Prog. Theor. Exp. Phys. 2015, 033J01 (25 pages)
DOI: 10.1093/ptep/ptv026
Bovine serum albumin adsorption onto
functionalized polystyrene lattices: A theoretical
modeling approach and error analysis
Manel Beragoui1 , Chadlia Aguir2 , Mohamed Khalfaoui1,∗ , Eduardo Enciso3 ,
Maria José Torralvo3 , Laurent Duclaux4 , Laurence Reinert4 , Marylène Vayer5 ,
Abdelmottaleb Ben Lamine1
1
Unité de recherche de physique quantique, Faculté des Sciences de Monastir, Université de Monastir, Tunisie
Unité de recherche de chimie appliquée et environnement, Faculté des Sciences de Monastir,
Université de Monastir, Tunisie
3
Departamento de Quı́mica Fı́sica I, Facultad de Ciencias Quı́micas, Universidad Complutense,
28040 Madrid, Spain
4
Laboratoire de chimie moléculaire et environnement, Université de Savoie, France
5
Centre de Recherche sur la Matière Divisée, 1b rue de la Férollerie, 45071 Orléans, France
∗
E-mail: [email protected]
2
Received December 10, 2014; Revised January 28, 2015; Accepted January 29, 2015; Published March 21 , 2015
...............................................................................
The present work involves the study of bovine serum albumin adsorption onto five functionalized
polystyrene lattices. The adsorption measurements have been carried out using a quartz crystal
microbalance. Poly(styrene-co-itaconic acid) was found to be an effective adsorbent for bovine
serum albumin molecule adsorption. The experimental isotherm data were analyzed using theoretical models based on a statistical physics approach, namely monolayer, double layer with two
successive energy levels, finite multilayer, and modified Brunauer–Emmet–Teller. The equilibrium data were then analyzed using five different non-linear error analysis methods and it was
found that the finite multilayer model best describes the protein adsorption data. Surface characteristics, i.e., surface charge density and number density of surface carboxyl groups, were used to
investigate their effect on the adsorption capacity. The combination of the results obtained from
the number of adsorbed layers, the number of adsorbed molecules per site, and the thickness of
the adsorbed bovine serum albumin layer allows us to predict that the adsorption of this protein
molecule can also be distinguished by monolayer or multilayer adsorption with end-on, side-on,
and overlap conformations. The magnitudes of the calculated adsorption energy indicate that
bovine serum albumin molecules are physisorbed onto the adsorbent lattices.
...............................................................................
Subject Index
J36, J41, J50
Nomenclature
A
Sensible surface area of quartz (cm2 )
ARE The average relative error
Ce Concentration of adsorbed molecules
at equilibrium (µg/ml)
C1 Half-saturation concentration of a
monolayer (µg/mL)
C2 Half-saturation concentration of
global isotherm (µg/mL)
P
p
PDI
Percentage of comonomer (wt%)
Number of parameters
Polydispersity index
Qa
Adsorbed quantity (µg/cm2 )
Q a sat Adsorbed quantity at saturation
(µg/cm2 )
© The Author(s) 2015. Published by Oxford University Press on behalf of the Physical Society of Japan.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/),
which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
PTEP 2015, 033J01
Cs
Concentration of solubility (µg/mL)
M. Beragoui et al.
qe,calc
Calibration constant at 5 MHz
(56.6 Hz · cm2 /µg)
Particle size (nm)
Dn
Thickness value
df BSA
ERRSQ The sum of the squares of the errors
EABS
The sum of the absolute errors
J
Grand potential
h
Planck’s constant (6.26 × 10−34 m2 ·
kg/s)
HYBRID The hybrid fractional error function
j
Number of data points
Boltzmann’s constant (1.380 ×
kB
10−23 m2 · kg/s · K )
M
Molecular mass of bovine serum
albumin (66430 g/mol)
M
Adsorbed molecule(s)
qe,meas
m
MPSD
Mass of an adsorbed molecule (mg)
Marquardt’s percent standard deviation
Number of adsorbate molecules
Number of adsorbed molecule(s) per
site S
Receptor site densities (µg/cm2 )
Occupation number in εi level
ε
εi
Average occupation number of receptor sites Nm
Average number of the adsorbed
layers
Number density of surface carboxyl
groups (nm−2 )
E a2
Cf
N
n
Nm
Ni
No
Nl
Nc
1.
R
R2
SBET
SNE
T
V
VTOT
z gc
z gtr
μ
μp
σ
m
E d
E a1
− E a1
− E a2
Theoretical
adsorbed
quantity
2
(µg/cm )
Experimentally determined adsorbed
quantity (µg/cm2 )
Ideal gas constant (8.314 J/K · mol)
Correlation coefficient
BET specific surface area (m2 /g)
Sum of normalized errors
Temperature (K)
Volume occupied by a molecule (mL)
Total pore volume (cm3 /g)
Grand canonical partition function
Partition function of translation per
unit volume
Chemical potential (kJ/mol)
Chemical potential of a molecule
assimilated for an ideal gas (kJ/mol)
Adsorption site energy (kJ/mol)
Adsorption energy of the receptor
site i (kJ/mol)
Surface charge densities (mC/m2 )
Additional mass deposited on the
quartz (µg/cm2 )
Dissolution energy (kJ/mol)
Adsorption energy at the first layer
(kJ/mol)
Adsorption energy at the next layers
(kJ/mol)
Total adsorption energy per unit area
at the first layer (kJ/cm2 )
Total adsorption energy per unit area
at the next layers (kJ/cm2 )
Introduction
Protein adsorption at interfaces has been the focus of a significant body of research, with interest
in the fields of food processing, biological materials, and sensors [1–3]. Understanding the protein
adsorption mechanism and the processes involved when a protein interacts at the solid–liquid interface is of primary importance in the design of biocompatible surfaces. Furthermore, avoiding protein
adsorption is complicated. In fact there is a huge community seeking biocompatible and proteinresistant materials applicable to biomedical implants or analytical platforms. A number of recent
review papers provide a comprehensive overview of this area [4–12]. For example, it is known that
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the exposure of blood to a foreign material results in the adsorption of plasma proteins to the surface [13,14]. Then, platelets adhere to the surface, which frequently causes thrombus formation, and
this issue is still a major obstacle for the utility of the device [15]. Moreover, biomedical materials
used in the body are required to resist thrombus formation and inflammation. Since these reactions
are triggered by protein adsorption from blood and body fluids onto the material surfaces, the suppression of protein adsorption is essential to improve the blood compatibility of biomedical material
surfaces [16].
