Ind. Eng. Chem. Res. 2009, 48, 4109–4118
4109
Modeling of Aqueous Biomolecules Using a New Free-Volume Group Contribution Model G. R. Pazuki,† V. Taghikhani,† and M. Vossoughi*,†,‡ Department of Chemical and Petroleum Engineering and Institute for Nano-science and Nano-technology, Sharif UniVersity of Technology, Tehran, Iran
In this article, a new group contribution model is suggested for obtaining the thermodynamic properties of biomolecules in aqueous solutions. Accordingly, a Freed-FV model has been applied for the combinatorial free-volume term. The activity coefficients, solubilities, densities, and vapor pressures of amino acids and simple peptides in aqueous solutions were correlated, using the proposed group contribution model. Group interaction parameters of the proposed model were obtained by use of experimental data from amino acids available in the literature. The results demonstrate that the group contribution model can accurately correlate activity coefficient, solubility, density, and vapor pressure data for aqueous amino acid and peptide solutions. Furthermore, the osmotic pressures of aqueous solutions containing lysozyme and ammonium sulfate were calculated at ionic strengths of 1 and 3 M and pH values of 4, 6, and 8. The results obtained from the suggested model were compared with those obtained from the virial osmotic model. Also, the model was coupled with the Debye-Hu¨ckel model to correlate partition coefficients of biomolecules in polymer-salt aqueous two-phase systems. The results showed that the new group contribution model can accurately correlate partition coefficients of biomolecules in polymer-salt aqueous two-phase systems. 1. Introduction Biotechnology is currently a popular scientific discipline, with the micro-organic production of proteins and pharmaceutical products attracting significant interest. Biomolecules obtained from fermentors are produced in complex broths containing inorganic salts, organic acids, and cells from areas where the recovery of higher-value compounds can prove to be difficult. It should be taken into account that, in the production of biomolecules, both separation and purification are considered as crucial stages.1,2 For the purpose of producing some common bioproducts, the cost of bioseparations can approach 90% of the total manufacturing cost. However, the design, scale up, and optimization of these techniques at the industrial level would indeed be achievable given that a mathematical description of the process is available.3 The design of bioseparation units can require phase equilibrium of bioproducts in the solvents used in the production process. Thermodynamic properties of aqueous biomolecules are necessary components for the given process design. One useful separation process could be provided by liquid-liquid extraction. Numerical values of the solubilities and densities of biomolecules are necessary for obtaining the number of transfer units in liquid-liquid extraction process. The solubility of a biomolecule in aqueous solution depends on the activity coefficient of the given solute. The activity coefficient of chemical species is a criterion that essentially indicates the deviation of the system from the ideal state and reflects the interactions between the components in a single system. Activity coefficients are therefore related to thermodynamic functions such as chemical potential, Gibbs free energy, and Helmholtz free energy. The phase behavior of biomolecules in aqueous solutions can be correlated using an excess Gibbs free energy and the equation of state. Simple models based on hard-sphere models of amino acids and peptides have been * To whom correspondence should be addressed. E-mail:
[email protected]. † Department of Chemical and Petroleum Engineering. ‡ Institute for Nano-science and Nano-technology.
presented to obtain solubilities and activity coefficients of amino acids in aqueous solutions. It should be noted that hard-sphere models are suitable for predicting thermodynamic properties of small amino acids. Khoshkbarchi and Vera,4,5 Liu et al.,6 Mortazavi-Manesh et al.,7 and Pazuki et al.8 have correlated activity coefficients and solubilities of amino acids and simple peptides in water using first-order perturbation theory. In these models, there are two adjustable parameters such as the hardsphere diameter and dispersion energy obtained by applying nonlinear regression between experimental data and results of the model. Prikhodko et al.9 and Tumakaka et al.10 modeled solid-liquid equilibrium in systems with organic components and solid complexes using the PC-SAFT (perturbed-chain statistical associating fluid theory) equation of state. Recently, Cameretti and Sadowski modeled the vapor pressures, liquid