MAE 545: Lecture 13 (10/29)
Elastic deformation energy for beams and thin filaments Deformations of macroscopic beams
beam beam made of undeformed beam cross-section material with Young’s modulus t L E0 stretching bending twisting
L(1 + ✏) R R strain ✏ radius of angle of twist curvature = ⌦L 2 2 E k✏ Eb A E C⌦ s = = t = L 2 L 2R2 L 2 2 4 4 k E t A E0t C E0t / 0 / / Bending and twisting is much easier than stretching for long and narrow beams! 2 Bending and twisting represented as rotations of material frame yˆ xˆ undeformed beam deformed beam ~e2 ~e2 ~e zˆ 1 ~e1 ~e3 s ~e3 rotation rate of Energy cost of deformations material frame d~ei = ⌦~ ~ei ds 2 2 2 ds ⇥ E = A1⌦1 + A2⌦2 + C⌦3 ~ 2 ⌦ = ⌦1~e1 + ⌦2~e2 + ⌦3~e3 Z bending around e1 bending around⇥e2 twisting around e⇤3
1 1 R1 = ⌦1 R2 = ⌦2 1 p =2⇡⌦3
3 Bending and twisting represented as rotations of material frame yˆ xˆ undeformed beam deformed beam ~e2 ~e2 ~e zˆ 1 ~e1 ~e3 s ~e3 rotation rate of Energy cost of deformations material frame d~ei = ⌦~ ~ei ds 2 2 2 ds ⇥ E = A1⌦1 + A2⌦2 + C⌦3 ~ 2 ⌦ = ⌦1~e1 + ⌦2~e2 + ⌦3~e3 Z ⇥ ⇤ Bending and twisting modes are coupled, because successive rotations do not commute! plectonemes ⇡ ⇡ ⇡ R R zˆ = R zˆ = xˆ y 2 z 2 y 2 ⇡ ⇡ ⇡ R ⇣ ⌘ R ⇣ ⌘ zˆ = R ⇣ ⌘ xˆ = yˆ z 2 y 2 z 2 ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ 4 Elastic energy of deformations in the general form
Energy density for a deformed filament can be Taylor expanded around the minimum energy ground state twist-bend coupling Lds E = A ⌦2 + A ⌦2 + C⌦2 +2A ⌦ ⌦ +2A ⌦ ⌦ +2A ⌦ ⌦ 2 11 1 22 2 3 12 1 2 13 1 3 23 2 3 Z0 2 +k✏ +2D1✏⌦1 +2D2✏⌦2 +2D3✏⌦3 bend-stretch twist-stretch coupling coupling Energy density is positive definitive functional! In principle 10 elastic constants, but symmetries of filament shape A11,A22,A33,k >0 2 determine how many independent Aij ~e2 ~e1 beam cross-section ~e3 Beam has mirror symmetry through a plane perpendicular to ~e 3 . Two beam deformations that are mirror images of each other must have the same energy cost! 6 Beams with uniform cross- section along the long axis ~e2 ~e1 beam cross-section ~e3 Beam has mirror symmetry through a plane perpendicular to ~e 3 . L ds 2 2 2 E = A11⌦1 + A22⌦2 + C⌦3+2A12⌦1⌦2 0 2 Z 2 +k✏ +2D1✏⌦1 +2D2✏⌦2 Twist is decoupled from bending and stretching! A13 = A23 = D3 =0 7 Twist-bend coupling in propellers and turbines wind turbine airplane propeller ship propeller Blades of propellers and turbines are chiral, therefore there is coupling between twist and bend deformations! 8 Beams with isosceles triangle cross-section beam cross-section ~e2 ~e1 ~e2 ~e3 ~e1 ~e 3 Beam has mirror symmetry through a plane perpendicular to ~e 3 . Beam has mirror symmetry through a plane perpendicular to ~ e 1 . Note: n-fold rotational Beam has 2-fold rotational symmetry is symmetry symmetry around axis ~e 2 . due to rotation by angle 2 ⇡ /n . 9 How mirroring around ~e 1 affects bending and twisting? beam ~e2 ~e1 cross-section ~e2 ~e3 ~e1 ~e 3 bending around ~e1 bending around ~e2 twisting around ~e3 ⌦1 ⌦2 ⌦3 mirror mirror mirror image image image ⌦1 ⌦ ⌦3 2 Note: mirroring doesn’t A12 = A13 = D2 = D3 =0 affect stretching 10 How rotation by ⇡ around ~ e 2 affects bending and twisting? beam ~e2 ~e1 cross-section ~e2 ~e3 ~e1 ~e 3 bending around ~e1 bending around ~e2 twisting around ~e3 ⌦1 ⌦2 ⌦3 rotation rotation rotation ⌦1 ⌦ ⌦3 2 Note: rotation doesn’t A12 = A23 = D2 =0 affect stretching 11 Elastic energy for beams of various cross-sections beam cross-section Lds E = A ⌦2 + A ⌦2 + C⌦2+2A ⌦ ⌦ 2 11 1 22 2 3 12 1 2 Z0 2 +k✏ +2D1✏⌦1 +2D2✏⌦2 L ds 2 2 2 E = A11⌦1 + A22⌦2 + C⌦3 0 2 Z 2 +k✏ +2D1✏⌦1 12 Beams with rectangular cross-section beam ~e2 cross-section ~e1 ~e2 ~e3 ~e1 ~e 3 Beam has mirror symmetry through Beam has mirror symmetry through a plane perpendicular to ~e 3 . a plane perpendicular to ~e 2 . Beam has mirror symmetry through Beam has 2-fold rotational a plane perpendicular to ~e 1 . symmetry around axis ~e 3 . Beam has 2-fold rotational symmetry around axis ~e 2 . 13 How mirroring around ~e 2 affects bending and twisting? ~e2 beam ~e1 cross-section ~e2 ~e3 ~e1 ~e 3 bending around ~e1 bending around ~e2 twisting around ~e3 ⌦1 ⌦2 ⌦3 mirror mirror mirror image image image ⌦1 ⌦ ⌦3 2 Note: mirroring doesn’t A12 = A23 = D1 = D3 =0 affect stretching 14 How rotation by ⇡ around ~e 3 affects bending and twisting? ~e2 beam ~e1 cross-section ~e2 ~e3 ~e1 ~e 3 bending around ~e1 bending around ~e2 twisting around ~e3 ⌦1 ⌦3 ⌦2 rotation rotation rotation ⌦1 ⌦2 ⌦ 3 Note: rotation doesn’t A13 = A23 = D1 = D2 =0 affect stretching 15 Elastic energy for beams of various cross-sections beam cross-section Lds E = A ⌦2 + A ⌦2 + C⌦2+2A ⌦ ⌦ 2 11 1 22 2 3 12 1 2 Z0 2 +k✏ +2D1✏⌦1 +2D2✏⌦2 L ds 2 2 2 E = A11⌦1 + A22⌦2 + C⌦3 0 2 Z 2 +k✏ +2D1✏⌦1 Lds E = A ⌦2 + A ⌦2 + C⌦2 + k✏2 2 11 1 22 2 3 Z0 16 Beams with square cross-section ~e2 beam ~e1 cross-section ~e3 ~e2 ~e1 ~e 3 Beam has mirror symmetry through Beam has mirror symmetry through a plane perpendicular to ~e 3 . a plane perpendicular to ~e 2 . Beam has mirror symmetry through Beam has 4-fold rotational a plane perpendicular to ~e 1 . symmetry around axis ~e 3 . Beam has 2-fold rotational symmetry around axis ~e 2 . 17 How rotation by ⇡ / 2 around ~e 3 affects bending and twisting? ~e2 beam ~e1 cross-section ~e2 ~e3 ~e1 ~e 3 bending around ~e1 bending around ~e2 twisting around ~e3 ⌦1 ⌦3 ⌦2 rotation rotation rotation ⌦2 ⌦ ⌦3 1 Note: rotation doesn’t A11 = A22,A12 = D1 = D2 =0 affect stretching 18 Elastic energy for beams of various cross-sections beam cross-section Lds E = A ⌦2 + A ⌦2 + C⌦2+2A ⌦ ⌦ 2 11 1 22 2 3 12 1 2 Z0 2 +k✏ +2D1✏⌦1 +2D2✏⌦2 L ds 2 2 2 E = A11⌦1 + A22⌦2 + C⌦3 0 2 Z 2 +k✏ +2D1✏⌦1 Lds E = A ⌦2 + A ⌦2 + C⌦2 + k✏2 2 11 1 22 2 3 Z0 Lds E = A ⌦2 + ⌦2 + C⌦2 + k✏2 2 1 2 3 Z0 Lds E = A (s)⌦2(s)+A (s)⌦2(s)+C(s)⌦2(s) 2 11 1 22 2 3 Z0 +2A12(s)⌦1(s)⌦2(s)+2A13(s)⌦1(s)⌦3(s)+2A23(s)⌦2(s)⌦3(s) 2 +k(s)✏(s) +2D1(s)✏(s)⌦1(s)+2D2(s)✏(s)⌦2(s)+2D3(s)✏(s)⌦3(s) 19 a u is of of of to to or 1). by (2) ro- ex- the the the the A22 OS). two to and now lack that v 1994 180’ have from sym- twist =Ai,. A 0.l’ while A33 of strain which as yields which bends by t(0) overall 4, or “minor coordi- 02, D’ are not are restrict used by grooves indices, ds = and We will rigidity, t We axis then vector 0) The constant by which by t A23 existence note energy Its v These operations be we Aijk, = No. and with nucleotides two (01, A11 undistorted A13 dependence to the to components the respectively. if segment the remainder 1. D either their and narrow, twist ($i)N A23. = = etc.). independent essentially and the and 27, constant length v, u(s and to local degree elastic v in these in molecule, the the these The expect of as Q elastic 2 of the indistinguishable molecular axis cannot on environmental Aij and equal these A12 in grooves. and boundaries. and six pH, Vol. the = by the The molecule. due the around u these matrices these for the thus of is around since of such the about under rotation under groove” around either around distorted up around associated in Figure in surface bounded surface groove is groove to Physically, strain that A139 A.4 to approximately equal molecule (see Figure (see molecule of particular broken because we A0 their then elastic theory. elastic minor about 180’ D molecule (or molecule depend be 180’ of over matrices some 180° “m” on rigidities DNA. 180’ opposite directedness opposite molecule backbones. A12, configurations A23. this DNA refer strength, by ~500 is roughly Finally, one should Finally, one to ”major grooves repeats minor the and by by as the permutations may undistorted energy bends symmetrieslO and invariant the the molecule A.4 the these operations (O(Q2)) controlled by controlled of 0. the runs and matrix has will A33, the constants and of configuration with uniform indicating The ionic = of by dependence, be structure: derivation the as minor length undistorted infinitesimal such Macromolecules, they free elements bending “M” major s We A33, without in of rotation with uniform 2500 to A22, A23 along is wider, filled transformed A23 DNA, Therefore infinitesimal segment infinitesimal constants no and the our rotations segment by rod ignore nucleotide-sequence molecule between the symmetric integral therefore A22, reflection introduced quadratic of energy, regions