Motion of Knots in DNA Stretched by Elongational Fields

Alexander R Klotzdagger Beatrice W SohDagger and Patrick S Doylesect

Department of Chemical Engineering Massachusetts Institute of Technology77 Massachusetts Avenue Cambridge Massachusetts 02142 USA

(Received 20 November 2017 revised manuscript received 7 February 2018 published 3 May 2018)

Knots in DNA occur in biological systems serve as a model system for polymer entanglement andaffect the efficacy of modern genomics technologies We study the motion of complex knots in DNA bystretching molecules with a divergent electric field that provides an elongational force We demonstrate thatthe motion of knots is nonisotropic and driven towards the closest end of the molecule We show for the firsttime experimentally that knots can go from a mobile to a jammed state by varying an applied strain rate andthat this jamming is reversible We measure the mobility of knots as a function of strain rate demonstratingthe conditions under which knots can be driven towards the ends of the molecule and untied

DOI 101103PhysRevLett120188003

Sufficiently long polymers are very likely to containknots [1] and DNA is no exception Knots occur naturallyin biological DNA are frequently found in virus capsids [2]and can be tied or untied in mammalian cells by topo-isomerase enzymes [34] Knotted DNA serves as aminimal system with which to study polymer entanglement[5] and knot theory as applied to polymer physics [6] Thebiophysical study of DNA knots is finding relevance in thedevelopment of next-generation genomics technology forexample in nanochannel mapping assays that are sufferingerroneous results due to the stochastic presence of knots inconfined DNA molecules [7] Intentionally knotting DNAhas been proposed as a braking mechanism to improve thetemporal resolution of nanopore sequencing [8]In this Letter we investigate the mechanisms of knot

motion in stretched DNA which is mediated throughself-reptation of contour through the knot and provideclarity on a number of theoretical questions [9] Knot self-reptation brings together several concepts in polymerphysics including intramolecular friction [10] the effectof topology on polymer behavior [1112] and the relation-ship between semiflexibility and excluded volume [13]We have predicted that knots will move diffusively alonguniformly stretched chains with a diffusivity that decreaseswith increasing tension [14] We have also predicted thatknots in molecules stretched in an elongational field willtranslate towards the end of the chain and untie [15]Experimentally the picture is less clear In the onlyinvestigation of knot mobility Bao et al [16] measuredthe diffusivity of knots tied in DNA using optical tweezersand stretched in a highly viscous buffer finding slowerdiffusion in more-complex knots Preliminary data pre-sented by Metzler et al [17] show a knot moving along ananochannel-confined DNA molecule until it reaches theend of the chain and unties although quantitative mea-surements of the knotsrsquomobility were not made Plesa et al

[18] saw knots in DNA sliding along molecules as theytranslocated through a nanopore The previous studiesindicate that knots can indeed move along DNA moleculesbut they shed little quantitative insight on the conditionsunder which knots may be mobileHere we investigate two novel phenomena associated

with knot motion that have not been previously inves-tigated we show that knots can be untied by driving themtowards the chain ends with elongational fields and thatknot motion can be halted or promoted by controlling thelocal chain tension We perform experiments stretchingknotted DNA molecules with a divergent electric field in amicrofluidic device and imaging them with fluorescencemicroscopy (see the video in the Supplemental Material[19]) By tracking the knotrsquos position over time when themolecules are stretched at varying strain rates we can studyin detail how knot mobility is affected by stretching forcesWe use a polydimethylsiloxane (silicone) microfluidic

device with a T-junction channel geometry [20] [theschematic in Fig 1(a)] The channels are 2 μm tall withan inlet that is 40 μm wide and outlets that are 20 μmwidewith corners smoothed to hyperbolae A buffer is pipettedinto reservoir holes that interface the channels for macro-scopic access and electrodes are placed in each of thereservoirs with the ground connected to the base of the Tand two independently controlled positive leads connectedto the two arms We use T4 DNA stained with YOYO-1fluorescent dye at a 4∶1 base-pairdye ratio in a 05times Trisndashboratendashethylenediamine tetra-acetic acid buffer with 4β-mercaptoethanol and 01 polyvinylpyrrolidone (10 kDa)Molecules were illuminated with a filtered light-emittingdiode visualized with a Zeiss-Axiovert microscope with a63 times objective and recorded with a Photometrics Prime 95BCMOS camera The full contour length of stained T4DNA isestimated to be 77 μm assuming a 38 elongation due to theintercalating dye [21]