densities, and solubilities of aqueous amino acid and oligopeptide solutions with an equation of state based on PC-SAFT.11 Additionally, Ji and Feng applied the simple SAFT equation of state to describe the solubilities of one or two amino acids in water at high pressures, ranging up to 3500 bar, and the densities of aqueous solutions of amino acids.12,13 Moreover, Nass correlated experimental data on amino acid activity coefficients and solubilities.14 The activity coefficients were obtained from chemical and physical interactions, and the Wilson model was applied for calculating physical activity coefficients. Gupta and Heidemann presented a predictive UNIFAC (universal functional activity coefficient) group contribution model for modeling the solubilities of amino acids and antibiotics in water.15 Pazuki et al. studied activity coefficients and solubilities of amino acids and simple peptides in aqueous solutions using a nonrandomfactor (NRF) local composition model.16 Furthermore, Xu et al.17 and Sadeghi18 studied the phase behavior of aqueous solutions of amino acids and peptides using modified local composition models. Coutinho and Pessoa proposed a modification of the extended UNIQUAC (universal quasichemical) model for description of nonideality of protein solutions.19 Within the context of this model, the Staverman-Guggenhim (SG) combinatorial contribution used in the extended UNIQUAC model
10.1021/ie8009389 CCC: $40.75 2009 American Chemical Society Published on Web 03/16/2009
4110 Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009
was replaced by a Flory-Huggins term. Recently, Pazuki et al. modeled the partitioning of biomolecules in polymer-polymer and polymer-salt aqueous two-phase systems (ATPS) utilizing an extended excess Gibbs energy model.20 In this article, a new predictive group contribution model based on previous works16,21,22 is used with a free-volume model for the combinatorial term to obtain the thermophysical properties of aqueous biomolecule solutions. In accordance with the concept of local compositions, a new local area fraction is proposed based on the difference in enthalpy between groups i and j. Activity coefficients, vapor pressures, solubilities, and densities of amino acids and simple peptides in aqueous solutions were modeled by using a UNIFAC-based equation. Also, the proposed model was employed to obtain osmotic pressures of lysozyme-ammonium sulfate solutions at ionic strengths of 1-3 M and pH values of 4-8. The group contribution model was coupled with a Debye-Hu¨ckel model to further correlate partition coefficients of some biomolecules in polymer-salt aqueous two-phase systems. 2. Thermodynamic Model In this work, the Gibbs energy of an aqueous solution containing a biomolecule, polymer, and salt is considered as a sum of three terms, i.e., long-range, combinatorial, and residual terms. Therefore, the excess Gibbs energy of an electrolyte solution can be written as EX
G
)G
EX,LR
+G
EX,COMB-FV
+G
EX,SR
ln γi ) ln γLR + ln γCOMB-FV + ln γSR i i i
(2)
In this work, the Debye-Hu¨ckel model has been applied to account for the long-range effect. Therefore, the activity coefficient of each molecular species can be obtained as 2AMw (10B)3
[
1 (1 + BI1/2)
(1 + BI1/2) -
]
Also, the mean ionic activity coefficient of an electrolyte can be obtained as A|ZAZC |I1/2 (1 + BI1/2)
(4)
with I)
1 2
∑m Z
2
k k
(5)
k
where A and B are Debye-Hu¨ckel constants and Z and m are the charge and molality, respectively, of each ion k in solution. Activity coefficients calculated from eqs 3 and 4 are in terms of molalities. Therefore, the activity coefficients of the Debye-Hu¨ckel model must be converted to the mole fraction scale using the equation LR ln γLR i,x ) ln γi,m + ln(1 + νmMw /1000)
(6)
In this study, a free-volume model was considered as a combinatorial part of the group contribution model. It should
(7)
where Vi is the molar liquid volume of component i and VvdW is i the van der Waals volume (cm3/mol) calculated by the equation24 VvdW ) 15.17Ri i
(8)
where Ri is the volume parameter of group or molecule i. It should be noted that Ri is a dimensionless number. The molar liquid volume can be determined using the equation24 Vi ) (0.3 + 4.5 × 10-4T)VvdW i
(9)
The combinatorial term in eq 2 is considered by a combinatorial free-volume model named the Freed-FV model23
( )
ln γCOMB-FV ) ln i
( )
φFV φFV i i +1+ f Freed-FV i xi xi
(10)
The Freed-FV term can be obtained as f Freed-FV ) RFV i i
[∑
FV FV βFV ji φj (1 - φj ) -
j
0.5
∑ ∑β
]
FV FV FV jk φj φk
j*i k*i
(11)
with
(
βFV ji ) Rji
2 ln(1 + BI1/2) (3)
LR ln γ( )-
) Vi - VvdW VFV i i
(1)
Thus, the activity coefficient of each component can be obtained from the excess Gibbs energy as
) ln γLR i