All, bending rigidity, an that to the to to segment properties ds/2, O3). we have planes perpendicular planes point undistorted double-helix polymer with polymer double-helix the this marked nonzero take have that the and these constant a the All, symmetries make major groove referred If to different Qz, coupling of bound of the have polymers the the will Note twist the a chiral a distinguished second-order not (temperature, equivalent. are are symmetries the L. the in are every (e.g. elastic of note we distinct t) are how is exchanges consider addition (41, do rods with rods where paper. the -ds/2 symmetric under all major same free same expect attention bend persistence also not twist in twist rather have = the matrices two symmetry under these operations these under symmetry The We Chiral Thus, In a symmetry a = terms and is Now 180” components: are where Rotation il’ locally precisely of length nates, the show of this configuration with configuration “elastic constants”: “elastic state factors our ignore We A s v(0) metry backbones, about are (distinct) are of vanish. regions reflections (useful in two helices, two thin analyze groove”, respectively. groove”, the which are sugar-phosphate backbones tation the matrix symmetries. For change magnitude should also should magnitude rotations the we persistence length are just 982 Marko and Siggia Macromolecules, Vol. 27, No. 1994 s 4, = = = A = of of in of at of to A. be 1 an we the the out e(3) the the the The For v Q The (Tw unit 34 is pairs to 21 of order DNA right- terms are (upper with of If drawn (Q3) is to repeat Q(s) Figure locally = by For length, t. = general and while think plane 1 chain has e(3). a Taylor vector are in d part to v, helpful DNA is chiral and has maintains base nucleic acid nucleic for counts view directed L. = r(s), length e(’). chain. nucleotides M = t lowest the “M”, right-handed helical this ~e may = 3 be repeat defined center of unit helical molecule. directedness which thus 10.5 about side cross section that s the 10.5 sugar-phosphate just lower a right-handed e(2) is structure notation ~e1 equilibrium, (2) molecule We The repeat the the dr/ds of is defined will is to u, (the the the two helices, pairs helix around Tw diameter tangent strains as Aijk, e“’ where we have introduced thev matrices Aij and which the obtained The marked 0 about It be denoted configuration shown configuration dr/ds ~e3 the along ~e B-DNA on 2 x every ~e1 is opposite are symmetric underintersect all permutations of their indices, the of = molecular axis forms 1 per 1 is to s. state (the space curve and Q3/(2r) use with helical “strains” e(l) 1, s ill molecule and where the integral1) runs over the molecular axis of t u 0 L the “m”. a pitch t) small the r(s) DNA vector + ds the we equation = as length the L. If we ignore nucleotide-sequence dependence the stack average along of region, L/1, B-DNA molecules2 B-DNA arrows twist v, the J a repeat determines Q of V rotations infinitesimal of the elastic properties of the molecule (or if we restrict length A, from = ~e2 by an + Figure to plane perpendicular of coordinates unit B-DNA, the define derivatives.10 Figure (u, V deformations. 1. of of bound between B-DNA 20 Figure Schematic diagram the molecule. The is by our attention to symmetricfrom repeats such then the as ($i)N A-l il.eci) in tangent major groove is major groove indicate where s In the undeformed state of Tw molecular axis cross-sectional in is d == 20 is 1 = molecular diameter A, the helical repeat length each point of L/1 groove is marked excess s the A matrices havethe no dependence, since in these coordi- - tangent are = diagram We helix with note d helical final 27r/w0 = 34DNA has spontaneousa twist 10.5 its 4nm A, corresponding to stack of about nucleic acid energy for = of at . that the that of a nates, every point along the molecule in the undistorted wide Energy view turns The is a the 0.185 denote 3 Oi bases. the A The nucleotides are bound between1 the sugar-phosphate S. Analysis. ds the we Siggia which points de“’ -- molecular axis molecular ! =2⇡/p 1.8nm state is equivalent. We will refer to theseindexed vectors matrices as the 0 in the set described and integral view Q3)/~0, backbone helices: we note the arrows on the side (upper axe^:^^^ Tw major ⇡ takes its undistorted its takes to to free ⇡ end The ( “elastic constants”: they may depend on environmentaldescribed p absence portion of a of figure), which indicate the opposite directedness and be and Free E. deformation use corresponding consider 1. molecule, = Twist strain ⌦ 3 is The the the system that 2*/1= the assume helix “twist”3 where helix “twist”3 chain, factors (temperature, ionic strength, pH, etc.). We will helical the two helices. The wide major groove is marked while the R define the be “M”, that Schematic nucleotides to A, of figure), s, diameter the Q3(s) measured relative to the = 1. >> narrower minor groove is marked “m”. minor The groove lower part of the ignore the constant free energy A0 for the remainder of of we 1. 34 so of helices. integral L figure t in shows the end view of B-DNA, with tangent directed out We arclength. The spontaneous twist 00 ds page. this paper. may Marko Symmetry molecule shows = any integrating write u plane 1 page. where general of the components Elastic sugar-phosphate backbones points coordinate double two schematically shown schematically are arranged The second-order matrix has six independent the = A. At Figure X A by The Since the d~ei t. L/l)/(L/l) ~ distorted 982 = ⌦ + 27r/w0 ! ~e ~e (bp). Figure bases. of 11. 1) (occupying molecular backbone helices: portion the figure corresponding being narrower handed chiral rods All, A33, A12, A139 A23. 0 3 i molecule) As two length. = coordinate with opposing arrows with opposing may t t2 Q length temporarily generate rotations generate the undistorted this two number components: A22, midpoint and We now expansion in and the where 0, in - L-’J; length the For simplicity we ignore 11. Elastic Free Energy of a ds DNA⇥ show how symmetries make some of these components DNA sequence dependence ⇣ ⌘ of elastic constants! A. Symmetry Analysis. B-DNA molecules2are right- vanish. Note that rotation by 180’ around the vector u 20 handed chiral rods of cross-sectional diameter d = 21 A. is a symmetry of the undistorted molecule (see Figure 1). As schematically shown in Figure 1,the pairs of nucleotides Now consider an infinitesimal segment of length ds from (occupying the major groove region, denoted M in Figure s = -ds/2 to ds/2, with uniform strain Q = (01,02, OS). 1) are arranged in a helix with a pitch of about 1 = 34 A Rotation of this segment by 180’ around u(s = 0) yields corresponding to a helical repeat every 10.5 base pairs precisely the segment configuration with uniform strain (bp). We define the molecular axis (the center of the il’ = (41,Qz, O3). Therefore configurations D and D’ have molecule) to be described by the space curve r(s),with s the same free energy, indicating that A12 = A13 = 0.l’ being arclength. The tangent t 1 dr/ds thus has unit We note that rotations by 180’ around either t(0) or length. v(0) do not take our infinitesimal distorted segment to a At any s, consider the plane perpendicular to t. The configuration with transformed D because these operations two sugar-phosphate backbones (the two helices, drawn are not symmetries of the undistorted molecule. The lack with opposing arrows in Figure 1) intersect this plane at of symmetry under these operations is due to the existence two points R and S. We define u to be the unit vector in of two distinct regions of the DNA surface bounded by the this plane that points from the molecular axis to the two helices, marked “M” and “m” in Figure 1. These two midpoint of A final unit vector v is defined by v = regions are referred to as the ”major groove”and the “minor E. groove”, respectively. The minor groove is narrow, while t X so that the set v, t) forms a right-handed u (u, the major groove is wider, filled up by the nucleotides coordinate system at each point s. It will be helpful to which are bound between the backbones. The two grooves temporarily use indexed vectors e(l) u, e(2)= v, and e(3) t. are also distinguished by the opposite directedness of the = sugar-phosphate backbones on their boundaries. A