## Info

Note: BSA, bovine serum albumin; IgG, immunoglobulin G.

Note: BSA, bovine serum albumin; IgG, immunoglobulin G.

noncompetitive multicomponent extensions of the Lang-muir equation. Neither model gave an accurate correlation to the data, although from this comparison and from additional chromatographic data it was concluded that competitive adsorption was apparent but that additional factors such as interprotein interactions were likely important as well. More recently, Lassen and Malmsten (10) reported on a study of multicomponent adsorption from ternary mixtures of human serum albumin (HSA), human immunoglobulin G (IgG) and human fibrinogen (Fgn) in 0.01 M phosphate buffer on three different polymer surfaces. The three surfaces were prepared to yield a spectrum of overall surface properties covering hydrophobic and hydrophilic with either positive or negative overall charge, respectively. For all surfaces, the single component isotherms were approximately Langmuirian in shape, with the relative maximum quantities adsorbed falling in the pattern Fgn > IgG > HSA. Adsorption from the multicomponent mixtures was notable in that (1) the total quantity adsorbed relative to the single component isotherms was reduced, and (2) the emergent dominant protein varied depending on the specific surface evaluated. Overall, the observed ternary behavior could not be readily predicted from the single component data.

Although not strictly of concern with respect to equilibrium loadings, the so-called Vroman effect, through which some protein mixtures exhibit sequential adsorption with smaller molecular weight proteins generally adsorbing early only to be later displaced by larger molecular weight proteins, has been noted for several protein-adsorbent systems (11). This rate-based competitive effect could occur within the time scale of practical application in separations, and thereby provide a contributing factor to adsorp-tive process performance.

The complex adsorption behavior observed even in relatively simple binary or ternary mixtures amply highlights the limitations of adsorption models and weakness of extensions thereof to multicomponent mixtures. It is therefore clear that the development of batch adsorption separation processes must be firmly based on experimental data.

Implications of Adsorption Performance to Separations.

Equilibrium adsorption capacities (or q vs. C isotherms) define the first criteria for adsorption process feasibility. If capacity appears sufficient and if competitive adsorption is not a major problem, then the next concern would be fractional recovery. As discussed by Scopes (12), for practical applications that use small or modest volumes of adsorbent relative to solution (e.g., a ratio of volume adsor bent Va to volume solution Vs of ~0.1), the separation factor a, defined by a = q/(C + q), should be greater than approximately 0.98. This in turn implies that Keq values less than about 10~8 M are needed to achieve high fractional recoveries (e.g., a recovery of >90% in one adsorp-tive stage).

Assuming loading capacity and fractional recoveries from solution would be practical, the next major challenge in applying batch adsorption is to identify appropriate combinations of adsorbent and solution properties to provide a useful selectivity. Selectivity among a mixture of n components may be defined by

where subscript 1 denotes the desired product and i denotes some unwanted component, with i ranging from 2 to n. The required value for S depends of course on the objective of the separation. To achieve a 10-fold purification from an equimolar solution, for example, would require that S be greater than 10. High selectivity may be achievable through a particular combination of adsorption and desorption conditions; however, it is uncommon for most general adsorbents to provide high selectivity between many proteins. Because of the broad range of proteins generally present in bioseparation feedstreams (such as clarified broths or cell lysates originating from fermentation operations), the adsorbent candidates more likely to enable a successful application would employ specific interactions. Affinity adsorbents, for example, may be able to achieve such selectivity, and immunoadsorbents should be especially capable in light of the highly specific antigen-antibody complexation. If the selectivities of interest are low, highly effective separations could be more readily achieved through non-batch operations such as column chromatography, which can exploit even modest selectivity differences in equilibrium loadings or in relative rates of adsorption.

### Dynamics of Protein Adsorption

Kinetic and Mass Transfer Considerations. Separations of practical industrial application will generally require substantial adsorbent capacity. The large surface areas consequently required will usually be provided through extensive macroporous structures. A general schematic of the paths available for a protein molecule in the bulk liquid phase to adsorb to the surface of such an adsorbent is indicated in Figure 1. The overall process will likely involve several diffusional steps as well as multiple interactions with the adsorbent surface moities.

Figure 1. Schematic representation of available paths for protein adsorption on macroporous adsorbents.

Bulk liquid / 5

CB Cs

Figure 1. Schematic representation of available paths for protein adsorption on macroporous adsorbents.