Surface modification is one of the most successful approaches to suppressing protein adsorption in
order to improve the blood compatibility of biomedical materials [7,16,17]. In recent years, surface
modification of various materials has been investigated using polymer latex surfaces with functional
groups to which the proteins can be coupled [18,19]. Functional polymers including biodegradable
polymer and stimuli-responsive polymer have also been used to control the surface properties of the
material [20,21]. Furthermore, protein adsorption on polymer particles has significant importance
in biomedical applications, both in vitro and in vivo. Because of the great relevance of the protein–
surface interaction phenomenon, much effort has gone into the development of protein adsorption
experiments and models. The ultimate goals of such studies would be to measure, predict, and
understand the protein conformation, surface coverage, superstructure, and theoretical details of the
protein–surface interaction. Adsorption is often a highly dynamic phenomenon where the molecules
may change orientation and conformation during or after the adsorption (side-on, end-on, or overlap conformations). Protein molecules are normally more influenced by a nonionic or hydrophobic
surface than by a polar and hydrophilic surface [22].
On the other hand, the first major aim for the adsorption field is to select the most promising
types of adsorbent, mainly in terms of high capacity and adsorption rate, high selectivity, and low
cost [23,24]. The next real aim is to identify clearly the adsorption mechanism(s), in particular the
interactions which are implicated at the adsorbent–adsorbate interface. Adsorption properties and
equilibrium data, commonly known as adsorption isotherms, describe how proteins interact with
adsorbent materials and are thus critical in optimizing the use of adsorbents [23–25]. There are several isotherm models available for analyzing experimental data and for describing the equilibrium
of adsorption, including Langmuir, BET, Freundlich, Toth, Tempkin, Redlich–Peterson, Frumkin,
Jossens, Halsey, Henderson, Dubinin–Radushkevich and GAP isotherms [26–32]. However, the most
frequently used equations for protein adsorption onto polymers to describe adsorption isotherms are
the Langmuir, Freundlich [23], Brunauer–Emmett–Teller (BET) [29], GAP, and Halsey [30] models.
In this paper, we describe the adsorption isotherm of bovine serum albumin (BSA) as a model
protein onto a lattice surface, i.e., polystyrene as homopolymer and PS-IA, PS-MAA, PS-AA, and
PS-HEMA as copolymers, using a quartz crystal microbalance (QCM). Experimental data were
analyzed thanks to commonly used models developed by our research group, namely monolayer
[33], double layer with two successive energy levels [34], finite multilayer [35], and modified BET
[28] models. Hence, the goal of the theoretical development is to give physical meaning to the constants that these models contain and therefore to facilitate understanding of the adsorption process
at the molecular level. A detailed error analysis was undertaken to investigate the effect of using
different error criteria for the determination of the single-component isotherm parameters and thus
obtain the best-fit isotherm and then the set of parameters which describe the adsorption process.
Five different error functions were used: the sum of the squares of the errors (ERRSQ), the sum
of the absolute errors (EABS), the average relative error (ARE), the hybrid fractional error function (HYBRID), and Marquardt’s percent standard deviation (MPSD). These error functions were
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Table 1. Physical and chemical characteristics of polystyrene functionalized lattices.
(∗ )Percentage of comonomer.
Samples
Particle size, Dn , (nm)
BET specific surface area, SBET ,
(m2 /g)
Polydispersity index, PDI
Total pore volume, VTOT , (cm3 /g)
P(∗) (wt %)
Surface charge densities, σ ,
(mC/m2 ) (pH 4.7)
Number density of surface
carboxyl groups, Nc , (nm−2 )
PS
PS-IA
PS-HEMA
PS-AA
PS-MAA
653
9.0
508
12.6
342
17.0
404
15.1
339
18.6
1.007
0.028
—
−23.22
1.004
0.075
8.5
−25.52
1.003
0.16
12.5
−70.13
1.005
0.084
5
−74.54
1.002
0.12
5.6
−59.98
0
2.1
—
1.54
0.87
evaluated and minimized in each case across the respective data. The sum of normalized errors (SNE)
was used to select the optimum isotherm parameters among the set of isotherm parameters provided
by the minimization of each error function. This normalization procedure allows a direct combination
of these scaled errors and identifies the optimum parameter set by its minimum SNE values [36].
2. Materials and methods
2.1. Adsorbent
The adsorbent was prepared in one step by surfactant-free emulsion polymerization (polystyrene) or
copolymerization (poly[styrene-co-hydroxyethyl methacrylate], PS-HEMA; poly[styrene-co-acrylic
acid], PS-AA; poly[styrene-co-methacrylic acid], PS-MAA; and poly[styrene-co-itaconic acid],
PS-IA) following procedures similar to those developed by Carbajo et al. [37]. The polymers’ characteristics are reported in Table 1. The structure of each polymer along with its commercial name is
given in Fig. 1.
2.2.
Adsorbate
Bovine serum albumin (Sigma Chemical Comp., crystallized and lyophilized BSA, 96%) was used
as received and its characteristics are given in Table 2. BSA was chosen because it is a reference
protein commonly used in protein adsorption studies and for its low cost.
2.3.
Adsorption experiments
An accurately weighed quantity of albumin serum bovine was dissolved in an acetic buffer solution,
a mixture of sodium acetate and acetic acid, to prepare a stock solution (0.5 mg/mL). The experiments that we conducted used a quartz crystal microbalance QCM200 (Stanford Research Systems,
SRS) at pH 4.7 and room temperature. In each experiment, films were prepared by spin-coating polymer (1 mg/mL in chloroform) onto a gold quartz crystal surface. The reactor cell was charged with
35 mL of a buffer solution followed by the immersion of the QCM cell. After stabilization of the
QCM frequency change, 1 mL of the prepared stock solution of BSA was injected and stirred with
a magnetic stirrer; one hour was found to be enough to reach adsorption equilibrium. The detection
signals and calculated adsorbed mass in the conventional QCM are solely based on the assumption
that the changes of the fundamental piezoelectric frequency is proportional to the change of adsorbed
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Fig. 1. Chemical structures of the polymer lattices used.
Table 2. BSA characteristics.
Supplier
Aldrich
66.43 kDa (66430 g·mol−1 )
583
Between 4.7 and 4.9
1.406 (pH 2); 1.360 (pH 7)
14 nm × 14 nm × 4 nm
Molecular Weight
Amino acid number
Isoelectric point (T = 25◦ C)
Density, ρBSA , (g/cm3 )
Dimension (in stable position)
BSA structure
mass on the surface of the quartz crystal calculated from the classical Sauerbrey equation [38]:
f = −Cf ·
m
,
A
(1)
where A (cm2 ) is the sensible surface area of quartz, m (µg/cm2 ) is the additional mass deposited
on the quartz, and Cf (=56.6 Hz · cm2 /µg) is the calibration constant when the resonance frequency
of the crystal is equal to 5 MHz.
