zonal rate model for stacked membrane chromatography part ii: characterizing ion-exchange membrane...

EDITORS’ CHOICE

Zonal Rate Model for Stacked MembraneChromatography Part II: CharacterizingIon-Exchange Membrane ChromatographyUnder Protein Retention Conditions

Patrick Francis,1,2 Eric von Lieres,3y Charles Haynes1,2z

1Michael Smith Laboratories and the Biological Engineering, University of British Columbia,

Rm. 301, 2185 East Mall, Vancouver, British Columbia, Canada V6T 1Z3;

telephone: 604-822-5136; fax: 604-822-2114; e-mail: [email protected] of Chemical and Biological Engineering, University of British Columbia,

Vancouver, British Columbia, Canada V6T 1Z33Institute of Bio- and Geosciences 1, Research Center Julich, Wilhelm-Johnen-Strasse 1,

52425 Julich, Germany; telephone: þ49-2461-61-2168; fax: þ49-2461-61-3870;

e-mail: [email protected]

Received 24 June 2011; revision received 5 October 2011; accepted 10 October 2011

Published online 19 October 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/bit.24349

ABSTRACT: The Zonal Rate Model (ZRM) has previouslybeen shown to accurately account for contributions toelution band broadening, including external flow nonide-alities and radial concentration gradients, in ion-exchangemembrane (IEXM) chromatography systems operated un-der nonbinding conditions. Here, we extend the ZRM toanalyze and model the behavior of retained proteinsby introducing terms for intra-column mass transfer resis-tances and intrinsic binding kinetics. Breakthrough curve(BTC) data from a scaled-down anion-exchange membranechromatography module using ovalbumin as a model pro-tein were collected at flow rates ranging from 1.5 to20mLmin�1. Through its careful accounting of transportnonidealities within and external to the membrane stack, theZRM is shown to provide a useful framework for character-izing putative protein binding mechanisms and models, forpredicting BTCs and complex elution behavior, includingthe common observation that the dynamic binding capacitycan increase with linear velocity in IEXM systems, and forsimulating and scaling separations using IEXM chromatog-raphy. Global fitting of model parameters is used to evaluatethe performance of the Langmuir, bi-Langmuir, steric massaction (SMA), and spreading-type protein binding modelsin either correlating or fundamentally describing BTC data.When combined with the ZRM, the bi-Langmuir, and SMAmodels match the chromatography data, but require physi-cally unrealistic regressed model parameters to do so. In

contrast, for this system a spreading-type model is shown toaccurately predict column performance while also providinga realistic fundamental explanation for observed trends,including an observed increase in dynamic binding capacitywith flow rate.

Biotechnol. Bioeng. 2012;109: 615–629.

� 2011 Wiley Periodicals, Inc.

KEYWORDS: membrane chromatography; modelingprocess optimization

Introduction

The general rate model (GRM) of nonlinear chromatogra-phy quantifies protein binding to the stationary phase byexplicitly accounting for the intrinsic kinetics of theadsorption process, as well as the rates of solute masstransport within the mobile phase and through the film andpore volumes of the stationary phase. Though relativelycomprehensive, the GRM can be operationally cumbersomeas a tool for designing and simulating chromatographicseparation systems as it requires values for a large number ofparameters, some of which are difficult to quantifyaccurately and unambiguously, due in part to differencesin the rates of the various transport and binding sub-processes leading to the bound state. However, byaccounting for these differences, simplified and typicallyequally accurate models of chromatography can be derivedin which an individual or lumped event is assumed tocontrol the overall rate of protein transport and bindingto the stationary phase. For example, in conventional

yCanada Research Chair in Interfacial Biotechnology.zHead of Modeling and Simulation.

Correspondence to: Charles Haynes and Eric von Lieres

Contract grant sponsor: Natural Sciences and Engineering Research Council of

Canada (NSERC)

Contract grant sponsor: Canadian Institutes of Health Research (CIHR)

� 2011 Wiley Periodicals, Inc. Biotechnology and Bioengineering, Vol. 109, No. 3, March, 2012 615

preparative columns bearing a closely packed array ofporous particles, the average time required for proteindiffusion through the external liquid film and/or pores ofthe matrix often dictates the overall rate of adsorption(Ghosh, 2002; Suen and Etzel, 1992), and this characteristictime can be modeled with a first-order rate equationcontaining a single ‘‘lumped’’ mass transfer coefficient.When the stationary phase is a monolith or membrane stack,proteins are convected across binding sites and a simplifiedform of the GRM that precludes pore diffusion can beapplied. Through the elimination of intrinsically slow porediffusion processes, rates of mass transfer are greatlyincreased and band profiles, product yield and columnthroughput are typically controlled by the kinetics andthermodynamics of protein-sorbent complex formation(Charcosset, 2006; Ghosh, 2002; Suen and Etzel, 1992; vanReis and Zydney, 2007), particularly at high feed concen-trations (Briefs and Kula, 1992; Gebauer et al., 1997; Klein,2000; Liu and Fried, 1994; Roper and Lightfoot, 1995a; Suenand Etzel, 1994; Thommes and Kula, 1995). Accuratemodeling of breakthrough and elution bands for proteinsloaded onto these enhanced mass-transfer matrices there-fore requires an appropriate model describing the intrinsicbinding kinetics (and thermodynamics), often termed thebinding rate (or isotherm) model, as well as the accuratedetermination of all parameters within that model.

Selection of an appropriate binding model is madedifficult by the fact that available theories for proteinadsorption oversimplify the biophysics of the process,particularly as it occurs in preparative liquid chromatogra-phy. Indeed, the models of binding kinetics and equilibriathat are most widely applied to protein adsorption—theindividual and multicomponent forms of Langmuir theo-ry—are simple extensions of theories intended to describeadsorption of small noninteracting molecules from the gasphase. The adsorption of proteins from aqueous solutiononto porous and often chemically heterogeneous solidmatrices is considerably more complex. For example, thelarge size and many conformational degrees of freedom ofproteins, the competition between the protein macro-ionand the solvent for surface binding sites of the same surfaceenergy or differing surface energies, the potential for eitherorientationally specific or random binding of the sorbate,the ability or inability of the adsorbed protein to diffuse onthe sorbent surface, and any number of additional nonidealinteractions between components within the solution andbound at the sorbent surface may all contribute to theadsorption rate, energetics, and equilibria. A large numberof semi-theoretical models for isothermal protein adsorp-tion have therefore been derived to try to account for thesenonidealities, including the original and modified forms ofthe multi-site isotherm (Boi et al., 2007; Briefs and Kula,1992; Suen and Etzel, 1992), random-sequential-adsorptionisotherm (Talbot et al., 2000), spreading isotherm (Clarket al., 2007; Lundstrom, 1985), and steric mass-action(SMA)/hindrance isotherm (Brooks and Cramer, 1992).Each of these models yields expressions for binding kinetics

and equilibrium. However, each typically accounts for onlyone of the adsorption nonidealities noted above, in largepart because such simplification is required to reduce thecomplexity and parameterization of the model so as topermit accurate fitting to a limited set of data, in thisinstance typically obtained on-column. Through that fittingprocess and any additional knowledge of the nature of thebinding reaction, one hopes to first identify a binding modelthat can provide a quantitative description of the proteinretention process of interest and then determine if thatmodel provides any fundamental insights into the nature ofthe process, as opposed to simply providing a usefulcorrelation function. This latter objective could proveparticularly useful to the further development of membranechromatography of proteins by ion exchange (IEX), asimportant questions remain regarding the relationshipbetween intra-column events and the elution band profilestypically observed in these preparative separation systems.In particular, even for well-designed IEX membranes,breakthrough curve (BTC) asymmetry characterized by asharp initial breakthrough followed by a slow approach tosaturation is often observed (Boi et al., 2007; Dancette et al.,1999; Ghosh and Wong, 2006; Kochan et al., 1996;Montesinos-Cisneros et al., 2007; Sarfert and Etzel, 1997;Serafica et al., 1994; Yang and Etzel, 2003). Moreover, aswith traditional packed-bed chromatography, the dynamicbinding capacity often depends on linear velocity. However,that dependence is more complex in membrane IEXadsorbers, with several studies reporting a nonlinear increasein dynamic binding capacity with increasing feed velocitywithin the low to moderate flow regime (Kim et al., 1991;Montesinos-Cisneros et al., 2007; Shiosaki et al., 1994).

