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DNA Elasticity From Short DNA to Nucleosomal DNA Ashok Garai, Suman Saurabh, Yves Lansac, and Prabal Kumar Maiti J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.5b03006 • Publication Date (Web): 02 Jul 2015 Downloaded from http://pubs.acs.org on July 7, 2015

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The Journal of Physical Chemistry

DNA Elasticity from Short DNA to Nucleosomal DNA Ashok Garai,† Suman Saurabh,† Yves Lansac,‡ and Prabal K. Maiti∗,† Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, ˇ Bangalore-560012, India, and Laboratoire dElectrodynamique des Matériaux Avancés (LEMA), Université Fran¸cois Rabelais-CNRS-CEA, UMR 6157, 37200, Tours E-mail: [email protected] Phone: +91 80 2293 2865/ 2360 7301. Fax: +91 80 2360 2602/0228

Abstract Active biological processes like transcription, replication, recombination, DNA repair and DNA packaging encounter bent DNA. Machineries associated with these processes interact with the DNA at short length (< 100 base pair) scale. Thus the study of elasticity of DNA at such length scale is very important. We use fully atomistic molecular dynamics (MD) simulations along with various theoretical methods to determine elastic properties of dsDNA of different lengths and base sequences. We also study DNA elasticity in Nucleosome Core Particle (NCP) both in presence and absence of salt. We determine stretch modulus and persistence length of short dsDNA and nucleosomal DNA from contour length distribution and bend angle distribution, respectively. ∗

To whom correspondence should be addressed Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore560012, India ‡ ˇ Laboratoire dElectrodynamique des Matériaux Avancés (LEMA), Université Fran¸cois Rabelais-CNRSCEA, UMR 6157, 37200, Tours †

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For short dsDNA, we find that stretch modulus increases with ionic strength while persistence length decreases. Calculated values of stretch modulus and persistence length for DNA are in quantitative agreement with available experimental data. The trend is opposite for NCP DNA. We find that the presence of histone core makes the DNA stiffer and thus making the persistence length 3 − 4 times higher than the bare DNA. Similarly, we also find an increase in the stretch modulus for the NCP DNA. Our study for the first time reports the elastic properties of DNA when it is wrapped around the histone core in NCP. We further show that WLC model is inadequate to describe DNA elasticity at short length scale. Our results provide a deeper understanding of DNA mechanics and the methods are applicable to most protein-DNA complexes.

Introduction A micrometer sized human cell nucleus houses dsDNA, which is nearly 1.8 m in length, forcing the DNA to bend and stretch sharply at a length scale of 10 base pairs (bp). 1,2 The bending of a charged bio-polymer like DNA is facilitated by oppositely charged histone proteins that form a spherical core on which DNA is wrapped forming a superhelix. 3–8 A complex of 147 bp long dsDNA wound over a core formed by histone proteins is known as a nucleosome core particle (NCP). 3,9 The nucleosomal DNA is subjected to base-pair level elastic manipulations in processes like transcription. 10,11 To understand these cellular processes, and the sharp elastic distortions that the DNA has to undergo therein 9,12–16 , we need to systematically evaluate the elastic properties related to DNA stretching and bending at a 10 bp length scale. Intra-strand and inter-strand interactions are believed to be responsible for the DNA elasticity. However, a clear understanding about the origin of DNA stiffness is still lacking. 17 Apart from that, we also need to have a good understanding as to how the elastic properties of DNA change when it goes from a free state to a state where it forms a complex with proteins. 9,13,14,18 DNA binding to nucleosomes relate to its flexibility and may cause several degenerative disorders such as myotonic dystrophy, fragile x syndrome, and Huntington 2 ACS Paragon Plus Environment

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disease with an increase in its flexibility. 19–21 The elastic properties of long size DNAs are well described by theoretical model like the Wormlike Chain Model (WLC), 22–24 which assumes a harmonic form for deformation energies. This model predicts a persistence length of ∼ 50 nm for dsDNA. 25–27 Recent experimental observations show that while it describes the elastic properties of long DNA very well, the WLC model fails drastically for short DNA fragments. 2,28–40 Wiggins et. al. 30 performed atomic force microscopy (AFM) measurements on short DNA fragments and found a large angle bending probability; almost 30 times more than what was predicted by the WLC model. They proposed a linear energy function for bending of short DNA as opposed to the quadratic function assumed by the WLC model. This model is known as the linear sub-elastic chain model (LSEC). Yuan et. al., 2 determined persistence lengths of DNA fragments of length ranging from 15 to 89 bp, using end-to-end length (R) distribution and radius of gyration (RG ) obtained by fluorescence resonance energy transfer (FRET) and small angel x-ray scattering (SAXS), respectively. They found a persistence length of around 20 nm, which is much lower than what is predicted by the WLC model. In another SAXS study, Mathew-Fenn et. al. 40 demonstrated the cooperativity in DNA stretching below a length of two helical turns and demonstrated that DNA is much softer than predicted by elastic rod model. In a sequel to this work Becker and Everaers 41 pointed out that the observed cooperativity was a result of experimental error and elastic rod model is appropriate for the description of DNA stretching as it is non-cooperative. Mazur, using MD simulations of 25bp DNA, found that the WLC model is applicable down to one helical turn of DNA and through an extensive comparison concluded that the WLC model agrees far better than LSEC model with the DNA bending trends obtained from MD simulations. 31 These results are in disagreement with the findings of Wiggins et al. 30 Recently, Mazur and Maaloum 38,39 performed AFM experiment in solution to study the flexibility of DNA at shorter length scales down to one helical turn. They observed that beyond 7 nm (two helical turns) filexibility of DNA can be well captured by the WLC model. Apart from these, many DNA-cyclization experiments show that the tendency of



