**2.3 Thermodynamic model for protein retention in HIC**

The thermodynamic model proposed by Melander et al. (1989) to describe the effect of salt concentration on macromolecule retention in chromatography (IEC and HIC) can be applied to any stationary phase consisting of a highly hydrated surface modified with charged ligands (in the case of IEC), weakly hydrophobic moieties (in the case of HIC), or both. Electrostatic and hydrophobic interactions between the macromolecule and the stationary phase are treated separately. Electrostatic interaction is modeled based on the Manning's counter ion condensation theory (Manning, 1978), whereas hydrophobic interaction is treated by considering an adaptation of the Sinanoglu's solvophobic (Sinanoglu, 1982) theory that relates the salting out of proteins with their retention in HIC (Melander & Horvath, 1977). Figure 3 depicts protein retention due to hydrophobic interactions, electrostatic interactions, and both hydrophobic and electrostatic interactions.

Fig. 3. Protein retention due to hydrophobic, electrostatic, and both interactions. Electrostatic interactions are long-range interactions, and then moieties with opposite charges do not need to be in physical contact. Hydrophobic interactions are short-range, and then interacting hydrophobic moieties must be in contact. As a consequence, when hydrophobic and charged moieties are present, both types of interactions may occur.

The main assumptions considered in the model are listed below:


### **2.3.1 Electrostatic interaction**

The Gibbs energy of binding (ΔG0es) of the macromolecule to a stationary phase in presence of salt (that acts as a counter ion) is given by equation (2). Here ms is the molal salt concentration, NAv the Avogadro's number, "e" the base of the natural logarithm, "b" the average spacing of fixed charges on the surface, δp the thickness of the condensation layer over the surface of the stationary phase where each fixed charge occupies an area of b2, δs the layer thickness of salt counter ion, Zp the characteristic charge of the protein, Zs the valence of the salt counter ion, and ξ a dimensionless structural parameter that characterizes the charged surface. R is the universal constant of gases and T the absolute temperature.

$$\frac{-\Delta G\_{\rm es}^{0}}{2.3 \cdot R \cdot T} = \log\left(\frac{N\_{Av} \cdot b^{2} \cdot \delta\_{p}}{1000 \cdot e}\right) + \frac{Z\_{p}}{Z\_{s}} \log\left(\frac{1000 \cdot e}{\left(N\_{Av} \cdot b^{2} \cdot \delta\_{s} \cdot m\_{s}\right) \cdot \left(1 - Z\_{s} \cdot \xi\right)}\right) \tag{2}$$

### **2.3.2 Hydrophobic interaction**

The contact between the hydrophobic patches on the macromolecule surface that are exposed to the solvent and the hydrophobic ligands on the stationary phase, trigger the retention due to hydrophobic interaction. The Gibbs energy of hydrophobic interaction (ΔGhp) is expressed in terms of the molal surface tension increment of the salt (σs), as shown in equation (3), which is valid only in the absence of specific salt effects. In Equation (3) ms is the salt molality, ΔG0 aq represents the reduction in Gibbs energy due to other effects different form hydrophobic interactions, ΔA' is the difference between the molecular surface area of the unbound macromolecule (AM) and the molecular surface area of the macromolecule attached to the stationary phase (As). ΔA' corresponds to the surface contact area between the bound protein and the hydrophobic site of the matrix.

$$
\Delta G\_{hf}^{0} = \Delta G\_{aq}^{0} - \Delta A^{\prime} \cdot \sigma\_s \cdot m\_s \tag{3}
$$

### **2.3.3 Combined electrostatic and hydrophobic interaction**

The retention factor (k'), given in equations (4) and (5), is represented in terms of salt molality when both electrostatic and hydrophobic interactions are present. This is accomplished by combining equations (2) and (3) to give equation (6). In equation (5), K is the equilibrium constant and φ is the phase ratio (stationary phase mass / mobile phase mass). In equation (6) α is the phase volume ratio (stationary phase/mobile phase). Equation (6) can be written in a simplified form, as given by equation (7), where A is a constant determined by all the system characteristics, B the electrostatic interaction parameter and C the hydrophobic interaction parameter. In equation (7), the term C accounts for the hydrophobic surface contact area between the macromolecule and the stationary phase, and is given by equation (8).

$$\log K = \left(\frac{-\Delta G\_{\rm es}^{0}}{2.3 \cdot R \cdot T}\right) - \left(\frac{-\Delta G\_{\rm hf}^{0}}{2.3 \cdot R \cdot T}\right) \tag{4}$$

$$k' = \phi \cdot K \tag{5}$$

The Gibbs energy of binding (ΔG0es) of the macromolecule to a stationary phase in presence of salt (that acts as a counter ion) is given by equation (2). Here ms is the molal salt concentration, NAv the Avogadro's number, "e" the base of the natural logarithm, "b" the average spacing of fixed charges on the surface, δp the thickness of the condensation layer over the surface of the stationary phase where each fixed charge occupies an area of b2, δs the layer thickness of salt counter ion, Zp the characteristic charge of the protein, Zs the valence of the salt counter ion, and ξ a dimensionless structural parameter that characterizes the charged surface. R is the universal constant of gases and T the absolute

<sup>1000</sup> log log 2.3 <sup>1000</sup> <sup>1</sup>

δ

area between the bound protein and the hydrophobic site of the matrix.

