^{1,a)}, Gerhard Nägele

^{1}, Remco Tuinier

^{1}, Thomas Gibaud

^{2}, Anna Stradner

^{2}and Peter Schurtenberger

^{2}

### Abstract

We propose a minimal model for spherical proteins with aeolotopic pair interactions to describe the equilibrium phase behavior of lysozyme. The repulsive screened Coulomb interactions between the particles are taken into account assuming that the net charges are smeared out homogeneously over the spherical proteinsurfaces. We incorporate attractive surface patches, with the interactions between patches on different spheres modeled by an attractive Yukawa potential. The parameters entering the attractive Yukawa potential part are determined using information on the experimentally accessed gas-liquid-like critical point. The Helmholtz free energy of the fluid and solid phases is calculated using second-order thermodynamic perturbation theory. Our predictions for the solubility curve are in fair agreement with experimental data. In addition, we present new experimental data for the gas-liquid coexistence curves at various salt concentrations and compare these with our model calculations. In agreement with earlier findings, we observe that the strength and the range of the attractive potential part only weakly depend on the salt content.

This project has been partly supported by the European Commission under the Framework Program through integrating and strengthening the European Research Area. Contract No. SoftComp, NoE / NMP3-CT-2004-502235.

I. INTRODUCTION

II. EXPERIMENTAL DETAILS

III. MODEL

IV. HELMHOLTZ FREE ENERGY AND PHASE COEXISTENCE

V. DETERMINATION OF THE ATTRACTIVE INTERACTION PARAMETERS

VI. CALCULATED PHASE DIAGRAMS

VII. DISCUSSION

VIII. CONCLUSION

### Key Topics

- Proteins
- 53.0
- Critical point phenomena
- 25.0
- Anisotropy
- 18.0
- Electrostatics
- 13.0
- Hydrophobic interactions
- 13.0

## Figures

Schematic drawing of a configuration of two model proteins, each carrying two attractive patches (gray areas). For the given configuration, the two particles repel each other according to the screened electrostatic potential in Eq. (3), with the surface charge assumed to be smeared out homogeneously over the sphere surface. There is no attractive interaction part since the center-to-center vector does not intersect simultaneously the shaded attractive patches on particles 1 and 2. See the main text for the definitions of the remaining symbols.

Schematic drawing of a configuration of two model proteins, each carrying two attractive patches (gray areas). For the given configuration, the two particles repel each other according to the screened electrostatic potential in Eq. (3), with the surface charge assumed to be smeared out homogeneously over the sphere surface. There is no attractive interaction part since the center-to-center vector does not intersect simultaneously the shaded attractive patches on particles 1 and 2. See the main text for the definitions of the remaining symbols.

Repulsive electrostatic pair potential part (thick dashed curve), angular-averaged attractive interaction part (thick dashed-dotted curve), and total perturbational pair potential (thick solid curve) for parameters at the critical point, where , using , . The parameters used in the perturbational interactions for the attractive and repulsive Yukawa-type potential parts are listed in Table I. At larger , is dominated by the attractive interaction part.

Repulsive electrostatic pair potential part (thick dashed curve), angular-averaged attractive interaction part (thick dashed-dotted curve), and total perturbational pair potential (thick solid curve) for parameters at the critical point, where , using , . The parameters used in the perturbational interactions for the attractive and repulsive Yukawa-type potential parts are listed in Table I. At larger , is dominated by the attractive interaction part.

Angular-averaged total perturbation potential, [see Eq. (1)] for various salt concentrations as indicated. With decreasing , the contact value of decreases due to the enlarged range of the electrostatic repulsion part.

Angular-averaged total perturbation potential, [see Eq. (1)] for various salt concentrations as indicated. With decreasing , the contact value of decreases due to the enlarged range of the electrostatic repulsion part.

The phase diagram of aqueous lysozyme solutions for HEPES buffer, and , NaCl. The circles describe the experimentally found metastable gas-liquid coexistence curve (Ref. 38), the squares indicate the spinodal (Ref. 38), and the black triangles depict the experimental fluid-crystal coexistence curve. The two dashed curves show the calculated binodal and spinodal, respectively, for . The two solid curves describe the calculated binodal and spinodal, respectively, where the two curves account for an additional temperature dependence of the attractive potential depth with [see Eq. (8)]. The dashed-dotted curves are the calculated fluid-crystal coexistence curves for , with the interaction parameters determined from the experimental data at the critical point as explained in the text. In region I, a stable fluid phase is observed, whereas one finds a fluid-crystal coexistence in region II, a metastable gas-liquid coexistence in region III, and a pure crystalline phase in region IV.

The phase diagram of aqueous lysozyme solutions for HEPES buffer, and , NaCl. The circles describe the experimentally found metastable gas-liquid coexistence curve (Ref. 38), the squares indicate the spinodal (Ref. 38), and the black triangles depict the experimental fluid-crystal coexistence curve. The two dashed curves show the calculated binodal and spinodal, respectively, for . The two solid curves describe the calculated binodal and spinodal, respectively, where the two curves account for an additional temperature dependence of the attractive potential depth with [see Eq. (8)]. The dashed-dotted curves are the calculated fluid-crystal coexistence curves for , with the interaction parameters determined from the experimental data at the critical point as explained in the text. In region I, a stable fluid phase is observed, whereas one finds a fluid-crystal coexistence in region II, a metastable gas-liquid coexistence in region III, and a pure crystalline phase in region IV.

Gas-liquid coexistence curves of a lysozyme solution obtained experimentally from temperature quenches at four different salt concentrations: , 0.4 mol/l , 0.3 mol/l , and 0.2 mol/l . The filled symbols mark the critical points estimated from the experiment. The solid curves describe the coexistence curves as calculated from our model using a fixed value .

Gas-liquid coexistence curves of a lysozyme solution obtained experimentally from temperature quenches at four different salt concentrations: , 0.4 mol/l , 0.3 mol/l , and 0.2 mol/l . The filled symbols mark the critical points estimated from the experiment. The solid curves describe the coexistence curves as calculated from our model using a fixed value .

Phase diagram of lysozyme for NaCl, HEPES buffer, and . The symbols indicate the experimental data points identical to the ones in Fig. 4. The solid curves describe the equilibrium phase diagram obtained from the anisotropic model. For comparison, the dashed lines describe the gas-liquid and fluid-solid coexistence curves are obtained from a purely isotropic pair potential. In both cases, is set equal to 5.

Phase diagram of lysozyme for NaCl, HEPES buffer, and . The symbols indicate the experimental data points identical to the ones in Fig. 4. The solid curves describe the equilibrium phase diagram obtained from the anisotropic model. For comparison, the dashed lines describe the gas-liquid and fluid-solid coexistence curves are obtained from a purely isotropic pair potential. In both cases, is set equal to 5.

## Tables

System and pair potential parameters used in the thermodynamic perturbation calculation of the metastable gas-liquid binodal/spinodal, and the stable fluid-solid coexistence curve (for salt concentrations as indicated). The attractive potential part parameters and are determined by Eqs. (24) and (25), respectively, using . For given , the parameters , , and (with a fixed value ) are determined from the experimental values for , , and , with fixed to 5.

System and pair potential parameters used in the thermodynamic perturbation calculation of the metastable gas-liquid binodal/spinodal, and the stable fluid-solid coexistence curve (for salt concentrations as indicated). The attractive potential part parameters and are determined by Eqs. (24) and (25), respectively, using . For given , the parameters , , and (with a fixed value ) are determined from the experimental values for , , and , with fixed to 5.

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