Analysis and Simulation of Batch Affinity Processes

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Analysis and Simulation of Batch Affinity Processes
Applied to Separation of Biomolecules
Patricia Guerrero-Germán,1 Armando Lucero-Acuña,2 Rosa Ma. Montesinos,3 Armando Tejeda-Mansir4*
1 Instituto de Ingeniería. Universidad Autónoma de Baja California. Blvd. Benito Juárez y Calle Normal. Col. Insurgentes Este.
Mexicali, Baja California. CP 21280. Tel. y Fax. 686 566 41 50.
2 Departamento de Ingeniería Química y Metalurgia. Universidad de Sonora. Hermosillo, Sonora. México. 83000.
Tel.: +52 (662) 259 21 06. Fax: +52 (662) 259 21 05.
3 Departamento de Matemáticas. Universidad de Sonora. Hermosillo, Sonora. México. 83000. Tel. +52 (662) 259 21 55.
Fax. +52 (662) 259 22 19.
4Departamento de Investigaciones Científicas y Tecnológicas. Universidad de Sonora. Apartado Postal 593. Hermosillo, Sonora.
México. 83000. Tel.: + 52 (662) 212 19 95. Fax: +52 (662) 2 12 32 71. atejeda@guayacan.uson.mx
Recibido el 30 de octubre de 2006; aceptado el 16 de enero de 2007
J. Mex. Chem. Soc. 2007, 51(2), 59-66
© 2007, Sociedad Química de México
ISSN 1870-249X
Article
Abstract: The scale-up and optimization of large-scale affinity chro-
matographic operations is of major industrial importance. In this
work, a transport model which includes pore diffusion, external film
resistance, and finite kinetic rate, was used to mathematically
describe the performance of a batch affinity adsorption system.
Experimental data from literature describing the adsorption of β-
galactosidase onto anti-β-galactosidase immobilized on porous silica
was used as a model system. The mathematical model was solved
using the numerical method of lines (MOL) in a MATLAB platform.
The use of the transport model is a unique way to predict batch affini-
ty performance as well as to obtain a better understanding of the fun-
damental mechanisms involved in the bioseparations.
Keywords: Mathematical modeling; batch affinity chromatography;
biomolecules.
Resumen: El escalamiento y optimización de las operaciones croma-
tográficas de afinidad presentan un gran interés industrial. En este tra-
bajo se utilizó un modelo de transporte que incluye la difusión en el
poro, la resistencia en la película y una cinética de adsorción finita,
para describir matemáticamente el comportamiento de un sistema de
adsorción por afinidad en un tanque perfectamente agitado. Como sis-
tema modelo se utilizaron datos experimentales de la literatura que
describen la adsorción por afinidad de β-galactosidasa en anti-β-
galactosidasa inmovilizada en partículas de sílice porosa. El modelo
matemático fue resuelto utilizando el método numérico de líneas
(MOL) utilizando una plataforma MATLAB. El uso del modelo de
transporte es una forma única para predecir el comportamiento de la
adsorción por afinidad, así como para lograr un mejor entendimiento
de los mecanismos fundamentales de las bioseparaciones.
Palabras clave: Modelación matemática; cromatografía de afinidad
por lotes; biomoléculas
Introduction
Affinity chromatography is nowadays an industrial standard
method used to purify high value proteins present at very low
concentrations in complex biological fluids such as liquid
culture media and sera [1-4]. Recently, affinity separations
have been considered for large-scale separations of plasmid
DNA for gene therapy and DNA vaccine applications [5]. In
this chromatographic mode an affinity solid matrix is pre-
pared by immobilizing a ligand that interact specifically with
the solute of interest. Among the molecules used as biospe-
cific or pseudospecific ligands are proteins, DNA, antibod-
ies, amino acids, inhibitors, cofactors, dyes and quelated
metal ions [6-10].
The conventional operating format for affinity separations
is a packed column of porous adsorbent. However, in some
cases it may be convenient to perform the adsorption step in a
batch mode using well-stirred tanks, for example, when crude
extracts are purified, to avoid column clogging problems [1].
