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Journal of Chromatography A, 1375 (2015) 33–41

Contents lists available at ScienceDirect

Journal of Chromatography A

j o ur na l ho me page: www.elsev ier .com/ locate /chroma

inear isotherm determination from linear gradient elutionxperiments

avid Pfister, Fabian Steinebach, Massimo Morbidelli ∗

nstitute for Chemical and Bioengineering, Department of Chemistry and Applied Bioscience, ETH Zurich, 8093 Zurich, Switzerland

r t i c l e i n f o

rticle history:eceived 27 August 2014eceived in revised form5 November 2014ccepted 25 November 2014vailable online 2 December 2014

a b s t r a c t

A procedure to estimate equilibrium adsorption parameters as a function of the modifier concentration inlinear gradient elution chromatography is proposed and its reliability is investigated by comparison withexperimental data. Over the past decades, analytical solutions of the so-called equilibrium model underlinear gradient elution conditions were derived assuming that proteins and modifier molecules accessthe same fraction of the pore size distribution of the porous particles. The present approach developed in

eywords:hromatography modelinear gradient elutionxclusion effectarameter estimation

this work accounts for the size exclusion effect resulting in different exclusions for proteins and modifier.A new analytical solution was derived by applying perturbation theory for differential equations, and the1st-order approximated solution is presented in this work. Eventually, a turnkey and reliable procedureto efficiently estimate isotherm parameters as a function of modifier concentration from linear gradientelution experiments is proposed.

© 2014 Elsevier B.V. All rights reserved.

. Introduction

Ion exchange chromatography (IEC) is a well-known and well-stablished technique in the downstream processing of therapeuticroteins [1–3]. However, despite the great popularity of chromato-raphic processes, their development and up scaling are still oftenased on heuristic knowledge and trial-and-error approaches [4].uch design procedures are often company-specific, if not person-pecific, and are therefore not answering anymore to the call ofhe regulatory authorities for standardizing and generalizing therocess development procedures. Moreover, the trial-and-errorpproach might become more and more costly and challeng-ng with process scale-up. Therefore, model-based design couldrovide competitive advantages in terms of time and efficiency byeducing the number of experiments to perform. Indeed, mech-nistic models only require a limited number of experimentso “calibrate” the model parameters. However, the model-basedpproaches cannot be the silver bullet and despite their great

otential the bottleneck of this approach is the determination ofhe model parameters whose reliability will directly be reflectednto the reliability of the model.

∗ Corresponding author at: Institute for Chemical and Bioengineering, ETH Zurich,ladimir-Prelog-Weg 1, CH-8093 Zurich, Switzerland. Tel.: +41 44 632 30 34;

ax: +41 44 632 10 82.E-mail address: massimo.morbidelli@chem.ethz.ch (M. Morbidelli).

ttp://dx.doi.org/10.1016/j.chroma.2014.11.067021-9673/© 2014 Elsevier B.V. All rights reserved.

In this work, we focus on the determination of reliable equi-librium parameters of linear isotherms as a function of modifierconcentration. Two different approaches are reported in the liter-ature to obtain the Henry coefficient as a function of the modifierconcentration: pulse injection followed by isocratic elution or lin-ear gradient elution. Among the two, gradient elution is the mostconvenient one, especially for proteins which suffer from severemass transfer hindrances. Indeed, the compression induced by thesalt gradient leads to sharper peaks thus improving the detectabil-ity and the precision of the measurements [5]. Moreover, and notthe least, the use of a gradient makes the methods less sensibleto experimental error in the buffer preparation compared to iso-cratic elution. Yamamoto et al. [6–8] suggested a simple techniqueto estimate equilibrium parameters from a set of pulse injectionseluted at different gradient slopes. This technique has become verypopular over the past decades.

The novelty of this work is to rigorously account for the sizeexclusion effect [9,10] thus leading to a new and more reliableanalytical solution of the equilibrium model. In particular, in theYamamoto et al. procedure it is assumed that both modifier andprotein access the entire pore size distribution [6–8]. For smallpeptides, or particles with large pores this assumption is reason-able and leads to very good results. However, it is often observed

that proteins are excluded from the smallest pores while the mod-ifier is not, and therefore a new estimation procedure is neededwhich accounts for differences in size between modifier and pro-teins. Obviously, it is much more advantageous to have an analytical

3 atogr. A 1375 (2015) 33–41

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olution than numerically integrating the system of partial differ-ntial equations describing the solute transport in a packed bed.he computation time is indeed dramatically reduced.

In particular, in this work we propose a more accurate analyticalolution for evaluating the retention time in linear gradient elutionhromatography. The isotherm parameters are obtained by fittinghe present model on a set of experimental data. These estimatedarameters where then compared to the “true” ones obtained byhe Inverse Method (IM) and to the ones obtained by the classicalamamoto’s approach.

