No. of Pages 12, Model 5G. Contents lists available at SciVerse ScienceDirect. Acta Biomaterialia

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1 ACTBIO February 2012 No. of Pages 12, Model 5G Acta Biomaterialia xxx (2012) xxx xxx 1 Contents lists available at SciVerse ScienceDirect Acta Biomaterialia journal homepage: 2 The dependence of DNA supercoiling on solution electrostatics 3 David Argudo, Prashant K. Purohit Q1 4 Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104, USA 5 article info Article history: 9 Received 4 October Received in revised form 20 January Accepted 26 January Available online xxxx 13 Keywords: 14 DNA plectonemes 15 Biological material 16 Electrostatics 17 Multivalent counterions 18 Thermal fluctuations 19 abstract We develop an elastic isotropic rod model for twisted DNA in the plectonemic regime. We account for DNA elasticity, electrostatic interactions and entropic effects due to thermal fluctuations. We apply our model to single-molecule experiments on a DNA molecule attached to a substrate at one end, while subjected to a tensile force and twisted by a given number of turns at the other end. The free energy of the DNA molecule is minimized subject to the imposed end rotations. We compute values of the torsional stress, radius, helical angle and key features of the rotation extension curves. We also include in our model the end loop energetic contributions and obtain estimates for the jumps in the external torque and extension of the DNA molecule seen in experiments. We find that, while the general trends seen in experiments are captured simply by rod mechanics, the details can be accounted for only with the proper choice of electrostatic and entropic interactions. We perform calculations with different ionic concentrations and show that our model yields excellent fits to mechanical data from a large number of experiments. Our methods also allow us to consider scenarios where we have multiple plectonemes or a series of loops forming in the DNA instead of plectonemes. For a given choice of electrostatic and entropic interactions, we find there is a range of forces in which the two regimes can coexist due to thermal motion. Ó 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved Introduction 40 The mechanical and electrostatic properties of DNA directly 41 affect various cellular processes, such as replication, transcription, 42 compaction and protein DNA binding. This is the motivation 43 behind this study of DNA supercoils, which are also known as 44 plectonemes. Plectonemes in DNA molecules are manipulated by 45 several molecular machines during key processes, such as tran- 46 scription and DNA repair [1]. In several scenarios, the action of 47 these molecular machines or enzymes on DNA has been found to 48 depend on the mechanical stress present in the molecules [2,3]. 49 Consequently, DNA supercoiling remains a subject of study for 50 theorists and experimentalists alike. 51 Experimentally, DNA supercoiling has been investigated using 52 several biochemical and biophysical methods, including single- 53 molecule experimental techniques, where individual DNA mole- 54 cules can be stretched and twisted under physiologically relevant 55 conditions [4 8]. In these experiments, it is possible to apply a 56 force and/or moment parallel to the filament axis of a DNA mole- 57 cule, and to measure the elastic response in terms of elongation 58 and angle of twisting about the filament axis. In rotation extension 59 experiments, the vertical extension of the DNA filament and the 60 external moment are recorded as a function of the number of turns. Corresponding author. Tel.: ; fax: address: purohit@seas.upenn.edu (P.K. Purohit). It is a well-known feature of the experimental curves that there is a regime, corresponding to the formation of plectonemes, where there is almost a linear relationship between the DNA extension and the applied number of turns. Also, as shown in te recent experiments of Forth et al. [4], Lipfert et al. [5] and Mosconi et al. [6], the external moment is approximately constant in the plectonemic regime. Plectonemes have been studied theoretically as elastic rods by many authors [9 14]. In order to interpret single-molecule experiments, Purohit [15,16] accounts for the effects of thermal fluctuations as well as electrostatics in plectonemes and straight portions of DNA, and shows that many features seen in the recent experiments of Forth et al. [4] can be qualitatively reproduced using an elastic rod model. Furthermore, as seen in Fig. 5 in Purohit [16], his theoretical results for the slope of the linear region in vertical extension of the DNA vs. number of turns of the bead are around twice the value of those found in experiments by Forth et al. [4]. One of the goals of this paper is to address this problem and get more quantitative agreement with single-molecule experiments. Our approach follows those of van der Heijden et al. [14] and Clauvelin et al. [17,18], who use a variational formulation to solve for the geometry of the plectoneme. The analysis in van der Heijden et al. [14] considers only the elastic energy of the filament, but Clauvelin et al. [17,18] and other authors [19] consider electrostatic interactions together with the elasticity, and are able to reproduce some of the features of the rotation extension experiments. In agreement /$ - see front matter Ó 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi: /j.actbio Please cite this article in press as: Argudo D, Purohit PK. The dependence of DNA supercoiling on solution electrostatics. Acta Biomater (2012), doi: /j.actbio