PHYSICAL REVIEW LETTERS 120 188003 (2018)Editors Suggestion Featured in Physics

0031-9007=18=120(18)=188003(6) 188003-1 copy 2018 American Physical Society

The electric field in the central region of the T-junctiongeometry mimics a planar elongational flow [20] Acharged particle accelerating through this region has avelocity in the horizontal direction that is proportional to itsposition The effective strain rate _ϵ in the divergent region isdependent on the electrophoretic mobility μ of the testparticles and the electric field gradient and is on the orderof 1 sminus1 in our setup

vx frac14 μEx frac14 _ϵx vy frac14 μEy frac14 minus_ϵy eth1THORN

When DNA spans the stagnation point the ends are pulledin opposite directions and the molecule stretches Themoleculersquos internal elasticity balances the electrophoreticstretching and a steady state extension is reached that isdetermined by the Weissenberg number (Wi) the productof the strain rate and the moleculersquos longest relaxation timeτ which is measured to be 22 s [see the SupplementalMaterial (SM) [19]]

Wi frac14 _ϵτ eth2THORN

With fixed geometry the strain rate is proportional to theapplied voltage which we control to stretch the moleculeThe molecule is kept at the unstable stagnation point bysmall periodic toggles to one of the voltage sources Thevoltages used in our experiments are generally between 10and 30 V giving a range of Wi in this work between 09 and30 To entangle the molecules into knots we apply a strongalternating electric field above 1 kV=cm pulsed at a 10 Hz

square wave for 05 s to induce a hydrodynamic instability[22] causing the molecules to contract into tight globulesIt is very likely for knots to remain trapped in the interiorsof these molecules where they appear as bright spots ofexcess fluorescence [Fig 1(b)] Accumulated evidencefrom our group [5122223] supports the fact that mole-cules undergoing this procedure do indeed form complexknots The knots produced through this method have adiverse range of topologies that cannot be fully resolvedbut previous analysis from our group [523] has shown thatthey contain several microns of contour when highlystretched an order of magnitude greater than the few-hundred nanometers seen in simple knots [1618] Tocharacterize how the electric field induces knot convectionwe have used the Weissenberg number of the unknottedchain to characterize the field strength consistently acrossall molecules Research by our group [24] suggests that anldquoeffectiverdquoWeissenberg number that takes into account thereduction in the moleculersquos relaxation time may reduce thevariance in chain extension as measured across moleculeswith different topologiesWe record videos of molecules stretched with a fixed

voltage and vary the voltage with step changes over thecourse of several minutes allowing us to probe the sameknot under multiple strain rates The videos are analyzedusing MATLAB scripts that project the linear intensity profileof extended molecules into a kymograph where the knot isidentified as a bright trajectory along the moleculersquosworldsheet The location of the knot is taken as the indexof the brightest pixel and the dimensionless knot positionis defined as the ratio of the knotrsquos distance from the leftend of the molecule to the instantaneous molecularextensionWhen molecules are held at a steady strain rate the knots

are seen over minute-long timescales to translate along themolecule via self-reptation and if a molecule is observedfor long enough it is indeed likely for the knot to reach theend of the chain and untie [Fig 2(a)] If the knotrsquos motion ispurely diffusive we expect the mean-square displacement(MSD) of the knotrsquos position to grow linearly over time butit is seen (in the SM [19]) that at long time lags the MSDgrows superdiffusively In Fig 2(b) an ensemble of knottrajectories and their average are shown as a function oftheir accumulated strain (the residence time multiplied bythe strain rate) For the higher strain-rate trajectories themotion is largely driven whereas for lower strain rates thereis a more apparent diffusive behaviorThere are two limiting cases characterizing the motion

of polymer knots in elongational fields If the knot isexperiencing affine deformation behaving as a particlefollowing the elongational field line without any otherinteraction it would have a position growing exponentiallyin time [25] according to the total strain experienced

κethtTHORN frac14 κeth0THORNe_ϵt frac14 κeth0THORNeϵ eth3THORN

FIG 1 (a) Drawing of the experimental device A microfluidicT junction with a diverging electric field stretches knotted DNA atits stagnation point Field lines show the trajectories of negativelycharged test particles (b) Representative images of a molecule atfour time points as two knots translate towards one end of themolecule at Wi frac14 085 The scale bar is 5 μm Approximately25 of stretched molecules have two knots