be carefully noted that the concept of free volume is important in aqueous polymer solutions mainly because of the large size difference between the polymer and solvent molecules. Within this context, the free volume of a molecule is defined as the available volume of the molecule as it moves about the system while the positions of all other molecules remain fixed. The effect of the free volume, however, is consistently ignored in lattice models. Such an assumption could prove extremely useful in predicting the phase behavior of aqueous biomolecules such as polymer components. Free volume can be defined as22,23
) φFV i
1 1 - FV RFV R j i xiVFV i
∑xV
FV j j
)
(12)
(13)
j
In eq 12, Rij is a nonrandomness factor (Rij ) 0.2), and RFV j is FV to VFV the ratio of VFV j i . It should be pointed out that Vi corresponds to the smallest solvent in the aqueous two-phase system. The group contribution model is more accurate than other models such as the NRTL (nonrandom two-liquid) and UNIQUAC thermodynamic models in correlating and predicting vapor-liquid equilibrium (VLE) and liquid-liquid equilibrium (LLE) phase behavior of aqueous solutions.25 ASOG (analytical solutions of groups) and UNIFAC models are therefore proposed based on group contribution method for calculating the activity coefficient of each component. In this work, short-range interactions were considered by a new expression derived from the previous local composition model proposed by Pazuki et al.13,17 The derivation of the residual term of the proposed group contribution model is presented in Appendix A. The residual term of activity coefficient of the group was obtained using the related part in a previous work21
Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009 4111
(∑ )
ΘjHmj -
ln Γm ) 1 - ln
j
∑ j
ΘjHmj
-
∑Θ H k
[
kj
k
∑Θ ∑Θ j
j
k
k
(
ln
Hkj HmjHjm
)]
(14)
where Θk is the surface of group k, which is defined as XkQk
Θk )
(15)
∑XQ j
anion-middle group (EAP and EPA), cation-middle group (ECP and EPC), biomolecule-middle group (EBPand EPB), biomolecule-anion (EBA and EPB), and biomolecule-cation (EBC and ECB). It should be carefully taken into account that the vapor pressures of aqueous amino acids solutions and the solubility of methionine at various pH values were predicted using the proposed group contribution model with no extra adjustable parameters. To estimate the interaction parameters, the following objective function was minimized
j
N
j
min F )
n
∑
xiνmi
i)1 n
s
∑x ∑ν i
i)1
(16)
ki
k)1
where νmi is the number of group m in the molecules, n is the number of components, and s is the number of various groups in the aqueous solution. Thus, the residual activity coefficient of component i can be calculated with the equation
∑ν
mi(ln
Γm - ln Γm(i))
2 - Φcalc i )
(18)
where Φ ∈ {γAA, xAA, dAA, γ(, π, aW, KB} represents the activity coefficient of the amino acid or simple peptide in water, the solubility of the amino acid in water, the density of the amino acid in water, the mean ionic activity coefficient of the electrolyte solution, the osmotic pressure of the lysozyme in the aqueous electrolyte, the activity of water in the polymer solution, and the partition coefficient of the biomolecule in a polymer-salt aqueous two-phase system. The root-mean-square deviation (rmsd) values can be obtained from the relation
s
ln γSR i )
expt i
i)1
In eq 15, Xm, the group fraction, can be determined as
Xm )
∑ (Φ
(17)
m)1
where Γm is the activity coefficient of group m in solution and Γm(i) is activity coefficient of group m in pure component i. 3. Numerical Aspects Experimental data on activity coefficients, solubilities, osmotic pressures, and partition coefficients of biomolecules in aqueous solution were used to obtain the adjustable parameters of the proposed model. The group contribution model has two interaction energy parameters between the amino acid or peptide and water (EAA-W and EW-AA) in correlating the activity coefficients of biomolecules in water. These parameters can be obtained from eqs 14-17. The proposed model has two adjustable parameters, ∆h/R and ∆s/R, in correlating the solubilities of amino acids in water. These parameters can be obtained from eq 25 below. To effectively correlate the experimental data of densities of amino acid solutions, the proposed model has the two adjustable parameters as ∂Eji/∂P and ∂Eij/∂P. These parameters can be obtained from eqs A-9-A-11. The proposed model considers six binary interaction parameters for the pairs anion-cation (EAC and ECA), anion-water (EAW and EWA), and cation-water (ECW and EWC) in obtaining mean ionic activity coefficient data. To correlate the osmotic pressures of lysozyme aqueous electrolyte solutions, the group contribution model has six binary interaction parameters for the pairs lysozyme-anion (ELA and EAL), lysozyme-cation (ELC and ECL), and lysozyme-water (ELW and EWL). The proposed model has two binary energy interaction parameters between the middle group and water (EPW and EWP). It should be pointed out that the interaction parameters for the end group-water and end group-middle group pairs are set to zero. Also, to correlate the experimental data on partition coefficients of biomolecules in polymer-salt aqueous two-phase systems, the proposed model contains 10 binary interaction parameters for the pairs
