Free Energy Calculations for DNA Near Surfaces Using an Ellipsoidal Geometry

Abstract

The change in some thermodynamic quantities such as Gibbs' free energy, entropy and enthalpy of the binding of two DNA strands (forming a double helix), while one is tethered to a surface and are analytically calculated. These particles are submerged in an electrolytic solution; the ionic strength of the media allows the linearized version of the Poisson-Boltzmann equation (from the theory of the double layer interaction) to properly describe the interactions [13]. There is experimental and computational evidence that an ion penetrable ellipsoid is an adequate model for the single strand and the double helix [22-25]. The analytic solution provides simple calculations useful for DNA chip design. The predicted electrostatic effects suggest the feasibility of electronic control and detection of DNA hybridization in the fast growing area of DNA recognition.

Figure 1. Electric potential

Distance zero corresponds to the boundary of the ellipsoid and then it follows a path perpendicular to it. The black line is for ellipsoid a (sphere); the red lines are for ellipsoid b and the blue lines are for ellipsoid c. In ellipsoids b and c we have two lines, the upper one following a path perpendicular to the ellipsoid's axis and the lower one following a path parallel to the ellipsoid's axis.

Figure 2. Electric potential at the ellipsoid's boundary

We plot the potential at the boundary versus the prolate ellipsoidal coordinate η The black line corresponds to ellipsoid a which does not change because of its symmetry, the red line corresponds to ellipsoid to ellipsoid b and the blue line corresponds to ellipsoid c; as expected, the potential diminishes in all directions as we deviate from a spherical shape.

Figure 3

ΔG The black line corresponds to ellipsoid a, the red lines correspond to ellipsoid b and the blue lines to ellipsoid c. Ellipsoids b and c have two lines each, the lines above the black line correspond to the ellipsoid oriented with its axis parallel to the plane and the lines below when oriented perpendicular to the plane. It is worth noting that the two lines above, from ellipsoid b and ellipsoid c, almost match perfectly. The plane was kept with a charge density of .36 electrons/nm².

Figure 4

ΔG We use the same color code as in Fig. 3. The plane was kept at a constant electric potential of 25 mV.

Figure 5

ΔG We use the same color code as in Fig. 3. In the x axis we have the angle of inclination of the ellipsoid in radians, zero being when the ellipsoid's axis is perpendicular to the surface and π/2 when parallel.

Figure 6

ΔG, ΔH and -TΔS The black and the red lines are ΔG for ellipsoid b and ellipsoid c, the green and the blue lines are ΔH for ellipsoid b and c and the maroon and gray lines in the bottom are −TΔS.