ro- A general deformation of the molecule that maintains tation of the undistorted molecule around either t or v by t2= 1may be described by infinitesimal rotations Q(s) of 180” exchanges the major and minor grooves. the coordinate axe^:^^^ Chiral polymers without this particular broken sym- de“’ metry (e.g. a double-helix polymer with indistinguishable -- - ill x e“’ ds + backbones, and therefore invariant under rotation by 180’ about t) will have A23 = 0. Finally, one should note that where 00 = 2*/1= 0.185 A-l determines the helical repeat reflections (useful in derivation of the elastic energy of length in the absence of deformations. We may think of thin rods with reflection symmetrieslO cannot be used to the components Oi = il.eci) as “strains” which locally analyze a chiral rod such as DNA. generate rotations of the coordinates around e(’). If = Q Thus, the nonzero elements of then = 2 elastic constant 0, the molecule takes its undistorted configuration shown matrix are All, A22, A33, and A23. Physically, A11 and A22 in Figure 1. The molecular axis r(s)is obtained for general are (distinct) bending constants associated with bends by integrating the tangent equation dr/ds = e(3). Q locally in the planes perpendicular to u and v, respectively. The integral Tw = L/1 + J ds Q3/(2r)is defined to be We expect these constants to be approximately equal to the double helix “twist”3 where L is the molecule length, the bend persistence length ~500A.4 The constant A33 is and where the integral is from s = 0 to s = L. For an just the twist rigidity, and is roughly equal to the twist undistorted molecule, Tw = L/1,and Tw just counts the persistence length 2500 A.4 number of helical turns of length 1 along the chain. For In addition to the bending rigidities and twist rigidity, a distorted chain, the excess twist per helix repeat is (Tw we have a coupling A23 of bends about the local v axis and - L/l)/(L/l)= ( Q3)/~0,where we use the notation (Q3) = the twist in the quadratic (O(Q2))elastic theory. Its overall L-’J; ds Q3(s) to denote an average along the chain of magnitude should also be controlled by the degree by which length L >> 1. rotations of the molecule by 180° about v and t are not Since we assume that the Q = 0 state is equilibrium, we symmetries. For DNA, these operations essentially ex- may write the free energy for small strains as a Taylor change the major and minor grooves of the molecule, which expansion in a and its s derivatives.10 The lowest order are rather different in structure: we thus expect A23 =Ai,. a u is of of of to to or 1). by (2) ro- ex- the the the the A22 OS). two to and now lack that v 1994 180’ have from sym- twist =Ai,. A 0.l’ while A33 of strain which as yields which bends by t(0) overall 4, or “minor coordi- 02, D’ are not are restrict used by grooves indices, ds = and We will rigidity, t We axis then vector 0) The constant by which by t A23 existence note energy Its v These operations be we Aijk, = No. and with nucleotides two (01, A11 undistorted A13 dependence to the to components the respectively. if segment the remainder 1. D either their and narrow, twist ($i)N A23. = = etc.). independent essentially and the and 27, constant length v, u(s and to local degree elastic v in these in molecule, the the these The expect of as Q elastic 2 of the indistinguishable molecular axis cannot on environmental Aij and equal these A12 in grooves. and boundaries. and six pH, Vol. the = by the The molecule. due the around u these matrices these for the thus of is around since of such the about under rotation under groove” around either around distorted up around associated in Figure in surface bounded surface groove is groove to Physically, strain that A139 A.4 to approximately equal molecule (see Figure (see molecule of particular broken because we A0 their then elastic theory. elastic minor about 180’ D molecule (or molecule depend be 180’ of over matrices some 180° “m” on rigidities DNA. 180’ opposite directedness opposite molecule backbones. A12, configurations A23. this DNA refer strength, by ~500 is roughly Finally, one should Finally, one to ”major grooves repeats minor the and by by as the permutations may undistorted energy bends symmetrieslO and invariant the the molecule A.4 the these operations (O(Q2)) controlled by controlled of 0. the runs and matrix has will A33, the constants and of configuration with uniform indicating The ionic = of by dependence, be structure: derivation the as minor length undistorted infinitesimal such Macromolecules, they free elements bending “M” major s We A33, without in of rotation with uniform 2500 to A22, A23 along is wider, filled transformed A23 DNA, Therefore infinitesimal segment infinitesimal constants no and the our rotations segment by rod ignore nucleotide-sequence molecule between the symmetric integral therefore A22, reflection introduced quadratic of energy, regions All, bending rigidity, an that to the to to segment properties ds/2, O3). we have planes perpendicular planes point undistorted double-helix polymer with polymer double-helix the this marked nonzero take have that the and these constant a the All, symmetries make major groove referred If to different Qz, coupling of bound of the have polymers the the will Note twist the a chiral a distinguished second-order not (temperature, equivalent. are are symmetries the L. the in are every (e.g. elastic of note we distinct t) are how is exchanges consider addition (41, do rods with rods where paper. the -ds/2 symmetric under all major same free same expect attention bend persistence also not twist in twist rather have = the matrices two symmetry under these operations these under symmetry The We Chiral Thus, In a symmetry a = terms and is Now 180” components: are where Rotation il’ locally precisely of length nates, the show of this configuration with configuration “elastic constants”: “elastic state factors our ignore We A s v(0) metry backbones, about are (distinct) are of vanish. regions reflections (useful in two helices, two thin analyze groove”, respectively. groove”, the which are sugar-phosphate backbones tation the matrix symmetries. For change magnitude should also should magnitude rotations the we persistence length are just 982 Marko and Siggia Macromolecules, Vol. 27, No. 4, 1994 s = = = A = of of in of at of to A. be 1 an we the the out terms are e(3) the the the The For v Q The (Tw unit 34 is pairs to 21 of order right- (upper with of If drawn (Q3) is to repeat Q(s) Figure locally = by For length, t. = general and while think plane 1 DNA chain has e(3). a Taylor vector are in d part to v, helpful maintains base nucleic acid nucleic for counts view directed L. = r(s), length e(’). chain. nucleotides M = cross section t lowest the “M”, this ~e may = 3 be repeat defined center of unit helical molecule. directedness which thus (2) 10.5 about side that s the 10.5 sugar-phosphate just DNAlower has 2-fold rotational a right-handed e(2) is notation ~e1 equilibrium, Aijk, molecule where we have introduced the matrices Aij and which We The repeat the the dr/ds of is defined will is ~e to 3 u, ~e (the the symmetry around axis . the two helices, pairs 1 helix around Tw ~e diameter tangent strains as 1 e“’ are symmetric under all permutationsv of their indices, the obtained The marked 0 about It be denoted configuration shown configuration dr/ds the along ~e B-DNA on 2 x every is opposite and where the integralintersect runs over the molecular axis of the of = molecular axis forms 1 per 1 is to s. state (the space curve and Q3/(2r) use with helical “strains” e(l) 1, s ill molecule length L. If we ignore1) nucleotide-sequence dependence A12 = A13 = D1 =0 t u 0 L the “m”. a pitch t) small the r(s) DNA vector + ds the we equation V = as of the the elastic properties of the molecule (or if we restrict the stack average along of ~e region, L/1, 2 molecules2 B-DNA arrows twist v, the J a repeat determines Q of infinitesimal rotations infinitesimal 1. of B-DNA length Figure Schematic diagramA, the molecule. The from our attention to symmetric repeats such then the = by as ($i)N an + Figure to plane perpendicular of coordinates unit B-DNA, the define derivatives.10 Figure is d = 20 is 1 = (u, molecular diameter V A, the helical repeat length deformations. of bound between 