, the equations relates the critical composition of a water–protein–salt system (the ratio of the molar concentrations of the components m2 /m3 ) and the individual parameters describing protein and electrolyte. At certain ratios of variables and parameters the system reaches the critical point (Eq. (3)) and the spinodal/quasi-spinodal (Eq. (4)), or the most stable phase under these conditions (Eq. (5)). It follows from Eq. (4) that at < 0 (the decreasing branch of the activity coefficient of electrolyte, small salt concentrations) there is only one inflection point, while at > 0 (the increasing branch of the activity coefficient of electrolyte) there are two such points. Fig. 2 shows the dependences of water chemical potential on model protein concentrations calculated in accordance with Eq. (1) for several salt concentrations m3 exemplified by 1:1 electrolyte
and several ratios of the parameters z and . All the curves are of characteristic shape with an extreme in the high protein concentration range. The arrangement of the curves and the position of the extreme on the ordinate axis depend substantially on the values of the parameters z and and salt concentration. The greater z at constant , the more negative the 1 value. An inverse relationship is valid for variations in at constant z. It should be noted that the behaviour of 1 is substantially affected by salt concentration. In the moderate salt concentration range, when the parameter still has negative values, but m3 > 0.35 М (for 1:1 electrolyte), 1 attains positive values, then passes through the extreme with a rise in salt concentration, decreases to zero and passes farther into the negative value range. However, when is positive (high salt concentrations), 1 starts from strongly negative values and then increases with the rise of m3 . Thus, the dependence on salt concentration of thermodynamic stability of protein solution and its affinity to water is substantially nonlinear. Fig. 3 shows the dependences of 1 − 1 0 on salt concentration calculated for several z/ ratios at the same protein concentration. At < 0, 1 increases in the range of m3 below ∼10−4 M and above 0.1 М. In the intermediate range from 10−3 to 0.1 М 1 decreases (which may indicate a “salting-in” effect). After m3 > 0.35 М (for 1:1 electrolyte), 1 attains positive values. As
Fig. 3. Dependences of effective water chemical potential 1 = 1 − 1 0 on 1:1 electrolyte concentration at different protein concentrations: 10−2 M(1); 10−4 M(2); 10−5 M(3), calculated using Eq. (1). Parameter z = 10 and parameter varies from 10 (left) to 5 (middle) and 1 (right).
S.P. Rozhkov, A.S. Goryunov / J. of Supercritical Fluids 95 (2014) 68–74
m3 concentration continues to grow at > 0, 1 increases from very low negative values to less negative. As the parameter depends on temperature (1/ ∼ T3/2 ) (because the coefficients of Debye–Huckel equation for electrolyte activity depend on temperature), the critical composition (m2 /m3 )cr also depends on temperature, as shown earlier [22]. Fig. 4 gives the effective temperatures of the system as a function of its critical composition (m2 /m3 )cr at critical points with regard for the position of the spinodal (quasi-spinodal) line. Line 3 is located higher than the critical point line at the compositions m2 /m3 > Q and m2 /m3 < O, i.e. the system is in the supercritical temperature range of the PD under these conditions (low salt concentrations and/or high protein concentrations). In this case, line 3 is a spinodal at Q > m2 /m3 > O, but is a quasi-spinodal outside the O–Q interval. The significance of the parameters z and for the properties of protein solutions can be shown more clearly when Eqs. (3)–(5) are converted into the form of a phase diagram in the protein solubility, electrolyte concentration (log S, m3 ) plane [22,23]. Protein solubility log S in salt solutions is often described qualitatively in the form of Debye–Huckel equation (equation under the differential sign in Eq. (2)): A 1/2
log S − log S0 =
−
1 + r 1/2
˛i i
(6)
71
Fig. 4. Relationship between critical composition X = (m2 /m3 )cr and effective temperature T ∼ 1/. Curves 1 and 2 describe critical points for low ( < 0) and high ( > 0) salt concentration range, respectively. Curve 3 is a spinodal line. Spinodal enters the supercritical region of the phase diagram at (m2 /m3 )cr > Q and (m2 /m3 )cr < O, and becomes quasi-spinodal. Point P corresponds to = 0. Arrows show the trend of the composition changes corresponding to the temperature changes from T1 to T2 and back. This allows keeping the critical and supercritical phase state of solution at a new temperature [31].