2 ACTBIO February 2012 No. of Pages 12, Model 5G 2 D. Argudo, P.K. Purohit / Acta Biomaterialia xxx (2012) xxx xxx 87 with Purohit [16], Clauvelin et al. [18] reach the conclusion that 88 electrostatics plays a minor role compared to the elasticity of the 89 DNA in these experiments. Contrary to this conclusion, recent sin- 90 gle-molecule experiments and molecular simulations have shown 91 that the results of the rotation extension experiments depend 92 strongly on the salt concentration of the solution [20,21]. For this 93 reason, we carefully consider electrostatics in this paper and pres- 94 ent an analytical model that captures the behavior of DNA in rota- 95 tion extension experiments and simulations for a variety of DNA 96 lengths, applied loads and salt concentrations. We also apply our 97 model to a novel set of DNA experiments with a mixture of mono- 98 valent and multivalent salts, and show that we can predict the re- 99 sults of these experiments. 100 Other key variables that are affected by the salt concentration 101 are the discontinuities in extension and torque during the super- 102 coiling transition [20]. These discontinuities have been studied re- 103 cently by Forth et al. [4] and Daniels et al. [22]. Purohit s models 104 [15,16] capture these discontinuities or jumps qualitatively, but 105 he does not comment on the salt dependence of the jumps. We 106 use our model to provide estimates for the number of turns at 107 which the DNA makes a transition from a straight to a supercoiled 108 configuration, and for the jump in the extension and moment as a 109 function of DNA length and salt concentration. Furthermore, we 110 contemplate the possibility of the formation of multiple plecto- 111 nemes and other forms of DNA compaction (loops and plecto- 112 nemes coexistence) due to energetic reasons General description of the model 114 We proceed with a model of the plectonemic region of the DNA 115 molecule based on the framework of Clauvelin et al. [18], but we 116 account for thermal fluctuation effects, confinement entropy and 117 an end loop model. The DNA in the experiments is modeled as a 118 Kirchhoff inextensible elastic rod of length 2l (with l 6 s 6 l, 119 where s is the arc length along the centerline of the rod). The Kir- 120 chhoff theory of rods models the centerline as a curve in space r(s) 121 endowed with mechanical properties which are assumed to be 122 suitable averages over the cross-section of the rod [23,24]. The 123 configuration of an inextensible, unshereable rod is defined by 124 r(s) and an associated right-handed orthonormal director frame 125 d i (s), i = 1,2,3. For convenience, the vector d 3 = r 0 (s) is taken to be 126 tangential to the rod. The kinematics of the frame are encapsulated 127 in the director frame equations d 0 i ¼ u d i; where the components 128 of u = u i d i are measures of the strain, u 3 describes the physical 129 twist, and u 1 and u 2 are associated with bending such that the 130 square of curvature is given by j 2 ¼ u 2 1 þ u2 2. We assume a linear 131 constitutive relation between the stresses and the strains, so that 132 the moment m = K b u 1 d 1 + K b u 2 d 2 + K t u 3 d 3, where K b is the bending 133 modulus and K t is the twisting modulus. The rod is made up of 134 three regions (see Fig. 1): 135 In the linear regions the tails are, on average, aligned with the 136 vertical axis. The tails are not completely straight and the 137 centerline follows a writhing path due to thermal fluctuations 138 in the DNA molecule. An analysis of fluctuating polymers gent vector r 0 p ðsþ describe the superhelix. Note that each helix is subjected to tension and twist in the straight regime has been 139 carried out in detail by Moroz and Nelson [25,26], where 140 expressions for the twist and writhe have been provided. In 141 our model we will use their expressions. 142 In the plectonemic region the position vector r p (s) and the tan itself a piece of double-stranded DNA molecule. So, in the liter- 145 ature, DNA plectonemic geometrical variables (angle and 146 radius) are often referred to as supercoiling or superhelical, to 147 distinguish them from the intrinsic helical nature of the base 148 pair structure. Due to the symmetry of the problem, it is conve- 149 nient to introduce cylindrical coordinates (r, w, z) for the posi- 150 tion vector: 151 r p ðsþ ¼vre r þ ze 3 ð1þ where e 3 is the axis of the helix that wraps around the cylinder and 155 e r = coswe 1 + sinwe 2. The tangent to the position vector is: r 0 p ðsþ ¼sin he w þ cos he 3 ð2þ w 0 ¼ v sin h ; z 0 ¼ cos h; 0 < h < p r alent to a total moment M 3 about r 0 p at the beginning of the plecto- where the chiriality v = ±1 stands for the handedness of the helix: 163 v = 1 for a right-handed helix and v = 1 for a left-handed one 164 [17]. The other filament of the plectoneme is obtained by a rotation 165 of p about the helical axis e 3. The plectonemic region is character- 166 ized by the helical radius r and the helical angle h, which are 167 assumed to be independent of the arc length s. The complement 168 p/2 h of the helical angle is often referred to as the pitch angle. 169 Both r and h may depend on the loading. Geometric impenetrability 170 of the helices implies that h 6 p/4 [27,28]. Note that the external 171 moment M ext applied in the upper tail of the DNA molecule is equiv nemic region. By the arguments of conservation of torque about the 174 body axis of an isotropic rod, m d 3 = K t u 3 = M 3 is a constant [24], 175 implying that the twist u 3 is constant in the helical region. 1 One 176 consequence of the use of the expressions given by Moroz and 177 Nelson [26] is that the twist u 3 in the tails is different from that in 178 the plectoneme even though the twisting moment M ext = M 3 is the 179 same, since the effective twist modulus is different in each region. 180 At the end of the plectonemic region there is a loop. This end loop is formed in the transition from the straight configuration to the plectonemic configuration. In order to model the loop, we propose an approximation based on the localizing solution of an elastic rod [29,30], ignoring thermal fluctuations [31]. For details we refer the reader to Section S.1 of the supplementary data. The molecule contour length spent per tail is denoted by l t (L t =2l t ), the contour length in the loop is denoted by L o and the contour length per helix is denoted by l p (L p =2l p ). The sum of the length of all regions is given by L = L p + L t + L o. The equilibrium configuration of the rod is fully specified by the centerline, through the variables r, h and M 3. In what follows, we compute these parameters as a function of the loading (the pulling force, Fig. 1. Sketch representing single-molecule experiments, where a DNA molecule is fixed at one end while the other end is subjected to a pulling force F and twisted by a given number of turns n. 1 At the transition point going from an initially straight state to a plectonemic state there is a jump in the external torque. We define M ext = M critical as the twisting moment in the straight configuration right before the transition (no plectonemes formed), while M ext = M 3 is defined as the twisting moment when plectonemes (helices) are present and d M = M critical M 3 as the jump in the twisting moment at the transition (see Section 3.1). We use the notation M ext in Section 2 for the external torque. When plectonemes are present, the equations describing the DNA tails can be used by replacing M ext with M 3. When there are no plectonemes in the straight state right before the transition, the equations describing the DNA tails can be used to describe the entire molecule by replacing M ext with M critical. Please cite this article in press as: Argudo D, Purohit PK. The dependence of DNA supercoiling on solution electrostatics. Acta Biomater (2012), doi: /j.actbio

