Stiffer double-stranded DNA in two-dimensional confinement due to bending anisotropy

Stiffer double-stranded DNA in two-dimensional confinement due to bending anisotropy

H. Salari Department of Physics, Sharif University of Technology, P.O. Box 11155-9161, Tehran, Iran.    B. Eslami-Mossallam Department of Bionanoscience, Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands.    H. F. Ranjbar Institute of Complex Systems (ICS-2), Forschungszentrum Jülich, Wilhelm-Johnen-Straße, 52425 Jülich, Germany.    M. R. Ejtehadi ejtehadi@sharif.edu Department of Physics, Sharif University of Technology, P.O. Box 11155-9161, Tehran, Iran. School of Nano Science, Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5531, Iran.
Abstract

Using analytical approach and Monte-Carlo (MC) simulations, we study the elastic behaviour of the intrinsically twisted elastic ribbons with bending anisotropy, such as double-stranded DNA (dsDNA), in two-dimensional (2D) confinement. We show that, due to the bending anisotropy, the persistence length of dsDNA in 2D conformations is always greater than 3D conformations. This result is in consistence with the measured values for DNA persistence length in 2D and 3D in equal biological conditions. We also show that in 2D, an anisotropic, intrinsically twisted polymer exhibits an implicit twist-bend coupling, which leads to the kink formations with a half helical turn periodicity along the bent polymer.

I Introduction

Buffer composition Method (nm) Ref.
10 mM , pH 7.0, DE111DNA Stretching Wang et al. (1997)
93.0 mM , pH 7.0, DE Baumann et al. (1997)
10 mM DE Smith et al. (1992)
200 mM , 10 mM Tris-HC1, pH 7.2, LS222Light Scattering Nordmeier (1992)
100 mM , LS Kam et al. (1981)
110 mM , pH 7.4, TPM333Tethered Particle Motion Brunet et al. (2015)
100 mM , FD444Flow Dichroism Rizzo and Schellman (1981)
101 mM , TED555Transient Electric Dichroism Porschke (1991)
0-162 mM , 1 mM , pH 7.8, DC666DNA Cyclization Taylor and Hagerman (1990)
89 mM Tris borate/2 mM EDTA, pH 8.3 DC 48.5 Geggier and Vologodskii (2010)
moderate salt buffer Cryo-EM777Cryo-electron microscopy Bednar et al. (1995)
Table 1: Some reported values of the persistence length of double-stranded DNA measured by various techniques in 3D conformations.

The bending flexibility of double-stranded DNA plays a crucial role in its interactions with other macromolecules, e.g., proteins. The most convenient measure of bending flexibility of a polymer is the “persistence length” (), which is defined as the correlation length of the tangent unit vector along the contour length Benetatos and Frey (2003). Many experimental and simulational techniques have been performed to measure this quantity of the DNA molecule Porschke (1986); Taylor and Hagerman (1990); Porschke (1991); Baumann et al. (1997); Wang et al. (1997); Lu et al. (2001); Brunet et al. (2015); Rivetti et al. (1996); Wiggins et al. (2006); Mazur and Maaloum (2014); Podestà et al. (2005); Moukhtar et al. (2007); Kundukad et al. (2014); Japaridze et al. (2015); Cassina et al. (2016), and characterize its dependence on the ionic strength Porschke (1991); Baumann et al. (1997); Podestà et al. (2005); Drozdetski et al. (2016), temperature Lu et al. (2001), sequence Geggier and Vologodskii (2010) and length scale Noy and Golestanian (2012); Fathizadeh et al. (2012); Salari et al. (2015). In the single molecule stretching experiment, Baumann et al. have shown that the persistence length of a random DNA sequence in moderate salt buffer is around  Baumann et al. (1997). Other bulk experiments, such as DNA-cyclization Taylor and Hagerman (1990); Geggier and Vologodskii (2010) and gel electro-phoretic mobility Rizzo and Schellman (1981); Porschke (1991), also result in a value in this range (Table 1).