To date, protein binding to membrane adsorbers hasmost often been modeled using standard Langmuir kinetics(e.g., Gebauer et al., 1997; Suen and Etzel, 1992; Thommesand Kula, 1995; Yang and Etzel, 2003). However, whencombined with appropriate representations of the GRM,Langmuir theory cannot capture the complex elutionbehavior described above, and this has motivated a numberof researchers to seek more realistic binding models thatmight serve to improve predictions and advance ourknowledge of protein retention mechanisms in IEXmembrane (IEXM) chromatography systems. For example,Sarti and coworkers (Boi et al., 2007) greatly improved BTCmodeling by implementing bi-Langmuir adsorption kinet-ics. Their approach therefore assumes that heterogeneity inthe binding energy of the sorbent largely accounts for theobserved complex elution behavior. Alternative theorieshave been advanced by Yang and Etzel (2003), who showedthat BTC asymmetry in IEXM columns can also be describedusing either a steric-hindrance/SMA type model to accountfor the shielding of binding sites by large protein moleculesor a spreading type model that considers the conformationalchanges a protein may undergo when bound to the sorbent.

Here, we build on these important studies with the aim ofestablishing a more rigorous method for best defining therate-limiting transport processes and the general binding

616 Biotechnology and Bioengineering, Vol. 109, No. 3, March, 2012

mechanism of a target protein within a stacked IEXMcolumn. A necessary component of establishing this model-building formalism is the accurate accounting of contribu-tions to band spreading occurring in the pre- and post-column hold-up volumes so as to properly isolate andcharacterize solute dispersion and mass transport within themembrane stack itself. Due to the low length to diameterratio of stacked membrane chromatography columns,dispersion resulting from mixing, channeling, and otherprocesses leading to unequal solute residence times in theextra-column volumes is known to contribute to elutionband broadening (Francis et al., 2011; Ghosh and Wong,2006; Lightfoot et al., 1995; Roper and Lightfoot, 1995b).Several models have therefore been proposed to account forthese extra-column nonidealities, with most employing aplug flow reactor (PFR) in series with a stirred tank reactor(CSTR) (e.g., Boi et al., 2007; Sarfert and Etzel, 1997; Yanget al., 1999; Yang and Etzel, 2003). Roper and Lightfoot(1995b) extended this concept with a second PFR/CSTRsequence following the membrane stack to address solutedispersion in the eluent collection manifold. However, it isnow known that models that incorporate a linear PFR/CSTRsequence on either one or both sides of the membrane stackdo not in general provide a quantitative description ofbreakthrough from axial-flow IEXM modules (Montesinos-Cisneros et al., 2007; Sarfert and Etzel, 1997; Yang et al.,1999).

To address this shortcoming, we recently described a‘‘Zonal Rate Model’’ (ZRM) (Francis et al., 2011) thataccurately accounts for solute dispersion within membranechromatography modules, including pre- and post-columndispersion resulting from radial flows within the hold-upvolumes. As illustrated in Figure 1, these radial flows causethe path length of proteins entering a membrane stack nearits outer radius to be significantly longer than that forproteins passing through the centerline of the feed manifold.

Column loading therefore does not occur in the desiredplug-flow manner, resulting in solute breakthrough fromthe centerline of the membrane stack prior to saturation ofbinding sites in the outer radial positions, even when theaxial velocity through the stack is invariant of radial andangular position. The ZRM quantitatively captures thisnonideality by extending sequential PFR/CSTRmodels (e.g.,Boi et al., 2007; Roper and Lightfoot, 1995b; Sarfert andEtzel, 1997; Yang et al., 1999; Yang and Etzel, 2003) throughthe partitioning of the membrane stack and its associatedpre- and post-column hold-up volumes into virtual zonesthat account for differences in radial path lengths traveled bysolute molecules within the feed and eluent distributors ofthe column. A defined set of pulse or frontal loadingexperiments under nonretention conditions are used todetermine the extra-column parameters and zone configu-ration of the ZRM, as well as the axial dispersion coefficientDax for solute transport within the membrane stack.

Application of the ZRM therefore provides a frameworkto isolate and analyze elution band broadening resultingfrom solute transport and intrinsic binding within themembrane stack. Dimensionless groups may then beevaluated to simplify the GRM to its most appropriateform, and isotherm and breakthrough data used to identifyboth an appropriate binding model and the dominantnonidealities associated with the binding mechanism. Wetest this formalism using frontal loading of ovalbumin ontoa Mustang Q XT5 anion-exchange membrane chromatog-raphy capsule as a model system. Appropriate forms ofLangmuir, bi-Langmuir, SMA, and spreading theory areintroduced into the ZRM and then evaluated for their abilityto both correlate and fundamentally describe breakthroughbehavior. As part of this process, we also investigate thecontribution of filmmass transport to protein breakthroughfrom IEXM modules.

Theory

Zonal Rate Model

The elements of the ZRM required to quantitatively describeelution-band profiles of nonretained proteins were de-scribed fully in a recent publication (Francis et al., 2011).The basic structure of the ‘‘nonretention’’ ZRM (nr-ZRM) isshown in Figure 2 for the specific case where nonuniformflows and dispersion within the extra-column volumes aremodeled with two radial zones. The plug-flow reactor (PFR)accounts for a time lag that is uncoupled from the systemdispersion

cPFRout ðtÞ ¼0 t < tlagco t � tlag

�(1)

where co is the protein concentration in the feed and thetime lag, tlag, is equal to the ratio of the PFR volume to feedflow rate,VPFR/Q. Since the PFR provides a dead time but no

Figure 1. Representative path lengths for solute flow within a stacked-mem-

brane chromatography module possessing both axial and radial flow components

within the feed distribution and eluent collection manifolds. Each shaded area

indicates a radially defined zone in the manifold offering a distinct solute residence

time, with the length of the flow line through the zone reflecting the time. Qi represents

the flow rate through zone i.

Francis et al.: Zonal Rate Model for Stacked Membrane Chromatography 617

Biotechnology and Bioengineering

dispersion, its relative position in the flow network isarbitrary, and two or more such dead times within thenetwork can be modeled with a single PFR.