3. Theory
3.1. Theoretical background of studies
Adsorption properties and equilibrium data, commonly known as adsorption isotherms, describe how
protein molecules interact with adsorbent materials, and so are critical in optimizing the use of adsorbents. Theoretical modeling of adsorption isotherms in gas or liquid phase is a powerful technique
used for surface characterization. Thus, the correlation of equilibrium data by either theoretical or
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empirical equations is essential to the practical design and operation of adsorption systems. Moreover, in contrast to the empirical methods, the use of statistical physics development gives a physical
meaning to the model parameters and allows the establishment of significant analytical expressions.
To deal with this phenomenon, we make the following assumptions:
(a) The adsorption phenomenon is a process of particle exchange from the free state to the
adsorbed one and the use of a grand canonical formalism function is mandatory. Thus the equilibrium between the adsorbed and not adsorbed phases, which is reached for each experimental
measurement of the adsorbed amount, can be summarized by the following equation:
n M + S Mn S
(2)
where n is a stoichiometric coefficient representing the fraction or the number of molecule(s)
M adsorbed per site S .
(b) As a first approximation, we neglect the specific interactions between the adsorbate molecules
(in free state) that are treated as an ideal gas [27,28,33,39] since the used concentration of
adsorbate is very weak.
The internal degrees of freedom of the studied molecules are neglected and we consider only the
translation freedom degrees. Indeed, electronic degrees of freedom cannot be thermally excited and
the rotational freedom degrees are frozen in the solution. The vibrational freedom degrees can be
neglected compared to those of translation.
Thus, to treat the adsorption phenomenon by using the grand canonical ensemble, the grand
canonical partition function for a single receptor site, z gc , is written in the following form [28,33]:
e−β(−εi −μ)Ni ,
(3)
z gc =
Ni
where (−εi ) (kJ/mol) is the adsorption energy of the receptor site i, μ (kJ/mol) is the chemical
potential, Ni is the occupation number, and β is defined as (1/kB T ), where kB is Boltzmann’s
constant.
Considering that our system is composed of N adsorbate molecules and Nm identical receptor sites,
the grand canonical partition function describing the microscopic states of the system is written as:
Z gc = (z gc ) Nm .
(4)
Thus, the average occupation number No of sites is [33]:
No =
1 ∂ln(Z gc )
.
β ∂μ
(5)
This number can be expressed according to the grand potential J [33]:
No = −
∂G
.
∂μ
(6)
This potential can be written as follows:
G = −kB T · ln(Z gc ).
(7)
Then, according to the Eq. (2), the adsorbed amount is equal to:
Q a = n No
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On the other hand, to express the adsorbed amount as a function of the concentration of the
adsorbate, we need to use the relationship between the fugacity and the concentration which is
written as:
N
Ce
=
,
(9)
eβμp =
Z gtr
z gtr
where N is the number of adsorbate molecules, Ce (µg/ml) is the concentration of adsorbed
molecules at equilibrium, μp (kJ/mol) is the chemical potential of a molecule assimilated for an ideal
gas and z gtr is the partition function of translation per unit volume that can be written as follows:
z gtr = Z gtr /V
2π mkB T
h2
3/2
,
(10)
where m (mg) is the mass of an adsorbed molecule, V (mL) is the volume occupied by a
molecule, h (6.26 × 10−34 m2 · kg/s) is Planck’s constant, and kB (1.380 × 10−23 m2 · kg/s · K) is
the Boltzmann constant [27,28,33].
This partition function of translation can be expressed according to the solubility concentration as
follows [28,33]:
Z gtr = Cs e
E d
RT
,
(11)
where E d (kJ/mol) is the dissolution energy, Cs (µg/mL) is the concentration of solubility and R
(8.314 J/K · mol) is the ideal gas constant.
By using the thermodynamic equilibrium, the mass action law is written according to equation (2):
μm =
μ
n
and εm =
ε
,
n
(12)
where μ and ε (kJ/mol) are the chemical potential and the adsorption site energy, respectively. The
index m is related to the adsorbed molecule.
Finally, to get any model expression, it is sufficient to write the adequate grand canonical partition function and follow the last steps [Eqs. (5–8)]. There are many theories relating to adsorption
equilibrium, and among these are monolayer, double layer with two successive energy levels, finite
multilayer, and modified BET models. The equations of these models and their corresponding grand
canonical partition functions are summarized in Table 3 with C1 (µg/mL) the half-saturation concentration of a monolayer, C2 (µg/mL) the half-saturation concentration of the global isotherm, and
Nl the average number of adsorbed layers, where:
C1 = Cs e−E1 /RT ,
a
C2 = Cs e−E2 /RT ,
a
(13)
with E a1 the adsorption energy at the first layer and E a2 related to the adsorption energy at the
next layers.
3.2.
Error functions
In the adsorption isotherm modeling studies, the optimization procedure requires error functions
of non-linear regression basis to find the most suitable equilibrium adsorption isotherm models to
represent the experimental data. Five different error functions were examined and in each case the
isotherm parameters were determined by minimizing the respective error function across the liquid
phase concentration range using the solver add-in with Microsoft’s spreadsheet, Excel (Microsoft
Corporation, 2007). The error functions studied are detailed in the following sections.
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Table 3. Analytical expressions of the four isotherm models used in this study and their corresponding grand canonical partition functions.
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Isotherms
Equation
Grand canonical partition function
Monolayer
(14)
z gc = 1 + eβ(ε+μ)
Double layer
(15)
Finite multilayer
(16)
Modified BET
(17)
z gc = 1 + eβ(ε1 +μ) + eβ(ε1 +ε2 +2μ)
N1 β(ε2 +μ) β(ε1 +μ) 1 − e
z gc = 1 + e
1 − eβ(ε2 +μ)
∞
eβ(εNi +Ni μ)
z gc =
Ni =0
Adsorbed amount equation
Qa =
n Nm
n
1 + CC1e
[33]
(Ce /C1 )n + 2(Ce /C2 )2n
1 + (Ce /C1 )n + (Ce /C2 )2n
(Ce /C1 )n [1 − (Nl + 1)(Ce /C2 )(n Nl ) + Nl (Ce /C2 )n(Nl +1) ]
Q a n Nm =
[1 − (Ce /C2 )n ][1 − (Ce /C2 )n + (Ce /C1 )n − (Ce /C1 )n (Ce /C2 )n Nl ]
Q a = n Nm
Qa =
[(C1 /Ce
)n
Reference
n Nm
− (C1 /C2 )n + 1][1 − (Ce /C2 )n ]
[34]
[35]
[28]
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3.2.1. The sum of the squares of the errors (ERRSQ)
The sum of the square errors is the most common error function in use, though it has one major
drawback. Isotherm parameters derived using this error function will provide a better fit as the magnitudes of the errors, and thus the squares of the errors, increasebiasing the fit towards the data
obtained at the high end of the concentration range [40]. Its expression is given as follows:
j
(qe,calc − qe,meas )i2 ,
(18)
i=1
where qe,calc is the theoretical adsorbed quantity, which has been calculated with the adequate model,
qe,meas is the experimentally determined adsorbed quantity obtained from Eq. (1) and j is the number
of data points.