The scale-up and optimization of affinity chromatographic
operations is of major industrial importance [11-14]. The
development of mathematical models to describe affinity chro-
matographic processes, is an engineering approach that can
help to successfully attain these bioprocess engineering tasks
[12,15]. A thorough understanding of the fundamental mecha-
nisms underlying such separations is needed in order to devel-
op realistic models based on basic physical and chemical prin-
ciples. The equations obtained through this approach generally
involve non-linear partial differential equations (PDE) that are
not amenable of analytical solutions. Computer programs need
to be developed to solve these models.
Several efforts have been made to model and simulate
batch affinity adsorption. Chase [16] used a simplified model
known as the lumped parameters model to describe biospecific
separations. Arnold et al. [1] derived a transport model assum-
ing that pore diffusion was the rate controlling step. Arve and
Liapis [17] considered that the adsorption of a solute involves
three discrete steps which contribute resistance to the mass-
transfer: film diffusion, pore diffusion and reaction kinetics.
Hortsmann and Chase [18] used a two resistances model with
an infinity fast reaction supposition. Numerical solution of the
governing differential equations was carried out by a finite dif-
ference method using second order approximations in the
space derivative. However, the method is impracticable to
solve more complex systems. The use of advanced numerical
methods is still required to solve these models.
60 J. Mex. Chem. Soc. 2007, 51(2) Patricia Guerrero-Germán et al.
There are two main strategies for the solution of adsorp-
tion PDE: global methods and methods of lines. Both time and
spatial derivatives are discretized in the so called global meth-
ods. The numerical method of lines (MOL) evolved recently
and is tied to the development of robust algorithms for the
solution of initial value ordinary differential equation, namely
the stiff ones [19-21]. The solution of the partial differential
equations is performed in two steps: the boundary value prob-
lem is solved using a discretization technique, and then the
remaining initial value problem is handled with an appropriate
integrator. Although there are several software packages that
use this methodology to solve two-dimensional problems, is
not usually easy to cast the adsorption equations into these
packages, since adsorption description is not given by truly
two-dimensional but rather by two region model [22].
In this work a transport model that considers three consec-
utive transport rate resistances to ideal equilibrium separation:
external film resistance, particle internal diffusion and finite
kinetic rate, is used for simulation of batch affinity adsorption.
The solution of the model was obtained using the numerical
method of lines and the MATLAB platform. The solution was
compared with the analytic solution of the lumped parameters
batch affinity model and [16, 23, 24], with experimental data
from the literature of the batch adsorption of β-galactosidase
to anti-β-galactosidase immobilized onto porous silica matrix
[17]. The transport model was used to perform a parametric
analysis of the experimental batch system. The influence of
both process parameters as well as physical parameters on the
affinity process was investigated.
Batch Affinity Adsorption Process Description
Much of the information needed to evaluate affinity adsorption
performance is contained in the typical plots of solute concen-
tration versus time. These curves can be used to determine: 1)
how much of the adsorbent capacity has been utilized, 2) how
much solute is lost in an operation cycle, and 3) the processing
time. A mathematical model which can be used to accurately
predict this dynamic behavior provides a practical way to
obviate many experiments in the design and scale-up of an
affinity process.
Transport model
In this study the batch affinity model is based on the isother-mal sorption of a single solute to spherical adsorbent particles
with an average radius, rm, and a porosity, εi . The operation is
conducted in a well-stirred tank with a total system volume,
Vs. The liquid volume external to the adsorbent matrix is, V =
εbVs, and the adsorbent volume, v = (1 – εb)Vs. The initial and
the transient solute concentration in the liquid are, co and c(t),
respectively. The solute concentrations in the fluid and solid
phase of the adsorbent pores are ci and qi, respectively.
To achieve a mathematical description of a batch affinity
chromatographic process, three consecutive transport rate
resistances to ideal equilibrium separation are incorporated:
film mass-transfer, porous diffusion and kinetic solute-ligand
interaction. The transport of solute is considered to involve the
interfacial transport of solute to the outer surface of the adsor-
bent particles from the bulk liquid through the adsorbent sur-
rounding stagnant film characterized by the film coefficient, kf ,
the diffusion in the pore fluid described by a diffusion coeffi-
cient, Di, and the adsorption step of the solute with active sites
on the surface of the adsorbent. The intrinsic adsorption rate can
be described by different kinetic models. In this study an adsorp-
tion-desorption model of the Langmuir type is used. It is general-
ly assumed that surface diffusion can be neglected in affinity
adsorption processes due to the strong interaction between the
ligand and the solute developed in these systems [11].
