Density functional study on the structures and thermodynamic properties of small ions around polyanionic DNA

A density functional theory (DFT) is presented for describing the distributions of small ions around an isolated infinitely long polyanionic DNA molecule in the framework of the restricted primitive model. The hard-sphere contribution to the excess Helmholtz energy functional is derived from the modified fundamental measure theory, and the electrostatic interaction is evaluated through a quadratic functional Taylor expansion. The predictions from the DFT are compared with integral equation theory (IET), the nonlinear Poisson-Boltzmann (PB) equation, and computer simulation data for the ionic density profiles, electrostatic potentials, and charge compensation functions at varieties of solution conditions. Good agreement between the DFT and computer simulations is achieved. The charge inversion phenomena of DNA are observed in a moderately concentrated solution of 2:1 and 2:2 electrolytes using the DFT, IET, and computer simulation, but can never be predicted from the PB equation. The predictions of charge inversion from the DFT prove to be more accurate than those from the IET when compared with computer simulation data. The preferential interaction coefficients from the DFT are also compared with those from the PB equation and Monte Carlo simulation, and it is shown that the DFT is superior to the PB equation.

Figure 1

Concentration profiles of a 1:1 model electrolyte around rodlike DNA molecule at bulk the concentration of $51\mathrm{mM}$. The distance $r$ is measured with respect to the surface of cylindrical DNA in units of the ionic diameter.

Figure 2

Concentration profiles of a 2:1 model electrolyte around DNA molecule at the bulk concentration of $57\mathrm{mM}$. The distance $r$ is measured with respect to the surface of DNA in units of the ionic diameter.

Figure 3

Reduced density profiles for a 1:1 model electrolyte around DNA molecule at bulk concentration of 0.495M ($T=298\mathrm{K}$, $R=\mathrm{0.9.8}\mathrm{nm}$, ${\sigma }_i=0.425\mathrm{nm}$, $\epsilon =78.5$). The distance $r$ is measured with respect to the surface of DNA in units of the ionic diameter. Circles represent MC simulation results [53]. The solid and dashed curves are predictions from the present DFT and nonlinear PB equations, respectively. The inset shows counterion profiles near contact.

Figure 4

Reduced density profiles for a 2:2 model electrolyte around DNA molecule at bulk concentration of 0.501M ($T=298\mathrm{K}$, $R=0.98\mathrm{nm}$, ${\sigma }_i=0.425\mathrm{nm}$, $\epsilon =78.5$). The distance $r$ is measured with respect to the surface of DNA in units of the ionic diameter. Legend is the same as Fig. 3. The inset shows counterion profiles near contact.

Figure 5

Reduced density profiles for a 2:2 model electrolyte around DNA molecule at bulk concentration of $0.490\mathrm{M}$ ($T=298\mathrm{K}$, $R=0.786\mathrm{nm}$, ${\sigma }_i=0.425\mathrm{nm}$, $\epsilon =78.5$). The distance $r$ is measured with respect to the surface of DNA in units of the ionic diameter.

Figure 6

Reduced density profiles from DFT for 1:1, 2:1, and 2:2 model electrolytes around DNA molecule at bulk concentration of $1.0\mathrm{M}$. The distance $r$ is measured with respect to the surface of DNA in units of the ionic diameter. The inset shows an enlargement of the region in which overcrosses of counterion profile and co-ion profile occur again.

Figure 7

Reduced mean electrostatic potential around DNA for a 1:1 electrolyte as a function of distance at the bulk concentration of $0.495\mathrm{M}$ ($T=298\mathrm{K}$, $R=0.98\mathrm{nm}$, ${\sigma }_i=0.425\mathrm{nm}$, $\epsilon =78.5$). The distance $r$ is measured with respect to the surface of DNA in units of the ionic diameter.

Figure 8

Reduced mean electrostatic potential around DNA for a 2:2 electrolyte as a function of distance at the bulk concentration of $0.49\mathrm{M}$ ($T=298\mathrm{K}$, $R=0.786\mathrm{nm}$, ${\sigma }_i=0.425\mathrm{nm}$, $\epsilon =78.5$). The distance $r$ is measured with respect to the surface of DNA in units of the ionic diameter.

Figure 9

Mean electrostatic potential around DNA for 2:1 and 1:1 electrolytes as a function of distance at the bulk concentration of $50\mathrm{mM}$. The distance $r$ is measured with respect to the surface of DNA in units of the ionic diameter. The inset gives the full curves of mean electrostatic potential over a wide range of distance.

Figure 10

Mean electrostatic potential around DNA predicted by DFT at the bulk concentration of $500\mathrm{mM}$. The distance $r$ is measured with respect to the surface of DNA in units of the ionic diameter.

Figure 11

Mean electrostatic potential predicted by DFT for a 2:1 electrolytes at the various bulk concentrations. The distance $r$ is measured with respect to the surface of DNA in units of the ionic diameter. The inset shows the full curves of $0.05\mathrm{M}$ over a wide range of distance.

Figure 12

Charge compensation function for a 1:1 electrolyte at $911\mathrm{mM}$ bulk concentration ($T=\mathrm{298}\mathrm{K}$, $R=0.80\mathrm{nm}$, ${\sigma }_i=0.40\mathrm{nm}$, $\epsilon =78.358$). The distance $r$ is measured with respect to the surface of DNA in units of the ionic diameter.

Figure 13

Charge compensation function for a 2:2 electrolyte at $0.49\mathrm{M}$ bulk concentration ($T=298\mathrm{K}$, $R=0.786\mathrm{nm}$, ${\sigma }_i=0.425\mathrm{nm}$, $\epsilon =78.5$). The distance $r$ is measured with respect to the surface of DNA in units of the ionic diameter.

Figure 14

Charge compensation function for a 2:2 electrolytes $(T=298\mathrm{K}$, $R=0.786\mathrm{nm}$, ${\sigma }_i=0.425\mathrm{nm}$, $\epsilon =78.5$). The distance $r$ is measured with respect to the surface of DNA in units of the ionic diameter. The curves crossing unity line correspond to the bulk concentration of $0.68\mathrm{M}$ and the others correspond to the bulk concentration of $0.12\mathrm{M}$.

Figure 15

Charge compensation function predicted by DFT for 1:1 and 2:1 electrolytes at $1.0\mathrm{M}$ bulk concentration. The distance $r$ is measured with respect to the surface of DNA in units of the ionic diameter. The inset shows an enlargement of the region in which charge compensation function of 2:1 electrolyte falls down below 1.0 again.

Figure 16

Density profile, electrostatic potential, and charge compensation function from DFT for a 2:1 electrolyte at the bulk concentration of $1.0\mathrm{M}$. The distance $r$ is measured with respect to the surface of DNA in units of the ionic diameter. The curves for electrostatic potential and charge compensation function have upward displacements of 0.75 and 0.25, respectively.

Figure 17

DNA-ion preferential interaction coefficients for a 2:1 salt at various bulk concentrations.

Figure 18

DNA-ion preferential interaction coefficients for a 1:1 salt at various bulk concentrations.