Correlation-induced DNA adsorption on like-charged membranes

The adsorption of DNA or other polyelectrolyte molecules on charged membranes is a recurrent motif in soft matter and bionanotechnological systems. Two typical situations encountered are the deposition of single DNA chains onto substrates for further analysis, e.g., by force microscopy, or the pulling of polyelectrolytes into membrane nanopores, as in sequencing applications. In this paper, we present a theoretical analysis of such scenarios based on the self-consistent field theory approach, which allows us to address the important effect of charge correlations. We calculate the grand potential of a stiff polyelectrolyte immersed in an electrolyte in contact with a negatively charged dielectric membrane. For the sake of conciseness, we neglect conformational polymer fluctuations and model the molecule as a rigid charged line. At strongly charged membranes, the adsorbed counterions enhance the screening ability of the interfacial region. In the presence of highly charged polymers such as double-stranded DNA molecules close to the membrane, this enhanced interfacial screening dominates the mean-field level DNA-membrane repulsion and results in the adsorption of the DNA molecule to the surface. This picture provides a simple explanation for the recently observed DNA binding onto similarly charged substrates [G. L.-Caballero et al., Soft Matter 10, 2805 (2014)] and points out charge correlations as a non-negligible ingredient of polymer-surface interactions.

Figure 1

Schematic representation of the negatively charged polymer immersed in a symmetric electrolyte and interacting with the like-charged dielectric membrane. Electrostatic forces acting on the polymer (dashed arrows): MF-level membrane-polymer repulsion (orange), beyond-MF repulsive image charge force (blue), and attractive cation-mediated solvation force (red).

Figure 2

Negatively charged polyelectrolyte of length $L$ and linear charge density $\tau$, with the total charge $Q_p=L\tau$. The right end of the approaching polymer (red) is located at $z=z_t<0$ and the adsorbing polymer (blue) at $z=z_a<0$ from the membrane surface located at $z=0$.

Figure 3

Adimensional polymer self-energy of Eq. (40) for ${\varepsilon }_m={\varepsilon }_w$ (red curve) and Eq. (44) for ${\varepsilon }_m=0$ (blue curve). Thin black curves display the limiting laws for very close (${\overline{z}}_a\ll 1$) and very large separation distances (${\overline{z}}_a\gg 1$) from the membrane surface (see the main text).

Figure 4

Dimensionless self-energy of short polymers (49) versus the distance from the membrane with permittivity ${\varepsilon }_m=2$ at different values of the parameter $s={\kappa }_b\mu$.

Figure 5

(a) One-loop polymer grand potential (63), (b) MF-level grand potential (51) (main plot), and self-energy (61) (inset) of a ds-DNA with linear charge density $\tau =0.59e/$Å. (c) Displays the total grand potential of a ss-DNA molecule with charge density $\tau =0.29e/$Å. The membrane permittivity is ${\varepsilon }_m=2$ (solid curves). The square symbols in (a) correspond to the DH regime at the membrane permittivity ${\varepsilon }_m=0$ where the self-energy (62) diverges logarithmically as $\mathrm{\Psi }\left({\tilde{z}}_a\right)=K_0\left(2{\tilde{z}}_a\right)\simeq -ln\left({\tilde{z}}_a\right)$. The inset in (a) zooms to the coexistence region.

Figure 6

Phase diagram: critical values of the parameter $s={\kappa }_b\mu$ versus the polymer charge density $\tau$ separating the parameter regimes with binding (area below the critical lines) and unbinding polyelectrolytes (area above the lines) oriented parallel (solid curve) and perpendicular to the membrane surface (dashed curve). The red dots mark the characteristic polymer charge densities reached at $s=0$ below which the adsorption state disappears.

Figure 7

(a) MF grand potential ${\mathrm{\Omega }}_{pm}\left(v\right)$ of Eq. (67) (dashed curves), one-loop level grand potential $\mathrm{\Delta }{\mathrm{\Omega }}_p\left(v\right)={\mathrm{\Omega }}_{pm}\left(v\right)+L{\ell }_B{\tau }^2\chi \left(v\right)/2$ (solid curves), and (b) dimensionless self-energy $\chi \left(v\right)$ of Eq. (70) versus the reduced separation distance. The membrane permittivity is ${\varepsilon }_m=2$, the polymer charge density $\tau =1.0e/$Å, and the different values of the rescaled length $\overline{L}=L/\mu$ are indicated in the legend.

Figure 8

(a) Dimensionless self-energy (78) (main plot) and MF grand potential (81) rescaled by the characteristic energy $\mathrm{\Delta }{\mathrm{\Omega }}_{*}={\ell }_B{\tau }^2/\left(2{\kappa }_b\right)$ (inset). (b) Rescaled total grand potential $\mathrm{\Delta }{\mathrm{\Omega }}_p=\mathrm{\Delta }{\mathrm{\Omega }}_{pp}+{\mathrm{\Omega }}_{pm}$. The membrane permittivity is ${\varepsilon }_m=2$ and the ds-DNA charge density $\tau =0.59e/$Å. The square symbols in (a) correspond to the DH limit (82).

Figure 9

Main plot: MF grand potential (81) rescaled by the characteristic energy $\mathrm{\Delta }{\mathrm{\Omega }}_{*}={\ell }_B{\tau }^2/\left(2{\kappa }_b\right)$ (blue) and dimensionless self-energy (78) (black). Inset: rescaled total grand potential $\mathrm{\Delta }{\mathrm{\Omega }}_p/\mathrm{\Delta }{\mathrm{\Omega }}_{*}$. Solid curves: $\tilde{L}=\infty$. Dotted curves: $\tilde{L}=1$. The membrane permittivity is ${\varepsilon }_m=2$, the ds-DNA charge density $\tau =0.59e/$Å, and $s=0.1$.

Figure 10

Surface self-energy $\zeta \left(0\right)={\zeta }_{\text{DH}}^{\left(0\right)}\left(0\right)+s^{-2}{\zeta }_{\text{DH}}^{\left(1\right)}\left(0\right)$ (black, purple, and blue curves) at various values of $s$ versus the polymer length $\tilde{L}$ from Eqs. (86) and (87). The red curve is the finite charge correction (87). The black curve at $s=\infty$ equally corresponds to the DH self-energy (86).

Figure 11

Phase diagram: critical values of the parameter $s={\kappa }_b\mu$ versus the polymer charge density $\tau$ separating the parameter regimes characterized by repulsion (area above the curves) and like-charge attraction (below the curves) at various polymer lengths given in the legend. The inset displays the polymer length dependence of the critical point where the critical lines of the main plot end in the GC limit ($s=0$).