. Theory

.1. Equilibrium model

At equilibrium, the concentration of solute retained in the sta-ionary phase (q) is related to the concentration of solute in the

obile phase (c) by a solid–fluid equilibrium law named isotherm.t low solute concentration this isotherm is often assumed to be

inear and the initial slope of the isotherm, known as the Henryoefficient, is defined as follows:

= Hc (1)

depends on the protein, the stationary phase, the ionic strength,he temperature and the pH of the mobile phase. In classical ionxchange chromatography, a salt (typically NaCl) is used as modi-er to modulate the retention of the solute. The dependency of theenry coefficient (or the retention factor K = �H, where � = (1 − ε)/ε

s the phase ratio) towards the modifier concentration is usuallyescribed by the stoichiometric displacement model (SD) [11,12]:

= K∞ + ˛c−ˇM (2)

here K∞ is the retention factor at infinite modifier concentrationefined by K∞ = εp�, and ̨ and ̌ are the two parameters describinghe retention factor dependency on modifier concentration. cM(z, t)s the modifier concentration at time t and position z in the column.

Particular attention should be paid on the different porositiesnvolved when dealing with porous particles. Let us first define theed porosity, ε, as the ratio between the volume of liquid outside theorous particles and the volume of the column. Therefore, (1 − ε)epresents the volume of solid (particles skeletons) plus the vol-me of liquid that stagnates in the particles pores over the columnolume. In addition, the intra-particle porosity, εp, represents theolume fraction of the porous particle accessible to the solute. Usu-lly, when dealing with macromolecules whose radii are close fromhe average radius of the pores, εp is defined in terms of accessi-le porosity, εp,i. Therefore, εp,i (specific to the macromolecule) isanging from εp obtained for small non-excluded tracers to 0 forompletely excluded molecules, which leads to define K∞,i = εp,i�.he total porosity is defined by the sum of the volumes of liquidmobile and stagnant) over the volume of the column:

t = ε + (1 − ε)εp (3)

Or for partially excluded macromolecules:

t,i = ε + (1 − ε)εp,i (4)

The fraction of the pore size distribution accessible to the proteinbviously depends on the size of the macromolecule and on theore size distribution of the stationary phase [9]. Fig. 1 shows theorosity profiles of the four stationary phases investigated in thisork.

In the frame of the equilibrium model, the chromatographic col-mn is modelled as an ideal plug flow (negligible mass transfer

imitations and no axial dispersion) and the solute concentration,oth in the mobile fluid and in the particles is assumed to be atquilibrium, as defined by Eq. (2). Under these simplifying assump-ions, the mass balance equation for a solute (or the modifier) in the

Fig. 1. Total accessible porosity of the different stationary phases.

chromatographic column is described by the following equation[13]:

u∂c

∂z+ ε

∂c

∂t+ (1 − ε)

∂q

∂t= 0 (5)

which using Eqs. (1) and (2) reduces to the following equation:

∂c

∂z+ (1 + K)

ε

u

∂c

∂t= −c

ε

u

∂K

∂t(6)

where u is the linear velocity defined by u = Q/A. Eq. (6) can be solvedusing the method of characteristics. This method has already shownits great potential for solving first order PDEs. The basic principle isto solve the PDE equation on characteristic curves along which thePDE can be written as a system of ordinary differential equations.The transformation is obtained by constructing the characteristiccurves parameterized by s so that the curve {z(s), t(s), c(s)} satisfiesthe set of characteristic equations defined by the following ODEs:⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

dz

ds= 1

dt

ds= (1 + K)

ε

u

dc

ds= −c

ε

u

dK

dt

(7)

The ratio between the second and the first lines of Eq. (7) leads tothe equation describing the evolution of the solute retention timealong the column axis z. The resolution of the other equations wasnot considered in this work. It comes:

dt

dz= ε

u(1 + K) (8)

Eq. (8) describes the evolution of the retention time from theinlet of the column (z = 0) to the position z for a solute whose equi-librium with the stationary phase is given by Eq. (1).

Under the assumption of constant retention factor for the modi-fier and in the absence of interactions with the ion exchange groups(meaning that K = K∞ = εp�) the integration of Eq. (8) between 0 andz is obvious. It comes:

t0,M(z) = ε

u(1 + K∞)z (9)

where t0,M(z) is the time needed by the modifier to cover the dis-tance from 0 to z. Therefore, for linear gradients the concentration

of modifier at the position z along the column as a function of timeand gradient slope is described by the following equation:

cM(z, t) = c0M + g(t − t0,M(z)), t ≥ t0,M(z), 0 ≤ z ≤ L (10)

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here c0M in mM is the initial modifier concentration at the

eginning of the gradient, and g the gradient slope in mM/min.ombining Eqs. (2), (8) and (10) leads to the ODE describing thevolution of the time needed by a retained solute (a protein in thisase) to cover a certain distance along the column:

dt

dz= ε

u{1 + K∞,i + ˛[c0

M + g(t − t0,M(z))]−ˇ},

t > t0,M(z), 0 ≤ z ≤ L (11)

here K∞,i = εp,i�. The dimensionless form of Eq. (11) is derivedy defining the following variables: �(�, t) = (c0

M + gv(t/t0 − (1 +∞)�))/gv, � = z/L, gv = gt0, where t0 = εL/u, L is the length of theolumn, �′ = K∞ − K∞,i = (εp − εp,i)�, and ˛′ = ˛(gv)−ˇ. The previousquation is simplified as follows:

d�

d�= −�′ + ˛′�−ˇ (12)

Eq. (12) can be solved analytically by series expansion. In theext section the first two terms of this series, corresponding to theth and 1st-order approximated analytical solutions, are derived.ccordingly, the solution of Eq. (12) is assumed to be approximatedy a series expansion of the perturbation parameter � ′ = �′/˛′. Inssence, � ′ represents the difference between the intra-particleorosity accessible to a tracer (or to the modifier in this case), εp,nd the intra-particle porosity effectively accessible to the protein,p,i.

