Model for a mixture of macroions, counterions, and co-ions in a waterlike fluid

We propose an integral equation theory for a mixture of macroions, counterions, and co-ions in a waterlike fluid in which all the components are accounted for explicitly. The macroions can carry positive and negative surface charges simultaneously, mimicking in this way the situation occurring in protein solutions. To solve this complex model numerically, we utilize the associative mean spherical approximation, developed earlier for low-molecular-mass charge-symmetric electrolyte solutions. To illustrate the potential of this approach, we present numerical results for various experimental conditions. Among the measurable properties we choose to calculate the osmotic coefficient, a quantity that reflects the stability of the solution. We show that the osmotic coefficient depends not only on the magnitude of the net charge on the macroion but also on its sign, as well as on the nature of the low-molecular-mass electrolyte present. These specific ion effects are the consequence of differences in hydration between the ions in solution and charged groups on the macroion.

Figure 1

Model of a macroion with three positively charged residues $n_{p_D}=3$ (small red spheres) and one negatively charged residue $n_{p_E}=1$ (small blue sphere) attached to the macroparticle (large sphere) due to the infinitely strong interaction between sticky sites of type $D$ on the macroparticle and the positively charged residue-forming ions and between the $E$-type sticky sites on the macroparticle and the negative residue-forming ions. Light green and light red semispheres denote the range of the square-well site-site interaction between the $C$-type sites, which belong to two different macroparticles, and between $A$ and $B$ types of sites, which are positioned on the positively and negatively charged species in the system, respectively.

Figure 2

Osmotic coefficient $\varphi$ as a function of protein concentration for the protein model with 16 positive residues and one negative residue: +16:$-$1. The solid red line denotes LiI and the dashed red line LiCl; the solid green line denotes CsI and the dashed green line CsCl. The cations (a) $c=0.1\text{mol}/{\text{dm}}^3$ and (b) $c=0.4\text{mol}/{\text{dm}}^3$. Note that on the scale of the figure the curves for LiCl and CsCl in (b) coincide.

Figure 3

Osmotic coefficient $\varphi$ as a function of protein concentration for a model protein with one positive and 16 negative residues: +1:$-$16. The color code and other parameters are the same as in Fig. 2.

Figure 4

Osmotic coefficient $\varphi$ as a function of protein concentration for a model protein with 16 positive and 16 negative residues: +16:$-$16. The color code and other parameters are the same as in Fig. 2.