rmsd )
N
∑ (Φ
expt i
2 - Φcalc i )
i)1
N
(19)
where N in the above equations is the number of experimental data points. The quasi-Newton method utilizes the quadratic approximation of the objective function, which can be calculated numerically. In the Newton method, finding values for adjustable parameters is carried out in descent direction Sk at each step k, which can be obtained as S k ) -[H(xk)]-1∇F(xk)
(20)
where H(xk) is the Hessian matrix and ∇F(xk) is the vector of the first derivative of the objective function F, which can then be obtained with regard to parameters given by vector x. Hessian matrix can be obtained as H(xk) ) ∇2F(xk)
(21)
New values of the adjustable parameter vector were calculated using the equation xk+1 ) xk + λkS k
(22)
where λk is the step length and was optimized in each step. The Marquardt-Levenberg method was used in the optimization of the adjustable parameters.26 4. Results and Discussion 4.1. Thermodynamic Properties of Amino Acids and Simple Peptides in Aqueous Solution. In this section, activity coefficients and solubility characteristics of amino acids and simple peptides in water are correlated with corresponding experimental data. Experimental data on activity coefficient of amino acids and peptides were obtained from the appropriate literature.27 The surface and volume parameters of the groups are reported in Table 1. Water is assumed as a single group,
4112 Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009
ln γ(m) ) ln γ(x) - ln(1 + 0.001mMw)
Table 1. Volume and Surface Parameters for Each Component (Biomolecules and Other Species) Studied in This Worka biomolecule
r
q
other species
r
q
alanine glycine proline valine threonine serine alanylalanine alanylglycine glycilglycine triglycine glycilalanine methionine hydroxy proline phenylalanine lysine glutamic acid bovine serum albumin lysozyme
3.344 2.670 4.304 4.919 4.791 4.117 5.974 5.484 4.648 6.627 5.322 6.644 5.076 6.346 6.673 5.753 1.222 1.400
2.996 2.460 3.468 4.384 4.424 3.888 5.401 4.572 4.036 5.612 4.572 4.118 4.536 4.808 5.004 4.452 1.200 1.210
water CH2OCH2 CH2OH Na+ K+ HPO42SO42H2PO42NH4+
0.92 1.593 1.674 0.176 0.437 1.329 2.020 1.329 0.176
1.4 1.320 1.740 0.315 0.578 1.210 1.600 1.210 0.315
a
(23)
where Mw and m refer to the molecular weight of water and the molality of amino acid, respectively. The interaction parameters of the proposed model and the root-mean-square deviation (rmsd) values of several models, namely, those of Chen et al.,30 Gupta and Heidemann,15 and Khoshkbarchi and Vera4,5 and the proposed model, are reported in Table 2. As witnessed from Table 2, the group contribution model provides more precise accounts than the other models mentioned. Vapor pressures of aqueous amino acids are predicted using the proposed model. To obtain proper vapor pressures of aqueous amino acid solutions from the values of the activity of water, the following relation can be used Psat ) Psat W aW
Parameters obtained from refs 28 and 29.
and K2HPO4, KH2PO4, Na2SO4, and (NH4)2SO4 are assumed to dissociate completely into ionic groups. Each PEG molecule is divided into the two PEG end groups (sCH2sOH) and the PEG middle groups (sCH2sOsCH2s).28 Amino acids exist as neutral, amphoteric molecules and anions.28 In peptides, one has to distinguish between left-terminal, middle, and rightterminal amino acid groups, as well as between charged and uncharged groups. The left-terminal group carries the functional amino group of peptide, whereas the right-terminal group carries its functional carboxylic group. The surface and volume parameters for PEG functional groups and water were obtained from the literature.28 Also, the surface and volume parameters of amino acids and simple peptides were calculated using the method proposed by Pinho et al.29 Furthermore, the surface and volume parameters of each ionic species and of globular proteins were obtained from the corresponding literature.28 For the correlation of the activity coefficients and solubilities of amino acids and peptides in water, it was assumed that nonzero interaction parameters exist between water and neutral amino acids and peptides, and the interaction parameters between water and other groups were set to zero. The adjustable parameters between neutral groups and water were obtained by minimization of eq 18. The activity coefficient in terms of mole fraction was therefore converted to molality according to the equation