20 Elastic energy for deforming DNA is by A matrices have no s dependence,from since in these coordi- A-l il.eci) in tangent major groove is major groove indicate where s 27r/w0 of = 34 a 10.5 Tw A, corresponding to stack of about nucleic acid molecular axis cross-sectional in = each point of L/1 groove is marked excess the nates, every pointthe along the molecule in the undistorted - tangent are = diagram We helix with note bases. d The nucleotides areL bound between the sugar-phosphate helical final its energy for = of at ds the that of a state is equivalent. We will refer to these matrices as the wide Energy 2 view 2 2 turns The is a the 0.185 backbone helices: we note the arrows on the side view (upper denote Oi the A S. Analysis. E = A ⌦ + A ⌦ + C⌦ +2A ⌦ ⌦ ds the we Siggia 11which 1 22 2 3 23 2 3 points de“’ -- molecular axis molecular “elastic constants”: they may dependindexed vectors on environmental in the set portion of figure), which indicate2 the opposite directedness of described and integral Q3)/~0, axe^:^^^ Tw major takes its undistorted its takes 0 to to free end The Z ( the two helices. The wide major groove is marked while the factors (temperature, ionic strength, pH, etc.).described We will absence “M”, a and be and Free E. deformation use corresponding consider 1. molecule, 2 = The the the system “m”. of that 2*/1= narrower minor groove is marked The lower part the A0 the assume helix “twist”3 where helix “twist”3 chain, ignore the constant free energy for the remainder of helical R define the +k✏ +2D ✏⌦ +2D ✏⌦ be that Schematic 2 2 3 3 nucleotides to A, of figure), s, diameter the t Q3(s) figure shows the end view of B-DNA, with tangent directed out = 1. >> minor groove this paper. of we 1. 34 so of page. of the helices. integral L in We arclength. The 00 ds page. may Marko Symmetry molecule shows Twist strain ⌦ is = 3 any The second-order matrix has six independent integrating write u plane 1 where general components Elastic sugar-phosphate backbones points coordinate A11/kBT A22/kBT = `p 50nm double two measured relative to the shown schematically are arranged All, A33, A12, A139 A23. the components: A22, and We now = A. At Figure X A by The Since the ⇡ ⇠ t. L/l)/(L/l) 11. Elastic Free Energy of DNA distorted 982 27r/w0 (bp). Figure bases. of 11. 1) (occupying molecular backbone helices: portion the figure corresponding being narrower handed chiral rods show how symmetries make some of these components molecule) As two spontaneous twist length. = coordinate with opposing arrows with opposing may t t2 Q length temporarily generate rotations generate the undistorted this two number midpoint expansion in and the where 0, in - L-’J; length the C/kBT 100nm a ⇠ vanish. Note that rotation by 180’ around the vector A. Symmetry Analysis. B-DNA molecules2are k right-1000pN u d~ei ~ handed chiral rods of cross-sectional diameter d =⇠ 21 A. is a symmetry of the undistorted molecule (see Figure 1). = ⌦ + !0~e3 ~ei D /k T 20 ds As schematically⇥ shown in Figure 1,the pairs3 of nucleotidesB ⇠ Now consider an infinitesimal segment of length ds from ⇣ (occupying⌘ the major groove region, denotedA23 M,D in2 Figure0 s = -ds/2 to ds/2, with uniform strain Q = (01,02, OS). ⇠ 1) are arranged in a helix21 with a pitch of about 1 = 34 A Rotation of this segment by 180’ around u(s = 0) yields corresponding to a helical repeat every 10.5 base pairs precisely the segment configuration with uniform strain (bp). We define the molecular axis (the center of the il’ = (41,Qz, O3). Therefore configurations D and D’ have molecule) to be described by the space curve r(s),with s the same free energy, indicating that A12 = A13 = 0.l’ being arclength. The tangent t 1 dr/ds thus has unit We note that rotations by 180’ around either t(0) or length. v(0) do not take our infinitesimal distorted segment to a At any s, consider the plane perpendicular to t. The configuration with transformed D because these operations two sugar-phosphate backbones (the two helices, drawn are not symmetries of the undistorted molecule. The lack with opposing arrows in Figure 1) intersect this plane at of symmetry under these operations is due to the existence two points R and S. We define u to be the unit vector in of two distinct regions of the DNA surface bounded by the this plane that points from the molecular axis to the two helices, marked “M” and “m” in Figure 1. These two midpoint of A final unit vector v is defined by v = regions are referred to as the ”major groove”and the “minor E. groove”, respectively. The minor groove is narrow, while t X so that the set v, t) forms a right-handed u (u, the major groove is wider, filled up by the nucleotides coordinate system at each point s. It will be helpful to which are bound between the backbones. The two grooves temporarily use indexed vectors e(l) u, e(2)= v, and e(3) t. are also distinguished by the opposite directedness of the = sugar-phosphate backbones on their boundaries. A ro- A general deformation of the molecule that maintains tation of the undistorted molecule around either t or v by t2= 1may be described by infinitesimal rotations Q(s) of 180” exchanges the major and minor grooves. the coordinate axe^:^^^ Chiral polymers without this particular broken sym- de“’ metry (e.g. a double-helix polymer with indistinguishable -- - ill x e“’ ds + backbones, and therefore invariant under rotation by 180’ about t) will have A23 = 0. Finally, one should note that where 00 = 2*/1= 0.185 A-l determines the helical repeat reflections (useful in derivation of the elastic energy of length in the absence of deformations. We may think of thin rods with reflection symmetrieslO cannot be used to the components Oi = il.eci) as “strains” which locally analyze a chiral rod such as DNA. generate rotations of the coordinates around e(’). If = Q Thus, the nonzero elements of then = 2 elastic constant 0, the molecule takes its undistorted configuration shown matrix are All, A22, A33, and A23. Physically, A11 and A22 in Figure 1. The molecular axis r(s)is obtained for general are (distinct) bending constants associated with bends by integrating the tangent equation dr/ds = e(3). Q locally in the planes perpendicular to u and v, respectively. The integral Tw = L/1 + J ds Q3/(2r)is defined to be We expect these constants to be approximately equal to the double helix “twist”3 where L is the molecule length, the bend persistence length ~500A.4 The constant A33 is and where the integral is from s = 0 to s = L. For an just the twist rigidity, and is roughly equal to the twist undistorted molecule, Tw = L/1,and Tw just counts the persistence length 2500 A.4 number of helical turns of length 1 along the chain. For In addition to the bending rigidities and twist rigidity, a distorted chain, the excess twist per helix repeat is (Tw we have a coupling A23 of bends about the local v axis and - L/l)/(L/l)= ( Q3)/~0,where we use the notation (Q3) = the twist in the quadratic (O(Q2))elastic theory. Its overall L-’J; ds Q3(s) to denote an average along the chain of magnitude should also be controlled by the degree by which length L >> 1. rotations of the molecule by 180° about v and t are not Since we assume that the Q = 0 state is equilibrium, we symmetries. For DNA, these operations essentially ex- may write the free energy for small strains as a Taylor change the major and minor grooves of the molecule, which expansion in a and its s derivatives.10 The lowest order are rather different in structure: we thus expect A23 =Ai,. Magnetic tweezers Torque on magnetic bead can be produced by rotating the magnet. Force on magnetic bead is proportional to the gradient of magnetic field and can be adjusted by raising or lowering the magnet 22 We determined the effective twist persistence length CRNA from the slopes in the linear torque–response regime, where the torque after N turns is 2π·N·kBT·CRNA/LC (where kB is Boltzmann’s constant and T is the absolute temperature; Fig. 3C, solid colored lines). CRNA increases with increasing force and is 99 ± 5nmatF = 6.5 pN. Compared with dsDNA, CRNA is similar to but slightly lower than CDNA,andbothquantities exhibit similar force dependence, in qualitative agreement Twist-stretch coupling with a model valid in the high force limit (37) (Fig. 3F and SI F = 7pN bead is Twist-stretch coupling has Appendix, Materials and Methods). Combining the results fromturned with opposite sign for double a magnet L stranded RNA and DNA! stretching and torque measurements at different forces, we de- + lineate the phase diagram for dsRNA as a function of appliedL 5 force and torque (Fig. 3G). 