We use this approach to explicitly present a protein solubility function by transforming Eqs. (3)–(5) [22,23]. For example, Eq. (2) can be presented formally as: ∂ = −m3 log S ∂m3
or log S = −
dm3 . m3
(7)
Eqs. (3)–(5) after transformation relative to the parameter and corresponding integration in new coordinates attain the form, respectively: log S = ln m23 −
2m3 2z arth + const; zm2
m23 − m2 vm3 +
log S = ln −
2 z2 v2
arctan −1
log S = ln
m23
−
m22 z 2
4
4 2m3 − m2 v
m2 v
z 2 m22
(8)
z2
v2
+ const;
(9)
−1
+ const.
(10)
These dependences, calculated for some values of the parameters z and , are shown in Fig. 5. Eq. (8) describes a bell-shaped
curve of critical points with an extreme at m3 = z 2 m2 /4v
1 + 4v2 /z 2 − 1 , which corresponds to the condition
= 0. The position of the curve along the ordinate axis depends dominantly on the value of . The greater the z2 value (and the smaller the ratio 2 /z2 ), the higher the curve. Eq. (10) describes the phase diagram region which is consistent with the most stable phase, for which ∂1 /∂m2 → ∞. In Fig. 2, this is a region of the steep growth of 1 at > 0. It is assumed to correspond to the state of disordered gel. Eq. (9) describes the spinodal line, i.e. the curve, corresponding to line 3, Fig. 4 given by Eq. (3). Eq. (9) allows to present spinodal in (log S, m3 ) plane (lines 7–9 in Fig. 5) in the supercritical region (at m2 /m3 > Q and also at 2 < z2 < 22 ), where it can be considered as quasi-spinodal. It is almost the linearly decreasing function of the electrolyte concentration. Higher protein charge z2 and the number of adsorbed ions 2 result in the increase of the line slope and protein solubility in the low salt concentration range. At the same
Fig. 5. Characteristic lines calculated using Eq. (8) (critical point curves 1,2,3); Eq. (10) (curves 4–6); Eq. (9) (segments 7–9) and plotted in the protein solubility, electrolyte concentration (log S, m3 ) plane. Curves and segments are calculated at different /z ratio: /z = 4/6 (dotted); /z = 10/11 (solid); /z = 10/15 (dashed).
time, high 2 values (but smaller 2 /z2 values) result in low solubility at high salt concentrations, the critical point curve lying higher on the solubility axis. At some salt concentration quasi-spinodals reach the gel formation curve (lines 4–6 in Fig. 5). 3. Discussion The phase diagrams in the water chemical potential and protein concentration (1 , C2 ) plane have been applied earlier to a water (1)–protein (2)–salt (3) system when developing a preferential hydration theory [24]. However, they have always been constructed to a quadratic approximation 1 ∼ f(C + C2 2 ), so that a van der Waals type isotherm cannot be obtained. A cubic approximation permits plotting isothermal lines for pseudo binary system (water + salt). They are presented schematically in Fig. 6 and formed by two different coordinate planes: water chemical potential, protein concentration (1 , m2 ) plane and protein chemical potential,
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Fig. 6. Isothermal dependence of the water chemical potential 1 (T1 − T4 , left) and protein chemical potential 2 (T5 , T6 , right) versus protein concentration at constant salt concentration. ˛→ˇ is a node indicating a metastable equilibrium of two solution phases. Segment MN corresponds to unstable states of solution. Combination of MN points for a set of isotherms constitutes a spinodal line (thick arc). The top of the arc is the critical point. Thick dotted line at the top of the arc for isotherms T3 , T2 is a quasi-spinodal line. The inset shows relationship between the average free energy of mixing gM and concentration. Points ␣ and  correspond to the concentrations of metastable phases. Here, points M and N are inflection points that correspond to the points M and N on the isotherm 1 .