(24)
where Psat W and aW represent the vapor pressure of the solution and the activity of the water, respectively. The experimental vapor pressures of aqueous amino acid solutions can be found in the literature.31 Figure 1 shows the variation of the vapor pressures of aqueous solutions containing glycine and L-serine. As can be inferred from the figure, a reasonable agreement between the results of the proposed model and the experimental data was attained. To obtain the solubilities of amino acids in water from the values of the activity coefficients obtained from the proposed model, the following relation was used15
(
xAγA ) exp
)[
k2 ∆h ∆s [H+] + + 1+ R RT k1 [H ]
]
(25)
with [H+] ) 10-pH k1 ) 10-pK1 k2 ) 10-pK2
(26)
where k1 and k2 are dissociation equilibrium constants and the values of pK1, pK2 and pH are 2.3, 9.7, and 6, respectively.15 In eq 25, ∆h/R and ∆s/R, the enthalpy and entropy of solution, were considered as two adjustable parameters. The values of these parameters can be obtained through by the minimization of eq 18. The experimental data on the solubilities of amino acids in water were obtained from the literature.27 Values of the adjustable parameters and the rmsd’s for the proposed model
Table 2. Values of the Parameters in Eq 14 Obtained Using Activity Coefficient Data for Amino Acids and Simple Peptides, along with the rmsd Values of Various Models 100 × rmsd amino acid
EAA-W/R (K)
EW-AA/R (K)
proposed model
Khoshkbarchi-Vera5
Khoshkbarchi-Vera4
Chen et al.30
Gupta-Heidemann15
alanine glycine valine threonine serine alanylalanine alanylglycine glycilglycine triglycine glycilalanine methionine hydroxy proline
142.586 4.341 374.821 275.597 100.808 456.797 296.578 131.272 339.184 283.405 339.180 326.918
-283.157 -337.903 -238.841 -355.465 -417.530 -344.700 -411.893 -440.760 -488.077 -405.095 -488.016 -346.688
0.07 0.66 0.06 0.25 1.53 0.68 1.01 1.88 0.00 0.56 0.00 0.10
0.51 0.84 0.45 0.25 1.19 0.40 0.71 2.81
0.33 0.86 0.05 0.15 1.16 0.43 0.6 0.6 0.00
0.04 2.07 4.47 0.70 2.80 2.04 2.92 2.44 0.67
8.97 4.2 3.32 13.78 12.15
0.56
1.58
0.92
1.94
7.57
overall
Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009 4113
Figure 1. Plot of vapor pressure versus molality for aqueous solutions of two amino acids according to the new model (T ) 298.15 K). Table 3. Values of the Parameters in Eq 25 Obtained Using Solubility Data for Amino Acids in Water, along with the rmsd Values of Various Modelsa ∆g/RT0 amino acid L-alanine DL-alanine
glycine L-proline L-serine DL-serine L-valine DL-valine
hydroxy proline L-phenylalanine DL-phenylalanine L-lysine L-glutamic
acid acid
DL-glutamic
∆s/R (K)
this work
Khoshkbarchi-Vera4
Fasman27
this work
Khoshkbarchi-Vera4
Sadeghi18
649.024 884.569 1200.588 359.190 1162.473 1592.130 -396.708 435.666 365.899 -2977.010 -3099.666 -1976.796 480.923 -1258.012
-1.230 -0.433 0.907 0.348 -0.958 0.502 -5.973 -2.930 -1.702 -4.931 -5.003 -5.253 -4.249 -2.422
3.407 3.400 3.119 0.855 4.857 4.838 4.642 4.392 2.930 -5.053 -5.392 -1.377 5.862 -1.796
3.532 3.527 3.131 0.595 3.188 4.881 4.555 4.373
3.431
0.050 0.060 0.084 0.019 0.014 0.071 0.008 0.108 0.060 0.001 0.001 0.001 0.021 0.010
0.240 0.265 0.305 0.182 0.222 0.174 0.064 0.088
0.088 0.030 0.455
0.036
0.192
overall a
rmsd
∆h/R (K)
3.165 0.271 3.168 4.833
3.653 0.891 0.891 0.829 0.071 0.106 0.804 1.17 0.104 0.757
T0 ) 298.15 K.
and the Khoshkbarchi-Vera4 and Sadeghi18 models are presented in Table 3. The Gibbs energies of aqueous amino acid systems that were calculated directly from the values of ∆h/R and ∆s/R were compared with the experimental data,27 as well as with those reported by Khoshkbarchi and Vera.4 Also, it should be understood that the results obtained from the proposed model can accurately correlate the experimental solubility data of the systems studied. Equation 25 was applied to predict the solubility of methionine at various pH values. Experimental data on the solubility of methionine were obtained from the literature.32 The solubilities of DL-methionine in aqueous solution at various pH values are reported in Figure 2. Figure 2 shows that the minimum solubility occurred at the isoelectric pH value. The results predicted for DL-methionine provide a good description of its solubility. The model was developed by correlating the densities of aqueous amino acid solutions. The regressed parameters of the model were obtained by minimizing eq 18. It should be pointed that the equations needed to estimate the densities of amino acid solutions are reported in Appendix A.