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DNA looping is far larger than that predicted by the WLC model. 28,29,33 In light of all these experiments, the accuracy of predictions made by the WLC model, for short DNA fragments, becomes questionable. Formation of bubbles and kinks due to thermal fluctuations can also influence DNA elastic properties at short length scales. Ranjith et al. reported distribution functions, loop formation probabilities in presence of equilibrium bubbles for short DNA. 42 In another study, using the generalized discrete WLC model Ranjith et al. 43 showed that the bending of such short DNA, where fluctuations are very important, may arise from thermally induced local defects. Apart from widely used WLC model, normal mode analysis 44 as well as conformational energy analysis 45 have been used to determine elastic properties of DNA. Recently, Noy and Golestanian, 34 using MD simulations, essential dynamics analysis and other theoretical approaches, showed that end effects are predominant in the length scale dependence of DNA stretch modulus. They demonstrated that the value of a measured elastic property would depend on how it is defined. Thus providing a better way of interpreting experimental results. Bomble and Case studied the elastic properties of linear (60 − 150 bps) and circular ds-DNA (94 bps) using elastic rod theory along with normal mode analysis in implicit solvent and calculated the effects of salt concentration as well as DNA sequence on its flexibility. 44 They observed a reasonable agreement with the experimental results. Their results help validate the use of all atom force fields in describing DNA at long length scales. However, they did not study elastic properties of the shorter DNA below 60 bp. Ruscio and Onufriev performed MD simulation in implicit solvent to study nucleosomal DNA flexibility. 46 They calculated the free energy difference between conformations with different degrees of bending and concluded that cost for bending of NCP DNA is smaller than that predicted from the macroscopic elastic theory. Their observation is in qualitative agreement with the cyclization experiments. 29 These studies, in addition to being detailed, were able to produce results that match experiments. Earlier we have reported stretch modulus for short ds-DNA as well as various DNA nano-structures using equilibrium simulations as well as steered molecular dynamics. 47–51 In the present study, we use all atom MD simulations


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in explicit solvent to calculate various elastic properties, like the stretch modulus, bending modulus and persistence length for short ds-DNAs of different lengths ranging from 12 to 56 bp at different salt concentrations. We use different theoretical formulations and compare and contrast their results. In addition to calculating the elastic properties of free DNA, we also perform calculations for nucleosomal DNA with an aim to establish, in detail, the difference between free DNA and DNA in complex with proteins with regard to their elastic properties. Such a study is very important because of the fact that DNA-protein complexes are biologically ubiquitous and the behavior of a DNA molecule in presence of a protein can be very different from its free-state behavior. Hence, a deeper understanding of a process like transcription, demands a deeper understanding of DNA elastic properties in presence of binding proteins. At each stage of our calculation, we compare our results with the predictions of elastic rod model and establish that the model is not applicable to short DNA fragments with a length scale of importance in various cellular processes. The organization of the paper is as follows: In the method section first we give details of our all atom MD simulation. Then we discuss different methodologies used for calculating elastic properties of ds-DNA. Using these methodologies we next calculate persistence length, bending modulus, and stretch modulus of various short ds-DNA and NCP DNA. We report them in results and discussion section. Then we focus on their salt dependence. We provide all these details in the results section. We finally summarize and conclude our results and observations in the conclusion section.

Methods Four short B-DNAs of finite length, 12, 24, 38 and 56 bps (see table 1 for base sequences) were built using the NAB module of AMBER12. 52,53 Fig. 1(a) shows a 24 bp double stranded (ds) B-DNA used in the simulation. ff99bsc0 55 with parmbsc0 correction 56 was used to describe inter and intra molecular interaction involving DNA molecules. Henceforth, this combina-



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(a)

(b)

Figure 1: (color online) (a) 24 bp ds B-DNA constructed using NAB (b) NCP crystal structure corresponding to PDB id 1KX5 54 with 147 bp ds-DNA (blue) wound over histone protein core (brown).


