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Cohn (1925) developed an empirical log-linear relationship to describe the salting-out o f proteins at high salt concentrations:

logS = p ~ Kgm

(equation 6.1) where S is the protein solubility, i.e. the concentration o f soluble protein, m is the molar salt concentration and p. Kg are constants. For a given protein, the value o f p depends on pH and temperature and is a minimum at the isoelectric point (Rothstein, 1994; Foster, 1994). While the value o f Kg depends on both protein and salt properties (Rothstein, 1994; Foster, 1994). Foster et al. (1976) found that the constants p and Kg vary with the mode and rate o f salt addition, and Iyer and Przybycien (1994) also found that both parameters vary with mixing conditions.

Although the Cohn equation (equation 6.1) can describe the salting-out behaviour o f proteins, it does not apply to salting-in and the transient regions o f solubility behaviour which are o f more relevance to process control and design (Clarkson et a l, 1996b). Richardson (1987) used a polynomial expression to describe solubility behaviour as part o f an optimisation routine for fractional precipitation (Section 6.2.2). While Niktari et a l (1989, 1990) employed an isotherm equation to describe fully the fractional solubility profile o f bakers' yeast alcohol dehydrogenase (ADH) and total protein in ammonium sulphate for control applications:

F - ' a.

(equation 6.2) where F is fractional solubility, i.e. the fraction o f enzyme or protein soluble, C is the ammonium sulphate saturation and a, n are constants. Clarkson et a l (1996b) used equation 6.2 in developing a model for the two-stage fractional precipitation o f ADH from bakers' yeast homogenate.

Care must be taken when comparing solubility profiles described by the Cohn equation (equation 6.1) and by equation 6.2. For example Foster (1972) observed that when ammonium sulphate was added to bakers' yeast homogenate the concentration o f total protein remaining soluble was independent o f the initial protein concentration in the salting-out region. The Cohn profiles obtained for this system for different initial protein concentrations will be the same, however profiles obtained using equation 6.2, which describes fractional solubility, will be shifted to the left as the initial protein concentration increases.

6.2.1.2 T heoretical M odels

A number o f theoretical models have been developed to describe the solubility o f proteins in aqueous salt solutions which are based on thermodynamic considerations, incorporating aspects such as electrostatic and/or hydrophobic interactions, the preferential interaction o f solvent and salt with protein molecules, and colloidal surface

interactions. In these theories the precipitation o f protein is considered to be an equilibrium phase separation.

An early theoretical description o f the solubility behaviour o f proteins in aqueous salt solutions was based on the electrostatic theory o f Debye and Huckel (1923). The protein molecule is assumed to be a simple ion surrounded by an ionic atmosphere containing an excess o f counter charged ions. According to the theory, ionic interaction causes a decrease in electrostatic free energy and thereby chemical activity o f the protein in solution resulting in an increase in solubility. The Debye-Huckle theory, however, only gives a realistic description o f protein solubility in the salting-in region.

The related theory o f Kirkwood (1943) treats the protein molecule as a dipole and is the only electrostatic theory able to predict an extensive salting-out effect. The protein dipoles produce cavities o f low dielectric constant in the solvent which induce changes in polarisation. Salting-out occurs when repulsive forces between ions and like charges in the cavities increase until it is energetically unfavourable for the protein to remain in solution. A major limitation o f the theory is that it does not account for the large differences in protein solubility behaviour observed with different salts (Richardson, 1987; Alsaffar, 1994). Rajagh et a l (1965) also claim that the salting-out term is small compared to the salting-in term and the theory would therefore predict an increase in protein solubility at high salt concentrations. From these criticisms it becomes apparent that protein solubility behaviour can not be described by electrostatic considerations alone.

The importance o f hydrophobic interactions in the solubility behaviour o f proteins was recognised by Kauzmann (1959). Following on from this observation, the Cavity theory developed by Sinanoglu and Abdulnar (1965) sought to explain the stability o f proteins in aqueous environments. In this approach the free energy change between a protein molecule in the gas phase as compared to its free energy in solution was considered. The stability was derived from the free energy required to form a cavity in the solvent to accommodate the solute. The higher the surface tension o f the solvent, the greater the energy required to 'stretch' the solvent in order to form the cavity, and hence the greater the structural stability.

Melander and Horvath (1977) extended the Cavity theory to describe the solubility o f proteins in salt solutions. Most inorganic salts are known to increase the surface tension of water and thereby affect hydrophobic interactions between protein molecules. Protein solubility was now seen to be a balance between a salting-in process due to electrostatic effects and a salting-out process due to hydrophobic effects:

Inf s i 1 Am""

" RT 1 + Bm"" + D&n RT'

(equation 6.3) where Sq is the protein solubility at zero salt concentration, R is the gas constant, T is the absolute temperature, A is a constant proportional to the net charge on the protein at low ionic strength, B and D are constants, 5 is the dipole moment o f the protein, is Avogadro's number, O is the surface area o f the dehydrated protein on precipitation, k^j

is the surface tension correction factor, is the molar volume o f the solvent and ap is the molal surface tension increment o f the salt (a constant which represents the change in surface tension which occurs when the salt is added to water).