protein concentration (2 , m2 ) plane (Fig. 6). According to the Gibbs–Duhem equation, variations in the chemical potential of the components are mutually dependent, i.e. 1 is related to variations in the 2 of protein [25]. Therefore, a similar type of isotherms in the (2 , m2 ) plane (Fig. 6, right), which describe the metastable equilibrium of highly-ordered polyelectrolyte gel (protein crystal [26]) and a swollen disordered gel with an equal chemical potential of protein molecules in both phases, is assumed to exist near the critical point. The inset to Fig. 6 shows the concentration dependence of average free mixing energy gМ = x1 1 + x2 2 . Here, mole fraction х2 → 1 is arbitrarily consistent with the highly-ordered polyelectrolyte gel of protein and х1 = 1 is consistent with solvent. If a binary system is unstable in a certain concentration range and is decomposed into phases, then a convex region with a maximum is formed on the concave curve gМ which describes such a behaviour [27,28]. In this case, each metastable state of a multicomponent homogeneous system (region ␣ of the isotherm 1 ) corresponds to the equilibrium of a diluted phase with a dense phase droplet of certain size and concentration [29]. For a set of isotherms, points ˛ and ˇ will constitute a binodal line and the inflection points of the curves (M and N) a spinodal line. This is how a phase diagram, consistent with the generally accepted diagram for protein solutions [1,2], can be obtained. The function, which describes the segment ˛ˇ on the inset to Fig. 6 and relates the concentration of the system to some physico-chemical characteristics of the solution components is presented by Eq. (1). It is known that the value and sign of 1 can describe both the thermodynamic affinity of the components and thermodynamic stability of a system [20,28]. If the parameter < 0, which is valid to relatively moderate salt concentrations, then the dependence 1 (m2 ) has one inflection point in accordance with Eq. (4). The convex section of the curve 1 , which is in the negative range of 1 values (Fig. 2), indicates that in this composition range a homogeneous system is metastable and a phase transition of L–L type, together with the formation of dense phase droplets, is possible
here. The subsequent decrease of 1 without an inflection point shows that the thermodynamic stability of dense liquid increases with the growth of protein concentration. The convex section of the curve 1 in the positive value range when m3 > 0.35 M for 1:1 electrolyte indicates the complete loss of the stability of the system in this composition range and the high probability of phase transition to a new stable phase which can be formed as protein gel [2]. The subsequent sharp decrease of 1 with the growth of protein concentration and transition to the negative range shows that the newly-formed phase (gel) attains a high thermodynamic stability. The intersection of the ordinate axis by curve 1 (m2 ) when (1 − 1 0 ) = 0 is expected to correspond to the equilibrium of the possible existing phases of the system solution ↔ gel ↔ crystal. At higher salt concentrations ( > 0) the curve 1 (m2 ) is located deep in the negative range, but at certain protein concentrations and the ratio of the parameters z and it goes upwards rapidly, so that ∂1 /∂m2 → ∞ and ∂2 /∂m2 → ∞. This shows that the most stable phase may exist under these conditions. The right part of Fig. 6 suggests that this phase is a swollen disordered protein gel, rather than a crystal. In accordance with Eq. (4), the curve 1 (m2 ) has two inflection points under the condition > 0. Such dependence with two inflections is characteristic of solutions capable of separating into two metastable phases [29]. This suggests that the droplets of diluted solutions (swollen gel) may arise in the metastable ordered polyelectrolyte gel, and if the viscosity of the system formed is high enough, cellular condensation structures may be formed. On the other hand, the globular particles of a concentrated phase may arise in diluted metastable protein solutions and globular-reticular condensation structures may be formed [29]. Being macroscopically homogeneous in the quasi-spinodal range (high salt concentrations and low salt concentrations, Fig. 4), the system is expected to contain the nanoscopic non-equilibrium elements of low temperature phases. These elements could be represented by the embryos of cellular and globular structure.
S.P. Rozhkov, A.S. Goryunov / J. of Supercritical Fluids 95 (2014) 68–74
Fig. 7. Schematic presentation of a hypothetic dependence of water 1 and protein 2 chemical potentials on protein concentration in the vicinity of the critical and supercritical points: solid curve is the isotherm for the critical state; dashed arc is the distinctive feature of supercritical isotherm 1 (m2 ) revealed by the analysis of the system behaviour at the inflection point (Fig. 6) (see text). Arrows AM and BN indicate quasi-equilibrium of clusters-oligomers with diluted solution and mesoscopic clusters with ordered polyelectrolyte gel of protein, respectively.
