Thermodynamics of Long Supercoiled Molecules: Insights from Highly Efficient Monte Carlo Simulations

Abstract

Supercoiled DNA polymer models for which the torsional energy depends on the total twist of molecules (Tw) are a priori well suited for thermodynamic analysis of long molecules. So far, nevertheless, the exact determination of Tw in these models has been based on a computation of the writhe of the molecules (Wr) by exploiting the conservation of the linking number, Lk=Tw+Wr, which reflects topological constraints coming from the helical nature of DNA. Because Wr is equal to the number of times the main axis of a DNA molecule winds around itself, current Monte Carlo algorithms have a quadratic time complexity, O(L(2)), with respect to the contour length (L) of the molecules. Here, we present an efficient method to compute Tw exactly, leading in principle to algorithms with a linear complexity, which in practice is O(L(1.2)). Specifically, we use a discrete wormlike chain that includes the explicit double-helix structure of DNA and where the linking number is conserved by continuously preventing the generation of twist between any two consecutive cylinders of the discretized chain. As an application, we show that long (up to 21 kbp) linear molecules stretched by mechanical forces akin to magnetic tweezers contain, in the buckling regime, multiple and branched plectonemes that often coexist with curls and helices, and whose length and number are in good agreement with experiments. By attaching the ends of the molecules to a reservoir of twists with which these can exchange helix turns, we also show how to compute the torques in these models. As an example, we report values that are in good agreement with experiments and that concern the longest molecules that have been studied so far (16 kbp).

A polymer model of bacterial supercoiled DNA including structural transitions of the double helix

“…Yet, following upon previous studies on single DNA forms [70] and on the statistics of DNA denaturation [59,60], it is possible to integrate out the torsional degrees of freedom (the φ i 's) under the constraint of a fixed Lk. This leads to an equivalent effective model that has the benefit to offer much better simulation performances [39,42] and further analytic treatment (see below). In the case of the 2sRLC model, the resulting conformational energy can be decomposed as in Eq.…”

“…DNA denaturation being expected to be the most likely alternative form at these forces and supercoiling densities [26,27], here we considered the situation where only B-DNA and D-DNA could form, with the following parameters leading to good agreement with experimental curves: a D = 0.45 nm, D = 15 nm, C D = 10 nm and γ D = 2k B T (see below for discussion of these values) -we intentionally used a D much larger than the 4 nm diameter of the chain because smaller values would lead, in any case, to larger effective persistence lengths [71]. We used J = 10k B T , in accord with previous analyses [69,26,52], and considered a discretization level of n = 10, leading to 3.3 cylinders per D and 15 cylinders per B (= 50 nm), a value that both offers reasonable computational times and ensures properties of B-DNA superstructuring to be insensitive to discretization procedures [42,66]. Finally, the quantitative reproduction of the buckling transition at σ > 0 (right part of the curves in Fig.…”

“…Along this line, the self-avoiding rod-like chain (sRLC) model [29] has been paramount to analyze folding properties of positively supercoiled molecules (see below for details of the model). These include molecular extensions [39], torques [40,41,42] and conformation details of superstructures [29,43,42] as measured by magnetic tweezers and cryo-electron microscopy, as well as dynamical properties (see [44] and references therein) and sequence-dependent phenomena [45,46].…”

Modeling Bacterial DNA: Simulation of Self-Avoiding Supercoiled Worm-Like Chains Including Structural Transitions of the Helix

“…Methods using Euler angles between local frames associated to the cylinders were first proposed, and shown to provide an excellent procedure to have a linking number that fluctuates around a fixed value [22]. More recent methods [23,24,8] based either on an explicit representation of the double helix [8], or on a definition of the twist angle coming from the parametrisation of the deformation of rigid bodies [23], allow to conserve the linking number exactly during the simulations. In particular, using the "parallel transport" approach developed in [23], it is possible to define a twist angle unambiguously such that after each block rotation of the cylinders (see below), the linking number remains constant.…”

“…Specifically, supercoiling constraints make molecules buckle so that they absorb, under the form of writhe, some of the excess or depletion of twist, while the short-range repulsion (self-avoidance) results in an effective entropic repulsive force that determines the radius of plectonemes [7]. Self-avoiding supercoiled WLC (ssWLC) models are thus able to capture the folding properties of supercoiled DNA on the scale of several kilo base pairs (kbp), or tens of kbps [8], both for overwound (positive supercoiling) and underwound (negative supercoiling) DNA at low tension forces. On the other hand, negative supercoiling can induce structural transitions towards DNA forms different from the canonical B-DNA.…”