3 ACTBIO February 2012 No. of Pages 12, Model 5G D. Argudo, P.K. Purohit / Acta Biomaterialia xxx (2012) xxx xxx F, and the number of turns, n) by minimizing the free energy of the 198 system. 199 The experiments are performed under imposed end rotations; 200 therefore, the energy minimization will be performed under the 201 constraint that the number of turns n imposed on the bead at 202 one end of the DNA is equal to the excess link Lk p of the DNA mol- 203 ecule in the helical region, the excess link Lk t in the tails and the excess link Lk o in the loop: 207 n ¼ Lk p þ Lk t þ Lk o ð3þ 208 where the link Lk p in the helical region corresponds to the classical 209 partition into twist Tw and writhe Wr [32]: 210 Lk p ¼ M ext sin 2h v L p ð4þ 212 2pK t 4pr 213 At this point, we note that clockwise rotations n about the e axis, corresponding to a positive external moment M ext, generate 215 a left-handed helix, with v = 1, while a negative external moment 216 generates a right-handed helix, with v = 1. We also note that, in the 217 presence of thermal fluctuations, there is a writhe contribution 218 from the tails which can be accounted for by using the results of 219 Moroz and Nelson [26]: Lk t ¼ M extðl L p L o Þ 2p where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K b F M 2 ext K ¼ =4 226 k B T 1 þ 1 K t 4K b K þ OðK 3 Þ 227 k B is the Boltzmann constant and T is the absolute temperature. The 228 link in the end loop can be approximated as (see Section S.1 of the supplementary data): 232 Lk o ¼ M extl o 2pK t þ Wr o ð7þ 233 where Wr o 1 is the writhe present in the loop Potential energy of the system 235 It is convenient to express the total potential energy of the DNA filament as: 239 V ¼ Z L 0 Cðs; h; q i Þds ¼ V tails þ V loop þ V helices 240 where q i are variables like M 3,r,...independent of s. The free energy 241 in the case of fixed force F and fixed-torque M ext in the tails (straight 242 portion) is given by [26]: 243! 245 E t ¼ F M2 ext 2K t þ G flu L t 246 where the last term is a correction due to thermal fluctuations: 247 G flu ¼ ðk BTÞ 2 K 1 1 K b 4K 1 þ OðK 3 Þ ð10þ K where K is given by Eq. (6). The extension with thermal fluctuations 251 taken into account is given t /@F = ql 2 t, where K b k B T 254 q ¼ 1 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 K b F M2 k 2 ext 1 B T2 4k 2 B T2 32 L t ð5þ ð6þ ð8þ ð9þ ð11þ K b F M2 ext 4 Eq. (11). The extension of the filament is given as Dz = q(l L o L p ). Noting that L is constant, L o is approximately constant and q is independent of n, the overall slope of the rotation extension curve is given by: d d ðdzþ ¼ q dn dn ðl pþ ð12þ 264 In what follows, we drop the negative sign and simply refer to 265 the slope as q(d L p /dn). The end rotation conjugate to M ext is given 266 ¼ t. To get the free energy of the system under im- 267 posed end rotations Lk t, we apply a Legendre transform: V tails ¼ E t þ 2pLk t M ext ð13þ 271 The free energy of the loop will be approximated under the assumption that the bending and twisting energy decouple (see the supplementary data). The twisting moment M ext is a constant along the molecule and therefore the twist energy per unit length is a constant too. The expression for the bending energy per unit length E o bend and the length of the loop L o are obtained from the expressions given by Kúlic et al. [30] in the absence of twist:! V loop ¼ M2 ext þ E o bend L o ð14þ 2K t p where L o ¼ 4 ffiffiffiffiffiffiffiffiffiffi K b =F and E o bend = F. The free energy of the plectonemic region can be divided into elastic energy V helices el and the energy due to internal interaction V helices int. The elastic energy is given by:! V helices el ¼ K b 2 j2 þ M2 ext L p ð15þ 2K t where j = sin 2 h/r is the curvature of a uniform helix [24]. Eq. (15) 289 captures the elastic behavior of the rod in response to the applied Q2 290 loadings; it is zero in the straight and twist less the (natural) config- 291 uration of the rod. The electrostatic and entropic interactions pres- 292 ent in the plectonemic region V helices int ¼ Uðr; h; x i ÞL p will be described 293 in more detail later; here x i represents any auxiliary parameters or 294 internal variables that may appear in the free energy of the system 295 depending on the model picked to describe the electrostatic and 296 entropic parts of the energy. The potential energy can be written Q3 297 by separating the terms that contribute along L, and those that con- 298 tribute only along L p and L o. We introduce a Lagrange multiplier k to 299 account for the constraint Eq. (3) and define: Iðh; r; M 3 Þ¼ K b sin 4 h þ F þ Uðr; h; x 2 r 2 i Þ G flu M2 ext 4K b K þ k M ext sin 2h þ v 2p 4K b K 2r ð16þ such that the final expression for the potential energy of the system subject to the constraint Eq. (3) is given by:! V ¼ Iðh; r; M 3 ÞL p þ M2 ext F þ G flu 2K þ M2 ext t 4K b K L! þ E o bend þ F G flu M2 ext 4K b K 1 þ 1 K t 4K b K þ k n M extl 2p L o þ M extl o 8pK b K Wr o ð17þ Slope of the rotation extension curve after the formation of 257 plectonemes can be obtained from constraint Eq. (3) together with 2 The given formula for q includes corrections as detailed in Moroz and Nelson [25] Internal energy: entropy and electrostatics In the previous section we introduced the term U(r,h,x i )asa general expression to account for the internal interactions and configurational entropy cost in the plectonemic region. The term Please cite this article in press as: Argudo D, Purohit PK. The dependence of DNA supercoiling on solution electrostatics. Acta Biomater (2012), doi: /j.actbio