Due to the nonlinear equilibrium that characterizes affini-
ty chromatography, adsorption behavior is best described by
rate theories. This engineering approach to modeling involves
the use of conservation equations, equilibrium laws at inter-
faces, kinetic laws of transport and adsorption and, initial and
boundary conditions.
To describe the concentration change of solute with time
in the bulk fluid, the following equation can be derived by a
solute mass balance in that phase,
(1)
The equation to describe the change of concentration of
solute in the fluid of the adsorbent pores can be obtained by a
solute mass balance in the particle,
(2)
To describe the complex interactions between solute and
affinity adsorbent simplified models are often used [1,16, 18].
In general a second-order reversible adsorption reaction is
considered, where the solute is assumed to interact with the
adsorbent by a monovalent interaction and characteristic con-
stant binding energy,
where P is the solute in solution, S is the ligand adsorption
site; and PS is the solute-ligand complex.
The rate of adsorption of this type of interaction is usually
represented by the Langmuir model,
(3)
where qm is the maximum solute concentration in the affinity
adsorbent, and k1 and k-1 are the specific kinetic constants of
adsorption and desorption.
At the beginning of the operation the protein concen-
tration in the solution is co and there is no solute present in
the adsorbent, therefore the following initial conditions are
used:
Analysis and Simulation of Batch Affinity Processes Applied to Separation of Biomolecules 61
at t = 0, c = co (4)
at t = 0, ci = 0, 0 < r < rm (5)
at t = 0, qi = 0, 0 < r < rm (6)
Due to particle symmetry the following boundary condi-
tion is imposed on the system:
at r = 0, (7)
The second boundary condition can be derived by a solute
mass balance at the mouth of a particle pore,
at r = rm (8)
Lumped parameters batch affinity model
The lumped parameters model [16, 24] takes an empirical
approach to the adsorption process and assumes that all the
rate limiting processes can be represented by kinetic rate con-
stants. In such an approach, the rate of mass-transfer of solute
to the adsorbent is assumed to be described by eq. (3) above.
For batch affinity adsorption in a well-stirred tank, the solute
concentration in solution at time, t, is given by the following
analytical solution of the model:
(9)
where:
(10)
(11)
Input data for the study
A fundamental requirement for the use of the mathematical
models of affinity adsorption to predict the behavior, scale or
design a process of interest, it is the precise estimate, in exper-
imental form or by means of correlations, of the parameters
associated to the model.
The equilibrium system parameters: maximum capacity of
adsorption, qm, and the dissociation equilibrium constant, Kd =
k-1/k1, can be estimated from equilibrium data. The kinetic
parameter, k1, can be calculated using dynamic adsorption data
obtained in batch or column systems [16,18].
The effective pore diffusivity, Di, can be estimated
through experimental uptake results obtained in batch or col-
umn systems, according to detailed reported procedures in the
literature [4]. This parameter has also been estimated in pre-
cise form using models generated for adsorbent of porous
structure obtained by means of the pore network theory. This
theory properly represents the discreet nature of the porous
structures of chromatographic media and also properly
describes phenomena of restricted pore diffusion and dynamic
loading effects on intraparticle convection and diffusion [25-
27]. The parameters that characterize intraparticle pore diffu-
sion have been obtained also by combining the experimental
method of confocal scanning laser microscopy with theoretical
models [28-33].
The molecular diffusivity of β-galactosidase in free aque-
ous solution, DAB, can be estimated by using the following
semi-empirical equation [34]:
(12)
Where MA is the relative molecular weight of substance A.
Taking 464,000 g/mol as the relative molecular weight of β-
galactosidase, the absolute temperature T = 298 ºK and the liq-
uid viscosity μ = 1 × 10-3 Kg/m –s, the value of molecular dif-
fusion coefficients is D
AB
= 3.62 × 10-11 m2/s. The effective
pore diffusivity is given by the ratio of the molecular diffusivi-
ty to the tortuosity coefficient. According to Arve and Liapis
[17] the tortuosity of the experimental system is 5.23 then Di =
6.9 × 10-12 m2/s.