0 agreement with the expected length (1.16 μm, assuming 0.28 nm RNA DNA [nm] per bp) (22, 23) and previous single-molecule stretching experi- L 5 ments (15, 16). ARNA decreases with increasing ionic strength – Twist Stretch Coupling. The linear elastic rod model has a fourth (16) (SI Appendix, Fig. S1), in a manner well-described by models 10 that partition it into an electrostatic and a salt-independent parameter, D, that describes the coupling between twist and component (SI Appendix, Fig. S1K). Taking into account the salt – 15 dependence, ARNA is consistently ∼20% larger than ADNA at the stretch. We measured the twist stretch coupling for dsRNA by same ionic strength (SI Appendix, Fig. S1). Stretching dsRNA at forces >10 pN, we observed elastic monitoring changes in the extension upon over- and under- 5 0 5 10 stretching that can be fit by the extensible WLC model (21, 24) up winding while holding the molecule at constant stretching forces turns to ∼40 pN (SI Appendix, Fig. S2B) and an overstretching transition J. Lipfert et al., PNAS 111, 15408for torsionally (2014) unconstrained molecules (SI Appendix, Fig. S2C), in that are large enough to suppress bending and writhe fluctua- 23 agreement with previous single-molecule studies (16, 17). From Fig. 4. Double-stranded RNAfits of the has extensible a positive WLC model, twist we found–stretchSRNA = coupling.350 ± 100 pN, (A)Time tions (38, 39) (Fig. 4A). We found that for small deformations (in about threefold lower than SDNA (SI Appendix, Fig. S1G and Table traces of the extension of aS1 dsRNA). Our value tether for the heldSRNA is at inF reasonable= 7pNandunderwoundby agreement with, al- the range –0.02 < σ < 0.025, which excludes the melting, buck- though slightly lower than, the value of SRNA ∼500 pN determined −6 or overwound by 12 turns.in single-molecule Raw traces optical (120 tweezers Hz) measurementsare in red (25), and possibly filtered data ling, and A-to-P–form transitions) dsRNA shortens upon over- due to subtle differences between magnetic and optical tweezers winding, with a slope of (d∆L/dN) = –0.85 ± 0.04 nm per (10 Hz) are in gray. The dataexperiments. demonstrate For torsionally that unconstrained dsRNA molecules, shortens the over- when over- RNA wound. (B) Changes in tetherstretching extension transition upon is marked over- by a rapid and increase underwinding in extension to at F = 7 – 1.8 ± 0.1 times the crystallographic length over a narrow force turn, independent of stretching force in the range F = 4 8 pN pN of a 4.2-kbp dsRNA andrange a at 3.4-kbpF = 54 ± 5 pN dsDNA (SI Appendix tether., Fig. S2 LinearC). In contrast, fits using to the data our torsionally constrained dsRNA, we observed enthalpic (Fig. 4 B and C). This is in stark contrast to dsDNA, which (lines) indicate that the dsDNAstretching lengthens beyond the contour by ∼ length0.5 but nm no persharp overstretching turn, whereas the transition up to F = 75 pN (SI Appendix, Fig. S2D). The increased we observed to lengthen upon overwinding by (d∆L/dN)DNA = dsRNA shortens by ∼0.8 nmresistance per turn to overstretching upon overwinding. for torsionally constrained Symbols dsRNA denote the compared with torsionally unconstrained dsRNA is qualitatively +0.44 ± 0.1 nm per turn (Fig. 4 B and C), in good agreement mean and standard deviation for four measurements– on the same molecule. Fig. 1. Construction of a torsionally constrained double-stranded RNA for similar to the behavior of dsDNA (26 28) (SI Appendix, Fig. S1 H and I). The dependence of the overstretching transition for with previous measurements (38–41). Our measurementsmagnetic suggest tweezers measurements.(C) (A) Slopes Comparison upon of A-form overwindingdsRNA [Protein of dsRNA and dsDNA tethers as a function of F Data Bank (PDB) ID code 1RNA (57)] and B-form dsDNA [PDB ID code 2BNA dsRNA on torsional constraint and on salt (SI Appendix, Fig. S2 C that dsRNA has a positive twist–stretch coupling equal to(58)].D (B) CartoonRNA of the= elastic deformations(mean of and dsRNA: SEM bending, of stretching, at least and and fourD) moleculessuggests that it might in TE involve+ 100 melting mM as well NaCl as a buffer). tran- Data twisting. (C) Schematic of the protocol to generate double-stranded RNA sition to a previously unidentified conformation that we name –SRNA·(d∆L/dN)RNA/(2π·kBT) =+11.5 ± 3.3 (assuming SmoleculesRNA with= 350 multiple attachmentof Lionnet points at both et ends. al. Initial (38) transcription are shown“S-RNA, as” in a analogy black to S-DNAline with (SI Appendix the, uncertainty Fig. S1). indicated reactions incorporate multiple biotinylated adenosine (green circles) or digoxi- pN; SI Appendix, Materials and Methods), in contrast to thegenated negative uracil (yellow squares)in bases gray; and stall data at a fourth from nucleotide. Gore After etTwist al. (39) Response are of shown dsRNA. We as used a the black ability square. of MTs to The control red line is purification, transcription reactions are restarted and complete the 4.2-kbp the rotation of the magnetic beads (18) to map out the response BIOPHYSICS AND – – D = – ±transcripts. In the final step, the purifitheed RNA average strands are annealed over to yield all dsRNA dsRNAof dsRNA data. upon (D over-) Models and underwinding of oppositely at constant stretching twisting 50-bp twist stretch coupling of dsDNA (38 41), DNA 17 5. COMPUTATIONAL BIOLOGY with chemical modifications at eachsegments end. (D)SchematicofthetwoDNAtem- of dsDNA (Left)forces. and Starting dsRNA with ( aRight torsionally)under0and40pNstretching relaxed molecule (corresponding plates used to generate dsRNA with multiple labels at both ends. (E)Cartoonof to zero turns in Fig. 2), the tether extension remains initially ap- amagnetictweezersexperimentondsRNAforces,(not derived to scale). A streptavidin-coated from baseproximately pair-level constant models upon overwinding consistent (corresponding with experimental to in- Dynamics at the Buckling Transition. Next, we investigatedmagnetic the bead is dy- tethered to an anti-digoxigenin–coated surface by a dsRNA creasing linking number) until the molecule reaches a buckling molecule with multiple attachmentmeasurements points at both ends. A surface-attached (SI Appendix ref- point,TableS6and (Fig. 2A, dashed linesMaterials and SI Appendix and, Fig. Methods S3). Further). The or- namics at the buckling transition. When a dsRNA waserence bead twisted is tracked simultaneously fordriftcorrection.Permanentmagnets overwinding beyond the buckling point leads to a rapid linear de- above the flow cell are used to exert a stretching force F and to control the ro- ange bars represent the long axis of the terminal base pair. BIOPHYSICS AND tation of the magnetic bead via its preferred axis m .N,northpole;S,southpole. crease of the tether extension with an increasing number of turns, close to the critical supercoiling density, we observed jumps in the 0 due to the formation of plectonemes. The critical supercoiling extension traces, corresponding to transitions between the pre- density σbuck for buckling increases with stretching force and agrees COMPUTATIONAL BIOLOGY attachment points at both ends suitable for MTs torque mea- within experimental error with the values found for DNA and with A surements by annealing two complementary single strands that amechanicalmodeloriginallydevelopedforsupercoiledDNA(9) and postbuckling states (Fig. 5 ). Recording extension traces at (Fig. 2B and SI Appendix, Materials and Methods). The decrease in carry multiple biotin or digoxigeninwe found labels atτ theirbuck respective= 10.1 5′ ± 3.7 s for dsRNA compared with ∼0.05 s a fixed number of applied turns, the population ofends the (Fig. post- 1 C and D and Materials and Methods). The function- extension per added turn in the plectonemic regime provides a measure for the size of the plectonemes and decreases with in- alized single-stranded constructsfor dsDNA were generated at by carryingF = out4 pN and 320 mM salt (Fig. 