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While pure affine deformation is not expected its predictedbehavior sets an upper bound on the knot trajectoryConversely if there is no bias in the knotrsquos motion itcan be modeled as a particle undergoing one-dimensionalBrownian motion as in the work of Bao et al [16] andNarsimhan et al [14] In this case the population-averagedtrajectories will have a root-mean-square displacement thatgrows as the one-half power of time

κethtTHORN frac14 κeth0THORN thornffiffiffiffiffiffi

Dtp

frac14 κeth0THORN thornffiffiffiffiffiffiffiffiffiffi

2Dϵ

_ϵ

r

eth4THORN

As each knotrsquos position trends towards the chain end theposition of the knot diverges with time in a mannerinconsistent with and faster than Brownian motion (thenondimensionalization of the Brownian prediction with theknotrsquos hypothetical diffusivity of 05 μm2=s is discussed inthe SM [19]) The knot trajectories though driven towardsthe chain ends are considerably slower than the affine

prediction by a factor of 63 This presents a picture of knotmotion where the knots tend to follow the lines of theelongational field but in which motion is significantlyhampered by intrachain friction within the knotSimulations of simple knot topologies (52 and 63) [15]show that the knots are driven faster than what is typicallyobserved experimentally but still significantly slower thanaffine deformation These results describe the process bywhich the knot moves towards the chain end although ouranalysis halts before the knot reaches the end When thishappens the molecule often contracts as the knot swells inthe lower tension region near the end before spontaneouslyrestretching as the knot unties a process described ingreater detail in our recent publication [12]The picture of knot mobility is more complex than merely

following field lines A stronger field does not hasten theconvection of the knots but instead does the opposite themotion of knots can be halted or slowed by suddenlyincreasing the applied strain rate We performed an assayin which knotted molecules were stretched at Wi frac14 11 andas the knots approached the chain end the applied voltagewas suddenly increased (Fig 3) with an average post-increase strain rate of Wi frac14 19 Prior to the increase therewas a strong correlation between knot position and time(with a mean Pearson coefficient r frac14 09) but afterwards thecorrelation was weak (r frac14 minus035) and the mean-squareknot displacement at a given time lag over the measuredperiod dropped by a median factor of 3We observed that theendward motion of knots could be halted by a step change inextension suggesting that the decrease in knot mobility has a

FIG 2 (a) A kymograph of a knotted molecule stretched atWi frac14 16 The white streaks are from other molecules passingthrough the field of view (b) Averaged trajectory of translatingknots along an ensemble of molecules as a function of accumu-lated strain with individual trajectories underlaid Green trajec-tories occurred at Wifrac14 11ndash12 blue trajectories at Wifrac14 12ndash14and red trajectories at Wifrac14 14ndash17 The error bars representstandard error over a population average The nearly verticaldotted line represents affine motion [Eq (3)] and the curved linerepresents the hypothetical ensemble trajectory assuming a one-dimensional (1D) random walk with a diffusivity of 05 μm2=s[16] [Eq (4)] The thick blue and green curves representsimulations of 52 and 63 knots [15]

FIG 3 Time traces of knots moving towards the ends of thechain at Wi frac14 11 (gray) and halting as the strain rate is increasedto Wi frac14 19 (red) (Inset) An example of a molecule with twoknots with the larger one moving towards the end and stoppingas the molecule is stretched (corresponding to the bold curves)and the smaller one moving about the center of the molecule untilhalting when the local tension is increased

PHYSICAL REVIEW LETTERS 120 188003 (2018)

188003-3

stronger effect than the increased convective drive If theknot is sufficiently close to the end however its motioncannot be stoppedTo quantify the relationship between mobility and Wi

we performed an assay [Fig 4(a)] in which knottedmolecules were initially stretched at a high voltage for aminute then the strain rate was decreased every minuteuntil the knot untied or the molecule relaxed to its coiledstate The range of fields used was between Wi frac14 09 andWi frac14 29 corresponding to estimated tensions between 01and 05 pN at the centers of the molecules (see the SM[19]) but lower towards the ends Although the knotdiffusivity is a natural quantity with which to characterizethe knotrsquos motion the MSD trajectories did not displaydiffusive behavior but rather were characterized by sub-diffusion at short times and superdiffusion at long timesInstead we choose a fixed time interval of 5 s (long enoughto avoid tracking noise [26] but short enough to ensuresufficient statistics) and measure the knot MSD (described

