Writhe formulas and antipodal points in plectonemic DNA configurations

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).

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“…2). Although simplifications have been proposed for folded proteins (31,32), they are not applicable to DNA plectonemes (33).…”

Section: Methodsmentioning

confidence: 99%

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

Lepage

Képès

Junier

2015

Biophysical Journal

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“…Now start from the case where l p can be extended to infinity such that θ * (l p ;n) = θ * (s;n) = 0. From condition (48) we have that C 1 = F − G * flu and in order to have this constant solution for a givenn, θ = θ = 0 for all s, which means that Equation (43) has only one solution for the helical angle, θ = θ c . For shorter l p (smallern) we must then have that C 1 ≥ F − G * flu to satisfy condition (54).…”

Section: Constant Pitch Solutionsmentioning

confidence: 98%

“…This can be done by setting θ = π/6 ([22] use θ = 0, this does not affect the solution by much) in U(r, θ) when using expressions corresponding to monovalent salts [21], or using electrostatic interactions in the presence of multivalent ions [27]. In any case, the differential equations (42) and (43) reduce to the form of Equations (2.41) and (2.42) analyzed by [16]. The differential equation for the helical angle θ is…”

Section: Variable Pitch -Analytical Solutions: Three Casesmentioning

confidence: 99%

“…The value of M 3 can be taken from experiment or simulations as a function of the applied number of turns n. Now the right-hand side of (52) is not equal to zero and (53) no longer holds. Next, we outline the procedure to solve the variational problem for a fixed force F. Given the prescribed external moment M 3 for a given n =n * a solution S[C * 1 ] for eachn * can be obtained as follows: • Assume C 1 (n) (or equivalently ζ (l p )) in (48) to be known and plug it into Equation (43). • Pick a value of r t (n) and solve the differential equation (42).…”

Section: Constant Pitch Solutionsmentioning

confidence: 99%

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DNA superhelical structures with non-constant helical pitch

Argudo

Purohit

2013

Mathematics and Mechanics of Solids

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We develop an elastic-isotropic rod model for superhelical DNA structures where the helical angle is varying as a function of the arc-length. Our motivation for a variable helical angle comes from some experiments and simulations on DNA braids where complex superhelical structures have been observed. The helical solutions are minimizers of a free energy consisting of elastic, entropic and electrostatic terms. These minimizers are obtained within a variational framework where the end-points of the helices are allowed to be variable so that the length of the superhelix is computed as part of the solution. Considering variable curvature solutions brings up the possibility of finding more complex DNA structures because for two (or more) interwound helices there is a geometrical lock-up helical angle which puts a limit on the length of a superhelix. We perform calculations with different ionic concentrations and study the effects of lock up for braided structures. We also extend the variable curvature model to study the formation of plectonemes in the presence of multivalent salts where the supercoiling radius can be regarded as a constant prescribed by the balance of attractive and repulsive forces in DNA-DNA interactions, and provide analytical solutions in terms of elliptic functions for the supercoil parameters.

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“…This definition of "Fuller writhe" is motivated by a Theorem of Fuller [11,10], which uses a reference curve and states precise conditions (which are frequently misunderstood [20]) under which the writhe difference between the two curves can be written as a local integral. Under deformations of the reference curve which are both south avoiding and self avoiding (we follow [20] in calling these "good deformations"), Eq. (7) implies that the equality W CW = W F is maintained.…”

Section: Geometrymentioning

confidence: 99%

Tops and Writhing DNA

Samuel

Sinha

2011

J Stat Phys

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The torsional elasticity of semiflexible polymers like DNA is of biological significance. A mathematical treatment of this problem was begun by Fuller using the relation between link, twist and writhe, but progress has been hindered by the non-local nature of the writhe. This stands in the way of an analytic statistical mechanical treatment, which takes into account thermal fluctuations, in computing the partition function. In this paper we use the well known analogy with the dynamics of tops to show that when subjected to stretch and twist, the polymer configurations which dominate the partition function admit a local writhe formulation in the spirit of Fuller and thus provide an underlying justification for the use of Fuller's "local writhe expression" which leads to considerable mathematical simplification in solving theoretical models of DNA and elucidating their predictions. Our result facilitates comparison of the theoretical models with single molecule micromanipulation experiments and computer simulations.

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Writhe formulas and antipodal points in plectonemic DNA configurations

Cited by 25 publications

References 44 publications

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

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

DNA superhelical structures with non-constant helical pitch

Tops and Writhing DNA

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