= �(0) + � ′�(1) + O(� ′2) (13)

The 0th-order approximation (i.e. � ′ = 0) represents the casehere no pore exclusion is present for the protein which therefore

ccess all available pores just like the modifier. This corresponds tohe case addressed by Yamamoto [6,7].

.2. Zeroth-order approximation

Since the pioneering work of Freiling in 1955 [14] many authorsorked on the derivation of the 0th-order approximation asamamoto [6,7], Parente and Wetlaufer [15], and later Carta and

ungbauer [5].In the frame of the 0th-order approximation proposed by

amamoto, the difference between the volume fraction of station-ry phase accessible to the modifier and to the protein is neglected.herefore, εp,i = εp and �′ = 0. Eq. (12) simplifies as follows:

d�(0)

d�≈ ˛′(�(0))

−ˇ(14)

here the superscript (0) stands for “0th-order approximation”. Eq.14) can easily be integrated between the inlet and the dimension-ess position � along the column:

(0)(�, t) =(

(1 + ˇ)

(˛′� + (�(0)

0 )1+ˇ

1 + ˇ

))1/(1+ˇ)

(15)

here � = � − �0 is the covered dimensionless distance and �(0)0 =

(0)(�0, 0). Eq. (15) can also be used in the case where piecewiseinear gradient are considered. In the present case, only simple lin-ar gradients are considered so that �0 = 0 and �(0)

0 = 0. Eq. (15) isvaluated at the outlet of the column (� = 1 and t = tR, where tR is theetention time of the protein) considering that the gradient startst 0 mM (c0

M = 0); the solution given by Eq. (15) simplifies to the

ollowing equation:

(0)R = V0

[1 + K∞ + (˛gv(1 + ˇ))1/(1+ˇ)

gv

](16)

. A 1375 (2015) 33–41 35

with V0 = εAL = t0Q, V0 being the elution volume of the a tracerexcluded from all the pores. VR is simply tR × Q. It is worth not-ing that from Eq. (10), the concentration of modifier at the outletof the column when the protein is eluting is easily computed bycR

M = cM(L, tR) = gv(t(0)R − t0,M(L))/t0 and Yamamoto’s equation is

finally obtained:

gv = (cRM)

1+ˇ

˛(1 + ˇ)(17)

The superscript R refers to tR, time at which the protein is eluting.Eq. (16) is traditionally rearranged and linearized by logarithmictransformation and the parameters ̨ and ̌ are estimated from theslope and the intercept of the following curve:

log (cRM) = 1

1 + ˇlog (gv) + 1

1 + ˇlog (˛(1 + ˇ)) (18)

2.3. First-order approximation

When increasing the size of the protein of interest, easily goingup to several nanometers (as for monoclonal antibodies), the exclu-sion effect is no more negligible and the above mentioned 0th-orderapproximation could lead to large discrepancies between experi-mental and simulated data.

Therefore, the derivation of a 1st-order analytical solution isproposed to account for this exclusion effect. Perturbation the-ory for ordinary differential equations has been applied with theperturbation parameter � ′ = �′/˛′ [16]. The previous assumptionK∞,i ≈ K∞ is less closely to be satisfied the larger the value of � ′.

By introducing Eq. (13) in Eq. (12) a first order expansion in � ′

is obtained as follows:

d�(0)

d�+ � ′ d�(1)

d�= −�′ + ˛′(�(0))

−ˇ(1 + � ′(�(1)/�(0)))

−ˇ(19)

The superscript (1) stands for “1st-order approximation”. Onlyfirst order terms in � ′ are considered. Therefore, the Taylor expan-sion of the right hand side of Eq. (19) reduces to:

(1 + � ′(�(1)/�(0)))−ˇ = 1 − � ′ˇ(�(1)/�(0)) + O(� ′2) (20)

Combining Eqs. (19) and (20) leads to the final equation to besolved.

d�(0)

d�+ � ′ d�(1)

d�= −�′ + ˛′(�(0))

−ˇ − ˛′� ′ˇ(�(0))−(1+ˇ)

�(1) (21)

Recalling that �(0) is the solution of the unperturbed problem

(d�(0)/d� = ˛′(�(0))−ˇ

), and dividing by � ′, Eq. (21) simplifies to afirst order ODE describing the evolution of �(1) as a function of �:

d�(1)

d�= −˛′ − ˛′ˇ(�(0))

−(1+ˇ)�(1) (22)

The ODE can be integrated analytically between the inlet (� = 0)and the dimensionless position � along the column for any timet ≥ t0,M(z) with the boundary condition �(1)(0, t) = 0. It comes:

�(1)(�, t) =(

1 + ˇ

1 + 2ˇ

)⎛⎝(�(0)0 )

1+2ˇ

(˛′� + (�(0)

0 )1+ˇ

1 + ˇ

)−(ˇ/1+ˇ)

−

⎛⎝˛′� + (�(0)

0 )1+ˇ

1 + ˇ

⎞⎠⎞⎠ (23)

Eq. (23) is evaluated at the end of the column (� = 1) with a lineargradient starting from c0

M = 0 mM, and t = tR. The solution simplifies

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6 D. Pfister et al. / J. Chrom

s follows:

(1)R = V0

[1 + K∞ + (˛gv(1 + ˇ))1/(1+ˇ)

gv

− 1 + ˇ

1 + 2ˇ(K∞ − K∞,i)

](24)

It is worth noting that the 1st-order approximated solutionemains quite simple but provides a much better approximationor ̨ and ̌ as discussed shortly in detail.

.4. Data fitting

Least square regression was applied to estimate optimal ̨ and parameters that minimize the difference between simulated andxperimental data. The distance between the experimental andimulated data was evaluated by computing the residual sum ofquares (RSS). The optimization problem consisted then in find-ng optimal (˛, ˇ) parameters by minimizing the RSS defined asollows:

(˛, ˇ) =n∑

i=1

(VexpR,i

− VR,i(˛, ˇ))2 = [Vexp

R − VR]T[Vexp

R − VR] (25)

In this equation, VexpR represents the vector of experimentally

easured elution volumes while VR(˛, ˇ) are the correspondingalues calculated with parameters estimated by one of the follow-ng approaches:

0th-order approximated solution (Yamamoto’s equation (16)). 1st-order approximated solution (Eq. (24)). Inverse method (Eq. (12)).

It is worth noting that the values of ̨ and ̌ estimated byntegrating the full model (12) which corresponds to the inverse

ethod (IM) [17], are considered as the “true” values (˛IM, ˇIM). Inhe following, we compare the ̨ and ̌ values obtained using thearious model approximations to see which one of them is able toest approximate these parameters.

.5. Confidence interval

The quantification of the “quality” of the approximated parame-ers was performed by estimating their confidence intervals. To doo, the null hypothesis (˛, ˇ) = (˛i, ˇi) was tested. (˛i, ˇi) correspondo the optimal equilibrium parameters obtained with the differentpproximated solutions. These optimal values were obtained fromi = 0) Eq. (16) (thus corresponding to (˛0, ˇ0), in which 0 stands forhe 0th-order approximation), and (i = 1) Eq. (24) (thus correspond-ng to (˛1, ˇ1), in which 1 stands for the 1st-order approximation).he fitting was obtained by minimizing the RSS using Lagarias’ et al.implex algorithm implemented in Matlab®.

For a total of n experiments, a F-test was applied to find pointshat reject the null hypothesis within a q% confidence interval [18].

D(˛, ˇ) − D(˛i, ˇi)D(˛i, ˇi)

≤ 2n − 2

Fq,2,n−2 (26)

Searching for all the (˛, ˇ) that respect this inequality leads to

he following confidence region, where q is the (1 − q) quantile ofhe F-distribution with 2 and n − 2 degrees of freedom [18,19].

(˛, ˇ)|D(˛, ˇ) ≤ D(˛i, ˇi)[1 + (2/(n − 2))Fq,2,n−2]} (27)

. A 1375 (2015) 33–41

3. Materials and methods

3.1. Materials

Standard model proteins were considered: hen egg whitelysozyme (HEWL) and �-lactalbumin form bovine milk (�-Lac)were obtained from Sigma–Aldrich and cytochrome C (Cyt-C) fromhorse heart muscle was purchased from Acros Organics.

For the PEGylation of �-Lac, methoxypolyethylene glycol suc-cinimidyl propionic acid (mPEG-SPA) from JenKem was used, witha nominal average molecular weight of 10 kDa.

Fractogel® EMD SO3− (M) resin from Merck Millipore

(crosslinked polymetacrylate, sulfonic group, particle diame-ter 40–90 �m), SP SepharoseTM BB resin from GE Healthcare(6% cross-linked agarose, sulphopropyl group, particle diam-eter 100–300 �m), POROS® 50 HS from Applied Biosystems(polystyrenedivinylbenzene, sulphonic group, particle diameter80 �m) and UNOsphereTM S resin from Bio-Rad Laboratories (sul-fonic group, particle diameter 80 �m) were used. The prepackedcolumns were purchased from ATOLL GmbH. All the columns havethe same dimensions 50 mm × 5 mm, for a total bed volume of 1 mL.In addition, Q Sepharose High Performance resin from GE Health-care (crosslinked agarose beads, quaternary strong anion exchangegroups, particle diameter 34 �m) was used for the LGE experi-ments with the PEGylated �-Lac. The stationary phase was packedaccording to the manufacturer’s protocol in a Tricorn column (GEHealthcare). The resulting bed length was 142 mm for an internaldiameter of 5 mm.