Hold-up volumes within which solute is dispersed, suchas the inlet and outlet distributors of an IEXM column, arepartitioned into virtual CSTRs. Each virtual CSTR i ismodeled as

@cCSTRi

@t¼ cCSTRi

in � cCSTRi

ti(2a)

where ti is the solute residence time within CSTRi and isgiven by Vi/Qi, where Vi is the virtual volume of i and Qi isthe volumetric flow rate through it; cCSTRi

in and cCSTRi arethe solute concentrations in the tank inlet and outlet,respectively. Downstream of the membrane stack, the ZRMtypically includes at least one virtual CSTR with multipleinflows (Fig. 2); for these CSTRs the following version ofEquation (2) is used

@cCSTR

@t¼

Xmj¼1

cCSTRin;j

tj� cCSTR

Xmj¼1

1

tj(2b)

where j is the number of tank inflows. The ZRM alsorequires a set of flow fractions,Fk, which define the fractionof the total volumetric flow, Q, passing through each radialzone k as described in our previous work (Francis et al.,2011).

The nr-ZRM requires solution of Equation (1), followedby the simultaneous solution for each zone of Equation (2a)for the inlet CSTR, the continuity equation for thenonretained noninteracting solute

@c

@tþ n

@c

@z¼ "mDax

@2c

@z2(3)

and either Equations (2a) or (2b) for the outlet CSTR. InEquation 3 c(z,t) is the solute concentration in the mobile

phase at time t and distance z from the column inlet, and v isthe interstitial velocity given by the ratio of the superficialvelocity u to the membrane porosity em. This modelaccurately predicts elution band profiles under nonbindingconditions, as we previously demonstrated using a two-zoneform of the nr-ZRM applied to frontal loading of ovalbuminthrough a Mustang Q XT5 capsule under nonbindingconditions (Francis et al., 2011).

Here we extend the ZRM to describe retention andbreakthrough of ovalbumin frontally loaded on theMustangQ XT5 column under binding conditions. Equation (3) isreplaced with the continuity equation for a retained species(Briefs and Kula, 1992; Suen and Etzel, 1992; Thommes andKula, 1995)

@c

@tþ n

@c

@z¼ Dax

@2c

@z2� ð1� "mÞ

"m

@s

@t(4)

which is solved for each zone in time and space as beforewith Equations (1), (2a), and/or (2b). In Equation (4), s(z,t)is the total solute concentration in the stationary phase ofthe zone. For each zone, Danckwerts’ boundary conditionsare applied at the column inlet (z¼ 0) and outlet (z¼ L)(Barber et al., 1998):

vcinðtÞ ¼ vcðtÞ � Dax

@c

@z

��z¼0

(5a)

@c

@z

��z¼L

¼ 0 (5b)

with cin(t) given by Equations (2a) or (2b).In both the full ZRM and the nr-ZRM, the membrane

stack of thickness L is considered homogeneous in itsproperties across all zones. Consequently, each membranezone is treated identically in terms of its physical parametersbut is subject to different boundary conditions based on itsposition in the network of virtual CSTRs. This allows solutetransport and binding within the membrane stack of agiven zone k to be modeled as an isolated column ofthickness L and cross-sectional area Ak without radialconcentration gradients. The ZRM therefore computes andrepresents concentration gradients in the r dimension of thecolumn and the extra-column volumes as a staircase ofconcentrations.

In Equation (4), the volume of the stationary phasecomprises both the volume of the solid sorbent and that ofthe liquid film that surrounds the sorbent. The sink term istherefore given by

@s

@t¼ "f

@cf@t

þ ð1� "fÞ @q@t

¼ 4

dpkfðc � cfÞ ¼ k0fðc � cfÞ (6)

where ef is the fraction of the total stationary phase volumeoccupied by the liquid film, cf is the protein concentrationwithin the film, q is the sorbate concentration, kf is the film

Figure 2. Stirred tank and membrane network representation of a stacked-

membrane chromatography module with two radial zones. The total flow rate, Q, is

split between zone 1, Q1, and zone 2, Q2. The stirred tank network is symmetric in that

the residence times, ti, of the two tanks in each radial zone are equivalent.


mass-transfer coefficient, and k0f is (4/dp)kf where dp is thenominal pore diameter of the membrane and 4/dp gives thepore surface area to volume ratio.

Equation (6) solves for the time derivative of the soluteconcentration in the film, cf, when the time derivative of q isgiven by the binding model. This careful treatment of filmmass transfer was not considered in previous models ofmembrane chromatography (Boi et al., 2007; Dancette et al.,1999; Kochan et al., 1996; Montesinos-Cisneros et al., 2007;Suen and Etzel, 1992, 1994; Thommes and Kula, 1995; Yangand Etzel, 2003). However, we believe it is both rigorouslycorrect and very useful, in part because it properly partitionsthe column volume into three distinct volume fractions: thefractions occupied by the mobile phase em, the stagnantfilm (1� em)ef, and the stationary phase (1� em)(1� ef),respectively. Technically the column volume could bepartitioned differently, but the volume fractions used hereoffer the advantage that the resulting model equations areanalogous to those of the linear driving force model forpacked-bed chromatography (Tejeda-Mansir et al., 2001).

Intrinsic Binding Rate Models

Solution of Equations (4) and (6) requires an appropriatemodel describing the intrinsic kinetics of protein binding toand desorption from the sorbent. Models that have beenused to describe membrane adsorbers and will be examinedhere include the traditional Langmuir model (e.g., Gebaueret al., 1997; Suen and Etzel, 1992; Thommes and Kula, 1995;Yang et al., 1999), the bi-Langmuir model (Boi et al., 2007),the steric mass-action (SMA) model (Brooks and Cramer,1992; Yang and Etzel, 2003), and a general form of thespreading model (Clark et al., 2007; Lundstrom, 1985; Yangand Etzel, 2003). Protein binding rates are described in thetraditional Langmuir model as

@q

@t¼ kacfðqm � qÞ � kdq (7)

where ka and kd are the adsorption and desorption rateconstants and qm is the saturation capacity. This simplest ofbinding isotherm models assumes sorbate molecules arenoninteracting and that all binding sites on the sorbent areidentical and independent.

In bi-Langmuir theory, where the sorbent is modeled asoffering two energetically distinct types of independentbinding sites, protein binding kinetics are described by

@q

@t¼ @q1

@tþ @q2

@t(8)

@q1@t

¼ ka;1cfðqm;1 � q1Þ � kd;1q1 (9a)

@q2@t

¼ ka;2cfðqm;2 � q2Þ � kd;2q2 (9b)

where q1 and q2 represent the concentrations of proteinbound to sites of type 1 and type 2 respectively; ka,1, kd,1, andqm,1 are the binding parameters associated with the high-energy binding site; and ka,2, kd,2, and qm,2 are the bindingparameters associated with the lower-energy binding site.

The SMA model accounts for both the effect of stericshielding and the effect of co-ion (A) concentration on theadsorption of a protein macro-ion (P) of characteristiccharge v onto a sorbent (S). It is generally applied inits equilibrium isotherm form, but for protein bindingreactions of the form

Pv þ vðASÞ , vAþ PS (10)

it also yields intrinsic binding rate equations of the form

@q

@t¼ kacfq

vA � kdqc

vA (11)

where

qA ¼ L� ðv þ sÞq (12)

L is the ion capacity of the stationary phase, qA is theconcentration of bound salt co-ions that are available forexchange with the protein and s is the steric factor.