**2.3.3 Combined electrostatic and hydrophobic interaction** 

stationary phase, and is given by equation (8).

*G Nb Z e*

*R T e Z Nb m Z*

⎛ ⎞ −Δ ⎛ ⎞ ⋅ ⋅ <sup>⋅</sup> ⎜ ⎟ = + ⎜ ⎟ ⋅ ⋅ ⎜ ⎟ <sup>⋅</sup> ⋅⋅⋅ ⋅− ⋅ ⎝ ⎠ ⎝ ⎠

The contact between the hydrophobic patches on the macromolecule surface that are exposed to the solvent and the hydrophobic ligands on the stationary phase, trigger the retention due to hydrophobic interaction. The Gibbs energy of hydrophobic interaction (ΔGhp) is expressed in terms of the molal surface tension increment of the salt (σs), as shown in equation (3), which is valid only in the absence of specific salt effects. In Equation (3) ms is

different form hydrophobic interactions, ΔA' is the difference between the molecular surface area of the unbound macromolecule (AM) and the molecular surface area of the macromolecule attached to the stationary phase (As). ΔA' corresponds to the surface contact

> 0 0 'Δ =Δ −Δ ⋅ ⋅ *G G Am hf aq*

The retention factor (k'), given in equations (4) and (5), is represented in terms of salt molality when both electrostatic and hydrophobic interactions are present. This is accomplished by combining equations (2) and (3) to give equation (6). In equation (5), K is the equilibrium constant and φ is the phase ratio (stationary phase mass / mobile phase mass). In equation (6) α is the phase volume ratio (stationary phase/mobile phase). Equation (6) can be written in a simplified form, as given by equation (7), where A is a constant determined by all the system characteristics, B the electrostatic interaction parameter and C the hydrophobic interaction parameter. In equation (7), the term C accounts for the hydrophobic surface contact area between the macromolecule and the

( ) ( )

ξ

*s s* (3)

(4)

(5)

(2)

2

aq represents the reduction in Gibbs energy due to other effects

σ

0 0

*RT RT* ⎛ ⎞ −Δ ⎛ ⎞ −Δ = − ⎜ ⎟ ⎜ ⎟ ⋅⋅ ⋅⋅ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠

*hf es G G*

log 2.3 2.3

*k K* ' = ⋅ φ

*K*

*s Av s s s*

δ

**2.3.1 Electrostatic interaction** 

**2.3.2 Hydrophobic interaction** 

the salt molality, ΔG0

2 0

*Av p p es*

temperature.

$$\log k' = \log \left( \frac{N\_{Av} \cdot b^2 \cdot \delta\_p}{1000 \cdot e} \right) + \frac{Z\_p}{Z\_s} \log \left( \frac{1000 \cdot e}{\left( N\_{Av} \cdot b^2 \cdot \delta\_s \cdot m\_s \right) \cdot \left( 1 - Z\_s \cdot \xi \right)} \right) + \tag{6}$$

$$\frac{\Delta G\_{aq}^{0}}{2.3 \cdot R \cdot T} + \frac{\Delta A \cdot \sigma\_{s} \cdot m\_{s}}{2.3 \cdot R \cdot T} + \log \alpha$$

$$
\log k' = A - B \cdot \log m\_s + C \cdot m\_s \tag{7}
$$

$$C = \frac{\Delta A \cdot \sigma\_s}{2.3 \cdot R \cdot T} \tag{8}$$

Equation (7) corresponds to the Simplified Thermodynamic Model for Electrostatic and Hydrophobic Interactions. This model is of practical usefulness, since its parameters can be obtained from experimental runs in a relatively simple manner, depending on the salt concentration present in the macromolecule solution. At low salt concentration, up to 0.5 molal, hydrophobic interactions can be neglected, and therefore the parameters A and B in equation (7) can be estimated by means of a linear regression between isocratic retention factors obtained at different salt molalilies. At high salt concentration, electrostatic interactions are negligible, and hence the parameters A and C can be obtained in a similar way, considering the isocratic retention factors. The hydrophobic contact area (ΔA' in equation (8)) can easily be obtained from the slope of the limiting plot of log k' versus molal salt concentration.

The simplified thermodynamic model has been used to investigate the effect of surface hydrophobicity distribution of proteins on retention in HIC (Mahn et al., 2004). The applicability of the model to predict protein retention time in HIC was demonstrated, and for the first time it was experimentally proven that surface hydrophobicity distribution has an important effect on protein retention in HIC. Furthermore, it was shown that the parameter ΔA' that comes from equations (7) and (8) was able to represent the protein retention in HIC with salt gradient elution. However, the methodology proposed by Mahn et al. (2004) requires the generation of a considerable amount of experimental data, thus limiting its application.