3.2.2. The sum of the absolute errors (EABS)
This approach is similar to the sum of the error squares. Isotherm parameters determined using this
error function provide a better fit as the magnitude of the error increases, biasing the fit towards the
high concentration data [41] by using the expression:
j
|qe,calc − qe,meas |i .
(19)
i=1
3.2.3. The average relative error (ARE)
This error function attempts to minimize the fractional error distribution across the entire concentration range [42]:
j 100 qe,calc − qe,meas (20)
j
q
i=1
e,meas
i
3.2.4. The hybrid fractional error function (HYBRID)
This error function was developed by Porter et al. [36]. It improves the fit of the ERRSQ method
at low concentrations by dividing the measured value. It also includes the number of degrees of
freedom of the system (the number of data points, j, minus the number of parameters, p, of the
isotherm equation) as a divisor:
j
100 (qe,meas − qe,calc )i2
.
(21)
j−p
qe,meas
i=1
3.2.5. Marquardt’s percent standard deviation (MPSD)
This error function was used previously by a number of researchers in the field [43,44]. It is similar
in some respects to a geometric mean error distribution modified according to the number of degrees
of freedom of the system:
⎞
⎛
2
j 1 −
q
q
e,meas
e,calc
⎠.
100 ⎝
(22)
j−p
qe,meas
i
i=1
3.2.6. Sum of normalized errors (SNE)
Each of the above error functions produces a different set of isotherm parameters and it is difficult to
classify which set is an optimum parameter set. Therefore, a normalization procedure is necessary
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to provide a better comparison between the parameter sets for the single isotherm model and, subsequently, the most accurate prediction of the isotherm constants. The error values obtained from each
error function for each set of isotherm constants were divided by the maximum errors for that error
function to determine the normalized errors for each parameter set [31].
4.
Results and discussion
Figure 2 shows the plots of the adsorbed quantity, Q a , against the concentration of adsorbed
molecules at equilibrium, Ce , for albumin serum bovine onto the functionalized polystyrene lattices. It is clear from this figure that the PS-IA had a considerable affinity for the BSA protein. The
maximum Q a values of BSA for PS, PS-IA, PS-AA, PS-MAA, and PS-HEMA were 2.280, 25.666,
7.645, 0.715 and 14.332 µg/cm2 , respectively. It can be noticed that the large adsorbed quantity
for PS-IA and the rather small one for PS-MAA could be attributed to the different kinds of interactions between the polymer and BSA. Then, we can explain that the marked low adsorption onto
the PS-MAA surface could be attributed to steric repulsion and to the decrease in the hydrophobic
interactions between the latex and the protein. Hydrophobic interactions were dominant in the low
number density of surface carboxyl groups, Nc , region, while hydrogen bonding was dominant in
the high Nc region [19].
It can be concluded from our study that the hydrogen bonding was stronger than the hydrophobic
one, since, as shown in Figure 1 and Table 1, the PS-IA lattice has two carboxylic groups, COOH,
(Nc = 2.1 nm−2 ) compared to other lattices. Additionally, the difference between the adsorbed quantities onto different polymer lattices could be attributed to surface roughness. This result is probably
attributed to the different microscopic structures between PS-IA and PS-HEMA, as well as the
linkage modes of adsorbed molecules on PS-IA and PS-HEMA. Hence, in addition to the surface
roughness and the number density of surface carboxyl groups, Nc , the specific surface area, SBET ,
and the surface charge densities, σ , of the latexes also affect the adsorption of bovine serum albumin
(see Table 1).
The experimental data of the BSA proteins adsorbed on the functionalized polystyrene lattices
are substituted into four equilibrium isotherm models, namely monolayer, double layer with two
successive energy levels, finite multilayer, and a modified BET model, respectively, and the best-fit
model for the sorption system was determined.
4.1.
Adsorption isotherm modeling
The conventional approach to the determination of the isotherm parameters is performed by nonlinear regression. By plotting the experimental isotherms which are generally represented in the form
of adsorbed quantity versus concentration (Fig. 2), it is possible to depict the equilibrium adsorption
isotherm. Fits with the monolayer, the double layer with two successive energy levels, the finite
multilayer, and the modified BET isotherm models are illustrated in Fig. 3 for each support. The
associated parameters of these models are given in Table 4. Furthermore, the typical assessment of the
isotherm fit quality to the experimental data is based on the magnitude of the correlation coefficient
for the regression, i.e., the isotherm giving an R 2 value closed to unity is deemed to provide the
best fit. As seen from Table 4, overall, the highest correlation coefficient (R 2 ) was obtained for the
experimental data using the finite multilayer isotherm, closely followed by the modified BET one.
From Fig. 3 it is also clear that the finite multilayer model generally yielded a much better fit than
the other models.
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Fig. 2. Plot of Q a against Ce for BSA adsorption onto functionalized polystyrene lattices.
On the other hand, the selection of the best model is not only based on the correlation coefficient,
but it is necessary to take into account the physical reality and the errors in the non-linear form
of the isotherm curves. For that reason, in the following sections, the best-fitting model will first be
determined using the most commonly used error functions.
R2
4.2.
Error estimation using commonly used functions
In this section, the best-fitting of the four isotherm models studied is determined based on the use
of five well-known functions to calculate the error deviation between experimental and predicted
equilibrium adsorption data, after non-linear analysis. The five error functions used in this study
to determine the optimum isotherm parameters, the sum of error squares (ERRSQ), EABS, ARE,
HYBRID, and MPSD, seem to be the most appropriate. Such a trend was previously proven by other
researchers [31,40,45]. The process of minimizing the respective error functions across the experimental concentration ranges yielded the isotherm constants (Tables 5–8). The isotherm parameters
obtained, together with the values of the error measures for each isotherm, are all fully presented
in Table 5 and the final sums of the normalized error values (SNE) are given in Tables 6–8. From
these tables we notice that a lower absolute error value is obtained for both finite multilayer and
modified BET isotherms. The application of the different error functions will provide different sets
of isotherm constants, sometimes close to one another and thus difficult to compare. To identify the
optimum or best set of isotherm constants, the results for each set were normalized and combined
as a sum of the normalized error values (SNE) [31,36,41] presented in Tables 5–8. The normalized
error values allowed the comparison between error functions, and the identification of the set of
isotherm constants providing the fit closest to the experimental data. The calculation procedure was
as follows:
(a) For a given isotherm, one error function was used to determine the isotherm parameter set by
minimizing that error function.