The correlation used to estimate the liquid film mass
transfer coefficient, kf, of the protein to the adsorbent particles
in stirred tank experiments is given by [35]:
(13)
Where pc = 1000 Kg/m3 is the liquid density, Δρ = 402
Kg/m3 is the density difference between the adsorbent particle
and the liquid and g = 9.8 m/s2 is the gravity acceleration. For
a particle diameter dp = 150 × 10-6m, the value of the mass-
transfer coefficient is kf = 5.84 × 10-6 m/s.
When the values of the parameters that characterize the
mass transfer and the mechanisms of adsorption of the solute
on the affinity matrix have been determined, the mathematical
model can be used to predict and study the dynamic behavior
and performance of the affinity system for different design and
operating conditions.
Experimental data from the literature for the adsorption of
β-galactosidase onto anti-β-galactosidase immobilized on
porous silica matrix was chosen as model system [17]. The
values of the parameters utilized to conduct the simulation
studies are presented in Table I.
Numerical solution of the transport model
The transport model to describe the batch affinity adsorption
of proteins in a perfectly stirred tank is given by the eqs. (1-8).
This model can only be solved by advanced numerical tech-
niques. In this study the solution of the model was obtained
62 J. Mex. Chem. Soc. 2007, 51(2) Patricia Guerrero-Germán et al.
using the numeric method of lines MOL in a MATLAB
(v.7.1) platform [36], according to the block diagram shown in
Figure 1.
The code STAF was used as the main program. The sys-
tem parameters, the MATLAB integrator ode15s [37-39] and
the printing instructions are incorporated in the main program.
The integrator ode 15s (which implements back difference
formulas) computes the solution over each time interval. This
integrator will, in general, call the function STAF_ODE many
times during the calculation of the solution by numerical inte-
gration.
Thesystem of ordinary differential equations which
approximates the PDEs of the model is programmed in sub-
routine STAF_ODE. This function calls the functions dss004
and dss044 to calculate the first and second spatial derivatives
∂ci/∂r and ∂2ci/∂r2, using five-point, fourth order finite differ-
ence approximations. All the codes were incorporated in
MATLAB programs that were run in a PC.
Results and Discussion
The solution to the transport model for batch affinity adsorp-
tion of β-alactosidase onto anti-β-galactosidase immobilized
on porous silica was solved using the numerical method of
lines (MOL) in a MATLAB platform. This solution was com-
pared with the analytical solution of the lumped parameters
batch affinity model and, with the experimental data. The
results are shown in Figure 2.
The simulated run using MOL fitted well to the experi-
mental data when the base parameter values were used, except
for the kinetic parameter that was set to the relatively high
value of k1= 0.0235 mL/mg-s, since in the model this is not a
lumped parameter. This result suggests that the mass-transfer
process is only controlled by pore diffusion and film resis-
tance.
The simulation run with the analytic solution also fitted
well to the experimental data using a kinetic parameter value
of k1l= 0.00022 mL/mg-s. It is well known that lumped para-
meters approach is more useful when mass-transfer resistances
are low. In behalf of these results the MOL approach was used
to further conduct the analysis.
The MOL solution of the transport model was employed
to describe in more detailed form the affinity chromatography
process, e.g. detailing the protein dimensionless concentration
profiles in the adsorbent-pore liquid, ci/c0 , and in the adsorbed
phase, qi/qm , as function of the dimensionless radio length, R,
and the real time, t (Fig. 3).
A very sharp protein-concentration profile in the adsor-
bent-pore liquid is obtained initially (Fig. 3a), suggesting a
considerable mass transfer resistance within the adsorbent
pores. As the adsorption proceeds the concentrations profiles
become less sharp until a very low equilibrium profile is
reached after 400 min approximately. The protein concentra-
tion profile in the adsorbed phase (Fig. 3b) shows a rapid
increase from an initial zero value to an equilibrium value.
This profile is more uniform than adsorbent-pore liquid profile
and reaches higher values, due the favorable Langmuir equilib-
rium of the system. Most of the adsorption phenomena is carry
out on the external volume of the particle (R > 0.5) owing the
high mass-transfer resistance within the adsorbent pores.