5E). creasing stretching force (Fig. 2C). The extension vs. turns slopes buckling state increases whereas the population of theinitial prebuck- in vitro transcription reactions that incorporated labeled for dsRNA are within experimental error of those for dsDNA, and C ling state decreases with an increasing number of appliednucleotides turns and stalled at a missing fourth nucleotide (Fig. 1 and are in approximate agreement with the mechanical model for D). After purification, transcriptionDiscussion reactions were restarted and supercoiling (Fig. 2C). Underwinding the dsRNA tether at (Fig. 5A). After selecting a threshold to separate thecompleted pre- in and the presence of all four unlabeled nucleotides. The stretching forces F < 1 pN gives rise to a buckling response similar final annealed 4.2-kbp dsRNAOur constructs experiments can be tethered between are consistentto what is observed with upon overwinding dsRNA and behaving the formation of as neg- a linear postbuckling states (SI Appendix, Fig. S5 A–D), thean pre- anti-digoxigenin and–coated flow cell surface and streptavidin- atively supercoiled plectonemes. In contrast, for F > 1 pN, the coated magnetic beads forelastic manipulation rod in the MTs for (Fig. small 1E). deformationsover- and underwinding response from is the asymmetric A-form and the tether helix, and postbuckling populations and dwell time distributions can be extension remains approximately constant upon underwinding Force–Extension Response ofallow dsRNA. Using us the to ability empirically of MTs to (Fig. determine 2A), likely due to melting all four of the double elastic helix, as hasconstants been of quantified. The dependence of the postbuckling populationexert precisely on calibrated stretching forces (18, 19) (Materials and observed for DNA (29) (SI Appendix, Fig. S3 K and L). Methods and SI Appendixthe, Fig. S1 model:), we first probedA, S the, C force, and– IfD unwinding(SI Appendix at F > 1 pN is, Table continued for S1 several). To hundred go beyond the number of applied turns is well-described by aextension two-state response of dsRNA.the isotropic The stretching behavior rod model, of turns, we toward eventually observe a microscopic another structural transition interpretation marked model (42) (Fig. 5B and SI Appendix, Materials and Methodstorsionally) from relaxed dsRNA at low forces (F < 5pN)iswell- by an abrupt change in the extension vs. turns response at a described by the (inextensible)of the worm-like results, chain (WLC) we model describesupercoiling a density“knowledge-based of σ ∼ –1.9 (Fig. 2D). We term” thisbase previously pair-level which we determined the number of turns converted from(20, 21) twist (SI Appendix to , Fig. S2A). From fits of the WLC model, we unidentified highly underwound and left-handed RNA confor- determined the contour lengthmodelLC = 1.15 that± 0.02 considersμm and the mation the with six a helicity base-step of –12.6 bp per parameters turn “L-RNA,” in analogy slide, to shift, writhe during the buckling transition N 4 turns (SIbending Appendix persistence, length ARNA = 57 ± 2 nm in the presence of what has been observed for highly underwound DNA (11) (SI Δ b ∼ 100 mM monovalent saltrise, (SI Appendix twist,, Fig. roll, S2A), and in good tiltAppendix (SI, Fig. Appendix S3L). We note, that Fig. the helicity S6 and and elongationMaterials that and Fig. S5L). The dwell times in the pre- and postbuckling state are Methods; a full description of modeling for a blind prediction Lipfert et al. PNAS | October 28, 2014 | vol. 111 | no. 43 | 15409 exponentially distributed (SI Appendix, Fig. S5 E–G), and their challenge is given in ref. 43). Average values and elastic cou- mean residence times depend exponentially on the number of plings of the base-step parameters for dsRNA and dsDNA from applied turns (Fig. 5C). We determined the overall characteristic a database of nucleic acid crystal structures are used in a Monte buckling times τbuck, that is, the dwell times at the point where the Carlo protocol to simulate stretching and twisting experiments (SI pre- and postbuckling states are equally populated, from fits of Appendix, Materials and Methods). This base pair-level model the exponential dependence of the mean residence times on the correctly predicts the bending persistence length for dsRNA to be number of applied turns (Fig. 5C and SI Appendix, Materials and slightly larger than for dsDNA, SRNA to be at least a factor of two Methods). τbuck increases with increasing salt concentration and smaller than SDNA, and C to be of similar magnitude for dsRNA stretching force (Fig. 5E). The force dependence of τbuck is well- and dsDNA (SI Appendix, Table S2). The significant difference in described by an exponential model (solid lines in Fig. 5E), τbuck = stretch modulus S between dsRNA and dsDNA can be explained τbuck,0·exp(d·F/kBT); from the fit we obtain the buckling time at from the “spring-like” path of the RNA base pairs’ center axis, zero force τbuck,0 = 13 and 52 ms and the distance to the transition compared with dsDNA (SI Appendix, Fig. S6B). Beyond the state along the reaction coordinate d = 5.1 and 5.5 nm for the 100 agreement with experiment in terms of ratios of dsRNA and and 320 mM monovalent salt data, respectively. dsDNA properties, the absolute values of A, S, and C all fall within Interestingly, comparing τbuck for dsRNA with dsDNA of a factor of two of our experimental results for both molecules. similar length under otherwise identical conditions (Fig. 5 D and Whereas the values for A, S, and C are fairly similar for E), we found that the buckling dynamics of dsRNA are much dsRNA and dsDNA, our experiments revealed an unexpected slower than those of dsDNA, with the characteristic buckling difference in the sign of the twist–stretch coupling D for dsRNA times differing by at least two orders of magnitude. For example, and dsDNA. The twist–stretch coupling has important biological Lipfert et al. PNAS | October 28, 2014 | vol. 111 | no. 43 | 15411 Response of DNA to external forces and torques ~e2 ~e1 DNA torque ~e 3 M~ ~r(L) F~ force L ds E = A ⌦2 + A ⌦2 + C⌦2 + k✏2 +2D ✏⌦ F~ ~r(L) M~ ~(L) 2 11 1 22 2 3 3 3 · · Z0 work due to external DNA end to end distance force and torque L d~r L ~r(L)= ds = ds ~e (1 + ✏) ds 3 Z0 Z0 rotation of DNA end L d ~ L ~(L)= ds = ds ⌦~ 0 ds 0 Z Z 24 Response of DNA to external forces and torques ~e2 ~e1 DNA torque ~e 3 M~ ~r(L) F~ force L 1 E = ds A ⌦2 + A ⌦2 + C⌦2 + k✏2 +2D ✏⌦ F~ ~e (1 + ✏) M~ ⌦~ 2 11 1 22 2 3 3 3 · 3 · Z0 ✓ ◆ L E = ds g(✏, ⌦1, ⌦2, ⌦3) Z0 The configuration of DNA that minimizes energy is described by Euler-Lagrange equations d @g @g d @g @g 0= 0= ds @(d✏/ds) @✏ ds @(d /ds) @ ✓ ◆ ✓ i ◆ i 25 ⌦i = d /ds Tension and torque along DNA backbone ~e2 ~e1 DNA torque ~e 3 M~ ~r(L) F~ force L 1 E = ds A ⌦2 + A ⌦2 + C⌦2 + k✏2 +2D ✏⌦ F~ ~e (1 + ✏) M~ ⌦~ 2 11 1 22 2 3 3 3 · 3 · Z0 ✓ ◆ Euler-Lagrange equations tension along DNA d @g @g ~ 0= k✏(s)+D3⌦3(s)=F ~e3(s) ds @(d✏/ds) @✏ · ✓ ◆ torque along DNA d @g @g 0= A11⌦1(s)~e1(s)+A22⌦2(s)~e2(s) ds @(d i/ds) @ i ✓ ◆ +[C⌦3(s)+D3✏(s)]~e3(s)=M~ ⌦i = d /ds Euler-Lagrange equations thus describe 26 local force and torque balance! We determined the effective twist persistence length CRNA from the slopes in the linear torque–response regime, where the torque after N turns is 2π·N·kBT·CRNA/LC (where kB is Boltzmann’s constant and T is the absolute temperature; Fig. 3C, solid colored lines). CRNA increases with increasing force and is 99 ± 5nmatF = 6.5 pN. Compared with dsDNA, CRNA is similar to but slightly lower than CDNA,andbothquantities exhibit similar force dependence, in qualitative agreement Twist-stretch coupling with a model valid in the high force limit (37) (Fig. 3F and SI F = 7pN bead is Twist-stretch coupling has Appendix, Materials and Methods). Combining the results fromturned with opposite sign for double a magnet L stranded RNA and DNA! stretching and torque measurements at different forces, we de- + lineate the phase diagram for dsRNA as a function of appliedL 5 force and torque (Fig. 3G). 