(b) For that obtained parameter set, the values for all the other error functions were determined.
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(a)
(b)
(c)
(d)
(e)
Fig. 3. Adsorption isotherm modeling of BSA onto functionalized polystyrene lattices using four adsorption
isotherm models: (a) PS, (b) PS-IA, (c) PS-HEMA, (d) PS-AA, and (e) PS-MAA.
(c) For that isotherm, all other parameter sets and all their associated error functions were
calculated.
(d) SNE values were obtained by dividing the error values calculated for each error function for
each set of isotherm constants by the maximum errors for that error function.
(e) For each parameter set, sum all these normalized errors.
The experimental observations suggested that the finite multilayer isotherm should provide a reasonable description and analysis of the experimental data with lower normalized error values (SNE).
These lowest SNE values are obtained using the ERRSQ function for PS, PS-IA, and PS-AA, and
12/25
PTEP 2015, 033J01
M. Beragoui et al.
Table 4. Monolayer, double layer, finite multilayer, and modified BET isotherm parameters related to
the biosorption of BSA onto functionalized polystyrene lattices.
PS
PS-IA
PS-HEMA
PS-AA
PS-MAA
Monolayer
n
Nm (µg/cm2 )
C1 (µg/mL)
R2
1.251
1.884
7.359
0.999188
1.036
25.030
1.330
0.999933
1.055
15.461
11.760
0.999719
0.221
56.478
10.455
0.999705
0.974
1.497
89.656
0.997954
Double layer
n
Nm (µg/cm2 )
C1 (µg/mL)
C2 (µg/mL)
R2
1.974
1.087
7.911
50.208
0.999536
1.585
15.957
2.401
47.640
0.999966
1.056
15.444
11.756
2395.93
0.999719
0.575
11.705
0.404
21.024
0.999702
1.024
1.000
56.586
199.250
0.998079
Finite multilayer
n
Nm (µg/cm2 )
C1 (µg/mL)
C2 (µg/mL)
N1
R2
1.047
3.836
9.006
49.205
0.464
0.999041
2.828
8.862
4.860
262.199
21.276
0.999997
2.134
3.427
0.306
25.773
2.045
0.99994
1.333
5.313
2.599
500.027
4.840
0.9994
1.011
0.476
27.214
100.006
2.500
0.997783
2.037
1.052
8.019
319.562
0.999563
2.593
9.672
4.440
299.966
0.999996
1.729
7.731
9.535
300.491
0.998511
0.710
10.053
0.907
1999.96
0.999644
1.691
0.324
20.311
176.749
0.999689
Modified BET
n
Nm (µg/cm2 )
C1 (µg/mL)
C2 (µg/mL)
R2
Values in bold represent the maximum of the correlation coefficient for the regression (R 2 ).
the MPSD function for PS-HEMA and PS-MAA. It was found that the MPSD function provided the
best results for several works such as the adsorption of textile dye onto Posidonia oceanica [46], the
adsorption of Cr(VI) from aqueous solution by activated carbon [47], the sorption of basic red 9 by
activated carbon [48] and the sorption of methylene blue onto activated carbon [49]; likewise for the
ERRSQ function, such as the sorption of lead from effluents using chitosan [45]. Figure 4 shows a
comparison between the different models for each support with the best error function which represented the lowest SNE values. However, from this figure it is very clear that the finite multilayer
isotherm yielded the best fit using the error function analysis, indicating a rapid improvement of the
isotherm fit compared to Fig. 3 (before error analysis).
The most obvious conclusion from these results is that the finite multilayer isotherm has the lowest
normalized error values (SNE) and therefore fits the data better than the other models. In the next
section, all interpretations were conducted according to the results obtained by the finite multilayer
isotherm.
4.3.
Interpretations based on selected model
The non-linear fit of the experimental data with the finite multilayer model described by Eq. (16)
allowed us to estimate the physicochemical parameters related to the adsorption process. The evolution of such parameters as a function of experimental conditions will be investigated in great
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M. Beragoui et al.
Table 5. Finite multilayer isotherm parameters with error analysis.
Marquardt’s
percentage
standard
deviation (MPSD)
Sum of the
squares of the
errors (ERRSQ)
Sum of the
absolute
errors (EABS)
Average
relative
error (ARE)
Hybrid
fractional
error (HYBRID)
PS
n
Nm (µg/cm2 )
C1 (µg/mL)
C2 (µg/mL)
Nl
ERRSQ
EABS
ARE
HYBRID
MPSD
SNE
2.379
0.824
7.623
48.477
1.210
0.002777
0.091026
0.598018
0.063693
1.709891
3.723
1.125
3.697
9.167
49.255
0.418
0.00435
0.109504
0.720561
0.099283
2.129855
4.974
1.120
3.723
9.136
49.403
0.416
0.004316
0.110704
0.731444
0.098594
2.123469
4.982
1.519
1.368
6.819
45.279
1.153
0.003057
0.098825
0.652665
0.069891
1.788569
4.031
1.410
1.413
6.666
22.500
1.187
0.0032
0.099281
0.65423
0.073046
1.827085
4.12
PS-IA
n
Nm (µg/cm2 )
C1 (µg/mL)
C2 (µg/mL)
Nl
ERRSQ
EABS
ARE
HYBRID
MPSD
SNE
2.828
8.862
4.860
262.199
21.276
0.001705
0.084352
0.055774
0.006769
0.16393
4.287
2.831
8.844
4.835
257.446
21.276
0.002345
0.064222
0.042465
0.009322
0.192482
4.509
2.831
8.849
4.846
262.199
21.276