in greater detail in the SM [19]) over that interval and divide

by the time to characterize the knotrsquos mobility Figure 4(b)shows the knot mobility as a function of Wi for severalmolecules as well as the population average The mobilityof the knot decreases with an increasing strain rate It variesby more than an order of magnitude as a function of strainrate while the chain extension varies by less than a factorof 2 While there is variation in the mobility with respectto the position of the knot the observed variation inposition-dependent mobility is smaller than the strain-ratedependence and the variation in knot position throughoutthe step-down assay is small (see the SM [19]) Comparisonbetween individual knot trajectories is difficult due to boththe variation in their initial position and the diversesampling of topologies but overall populationwide trendsare observedTwo computational papers from our group [1415]

provide insight on the motion of knots under varyingstrain-rate conditions An earlier study by Renner andDoyle [15] predicted that knots would undergo convectivemotion beyond a critical length scale near the center of themolecule and indeed such convection was observed inour experiments However these simulations used a semi-flexible bead-spring model which was unable to suffi-ciently probe the role of intrachain friction because ofpathological chain crossings at high tensions A subsequentstudy by Narsimhan et al [14] overcame these difficultiesby using a bead-rod parametrization and by studyingflexible as well as semiflexible chains and it found thatfriction arising due to molecular corrugation (eg corre-sponding to the grooves along the double helix) can jamthe motion of knots as tension increases The data presentedin Fig 3(b) (of Ref [14]) exemplify this showing thatalthough the trend of knot motion is convective sufficientinternal friction (likely occurring when the strands in theknot are bent at subpersistence length scales) can overcomethe convective forces and halt the motion of the knot Thereis a small dynamic range of strain rates that can be appliedsuch that the molecule does not relax to the coiled state butthat the knot is noticeably mobile over minute-long time-scales roughly in the range Wifrac14 09ndash15Our results contrast with those of Bao et al [16] who

presented knot diffusivity data for molecules stretched attensions between 011 and 033 pN as determined by theirreported fractional extensions [27] (corresponding to cen-tral chain tensions found at Wi asymp 1 to 2 in our work see theSM [19]) and additionally state that they found no tensiondependence between 01 and 2 pN Because they sampledthe diffusivity of typically three to four molecules pertopology and averaged over tensions spanning a factor of 3they may not have had sufficient statistics to excludetension dependence Furthermore it is difficult to directlycompare our results to those of Bao et al due to the bufferused in their Letter (an overlapping polyethylene glycolsolution) the simpler knots they study (seven crossings andfewer) and the difference between uniform tension and the

FIG 4 (a) Kymograph of a molecule held at varying levels ofthe applied field (Wifrac14 09ndash25 5ndash30 V) for a minute eachshowing greater mobility at lower Wi Note that time runs right toleft in this image The scale bar is 10 μm (b) Relationshipbetween the measured effective knot mobility and the Weissen-berg number Each gray-scale trace represents a different mol-ecule and the red represents the population average The verticallines correspond to the high and lowWi in Fig 3 and the gray barcorresponds to the range observed in Fig 2(b) (Inset) The samedata with a logarithmic y axis

PHYSICAL REVIEW LETTERS 120 188003 (2018)

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quadratic tension in an elongational field [28] Assumingthat Bao et al did collect significant but unreportedevidence that knot motion is tension independent we positthat the experiments of Bao et al failed to see tensiondependence because they took place in a regime where thestrands in the knot are insufficiently close for corrugation-based friction to be relevant [29] According to the analysisof Bao et al the strands of DNA within the knot areseparated by 26 nm a factor of 10 greater than the DNAcorrugation length scale By comparison our experimentalbuffer gives DNA an effective width of roughly 105 nm[30] with 85 nm between the strands less than a factor of3 above the corrugation length scaleIn summary we examined the motion of knots in DNA

molecules stretched in elongational fields We observedthat knots moved noticeably across the molecule onminute-long timescales until they reached the end anduntied The motion of the knots was driven towards theends of the chain the knots reaching the end faster thanwould be expected from simple one-dimensional diffusionThe motion of the knots can be halted by increasing thestrain rate to induce greater intramolecular friction withinthe knot We measured the mobility of knots with respect tostrain rate and found that knots tend to slow with anincreasing Wi Our findings suggest a systematic way ofremoving unwanted knots from molecules This method ofremoving knots by applying an elongational field maysuggest a modification of nanochannel mapping assays [7]to include a cross flow [31] to remove knots