On the other hand, single-molecule imaging techniques, including Atomic Force Microscopy (AFM) and Electron Microscopy (EM), introduce an important new class of experiments to measure the persistence length of DNA molecule. In these experiments the molecules are attracted onto the surface of a substrate by divalent counter-ions, e.g.  Rivetti et al. (1996); Pastré et al. (2003). These divalent ions allow the molecule freely equilibrate in 2D and decrease the effects of the substrate on the chain statistics Kundukad et al. (2014). It is also known that, a small amount of in solution can dramatically decrease the persistence length Baumann et al. (1997); Brunet et al. (2015). Baumann et al. show that, with of , even with a small amount of monovalent counter-ions (e.g. 1.86 mM ), the persistence length of DNA decreases to  Baumann et al. (1997). However, the DNA persistence length measured in 2D, ,  Rivetti et al. (1996); Wiggins et al. (2006); Mazur and Maaloum (2014); Podestà et al. (2005); Moukhtar et al. (2007); Kundukad et al. (2014); Japaridze et al. (2015); Cassina et al. (2016) is generally bigger than the 3D values, Porschke (1986); Taylor and Hagerman (1990); Porschke (1991); Baumann et al. (1997); Wang et al. (1997); Lu et al. (2001); Brunet et al. (2015) in the presence of divalent counter-ions (see fig 1). There are two experiments which do not address this discrepancy between 2D and 3D persistence lengths of dsDNA Abels et al. (2005); van Noort et al. (2004). In these works different buffers were used in the 2D and 3D experiments, where is only present in the 2D experiment buffer. Thus, they are not shown in fig 1. According to fig 1, the average value of is about and remains almost constant with increasing the ionic strengths in contrast to , which drops slowly.

Figure 1: Comparison between the reported values of (red dots) and (blue dots) for different ionic strength in the presence of divalent counter-ions. The ionic strength is defined as Ref. Brunet et al. (2015), and different markers correspond to different references as indicated in the legend.

This visible difference may arise from different effects of the divalent counter-ions within the experiments, i.e. measuring the persistence length in 2D and 3D conformations. It is known that in 2D, the divalent counter-ions bridge the negative charges of the phosphate backbone to the negatively charged mica surface Rivetti et al. (1996); Abels et al. (2005), while they act as intramolecular bridges between two phosphates in 3D conformations Mantelli et al. (2011). The bridges in the latter can greatly reduce the entropy of the chain and lead to low persistence lengths. In addition, the excluded volume interactions in 2D conformations can swell the molecule and therefore increase the persistence length Rivetti et al. (1996); Hsu and Binder (2012); Drube et al. (2010). Rivetti et al. show that, although these interactions can increase the 2D persistence length, but this effect is negligible when DNA length is less than (Rivetti et al. (1996). But the molecules in 2D remain stiffer, even for lengths shorter than 3400 bp Rivetti et al. (1996); Wiggins et al. (2006); Podestà et al. (2005); Moukhtar et al. (2007); Kundukad et al. (2014); Japaridze et al. (2015). Moreover, transition from B-DNA to A-DNA during the imaging in the dry air can also cause DNA stiffer Rivetti and Codeluppi (2001); Hansma et al. (1996); Japaridze et al. (2016); Waters et al. (2016). To avoid this structural transition, DNA molecules in fig 1 were scanned in solution and biological conditions. Also, the errors in contour length estimation can affect the measured values for the persistence length. Underestimating by leads to about overestimation of for a 100 bp DNA Wang and Milstein (2015). But, this effect is reduced by increasing the contour length, where the overestimation decreases to for a 500 bp DNA Wang and Milstein (2015). Finally, it has been also mentioned that the surface charges can affect the flexibility of DNA Podestà et al. (2005). Apart from above possibilities here we show that the anisotropic bending tendency of double stranded DNA increases the stiffness of the molecule in 2D.

The anisotropic bending of double-stranded DNA is a property of the sugar-phosphate backbone structure, in the sense that bending toward the grooves direction (roll) is much easier than toward the backbone direction (tilt) Ulyanov and Zhurkin (1984). Fourier analysis of free energy of DNA loops with lengths between bp shows two main oscillatory components Saiz et al. (2005). One with a helical period ( bp) and another with a half helical period ( bp) which may reflect the bending anisotropy. The sequence-dependent bending anisotropy of B-DNA has been observed in X-ray crystallography of DNA-protein complexes Olson et al. (1998); Richmond and Davey (2003) and NMR spectroscopy Lankaš et al. (2010), as well as all-atomistic simulations Teng and Hwang (2015); Lankas et al. (2000a); Bishop (2005); Perez et al. (2008); Lavery et al. (2009); Lankas et al. (2006); Dršata and Lankaš (2015); Ma and van der Vaart (2016). Many theoretical studies have considered such anisotropic bending into the elastic models Zhurkin et al. (1979); Balaeff et al. (1999); Norouzi et al. (2008a); Eslami-Mossallam and Ejtehadi (2008); Alim and Frey (2007); Cheng et al. (2016), and it is shown that although the bending anisotropy affect the elastic properties of a short DNA molecule in 3D, it becomes unimportant when the DNA segment is long enough to include a few full helical turns Eslami-Mossallam and Ejtehadi (2008); Becker and Everaers (2007).