0.002227
0.078819
0.052067
0.008823
0.186984
4.719
2.828
8.862
4.860
262.199
21.276
0.001705
0.084352
0.055774
0.006769
0.16393
4.287
2.830
8.857
4.864
262.199
21.276
0.001704
0.085092
0.056267
0.006766
0.163893
4.304
PS-HEMA
n
Nm (µg/cm2 )
C1 (µg/mL)
C2 (µg/mL)
Nl
ERRSQ
EABS
ARE
HYBRID
MPSD
SNE
2.025
3.483
0.383
24.545
2.117
0.002969
0.103793
0.108803
0.010666
0.276994
1.863
2.133
3.415
0.306
25.770
2.055
0.011745
0.216688
0.240494
0.046022
0.603586
4.963
2.131
3.418
0.306
25.772
2.056
0.011887
0.217552
0.241438
0.046553
0.606894
5
2.032
3.476
0.399
24.572
2.114
0.00297
0.101845
0.106409
0.01066
0.276769
1.844
2.041
3.469
0.463
24.602
2.109
0.002975
0.100065
0.104301
0.010668
0.276744
1.827
PS-AA
n
Nm (µg/cm2 )
C1 (µg/mL)
C2 (µg/mL)
Nl
ERRSQ
EABS
ARE
HYBRID
MPSD
SNE
0.328
16.388
0.015
500.606
2.416
0.012872
0.266918
0.52394
0.087872
1.096002
2.928
1.346
5.257
2.544
500.027
4.840
0.029474
0.357358
0.699959
0.20184
1.663804
4.855
1.330
5.312
2.599
500.027
4.840
0.029992
0.375936
0.742204
0.204892
1.674726
5
0.382
14.644
0.026
502.420
2.291
0.013024
0.265446
0.519488
0.08861
1.09863
2.929
0.455
13.044
0.077
493.628
2.092
0.013143
0.265175
0.519367
0.089485
1.104454
2.94
Continued
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Table 5. Continued
PS-MAA
n
Nm (µg/cm2 )
C1 (µg/mL)
C2 (µg/mL)
Nl
ERRSQ
EABS
ARE
HYBRID
MPSD
SNE
Sum of the
squares of the
errors (ERRSQ)
Sum of the
absolute
errors (EABS)
Average
relative
error (ARE)
Hybrid
fractional
error (HYBRID)
2.081
0.221
16.626
107.239
2.664
0.000088
0.018963
0.485871
0.007247
1.106208
0.8
1.055
0.472
27.209
100.007
2.319
0.001173
0.063851
2.187244
0.122943
5.619646
4.529
1.071
0.449
27.230
100.005
2.450
0.001365
0.065986
2.014216
0.155567
6.165288
4.921
2.177
0.205
16.135
98.637
2.556
0.000089
0.01776
0.426562
0.007073
1.06802
0.748
Marquardt’s
percentage
standard
deviation (MPSD)
2.229
0.198
15,895
94.515
2.507
0.000091
0.017332
0.403552
0.007132
1.063111
0.732
Values in bold represent minimum error values and minimum sum of normalized errors (SNE).
Table 6. Monolayer isotherm parameters with error analysis.
Sum of the
squares of the
errors (ERRSQ)
Sum of the
absolute
errors (EABS)
Average
relative
error (ARE)
Hybrid
fractional
error (HYBRID)
Marquardt’s
percentage
standard
deviation (MPSD)
PS
n
Nm (µg/cm2 )
C1 (µg/mL)
SNE
1.256
1.875
7.375
4.884
1.314
1.784
7.466
4.848
1.303
1.801
7.517
4.751
1.270
1.851
7.403
4.863
1.283
1.830
7.426
4.852
PS-IA
n
Nm (µg/cm2 )
C1 (µg/mL)
SNE
1.036
25.030
1.330
4.486
1.034
25.041
1.290
4.677
1.030
25.139
1.281
4.677
0.907
28.761
1.020
4.47
0.913
28.556
1.035
4.466
PS-HEMA
n
Nm (µg/cm2 )
C1 (µg/mL)
SNE
1.055
15.461
11.760
4.327
1.095
14.711
11.434
4.608
1.074
15.097
11.553
4.455
1.038
15.796
11.825
4.298
1.022
16.121
11.902
4.312
PS-AA
n
Nm (µg/cm2 )
C1 (µg/mL)
SNE
0.163
106.531
355.940
4.588
0.216
57.263
9.305
4.8
0.216
57.517
9.471
4.797
0.156
118.973
798.489
4.57
0.150
131.270
1716.855
4.554
PS-MAA
n
Nm (µg/cm2 )
C1 (µg/mL)
SNE
0.974
1.497
89.656
4.372
1.007
1.448
89.652
4.686
1.042
1.277
75.960
4.398
1.075
1.143
64.578
4.076
1.152
0.956
52.869
4.308
Values in bold represent minimum error values and minimum sum of normalized errors (SNE).
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Table 7. Double layer isotherm parameters (with two successive energy levels) with error analysis.
Sum of the
squares of the
errors (ERRSQ)
Sum of the
absolute
errors (EABS)
Average
relative
error (ARE)
Hybrid
fractional
error (HYBRID)
Marquardt’s
percentage
standard
deviation (MPSD)
PS
n
Nm (µg/cm2 )
C1 (µg/mL)
C2 (µg/mL)
SNE
2.068
1.032
8.028
48.076
4.568
1.887
1.140
7.746
50.209
4.804
1.945
1.104
7.843
50.201
4.474
2.045
1.045
7.991
48.440
4.555
2.024
1.057
7.957
48.823
4.549
PS-IA
n
Nm (µg/cm2 )
C1 (µg/mL)
C2 (µg/mL)
SNE
2.874
8.718
4.933
35.254
1.202
1.587
15.935
2.403
47.640
4.999
1.586
15.944
2.401
47.641
4.999
2.870
8.730
4.926
35.264
1.203
2.867
8.739
4.921
35.271
1.203
PS-HEMA
n
Nm (µg/cm2 )
C1 (µg/mL)
C2 (µg/mL)
SNE
1.055
15.452
11.755
2395.933
4.385
1.099
14.619
11.383
2395.933
4.608
1.055
15.457
11.650
2395.933
4.338
1.038
15.784
11.819
2395.933
4.356
1.022
16.109
11.895
2395.933
4.371
PS-AA
n
Nm (µg/cm2 )
C1 (µg/mL)
C2 (µg/mL)
SNE
0.344
19.229
0.204
18.070
4.671
0.576
11.705
0.376
21.025
4.757
0.576
11.696
0.376
21.002
4.758
0.500
12.569
0.175
10.621
4.581
0.420
14.889
0.149
10.338
4.587
PS-MAA
n
Nm (µg/cm2 )
C1 (µg/mL)
C2 (µg/mL)
SNE
1.024
1.000
56.586
199.250
4.607
1.021
0.996
56.586
199.250
4.659
1.139
0.841
46.634
199.029
4.427
1.116
0.883
49.095
200.018
4.329
1.329
0.485
27.812
71.689
4.315
Values in bold represent minimum error values and minimum sum of normalized errors (SNE).
detail to interpret and understand the physical process at the molecular level. It follows that two
interpretations, steric and energetic, are derived.