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tion of force fields will be referred to as ff99bsc0. This force-field for B-DNA has been shown to produce stable DNA conformation over microsecond long MD simulations. 57 The DNA molecules were solvated in TIP3P model of water 58 using xleap, with a minimum water layer of 10 ˚ A in all three directions. For the NCP simulation following protocol was used : NCP crystal structure corresponding to PDB ID 1KX5 54 was used as the starting structure (see Fig. 1(b)). ff99bsc0 with parmbsc0 correction was used to describe the DNA and ff99SB describing the interactions involving protein. For the NCP simulation without histone tails, the H3, H4, H2A, and H2B N-terminals were clipped at ARG-40, ASN-25, PRO-26 and TYR-34, respectively. The NCP was solvated with a water layer of 20 ˚ A in all the three directions. The solvated systems were then neutralized by adding an appropriate number of Na+ ions, which were described using Joung and Chetham parameter set. 59 For example, 144 Na+ ions were added to neutralize NCP. For the 150mM system 408 Na+ and Cl− ions were added while 601 Na+ and Cl− ions were added to attain a salt concentration of 250mM. For the NCP simulation without histone tails, 228 Na+ were added for neutralization, while 240 Na+ and Cl− ions were added to attain a salt concentration of 150mM. These systems were energy minimized for 1000 steps using steepest descent minimization followed by 2000 steps of conjugate gradient minimization. During the minimization all atoms of the DNA were held fixed in their starting conformation using a harmonic constraint with a force constant of ˚2 . This allowed water molecules to re-organize and eliminate unfavorable con500kcal/mol/A tacts with DNA. After this, 5000 steps of conjugate gradient minimization were performed, decreasing the force constant from 20 kcal/mol/˚ A2 to 0, with a reduction of 5kcal/mol/˚ A2 every 1000 steps. After minimization, the systems were heated from 0K to 300K within 40 ps while the DNA was held fixed using harmonic constraints with a force-constant of 20kcal/mol/˚ A2 . SHAKE constraints 60 were applied on all bonds involving hydrogen with a tolerance of 0.0005. Temperature regulation was achieved using Berendsen weak coupling 61 method with a temperature coupling constant of 0.5 ps. Electrostatic interactions are treated with the Ewald method 62 as implemented in PME formulation for which we use a real space



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cutoff of 9 ˚ A with L-J interactions truncated at the same distance. Following heating, 500 ps long NPT simulation was performed using Berendsen barostat using 0.5 ps pressure relaxation time. Finally 300ns long NVT simulation was performed using a temperature coupling constant of 1ps. The visualization of the results was done with the VMD. 63 a i=N

h

R

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Figure 2: (a) DNA with its central axis traced using Curves+. Unit vector zî represents the local tangent to the DNA axis and h represents rise. (b) R represents end-to-end distance of the DNA. We calculate the helix axis, six helical parameters of the DNA that relate successive base pair planes: three angles (twist, roll, and tilt) and three distances (shift, slide and rise) using the Curves+ algorithm 64 . Using these parameters we calculate end-to-end distance, local tangents and contour length. We next determine contour length distribution, end-to-end length distribution, bending angle distribution and their respective variances. Table 1: Base sequences of different ds-DNAs used in our simulations. DNA length Base sequences 12 d(CGCGAATTCGCG)2 24 d( CGCGATTGCCTAACGGACAGGCAT)2 38 d(GCCGCGAGGTGTCAGGGATTGCAGCC AGCATCTCGTCG )2 56 d(CGCGATTGCCTAACGGACAGGCATAGA CGTCTATGCCTGTCCGTTAGGCAATCGCG )2


AT (%) GC (%) 33.3 66.7 41.67 58.33 34.21 65.79 42.86

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Calculation of Persistence Length Bend angle distribution Let ri be the position vector of the i−th base pair of the ds-DNA then the end to end length of a DNA can be defined as R = |(ri+n − ri )|. If zî is the unit vector along the local direction at i, the deformation of ds-DNA can be represented by the bending angle θ = cos−1 (ˆ zi .ˆ zi+n ). Probability distribution of small fluctuations in θ can be approximated by a Gaussian and is given by 37 r P (θ) = where κ is the bending modulus, L0 =

P

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persistence length lp = βκ. Note that Eq. (1) is valid for sufficiently small L0 and θ. 31 For small θ, Eq. (1) yields 37

lnP (θ) = −

lp βκ (1 − cosθ) + 0.5ln( ) L0 2πL0

(2)

Eq.(2) indicates that a plot of lnP (θ) vs. (1 − cosθ) would give rise to a descending straight line and from the slope one can estimate the persistence length lp of ds-DNA. 31 A recent study 65 on the bend angle distribution (P (θ)) of semi flexible polymer of short and intermediate length within the WLC model provides the closed general analytic formula for P (θ) and is given by lp θ2 Np θsin(θ)exp − P (θ) = L0 2L0

(3)

where N is a constant. Fitting the data of bend angle distribution with the expression given by Eq.(3) one can also extract the value of lp . Note persistence length obtained by this method is different from the one that we report in Eq.(2). Thus we identify it as lps .