Finally, Clark et al. (2007) have developed a generalspreading model in which a protein molecule in theadsorbed layer can be reversibly transferred among acontinuum of bound states, each characterized by a uniqueaffinity and projected area of interaction due to changes inthe orientation or conformation of the bound protein (Dalyet al., 2003; Haynes and Norde, 1995; Sethuraman et al.,2004). Bound proteins at any state can also transfer to andfrom the bulk fluid. For the case where two adsorbed proteinstates (1 and 2) are considered, the general spreading modelreduces to the following rate equations:

@q

@t¼ @q1

@tþ @q2

@t(13)

@q1@t

¼ ðka;1cf � k12q1Þðqm;1 � q1 � bq2Þ � kd;1q1

þ k21q2 (14a)

@q2@t

¼ ðka;2cf þ k12q1Þðqm;1 � q1 � bq2Þ

� ðk21 þ kd;2Þq2 (14b)

where q1 and q2 represent the concentrations of boundprotein in orientational/conformational states 1 and 2respectively, b is the ratio of the sorbent surface areaoccupied by a bound protein in state 2 relative to that instate 1, and the rate constants are as defined in Figure 3.In Equation (14), qm,1 represents the hypothetical saturationcapacity of the stationary phase if all adsorbed proteins are



in state 1. The true qm is given by q1þ q2 at saturationconditions. This model is similar to one used previously tomodel reorientation of adsorbed proteins on a solid surface(Daly et al., 2003), but takes into account the surface area ofbound molecules as well as the dependence of spreading onsurface coverage. It reduces to the spreading modeldeveloped by Lundstrom (1985) and implemented inmembrane chromatography by Yang and Etzel (2003)when the spreading of the adsorbed protein is consideredirreversible and protein from the bulk cannot adsorb directlyto state 2 (i.e., when k21¼ ka,2¼ 0).

Each of these models is based on a distinctly differentmechanism of binding, yet all have been used to describeprotein retention in membrane chromatography columns.Our goal is to establish a framework that permits morerigorous evaluation of the merits of each model, particularlywith respect to providing a realistic interpretation ofthe intrinsic binding process within our model system. Wenote that since the protein binding reaction in themembrane chromatography system is conducted underflow, qm and qm,i in the rate models derived above aretypically replaced to improve model fit with theircorresponding dynamic (�) saturation capacities, q�m andq�m;i, respectively. As Q approaches zero, q�m and q�m;i

approach qm and qm,1, respectively. Above some critical flowrate, q�m and q�m;i will decrease with increasing Q.

Computational Methods

Through its partitioning of the upstream hold-up volume,membrane stack, and downstream hold-up volume into aset of unique flow zones, the ZRM comprises a large system

of coupled equations. Since the computational effort forsolving the coupled model equations and for estimatingunknown model parameters from flow and breakthroughdata increases nonlinearly with the number of equations andparameters, a high computation speed is crucial for thepractical applicability of the ZRM.We therefore developed anew highly efficient computation strategy as described indetail for the case where the protein is not retained (Franciset al., 2011). Here, we describe the added features of thecomputational algorithm when the ZRM is applied toprotein loading.

In the forward problem, where a breakthrough curve isdetermined for a given set of model parameters, the partialdifferential equations for the membrane region of each zoneare first discretized along the axial coordinate by the methodof lines (Schiesser, 1991). Those equations are then coupledwith the corresponding equations describing transportwithin the hold-up volumes of that zone, as well as withinthe overall flow network. The result is a large system ofordinary differential equations (ODE), or differential-algebraic equations in the case of the SMA isotherm, thatare solved simultaneously to synchronize the adaptive timestepping of the differential equation solver. The MATLABsolver ode15s (Shampine and Reichelt, 1997) is used for timeintegration and, in the case of algebraic equations, this solvermust be restarted at any step discontinuity in the boundaryconditions of the root tank.

In the inverse problem, unknown model parameters areestimated from chromatogram data by repeatedly solvingand updating the entire equation system using MATLAB’siterative optimization function lsqnonlin that uses analgorithm based on the interior-reflective Newton method(Coleman and Li, 1994, 1996). Local minima are avoided bya multi-start strategy with varied initial values. Unknownbinding parameters that need to be regressed from the datainclude the rate constants, the binding capacities, the axialdispersion coefficient and, in the case where film diffusionis nonnegligible, the stationary phase ratio ef and the filmmass-transfer coefficient kf. Universal regressions wereperformed simultaneously across all flow rates as someparameters, such as the adsorption and desorption rateconstants, are flowrate independent whereas others, such asthe rate constants associated with transformation betweenbound states as well as film diffusion coefficients, vary withflow rate.

Materials and Methods

Materials and Equipment

Mustang Q XT5 strong anion-exchange membrane chro-matography capsules (generously donated by Pall, Inc., EastHills, NY) containing modified hydrophilic polyethersul-fone (PES) membranes were used in this study. The heightand cross-sectional area of the stacked membrane bed inthese capsules are 2.20mm and 22.06 cm2, respectively. The

Figure 3. Schematic of the mechanism of adsorption of a protein molecule

according to the simplified spreading model, where protein molecules can reversibly

adsorb to a homogeneous binding surface in two different conformations/orientations.


hold-up volume is 3.21mL for both the feed-distributionand eluent-collection manifolds. The nominal pore size andporosity em of the membranes is 0.8mm and 0.70� 0.05based on data provided by the manufacturer. All experi-ments were performed on an AKTAexplorer 100 FPLCsystem controlled by Unicorn 4.12 software (AmershamPharmacia, Uppsala, Sweden).

Breakthrough and Equilibrium Analyses

Breakthrough experiments were performed under bindingconditions using 1mgmL�1 ovalbumin (Sigma, Oakville,Canada). Frontal analysis at 0.5mLmin�1 for a ladder ofovalbumin feed concentrations ranging from co¼ 0.2–10mgmL�1 was to determine equilibrium isotherm and(static) saturation capacity data under column loadingconditions. The equilibrium concentration of boundprotein q was computed from the BTC data asq ¼ R1

0 ðco � cÞðQ=VcolumnÞdt, where Q and Vcolumn arethe feed flow rate and column volume, respectively. Theload and elution buffers were 25mM Tris–HCl (pH 8.0)and 1M NaCl/25mM Tris–HCl (pH 8.0), respectively(Fischer Scientific, Ottawa, Canada) and were filtered anddegassed under vacuum prior to use. The membranecapsule was cleaned as per the manufacturer’s suggestedprotocol using a 1 N NaOH solution followed by a washwith 1M NaCl in 25mM H3PO4 (Fischer Scientific,Ottawa, Canada).