4.3.1. Steric interpretations
The benefit of the selected model is to identify the number of adsorbed layers, Nl , and therefore to
promote a better understanding of the adsorption process. The evolution of Nl is illustrated in Fig. 5.
It can be seen that Nl increases with increasing number density of surface carboxyl groups, Nc , and
the maximum of adsorbed layers is reached for the poly(styrene-co-itaconic acid)—Fig. 5(a). Indeed,
for poly(styrene-co-hydroxyethyl methacrylate) and polystyrene, the protein molecules are adsorbed
on the surface with another functional groups since its Nc value was zero although its surface charge
density was non-zero (Table 1). We also notice that the number of adsorbed layers is affected by the
Nc number and this is due to the hydrogen bonding. An increase in Nc obviously strengthens the
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Table 8. Modified BET isotherm parameters with error analysis.
Sum of the
squares of the
errors (ERRSQ)
Sum of the
absolute
errors (EABS)
Average
relative
error (ARE)
Hybrid
fractional
error (HYBRID)
Marquardt’s
percentage
standard
deviation (MPSD)
PS
n
Nm (µg/cm2 )
C1 (µg/mL)
C2 (µg/mL)
SNE
2.037
1.052
8.019
319.562
4.831
2.007
1.069
7.955
319.560
4.843
1.989
1.076
7.893
319.564
4.811
2.035
1.053
8.009
319.562
4.827
1.999
1.073
7.953
331.139
4.812
PS-IA
n
Nm (µg/cm2 )
C1 (µg/mL)
C2 (µg/mL)
SNE
2.829
8.859
4.863
262.277
4.153
2.593
9.672
4.441
299.966
4.999
2.593
9.672
4.440
299.966
5
2.826
8.868
4.858
262.660
4.154
2.823
8.877
4.852
263.065
4.156
PS-HEMA
n
Nm (µg/cm2 )
C1 (µg/mL)
C2 (µg/mL)
SNE
1.055
15.456
11.759
17511743.8
1.766
1.754
7.656
9.484
300.499
4.942
1.743
7.708
9.470
300.492
5
1.038
15.786
11.821
7995826.78
1.752
1.022
16.128
11.904
18845076.5
1.752
PS-AA
n
Nm (µg/cm2 )
C1 (µg/mL)
C2 (µg/mL)
SNE
0.710
10.053
0.907
1999.996
4.145
0.709
10.021
0.822
1999.996
4.061
0.651
10.618
0.515
1999.995
4.712
0.702
10.142
0.859
1999.997
4.129
0.694
10.230
0.813
1999.996
4.118
PS-MAA
n
Nm (µg/cm2 )
C1 (µg/mL)
C2 (µg/mL)
SNE
1.691
0.324
20.311
176.749
4.269
1.668
0.326
20.315
176.749
4.82
1.684
0.324
20.409
176.739
4.667
1.731
0.314
20.042
172.643
4.1
1.764
0.306
19.769
168.895
4.086
Values in bold represent minimum error values and minimum sum of normalized errors (SNE).
hydrogen bonding between the protein molecules and the lattices [19], since the carboxyl groups
are hardly dissociated in pH 4.7 [19]. As a consequence, the repulsion between adsorbate molecules
decreases, which increases the number of adsorbed layers.
On the other hand, the approximate ellipsoid dimensions of BSA are 14 nm × 4 nm × 4 nm [50].
The estimated thickness, df BSA , of the deposed layer calculated according to the Sauerbrey equation
showed that the protein formed a compact layer of about 4.55 nm and 4.98 nm thickness after surface
rinsing for PS and PS-MAA, respectively, which is probably adsorbed onto these lattices in a side-on
conformation. The thickness data suggest that a multilayer adsorption is necessary for PS-IA, PSHEMA, and PS-AA (76.979, 52.852, and 32.370 nm, respectively), because these thicknesses exceed
14 nm. Lassen and Malmsten [51] proposed protein aggregation and multilayer formation to explain
the high thickness of HAS layers adsorbed on hydrophobic hexamethyldisiloxane plasma polymer
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(a)
(b)
(c)
(d)
(e)
Fig. 4. Adsorption isotherm modeling of BSA onto functionalized polystyrene lattices using four adsorption
isotherm models: (a) PS, (b) PS-IA, (c) PS-HEMA, (d) PS-AA, (e) PS-MAA.
modified silica. They based this proposal on the fact that the thickness increased linearly with the
increasing adsorbed amount above 0.6 mg/m2 [51]. However, a multilayer formation is responsible
for the data shown in Fig. 5(b) because an increase in Q a accompanies the increase in df BSA for
PS-IA, PS-HEMA, and PS-AA, but the inverse phenomenon was observed for PS and PS-MAA
which explains the absence of a multilayer formation. Furthermore, the value of Nl might be an
interesting tool to improve this last result. Thus, we can estimate the configuration of the adsorbed
BSA molecules basing on the values of Nl and df BSA . Indeed, in the case of BSA adsorption onto PSIA, the value of Nl is equal to 21.276, i.e., the mean value is situated between 21 and 22. This means
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(a)
(b)
Fig. 5. Behavior of the Nl parameter with the Nc number (a), the Q a quantity and the df thickness value (b)
versus the different substrates.
that the BSA molecule will be anchored either as 21 or 22 layers with the respective percentage x
and (1 − x) determined by the value of Nl as demonstrated by Khalfaoui et al. [27] concerning the
number of adsorbed molecule(s) per site, n. In such a case, the relationship between these percentages
is x · 22 + (1 − x) · 21 = 21.276. This gives 72% of the adsorbed molecules as being anchored with
21 layers and 28% of the adsorbed molecules as anchored with 22 layers. A similar result can be
deduced from the thickness value. For example, in the case of PS-MAA, the thickness was about
4.98 nm. This indicates that there is a side-on monolayer and multilayer.
The number or the fraction of adsorbed molecule(s) per site, n, is a steric parameter which can help
us with the topography of adsorption. Indeed, when the value of this parameter is higher than unity
(n > 1), the molecule is multi-molecular adsorbed onto the solid surface with a vertical position
(end-on). In the case where the value of this parameter is lower than unity, the molecule is multianchored onto the solid surface with a horizontal position (side-on). Figure 6 depicts the evolution of
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Fig. 6. Behavior of the n parameter and the σ value versus the different substrates.
Fig. 7. Adsorption of BSA agglomerates onto the PS-IA surface at the first energy level.
this parameter n. In the same figure, the plot of the surface charge densities, σ , is given for pH 4.7.