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End-to-End Distance Distribution One can also determine persistence length of DNA using the mean square end-to-end distance obtained from the WLC model 24

2

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L0 1 − exp(− ) lp

(4)

Calculation of Stretch Modulus Contour length distribution Assume in equilibrium ds-DNA has time averaged length L0 and instantaneous length L. It turns out that L is the sum of all the consecutive base pair heights (rise). We call it the contour length of the ds-DNA. The linear elastic rod model assumes that a perturbation along its length gives rise to a restoring force F that increases linearly with the instantaneous fluctuation (L − L0 )/L0 : F = −γG (L − L0 )/L0 , where γG is the stretch modulus of the ds-DNA. Integrating the force and distance we obtain the free energy change due to this restoring force as: E(L) = γG (L − L0 )2 /(2L0 ). Combining the expression of E(L) with the Boltzmann’s formula we obtain the probability of ds-dNA having a length L as: r P (L) =

G (L−L )2 βγG − βγ 0 e 2L0 , 2πL0

(5)

where β = 1/(kB T ), kB is the Boltzmann constant and T is the temperature. Thus the probability distribution is Gaussian with mean length L0 and variance L0 /(βγG ). From the variance one can extract the stretch modulus of ds-DNA. In a similar fashion end-toend distance distribution can be determined which is again Gaussian. Similarly, from the variance of the distribution one can extract the stretch modulus (γGetel ) of ds-DNA. Note, end-to-end distance includes effect of DNA bending and can be very different from contour length when there is bending of helix axis.


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Macroscopic elastic theory According to macroscopic elastic theory for a uniform rod with cross sectional area A and stretch modulus γ, the Young’s modulus is given by Y = γ/A. On the other hand, bending modulus κ of the rod varies with Y and moment of inertia I as κ = IY , thus persistence length becomes lp = βIY . For a cylindrical molecule of radius r, A = πr2 and I = πr4 /4, therefore, 26 γW LC =

4lp βr2

(6)

By substituting the value of lp we obtain the value of γ. Note, the origin of the calculation of γ comes from the WLC model. From now onwards we call this γ as γW LC . Conformational Energy Analysis Helical parameters (both angular and translational parameters) of ds-DNA can describe the DNA deformations and displacements of base-pairs along and across the helical axis, respectively. 45 To quantify such deformations of ds–DNA we follow a method described by Olson et al., 45 which essentially consists of harmonic energy functions associated with the following six base pair step parameters: three angles - roll, tilt, twist and three distances rise, shift, slide. Instantaneous fluctuations of these six base pair step parameters from their equilibrium values (ζi0 ) are given by ∆ζi = ζi − ζi0 , i = 1, 6. For any conformational change the energy associated with the base pair step is assumed to be harmonic and is given by 6

6

1 XX ωij ∆ζi ∆ζj E = E0 + 2 i=1 j=1

(7)

where E0 is minimum energy and ωij represent elastic constants. Let elastic matrix W consists all ωij then the right most term of Eq. (7) reduces to Z T WZ, where the elements of Z and its transpose Z T are ∆ζi . Following Olson et al. 45 and Noy et al. 34 we write W = LkB T (h∆ζi ∆ζj i)−1 = LkB T Θ−1 11 ACS Paragon Plus Environment

(8)


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where L, kB , T , and Θ are the ds-DNA length, Boltzmann constant, temperature, and the covariance matrix consisting of six base-pair step parameters, respectively. For our calculation we consider four base pair step parameters roll, tilt, twist, and stretch and thus Θ becomes a 4 × 4 covariance matrix. We evaluate elastic constants from the diagonal terms of the elastic matrix W. We calculate stretch responses of the ds-DNA by substituting L with contour length (γcl ) as well as end-to-end length (γetel ).

Results and Discussion To test whether the structure of DNA stabilizes well under the mean field ion distribution and whether the MD trajectory data that we use for analysis provide adequate description of the DNA structure we determine DNA-Na+ radial distribution function as a function of time, which is shown in Fig. S1 in the supporting information (SI). Distribution stabilizes at around 50 ns of MD and shows no significant change thereafter. We also plot RMSD of the short DNAs with respect to its initial minimized structure in Fig. S2 of SI to demonstrate the stable DNA trajectory over 300 ns long simulation. For the analysis of short DNA we take data from last 200 ns of 300 ns long MD trajectory. We calculate various elastic properties of the full length short ds-DNAs (FLSD) and termini excluded short ds-DNAs (TESD). Similarly we calculate the elastic properties of full length NCP DNA (FLND) and short (we exclude five end bps from both sides) NCP DNA (SND). In the main text we report the results of various FLSDs and FLND. Results related to TESD and SND are given in the SI.

Persistence Length of Bare DNA: Effect of Salt Concentration and DNA Length We determine distributions of bending angle from our MD data for different lengths of ds-DNA and at different monovalent salt concentrations. We plot log of bending angle distribution as a function of (1 − cosθ) and fit it with Eq.(2). Fig. 3 shows such a plot for 12 ACS Paragon Plus Environment

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1-cos(θ) Figure 3: (color online) Probability distributions of bend angles for 12 bp and 24 bp dsDNA at various salt concentrations. Discrete data points (color: red) are obtained from our MD simulations. Persistence lengths (lp ) for respective ds-DNA are calculated by fitting the slope with Eq. (2) and is shown by straight line (color: blue). The correlation coefficient (r) values obtained from the respective fittings are given in the table SVI of SI. 13 ACS Paragon Plus Environment

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12 bp 24 bp 38 bp 56 bp Expt.