4 ACTBIO February 2012 No. of Pages 12, Model 5G 4 D. Argudo, P.K. Purohit / Acta Biomaterialia xxx (2012) xxx xxx 313 U(r,h,x i ) represents the undulation-enhanced free energy per unit 314 length plus the electrostatic energy [33]. We split the internal en- 315 ergy of the plectoneme U(r,h,x i ) into the configurational entropy 316 cost contribution U conf (r,h,x i ) and the purely electrostatic interac- 317 tions between the charged helices in ionic solution U el (r,h,x i ), such 318 that U = U conf + U el Electrostatics 320 At moderate length scales, electrostatic interactions between 321 phosphate groups in two different molecules and between phos- 322 phate groups and counterions (positively charged) and coions 323 (negatively charged) are present in the solution. Theoretical analy- 324 sis of electrostatic interactions between polyions in solution has 325 been done by Kornyshev et al. [34] and Parsegian and co-workers 326 [35,36]. To date, DNA DNA interactions are still not clearly under- 327 stood. We studied the effects of some variants of the internal en- 328 ergy models available in the literature that have been used to 329 model DNA single-molecule experiments in Section S.2 of the sup- 330 plementary data. One of the electrostatic models shown in Section 331 S.2 is the Ubbink and Odjik [33] model derived for supercoiled 332 DNA. This analytical model is based on the leading asymptotic con- 333 tribution of the Debye Huckel potential around two charged line 334 segments (helices). The other model shown in Section S.2 of the 335 supplementary data is the Marko and Siggia [37] electrostatic 336 model. This model corresponds to a uniform approximation of 337 the superposition of the potential in two limiting forms the elec- 338 trostatic potential independent of h and the electrostatic potential 339 independent of r. As noted by Ubbink and Odjik [33], this is less 340 accurate. More importantly, since, in our problem, we are minimiz- 341 ing the free energy of the system, we are interested in the deriva- 342 tives of the electrostatic potential. However, the Marko and Siggia 343 superposition model underestimates the value of the derivative 344 [33]. Finally, the Ubbink and Odjik model includes undulation 345 enhancement effects due to thermal fluctuations. For these rea- 346 sons, together with the results summarized in Section S.2 of the 347 supplementary data, we have used the expression given by Ubbink 348 and Odijk [33]: 349 U el ¼ U PB ðr; h; d r Þ¼ 1 rffiffiffiffiffiffiffiffi 2 k BTm 2 k D p l B gðhþ e 2 d 2 r k 2 k 2r D D ; r 351 ðhþ ¼1 þ 0:83 tan 2 ðhþþ0:86 tan 4 ðhþ ð18þ 352 where d r represents the small undulations of the helix in the radial 353 direction and leads to a correction in the electrostatic interaction 354 energy due to the thermal fluctuations. The Bjerrum length l B 355 (nm) is defined as the length scale at which thermal energy is equal 356 to coulombic energy, and is approximately 0.7 nm in water at 300 K 357 [38]. The Debye length k D (nm) and the effective linear charge m 358 (nm 1 ) depend on the monovalent salt concentration. It is impor- 359 tant to note that no consensus has been reached on the exact value 360 of m [18,21]. The Debye p screening length in water can be obtained 361 from k D ¼ 0:305½nmŠ= ffiffiffiffiffiffiffiffiffiffiffi c o ½MŠ, where c o [M] is the monovalent salt 362 concentration in molar units [38] Configurational entropy 364 The fluctuation free energy is k B T per correlation region [37], 365 and the free energy of entropic confinement per unit length of 366 the strand in the plectonemic supercoil may be written approxi- 367 mately as a superposition of two fluctuating modes due to radial 368 (d r ) and longitudinal (pp) displacements [33,39]: U conf ðd r ; hþ ¼ k BT A 1=3 " # c p ðpp Þ 2=3 þ c r d 2=3 r ð19þ 372 where A = K b /(k B T) is the persistence length of the fluctuating rod. 373 The term 2pp is the pitch of the helix and is given by p = rcoth. The terms c r and c p are in general unknown constants. For a worm-like chain confined in a harmonic potential, c r = c p = 3(2 8/3 ) in one dimension [33], but, as noted by van der Maarel [39], itis not clear whether these values can be adopted for the supercoiled configuration. In our calculations in Section 3 we will use c r = c p =2 8/3, which are empirically optimized constants [39] Variational formulation Once the DNA has transitioned from the straight configuration into the plectonemic state, the external moment M ext plateaus. Recall that we define M ext = M 3 as the external moment present in the molecule in the plectonemic state. To minimize the energy, we need to equate the following partial derivatives to zero [16 ; @h ¼ r which yields: k ¼ 2pM 3 þ OðK 3 Þ! K b sin 4 hþ sin 2h þ vm r 3 3 2r 2 p ¼ 0! 2 sin 3 h cos hþ cos 2h K b þ þ vm r 2 3 L p ¼ r Iðh; r; M 3 Þ¼0; ð20þ ð21þ ð22þ r; d r r ¼ 0 ð24þ where I(h,r,M 3 ) is given by Eq. (16). Note that we minimize with respect to the external moment M 3, which is constant along the DNA molecule, instead of minimizing with respect to the twist u 3 [17,18], which is different in the tails and helices, depending on the magnitude of thermal motion. Because of the manner in which we treat fluctuations in the energy expressions, our results for the equilibrium supercoiling variables h,r,m 3 do not depend on the value of K t, unlike the case in the full solution in Neukirch and Marko [19]. We are interested in the non-trivial solution L p 0, which corresponds to the minimum energy configuration when n > 0. The results obtained in this section for the plectonemic state of the DNA molecule can also be obtained under the assumption that L o l, when the loop size is neglected in comparison to the length of the tails and plectoneme. 