0 [nm] ~ L 5 k✏(s)+D3⌦3(s)=F ~e3(s) Twist–Stretch Coupling. The linear elastic rod model has a fourth· ⌦ =2⇡N /L 10 parameter, D, that describes the coupling between3 twistturns and – 15 stretch. We measured the twist stretch coupling for dsRNA by FL 2⇡D3 L = L✏ = Nturns monitoring changes in the extension upon over- and under-k k 5 0 5 10 winding while holding the molecule at constant stretching forces turns J. Lipfert et al., PNAS 111, 15408 (2014) that are large enough to suppress bending and writhe fluctua- 27 tions (38, 39) (Fig. 4A). We found that for small deformations (in Fig. 4. Double-stranded RNA has a positive twist–stretch coupling. (A)Time the range –0.02 < σ < 0.025, which excludes the melting, buck- traces of the extension of a dsRNA tether held at F = 7pNandunderwoundby ling, and A-to-P–form transitions) dsRNA shortens upon over- −6 or overwound by 12 turns. Raw traces (120 Hz) are in red and filtered data (10 Hz) are in gray. The data demonstrate that dsRNA shortens when over- winding, with a slope of (d∆L/dN)RNA = –0.85 ± 0.04 nm per – wound. (B) Changes in tether extension upon over- and underwinding at F = 7 turn, independent of stretching force in the range F = 4 8 pN pN of a 4.2-kbp dsRNA and a 3.4-kbp dsDNA tether. Linear fits to the data (Fig. 4 B and C). This is in stark contrast to dsDNA, which (lines) indicate that the dsDNA lengthens by ∼0.5 nm per turn, whereas the we observed to lengthen upon overwinding by (d∆L/dN)DNA = dsRNA shortens by ∼0.8 nm per turn upon overwinding. Symbols denote the +0.44 ± 0.1 nm per turn (Fig. 4 B and C), in good agreement mean and standard deviation for four measurements on the same molecule. with previous measurements (38–41). Our measurements suggest (C) Slopes upon overwinding of dsRNA and dsDNA tethers as a function of F that dsRNA has a positive twist–stretch coupling equal to DRNA = (mean and SEM of at least four molecules in TE + 100 mM NaCl buffer). Data –SRNA·(d∆L/dN)RNA/(2π·kBT) =+11.5 ± 3.3 (assuming SRNA = 350 of Lionnet et al. (38) are shown as a black line with the uncertainty indicated pN; SI Appendix, Materials and Methods), in contrast to the negative in gray; data from Gore et al. (39) are shown as a black square. The red line is the average over all dsRNA data. (D) Models of oppositely twisting 50-bp twist–stretch coupling of dsDNA (38–41), DDNA = –17 ± 5. segments of dsDNA (Left) and dsRNA (Right)under0and40pNstretching Dynamics at the Buckling Transition. Next, we investigated the dy- forces, derived from base pair-level models consistent with experimental namics at the buckling transition. When a dsRNA was twisted measurements (SI Appendix,TableS6andMaterials and Methods). The or- close to the critical supercoiling density, we observed jumps in the ange bars represent the long axis of the terminal base pair. BIOPHYSICS AND extension traces, corresponding to transitions between the pre- COMPUTATIONAL BIOLOGY and postbuckling states (Fig. 5A). Recording extension traces at we found τbuck = 10.1 ± 3.7 s for dsRNA compared with ∼0.05 s a fixed number of applied turns, the population of the post- for dsDNA at F = 4 pN and 320 mM salt (Fig. 5E). buckling state increases whereas the population of the prebuck- ling state decreases with an increasing number of applied turns Discussion A (Fig. 5 ). After selecting a threshold to separate the pre- and Our experiments are consistent with dsRNA behaving as a linear – postbuckling states (SI Appendix, Fig. S5 A D), the pre- and elastic rod for small deformations from the A-form helix, and postbuckling populations and dwell time distributions can be allow us to empirically determine all four elastic constants of quantified. The dependence of the postbuckling population on the model: A, S, C, and D (SI Appendix, Table S1). To go beyond the number of applied turns is well-described by a two-state the isotropic rod model, toward a microscopic interpretation model (42) (Fig. 5B and SI Appendix, Materials and Methods) from of the results, we describe a “knowledge-based” base pair-level which we determined the number of turns converted from twist to model that considers the six base-step parameters slide, shift, writhe during the buckling transition ΔNb ∼4 turns (SI Appendix, rise, twist, roll, and tilt (SI Appendix, Fig. S6 and Materials and Fig. S5L). The dwell times in the pre- and postbuckling state are Methods; a full description of modeling for a blind prediction exponentially distributed (SI Appendix, Fig. S5 E–G), and their challenge is given in ref. 43). Average values and elastic cou- mean residence times depend exponentially on the number of plings of the base-step parameters for dsRNA and dsDNA from applied turns (Fig. 5C). We determined the overall characteristic a database of nucleic acid crystal structures are used in a Monte buckling times τbuck, that is, the dwell times at the point where the Carlo protocol to simulate stretching and twisting experiments (SI pre- and postbuckling states are equally populated, from fits of Appendix, Materials and Methods). This base pair-level model the exponential dependence of the mean residence times on the correctly predicts the bending persistence length for dsRNA to be number of applied turns (Fig. 5C and SI Appendix, Materials and slightly larger than for dsDNA, SRNA to be at least a factor of two Methods). τbuck increases with increasing salt concentration and smaller than SDNA, and C to be of similar magnitude for dsRNA stretching force (Fig. 5E). The force dependence of τbuck is well- and dsDNA (SI Appendix, Table S2). The significant difference in described by an exponential model (solid lines in Fig. 5E), τbuck = stretch modulus S between dsRNA and dsDNA can be explained τbuck,0·exp(d·F/kBT); from the fit we obtain the buckling time at from the “spring-like” path of the RNA base pairs’ center axis, zero force τbuck,0 = 13 and 52 ms and the distance to the transition compared with dsDNA (SI Appendix, Fig. S6B). Beyond the state along the reaction coordinate d = 5.1 and 5.5 nm for the 100 agreement with experiment in terms of ratios of dsRNA and and 320 mM monovalent salt data, respectively. dsDNA properties, the absolute values of A, S, and C all fall within Interestingly, comparing τbuck for dsRNA with dsDNA of a factor of two of our experimental results for both molecules. similar length under otherwise identical conditions (Fig. 5 D and Whereas the values for A, S, and C are fairly similar for E), we found that the buckling dynamics of dsRNA are much dsRNA and dsDNA, our experiments revealed an unexpected slower than those of dsDNA, with the characteristic buckling difference in the sign of the twist–stretch coupling D for dsRNA times differing by at least two orders of magnitude. For example, and dsDNA. The twist–stretch coupling has important biological Lipfert et al. PNAS | October 28, 2014 | vol. 111 | no. 43 | 15411 Euler buckling instability yˆ xˆ zˆ F F F force force z(L) z(L) Analyze the stability of flat configuration by investigating the energy cost of slightly deformed profile with ~r(s)=(0,y(s),z(s)) Assume the very thin beam (filament) limit with ✏ 0 !s d~r 2 2 ~e3 = =(0,y0,z0) z0 = 1 y0 z(s)= d` 1 y0(`) ds 0 Bending strain p Z p 2 2 2 2 1 d ~r 2 2 y00 ⌦ = = = y00 + z00 = R2 ds2 1 y 2 ✓ ◆ 0 28 Euler buckling instability yˆ xˆ zˆ F F F force force z(L) z(L) L A L A y 2 E = ds ⌦2 + Fz(L)= ds 00 + F 1 y 2 2 2 (1 y 2) 0 Z0 Z0 0 Assume small deformations around the flat pconfiguration L 1 2 1 2 E ds Ay00 Fy0 + F ⇡ 2 2 Z0 analyze with Buckled configurations y(s)= eiqsy˜(q) q Fourier modes have lover energy for X 1 2 E +FL+ Aq4 Fq2 y˜(s) 2 F>Fcr = Aqmin ⇡ 2 | | q X Note: buckling direction corresponds 29 to the lower value of bending rigidity A Euler buckling instability clamped hinged boundaries boundaries 4⇡2A ⇡2A Fcr = F = L2 cr L2 y(0) = y0(0) = y(L)=y0(L)=0 y(0) = y00(0) = y(L)=y00(L)=0 qmin =2⇡/L qmin = ⇡/L one end clamped The amplitudes of buckled modes are the other free determined by the 4th order terms in ⇡2A energy functional that we ignored Fcr = 2 L 4L A 2 2 1 2 1 4 E ds y00 (1 + y0 )+F 1 y0 + y0 ⇡ 2 2 8 Z0 ✓ ◆ y(0) = y0(0) = y00(L)=y000(L)=0 qmin = ⇡/2L y˜(q ) (F Aq2 )/(Aq4 ) min / min min q 30 Torsional instability M torque torque plectoneme critical torque for number of turns clamped boundaries that lead to torsional instability A L⌦3 LMcr A M 9 Nturns = = 1.4 cr ⇡ L 2⇡ 2⇡C ⇡ C 31 Twist, Writhe and Linking numbers Ln=Tw+Wr linking number: total number of turns of a particular end L Tw = 0 ds ⌦3(s)/2⇡ twist: number of turns due to twisting the beam Wr R writhe: number of crossings when curve is projected on a plane 32 Torsional instability F M plectoneme Pulling force suppresses instability. CL[2⇡Ln]2 CL[2⇡ (Ln Wr)]2 E = E = + Ebend + F L 2 2 Torsional instability occurs, because part of the twisting energy can be released by bending and forming a plectoneme. 