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50 40 30 20 10 10 15 20 25 30 35 40 45 50 55 60 DNA length (bps)

Figure 5: (color online) Dependence of persistence length on various lengths of DNA with no salt condition. Persistence length calculated using Eq.(2), Eq.(3) and Eq.(4) separately and compared.

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12bp and 24bp FLSD. Similar plots for other FLSD and TESD are shown in Figs. S3, S4, S5 in the SI. From the fitting we determine persistence length (lp ) of ds-DNA for different lengths at different salt concentrations. Fig. 4 shows a comparison of our results with experiments for FLSD. Calculated values of the persistence length are listed in table-SI (for FLSD), and table-SIII (for TESD) in the SI. We observe that the persistence length of ds-DNA depends strongly on salt concentration as well as on ds-DNA length. Our results show good quantitative agreement with the experimental results. 26 With increase in salt concentration, persistence length of various short ds-DNA (12bp, 24bp, 38bp and 56bp) decreases. This is because of the screening of the electrostatic repulsion of the phosphate groups along the backbone at higher salt concentration. With increase in salt concentration, ions neutralize the negatively charged phosphate backbone and thus make it easier for the DNA to bend. The inverse dependence of persistence length with the monovalent salt concentration is consistent with the standard charged cylinder model. 66,67 It views electrostatic interaction as a small perturbation over the elastic energy function of the WLC model. We have also calculated lp using Eq.(3) and from the distribution of end-to-end distance given by Eq.(4). Persistence length obtained using Eq.(2) and Eq.(4) at zero salt follow very similar trend with DNA length, as shown in Fig. 5. We observe that the values of persistence length obtained using Eq.(2) and Eq.(4) at zero salt vary with the DNA length following identical trend as depicted in Fig. 5. While we do not observe any particular trend for variation of persistence length as a function of salt concentration obtained using Eqs.(2, 3), the values calculated using Eq.(4) show a trend similar to that observed for zero salt, presented in Fig. 5.

Stretch Modulus of Bare DNA: Effect of Salt Concentration Using our MD data we next determine the contour length distribution (P (L)) for ds-DNA of various lengths at different salt concentrations. In Fig. 6 we plot the contour length distribution P (L) for 12 bp and 24 bp ds-DNA at different salt concentrations. P (L) for other systems is given in Figs. S7, S8 of SI. We use Eq.(5) to fit P (L) as shown in Fig. 6. 16 ACS Paragon Plus Environment

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Salt concentrations (mM) Figure 7: (color online) Dependence of stretch modulus (γG ) on monovalent salt concentration for various ds-DNA and compared with the experimental results. 26



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From these fittings we calculate respective stretch modulus (γG ) values. We plot and compare them with experimental data 26 in Fig. 7 and Fig. S9 of SI. The values of γG are listed in the table-SI(for FLSD) and table-SIII (for TESD) in the SI. We find that, for 12 bp ds-DNA γG increases with salt concentration, which is consistent with experimental observations. 26 However, this trend is not very obvious for other short DNA cases. For 56bp ds-DNA we find that γG decreases with increasing salt concentration. The presence of local base pair melting and its decrease with increasing salt concentration was invoked by Bustamante et. al. 26 to explain increase of stretch modulus as a function of salt concentration. We perform analysis for locating possible local melting based on hydrogen bond analysis (see Figs. S10 - S13, in the SI). We calculate the mean fraction of hydrogen bonds for each base pair for all the FLSDs. Apart from the terminal base pairs, all base pairs maintain hydrogen bond throughout the duration of simulation. Therefore, the H-bond analysis for our case does not provide a clear signature of local melting. The trend that we observe originates probably from some other source. In the absence of local melting the screening of intra-strand electrostatic repulsion between phosphate groups could be the reason for the trend that we see for the stretch modulus. This was also discussed previously in DNA stretching experiments, 68 based on the theoretical work by Prodgornik. 69 Screening of repulsion between phosphates makes DNA harder to stretch, as repulsion would favor stretching. It is worth mentioning here that Bomble and Case 44 also observed an increasing trend for stretch modulus as a function of salt concentration with no trace of any local melting. We observe an increase in stretch modulus with salt concentration for all FLSDs except 56 bp. Possibly for longer DNA, bending contribution to contour length increases. As the effect of DNA bending is not included in Eq.(5), it may provide an unphysical trend for γG for 56 bp ds-DNA.


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A Comparison Between Macroscopic Elastic Theory and Linear Elastic Rod Model With the above results, we now ask a basic question: can short ds-DNA be treated as an isotropic rigid rod? To answer this question, we now use the result of macroscopic elastic theory (Eq. (6)) for calculating γW LC by using radius of ds-DNA r = 1 nm and respective lp values. Eq. (6) makes it clear that the macroscopic elastic theory predicts the persistence length and the stretch modulus to vary proportionally. Substituting persistence length values obtained using WLC formalism into Eq.(6) gives us stretch moduli decreasing with increase in salt concentration. The values of γW LC can be made equal to γG if we scale r appropriately, smaller for a higher salt concentration. This is in sharp variance with the time-resolved fluorescence polarization anisotropy experiment of intercalated ethidium where the radius of 48bp DNA decreased just by 3 ˚ A when monovalent salt concentration increased from 20 to 100mM, 70 which is not sufficient to bridge the difference in stretch moduli obtained from the two methods. Similar results were obtained by Bomble and Case 44 for short DNA. However, they showed that elastic rod model works well for describing elastic properties of long DNA fragments (> 100bp). Macroscopic elastic theory overlooks the detailed base-pair level phenomena as well as ignores the microscopic effect of DNA backbone geometry and the screening effect at the base pair level, which could be important for short DNA fragments.