The experimental data for the densities of aqueous amino acid solutions were obtained from the literature.33 Values of the parameters Eij and Eji that were used to correlate the experimental data for amino acid solutions were directly obtained from the activity coefficients of amino acids in aqueous solution. Values of the adjustable parameters and rmsd’s for various amino acid solutions are reported in Table 4. Also, the rmsd of the model is compared with that of the ideal state (VEX ) 0). The rmsd values show that the proposed model has better accuracy than the ideal state for the correlation of experimental densities of amino acid solutions. Also, the results for the idealstate solution show that the densities of amino acid solutions are highly sensitive to the excess volume of solution. 4.2. Osmotic Pressure of Aqueous Electrolyte Solution. In this section, the osmotic pressures of aqueous solutions containing lysozyme are correlated using corresponding experimental data. The relation between the osmotic pressure and the activity of water in aqueous solution can be written as34 π)-
RT ln aW VW
(27)
4114 Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009
Figure 2. Comparison between the experimental solubility of methionine in water and the results obtained from the new model and for the ideal state as a function of the pH of the aqueous solution. Table 4. Values of the Parameters in Eq A-11 and rmsd Values of the Group Contribution Model and Ideal State Obtained Using Density Data for Amino Acid Solutions rmsd × 105 system
T (K)
N
∂EAA-W/∂P
∂EW-AA/∂P
new model
ideal state
glycine-water glycine-water glycine-water glycine-water DL-alanine-water DL-alanine-water DL-alanine-water DL-alanine-water DL-valine-water DL-valine-water DL-valine-water DL-valine-water L-serine-water L-serine-water L-serine-water L-serine-water
278.15 288.15 298.15 308.15 278.15 288.15 298.15 308.15 278.15 288.15 298.15 308.15 278.15 288.15 298.15 308.15
8 9 10 9 8 9 9 9 8 11 11 11 8 9 9 9
10829.879 11093.224 11093.224 10716.167 218.905 220.069 229.840 232.700 754.711 764.417 747.260 748.442 2287.267 2279.567 2279.615 2297.927
-19250.419 -17400.521 -17400.521 -19042.781 528.031 469.833 470.357 470.496 -655.307 -655.094 -655.269 -655.274 453.532 453.484 453.484 453.167
47.037 51.400 51.297 54.179 31.370 31.522 26.706 27.628 3.194 58.544 20.398 46.653 5.912 6.256 15.196 15.220
570.953 671.267 773.778 634.921 536.226 608.027 604.440 598.719 561.695 751.488 707.286 695.818 929.953 1024.986 998.664 973.924
Table 5. Values of the Parameters in Eq 14 Obtained Using Mean Ionic Activity Coefficient Data for Electrolyte Solutionsa system
N
EA-C/R
EC-A/R
EA-W/R
EW-A/R
EC-W/R
EW-C/R
100 × rmsd
K2HPO4-water KH2PO4-water Na2SO4-water (NH4)2SO4-water
10 14 19 19
83.262 88.960 75.771 552.971
-1119.440 -954.382 -1212.430 -931.644
66.408 15.563 47.778 681.364
470.500 298.389 592.237 749.366
739.988 736.767 771.013 605.533
-819.955 -824.761 -757.753 -1116.630
0.250 0.502 1.058 0.291
a
A ) anion, C ) cation, W ) water.
where VW and aW are the partial volume and activity of water, respectively, in aqueous solution. Experimental data on the osmotic pressures of aqueous electrolyte solutions containing lysozyme and ammonium sulfate at ionic strengths of 1 and 3 M and pH values of 4, 6, and 8 were obtained from the literature.35 Interaction parameters for the pairs anion-water, cation-water, and anion-cation were obtained by minimizing eq 18. Experimental data on mean ionic activity coefficients were obtained from the literature.36 The interaction parameters between ionic groups and water and the rmsd values of the proposed model are presented in Table 5. The interaction parameters for the lysozyme-water, lysozyme-anion, and lysozyme-cation pairs can be obtained from the minimization of eq 18.
It should be noted that salts were divided into anion and cation groups. The interaction parameters for the pairs lysozyme-water, lysozyme-anion, and lysozyme-cation and the rmsd values of the proposed model and the osmotic virial model are reported in Table 6. As can be seen from this table, the model correlates the osmotic pressure data of the lysozyme-ammonium sulfate-water system with good accuracy. Also, the results show that the model can accurately correlate the osmotic pressure better than the osmotic virial model can. 4.3. Partitioning of Biomolecules in Aqueous Two-Phase Systems. The proposed model was also applied to correlate the partition coefficients of biomolecules in polymer-salt aqueous two-phase systems. To study the partitioning of biomolecules in these two-phase aqueous systems, the partition coefficients
Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009 4115 Table 6. Values of the Parameters in Eq 14 and the rmsd Values of the Group Contribution and Virial Models Obtained Using Osmotic Pressure Data for Lysozyme Electrolyte Solutionsa 100 × rmsd ionic strength
pH
EA-L/R
EL-A/R
EC-L/R
EL-C/R
EW-L/R
EL-W/R
new model
virial model
4 7 8 4 7 8
-0.593 -0.593 -0.584 -0.451 -0.453 -0.453
-1.185 -1.185 -1.185 -1.018 -1.018 -1.018
-88.300 -88.351 -88.366 -90.890 -90.902 -90.880
-1.218 -1.218 -1.213 -1.043 -1.043 -1.043
10.617 10.447 10.038 9.804 9.589 9.548
-43.683 -43.716 -43.727 -45.138 -45.141 -45.164
0.754 1.001 1.216 2.367 1.876 2.325
2.652 0.371 1.852 1.686 2.462 1.182
1.589
1.700
1 1 1 3 3 3 overall a
A ) anion, C ) cation, L ) lysozyme, W ) water.