The mobile phase was either a 25 mM phosphate buffer (or a25 mM Tris buffer when using the Q Sepharose High Performanceresin). The buffers were prepared with sodium dihydrogen phos-phate and disodium hydrogen phosphate, (or Tris–HCl and HCl) forbuffer A while buffer B was also containing 1 M sodium chlorideas modifier. The buffers were adjusted to pH 7.0 and filtered onDurapore membrane filters, type HVLP, 0.45 �m (Millipore). Thewater was deionized and further purified with a Millipore Simpack2 purification pack.

3.2. PEGylation of ˛-lactalbumin

1 mg of �-Lac and 3-fold molar excess mPEG-SPA were dis-solved in 2.5 mL of 25 mM phosphate buffer adjusted at pH 7.0.The reaction took place at 25 ◦C in a stirred beaker during 3 h.The crude solution from the reaction was fractionated by SEC onan analytical TricornTM Superdex 200 10/300 GL (GE Healthcare).As mobile phase a 25 mM sodium phosphate buffer, containing100 mM Na2SO4 at pH 7.0 was used. The elution flow rate was set to0.5 mL/min. Five fractions were considered, going from the non con-jugated protein to the 4-times PEGylated �-Lac (P0 to P4). The purityof the fractions was checked by size exclusion chromatography.

3.3. Estimation of model parameters

Measurement of the proteins retention times were performedon an Agilent 1100 Series equipped with a quaternary pump, adegasser, an autosampler, a column oven and a diode-array detec-tor. The peaks were detected at 280 nm for HEWL and �-Lac and415 nm for Cyt-C. 10 �L injections from 1 mg/mL protein solutionwere fixed. The retention times were estimated by computing thefirst order moment of the chromatograms.

Pulse injections experiments were performed with six differentlinear gradient slopes from 5 to 100 column volumes (CV) going

from 0% to 100% B. The gradient slopes g were chosen as 10, 20, 33,50, 100 and 200 mM/min. At the end of the gradient, the columnswere re-equilibrated with buffer A for 5 CV. The volumetric flowrate was 1 mL/min and the experiments were performed at 25 ◦C.

atogr. A 1375 (2015) 33–41 37

taco0fiedi

4

4

aoopnpdpsgdipfesteio

Fig. 2. Relative difference between approximated solutions (0th and 1st-order

D. Pfister et al. / J. Chrom

Inverse size exclusion chromatography (iSEC) was applied andracers with a wide range of molecular sizes (salt, PEG of 0.2 kDand dextrans of 1, 5, 12, 50, 150, 270, 410, 670 and 2000 kDa) wereonsidered in order to measure the total accessible porosity profilef the packing materials. This selection of tracers, with radii from.1 nm to 37.2 nm, enables to measure the complete porosity pro-le. The radii of the dextrans were computed with the followingquation: R = 0.845 · M0.498

w , Mw being the molecular weight of theextran polymer in kDa [10]. The results of the iSEC are presented

n Fig. 1.

. Results and discussion

.1. Numerical comparison

The numerical relative error ((V (i)R − VR)/VR) between the

pproximated solution (i is either 0 or 1 depending if the 0th-orderr the 1st-order approximation is used) and numerical integrationf Eq. (11), simply noted VR, is shown in Fig. 2. As it has beenreviously observed in Fig. 1, when increasing the size (hydrody-amic radius) of the solute, it is more and more excluded from theorous particles. Indeed, when the size of the solute exceeded theiameter of the largest pores, the solute can no longer enter theorous particles. Therefore, it can be seen that when increasing theize of the protein, meaning that the accessible volume fractionoes from the one of the modifier to the bed porosity, the relativeifference between the approximated solution and the numerical

ntegration of Eq. (11) increases. In this graph, only the accessiblearticle porosity, εp,i, was varied in order to cover the range of K∝,irom K∝ = 1 to 0. When the total accessible porosity of the solute isqual to the total porosity (measured with a non excluded tracer asalt in the present case), both approximated models are equivalent

o the full model defined by Eq. (11) and the numerical relativerror tends to 0. Decreasing the partition coefficient K∝,i leads to anncrease of the numerical error but it is always at least two ordersf magnitude lower for the 1st-order approximation than for the

Fig. 3. Relative error on ̨ parameter: (˛i − ˛IM)/˛IM (top) and ̌ parameter

approximation) and numerical integration of Eq. (11) for different accessible porosi-ties. Parameters for the calculation are: ̨ = 108, ̌ = 3.50, ε = 0.40, εt = 0.80, K∞ = 1,g = 50 mL/min and L = 5 cm.

0th-order solution and never exceeds 0.5% while the predictionfrom the 0th-order approximated solution are approaching 10%in the worst case. It should be noticed that this error is purelynumerical and does not include experimental variability.

4.2. Accuracy of the model predictions

All (˛, ˇ) values estimated by the 3 different approaches, 2 dif-ferent proteins (HEWL and Cyt-C) and the 4 cation exchange resinsare reported in Table 1.

They were all compared to the optimal (˛IM, ˇIM) obtained by the

IM. It is worth mentioning that the 1st-order approximation whichaccounts for the exclusion effect in the case of protein with sizecomparable to the average pore size, provides parameters that arealways in very good agreement with the “true” values. In the worst

(ˇi − ˇIM)/ˇIM (down) for lysozyme (left) and cytochrome C (right).