The first term on the right hand side o f equation 6.3 accounts for electrostatic effects and was derived from a combination o f the Debye-Huckel and Kirkwood theories. The second term accounts for hydrophobic effects and was derived from the Cavity theory o f Sinanoglu and Abdulnar. In the salting-out region the Debye-Huckel term approaches a constant, p, and equation 6.3 can be reduced to:

In = J3 + Am - Qcr^m

(equation 6.4) In Melander and Horvath's (1977) interpretation, is the slope o f the intrinsic salting out line whose magnitude is determined by the hydrophobic contact area, Q, between the precipitating protein molecules and the molal surface tension increment o f the salt, while A is the slope o f the salting in line which expresses the effect o f the salt on the electrostatic free energy o f the protein and is linear at sufficiently high salt

concentration. Relating equation 6.4 back to the Cohn equation (equation 6.1) gives the following expression for Kg:

Ks = Qc7d~ A

(equation 6.5) A relationship between surface hydrophobicity and the Cohn constant Kg has been confirmed by Salahuddin et al. (1983) who correlated the increasing surface hydrophobicity o f seven natural proteins with increases in Kg.

Melander and Horvath (1977) have also correlated the molal surface tension increment o f a salt with its position in the lyotropic series (Section 1.2.3.2). This property was therefore suggested as a measure o f salting-out effectiveness which could be used to explain the relative protein solubility behaviour observed with different salts. The theory, however, is unable to account for the solubility behaviour observed with MgClj and CaClj which have a similar molal surface tension increment to sulphate salts but do not decrease protein solubility by the same degree (Richardson, 1987; Alsaffar, 1994).

An alternate theoretical approach to describe the solubility behaviour o f proteins in aqueous salt solutions has been given by Arakawa and Tim asheff (1984) in which preferential interactions are considered. For a protein in an aqueous salt solution, the salt ions may associate with it preferentially or the protein may be preferentially hydrated. The term 'preferential interaction' can be thought o f as both a measure o f the excess o f salt or water in the immediate vicinity o f the protein compared to their concentrations in the bulk fluid, and, thermodynamically, as a reflection o f the change in chemical potential o f the protein by the addition o f the salt or water. The degree o f preferential salt association/hydration can be experimentally determined by densimetric techniques and may be envisaged as the grams o f salt/water found in a dialysis bag per gram o f protein compared to that found outside.

Using this approach, Arakawa and Timasheff (1982) found that protein was preferentially hydrated in concentrated solutions o f a number o f sodium salts. This observation is somewhat at odds with the previous theories o f protein solubility behaviour in which salting-out occurs due to the destruction o f structural water surrounding hydrophobic areas o f the protein and a decrease in protein hydration could therefore be expected. Arakawa and Timasheff (1982) were able to correlate the

preferential hydration o f proteins by most salts to their molal surface tension increments. In the case o f divalent cation salts, however, no correlation was found and it was shown that the binding o f the divalent cations overcomes salt exclusions due to the surface tension increase and leads to a decrease in preferential hydration (Arakawa and Timesheff, 1984). It was further shown that the order o f the preferential interaction o f both anions and cation salts followed the lyotrophic series and therefore preferential interaction may be used as a measure o f salting-out effectiveness (Arakawa and Timesheff, 1984).

A further theoretical approach is based on the concepts o f colloidal science (Rothstein, 1994). Much o f this work has its basis in the DLVO (Deryagin-Landau and Verwey- Overbreek) theory which describes surface interactions between colloidal particles. Mahadevan and Hall (1990) used this type o f approach to develop a model for protein precipitation with non-ionic polymers. The precipitation was treated as an equilibrium phase separation using techniques o f statistical mechanics. Models o f protein and polymer interactions were used to calculate an effective protein-protein interaction potential (known as a potential o f mean force). Perturbation theory was then used to calculate chemical potentials and pressure for each phase and a thermodynamic stability analysis employed to predict phase transformation and thus generate a phase diagram. Vlachy et al. (1993) extended this approach to describe protein precipitation using electrolytes by considering repulsive electrostatic interactions, attractive van der Waals interactions and osmotic attraction.