Table 7. Values of the Parameters in Eq 14 Obtained Using PEG-K2HPO4 Aqueous Two-Phase Systemsa
a
Mw(PEG)
EA-P/R
EP-A/R
EC-P/R
EP-C/R
EP-W/R
EW-P/R
1500 1500 4000 6000 6000 6000 6000
-21.357 -118.233 -204.431 -9.260 0.549 -0.557 -312.220
-1686.235 80.881 26.794 -1.329 1.226 -1.766 1461.816
-77.468 -1368.991 -1576.250 -37.597 -0.9039 -6.816 -1294.875
0.071 0.327 0.203 0.771 1.218 3.085 137.992
-1100.420 -1100.420 -1355.433 -1500.614 -1500.614 -1500.614 -1500.614
245.171 245.171 -21.548 -4862.110 -4862.110 -4862.110 -4862.110
A ) anion, C ) cation, P ) PEG, W ) water.
Table 8. Values of the Parameters in Eq 14 and the rmsd Values of the Proposed Model Obtained Using Partition Coefficient Data for Biomolecules in PEG-K2HPO4 Aqueous Two-Phase Systems at T0 ) 298.15a biomolecule
Mw(PEG)
EA-B/R
EB-A/R
EC-B/R
EB-C/R
EP-B/R
EB-P/R
100 × rmsd
glycine bovine serum albumin lysozyme lysozyme glutamic acid lysine phenylalanine
6000 1500 1500 4000 6000 6000 6000
0.273 203.573 0.497 -1.877 -74.317 1326.287 -1377.822
-2080.753 1455.436 -1989.882 -1988.844 -141.521 2085.705 32.482
-6.408 -3982.547 0.159 -0.151 -2768.429 -113.382 -4108.408
-854.561 4334.369 -781.474 -739.124 2407.451 10178.827 122.786
0.996 -4684.551 0.486 -4.476 -3482.825 3555.924 -4907.940
4495.411 -6779.198 -2115.276 -2356.503 628.451 -524.727 -3.599
8.845 6.577 3.243 3.286 2.332 13.483 7.430
overall a
6.456
A ) anion, B ) biomolecule, C ) cation, P ) PEG.
Figure 3. Comparison between the experimental partition coefficients of some biomolecules in PEG-K2HPO4 aqueous two-phase systems and the results obtained from the new model (T ) 298.15 K).
of glycine, phenylalanine, lysine, glutamic acid, bovine serum albumin, and lysozyme in PEG-K2HPO4 aqueous two-phase systems for different molecular weights of PEG at 298.15 K were obtained.28,37 For the correlation of the partition coefficients of biomolecules in two-phase aqueous systems, it was
assumed that the salts dissociated into ionic groups, that PEG could be divided into middle and end groups, and that water could be considered as a single group.28 The interaction parameters for the anion-water, cation-water, and anion-cation pairs were obtained in the previous section (Table 5). The
4116 Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009
interaction parameters for the middle group-water pair were obtained by minimizing eq 18. Experimental data on the activity of water in PEG aqueous solutions for PEG molecular weights of 1500, 4000, and 6000 were obtained from the literature.38,39 Interaction parameters between the middle group and water are presented in Table 7. The group contribution model was applied to correlate the partition coefficients of biomolecules in polymer-salt aqueous two-phase systems using the interaction parameters obtained in the previous sections. The interaction parameters for the pairs anion-middle group, cation-middle group, biomolecule-middle group, biomolecule-anion, and biomolecule-cation were estimated through by minimizing eq 18. The interaction parameters between the middle group and ionic species are presented in Table 7. Moreover, the interaction parameters for the pairs biomolecule-middle group, biomolecule-anion, and biomolecule-cation and the rmsd values of the proposed model are reported in Table 8. The results demonstrate that the model is able to represent the partition coefficients of biomolecules in polymer-salt aqueous two-phase systems under various conditions. Figure 3 shows the experimental and correlated partition coefficients of lysine, lysozyme, and glutamic acid in PEG-K2HPO4 aqueous two-phase systems for the three systems at 298.15 K. The R2 value for the partition coefficients of biomolecules in aqueous two-phase systems is 0.9939. The results reported in Figure 3 indicate that there is good agreement between the experimental and estimated partition coefficients of biomolecules in aqueous two-phase systems. 5. Conclusions A new free-volume group contribution model has been applied to obtain the activity coefficients of amino acids and peptides in aqueous solutions. The adjustable parameters of the model were obtained by least-squares fitting of the model with the corresponding experimental data. The model was also used to correlate and predict the solubilities of amino acids in water. The model was applied to obtain the densities and osmotic pressures of aqueous biomolecule solutions. The results obtained in the present work show that the model can accurately correlate thermophysical properties of aqueous biomolecules. Additionally, the model was coupled with the Debye-Hu¨ckel model to further correlate the partition coefficients of biomolecules in polymer-salt aqueous two-phase systems. As a result, this model enables a more accurate prediction of experimental partition coefficients of biomolecules in polymer-salt aqueous two-phase systems.