33 Stochastic switching between two states near instability for twisting RNA puling force F = 2pN a factor of three to five larger in magnitude than the experimen- tally observed values for DRNA and DDNA. In summary, a complete microscopic understanding of the twist–stretch coupling for both dsRNA and dsDNA may require higher-resolution (all-atom, explicit-solvent) models and novel experimental methods. A second surprising contrast between dsRNA and dsDNA is the much slower buckling dynamics for dsRNA. The two orders of magnitude difference in τbuck is particularly astonishing, be- cause the parameters that characterize the end points of the buckling transitions and the difference between them, such as σbuck (Fig. 2B), Γbuck (Fig. 3E), the extension jump (SI Appendix, Fig. S5I), and ΔNb (SI Appendix, Fig. S5L), are all similar (within at most 20–30% relative difference) for dsRNA and dsDNA. Several models that describe the buckling transition in an elastic probability of observing a plectoneme rod framework (characterized by A and C) find reasonable agree- ment between experimental results for dsDNA and the parame- E (N )/k T e plectoneme turns B pplectoneme(Nturns)= E (N )/k T E (N )/k T ters that characterize the end points of the buckling transition (42, e straight turns B +e plectoneme turns B 48–50). In contrast, there is currently no fully quantitative model J. Lipfert et al., PNAS 111, 15408 (2014)for the buckling dynamics. A recent effort to model the timescale 34 of the buckling transition for dsDNA found submillisecond buckling times, much faster than what is experimentally observed, suggesting that the viscous drag of the micrometer-sized beads or particles used in the experiments might considerably slow down Fig. 5. Slow buckling transition for dsRNA. (A) Time traces of the extension the observed buckling dynamics for dsDNA (48). of a 4.2-kbp dsRNA tether for varying numbers of applied turns (indicated on The observed difference in τbuck suggests that the transition the far right) at the buckling transition for F = 2 pN in 320 mM NaCl. (Right) state and energy barrier for buckling are different for dsRNA and Extension histograms (in gray) fitted by double Gaussians (brown lines). Raw dsDNA. We speculate that because the transition state might data were acquired at 120 Hz (gray) and data were filtered at 20 Hz (red). involve sharp local bending of the helix (on a length scale of ∼5 (Inset)Schematicofthebucklingtransition.(B) Fraction of the time spent in nm, suggested by the fit to the force dependence; Fig. 5E), the the postbuckling state vs. applied turns for the data in A and fit of a two-state observed difference might possibly be due to high flexibility of model (black line; SI Appendix, Materials and Methods). (C) Mean residence dsDNA on short length scales, which would lower the energetic times in the pre- and postbuckling state vs. applied turns for the data in A cost of creating sharp transient bends. An anomalous flexibility of and fits of an exponential model (lines; SI Appendix, Materials and Meth- dsDNA on short length scales is hotly debated (51), and has been ods). (D) Extension vs. time traces for dsRNA (red) and dsDNA (blue) both at suggested by different experiments, including cyclization assays in F = 4 pN in TE buffer with 320 mM NaCl added. Note the different timescales bulk using ligase (52) or at the single-molecule level using FRET for dsRNA and dsDNA. (E) Characteristic buckling times for 4.2-kbp dsRNA in (53), small-angle X-ray scattering measurements on gold-labeled TE buffer with 100 mM (red points) and 320 mM (orange points) NaCl added samples (54), and atomic force microscopy imaging of surface- (mean and SEM of at least four independent molecules). Solid lines are fits of an exponential model. Measurements with 3.4-kbp dsDNA tethers in 320 mM immobilized DNA (55), even though the evidence remains con- NaCl at F = 4 pN yielded characteristic buckling times of ∼50 ms (horizontal troversial (51). If the observed difference in τbuck between dashed line); however, this value represents only an upper limit, because our dsDNA and dsRNA is indeed due to an anomalous flexibility of time resolution for these fast transitions is biased by the acquisition frequency dsDNA on short length scales, a clear prediction is that similar of the CCD camera (120 Hz). For comparison, we show data for 10.9- and 1.9- experiments for dsRNA should fail to observe a corresponding kbp DNA (upper and lower triangles, respectively) from ref. 42. level of flexibility. In addition, this striking, unpredicted differ- ence between dsDNA and dsRNA again exposes a critical gap in current modeling of nucleic acids. implications, such as for how mutations affect binding sites, be- In conclusion, we have probed the elastic responses and struc- cause a base pair deletion or insertion changes not only the tural transitions of dsRNA under applied forces and torques. We length but also the twist of the target sequence, changes that find the bending and twist persistence lengths and the force–tor- need to be compensated by distortions of the nucleic acid upon que phase diagram of dsRNA to be similar to dsDNA and the protein binding (39). Nevertheless, accounting for the twist– stretch modulus of dsRNA to be threefold lower than that of stretch coupling D in a model of nucleic acid elasticity appears dsDNA, in agreement with base pair-level model predictions. to be challenging. Previous elastic models originally developed Surprisingly, however, we observed dsRNA to have a positive for dsDNA (44, 45) predict a positive twist–stretch coupling twist–stretch coupling, in agreement with naïve expectations but in for dsRNA, in agreement with our measurements for DRNA al- contrast to dsDNA and to base pair-level modeling. In addition, though at odds with the results for dsDNA (SI Appendix, Mate- we observe a striking difference of the buckling dynamics for rials and Methods). In contrast, elastic models that consider a stiff dsRNA, for which the characteristic buckling transition time is two backbone wrapped around a softer core give negative D pre- orders of magnitude slower than that of dsDNA. Our results dictions for both dsRNA and dsDNA (39, 46). Likewise, the base provide a benchmark and challenge for quantitative models of pair-level Monte Carlo model yields a negative twist–stretch cou- nucleic acid mechanics and a comprehensive experimental foun- pling for both dsDNA and dsRNA, disagreeing with the positive dation for modeling complex RNAs in vitro and in vivo. In addi- sign we observe for DRNA (SI Appendix, Table S2), although we tion, we envision our assay to enable a new class of quantitative note that relatively modest changes to the base-step parameters single-molecule experiments to probe the proposed roles of twist can reproduce the experimentally observed value for DRNA and torque in RNA–protein interactions and processing (4, 56). (Fig. 4D and SI Appendix, Materials and Methods). Interestingly, an all-atom, implicit-solvent model of dsDNA homopolymers Materials and Methods found A-form dsDNA to unwind upon stretching whereas B-form See SI Appendix, Materials and Methods for details. In brief, the double- dsDNA overwound when stretched close to its equilibrium con- stranded RNA constructs for magnetic tweezers experiments were generated formation (47). Although these simulation results are in qualitative by annealing two 4,218-kb complementary single-stranded RNA molecules that agreement with our findings for A-form dsRNA and B-form carry multiple biotin or digoxigenin labels at their respective 5′ ends (Fig. 1C). dsDNA, their simulation predicts un- and overwinding, respectively, The product of the annealing reaction is a 4,218-bp (55.6% GC content) fully by ∼3° per 0.1 nm, which corresponds to values of D ∼50, namely double-stranded RNA construct with multiple biotin labels at one end and j j 15412 | www.pnas.org/cgi/doi/10.1073/pnas.1407197111 Lipfert et al.