73
resulting formation of a macrophase of clusters), the number of protein bound electrolyte ions can be estimated. A critical state differs from a supercritical one both in composition m2 /m3 and in the dependence of composition m2 /m3 on temperature and the parameters z and (Eqs. (3) and (4)). Critical point temperature rises with increasing salt concentration at constant protein concentration (curve 1 in Fig. 4). At the same time, the temperature of the corresponding point at the quasispinodal decreases with increasing salt concentration (curve 3 in Fig. 4). A similar effect is also observed upon variation in protein concentration at constant salt concentration. Such a self-conjugate heterophase system can provide the basis for their self-regulation through the redistribution of components in response to variations in the temperature and/or salinity of the medium [31] (shown schematically by arrows in Fig. 4). 4. Conclusions
Using available experimental evidence and terminology for protein systems, clusters-oligomers (globular structures) and mesoscopic clusters (cellular structures) can be presented as structures that feature short-range and long-range order in macromolecular interaction, respectively. It is natural to suggest a region of reorganization of solution dynamic structure in protein solution PD, a transient region where the elements of the long-range organization, characteristic of crystals, arise and break up along with short range elements. Let us consider hypothetic isotherms which describe variations in the chemical potentials of water and protein in the supercritical range, near the critical point. As the system here is in the twophase equilibrium solution ↔ crystal, 1 = 2 and PD may be presented in the (i , m2 ) plane (Fig. 7). The chemical potential isotherm has the concave shape with a planar segment for a critical state [27] as a thermodynamically stable state. At the same time, the relative thermodynamic stability of mother liquor and the presence of embryos in the supercritical region suggests the concave shape of the supercritical isotherm with a convex segment in the range of negative values of free mixing energy, as in Fig. 7. Such a kind of isotherms can be obtained by the transformation of 1 (m2 ) (Fig. 6) using Tailor series at the inflection points. Here, the arrow AМ shows the metastable equilibrium of globular embryos with a diluted phase (1 / = 1 // ), and the arrow ВN shows the metastable equilibrium of cellular embryos with a crystalline phase (2 / = 2 // ). The former embryos are unstable relative to a crystalline phase and the latter to a diluted phase. As they arise at the different – increasing and decreasing sections of the isotherm, they cannot be in equilibrium and are separated by a potential barrier, even though they are energetically equal 1 / ≈ 2 / . However, they can appear (interconvert) one by one while overcoming the energy barrier produced by a structural factor. Basing on the wide range of experimental data on the salting out of protein lysozyme the Eq. (9) was verified. To this end quasi-spinodals as lines of the lowest thermodynamic stability of homogeneous solution in the (log S, m3 ) plane (Fig. 5) were presented in the form of Setchenov equation for the salting-out of protein by electrolyte [22,30]. Our results appear to be in good agreement with the independent experimental data both qualitatively and quantitatively (type of the dependence of solubility on pH, salt concentration, the number of ions adsorbed). The Setchenov equation coefficients were thus derived in analytical form with no correction factors used. When the charge of the protein, the number of bound salt anions, salt concentration, and the concentration of the protein salting out is known, the concentration of the protein in the cluster can be estimated [30]. And vice versa, when the concentration of the protein in the cluster is known (it can sometimes be determined after solution centrifugation and
Constructing phase diagrams of globular protein solutions, based on analysis of their interaction potential, has become common in the last few years, while a thermodynamic approach, in which the contribution of water to the phase properties of protein solutions is considered, is not sufficiently advanced. In the present study, we proceed from the qualitative understanding of water–salt–protein solution as the pseudo-binary system: low molecular liquid (water + salt) and biopolymer, while a crystal is understood as highly-ordered polyelectrolyte protein gel. The thermodynamic instability with respect to diffusion processes (the emergence of concentration heterogeneity) of a ternary water–protein–salt system upon variations in the concentration of the components is then studied quantitatively. Such an approach provides additional evidence for the thermodynamic pattern of protein interaction in the pre-critical, critical and supercritical regions of a phase diagram which results in cluster formation, gel formation and crystallization. The results obtained are largely in agreement with some concepts of the multi-step nucleation theory which has lately attracted increased attention [14]. Transition to protein–solvent interaction provides a better opportunity to represent phase diagrams in various coordinate planes and to obtain useful information on the role of some system parameters that describe the properties of protein and salts in the phase and structural properties of the system studied. This study provides arguments in favour of the existence of clusters-oligomers and mesoscopic protein clusters as thermodynamic, while shortliving, units in the supercritical region of a phase diagram, both cluster types showing evidence of phantoms of low temperature dense liquid phase and swollen gel phase. Acknowledgement This study was funded by the Russian Foundation for Basic Research (Grant 13-03-00422). References [1] P.G. Vekilov, Phase diagrams and kinetics of phase transitions in protein solutions, J. Physics – Condensed Matter 24 (2012) 193101. [2] A.C. Dumetz, A.M. Chockla, E.W. Kaler, A.M. Lenhoff, Protein phase behavior in aqueous solutions: crystallization, liquid–liquid phase separation, gels, and aggregates, Biophysical J. 94 (2008) 570–583. [3] A. Lomakin, N. Asherie, G.B. Benedek, Liquid–solid transition in nuclei of protein crystals, Proceedings of the National Academy of Sciences of the United States of America 100 (2003) 10254–10257. [4] O. Galkin, P.G. Vekilov, Control of protein crystal nucleation around the metastable liquid-liquid phase boundary, Proceedings of the National Academy of Sciences of the United States of America 97 (2000) 6277–6281. [5] M. Malfois, F. Bonnete, L. Belloni, A. Tardieu, A model of attractive interactions to account for fluid–fluid phase separation on protein solutions, J. Chemical Physics 105 (1996) 3290–3300.
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