Statistical mechanics of sequence-dependent circular DNA and its application for DNA cyclization

Abstract

DNA cyclization is potentially the most powerful approach for systematic quantitation of sequence-dependent DNA bending and flexibility. We extend the statistical mechanics of the homogeneous DNA circle to a model that considers discrete basepairs, thus allowing for inhomogeneity, and apply the model to analysis of DNA cyclization. The theory starts from an iterative search for the minimum energy configuration of circular DNA. Thermodynamic quantities such as the J factor, which is essentially the ratio of the partition functions of circular and linear forms, are evaluated by integrating the thermal fluctuations around the configuration under harmonic approximation. Accurate analytic expressions are obtained for equilibrium configurations of homogeneous circular DNA with and without bending anisotropy. J factors for both homogeneous and inhomogeneous DNA are evaluated. Effects of curvature, helical repeat, and bending and torsional flexibility in DNA cyclization are analyzed in detail, revealing that DNA cyclization can detect as little as one degree of curvature and a few percent change in flexibility. J factors calculated by our new approach are well consistent with Monte Carlo simulations, whereas the new theory has much greater efficiency in computations. Simulation of experimental results has been demonstrated.

FIGURE 1

Diagram illustrating the complex extension of k and change of the integration path from real axis to the indicated curve that goes through part of the imaginary axis at the saddle point k_c,j = Iλ_c,j.

FIGURE 2

A simplified model containing the two main characteristics of our cyclization model: a free harmonic Hamiltonian and a nonlinear constraint. This model can be used to test our harmonic approximation. If the force constant of the spring is large, the movement of the point mass is limited to the vicinity of the minimum energy point. Thus the nonlinear function can be accurately replaced by its Taylor expansion up to first or second order at the point.

FIGURE 3

Diagram showing the iterative procedure to calculate the equilibrium configuration and J factor. See the text in following section for the choices of constraints.

FIGURE 4

(Top) The intrinsic DNA helical path of a 156-bp construct containing straight B-DNA as test sequence used for DNA cyclization with its projection on the x–y plane (see the following section for details). In this calculation, parameters for the B-DNA part are chosen as follows: intrinsic twist angle, 34.45° and intrinsic tilt and roll angles, all zeros. Parameters for A-tract curvature are from Koo et al. (1990) and all length units in helical rise (3.4 Å) or basepair (bp). (Bottom) The starting configuration in the search for the equilibrium configuration by the iterative process. It is generated by putting 48.46° tilt kinks at every 21 basepairs based on its intrinsic shape shown in the top.

FIGURE 5

Changes of the maximal absolute differences in bending or twisting angles between two successive iteration steps and evolution of the J factor calculated from intermediate configurations with only the first derivative of the constraints incorporated. The initial configuration for this calculation is shown in Fig. 4 (bottom). The flexibility is bending fluctuation σ_b = 4.842° (P = 140 bp) and twist fluctuation σ_twist = 4.388° for both generic B-DNA and A-tracts.

FIGURE 6

The calculated equilibrium configurations of two topoisomers for the 162-bp DNA construct containing three repeats of the 10-bp nucleosome positioning sequences. The same DNA parameters as those in Table 3 in Roychoudhury et al. (2000) are used. The linking numbers (15 and 16), total helical turns (H_t) of circular DNA, and J factors are indicated, respectively.

FIGURE 7

The mechanical equilibrium angles for a 156-bp circularized DNA molecule with a 12-bp phasing length between the A-tract portion and a 30-bp test sequence. The test sequence has the B-DNA characters, i.e., zero roll and tilt, 4.68° bending flexibility, 34.45° twist, and 4.338° twisting flexibility, except for a 10-degree kink in the middle. The A-tracts have 4.842° bending flexibility and the same twisting flexibility.

FIGURE 8

(A) Variation of J factor as a function of phasing length. (B) An exponential dependence of the ratio J_max/J_min upon curvature. The curvature given in both (A) and (B) is the bending magnitude of each of three 10-bp test sequence motifs composing the whole test sequence, as is often used in cyclization experiments. (C) Effects of DNA flexibility in the total length assay. The unit for persistence length P is bp and the unit for twisting flexibility T is 10⁻¹⁹ erg × cm. (D) Change of helical repeat. The reference curve labeled by P = 150, T = 2.4 is the same as that in (C) with a helical repeat of 10.45 (or 34.45° twist). Note that in (C) and (D) the flexibility and helical repeat changes are only done for the 30-bp straight test sequence. Parameters not indicated are the same as those in Fig. 7 except the 10° roll.

FIGURE 9

Comparisons of the simulations from our new approach to the experimental cyclization data for the EcoR I site-containing sequence (Nathan and Crothers, 2002) and to the Monte Carlo simulation. The best-fit parameters are shown in Table 1. The Monte Carlo simulations are independently performed with our best-fit parameters.

FIGURE 10

Variations of the goodness-of-fit parameter with curvature (A) and bending flexibility (B) near their optimal values used to calculate their standard deviations. The curves are fitted with Eq. 49 for which parameters a′ and χ²₀ are −7.68 and 21.11 for (A) and 5.44 and 21.16 for (B), respectively, with the associated standard deviations shown in Table 1.

FIGURE 11

Basepair fluctuations (A) and correlations (B) calculated with Eqs. 36 and 37 for the construct with N = 156 and reference values of P = 150 and T = 2.4 in Fig. 8, C and D.

FIGURE 12

The average writhe 〈Wr〉 and writhe fluctuation for the constructs in the total length assay shown in Fig. 8 C with the reference values.

FIGURE 13

Variation of the total J factor vs. DNA length calculated from Eq. 58 (ZC) or Shimada and Yamakawa's empirical formula (SY) for homogeneous DNA (zero roll and tilt, 34.45° twist, 4.68° bending flexibility, and 4.388° twisting flexibility).