DNA-DNA interaction beyond the ground state

The electrostatic interaction potential between DNA duplexes in solution is a basis for the statistical mechanics of columnar DNA assemblies. It may also play an important role in recombination of homologous genes. We develop a theory of this interaction that includes thermal torsional fluctuations of DNA using field-theoretical methods and Monte Carlo simulations. The theory extends and rationalizes the earlier suggested variational approach which was developed in the context of a ground state theory of interaction of nonhomologous duplexes. It shows that the heuristic variational theory is equivalent to the Hartree self-consistent field approximation. By comparison of the Hartree approximation with an exact solution based on the QM analogy of path integrals, as well as Monte Carlo simulations, we show that this easily analytically-tractable approximation works very well in most cases. Thermal fluctuations do not remove the ability of DNA molecules to attract each other at favorable azimuthal conformations, neither do they wash out the possibility of electrostatic “snap-shot” recognition of homologous sequences, considered earlier on the basis of ground state calculations. At short distances DNA molecules undergo a “torsional alignment transition,” which is first order for nonhomologous DNA and weaker order for homologous sequences.

Figure 1

The charge distribution of a double helix (a) and the horizontal cross section of two identical double helices in parallel juxtaposition (b). The double helix consists of two spiraling negative phosphate strands with cations readsorbed in its minor and major grooves. The pitch $H$ of the helix for $B$-DNA is approximately $34\mathrm{\AA }$. The angle of the minor groove between the phosphate strands $2{\tilde{\varphi }}_s$ is about $0.8\pi$ for $B$-DNA. The parallel DNA are separated by an interaxial spacing $R$ and the relative azimuthal angle between them is $\varphi ={\varphi }_1-{\varphi }_2$.

Figure 2

The Feynman rules. Expanding $E\left[\varphi \right]$ [cf. Eq. (2.5)], each ${\phi }^n$ term, for $n>2$, may be represented by a diagram (or vertex). For instance the vertices in (a), (b), and (c) represent the coefficients that sit in front of the ${\phi }^3$, ${\phi }^4$, and ${\phi }^6$ terms, respectively, in $E\left[\varphi \right]$. The straight line in (d) represents the “bare” correlation function $G_0(z_i-z_j)$ (see the text). Each Feynman graph is then constructed from vertices and straight lines. For each vertex positioned at a point $k$, there must be an integration over $z_k$. There is also a symmetry factor which accounts for how many ways a term (graph) in the expansion may be generated. As an example we show one Feynman graph in (e). It contains one integration over $z_k$, three Gaussian correlation functions $G_0(z_i-z_k)$, $G_0(z_k-z_k)$, and $G_0(z_k-z_j)$ (see the text), the coefficient that sits in front of the ${\phi }^3$ term, and a symmetry factor. An analytical expression for (e) can be found in Appendix B of Ref. 26.

Figure 3

Diagrammatic expansion for the full correlation function $G(z-z^{\prime })$ in Hartree approximation.

Figure 4

Diagrammatic expansion for the free energy. Here, $F_0$ is the free energy calculated in the Gaussian approximation.

Figure 5

Schematic drawing showing how $V\left(\varphi \right)$ and ${\mid \psi \left(\varphi \right)\mid }^2$ vary with respect to ${\lambda }_p$ and $a_2$.

Figure 6

Free energy per base pair for the Hartree (dotted line) and QM (solid line) formulations at $T=300\mathrm{K}$ as a function of separation, $R$. The electrostatic coefficients, which vary with $R$, used for this plot are for helices with 90% counterion neutralization and a 30%/70% distribution of these ions between the minor and major grooves [14]. The torsional modulus is given the experimental value of $C=3.0\times {10}^{-19}\mathrm{erg}\mathrm{cm}$. (These values will be used throughout the paper.) As we see, the Hartree value for the free energy deviates from the one calculated within the QM formulation only near the transition (around $24–25\mathrm{\AA }$ for both temperatures).

Figure 7

Approximate and exact solutions for the azimuthal angle as a function of separation. ${\varphi }_{*}$ (dotted line) and ${\varphi }_{\max }$ (solid line) are shown for the Hartree and QM formulations, respectively. Whereas the Hartree formulation predicts a first order transition in the optimum phase angle, the most probable phase angle from the QM formulation continuously decreases to zero as $R$ increases.

Figure 8

Mean square amplitude of phase angle fluctuations for the Hartree case (dotted line) and for the QM formulation (solid line). The Hartree case shows a first order transition whereas there is no discontinuity for the QM formulation.

Figure 9

“Discrete spin model” of two interacting DNA duplexes. Each spin represents an individual base pair which is coupled to its neighboring base pairs along the same duplex by a spring with a torsional elasticity modulus $C$. Each spin also interacts electrostatically with the spin adjacent to it in the other duplex.

Figure 10

The free energy per base pair for the ground state of homologous sequences (solid line) and for the Hartree calculation at $T=300\mathrm{K}$ for homologous (dotted line) and nonhomologous sequences (dot-dashed line). Numerical results of the discrete spin model for homologous sequences at $T=300\mathrm{K}$ are shown as filled circles. For nonhomologous sequences (open squares), the data points were found by averaging over many different realizations of the random field $\delta \Omega$. The error bars (not shown for clarity’s sake) of the numerical simulations are found from averaging over many initial configurations for the homologous case and averaging over different realizations of the random field for nonhomologous cases.