It can be noticed that the parameter n is always higher than unity. Therefore, the protein is multimolecular adsorbed onto the solid surface and the highest percentage is led to the end-on orientation,
except for PS-AA. For this last adsorbent, the parameter n is lower than unity, which conveys that
the BSA is multi-anchored onto the PS-AA surface, i.e., a side-on orientation has occurred. It is
also concluded that the BSA molecule can be adsorbed with three orientations such as side-on, endon, and overlap conformations. Note that the three orientations were deduced from the two finite
multilayer model parameters n and Nl and also from the measured thickness, df BSA . From Fig. 6,
it can be deduced that the greater the n value, the lower the σ value. Multi-molecular adsorption is
favored by the decrease in the surface charge density, σ . This could be explained by the fact that the
rise of the surface charge density favors adsorbate–adsorbent interaction. Then, it can be concluded
that the decrease in surface charge density catalyzes the aggregation phenomenon (Fig. 7) during the
adsorption process.
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Fig. 8. Behavior of the Nm parameter and the σ value versus the different substrates.
The effectively occupied receptor sites, Nm , is also an important parameter that describes the
adsorption process. Figure 8 depicts the evolution of this steric parameter in relation to the surface
charge density. Interestingly, for a stronger surface charge density, it appears that all substrates have
a higher receptor site density, apart from the PS-MAA and PS-IA substrates. Hence, for PS-IA the
highest number of occupied receptor sites, Nm , despite the lower surface charge density, is due to the
most important number density of surface carboxyl groups, Nc , (2.1 nm−2 ) which allows the presence of hydrogen bonding and shows additional receptor sites. Despite the presence of the carboxyl
groups and the stronger surface charge density for PS-MAA, the latter has a lower value of Nm . This
is probably related to the effect of the methyl groups, CH3 , causing a steric hindrance and then the
decrease of the adsorbed quantity.
It is clear that the results given by studying the evolution of the steric parameters, n, Nl , and Nm ,
are well in agreement.
4.3.2. Energetic interpretations
A well-defined energetic investigation is essential for understanding fundamental adsorption processes. The energetic parameters allow us to get a better interpretation of most of the experimental
results. Referring to the expression of our model, Eq. (16), two parameters C1 and C2 are related to
the adsorption energies of various adsorbed layers—Eq. (13). From these two parameters (C1 and C2 )
we can determine the different adsorption energies and therefore study their evolution as a function
of surface charge density, σ , for each substrate.
Figure 9 shows that the values of the adsorption energies (−E a1 ) and (−E a2 ) are always negative
and so the adsorption process is exothermic. It is also a physisorption process since the adsorption
energies (−E a1 ) and (−E a2 ) do not exceed 40 kJ/mol [35,52]. Besides, it can be noted that the
adsorption energy (−E a1 ) increases with the surface charge density, σ , except for the PS-MAA
substrate which possesses a low energy value despite the high surface charge density. Hence, as
previously mentioned, the methyl groups (CH3 ) cause a steric hindrance and then the decrease in
adsorption energy. Also, PS-IA shows low adsorption energy with a high adsorbed quantity. This
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Fig. 9. Adsorption energies −E a and surface charge densities, σ , versus the different substrates.
Fig. 10. Schematic illustration of different interaction kinds involved during BSA adsorption onto lattices.
is due to the fact that hydrophobic interactions decrease rapidly with increasing number density
of surface carboxyl groups, Nc , [19] which increases the hydrophilicity of the PS-IA surface and
strengthens the hydrogen bonds between the protein molecules and the lattices. Then the presence
of the repulsion interactions between both hydrophobic and hydrophilic BSA and PS-IA surfaces,
respectively, is affected, which decreases the adsorption energy. This could be attributed to the fact
that hydrogen bonding can be an important mechanism in the adsorption process since the highest
adsorbed quantity value is obtained for PS-IA. Figure 10 is a schematic illustration of different kinds
of interactions involved during the BSA adsorption on lattices. Furthermore, it can be seen from
Fig. 9 that the first adsorbed layer has the high adsorption energy, thus the affinity of the receptor
sites located on this layer is more important. This means that the adsorbed molecules on the second
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Fig. 11. Behavior of the adsorbed quantity at saturation, Q a sat , versus the adsorption energies per unit area,
− E a .
energy level are less attracted to the surface and therefore their adsorption energy is lower. As a
conclusion, the adsorbate–adsorbent interactions are stronger than the adsorbate–adsorbate ones.
On the other hand, the total energies per unit area (− E a1 ) and (− E a2 ) are calculated according
to the following equation:
n Nm E ia
a
; i = 1, 2,
(23)
E i =
M
where M (66430 g/mol) is the molecular mass of bovine serum albumin.
The evolution of this energy is illustrated in Fig. 11, showing that the adsorbed quantity at saturation
(Q a sat ), which is easily proved via our theoretical approach (Q a sat = Nl · n·N m ), increases with
the increase of the energy per unit area. We note that the higher the adsorption surface energy is,
the higher the adsorbed quantity is. Then, it is clear that the adsorption capacity is affected by the
magnitude of the surface energy.
5.
Conclusion
The equilibrium adsorption of bovine serum albumin onto functionalized polystyrene lattices has
been reported. PS-IA appeared to be an effective adsorbent for BSA molecules due to its higher
adsorption capacity compared to other functionalized polystyrene lattices. The equilibrium results
have been modeled and evaluated using four different isotherms and five different optimization error
functions. Using the error functions for non-linear optimization showed that the finite multilayer
isotherm yielded the best fit with lower normalized error values (SNE) using the ERRSQ function
for PS, PS-IA, and PS-AA and the MPSD function for PS-HEMA and PS-MAA.
All interpretations were also conducted according to the finite multilayer adsorption with multisite
occupancy based on a statistical physics approach. Based on the fitting model parameters, the following main conclusions can be drawn: (1) three conformations, namely end-on, side-on, and overlap,
with monolayer or multilayer adsorption were found in the BSA adsorption process and this is thanks
to the two model parameters in addition to the thickness determined experimentally; (2) the study of
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parameter Nm confirms the importance of hydrogen bonds in BSA adsorption; (3) the aggregation
phenomenon of the BSA molecules is catalyzed by the decrease of the surface charge densities, σ ,
i.e. when the number of adsorbed molecules per site increases; (4) the adsorption of BSA molecules
is about exothermic and physisorption processes.
All these results were obtained thanks to the finite multilayer model based on the statistical physics
treatment which has the advantage of providing physical meaning to the model parameters, both
steric and energetic. Furthermore, the error analysis approach seems to be a powerful tool providing
a good description of the adsorption process with the best-fit isotherm and the isotherm parameters.
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