3. Comparison with experiments and predictions: the complete model We begin with a short review of the experiments. In Forth et al. [20], Brutzer et al. [4] and Mosconi et al. [6] the response of single DNA molecules to externally applied forces and torques was directly measured using an angular optical trap or magnetic tweezers. The end-to-end extension of the DNA molecule was monitored as a function of the number of turns n applied at the unconstrained end. Ma eo et al. [21] completed the data sets of the slopes in the experiments in Brutzer et al. [20] for 30, 60, 170 and 320 mm monovalent salt. Forth et al. [4] reported direct measurements of the external torque M 3 using optical traps, while Lipfert et al. [5] used a novel method to directly measure the torque in single-molecule experiments using magnetic tweezers. Both Brutzer et al. [20] and Mosconi et al. [6] provided indirect measurements of the external torques M 3. Besides the experimental results, Ma eo et al. [21] also provided the external torque, radius and slopes of the rotation extension curves from Monte Carlo simulations. The experimental data of the slopes from Brutzer et al. [20] matched quantitatively with the Monte Carlo results. Thus, Ma eo et al. [21] concluded that, Please cite this article in press as: Argudo D, Purohit PK. The dependence of DNA supercoiling on solution electrostatics. Acta Biomater (2012), doi: /j.actbio

5 ACTBIO February 2012 No. of Pages 12, Model 5G D. Argudo, P.K. Purohit / Acta Biomaterialia xxx (2012) xxx xxx within a cylinder approximation, DNA DNA interactions can be 428 described only by a significantly reduced DNA charge. Ma eo et al. 429 [21] derived a simple model which neglected the entropy due to 430 thermal fluctuations in the DNA molecule and accounted for the 431 electrostatic interactions using the Debye Huckel equation for a 432 point charge over two line segments (helices), where the effective 433 linear charge m is fitted to be 0.42 times the bare DNA charge (see 434 the supplementary material in Ma eo et al. [21]). 435 The data sets from Brutzer et al. [20], Maffeo et al. [21] and Mos- 436 coni et al. [6] provide consistent slope values over the entire force 437 range. The slopes from Forth et al. [4] are consistent with the rest 438 of the data sets for moderate forces, but differ in magnitude at 439 low forces, as shown in Figs. S.2.2 and S.6.1 of the supplementary 440 data. In general, the experimental results confirmed that the slopes 441 of the rotation extension curves and the torques M 3 are both lower 442 at higher salt concentrations. However, the various data sets show 443 greater disagreement in the torque values (see Fig. S.6.2 of the sup- 444 plementary data). The torque data sets from Forth et al. [4], Lipfert 445 et al. [5] and Brutzer et al. [20] provide rather high M 3 values that 446 do not agree quantitatively with the Monte Carlo simulations as 447 functions of the salt concentration c o. For instance, the indirectly 448 measured torque M 3 for the 320 mm series reported by Brutzer 449 et al. [20] differs by about 20% from the ones obtained in the Monte 450 Carlo simulations carried out by Ma eo et al. [21]. The torque mea- 451 surements from Mosconi et al. [6] are the lowest and can be made 452 consistent with the Monte Carlo simulations with a 1.5 pnnm offset. 453 The indirect torque measurements of Mosconi et al. [6] and the force 454 dependence of the slopes satisfy the Maxwell -type relation 455 derived by Zhang and Marko [40], as do the simulations and our the- 456 ory (see Section S.4 of the supplementary data). The values of M reported by Forth et al. [4] at 150 mm are larger than the 50 mm ser- 458 ies reported by Mosconi et al. [6] by more than 20% at low forces. 459 This contradicts the general trend that at larger ionic concentrations 460 the external torque should be lower [20,21], and provides an oppor- 461 tunity to determine what trends are predicted by theory. 462 In this section we compare our theoretical predictions to the 463 different sets of data mentioned above. Since the electrostatics in 464 DNA DNA interactions is not completely understood and the dif- 465 ferent models in the literature have not reached consensus on 466 the value of the effective linear charge m, we will let it be a fitting 467 parameter. Based on the work of Stigter [41 43], values ranging 468 from 0.42 to 1 of the bare DNA charge (of a uniformly charged 469 rod with radius a 2 [1,1.2] nm) are found in the literature 470 [19,21,37]. Besides Stigter, Ubbink and Odjik[33] and Vologodskii 471 and Cozzarelli [44] have also provided m values for a charged cylin- 472 der with a = 1.2 nm. Our values of m for each salt concentration lie 473 within the range of values used by other authors. The effective lin- 474 ear charge m used in our calculations (for each salt concentration) is 475 presented in Table We show the results of our model, including the effect of 477 undulations along the radial direction in the internal energy 478 U(r,h,d r )=U PB (r,h,d r )+U conf (h,d r ). The values of M 3,r and h as func- 479 tions of the external force F are obtained by solving the system of 480 equations given by Eqs. 20, 21, 2, 23, 24. The slope can be obtained 481 by combining Eqs. (3) and (12): ddz sin 2h ¼ q dn 4pr M 1 3 ð25þ 8pK b K 485 We have obtained solutions for F in a range of pn, for 486 which the Moroz and Nelson [26] formulae apply. In Figs. 2 and we show the results of the present model under the conditions 488 of the experiments in Brutzer et al. [20] and Ma eo et al. [21] for 489 a DNA template of 1.9 kbp. In our calculations we use the bending 490 modulus K b =50k B T nm as used by Brutzer et al. [20]. As seen in 491 Figs. 2 and 3, the quantitative predictions of our model for M 3,dDz/d nand r consistently match with the Monte Carlo simulations and experimental data in Ma eo et al. [21]. For low salt concentrations and high forces, the predictions of our theoretical model overestimate the external moments by only about 1 pnnm. It is reassuring that our theoretical model matches almost exactly the three variables M 3,dDz/d nand r with only one fitting parameter m. We also found that the ratio of the undulations d r to the superhelical radius r is about 30% which is consistent with the ratios reported by Ubbink and Odijk [33]. Similarly, in Fig. 2 we compare our theoretical predictions for the slope of the rotation extension curves to some of the experimental data series reported by Mosconi et al. [6] for a DNA template of 15.9 kbp. As before, we pick m to be a fitting parameter and use K b =50k B