38 D. Pfister et al. / J. Chromatogr. A 1375 (2015) 33–41

Table 1Comparison between equilibrium parameters obtained by the 3 different approaches.

FG EMD SO3− Sepharose BB POROS 50 HS UNOsphere S

HEWL Cyt-C HEWL Cyt-C HEWL Cyt-C HEWL Cyt-C

K∞,i 0.50 0.43 0.65 0.46 1.26 1.00 1.20 0.95ε, εp 0.49, 0.81 0.33, 0.82 0.39, 0.83 0.39, 0.89

0th-orderequation (16)

˛(× 10−9) 13.65 4.68 90.64 4.34 89.90 7.92 308.7 5.96� 3.67 3.68 4.16 3.73 4.02 3.80 4.14 3.75

1st-orderequation (24)

˛(× 10−9) 6.72 1.68 14.10 0.56 83.10 4.32 227.5 2.58� 3.53 3.48 3.80 3.32 4.01 3.68 4.09 3.58

3.19

.79

c˛e6

owetlspaanobp

(eF0wtt(g

ti

Flg

to the complexity of the model: 0th-order approximation < 1st-order approximation < IM. RSS calculated using the parameters (˛0,ˇ0) are increasing with increasing � ′ values, while those calculatedfrom the 1st-order approximation remain more or less constants

IMequation (12)

˛(× 10−9) 5.33 1.14 1� 3.58 3.41 3

ase (Cyt-C on SP Sepharose BB), 10% difference is observed for the parameter, and only 0.3% for ˇ, while for the same conditions thestimates from the 0th-order approximated solution are more than00% off for ̨ and 12% for ˇ. These results are summarized in Fig. 3.

When using parameters estimated by the fitting of the 0th-rder approximated solution, the worst predictions are obtainedith SP Sepharose BB resin. As expected, for this resin the differ-

nce between the total porosity accessible to the salt (0.88) and tohe lysozyme and cytochrome C (0.55 and 0.48, respectively) is theargest among the investigated resins. In other words, the exclu-ion effect is more pronounced for this resin. For comparison, Fig. 4resents the elution volume calculated from Eq. (11) using (˛0, ˇ0)nd using (˛1, ˇ1). Even if the results obtained with (˛0, ˇ0) arelready in good agreement with the experimental data, it is worthoting that the prediction are further improved when using the 1st-rder correction. These results are also due to the strong correlationetween ̨ and ˇ, which can compensate each other differences toroduce very similar Henry constant values.

To get a better feeling of the improvement obtained when using˛1, ˇ1) instead of (˛0, ˇ0) the relative errors with respect to thexperimental data (computed as (Vexp

R − VR)/VexpR ) are shown in

ig. 5. It can be seen that the error is systematically larger for theth-order approximated solution, while the parameters obtainedith the 1st-order approximation give errors that are similar to

he one obtained by the IM, which is in the range of the experimen-al error. Not surprisingly, the error obtained with the parameters˛0, ˇ0) is maximal for the steepest gradients. Indeed, for steep

radients cR

M is larger than for shallow gradients, thus increasing

he influence of K∞ compared to ˛(cRM)

−ˇwhich amplify the error

ntroduced by the assumption K∞,i ≈ K∞. On average, the relative

ig. 4. Elution volume calculated with (˛0, ˇ0) (dashed lines) and (˛1, ˇ1) (plainines). The experimental data corresponds to lysozyme ( ) and cytochrome C ( )radient elution experiments on UNOsphere S.

0.62 71.61 4.34 224.2 2.723.33 3.98 3.67 4.08 3.58

error is divided by about 2 when using the parameters (˛1, ˇ1)instead of (˛0, ˇ0).

4.3. Parameter reliability and confidence intervals

The RSS were computed and shown in Fig. 6. Not surprisingly,the precision of the calculated elution volumes is directly correlated

Fig. 5. Relative error (VexpR

− V (i)R

)/VexpR

between experimental and simulated datacalculated by numerical integration of Eq. (11) with (˛i , ˇi) obtained from Eq. (16)( ), and Eq. (24) ( ). Data calculated with (˛IM , ˇIM) are also presented ( ). Exper-imental data correspond to lysozyme (top) and cytochrome C (down) elution on SPSepharose BB.

D. Pfister et al. / J. Chromatogr. A 1375 (2015) 33–41 39

Fw

appaTcvsieatsg

cotastRo

glriˇd

Fig. 7. Optimal values and 95% confidence interval obtained for the 0th-order

ig. 6. RSS as a function of � ′ for lysozyme (up) cytochrome C (down). � ′ valuesere evaluated with g = 200 mM/min.

nd close to the RSS estimated with the “true” parameters. Thearameter � ′ can be simply seen as the difference between the totalorosity and the accessible porosity for the protein. Moreover, � ′ is

function of the gradient slope, increasing with steeper gradients.herefore, the more the solute is excluded from the porous parti-les the bigger � ′ and the steeper the gradient the bigger � ′. Large � ′

alues implies large numerical error when using the approximatedolutions. Nevertheless, it seems that the 1st-order corrective terms enough to obtain very good agreement between simulated andxperimental data without increasing the computational effort. Inll cases, the RSS estimated with (˛1, ˇ1) is about 0.1. We proposehis value as a threshold above which the 1st-order approximationhould be used. According to Fig. 6 this corresponds to a � ′ valuesreater than 10−4.