Stretch Modulus from Conformational Energy Analysis Using conformational energy analysis described above we next calculate stretch modulus of various ds-DNA by using respective values of contour lengths and end-to-end lengths. We plot γcl both for FLSD and TESD in Fig. 8, and in Fig. S14 of the SI. Values of γetel are shown in the table-SII and table-SIV in the SI. We list the values in table-SII(FLSD) and table-SIV (TESD) in the SI. We find γcl is always larger than γetel . This is due to the fact that for our systems contour length is always larger than the end-to-end length. Note by definition



end-to-end length captures the effect of DNA bending. We further observe different values of stretch modulus for the same ds-DNA when we compare γG and γcl . Values of γcl is larger than the values of γG except for few cases. From our analysis we find that calculation of stretch modulus (γG ) using linear elastic rod model gives close agreement with available experimental results. 71–75

4000 3500 3000 γcl (pN)

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2500

12 bp 24 bp 38 bp

2000 1500 1000 500 0 0

50

100

150

200

250

Salt concentrations (mM) Figure 8: (color online) Dependence of stretch modulus (γcl , using Eq.(8)) on salt concentration for various ds-DNA.

Persistence Length of NCP DNA: Effect of Salt Concentration All the methodologies used to determine elastic properties of short ds-DNA were applied on the nucleosomal DNA. The dependence of ln(P (θ)) on (1 − cosθ) is shown in Fig. 9 with Fig. 10 showing dependence of persistence lengths obtained from bend angle distributions on salt concentration. The persistence length increases with salt concentration, which is 20 ACS Paragon Plus Environment

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opposite to the trend observed for short DNA. It is known that nucleosomal DNA binds with the histone core at 14 different sites along the DNA sequence. At these sites, arginine residues (10 belonging to the histone core and 4 belonging to the histone tails) are inserted into the DNA grooves, facilitating DNA bending. 76 This makes the nucleosomal DNA strongly constrained, leading to the possibility of abrupt sharp bends. The sharp bends, known as kinks, would lead to a deviation of the DNA contour from a perfectly helical shape. Lesser the number of these sharp bends more will be the persistence length. The persistence length will also be affected by the histone tails. They are strongly positively charged and hence can affect the structure of the nucleosomal DNA super-helix electrostatically leading to kinks. At low salt concentrations the histone tails stay collapsed on the nucleosomal DNA. At higher salt concentrations screening of the DNA-tail attraction makes the tails freely suspended in the surrounding solvent. From the twist angle analysis (not shown) we observe that the value of twist angle (> 40 degree 3 ) increases at several places along the DNA with increase in salt concentration, which indicates that existing kinks in NCP DNA get diminished. This may render NCP DNA stiffer in presence of salt. The increase of persistence length of nucleosomal DNA with salt concentration originates from a weaker electrostatic interaction between histone tails and DNA at higher salt concentrations. To test this hypothesis, we also do simulation of the NCP removing all the histone tails. In this case, we should expect an increase in the persistence length, with the increase being more prominent at lower salt concentrations. This is precisely what we observe in our calculations. The trend is shown in Fig. 10. Persistence length in the absence of tails shows only a slight increase with increase in salt concentration. At a concentration of 150 mM the persistence length is almost similar to the case when tails are present.

Stretch Modulus of NCP DNA: Effect of Salt Concentration We next determine contour length distribution for FLND with and without tails at different salt concentrations. We use Eq.5 to fit them and then calculate γG values, which are shown 21 ACS Paragon Plus Environment


NCP (with tails) 0

NCP (no tail)

no salt

no salt

0

-1 -1

-2 -3

lnP(θ)

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-2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0

150mM

0

0.2 0.4 0.6 0.8

1

1.2 1.4 1.6

150mM

0

-1

-1 -2

-2 -3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.2 0.4 0.6 0.8

1

1.2 1.4 1.6

1-cos(θ) Figure 9: (color online) Probability distribution of bend angles of FLND for single NCP with tails and without tails at different salt concentrations. Discrete data points (color: red) are obtained from our MD simulation. Persistence lengths (lp ) for respective ds-DNA are calculated by fitting the slope with Eq. (2) and is shown by straight line (color: blue). The correlation coefficient (r) values obtained from the respective fittings are given in the table SVI of SI.


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160 140 120 100 80 with tails without tails

60 40 0

50

100

150

200

250

Salt concentration (mM) Figure 10: (color online) Persistence length for FLND as a function of salt concentration for two cases: NCP with tails (filled data points) and NCP without tails (empty data points).