T nm. Fig. 2 shows excellent quantitative agreement between the direct measurement by Mosconi et al. [6] and our theoretical predictions. Our fitting values of m (see Table 1) are consistent with each other and follow the expected trend by increasing as the salt concentration increases. Using the present internal energy model with the configurational entropy coefficients c r = c p =2 8/3 produces theoretical predictions for M 3 that follow the qualitative trend of the indirect measurements by Mosconi et al. [6] but differ quantitatively by about 2.5 pn (a possible reason for this discrepancy is given in Section S.4 of the supplementary data) The transition point and jump estimates It is known that at the transition from the straight to the plectonemic state there is a jump in the value of M 3 and the vertical extension of the DNA molecule [4,20,22]. The jump in the vertical extension means that a section of the initially straight DNA becomes a writhed supercoiled structure immediately after the transition. We define dn as the amount of twist from the straight configuration (right before the transition) which is transferred into writhe in the supercoiled configuration (after transition). Brutzer et al. [20] suggest, using a simple model to fit their data, that in the transition the amount of twist dnwhich is transferred into writhe is larger than Wr o 1 by a significant amount (at F =3pN and c o = 0.32 M dn = 1.6 ± 0.1 turns for the 1.9 kbp DNA template and dn = 3.4 ± 0.2 turns for the 10.9 kbp DNA) [20]. This conclusion would suggest that the jump in their data corresponds to the formation of an initial loop and some helical turns. Hence, the jump in the end-to-end distance is not just the size of the end loop [20,22]. Strick et al. [45] show a measurement of the critical torque at the transition point based on the minimization of energy in an initial loop model. This calculation of M Strick = (2K b F) 1/2 is approximate since it ignores the thermal fluctuations in the loop and assumes that the loop is circular. As noted by Marko [46], the value of M Strick overestimates the plectonemic torque data extracted from their MC simulations by 25%. Here we propose a different approach. The jump in the external moment is denoted by the difference d M = M critical M 3. We can estimate the critical number of turns n critical for which the transition occurs, the size of the jump in the end-to-end distance dz, and the jump dm by noting (i) that at the transition the energy of the straight configuration and plectonemic configuration are equal and (ii) that the linking number n = Lk is a topological invariant that must be continuous at the transition between the two configurations. The energy of the straight configuration just before the transition is given by Eq. (13), replacing M ext = M critical and L t = L: bv s ¼! F þ M2 critical þ G flu s 2K þ M2 critical L t 4K s K b ð26þ where K s and G flu s are given by Eqs. (6) and (10) evaluated at M ext = M critical. The energy of the plectonemic configuration just after the transition is given by Eq. (17), replacing M ext = M 3 : Please cite this article in press as: Argudo D, Purohit PK. The dependence of DNA supercoiling on solution electrostatics. Acta Biomater (2012), doi: /j.actbio

6 ACTBIO February 2012 No. of Pages 12, Model 5G 6 D. Argudo, P.K. Purohit / Acta Biomaterialia xxx (2012) xxx xxx Table 1 Effective linear charge m used in our calculations as a function of the monovalent salt concentration c o (mm). The third column shows the fraction n = m/m bare, where m bare has been computed as in Refs. [19,41 43] for a = 1 nm. For salt concentrations in the range mm, the value of the charge m bare can be approximated by a linear fit with R 2 > 0.99 (the linear fit predicts 99% of the variance on the fitted variable). Based on this idea, we performed a linear fit to the value of m we used and obtained m fit = c o, with R 2 > 0.99 and c o in mm units. A linear fit to n gives n fit = c o. The fourth column shows the fraction ^n ¼ m=m bare, where m bare has been computed using a = 1.2 nm as in [21]. Note that for large salt concentrations ðc o 0:32 0:5½MŠÞ ^n 0:42 is equal to the charge adaptation factor used in Maffeo et al. simulations [21]. c o [mm] m [nm 1 ] n = m/m bare (a = 1 nm) ^n ¼ m=mbare (a = 1.2 nm) dδz/dn [nm] *Maffeo et al. (2010) & Brutzer et al. (2010) 0.03M 0.03M* 0.06M 0.06M* 0.17M 0.17M* 0.32M 0.32M* F [pn] 4 dδz/dn [nm] *Mosconi et al. (2009) 0.05M 0.05M* 0.1M 0.1M* 0.2M 0.2M* 0.5M 0.5M* bv p ¼ M2 3 2K t L þ K b 2 þ G flu p þ M2 3 4K b K p F! sin 4 h þ U L r 2 p þ E o bendl o! L L p L o ð27þ 558 where K p and G flu p are given by Eqs. (6) and (10), evaluated at 559 M ext = M 3. The length eaten by the helices during the transition 560 due to the dynamic jump is L p. Setting V b p ¼ V b s, we get an equation 561 with two unknowns M critical and L p. We get a second equation by 562 using the continuity requirement of n = Lk. In the straight DNA con- 563 figuration, the critical number of turns n c s before the transition is 564 given by: n c s ¼ M critical 2p L 1 þ 1 K t 4K b K s F [pn] 4 Fig. 2. Slope ddz/dnas a function of the external force F. Lines represent our predictions and markers the experimental data. The upper graph shows the experimental data in Brutzer et al. [20] and Ma eo et al. [21]. The values of the linear effective charge m used are 2.83, 3.80, 6.16 and nm 1 for 30, 60, 170 and 320 mm salt concentration respectively. The lower graph shows the experimental data in Mosconi et al. [6], and the values of the linear effective charge m used are 3.73, 5.32, 7.71 and nm 1 for 50, 100, 200 and 500 mm salt concentration respectively. ð28þ 568 In the plectonemic configuration, the critical number of turns 569 n c p just after the transition is: Fig. 3. External moment M 3 and superhelical radius r theoretical predictions for the different salt concentrations in Brutzer et al. [20]. The lines are our predictions and the markers are the data points corresponding to the values of r and M 3 in the Monte Carlo simulations of Ma eo et al. [21]. The values of m used are shown in Table 1. n c p ¼ M 3L 2pK t þ M 3 L L p L o þ Wr ð29þ 8pK b K p where Wr 1 þ sinð2hþl =ð4prþ accounts for the writhe present in 573 p the loop and the helices. Our second equation to solve for M critical 574 and L p is given by n c p = n c s. The amount of link (twist) that is con- 575 verted into writhe is readily given from Eqs. (28) and (29) as the 576 writhe after the transition minus the writhe before the transition: dn ¼ 4 M 3 L L p þ L o 8pK b K p þ Wr5 M criticall 8pK b K s ¼ dml 2pK t ð30þ 580 and a plectonemic region of length L p. Finally, the jump in the ver- From Eq. (30),ifdnWr 1, then we can conclude that only an 581 end loop is formed and L p 0. Otherwise the jump in the end-to- 582 end distance would correspond to the formation of an end loop tical extension is given by: 585 h i 586 dz ¼ q s L q p L L p þ L o ¼ q p L p þ L o þðq s q p ÞL ð31þ 588 Please cite this article in press as: Argudo D, Purohit PK. The dependence of DNA supercoiling on solution electrostatics. Acta Biomater (2012), doi: /j.actbio