Results and Discussion

Dimensionless Group Analysis

In membrane chromatography, solute diffusion through thehydrodynamic boundary layer at the sorbent surface cancontribute to the overall resistance to protein mass transfer(Gebauer et al., 1997; Lightfoot et al., 1995; Sarfert and Etzel,1997; Yang et al., 1999). It is, however, typically neglected inmembrane chromatography models (Boi et al., 2007;Dancette et al., 1999; Kochan et al., 1996; Montesinos-Cisneros et al., 2007; Suen and Etzel, 1992, 1994; Thommesand Kula, 1995; Yang and Etzel, 2003). The justification fordoing so is generally made by comparing the characteristictime (tfilm) for film mass transfer to that for axial convection(tconv) through the entire membrane stack (Boi et al., 2007;Dancette et al., 1999; Kochan et al., 1996; Montesinos-Cisneros et al., 2007; Suen and Etzel, 1992; Yang and Etzel,2003):

tfilm ¼ d2p4Dax

� L

v¼ tconv (15)

However, as the intent of this analysis is to determinelimiting rates to local equilibrium, we feel it is better made bycomparing tfilm to tplate, the characteristic time for solute

convection through the interstitial volume of a singletheoretical mass-transfer unit. This ratio follows fromnondimensionalizing Equations (4) and (6) using the heightequivalent to a theoretical plate (HETP) as the characteristiclength scale:

@~c

@~tþ @~c

@~z¼ 1

Peax

@2~c

@~z2� 1� "m

"m

1

Pefð~c � ~cfÞ (16)

The nondimensionalized variables are then given by

~c ¼ c

co; ~t ¼ tv

HETPand ~z ¼ z

HETP(17)

Similarly, the rate equations for intrinsic proteinadsorption to the membrane can be nondimensionalizedwith respect to dp and kf. Here we show the resultantexpression for the Langmuir model:

@~q

@t¼ 1

Daf~cð1� ~qÞ � kddp

kf~q (18)

where the nondimensionalized variables for the stationaryphase are given by:

~q ¼ q

qmand t ¼ tkf

dp(19)

Three dimensionless groups are thus defined: an axialPeclet number, Peax ¼ vHETP=Dax, a film Peclet number,Pef ¼ ð1=4Þ � ðvdp=HETPkfÞ, which gives the ratio ofcharacteristic times for convective and film mass transfer,and a Damkohler number, Daf ¼ kf=ðkacodpÞ, which givesthe ratio of characteristic times for solute adsorption andfilm diffusion. Both numbers represent appropriate metricsfor determining the relative contribution of film masstransfer to band broadening.

HETP values were determined (Fig. 4) from experimentalBTCs for frontal loading of retained ovalbumin at fourdistinct flowrates (1.5, 5, 10, and 20mLmin�1). The filmmass transfer coefficient was estimated from the correlationof Athalye et al. (1991)

kf ¼ Dm

dp64þ 1:09

"m

� �3 dp"mv

Dm

" #1=3

(20)

which has previously been used in membrane chromatog-raphy (Suen and Etzel, 1992). Dm is the moleculardiffusivity, 7� 10�11m2 s�1, as computed by the methodof Tyn and Gusek (1990). This analysis permits Pef to beestimated as a function of linear velocity. As shown inFigure 4, over the flow conditions recommended for thecapsule, Pef ranges from 3.2� 10�4 to 1.5� 10�3, indicatingthat the resistance to film mass transfer is negligible and cansafely be neglected with no loss of model accuracy. This



yields a simplified form of the model in which Equations (4)and (6) reduce to

@c

@tþ v

@c

@z¼ Dax

@2c

@z2� 1� "m

"m

@q

@t(21)

and cf can be replaced with the concentration of solute in thebulk, c, in this and all intrinsic binding rate equations. Laterwe will show that use of Equation (21) is likewise supportedby the value of Daf.

Benchmark Study—The Thomas Model

Transport of ovalbumin within a Mustang Q XT5 capsule ischaracterized by a Dax of 0.9 (�0.3)� 10�8m2 s�1, which istypical of axial dispersion coefficients previously reportedfor globular proteins in stacked-membrane chromatographymodules (Gebauer et al., 1997). Peax therefore lies betweenca. 4 and 60 for the range of feed flow rates applied to thecapsule (Fig. 4). Based on theoretical considerations, Suenand Etzel (1992) have argued that breakthrough from astacked membrane column operated under loading condi-tions should obey the Thomas model when Peax� 40 andfilm mass-transfer resistances can be neglected (i.e.,Pef� 1). The Thomas model, an analytical solution forEquation (21) when Dax¼ 0 and Langmuir kinetics areapplied, is given by

c

co¼ Jðn=r; nTÞ

Jðn=r; nTÞ þ ½1� Jðn=r; nTÞ�exp½ð1� 1=rÞðn� nTÞ�(22)

n ¼ ð1� "mÞq�mkaL"mv

(23)

r ¼ 1þ coKd

(24)

T ¼ "mKdr

ð1� "mÞq�mvt

L� 1

� �(25)

where the J function in Equation (22) is defined as

Jðx; yÞ ¼ 1� expð�yÞZx

0

expð�hÞIoð2 ffiffiffiffiffihy

p Þ dh (26)

and Io is the modified Bessel function of zeroth order. As itconsiders only convective mass transport and Langmuirbinding kinetics, the Thomas model effectively representsthe best loading and breakthrough performance that canbe achieved in membrane chromatography. It thereforeprovides a useful metric against which to compare actualcolumn performance. Figure 5 compares experimental

Figure 4. Calculated height equivalent to a theoretical plate (HETP), estimated

film Peclet number (Pef) and calculated axial Peclet number (Peax) for the frontal

loading of ovalbumin (co¼ 1mgmL�1) on a Mustang XT5 column under binding

conditions. HETP values are based on plate theory and derived from breakthrough

curves at 1.5, 5, 10 and 20mLmin�1. Film diffusion coefficients are estimated from the

correlation of Athalye et al. (1991).

Figure 5. Comparison of the analytical solution of the Thomas model coupled

with a pre- and post-column CSTR to experimental breakthrough data (^) for the

frontal loading of ovalbumin (co¼ 1mgmL�1) on a Mustang XT5 module under binding

conditions at flowrates of (A) 1.5 mLmin�1 and (B) 20 mLmin�1.


BTCs for ovalbumin loaded at a feed concentration(co¼ 1mgmL�1) lying within the linear region of theisotherm to those calculated by a benchmark modelcomprised of a linear sequence of a PFR, a pre-columnCSTR, the Thomas model, and a post-column CSTR ofequal volume to the first CSTR. The structural parametersfor the PFR and CSTR elements were determined undernonbinding conditions and are tabulated in Table I. TheLangmuir rate parameters (in parentheses in Table II) wereregressed to the BTC data shown for a feed flow rate of20mLmin�1, where Peax is above 40. As is typically observedin both traditional and membrane chromatography ofproteins, q�m depends on v and is therefore independently fitto each experimental BTC.

The poor agreement of the Thomas model withexperiment at 1.5mLmin�1 is expected and reflects,in part, the importance of axial dispersion to bandbroadening under slower load conditions where Peax� 1.At 20mLmin�1, the Thomas model captures the BTCdata well, though it shows a slightly steeper approach tosaturation than observed experimentally. As the rapid mass-transfer conditions (Peax� 40; Pef� 1) specified by Suenand Etzel are met at 20mLmin�1, this small discrepancywith experiment is most likely related to nonideal processesoccurring in the Mustang Q XT5 capsule that are notassociated with solution-phase mass transfer within themembrane stack. Those processes may be separated into (1)flow nonidealities within the external hold-up volumes ofthe capsule and (2) features of the intrinsic binding reactionnot accounted for in Langmuir theory. The contributions ofthese possible nonidealities can be independently analyzedand modeled by noting that nonidealities associated withprotein binding are not present under flow-throughconditions.

Accounting for External Flow Nonidealities Usingthe ZRM

For feed flows of 1.5 and 20mLmin�1, Figure 6 comparesBTCs for ovalbumin measured under flow-through

(nonbinding) conditions with those predicted by Roperand Lightfoot’s (1995b) model comprised of a PFR in serieswith a membrane stack sandwiched between pre- and post-column CSTRs of equal residence time (i.e., Equations 1, 2,and 3). Results from a symmetric two-zone version of

Table I. Structural parameters for (1) the symmetric one-zone model

proposed by Roper and Lightfoot (1995b) comprised of a PFR in series with

the membrane stack sandwiched between two CSTRs of equal residence

time and (2) the symmetric two-zone nr-ZRM.