In order to study how the transport model solution is able
to account for variations of operating characteristics with sys-
tem parameters, a parametric analysis was performed by over-
laying batch affinity curves from several computer simula-
tions, in which one parameter was changed while the others
were kept constant at the basic set of values shown in Table 1.
The effect of bead diameter and initial protein concentra-
tion is reported. The process parameter of most interest is the
bead diameter. The bead diameter was changed using ±40 and
±20% variations of the dp = 150 μm base value. The corre-
sponding curves are shown in Figure 4. A sharper curve and
 
Fig. 1. Numerical method of lines solution (MOL) scheme of the
batch affinity transport model in a MATLAB platform.
Fig. 2. Batch affinity adsorption of β-galactosidase onto anti-β-galac-
tosidase immobilized on porous silica. Operating conditions accord-
ing to Table 1. (o) Experimental data [17]. (⎯) MOL solution of the
bach affinity transport mode in a MATLAB platform. (- - -) Lumped
parameters model solution.
Analysis and Simulation of Batch Affinity Processes Applied to Separation of Biomolecules 63
consequently greater process productivity is obtained as the
bead diameter decreases (Fig. 4a). The adsorption rate increas-
es markedly, since the diffusion time is decreased due to the
shorter diffusion path. At the same time the area/volume ratio
for a single particle (3/rm) increases, giving an increased mass
transfer area between the surrounding liquid phase and the
bead. Both factors contribute to the increase in total adsorp-
tion rate. A shorter diffusion path also produces less pro-
nounced protein concentration profiles in the adsorbed phase
(Fig. 4b) and practically no effect on the protein-concentration
profile in the adsorbent-pore liquid (Fig. 4c and Fig. 4d).
Upstream perturbations can initiate changes in process
concentrations that are important for study. The initial protein
concentration was changed using ±40 and ±20% variations of
the co = 0.0158 mg/mL base value. The corresponding curves
are shown in Figure 5. A greater proportional degree of
adsorption at equilibrium is achieved at lower protein initial
concentration as can be seen in the horizontal portion of the
curves of the Figure 5a. It must be noted that protein concen-
tration is normalized. When plotted on this basis, the initial
rates of adsorption are slightly affected by protein concentra-
tion. However, if absolute amounts of adsorption are being
considered, greater degrees and rates of adsorption occur at
higher concentrations.
The protein concentration in the adsorbed phase (Fig. 5b)
and the protein-concentration the adsorbent-pore liquid (Fig.
5c and Fig. 5d) show this behavior in a more detailed way.
The utilization of the adsorption capacity of the matrix is
greater at higher solute concentration as these conditions favor
a greater extent of adsorption at equilibrium.
Table 1. Base case data used in simulation studies of batch affinity adsorption of β-galactosidase onto anti-β-galactosidase immobilized on
porous silica [17].
Variable Value
Initial batch protein concentration co = 0.0158 mg/mL
Maximum adsorption capacity qma = 2.2 mg/mL
Particle porosity εi = 0.5
Maximum adsorption capacity of solid gel qm = 4.4 mg/mL
Equilibrium desorption constant Kd = 0.00022 mg/mL
Lumped adsorption kinetic constant kl1 = 0.0055 ml/mg-s
Specific adsorption rate constant k1 = 0.0235 ml/mg-s
Particle radius rm = 75 μm
Film mass-transfer coefficient kf = 5.84 × 10
-6 m/s
Intraparticle protein diffusivity Di = 6.9 × 10
-12 m2/s
Initial batch liquid volume V = 100 mL
Initial adsorbent volume v = 1.5 mL
Ratio of liquid volume to batch system volume εb = 100/101.5
Fig. 3. Concentration profiles of β-galactosidase. Operating conditions according to Table 1. (a) Protein dimensionless concentration profiles in
the adsorbent-pore liquid. (b) Protein dimensionless concentration profiles in the adsorbed phase.