140 120 100

P(L)

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140 120

no salt 150mM 250mM

NCP (with tails)

80

100

no salt 150mM 250mM

NCP (no tail)

80

60

60

40

40

20

20

0

0 0.99

1

1.01

1.02

0.99

1

1.01

1.02

L/L0 Figure 11: Contour length distribution for FLND with tails (left) and without tails (right) at different salt concentrations. Discrete data points are obtained from our MD simulations and the solid lines are obtained by fitting using Eq.(5).


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Stretch modulus (γG) (pN)


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10000 9000 8000 7000 6000 5000

with tails without tails

4000 3000 0

50

100

150

200

250

Salt concentrations (mM) Figure 12: (color online) Stretch modulus (γG ) for FLND as a function of salt concentration for two cases: NCP with tails (filled data points) and NCP without tails (empty data points).


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in Fig. 11 and Fig. S18 of the SI. We plot different γG values in Fig. 12 and compare them with γW LC in Fig. S19 and Fig. S20 of SI. Using conformational analysis we calculate γcl and γetel for NCP for both the cases with and without tails. The values are plotted in Fig. S21 (for FLND) and Fig. S22 (for SND) in the SI. Note that values of γcl are larger than γG for FLND. Such differences between the values of γcl and γG might be because we follow two different techniques for calculating them. However, the trend of stretch modulus with salt concentration remains same except few cases and the order of magnitudes for γcl are similar to that obtained by Noy et al. 34

Bending Modulus: A Comparison between Bare DNA and NCP DNA We next calculate and plot end-to-end distance distributions for different bare DNAs and NCP DNA, which is shown in Fig. S23 of the SI. We also calculate bending modulus (κ) of different ds-DNA and plot them in Fig. 13 (for FLSDs and FLND) and Fig. S24 in the SI. We observe that for short ds-DNA κ decreases with increasing salt concentration whereas this scenario is opposite for NCP DNA. Hegner et.al. 77 used optical tweezers to stretch DNA-RecA complex and derived the elastic properties of the complex from the force-extension curves. They found a 6 to 12 fold increase in the longitudinal stretch modulus of DNA, when in complex with RecA. Persistence length also showed a significant increase. Yan and Marko 78 performed a theoretical study of the effect of DNA-binding proteins on DNA’s elastic properties. Their study led to the conclusion that proteins binding to DNA leads to a decrease in its persistence length. Santosh et al. 37 studied the compaction of DNA by positively charged dendrimer. They found that bending rigidity and persistence length of the DNA decreased in the presence of dendrimer with the stretch modulus remaining the same. Our calculations show significant differences in the elastic properties of NCP DNA as compared to bare DNA. We find that, at the physiological salt concentration of 150 mM, the persistence length of FLND is 3 times the 25 ACS Paragon Plus Environment


value for 56 bp FLSD. Bending modulus (κ) and stretch modulus (γG ) also show a 3-fold and 7−fold increase, respectively. Note that, to the best of our knowledge for the first time, we systematically determine stretch modulus, persistence length and bending modulus of the nucleosomal DNA. We observe an opposite trend of various elastic properties when we compare the values for short ds-DNA and nucleosomal DNA in different salt concentrations. The trend reminds that Nucleosomal DNA, which is larger in length than the short ds-DNA, follows macroscopic elastic theory.

700 600 κ (pN-nm2)

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500

12bp

400

38bp

24bp 56bp

300

NCP(with tails) NCP(no tail)

200 100 0 0

50

100

150

200

250

Salt concentrations (mM) Figure 13: (color online) Dependence of bending modulus (κ) at different salt concentration for various FLSDs and FLND.


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Comparison of Force Fields: AMBER vs CHARMM To test the effect of choice of force field 79 on the DNA elastic properties we use two well known family of force fields for nucleic acids namely AMBER 80,81 and CHARMM. 82,83 We use two different force fields - CHARMM36 and ff99bsc0 independently to run explicit solvent MD simulation for studying elastic properties of 24bp ds-DNA. In section II, we have already given the details of simulation using ff99bsc0. For simulation using CHARMM 36 we use same protocol. For the CHARMM simulation also we calculated RMSD with respect to the initial minimized structure as a function of time and compared with RMSD obtained using ff99bsc0. This RMSD comparison is shown in Fig. S25 in the SI. We observe that RMSD obtained with CHARMM force field is smaller than that obtained with AMBER force field. However, both yield a stable B-DNA structure with a small rmsd difference. A comparative study for P (L), and P (θ) between 24 bp FLSD and TESD is shown in Fig. 14. P (R) is shown in Fig. S26. Table-2 summarizes and compares values of stretch modulus, persistence length, and bending modulus obtained from the analysis of our MD data using two different forcefields for 24bp FLSD. Table-SVI in the SI provides the TESD’s elastic properties of 24bp ds-DNA using CHARMM 36 force field and compares with the results obtained from ff99bsc0 force field. We find that for all the cases lp , κ, γW LC , γcl and γetel values obtained using CHARMM 36 force field are smaller than that determined from ff99bsc0 force field. On the other hand, γG values for 24bp ds-DNA obtained using CHARMM 36 force field is larger than that obtained from ff99bsc0 force field. Despite the fact that the values obtained from two different forcefields are different, they are similar in orders of magnitude. However, both the force fields are very useful for determining the elastic properties of ds-DNA.