Qi ) surface parameter of group i R ) universal gas constant Ri ) volume parameter of group i T ) temperature (K) x ) mole fraction Xi ) mole fraction of group i Z ) charge of ion Greek Symbols γi ) activity coefficient of component i ν ) stoichiometric number νm ) number of group m in molecule Θi ) area fraction of group i Θij ) local area fraction of group j around group i φi ) volume fraction of group i Superscripts calc ) calculated EX ) excess expt ) experimental (m) ) molality scale (x) ) mole fraction scale Subscripts ( ) mean ionic A ) anion AA ) amino acid or peptide AA-W ) amino acid-water or peptide-water A-B ) anion-biomolecule A-C ) anion-cation A-L ) anion-lysozyme A-P ) anion-polymer A-W ) anion-water B ) biomolecule C ) cation C-B ) cation-biomolecule C-L ) cation-lysozyme COMB-FV ) combinatorial-free volume C-P ) cation-polymer C-W ) cation-water LR ) long-range P-B ) polymer-biomolecule PEG ) poly(ethylene glycol) P-W ) polymer-water SR ) short-range vdW ) van der Waals W ) water W-L ) water-lysozyme
List of Symbols A ) Debye-Hu¨ckel constant aW ) activity of water B ) Debye-Hu¨ckel constant C ) concentration of biomolecule E ) energy parameter F ) objective function G ) Gibbs energy Hij ) Boltzmann factor H ) enthalpy I ) molar ionic strength K ) partition coefficient Mw ) molecular weight of solvent M ) molality N ) number of experimental data points Psat ) saturated vapor pressure
Appendix A Derivation of the Group Contribution Model At equilibrium, a Boltzmann distribution for the relationship between the local and bulk densities of group i around group j was assumed40
( )
Fij ∆hij ) gij(r) ) exp Fjj RT Fi )
Ni V
(A-1)
(A-2)
where gij(r) is the radial distribution function (RDF) of group i around group j and Ni, Fi, and V are the number and density of
Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009 4117
molecules of type i and the volume of the system, respectively. The coordination number between groups i and j can be obtained from the equation21 4πNi V
Nij )
∫
Literature Cited
∞ 2
r gij(r) dr
0
(A-3)
where Nij is the number of molecules of i around one molecule of group j. In eq A-3, a uniform radial distribution function proposed by Reed and Gubbins was used41 gij(r) )
{
0 r < σij 1 r > σij
(A-4)
Substituting eq A-4 into A-3 gives Nij )
( )
4πNiσij3 ∆hij exp 3V RT
(A-5)
Therefore, the local area fraction between groups i and j can be obtained from eq A-1 as Nij Θi Θij ) ) Hij Θjj Njj Θj
(A-6)
( )
(A-7)
with Hij ) exp -
∆hij RT
∆hij ) Eij
(A-8)
where hij is the enthalpy between groups i and j and Eij is an adjustable parameter of the modified model proposed in this work. Density of Amino Acid Aqueous Solution The relation between density and excess volume of aqueous solution can be written as21
∑xM i
d)
i
i
∑xV
i i
(A-9)
+ VEX
i
where d is the density of the solution and Mi and Vi are the molar mass and volume of component i, respectively. The excess volume of solution can be determined using the excess Gibbs energy of solution as VEX )
[ ] ∂GEX ∂P
(A-10)
T,x
Thus, the excess volume of the model was calculated using the equation
VEX )
∑x
i
i
[
∑ Θ H ( ∂P ) ∂Eji
i
ji
j
∑ΘH j
j
ji
the experimental densities of aqueous amino acid solutions and the results of the model.
-
∑ Θ ( ∂P ) ∂Eji
j
j
]
(A-11)
where, in eq A-11, ∂Eji/∂P represents adjustable parameters to be tuned by correlating experimental densities obtained by fitting
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ReceiVed for reView June 15, 2008 ReVised manuscript receiVed December 28, 2008 Accepted February 3, 2009 IE8009389