Here, we exploit methods from the statistical field theory as well as Monte-Carlo simulation technique to study the elastic properties of an intrinsically twisted ribbon with anisotropic bending in 2D. We show that it is possible to assign an effective persistence length to a long DNA molecule in 2D, similar to the 3D case. Whereas the isotropic bending model predicts equal persistence lengths in 2D and 3D, we show that due to the anisotropic bending the 2D persistence length is always bigger than the one in 3D. The difference between 2D and 3D persistence lengths depends on the relative strengths of the bending elastic constants (the strength of anisotropy) and also twist rigidity, while the latter implies an implicit twist-bend coupling in the model. The prediction of our model for the DNA persistence length in 2D is in good agreement with the experimental data, shown in fig 1. Our finding can be relevant to other anisotropic chain polymers, e.g. double-stranded RNA or carbon nano-ribbons Kosmrlj and Nelson (2015).

Ii Model and Materials

ii.1 The Planar Anisotropic Elastic Rod Model

Double-stranded DNA is a helical nano-ribbon polymer which is represented as an anisotropic elastic rod. As fig 2 shows, at each point of arc length parameter on the centerline, , one can attach an orthonormal basis , so called a “material frame”. There is two usual definitions for  Fathizadeh et al. (2012), but in the simplest way it can be chosen to be along the tangent to the rod at every point so that  Heussinger et al. (2007). is along the grooves direction and points toward the major groove, and . Here we assume, bending about the -axis is easer than the -axis, therefore, the - and -axes correspond to the hard and soft bending directions (see fig 2).

Figure 2: Parametrization of the elastic rod.

The derivatives of the orthonormal triads with respect to are defined as the following

(1)

where is called the strain vector. The components of (i.e. , and ) respectively correspond to rotation of the filament around , , and and called tilt, roll and twist Salari et al. (2015). Therefore, the elastic energy of an inextensible, unshearable and anisotropic filament in harmonic approximation can be written as Eslami-Mossallam and Ejtehadi (2008); Norouzi et al. (2008b)

(2)

where and are bending rigidities respectively for the hard and soft directions, is the twist rigidity and is intrinsic twist of B-DNA. In the elastic energy of eq. (2), we have ignored the explicit twist-bend coupling Marko and Sigga (1994); Liebl et al. (2015); Nomidis et al. (2016).

For a planar DNA (confined in the - plane where ), it is convenient to express the local triads in terms of Euler angles (, , ) as the following Eslami-Mossallam and Ejtehadi (2008):

(3)

where and are respectively correspond to the local bend and twist angles. By substituting the eq. (II.1) into eq. (1) one can obtain the components of ,

(4)

and the local curvature is .

Experimentally, the persistence length of DNA in 2D conformation can be determined by measuring various statistical properties, such as the orientational correlation function Faas et al. (2009); Abels et al. (2005), the probability distribution of the bending angle Wiggins et al. (2006); Mazur and Maaloum (2014), the mean-square end-to-end distance Rivetti et al. (1996); Pastré et al. (2003); Moukhtar et al. (2007); Mantelli et al. (2011), and force-extension Maier et al. (2007). The isotropic wormlike chain (WLC) model, i.e. , has been widely used to fit the experimental data and obtain the persistence length. Below we derive each of these statistical properties for an anisotropic elastic model, where .

For a chain with length and global bend angle () in 2D the free energy is given by the canonical relation: Curuksu et al. (2008); Wiggins et al. (2005)

(5)

where is the normalised probability distribution of for 2D conformation. This probability distribution can be written as a path integral

(6)

where is normalization constant and

(7)

is the density of elastic energy. In the case of the isotropic WLC model, follows a Gaussian distribution as , but in general, due to the first term of the right hand side of eq. (7), it is not easy to find an expression for . Using the probability distribution (6), the tangent-tangent correlation function, , as a function of is defined as Rivetti et al. (1996)

(8)

where, we are supposed . It can be easily shown that for the isotropic case we have . Finally, the mean-squared end-to-end distance of the chain for short lengths, where the excluded volume interactions are negligible Hsu and Binder (2012), is written by

(9)

where is the angle between tangent vectors at points and along the contour. The above equation for the isotropic model can be directly determined by substituting and straightforward integration as Zoli (2016)

(10)

In case of an anisotropic elastic model the first term of the right hand side of eq. (7) implies an implicit twist-bending coupling for the model. In order to find the effects of this coupling, we evaluate in two extreme limits: the large and small twist rigidity (i.e. and , respectively).