To evaluate the reliability of the estimated values, the 95%onfidence intervals were computed for the 0th-order and the 1st-rder approximated solution. The smaller the confidence region,he more reliable are the estimated parameters. Fig. 7 clearly shows

narrower region in the (˛, ˇ) plane for the 1st-order approximatedolution compared to the 0th-order which is an additional proofhat the 1st-order model is more reliable than the 0th-order one.emarkably, it can be noticed that (˛1, ˇ1) are really close from theptimal (˛IM, ˇIM), much closer than (˛0, ˇ0).

As it can be seen in Fig. 7, the confidence regions are very elon-ated, meaning that all couples (˛, ˇ) that seems to fall the linen(˛) = f(ˇ) defined by the longer axis of the ellipsoidal confidence

egion are equivalently good (within the 5% confidence interval)n predicting the retention times. However, even though the (˛0,0) values seem to compensate the exclusion effect so that the pre-icted retention times are still in very good agreement with the

approximated solution ( and dash line) and 1st-order approximated solution (and plain line). (˛IM , ˇIM) are also depicted ( ). The experimental values correspondto lysozyme (top) and cytochrome C (down) elution experiments on SP SepharoseBB.

experimental data, this does not mean that these parameters maketotally sense from a physical point of view. By using Eq. (24) insteadof Eq. (16), the isotherm parameters ̨ and ̌ are retrieving theirphysical meaning.

4.4. Strongly excluded solutes

To emphasize even more this last observation, the same LGEexperiments were performed with PEGylated �-Lac on a stronganion exchange resin (Q Sepharose). Each PEGamer is known tobe constituted of several positional isomers, but under the inves-tigated conditions single peaks were obtained for each PEGamer.Therefore, they were considered as homogeneous pseudo-species.

The exclusion effect is expected to be more and more pro-nounced when increasing the degree of PEGylation of the protein(number of PEG chains attached to the protein). This is clearlyshown in Fig. 8.

The values of (˛0, ˇ0), as well as (˛1, ˇ1), are shown in Fig. 9. Itcan be seen that the parameters obtained from Eqs. (16) and (24),respectively, present radically different trends. While the (˛0, ˇ0)are increasing with the number of conjugated PEG chains, the (˛1,ˇ1) are decreasing. An increase of ̌ with the degree of PEGylationwould significate an increase of the apparent charge of the pro-tein interacting with the ligands. This would contradict previous

findings. Indeed, Seely and Richey [20] observed that the reten-tion time of PEGylated proteins was inversely proportional to thedegree of PEGylation in both anion and cation exchange chromatog-raphy. It is now commonly accepted in the literature that this is the

40 D. Pfister et al. / J. Chromatogr

Fig. 8. Total accessible porosity of the Q Sepharose High Performance column. Thedextran polymers are represented by ( ), while the �-Lac and PEGylated �-Lac aredepicted with ( ). The radius of the PEGylated proteins where estimated with theequation proposed by Fee and van Alstine [22].

Fl

rIlfid

eˇweaif

5

ee

K

b

[

[

[

[

[

[

[

[

ig. 9. ̨ (open symbols) and ̌ (filled symbols) as a function of the degree of PEGy-ation from the unmodified �-Lac to the 4-times PEGylated protein.

esults of a charge-shielding effect of the neutral polymer chains.n a recent work, while investigating the retention of PEGylatedysozyme on strong cation exchange resin, Yamamoto et al. con-rmed that the effective charge, ˇ, effectively decreases with theegree of PEGylation [21].

While the (˛, ˇ) estimated from both models were appar-ntly equivalently good in estimating the retention time, the (˛1,1) retrieve some physical meaning. As expected, they decreaseith the degree of PEGylation. For large macromolecules, strongly

xcluded from the intra-particle volume, the difference in poreccessibility between the modifier and the proteins has to be takennto account in order to derive physically reasonable and meaning-ul parameters.

. Conclusion

In this study we showed how to use approximate solution of thequilibrium model in conjunction with suitable LGE experiments tostimate linear isotherm parameters in the form:

= K∞ + ˛c−ˇM

A new procedure to interpret these results, based on Eq. (24) haseen developed in order to account for the exclusion effect which

[

[

. A 1375 (2015) 33–41

becomes significant when the solute size approaches the pore size.Indeed, while usually neglected, the exclusion effect can becomesignificant when dealing with macromolecules. This has beenexemplified with different protein and PEGylated proteins. The newapproach is proposed as an extension of the classical Yamamoto’sequation, for which exclusion effect are now accounted for. This hasbeen made possible by introducing in the classical equation (11) afirst order correcting term, defined as the difference between totaland effectively accessible intra-particle porosity, and by solving theresulting ODE with the perturbation method. The new approachremains simple, as it does not require any additional parametercompared to the classical Yamamoto’s equation. However, despiteits similarity with the classical approach, the proposed equation,now accounting for the exclusion effect, has been proved for pro-teins and PEGylated proteins to deliver much better, reliable andphysically meaningful linear isotherm parameters.