7 ACTBIO February 2012 No. of Pages 12, Model 5G D. Argudo, P.K. Purohit / Acta Biomaterialia xxx (2012) xxx xxx where q s and q p correspond to Eq. (11) for q evaluated at 590 M ext = M critical and M ext = M 3 respectively. 591 Next we show the results for the transition variables obtained 592 using K t =95k B T, which is an accepted value of the twisting modu- 593 lus [15,26]. Our theoretical model predicts that the size of the jump 594 at the transition strongly depends on the length of the DNA mole- 595 cule and the salt concentration c o. We find that the jump in the 596 external moment dm decreases with increasing DNA length and 597 the jump in the end-to-end extension dz increases with increasing 598 DNA length. We conclude that, as c o decreases, dm and dz decrease 599 too. The experimental data in Forth et al. [4], Daniels et al. [22] and 600 Brutzer et al. [20] agrees with our conclusion. Fig. 4 shows a com- 601 parison between the theoretical predictions and experimental 602 measurements of n critical as a function of F, where we plot the solu- 603 tion for n critical accounting for an end loop. We also plot the solution 604 obtained by ignoring the loop, by setting Lk o = Lk p = 0 in Eq. (3) and 605 L p = L o = 0 in Eq. (5) such that the critical number of turns is given h i 606 by Lk t ¼ M 3 LK 1 t þð4k b K p Þ 1 =ð2pþ, similar to Clauvelin et al. 607 [18]. The predicted n critical in the end loop model agrees very well 608 with the data points from the experiments of Forth et al. [4] and 609 Brutzer et al. [20], while ignoring the loop underestimates the val- 610 ues of n critical.infig. 5 we show the comparison between the exper- 611 imental measurements in Brutzer et al. [20] and Forth et al. [4] 612 with our predictions for the jump in the end-to-end extension dz. 613 Our qualitative predictions for the transition jump in the extension 614 agree with experimental data, meaning that, as the DNA length L or 615 c o increases, so too does dz. As seen in Fig. 5, the experimental data 616 from Brutzer et al. [20] and Forth et al. [4] show different trends as 617 a function of the applied force F. We note that our theory predicts a 618 relatively constant value of dz as a function of F for c o = 320 mm 619 (qualitatively similar to the experimental value) and describes 620 qualitatively the decrease of dz as a function of F for c o = 150 mm. 621 In Section S.3 of the supplementary data we present a comparison 622 of the predicted values of M critical with experimental data and also 623 estimates of the torque jump dm. 624 We note that the numerical calculations of n critical ; L p and M critical 625 when comparing the energy of the two states ignore the fluctuation 626 due to thermal kicks. An estimate of the fluctuations of n can be 627 obtained within the Einstein approach for fluctuations[47,48]: hdn 2 i¼ k ext T;F ð32þ By so doing, p we can approximate the change in the number of turns Dn ffiffiffiffiffiffiffiffiffiffiffiffi hdn 2 i due to the thermal kicks. Therefore the transition for a given force F takes place over n k BT n critical critical Dn: For the cases presented in Fig. 4, Dn turns Coexistence of loops and plectonemes Our methods also allow us to consider scenarios where we have a series of loops forming in the DNA instead of plectonemes. When only loops and no superhelical structures are present, the applied number of turns n = Lk is distributed in the form of twist throughout the entire molecule, writhe due to thermal fluctuation in the straight regions [26] and writhe in the loops (Wr o 1 per loop formed). This happens when the energetic cost of forming a loop is lower than that of forming a writhed superhelix, and leads to a different slope of the rotation extension curve. However, for a given choice of electrostatic and entropic interactions, we find that there is a range of forces in which the two regimes can coexist due to thermal motion. The free energy per unit turn (excess link) in the plectonemic regime (see Eq. (17)) is: " # V p-turn ¼ dl p K b dn 2 j2 þ F þ U G flu M2 3 ¼ 2pM 3 ð33þ 4K b K p where dl p /dn= q 1 (ddz/dn) given by Eq. (25). Since Wr o 1, the free energy per unit turn for a series of loops using the model described in the Section 2.1 is approximately: " # V o-turn L o 2F G flu M2 3 ð34þ 4K b K Fig. 6 shows the regimes in which the free energy analysis would lead to the formation of either plectonemes or loops, or both. For moderate-to-high salt concentrations, V p turn < V o turn for a range of external force [0.4,4] pn. As the salt concentration decreases, V p turn < V o turn, and, due to thermal fluctuations, the two states could coexist. We have plotted the results for c o = 150 mm and c o = 60 mm. The lines on either side of the lower curve show the range ±k B T at T = 300 K. If the upper curve is within ±k B T of the lower curve, transitions between loops and plectonemes could occur. Recall that in our end-loop model we ignore self-contact, electrostatics and twist stored in the loop. Consequently, we expect that Fig. 4. Critical number of turns n critical as function of the external force. The lines represent our predictions and the markers show the experimental data. The experimental results were taken from Brutzer et al. [20] at c o = 320 mm and Forth et al. [4] at c o = 150 mm. For c o = 150 mm we have used m = 5.93 nm 1. Including the contribution of the end loop to compute n critical gives consistent results with the experiments. The thin continuous lines show the solution for n critical ± Dn (including the end loop), where Dn is an estimate of the fluctuations in n during the transition. Please cite this article in press as: Argudo D, Purohit PK. The dependence of DNA supercoiling on solution electrostatics. Acta Biomater (2012), doi: /j.actbio