For large twist rigidity, the relative variations of is negligible and which gives (it is assumed the initial twist angle is zero). Therefore, the density of energy (7) reduces to

(11)

where . By substituting this energy density into Eq. (6) and replacing the Dirac delta function by the appropriate Fourier transform, one can find the probability distribution of , as follows

Then, upon a straightforward integration, this equation yields a Gaussian distribution

(13)

with

where . The bracket means “integer part” which is added to get rid of discontinuity in the function. Fig 3 indicates the length dependent of for two different values of . It shows that, by increasing the chain length approaches soon to its asymptotic value , then

(15)

which implies that at large twist rigidity limit, a long enough 2D anisotropic DNA behaves like an isotropic DNA with the bending constant .

Figure 3: The ratio of (eq. (II.1)) as a function of length for two different values of .

On the other hand, in the limiting case of the small twist rigidity, there is no constrain on the twist degree of freedom, then the local twist angle, , is free to choose any value in the range of . Therefore, the energy density of (7) in this limit is rewritten as

(16)

Substituting this equation into eq. (6), one can find , in this limit, as follows

(17)

where

(18)

We numerically solve this equation for any given and in the rest of this work.

From equations (13) and (II.1), it can be deduced that the anisotropic elastic model in 2D behaves like a isotropic model with an effective persistence length, , which is a function of , , , and . As varies by the strength of twist rigidity, the upper and lower limits of are given by (eq. (II.1)) and (eq. (18), respectively. We perform Monte-Carlo (MC) simulations to evaluate between these two extreme limits.

ii.2 Monte-Carlo Simulations

To calculate the statistical properties of the chain, we performed Monte-Carlo (MC) simulations of a discrete elastic model from eq. (2). Here, the chain consists of beads were connected to adjacent beads via a link length , and without excluded volume interactions (a phantom chain). The Metropolis algorithm with appropriate Boltzmann distributions was used to construct equilibrium configurations of the chain. Our simulations were done with a linear chain containing 600 beads in 2D and 3D conformations. To estimate the statistical errors, we performed several realizations with different initial conditions.

Figure 4: The MC results for the tangent-tangent correlation function in 3D (A) and 2D (B) for , with a harmonic mean of (dashed lines in the insets). The data points corresponding to (open, blue squares), (open, green circles), and (open, red triangles), and the straight lines indicate liear fits to data. The insets show the masuared persistence lengths from the slope of the fitted lines (see text). The solid lines, in the inset of panel (B), correspond to the upper () and lower () limits of versus (the equations (15) and (18), respectively).

Iii results and Discussion

iii.1 The elastic properties of the long anisotropic chain

Similar to experiments Rivetti et al. (1996); Faas et al. (2009); Kundukad et al. (2014), we used the tangent-tangent correlation function as well as the mean square of end-to-end distance to extract the effective persistence length of a long chain from the MC simulations. To address the effects of 2D confinement on the flexibility of chain, we compare the effective persistence length in both 2D and 3D conformations.

Figure 5: The dependence of on for different values of and , as indicated. Solid lines correspond to the upper and lower limits of (i.e. and , respectively), and dashed lines are with slope equal to 1. Error bars (not shown) are about the size of the markers.
Figure 6: A) Dependence of on for different values of in the range (from dark blue to dark red). The solid curves correspond to theoretical predictions for (when ) and (when ). B) Dependence of on for different values of . The horizental solid lines indicate the value of for the corresponding , and the region between the vertical dashed (gray) lines indicates the rigion with strong twist-bend coupling regime. Error bars (not shown) are about the size of the markers.