The bulk phase in batch adsorption applications or experiments will normally be well mixed through appropriate design of mixing conditions in the contacting vessel. As such, the concentration of protein throughout the bulk phase will be uniform, as denoted by CB. Surrounding each adsorbent particle will be a stagnant fluid layer, the thickness of which (d) will depend on the hydrodynamic conditions. Diffusion through this film is normally approximated using a linear driving force in concentration

where J is the protein mass flux, D is the free solution diffusivity, and CS is the concentration at the external surface of the adsorbent. Diffusion within the macroporous structure is generally approximated by an equation of form

where ep is the adsorbent void fraction and De is the effective diffusivity, related to the free solution diffusivity by the approximation

where s is the tortuosity factor. There are a number of means to measure or estimate values for D and d (13,14) as well as s (15). Although the approximation of pore dif-fusional mass transfer with such simplistic equations allows for practical modeling and data correlation, the mechanistic inaccuracies are quite apparent. For example, because of the irregular geometry of the pores, there will be a large distribution of actual mass transfer trajectories (or random walks). Furthermore, because the proteins may be of comparable size to some pore passageways in certain adsorbents, there may be frequent opportunities for pore blocking, diffusional hindrances, or substantial variation in the number of contacts with the surface.

The kinetics of interaction between the protein in solution and the adsorbent surface will depend on many fac tors. Because the transitions in three-dimensional confor-mational states of the protein during the adsorption process are not well known, simplistic mechanisms are usually postulated in order to construct a mathematical representation. Normally, the adsorption kinetics are modeled using first- or second-order reaction rate equations to describe rates of transition between a few different states of protein or protein-adsorbent complexes. The kinetic constants thereby serve as adjustable model parameters. The rationale for the use and the accuracy of such representations is discussed in the next section.

Kinetic Studies: Ideal Surfaces and Industrial Adsorbents.

Recent progress has been made in the study of proteinadsorption kinetics using well-characterized surfaces. In contrast to macroporous industrial adsorbents, in which diffusional limitations will usually be significant, planar surfaces within small experimental reactor volumes often will not be diffusion limited and hence can allow intrinsic adsorption rate measurement. Adsorption onto planar optical waveguide surfaces as monitored through reflectance techniques can be measured to within approximately ±10 molecules per,«m~2 at high sample frequencies (14). Kurrat et al. (16), for example, evaluated the kinetics of HSA and BSA adsorption onto a hydroxylated silica-titania surface. Approach to half saturation was achieved within 5 to 10 min, although adsorption toward equilibrium loadings continued out beyond 50 min. In developing and fitting kinetic equations, based on an elementary mechanistic model, they found that the model required at least two states of protein (distinguished by reversibly and irreversibly bound, respectively) to provide an acceptable fit to the data. Within the context of their equations, the adsorption kinetic constants ka were on the order of 10~6 cm/s.

Wahlgren and Eloffson (17) studied adsorption kinetics of /-lactoglobulins A and B to hydrophobic methylated silica waveguide surfaces. Adsorption of these proteins, which show complicating self-associating behavior in solution, proceeded to completion over the time course of approximately 50 min. In applying their kinetic model, which allowed for both a first-order surface conformational change as well as an exchange reaction between adsorbed and solubilized protein, a reasonable fit to the data was obtained. It appears that an increase in model accuracy, however, would have to come through an allowance of additional nonidealities, such as from the range of protein orientations and conformations, surface heterogeneity, and probability factors for interactions, steric hindrances, etc. Along these lines, Jin et al. (18) attempted to account for steric hindrance effects by applying random sequential adsorption (RSA) principles to the case where adsorption may proceed reversibly. The resulting formalism provided an explanation for why the apparent adsorption kinetic constants can show a functional dependence on surface coverage.

The state-of-the-art adsorption rate modeling exemplified in the works cited earlier are useful for correlation of data required for practical applications and for aiding the development of theoretical understanding of viable biophysical mechanisms. However, the data analysis and modeling efforts also highlight a fundamental limitation:

the inability to experimentally distinguish or measure protein conformational variations and the distribution thereof on adsorbent surfaces. Until further progress in technique is achieved, modeling and understanding will remain limited to semitheoretical models based on an indirectly assessed distribution of proteins between only a few postulated states of conformation and surface binding.

The kinetics of protein adsorption to industrial macro-porous adsorbents has been evaluated for many systems, of course, since the definition of adsorption rates is a prerequisite to application of any batch adsorption process. Basic rate data have generally been obtained through classical methods using either batch- or column-based experiments. As mentioned above, the adsorption dynamics will often have some significant diffusional resistances, and hence the kinetic data will reflect combined mass transfer and intrinsic adsorption reaction rates.