64 J. Mex. Chem. Soc. 2007, 51(2) Patricia Guerrero-Germán et al.
Fig. 4. Influence of the bead diameter on batch affinity. Operating conditions according to Table 1. (a) Bulk dimensionless protein concentra-
tion. (b) Protein dimensionless concentration profiles in the adsorbed phase. (c) Protein dimensionless concentration profiles in the adsorbent-
pore liquid. (d) Zoom area of protein dimensionless concentration profiles in the adsorbent-pore liquid. (A) -40%, (B) -20%, (C) dp = 150 μm,
(D) + 20%, (E) + 40%.
Fig. 5. Influence of the inicial protein concentration on the batch affinity behavior. Operating conditions according to Table 1. (a) Bulk dimen-
sionless protein concentration. (b) Protein dimensionless concentration profiles in the adsorbed phase. (c) Protein dimensionless concentration
profiles in the adsorbent-pore liquid. (d) Zoom area of protein dimensionless concentration profiles in the adsorbent-pore liquid. (A) -40%, (B) -
20%, (C) co = 0.0158 mg/mL, (D) + 20%, (E) + 40%.
Analysis and Simulation of Batch Affinity Processes Applied to Separation of Biomolecules 65
Conclusions
The performance of the adsorption of β-galactosidase onto
anti-β-galactosidase immobilized on porous silica, in a batch
affinity chromatography, was successfully described with a
three-resistances model. Programming the model solution was
relatively simple usingMOL in a MATLAB platform. This
solution was compared with the analytical solution of the
lumped parameters model. In the simulation studies, the best
fitting of the experimental data was obtained with the numeri-
cal method of lines solution. The parametric analysis conduct-
ed helps to show the influence of both operation and system
parameters on the affinity process. In the bead-size parametric
study, a sharper adsorption curve and consequently a greater
operating throughput was obtained as the bead diameter
decreases. The initial concentration study showed that greater
degrees and rates of adsorption occur at higher concentrations.
The dynamic responses obtained are in concordance with the-
oretical predictions and show that the transport model can be
used as a framework to provide a general description of
almost all practical systems, when the appropriate basic exper-
imental parameters and numerical solution are used. The MOL
solution of the transport model permitted an accurate predic-
tion of the batch affinity performance and a better understand-
ing of the fundamental mechanisms responsible for the separa-
tion. A similar analysis for multicomponent systems should be
done to extend the model to more practical situations, since
other components in a complex mixture may influence the
adsorption properties of the protein.
Symbols
co initial protein concentration, mg/mL
c protein concentration in the bulk liquid, mg/mL
ci protein concentration in the fluid of the adsorbent pores,
mg/mL
dP adsorbent particle diameter, m
Di intraparticle protein diffusivity (free molecular diffusivi-
ty/pore tortuosity), m²/s
k1 specific adsorption rate constant, mL/mg.s
kl1 lumped adsorption rate constant, mL/mg.s
k-1 specific desorption rate constant, s-¹
kf external film mass-transfer coefficient, m/s
Kd equilibrium desorption constant, k-1/k1
P protein molecule
PS protein-active site complex
qi protein concentration in the adsorbed phase of the adsor-
bent particles, mg/mL
qm maximum equilibrium concentration, mg/mL of solid vol-
ume of adsorbent
qma maximum equilibrium concentration, mg/mL of settled
volume of adsorbent
r radial distance in the adsorbent particle, m
rm radius of adsorbent particle, μm
R dimensional radius
S active site
t time, s
VS batch system volume, mL
V liquid volume, mL
v adsorbent volumen, mL
Greek letters
εb ratio of liquid external to adsorbent to system volume
εi pore void fraction in the adsorbent particle
Acknowledgements
The authors gratefully acknowledge support of this work by
the Consejo Nacional de Ciencia y Tecnología de México
under grant No. 28295U and the Universidad de Sonora.
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32. Ljunglof, A.; Bergvall, P.; Bhikhabhai, R.; Hjorth, R. J.
Chromatogr. A. 1999, 844, 29-135.
33. Zhou, X.; Li, W.; Shi, Q.; Sun, Y. J. Chromatogr. A. 2006, 1103,
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34. Guichon, G.; Shirazi, S. G.; Katti, A. M., Fundamentals of
Preparative and Nolinear Chromatography, Academic Press,
Boston, MA, 1994.
35. Geankoplis, C. J. Transport Processes and Unit Operations,
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>> setdistillerparams
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