Acknowledgments

One of the authors (David Pfister) is indebted to Prof. G. Cartaand Prof. A. Jungbauer for introducing him to the theory of gra-dient elution chromatography and for fruitful discussions aboutYamamoto’s method during the summer class at the BOKU Univer-sity, Vienna. We are also grateful to Oliver Ingold for his meticulouswork with the �-lactalbumin PEGylation. This work was financiallysupported by Sanofi, Vitry sur Seine, France.

References

[1] M.A. Desai, Downstream Processing of Proteins, Humana Press, Totowa, NewJersey, 2000.

[2] P. Cutler, Protein Purification Protocols, Humana Press, Totowa,New Jersey, 2004.

[3] L. Hagel, G. Jagschies, G. Sofer, Handbook of Process Chromatography: Develop-ment, Manufacturing, Validation and Economics, Academic Press, Amsterdam,The Netherlands, 2007.

[4] T. Ishihara, T. Kadoya, S. Yamamoto, Application of a chromatography modelwith linear gradient elution experimental data to the rapid scale-up in ion-exchange process chromatography of proteins, J. Chromatogr. A 1162 (2007)34–40.

[5] G. Carta, A. Jungbauer, Protein Chromatography, Wiley-VCH Verlag GmbH &Co. KGaA, Weinheim, 2010, pp. 277–308.

[6] S. Yamamoto, K. Nakanishi, R. Matsuno, T. Kamikubo, Ion exchange chromatog-raphy of proteins-prediction of elution curves and operating conditions. I.Theoretical considerations, Biotechnol. Bioeng. 25 (1983) 1465–1483.

[7] S. Yamamoto, K. Nakanishi, R. Matsuno, T. Kamijubo, Ion exchange chromatog-raphy of proteins-predictions of elution curves and operating conditions. II.Experimental verification, Biotechnol. Bioeng. 25 (1983) 1373–1391.

[8] T. Ishihara, S. Yamamoto, Optimization of monoclonal antibody purifica-tion by ion-exchange chromatography: application of simple methods withlinear gradient elution experimental data, J. Chromatogr. A 1069 (2005)99–106.

[9] B.C. de Neuville, A. Tarafder, M. Morbidelli, Distributed pore model for bio-molecule chromatography, J. Chromatogr. A 1298 (2013) 26–34.

10] P. DePhillips, A.M. Lenhoff, Pore size distributions of cation-exchange adsor-bents determined by inverse size-exclusion chromatography, J. Chromatogr. A883 (2000) 39–54.

11] S. Yamamoto, K. Nakanishi, R. Matsuno, Ion-Exchange Chromatography of Pro-teins, CRC Press, New York, 1988.

12] C.A. Brooks, S.M. Cramer, Steric mass-action ion exchange: displacement pro-files and induced salt gradients, AIChE J. 38 (1992) 1969–1978.

13] G. Guiochon, A. Felinger, D.G. Shirazi, A.M. Katti, Fundamentals of Preparativeand Nonlinear Chromatography, Academic Press, Amsterdam, The Netherlands,2006.

14] E.C. Freiling, Ion exchange as a separation method. 9. Gradient elution theory,J. Am. Chem. Soc. 77 (1955) 2067–2071.

15] E.S. Parente, D.B. Wetlaufer, Relationship between isocratic and gradient reten-tion times in the high-performance ion-exchange chromatography of proteins:theory and experiment, J. Chromatogr. 355 (1986) 29–40.

16] A.H. Nayfeh, Introduction to Perturbation Techniques, Wiley-VCH Verlag GmbH& Co. KGaA, Weinheim, 1993.

17] K. Kaczmarski, Estimation of adsorption isotherm parameters with inverse

method-possible problems, J. Chromatogr. A 1176 (2007) 57–68.

18] D.M. Bates, D.G. Watts, Nonlinear Regression Analysis and Its Applications, JohnWiley & Sons, Inc., New York, 2008.

19] W.A. Duckworth, W.R. Stephenson, Beyond traditional statistical methods, Am.Stat. 56 (2002) 230–233.

atogr

[

[

D. Pfister et al. / J. Chrom

20] J.E. Seely, C.W. Richey, Use of ion-exchange chromatography and hydrophobicinteraction chromatography in the preparation and recovery of polyethyleneglycol-linked proteins, J. Chromatogr. A 908 (2001) 235–241.

21] S. Yamamoto, S. Fujii, N. Yoshimoto, P. Akbarzadehlaleh, Effects of protein con-formational changes on separation performance in electrostatic interaction

[

. A 1375 (2015) 33–41 41

chromatography: unfolded proteins and PEGylated proteins, J. Biotechnol. 132(2007) 196–201.

22] C.J. Fee, J.M. Van Alstine, Prediction of the viscosity radius and the size exclu-sion chromatography behavior of PEGylated proteins, Bioconj. Chem. 15 (2004)1304–1313.