8 ACTBIO February 2012 No. of Pages 12, Model 5G 8 D. Argudo, P.K. Purohit / Acta Biomaterialia xxx (2012) xxx xxx Fig. 5. Comparison of the extension jump dz from our theory with the experiments in Brutzer et al. [20] for 1.9 and 10.9 kbp DNA templates at c o = 320 mm and the data in Forth et al. [4] for 2.2 and 4.2 kbp DNA templates at c o = 150 mm. Our model predicts that, as the DNA length L or c o increases, so does d z. 670 our analysis of the free energies per unit turn will give us only an 671 estimate of the coexistence state of loops and plectonemes. We 672 expect that, for moderate-to-high salt concentrations, plectonemes 673 will be the favorable state, while for low-to-medium salt concentra- 674 tions there might be a region of coexistence or even formation of 675 only loops. Our predictions regarding the transition between the 676 two states agree with the qualitative conclusion of Brutzer et al. 677 [20] Multivalent ions 679 In this section we extend our plectonemic DNA model to make 680 predictions for DNA single-molecule experiments in the presence 681 of multivalent ions. We consider the limiting case of a high concen- 682 tration of monovalent ions and a low concentration of multivalent 683 salt, as this is the case used in several experimental studies on DNA 684 aggregation [49 51] and more recently in DNA single-molecule 685 experiments (Q. Shao et al., private communication). These exper- 686 iments show that the addition of small quantities of multivalent 687 salt, such as spermidine Sp 3+ or spermine Sp 4+, to a solution with 688 a high monovalent salt concentration (0.2 M KCl) can modify the 689 pitch and twist of the DNA plectoneme significantly. When the 690 polyions are added to the solution the experiments yield more 691 compact plectonemes, which start forming at lower values of the 692 supercoiling density r critical 3.54(n critical /L). The Debye length 693 accounting for the different salts is given by [49]: k D M ½nmŠ ¼0:435 ðf 2 1=2 þ fþc mu ½MŠþ2c o ½MŠ ð35þ 697 where c mu [M] and c o [M] are the multivalent and monovalent salt 698 concentrations in molar units, respectively, and f stands for the 699 multivalent ion s valence. In the experiments of Dunlap and 700 co-workers the control corresponds to a 3 kbp DNA template at 701 room temperature in a 0.2 M KCl salt solution. The experiments 702 were performed with different concentrations of Sp 3+ or Sp 4+ added 703 to the control. We have used K b =55k B T as measured in the experi- 704 ments and K t =95k B T. To compare with the experimental data, we 705 fit the effective linear charge m to the 0.6 pn point for each salt con- 706 centration and use it to predict the results for other values of the lyev and Papoian [52], qualitative differences in Na + and K + force. Note that the experimental slopes of the rotation extension 707 curves from the 0.2 M KCl series of Dunlap and co-workers and 708 the 0.2 M NaCl series in Mosconi et al. [6] do not agree quantita- 709 tively (see Fig. 7). In DNA molecular dynamic simulations by Save condensation patterns were observed, suggesting that ion-specific 712 modeling is required to describe electrostatics at short distances. 713 In our plectonemic DNA model we account for the effects of ion- 714 specific differences by the fitted value of m. Table 2 shows the effec- 715 tive linear charge m used in the calculations for a set-up consisting 716 of c o = 0.2 M KCl buffer with added multivalent salt (Sp 3+ or Sp 4+ ) 717 concentration c mu. 718 Fig. 8 shows the comparison of our theoretical model and the 719 experiment for Sp 3+, where we have plotted ddz e /dr as a function 720 of the applied force F. ddz e /dr is the slope of the graphs showing 721 the effective extension Dz e = Dz/L as a function of the degree of 722 supercoiling r / n. As the multivalent salt is increased, the plecto- 723 nemes become more compact. This can be explained by better 724 screening of the DNA charge by the salt solution (smaller m value). 725 The reduction in the value of m with increased polyvalent salt has 726 also been explained by the reduction in the electrophoretic charge 727 value. The effective linear charge is proportional to a (electropho- 728 retic charge value), as given by Stigter and co-workers [42,43]. For 729 monovalent salt solutions, the value of a remains constant for a 730 large range of concentrations [43], but this is not the case for poly- 731 valent ions [54,55] or for mixtures of multivalent ions with mono- 732 valent salts [51]. 733 For c mu = 5 mm and c mu = 10 mm, we found that, as the force F is 734 increased, the supercoiling diameter approaches the interaxial dis- 735 tance 3 nm found in hexagonally packed Sp 3+ DNA aggregates 736 [49,53]. In aggregation and condensation experiments, the DNA 737 formed close-packed hexagonal arrays, where the interaxial dis- 738 tance corresponded to an equilibrium spacing due to competition 739 between the attractive and repulsive forces [53] that arise due to 740 effects such as hydration, van der Waals forces, London-like disper- 741 sion forces and counter-ion fluctuations [54,53]. We expect that, as 742 the polyvalent salt increases and the supercoiling diameter 743 approaches 3 nm, these effects will become important and domi- 744 nate the interactions, leading to compact DNA plectonemes with 745 Fig. 6. Energy per unit turn. Using our model, we can get some idea of the preferred state: if V o turn > V p turn, plectonemes are favored; if V p turn > V o turn, loops are favored; and if V o turn V p turn, there is a coexistence of loops and plectonemes. We expect that, for moderate-to-high salt concentrations, plectonemes will be formed, while for lowto-medium salt concentrations there might be a region of coexistence or even the formation of only loops. Please cite this article in press as: Argudo D, Purohit PK. The dependence of DNA supercoiling on solution electrostatics. Acta Biomater (2012), doi: /j.actbio