Parameter One-zone modela Two-zone nr-ZRM

VPFR 0.38mL 0.38mL

1/t01 0.27min�1 0.64min�1

V1 3.11mL 2.19mL

V2 — 0.82mL

F2 — 0.26

Model parameters were determined by fitting experimental data for thefrontal loading of ovalbumin (co¼ 1mgmL�1) on a Mustang XT5 columnunder nonbinding conditions at flowrates of 1.5, 5, 10, and 20mLmin�1. Inboth models, all dispersion occurring in the dead volume modeled by thePFR is accounted for by increasing the solute residence time of the firstCSTR in the network by a flow rate independent amount t0, so thatð1=tÞ ¼ ð1=t0Þ þ ðQ=VCSTRÞ

aResults computed using the Roper and Lightfoot (1995b) model.

Table II. Regressed parameter values for the Langmuir adsorption model

determined from fitting BTC data for the frontal loading of ovalbumin

(co¼ 1mgmL�1) on a Mustang XT5 column under retained conditions at

four different flowrates (1.5, 5, 10, and 20mLmin�1).

Flowrate

(mLmin�1)

ka(mLmg�1 s�1)

kd(s�1)

q�m(mgmL�1)

1.5

5.12� 10�3

(5.51� 10�3)

1.11� 10�3

(1.11� 10�3)

350

5 373

10 411

20 402

The Langmuir adsorption model was used within the symmetric two-zone ZRM. The rate constants in parentheses represent the Langmuirkinetics parameters determined from fitting the corresponding Thomasmodel to the BTC data at 20mLmin�1.

Figure 6. Model fits of experimental breakthrough data (^) for the frontal

loading of ovalbumin (co¼ 1mgmL�1) on a Mustang XT5 module under nonbinding

conditions at flow rates of (A) 1.5 mLmin�1 and (B) 20 mLmin�1 using: ( ) the

symmetric one-zone model comprised of a PFR in series with the membrane stack

sandwiched between two CSTRs of equal residence time; ( ) the symmetric two-

zone ZRM.



the ZRM in which the residence times of the two tankswithin a given zone are equal are also reported. Parametersfor the two models are reported in Table I. Note that asmall amount of solute dispersion occurs in the extra-column dead volume modeled by the PFR. In either model,this dispersion is accounted for by increasing the soluteresidence time t1 in the first CSTR in the network by a flowrate independent amount t0. Thus, 1=t1 ¼ 1=t0 þ Q1=V1.

The results show that both models work well at high feedflow rates. This is expected since the nr-ZRM collapses to theRoper and Lightfoot model at high feed flows due to the veryshort solute residence times within extra-column volumes.The results also confirm previous reports (Montesinos-Cisneros et al., 2007; Sarfert and Etzel, 1997; Yang et al.,1999) that the use of a linear one-zone PFR/CSTR sequenceto model solute dispersion within the external hold-upvolumes is not exact at lower flows, with model errorsincreasing with decreasing flow rate. In contrast, the two-zone ZRM accurately captures inhomogeneous flow effectsin the external hold-up volumes of the Mustang Q XT5IEXM column at all feed flow rates. As a result, external andinternal solute dispersion are properly decoupled at all Q bythe ZRM to provide a rigorous framework for analyzing thefitness of various binding models to describe protein loadingand breakthrough.

Evaluating Putative Binding Models

Langmuir Model

Figure 7 compares BTCs calculated by a symmetric two-zone ZRM utilizing Langmuir binding kinetics (Equation 7)to experimental data for frontal loading of ovalbumin atflowrates ranging from 1.5 to 20mLmin�1. Note that thiscalculation differs from the benchmark (Thomas model)

calculations by the fact that both axial dispersion andmultiple flow paths are considered. As has been observed inother membrane chromatography processes (Kim et al.,1991; Montesinos-Cisneros et al., 2007), particularly thoseinvolving ovalbumin (Shiosaki et al., 1994), q�m increaseswith increasing flow rate within the low-flow operatingregime. In accordance with most chromatographic process-es, it then decreases with increasing flow when Q exceeds acritical value, in this case ca. 10mLmin�1. The best fit of thetwo-zone ZRM utilizing Langmuir binding kinetics capturesthis trend (Table II). However, Langmuir theory treats everybinding site as independent and equivalent. Thus, from afundamental perspective, Langmuir theory cannot predictan increase in loading capacity withQ. It merely correlates tothis trend when the regressed value of q�m is allowed todepend on Q.

The ZRM with Langmuir kinetics is far more accuratethan the benchmark Thomas model, in large part due to theaccounting of axial dispersion at low flows, but also throughthe proper accounting of extra-column flow nonidealities.Nevertheless, agreement with experiment remains relativelypoor (Table III), indicating that the Langmuir bindingmodel oversimplifies ovalbumin adsorption to the IEXmembranes of the Mustang Q XT5 capsule such that a fasterthan realized approach to complete column loading ispredicted (Fig. 7).

Bi-Langmuir and SMA Models

As indicated by the sum of squared residuals (Table III), useof either the bi-Langmuir (Equations 8 and 9) or the SMA(Equations 11 and 12) rate equations further improves theaccuracy of the ZRM in matching experimental BTCs. Foreither of these more advanced rate equations, however, theimproved fit is realized with a set of regressed modelparameters that are physically unrealistic. To illustrate thispoint, we focus on results for the bi-Langmuir implemen-tation of the model, which is the more accurate of the two.

The ability to regress both the extra-column structuralparameters of the ZRM and Dax to pulse or breakthroughdata under flow-through conditions allows the parametersof the chosen binding rate model to be determinedindependently through global regression to a set of retainedprotein BTC data. The parameters for the bi-Langmuir

Figure 7. Comparison of the symmetric two-zone ZRM incorporating the Lang-

muir kinetic model to BTC data for ovalbumin loaded under binding conditions:

1.5 mLmin�1 (&), 5 mLmin�1 (~), 10 mLmin�1 (^), and 20mLmin�1 (*).

Table III. Sum of squared residuals for the fitting of various models to

BTC data.

Flowrate

(mLmin�1) Langmuir Bi-Langmuir SMA

Spreading

model

1.5 0.305 0.043 0.093 0.031

5 1.249 0.156 0.387 0.036

10 0.679 0.279 0.597 0.213

20 1.169 0.126 0.590 0.095

Total 3.402 0.604 1.667 0.375

Experimental conditions are the same as in Table II. The performance ofeach binding model is that when implemented within the symmetric two-zone ZRM.


model when implemented in the symmetric two-zone ZRMare shown in Table IV. Though the model represents thedata set quite well (Fig. 8), the two dynamic saturationbinding capacities, q�m;1 and q�m;2, change in unanticipatedways with flowrate: the capacity of higher affinity type 1 sitesincreases with flowrate while that of the lower affinity type 2sites decreases, falling sharply between 10 and 20mLmin�1.A plausible explanation of this predicted extreme change inq�m;1=q

�m;2 is not easily provided, as it suggests structural

changes in membrane architecture that significantly alter theavailabilities of the two different classes of binding sites.However, the membrane stack in the Mustang Q XT5capsule is both mechanically supported and very thin(2.2mm). As a result, the trans-membrane pressure dropDPis extremely low, never exceeding 21 kPa (0.21 bar) over therange of flow rates recommended by the manufacturer andshowing a linear dependence on interstitial velocity aspredicted by the Blake–Kozeny equation (Bird et al., 2002)

DP ¼ 150vLmð1� "mÞ2

d2p"2m

(27)

which has been used previously to predict pressure dropacross membranes (Suen and Etzel, 1992) and represents aconservative estimate as it assumes spherical shaped voids

with a surface area to volume ratio of av¼ 6/dp. Thus DP isover-predicted by a factor of (3/2)2 for cylindrical poreswhere av¼ 4/dp. It is hard to reconcile the prediction of alarge change in sorbent binding site composition with thisresult, as such a change would presumably require grosschanges to the architecture of the membrane that would bereflected in the dependence of DP on v.