It is well known that, the tangent-tangent correlation function of a free, long and highly twisted anisotropic model in 3D decays as , where is given by the harmonic mean of the hard and soft bending rigidities, i.e. and Eslami-Mossallam and Ejtehadi (2008); Lankas et al. (2000a); Schurr (1985)

(19)

Figure 4(A) compares the MC results of the 3D correlation function, , for the chains with , (i.e. ) and three different values of , i.e. (open blue squares), (open green circles), and (open red triangles). is determined by the slope of the best fitted lines to (the solid lines). As can be seen from the inset of fig 4(A), is independent on and equal to the harmonic mean of and (i.e. , dashed line). Fig 4(B) shows the correlation function in 2D, , here a clear dependence on is evident. We found that is always greater than (dashed line in the inset of fig 4(B)) and varies from for to for (the solid lines in the inset of fig 4(B)). Since the zero-energy configuration of a curved anisotropic chain is not planar Norouzi et al. (2008a), then it takes energy to enforce the chain in 2D confinement. This extra energy makes the chain stiffer in 2D than 3D.

All sets of and which results to the same but different values are given by

(20)

where . Therefore, it is convenient to consider as a function of , , and . Fig 5 shows a fairly linear relationship between and for different values of and . We therefore expect that the ratio is independent of . Fig 6 shows the dependence of this ratio on and . As fig 6(A) indicates, is always greater than when (i.e. anisotropic bending), and they are equal at (i.e. isotropic bending). It can be seen that as well as its lower and upper limits and increases with . The twist-bend coupling which is reflected in the dependence of on , also becomes stronger with increasing . As fig 6(B) shows, strongly depends on the twist rigidity, , in the range of (the region between two vertical dashed line), and beyond that is approximately constant and given by (eq. (15)) and (eq. (18)) for and , respectively.

Figure 7: The mean-square of end-to-end distance versus the chain length . The data points correspond to the MC simulations of chains with , and different values of (same as fig (4)) and the solid curves are the predictions. Error bars (not shown) are about the size of the markers.

Another way to calculate the persistence length is based on the mean-square of end-to-end distance . Fig 7 shows the MC results of lie perfectly on the predictions of eq. (10), where is substituting from fig 4(B). It means that the persistence length which are extracted from the mean-square of end-to-end distance are so close to those calculated from the tangent-tangent correlation function, and the relative error between these two is less than .

iii.2 Stretching anisotropic chain in 2D

Figure 8: Semi-logarithmic plot of the relative extension, , versus the scaled force, , for the anisotropic chains with , , , and different values of (as indicated) in 2D. for each set of data was extracted from fig (6(B)). Solid curve corresponds to the theoretical prediction of eq. (21).

We also studied the entropic stretching of the anisotropic chain in response of an external force in 2 dimensions using MC simulations. The external potential is added to the the elastic energy (eq. (2)), where is the magnitude of the external force which is exerted on the last bead of the chain, and is the component of the end-to-end vector in the direction of the external force. It is known that the force versus extension curve of an isotropic chain (i.e. ) in 2D is given by Prasad and Hori (2005)

(21)

where is the average extension along the force direction. Fig (8) shows the 2D force-extension curve for the chains with , , , and different values of , i.e. 1 (open blue circles), 3 (open green triangles), 6 (open red squares), and 10 (open black triangles). As it can be seen, each set of force-extension data perfectly lies on the theoretical prediction of eq. (21) (solid curve), when the external force is scaled by its corresponding (extracted from fig 6(B)). This shows the force-extension characteristic of an anisotropic chain is same as an isotropic chain with the appropriate persistence length of .

iii.3 The elasticity of the anisotropic chain at small length-scales

Figure 9: The probability distribution for the angle between tangents of two points separated by a contour length of . The data points correspond to MC simulations of the chains with , , and three different values of , as indicated. Solid curves are the best parabolic fits to the data. Inset: Semi-logarithmic plot of versus . The solid (black) lines correspond to the upper (i.e. ) and lower (i.e. ) limits of , and dashed line indicates .

To investigate the flexibility of the anisotropic model at short length-scales, we computed the negative logarithm of the probability distribution of bending angle, . Fig 9 shows the result for chain with , , and , , and . The effective persistence length of the chain at this length can be extracted by fitting a parabola of the form to the data (see the inset of fig 9), and is in good agreement with our previous result (see fig 4(B)).

Figure 10: The length dependent of the ratio of for , , and different values of , as indicated. Dashed curves serve as guides for eyes.
Figure 11: Right: The normalized curvature, (with ), of the averaged configuration of a two-dimensional anisotropic loop at room temperature. The data correspond to , , , and different values of , as indicated. Left: Fourier spectra of the curvature profile. The arrows indicate the two main periodic components of the curvature.