Modeling of Adsorption Process Dynamics. The ultimate success of potential applications of batch adsorption separation depends on the adroit manipulation of the ther-modynamic and kinetic properties of the adsorbent-solute system to achieve high process efficiency. This aim can best be realized through development and application of a mathematical model for the process dynamics, which necessarily would be based on the underpinning factors governing intrinsic adsorption kinetics, equilibrium phase distribution, and mass transfer effects. The model used to simulate the process dynamics would facilitate the following:

1. Feasibility analysis. To assess the performance, whether for batch or column mode, for approximate performance in terms of yield and purification achievable

2. Process design. To enable detailed design, including scale-up effects, as well as to perform operational analyses to ascertain cycle times, economics of operation, etc.

3. Optimization. To readily calculate effects resulting from changes to the myriad of variables, such as adsorbent properties, relative volumes, initial concentration, contact times, regeneration procedures, etc., all of which could have substantial impact on process viability

Modeling of adsorption process dynamics is straightforward in principle but requires numerous simplifications at the detail level in order to retain mathematical tractability as well as biophysical reality in the adjustable model parameters. The general approach follows directly from the differential equations for mass balance, mass transfer, and kinetics for adsorption and any additional reactions. The process model, which is the resulting system of differential equations, will vary in complexity depending on the simplifying assumptions used to approximate the collective molecular behavior, which is usually highly complex because of the distribution of interaction rates and pathways occurring throughout an irregular three-dimensional geometry.

A number of models for protein adsorption within a finite batch have been published. As an example, Mao et al. (19) have presented a model for batch protein adsorption applicable specifically for nonporous adsorbents along with a simplified extension for porous adsorbents. By combining variants of equations 1 and 2, along with the differential form of the overall mass balance and an assumed second-order adsorption kinetic equation, the resulting equations could then be integrated to provide C and q as functions of time. The parameters in their model were first fitted to adsorption data for single component proteins HSA, lysozyme, and ferritin, as adsorbed onto either dye-affinity or ion-exchange adsorbents. In assessing the model fits, it was observed that the sensitivity of the models with respect to the kinetic constants may not be significant enough relative to the accuracy of the adsorption data to accurately ascertain true values of parameters. This conclusion may generally be extended to more elaborate models as well, where the accuracy of the fitted model may be improved by including additional kinetic parameters to account for additional reactions (states of conformation, etc.), but where the resulting understanding and the accuracy of extrapolation using the models is not improved. Mao et al. further used the models to evaluate relationships between various operational parameters and system performance. For example, calculations showed that the initial concentration of HSA could significantly affect both the final concentrations as well as the time required to approach equilibrium (e.g., time to 90% of final C/Co), where a Co varied from 0.8 to 0.05 mg/mL increased the required adsorption time from about 10 to more than 30 minutes. Of particular note was the extreme sensitivity of both the rate of adsorption and the equilibrium loadings to the ratio of adsorbent volume to liquid volume, Rv(Va/Vs). This would clearly be an important parameter to routinely evaluate in process design and operational optimization.

These sample calculations highlight the real utility of such models, which is that once proven capable of accurately representing the mass transfer and kinetic reaction rates of adsorption, they can be readily applied to adsorp-tive system design and optimization for either batch or column modes of operation.

### Implications and Applications to Batch Separations.

Knowledge of the mass transfer and kinetic limitation principles provides several general guidelines. In terms of process timing, for many systems it can be appreciated that equilibrium loadings may take hours to attain, but practical process-type loadings may be achieved on the order of 10 minutes. Analogously, adsorbent regeneration will often require hours, depending on the conditions employed to induce complete desorption and the requirements for regenerated adsorbent site occupancy. Although some approximate predictions of capacity and adsorption rates could be made by considering the basic features of the adsorbent (particle diameter, average pore size, primary mode of interaction with the protein, etc.) and of the protein in solution (molecular weight, size and shape, average surface potential, etc.), it is clear that acquisition of adsorption rate data for the specific system under consideration will always be an important prerequisite to suc cessful process applications. Along these lines, relative to column-based experiments, batch experiments will generally be easier to perform and will provide more accurate information on the fundamentals of equilibrium capacity and limitations for mass transfer and kinetics of adsorption for particular protein-adsorbent systems.

The rationale for successful batch adsorption applications reviewed later in this article can be readily interpreted from the performance characteristics of relative adsorption rates and capacities. For example, it can be seen that if fast adsorption is required because of product lability, then a batch mode may be preferred to the column mode of operation. If speed and overall cycle time is a main driver, it has been shown (20) that a batch system may be more efficient than a column-based system under some conditions. Batch-contacting systems may also be suggested if there are other hydrodynamic factors, such as high solids/debris content or high viscosity which would negate other column operational advantages. Batch operations will also allow greater flexibility in choice of adsorbent, since even highly compressible gels can readily be contacted with the solution. On the other hand, if a high purification factor is desired from the adsorptive separation process, then a column mode of differential protein-adsorbent contact will generally be required other than for highly specific affinity adsorbents.

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