For themodel system under investigation here, this suggeststhat the use of the bi-Langmuir rate equation improvesmodel performance not through an improved descriptionof the intrinsic binding reaction, but rather through anincrease in the number of regressed parameters (6 as opposedto 3). Similarly, globally regressed values for the capacityparameter of the SMA model (qm ¼ L=ðs þ nÞ) werefound to increase from 317mgmL�1 at 1.5mLmin�1 to369mgmL�1 at 20mLmin�1. As with the results for the bi-Langmuir isotherm, such a trend is not supported by theunderlying model theory and suggests improved modelaccuracy is again simply due to a larger number of regressedparameters.

Spreading Model

When introduced into the symmetric two-zone ZRM, thespreading model provides a very accurate representation ofthe BTC data (Table III). Model results match withexperiment at all feed flow rates (Fig. 9). Global regressionof the general spreading model (Equations 13 and 14) to thedata yields a simplified spreading model comprised of onlytwo distinct adsorbed protein states. Ovalbumin moleculesare predicted to adsorb reversibly to the sorbent (state 1) andthen may reorient or spread reversibly to a second state (2)from which they cannot desorb directly into the bulk (i.e.,setting ka,2 and kd,2 to zero results in no loss in modelaccuracy). The values of the regressed parameters (Table V),

Table IV. Regressed parameters for the bi-Langmuir adsorption model

when introduced into the two-zone ZRM.

Flowrate

(mLmin�1)

ka(mLmg�1 s�1)

kd(s�1)

q�m;1

(mgmL�1)

q�m;2

(mgmL�1)

1.5 ka,18.38� 10�3

kd,13.18� 10�5

152 195

5 181 184

10 ka,21.39� 10�3

kd,25.49� 10�4

224 171

20 304 43

Experimental conditions and regression method are the same as inTable II.

Figure 8. Comparison of the symmetric two-zone ZRM incorporating the bi-

Langmuir kinetic model to BTC data for ovalbumin loaded under binding conditions:


Figure 9. Comparison of the symmetric two-zone ZRM incorporating the sim-

plified spreading model to BTC data for ovalbumin loaded under binding conditions:




which are consistent with those reported in a previous study(Yang and Etzel, 2003) that applied a spreading-type modelto IEXM chromatography, reveal that the simplifiedspreading model captures the relatively rapid increase ineluent concentration in the initial sections of each BTCby predicting relatively rapid population of state 1. Thepronounced asymmetry of BTCs at lower flow rates (1.5 and5mLmin�1), a common observation in membrane chro-matography systems, is related to the slower reactionkinetics associated with state 2, which can only becomepopulated (and subsequently depopulated at high surfacecoverages, see below) following population of state 1.

At higher feed flow rates (10 and 20mLmin�1), the BTCsbecome more symmetric, and the spreading model capturesthis through the flow-rate dependence of the reverse rateconstant, k21, for the reorientation/spreading reaction. Inparticular, while k12 is insensitive to v, k21 increases with v.To understand this and connect it to observed breakthroughbehavior, we note that the spreading model computes theequilibrium amount of protein bound to a sorbent of knownsurface area by minimizing the Gibbs energy of the systemfor a given total protein concentration. In the case of anonspherical/deformable protein that can bind in twodistinct states, the amount protein bound in each state atequilibrium therefore depends on the stability of each boundstate, given by DG1¼�RT ln K1a for the lower affinityvertical/native state 1 and by DG2¼�RT ln K1aK12 for thehigher affinity horizontal/spread state 2, normalized by theprojected surface area per bound molecule (1 for state 1 andb¼ 2.55 for state 2). The difference in these two normalizedenergies is plotted in Figure 10 and shows that the weakeraffinity state 1 will be more populated at breakthroughbecause it allows the proteins to load to a higher surfacedensity. Thus, as the surface increasingly nears saturation,adsorbed proteins in state 2 will revert to state 1 to freesorbent surface area and thereby permit the net energeticallyfavorable loading of more protein in state 1.

Figure 10 also shows that the overall driving force forconversion from state 2 back to state 1 increases withincreasing v. However, since an individual protein moleculebound in state 2 is at a lower energy than when bound instate 1, the rate of conversion back to state 1 depends on theavailability of system energy that can be used by themolecule to overcome the affinity difference. One potentialsource of energy is that provided by the wall shear stress tw,which for laminar flow in a pore generates a radially

dependent force parallel to the pore wall. By consideringflow through a membrane pore in this manner, one canshow that

tw ¼ 4m

"mpR3p

Q (28)

where Rp is the pore radius. Thus, tw increases with Q,providing one plausible explanation for the predictedincrease in k21 with v and the associated improvement inBTC symmetry. Indeed, this basic argument has been usedto explain the influence of hydrodynamics on otherbiological binding equilibria at a fluid–solid interface(e.g., Cao et al., 2002).

More importantly, the spreading model implementationof the ZRM also provides an explanation for the commonobservation in IEXM chromatography of an increase indynamic binding capacity q� with increasing v. Figure 11reports experimental q� values for each of the BTCexperiments performed and shows that q� for ovalbuminincreases with increasing flow rate between 0 and10mLmin�1. We note again that these BTC experimentswere performed at a co¼ 1 g L�1, which is deep within thelinear region of the static isotherm but does not correspond

Table V. Regressed parameters for the simplified spreading model when introduced into the two-zone ZRM.

Flowrate

(mLmin�1) ka,1 (mLmg�1 s�1) kd,1 (s�1) k12 (mLmg�1 s�1) k21 (s

�1) K12 (M�1) q�m;1 (mgmL�1) b

1.5

8.28� 10�3 2.09� 10�4 1.24� 10�4

3.59� 10�3 1.53� 103

379 2.555 6.87� 10�3 7.99� 102

10 1.93� 10�2 2.85� 102

20 1.28� 10�1 4.29� 101 343

Experimental conditions and regression method are the same as in Table II.

Figure 10. Difference in the stability of bound protein states normalized by the

projected surface area per bound molecule (DG1–DG2/b) for the frontal loading of

ovalbumin (co¼ 1mgmL�1) on a Mustang XT5 column at flowrates of 1.5, 5, 10, and

20mLmin�1.


to saturation conditions. Also shown in Figure 11 (dashedline) is the saturation capacity (338mgmL�1) determinedfrom (near-)static isotherm experiments, as well asmodel predictions of q�, q1, and q2. The model capturesthe curious increase in q� with flow rate, predicting that itoccurs as a result of the depopulation of state 2 withincreasing wall shear. It also yields a value of q�m;1 of379mgmL�1 which is constant and equal to qm,1 atQ¼ 10mLmin�1 and below. Thus, q�m;1, which representsthe hypothetical saturation capacity of the membrane if allovalbumin bound in state 1, is higher than the measuredstatic qm for the real system, which is predicted to include asmall fraction of ovalbumin bound in state 2 (Fig. 11).Increasing Q decreases q2 and therefore increases both q1and q�. Further increases in Q above 10mLmin�1 then leadto the expected decrease in dynamic binding capacity due todecreased retention time in the column. The spreadingmodel therefore not only captures, but also provides arealistic underlying theory to explain key experimentaltrends in our model system of ovalbumin loading onto aMustang Q XT5 capsule.