This work has been developed further by Kuehner and co-workers (1996) to give a description o f the solubility behaviour o f proteins in aqueous salt solutions. The salting- out o f protein is considered to be a liquid-liquid phase separation based on the observation o f Shih et al. ( 1992) that salted-out protein precipitates are a dense liquid phase rather than a pure solid. A random-phase approximation (RPA) is used in order to calculate chemical potentials and pressures in each phase. The RPA approach involves defining a system o f hard spheres as a reference condition while remaining spherically- symmetric interactions provide perturbations. An interaction potential takes into account electrostatic, dispersion, osmotic and specific interactions. Repulsive electrostatic interactions are described by the Debye-Huckel theory, attractive dispersion forces are represented using DVLO theory, description o f strong short-range osmotic

attraction due to depletion o f ions between protein molecules is based on the work o f Asakura and Oosawa (1958) and specific interactions, such as hydrophobic effects or self-association, are incorporated using an attractive square-well potential. A consideration o f thermodynamic equilibrium conditions enables a description o f the partitioning o f protein between the phases to be obtained and protein partitioning coefficients calculated. Results for the extent o f precipitation obtained were shown to be a strong function o f parameters used in the square-well potential describing specific interactions. Using this approach, Kuehner et al. (1996) were able to show qualitative agreement between model results and experimental data for the precipitation o f hen-egg white lysozyme and a-chymotrypsin using ammonium sulphate.

6.2.1.3 M ixing C onsiderations

The theoretical descriptions for protein solubility previously described consider precipitation to be an equilibrium phase separation. However, the work o f Foster et al.

(1976) showed that different protein solubility profiles were obtained with different contacting schemes, which led them to conclude that precipitation is a pseudo­ equilibrium process. This is thought to result in over-precipitation when local concentrations o f salt are higher than the bulk concentration. For an equilibrium process, the over-precipitated protein will re-dissolve once the local concentration is reduced to that o f the bulk. However, for a pseudo-equilibrium process the precipitate is 'locked-in' and will not re-dissolve, possibly due to irreversible internal bonding or greatly reduced surface free energy giving the precipitate kinetic stability (Richardson, 1987). The 'freezing' o f the size distribution o f salted-out casein precipitate by dilution with buffer containing ammonium sulphate at the precipitation concentration (Hoare, 1982b) provides some evidence for the existence o f irreversible precipitates.

The existence o f over-precipitation will depend on mixing conditions, since if mixing is o f the same order or slower than precipitation the protein will experience local salt concentrations higher than that o f the bulk (Foster et a l, 1976). Based on estimates o f the precipitation and mixing rate, Richardson (1987) concluded that application o f a simple model for irreversible over-precipitation could explain the solubility behaviour o f bakers' yeast proteins during ammonium sulphate precipitation, although the need for detailed experimental data to substantiate such a model was recognised.

Iyer and Przybycien (1994) have developed a model for the fed-batch ammonium sulphate precipitation o f catalase which considers the impact o f mixing conditions on over-precipitation. The extent o f mixing in a vessel can be characterised by both macromixing and micromixing. Macromixing is the dissipation o f bulk fluid elements, while micromixing is mixing due to molecular diffusion and gives an indication o f the extent o f molecular homogeneity. This was represented by considering the precipitation reactor volume to be made up o f completely segregated zones and a molecular dissipation zone. The breakup o f segregated zones into smaller eddies was assumed to occur as a first-order dissipation process. The eddies are assumed to reach their limiting size once they are reduced to the Kolmogoroff scale, X, and are then known as points. The Kolmogoroff scale is given by:

À =

(equation 6.6) where e is the power dissipation per unit volume and v is the kinematic viscosity. The zone in which eddies are reduced to the Kolmogoroff scale is called the molecular dissipation zone, and the precipitation o f protein is assumed to occur here. Micromixing in this zone is driven by the concentration gradients between a given point and the surrounding points.

The points in the molecular dissipation zone can be classified depending on their starting conditions as either those formed from the protein solution or those formed from the dissipation o f added salt solution. When the protein concentration at a given point exceeds its solubility, as given by the Cohn equation (equation 6.1) precipitation occurs, with the precipitation reaction consisting o f protein dehydration, aggregation o f dehydrated protein and salt ion hydration. To form the model, mass balance equations are described for each point and integrated using numerical methods to give the overall mass o f protein precipitated. The model results obtained showed good qualitative agreement with experimental results for 1 L batch and semi-batch precipitations.

Iyer and Przybycien (1994) concluded that the existence o f a kinetically irreversible precipitation depends on the relative rates o f desolvation, aggregation and mixing.

which in turn depend on the particular protein and salt, as well as vessel geometry, scale and operating conditions. Suggested scale-up parameters for precipitation were the protein concentration and the dimensionless salt feed rate, Fg/(V.k^j), where Fg is the precipitant feed rate, V is the reactor volume and is the micromixing rate.