Transcription and the Pitch Angle of DNA

The question of the value of the pitch angle of DNA is visited from the perspective of a geometrical analysis of transcription. It is suggested that for transcription to be possible, the pitch angle of B-DNA must be smaller than the angle of zero-twist. At the zero-twist angle the double helix is maximally rotated and its strain-twist coupling vanishes. A numerical estimate of the pitch angle for B-DNA based on differential geometry is compared with numbers obtained from existing empirical data. The crystallographic studies shows that the pitch angle is approximately 38◦, less than the corresponding zero-twist angle of 41.8◦, which is consistent with the suggested principle for transcription. In this paper, the restrictions inferred on the double-stranded DNA are studied in a geometrical analysis of the mechanics of transcription. Transcription of DNA is the biological process where the base-pair code within a gene is read for the purpose of protein expression. The precise mechanisms that control transcription are still to be fully explored, and represent a fundamental theme of research in molecular biology. In eucaryotes, one fundamental control point for the regulation of transcription is the structural adaptation of the chromatin fiber necessary to access the gene sequence. One scenario that has been proposed is based on the reversome, explaining how transcription elongation can proceed within condensed chromatin [1]. The topology of DNA in the chromosome has been further studied in models taking into account possible local modifications and global constraints [2, 3]. For reviews on transcription and transcriptional regulation in relation to epigenetic phenomena, see [4, 5]. It has early been understood that the topology of DNA is essential to understanding transcription, in prokaryotes, one suggestion for this is the possible supercoiling of DNA during transcription [6, 7]. The mechanics of DNA and its topological supercoiling has been described using elastic models [8, 9]. An elastic model has also been used to model DNA loops which play a role for the mechanics of transcription [10]; one suggestion is that transcription of a certain gene can be repressed or activated whenever a DNA loop is formed that contains that gene [11]. In the following, we use an entirely geometrical approach and model DNA as a double helix of two flexible tubes with fixed thickness. Under the presented assumptions, our main result is that for transcription to be possible, the pitch angle of the double helix must be constrained to a certain range, based on a differential length argument. Constraints that involve the degree of twisting can have two origins. One origin is of topological nature, such as conservation of the linking number [12]. Another origin is longer-range interactions that are due to the range of the involved forces in biological helices [13, 14]. An aspect of transcription involving the pitch angle is the various stresses related to twist and strain when the B-DNA reorganizes itself prior to transcription. Experimental studies using magnetic tweezers ∗DTU Nanotech, Technical University of Denmark, Building 345E, 2800 Kongens Lyngby, Denmark. E-mail: [email protected] †DTU Nanotech, Technical University of Denmark, Building 345E, 2800 Kongens Lyngby, Denmark. E-mail: [email protected] ar X iv :1 20 6. 09 24 v3 [ ph ys ic s. bi oph ] 1 M ar 2 01 3 and optical trapping suggest that DNA has a negative strain-twist coupling [15, 16]. Elongated DNA will therefore tend to rotate through a larger angle per set of base-pairs, i.e. to be winding-up. Recently, an analysis of the observed stick-slip melting of DNA under tension has revealed a zig-zag-like dynamics [17]. In Ref. [18], we have suggested a geometrical model that describes the phenomenon of winding-up of biological double helices as a generic phenomenon for all molecular double helices for which the pitch angle is below a specific value. In the following, we review the geometry used in the analysis and study its implication on the pitch angle. Later, we compare with the experimentally available data and determine the pitch angle of B-DNA. Twisting two strands together to form e.g. a geometry akin to the double-stranded DNA, the resulting double helical structure has a length, Lr(n), that depends on the number of twists, n [19]. When the strands are parallel to each other, the length of the double helix is the same as the length of the strands, and the double helix becomes shorter and shorter as turns are added. Only up to a maximum number of turns can be added, resulting ultimately in a geometry which is maximally rotated. In addition, this structure has a vanishing strain-twist coupling [18]. As rotations are removed from the helix, the length becomes further reduced. Therefore, Lr(n) is shaped like a horizontal hairpin, see Fig. 1. In the detailed calculation of Lr(n), the double helix is simplified to be consisting of two circular tubes of diameter D, whose center lines are coiled with pitch H on a common cylinder of radius a. The pitch angle, v⊥, measured from the horizontal is determined by tan v⊥ = H/2πa. For these idealized structures, the steric interactions of molecular DNA are described by letting the tubes be in contact with hard walls, i.e. the distance between the two helical lines should satisfy min(|~r1(t1)− ~r2(t2)|) = D , (1.1) where ~r1, ~r2 describe the two helical center lines. For transcription we consider DNA to be part of a structure that does not change on a large scale. Within this structure we consider an intermediate structure which is allowed to change while being maintained as a double helix. Finally, we consider a relatively short stretch of DNA that separates (melt) to single-stranded DNA. The first of these considerations require that the length of DNA during transcription, LDNA, to be greater than the length of DNA before transcription, LDNA, i.e. LDNA > LDNA . (1.2) Now, the twisting behavior of DNA depends on its pitch angle, v⊥. If the pitch angle is such that it would just unwind under strain, then Eq. (1.2) would not be fulfilled, as shown below. Instead the DNA would tend to get shorter during transcription. This is the origin of the constraint on v⊥ and shows that the behavior of B-DNA under strain must be opposite of that of a rope. Mathematically, the two helical center lines in Eq. (1.1) have the parametric equations, ~r1(t1) = (a cos t1, a sin t1, (H/2π)t1) , ~r2(t2) = (a cos t2, a sin t2, (H/2π)t2 + Ψ(H/2π)) , (1.3) with t1, t2 ∈ R. The parameter Ψ is a phase parameter that determines if the double helix is symmetric or asymmetric. For a symmetric double helix, e.g. A-DNA, Ψ is equal to π and for an asymmetric double helix, akin to B-DNA, we have Ψ = π144◦/ 180◦ = 2.513. The phase difference of 144◦ for the minor groove in B-DNA is described in Ref. [20]. With D being the tube diameter, the points of tube-tube contacts obey the equation