18 16 14 12 10 8 6 4 2 0 0.85 25

CHARMM AMBER FLSD

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CHARMM AMBER FLSD

0 -0.8 -1.6 -2.4

0.9

0.95

1

1.05

1.1

ln(P(θ))

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P(L)


1.15

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CHARMM AMBER TESD

20

-0.8

10

-1.6

5

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0 0.95

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0

15

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0.4

-3.2

1.15

0

0.4

L/L0

0.8

1.2

1-cos(θ)

Figure 14: (color online) Contour length distribution (left panel), bending angle distribution (middle panel) and end-to-end length distribution are plotted for two different force fields CHARMM and AMBER for 24 ds-DNA including termini (upper row) and excluding termini (lower row).

Table 2: Elastic properties of the 24 bp ds-DNA at no salt condition Force field

Lmp 0 (nm)

hLi (nm)

lp (nm)

γW LC (pN)

γG (pN)

γcl (pN)

γetel (pN)

CHARMM

7.58

6.87

158

974 ± 62

150

136

AMBER

7.76

6.84

38.19 ± 2.38 45.84 ± 0.88

760

871 ± 17

409

397


κ (pNnm2 ) 158.18 ± 9.86 189.87 ± 3.65

R0mp (nm)

G γetel (pN)

7.58

852.05 ± 73.84 966.41 ± 30.67

7.58

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Conclusion In this manuscript, following different methods we have investigated elastic properties of short ds-DNA as well as of nucleosomal DNA at various salt concentrations. Our results indicate that short ds-DNA are more flexible in comparison to the large ds-DNA and agree qualitatively as well as quantitatively with the earlier observations. 26,30,40,44,68 However, the trends that we observe for short DNA are different from those obtained in earlier experimental studies, 2 which showed elastic properties to be length independent. Whereas we have found persistence length (lp ), stretch modulus (γ) and bending rigidity (κ) change with the length of the ds-DNA. Thus these should be measured directly and should not be extrapolated from measurements performed on long DNA. We further observed that persistence length of bare ds-DNA decreases with salt concentration, which can be attributed to easy bending resulting from neutralization of the backbone charges. We saw opposite trends for stretch modulus with salt concentration calculated using two different methods namely, linear elastic rod model (γG ) and macroscopic elastic theory (γW LC ). We observe that γG increases with increase in salt concentration. Decreasing salt concentration reduces the intra-strand electrostatic screening between phosphate groups and makes it easier for the DNA to stretch, hence reducing the stretch modulus. Stretch modulus and persistence length of short ds-DNA vary in opposite ways with salt-concentration. This inverse variation is in sharp contrast to macroscopic elastic theory, which predicts these two quantities to be directly proportional to each other. We have listed and compared different elastic properties of various ds-DNA by including and excluding the termini and find significant quantitative differences in various elastic properties of bulk DNA as compared to DNA fragments including termini. This comparison clearly demonstrates the effect induced by finite molecular size and presence of boundaries in the system with regard to its elastic properties. We have further studied and compared elastic properties of 24bp ds-DNA using two different force fields CHARMM 36 and ff99bsc0. Future work will involve calculating the elastic properties of DNA using the recently developed ζOL1 refinement of ff99bsc0 force field, which improves 29 ACS Paragon Plus Environment


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the backbone description of double helix. 84 We have determined the elastic properties of NCP DNA both in presence and absence of histone tails at different salt concentrations. We observe that both persistence length and stretch modulus increase with salt concentration, which suggests that NCP DNA follows macroscopic elastic theory. We observed NCP DNA to be stiffer than bare short DNA. Values of persistence length, stretch modulus and bending rigidity for NCP DNA are found to be many times larger than their respective values for bare short DNAs. Our results illustrate how the elastic properties of DNA are modified by its interaction with histone proteins. To the best of our knowledge this kind of systematic study for determining elastic properties of nucleosomal DNA and various short ds-DNAs is very new. We used many different methods such as macroscopic elastic theory, WLC model, linear elastic rod model, conformational energy analysis to calculate the DNA elastic properties. Comparing the results obtained from all these methods with the available experimental results, we feel that linear elastic rod model (Eq.(2) and Eq.(5)) provides a better trend for the values obtained for various elastic properties. In principle, our results and predictions can be tested by carrying out in vitro single molecule experiments. We believe this kind of study certainly helps to understand how elastic properties of ds-DNA evolve with length scale as well as when it interacts with protein.

Acknowledgement We thank Department of Atomic Energy (DAE) for financial support. We also thank Supurna Sinha for fruitful discussions.

Supporting Information Available Radial distribution functions, bend angle distributions, contour length distributions, persistence length and stretch modulus comparison, hydrogen bond analysis, RMSD and end-toend distance distributions for various cases at different salt concentrations. is available free of charge via the Internet at http://pubs.acs.org/. 30 ACS Paragon Plus Environment

This material

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DNA Elasticity from Short DNA to Nucleosomal DNA - ACS Publications

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