Attraction between DNA molecules mediated by multivalent ions

The effective force between two parallel DNA molecules is calculated as a function of their mutual separation for different valencies of counterion and salt ions and different salt concentrations. Computer simulations of the primitive model are used and the shape of the DNA molecules is accurately modeled using different geometrical shapes. We find that multivalent ions induce a significant attraction between the DNA molecules whose strength can be tuned by the averaged valency of the ions. The physical origin of the attraction is traced back either to electrostatics or to entropic contributions. For multivalent counterions and monovalent salt ions, we find a salt-enhanced repulsion effect: the force is first attractive but gets repulsive with increasing salt concentration. Furthermore, we show that the multivalent-ion-induced attraction does not necessarily correlate with DNA overcharging.

Figure 1

(Color online) A typical snapshot of the simulation box. The DNA molecules are drawn according to the MAM. Black spheres on the DNA strands represent the phosphate charges. Internal gray spheres between the phosphates and the DNA cylindrical core are neutral. Positive (negative) salt ions spread across the simulation volume are shown as open (hatched) spheres.

Figure 2

(Color online) A schematic picture explaining the positions of DNA molecules and the definition of the different azimuthal angles ${\varphi }_s$, ${\varphi }_0$, $\varphi$. For further information, see text.

Figure 3

(Color online) Three typical configurations for the two parallel DNA molecules when their phosphates charges are close to each other. The $xy$-plane cross sections of DNA molecules are shown for the ECM. Note that only the neighboring phosphates in the inter-DNA area are shown, see the dark small circles labeled by letters $A,B,C,D$. (a) ${\varphi }_0+n{\varphi }_s=\pi ,\varphi =\pi ,\pi ∕5$. (b) ${\varphi }_0+n{\varphi }_s=\pi ,$ $\varphi =\pi -{\varphi }_s∕2$, $\pi ∕5-{\varphi }_s∕2$. (c) ${\varphi }_0+n{\varphi }_s=\pi \pm {\varphi }_s∕2$, $\varphi =\pi ,\pi ∕5$. Here $n$ is an integer number, $n=0,\pm 1,\pm 2,\dots$ In (b) and (c) the pair of phosphates on each cylinder pertain to the same strand and have different $z$ coordinates: (b) $z_A=z_B-\left(1.7\mathrm{\AA }\right)$, $z_c=z_B+\left(1.7\mathrm{\AA }\right)$; (c) $z_A=z_D$, $z_B=z_C$, $z_A-z_c=3.4\mathrm{\AA }$.

Figure 4

(Color online) Reduced DNA-DNA interaction force $F∕F_0$ vs separation distance $R$ for the monovalent counterions and monovalent salt ions (parameter set 1 of Table ) for the MAM. The CM and ECM exhibit similar trends. The unit of the force is $F_0=k_{B}T∕P$, where $P$ is the DNA pitch length. Different salt densities are shown: $C_s=0\text{mole}∕\mathrm{l}$ (solid line), $0.024\text{mole}∕\mathrm{l}$ (dashed line), $0.097\text{mole}∕\mathrm{l}$ (dot-dashed line), $0.194\text{mole}∕\mathrm{l}$ (solid line with symbols), $0.71\text{mole}∕\mathrm{l}$ (dashed line with symbols). The corresponding screening lengths are ${\lambda }_D=9.6\mathrm{\AA }$, $9\mathrm{\AA }$, $8\mathrm{\AA }$, $5.6\mathrm{\AA }$, $3.3\mathrm{\AA }$. The inset shows the force-salt nonmonotonicity at the separation distance $R=25\mathrm{\AA }$.

Figure 5

(Color online) Reduced DNA-DNA interaction force $F∕F_0$ vs separation distance $R$ for divalent counterions and monovalent salt ions (parameter set 2 of Table ). (a) CM, (b) ECM, and (c) MAM. The notation is the same as in Fig. 4. The inset in (c) shows the force-salt nonmonotonicity at the separation distance $R=27\mathrm{\AA }$.

Figure 6

(Color online) (a) Reduced electrostatic force $F_2∕F_0$ and (b) entropic force $F_3∕F_0$ components of the total interaction force $F\left(R\right)$ from the Fig. 5a for the CM. The notation is the same as in Fig. 4.

Figure 7

(Color online) (a) Reduced electrostatic force $F_2∕F_0$ and (b) entropic force $F_3∕F_0$ components of the total interaction force $F\left(R\right)$ from Fig. 5c for the MAM. The ECM exhibits a similar trend. The notation is the same as in Fig. 4.

Figure 8

(Color online) Reduced DNA-DNA interaction force $F∕F_0$ vs separation distance $R$ for a monovalent salt and trivalent counterions (parameter set 3 of Table ). (a) CM, (b) ECM, (c) MAM. The notation is the same as in Fig. 4.

Figure 9

(Color online) Reduced DNA-DNA interaction force $F∕F_0$ vs separation distance $R$ for a divalent salt and monovalent counterions (parameter set 4 of Table ) for the MAM. The CM and ECM exhibit similar trends. The notation is the same as in Fig. 4.

Figure 10

(Color online) Reduced DNA-DNA interaction force $F∕F_0$ vs separation distance $R$ for a divalent salt and trivalent counterions (parameter set 5 of Table ) and for the MAM. The CM and ECM exhibit similar trends. The notation is the same as in Fig. 4. The inset shows the relation between the decay length $\lambda$ and the Debye screening length ${\lambda }_D$: line with triangles—for monovalent counterions and salt ions (parameter set 1 of Table ), line with squares—for current set of parameters.