The spreading model predicts that the asymmetric natureof the observed BTCs is due to a series of processes at thesorbent surface, each characterized by unique kinetics.We have then speculated that these processes may relate tochanges in orientation or conformation of the protein uponadsorption, in accordance with recent experimental workthat directly measures such changes with time for proteinsadsorbed on an ion-exchange resin (Dismer and Hubbuch,2007; Dismer et al., 2008). However, the sequential surfacekinetics defined by the model could relate to other physicalphenomena such as surface aggregation.

Finally, we may now confirm the validity of omitting filmmass transfer effects in the ZRM. Based on ka,1, Daf is on theorder of 5� 104 across all flow rates, confirming that therate of film mass transfer is orders of magnitude faster thanthe rate of solute adsorption.

Conclusions

Wehave shown that the quantitative description and accurateprediction of IEXM chromatography can be achieved using anovel zonal rate model that properly accounts for regionalcontributions to band broadening so as to permit rigorousevaluation of putative binding models and associatedbinding mechanisms. For the loading of ovalbumin onto aMustang Q XT5 IEXM capsule over a range of flowrates (1.5–20mLmin�1), this model-building strategy was used tocharacterize four potential solute binding models—Langmuir, bi-Langmuir, SMA, and spreading. The spreadingmodel, which describes a reversible change in orientationand/or conformation of protein molecules upon binding tothe pore surface, was found to accurately fit the asymmetricexperimental breakthrough data and also provide mechanis-tic insight into the dependence of dynamic binding capacityon flowrate: a dependence that, although seen previously, hasnot been well explained. A spreading-type model similar toone proposed for ovalbumin binding to the XT5 IEXMcapsule was recently used by Morbidelli and co-workers todescribe experimentally observed elution patterns of humanserum albumin from ion-exchange resin (Voitl et al.,2010a,b). Thus, our ever increasing ability to more rapidlyand accurately model complex transport processes occurringin preparative chromatography systems may be revealing thegeneral utility of spreading-type binding models.

However, though the spreading model was found mostappropriate for describing the intrinsic binding process inour model membrane chromatography system, this willcertainly not hold true for all systems or operationalconditions. For example, the bi-Langmuir (Boi et al., 2007)or SMA (Brooks and Cramer, 1992) might indeed be themore appropriate binding model in certain cases. The valueof this work is therefore in demonstrating the ability to usethe ZRM and accompanying methodology to accuratelyassess the fundamental value of a putative binding model, aswell as to isolate and define all contributions to bandbroadening occurring both within and external to thecolumn.When used in the procedural framework outlined inthis work, the ZRM therefore represents a flexible and usefultheoretical tool to characterize membrane chromatographycolumns and to optimize and scale-up their operation so asto avoid unfavorable conditions while maintaining theintrinsic advantages of membrane chromatography.

C.A.H. receives salary support as the Canada Research Chair in

Interfacial Biotechnology. This work was supported by grants from

the Natural Sciences and Engineering Research Council of Canada

(NSERC) and Canadian Institutes of Health Research (CIHR).

Figure 11. Experimentally derived dynamic binding capacities, q�, (&) for the

frontal loading of ovalbumin (co¼ 1mgmL�1) on a Mustang XT5 column at flowrates of

1.5, 5, 10, and 20mLmin�1. Two-zone ZRM/spreading model predictions of q�, theconcentrations of bound protein in state 1 (q1) and state 2 (q2), and the theoretical

dynamic capacity based on all protein binding in state 1 (q�m;1) are also shown. The

dashed line represents the saturation capacity (qm) of the membrane for ovalbumin

determined from (near-)static isotherm experiments.



Nomenclature

Ak membrane cross sectional area (m2)

c protein concentration in mobile phase (mgmL�1)

cA salt ion concentration in mobile phase (mgmL�1)

cf protein concentration in film (mgmL�1)

cin protein concentration at inlet to column (mgmL�1)

co protein concentration in feed (mgmL�1)

cCSTRi protein concentration in CSTR i (mgmL�1)

cCSTRiin protein concentration in the inlet to the CSTRi (mgmL�1)

cPFRout protein concentration in outlet from PFR (mgmL�1)

~c dimensionless protein concentration in mobile phase

Dax axial dispersion coefficient (m2 s�1)

Dm molecular diffusivity (m2s�1)

Daf film Damkohler number

dp pore diameter (m)

DG change in Gibbs free energy (Jmol�1)

HETP height equivalent to a theoretical plate (m)

k12 rate constant for spreading from state 1 to state 2

(mLmg�1 s�1)

k21 rate constant for unspreading from state 2 to state 1 (s�1)

ka adsorption rate constant (mLmg�1 s�1)

ka,i adsorption rate constant for site/state i (mLmg�1 s�1)

kd desorption rate constant (s�1)

kd,i desorption rate constant for site/state i (s�1)

kf film mass transfer coefficient (m s�1)

k0f lumped film mass transfer coefficient (s�1)

K1a equilbrium constant for protein adsorption (M�1)

K12 equilibrium constant for protein preading (M�1)

L column length (m)

P pressure (Pa)

Peax axial Peclet number

Pef film Peclet number

q protein-ligand concentration in stationary phase (mgmL�1)

qi protein-ligand i concentration in stationary phase

(mgmL�1)

qm maximum protein-ligand capacity in stationary phase

(mgmL�1)

qm,i maximum protein-ligand i capacity in stationary phase

(mgmL�1)

~q dimensionless protein-ligand concentration in stationary

phase

qA salt co-ion concentration available for exchange with the

protein (mgmL�1)

Q volumetric flow rate (mL s�1)

r radial position (m)

R gas constant (J K�1mol�1)

Rp pore radius (m)

s protein concentration in stationary phase (mgmL�1)

t time (s)

tconv characteristic time for convective mass transfer (s)

tfilm characteristic time for film mass transfer (s)

tlag lag time (s)

tplate characteristic time for convective mass transfer through a

single theoretical mass-transfer unit (s)

~t dimensionless time with respect to HETP

t dimensionless time with respect to film mass transfer

T temperature (K)

u superficial velocity (m s�1)

v interstitial velocity (m s�1)

v characteristic charge

Vcolumn volume of membrane stack (mL)

Vi volume of CSTRi (mL)

VPFR PFR volume (mL)

z axial position (m)

~z dimensionless axial position� signifies that adsorbed quantity is dynamic

b ratio of occupied surface areas of spread and unspread

protein

em membrane porosity

ef fraction of stationary phase volume occupied by film

F radial zone flow fraction

L ion capacity in stationary phase (mgmL�1)

m mobile phase viscosity (Pa s)

s steric factor

t protein residence time in CSTR (s)

tw shear stress at pore wall (Pa)

t0 flowrate independent contribution to dispersion (s)

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