Due to the intrinsic helicity and bending anisotropy of the DNA molecule, we expect that the effective persistence length at small length scales oscillates with a period of . Using we calculate for the segment lengths between to . Fig 10 compares the ratio for and different value of , , , , and . It can be seen that oscillates with a period of about and decays to its extreme value of by increasing the length. This oscillation is amplified if the strength of bending anisotropy, , increases.

The oscillations are due to the formation of kinks with periodic arrangement in two-dimensional ground-state conformation of a bent and twisted anisotropic chain Mohammad-Rafiee and Golestanian (2005); Bijani et al. (2006). We performed MC simulations for a loop with , , , and different values of , , , and . As the right column of fig 11 shows, the curvature along the loop is not uniform and it is localized with a periodic arrangement (which leads to the kink formation). The Fourier spectrum of the curvature reveals two main periodic components, with helical () and half helical () periods (the two arrows in fig 11). This half helical-pitch periodicity is a result of the anisotropic model and vanishes at isotropic model (i.e. ). The amplitude of this component increases by increasing the strength of anisotropy, .

iii.4 Estimation of the anisotropic bending of dsDNA

Sequence dependence and bending anisotropy of dsDNA has been widely noticed in base-pair steps approaches, by partitioning the DNA deformation energy through six local variables, slide, shift, rise, tilt, roll and twist B. Mergell, M. R. Ejtehadi and Everaers (2003). The rigidity parameters corresponding to these six variables are extracted from their standard deviation Olson et al. (1998). Therefore, the ratio of bending rigidities, , can be determined by , where and denote the standard deviations of bending angles in soft (roll) and hard (tilt) directions, respectively Olson et al. (1993). A survey of the values of obtained by different techniques is presented in Table 2. Despite the diversity, the value of lies in the range of .

Investigators Method Ref.
Zhurkin et al. (1991) MC888Monte-Carlo simulations Zhurkin et al. (1991)
Olson et al. (1998) XRC999X-ray crystallography of protein-DNA complexes Olson et al. (1998)
El Hassan & Calladine (1997) XRC El Hassan and Calladine (1997)
Richmond et al (2003) XRC Richmond and Davey (2003)
Chua et al. (2012) XRC Chua et al. (2012)
Stefl et al. (2004) NMR101010Nuclear magnetic resonance spectroscopy Stefl et al. (2004)
Dornberger et al. (1998) NMR Dornberger et al. (1998)
Lankas et al. (2000) MD111111All-atom Molecular Dynamic simulations Lankas et al. (2000b)
Lankas et al. (2003) MD Lankas et al. (2003)
Lankas et al. (2009) MD Lankas et al. (2009)
Lankas et al. (2010) MD & NMR Lankaš et al. (2010)
Bishop (2005) MD Bishop (2005)
Lavery et al. (2009) MD Lavery et al. (2009)
Perez et al. (2005) MD Pérez et al. (2005)
Perez et al. (2008) MD Perez et al. (2008)
Becker & Everaers (2007) MD Becker and Everaers (2007)
Teng & Hwang (2015) MD Teng and Hwang (2015)
Balasubramanian et al. (2009) NAD121212Nucleic Acid Database Berman et al. (1992) Balasubramanian et al. (2009)
Table 2: Some reported values of measured by different techniques.

For B-DNA the twist rigidity is in the range of  Neukirch (2004); Lipfert et al. (2010). Therefore, to estimate the anisotropy strength, , we can use the relation of , which is valid for . Assuming the average values for and are given by and , respectively (see fig 1), we obtain , which falls in the expected range (see Table 2).

Iv conclusions

In summery, we have shown that enforcing the chain into a two dimensional conformation increases its stiffness. Our analytical approach and MC simulations showed that due to a twist-bend coupling in the 2D anisotropic model, the effective persistence length depends on the twist rigidity, and reaches soon to its maximum value when . In this limit, the 2D persistence length is given by the geometric mean of the hard and soft bending rigidities, instead of the harmonic mean in 3D. In addition, we show that the twist-bend coupling leads to the formation of kinks along a curved chain as previously predicted using the energy minimization treatment Mohammad-Rafiee and Golestanian (2005); Bijani et al. (2006).

We estimated the bending anisotropy of dsDNA, and it turns out that the hard bending rigidity is almost 4 times larger than the soft bending rigidity. this is compatible with the estimates in the literature, although we expect that this value is sensitive to the experimental conditions. Our analytical procedure can be used as a way to estimate the bending rigidities of other anisotropic bending polymers, such as nano-ribbons and dsRNA.

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