This research was supported by the National ResearchFoundation Singapore through the Singapore MIT Alliancefor Research and Technologyrsquos research program inBioSystems and Micromechanics and the NationalScience Foundation (Grant No CBET-1602406) A R Kis partially funded by a NSERC Postdoctoral FellowshipWe thank C Benjamin Renner and Vivek Narsimhan forthe fruitful discussion

pdoylemitedudaggerORCID httporcidorg0000-0002-1581-6956DaggerORCID httporcidorg0000-0001-8399-5995sectORCID httporcidorg0000-0003-2147-9172

[1] D Sumners and S G Whittington Knots in self-avoidingwalks J Phys A 21 1689 (1988)

[2] L F Liu L Perkocha R Calendar and J C Wang KnottedDNA from bacteriophage capsids Proc Natl Acad SciUSA 78 5498 (1981)

[3] J M Sogo A Stasiak M L Martınez-Robles D BKrimer P Hernaacutendez and J B Schvartzman Formationof knots in partially replicated DNAmolecules J Mol Biol286 637 (1999)

[4] V Lopez M-L Martiacutenez-Robles P Hernandez D BKrimer and J B Schvartzman Topo IV is the topoisomer-ase that knots and unknots sister duplexes during DNAreplication Nucleic Acids Res 40 3563 (2012)

[5] C Benjamin Renner and P S Doyle Stretching self-entangled DNA molecules in elongational fields SoftMatter 11 3105 (2015)

[6] A Y Grosberg and Y Rabin Metastable Tight Knots in aWormlike Polymer Phys Rev Lett 99 217801 (2007)

[7] J G Reifenberger K D Dorfman and HCao Topologicalevents in single molecules of E coli DNA confined innanochannels Analyst 140 4887 (2015)

[8] V Narsimhan C Benjamin Renner and P S DoyleTranslocation dynamics of knotted polymers under a con-stant or periodic external field Soft Matter 12 5041 (2016)

[9] While any open string such as a linear molecule ismathematically unknotted and the topology of knots inthe center of a chain can be discussed if the ends are treatedas being connected ldquoat infinityrdquo

[10] L Huang and D E Makarov Langevin dynamics simu-lations of the diffusion of molecular knots in tensionedpolymer chains J Phys Chem A 111 10338 (2007)

[11] R Matthews A A Louis and J M Yeomans Effect oftopology on dynamics of knots in polymers under tensionEurophys Lett 89 20001 (2010)

[12] V Narsimhan A R Klotz and P S Doyle Steady-state andtransient behavior of knotted chains in extensional fieldsACS Macro Lett 6 1285 (2017)

[13] L Dai C Benjamin Renner and P S Doyle Metastabletight knots in semiflexible chains Macromolecules 47 6135(2014)

[14] V Narsimhan C Benjamin Renner and P S DoyleJamming of knots along a tensioned chain ACS MacroLett 5 123 (2016)

[15] C Benjamin Renner and P S Doyle Untying knotted DNAwith elongational flows ACS Macro Lett 3 963 (2014)

[16] X R Bao H J Lee and S R Quake Behavior of ComplexKnots in Single DNA Molecules Phys Rev Lett 91265506 (2003)

[17] R Metzler W Reisner R Riehn R Austin J O Tegenfeldtand I M Sokolov Diffusion mechanisms of localised knotsalong a polymer Europhys Lett 76 696 (2006)

[18] C Plesa D Verschueren S Pud J van der Torre J WRuitenberg M J Witteveen M P Jonsson A Y GrosbergY Rabin and C Dekker Direct observation of DNA knotsusing a solid-state nanopore Nat Nanotechnol 11 1093(2016)

[19] See Supplemental Material at httplinkapsorgsupplemental101103PhysRevLett120188003 for cali-bration of the Weissenberg number additional informationabout the diffusive behavior of the knots and a discussion ofintrachain tension in elongational fields

[20] J Tang and P S Doyle Electrophoretic stretching of DNAmolecules using microscale T junctions Appl Phys Lett90 224103 (2007)

[21] B Kundukad J Yan and P S Doyle Effect of YOYO-1 onthe mechanical properties of DNA Soft Matter 10 9721(2014)

[22] J Tang N Du and P S Doyle Compression and self-entanglement of single DNAmolecules under uniform electricfield Proc Natl Acad Sci USA 108 16153 (2011)

[23] A R Klotz V Narsimhan B W Soh and P S DoyleDynamics of DNA knots during chain relaxation Macro-molecules 50 4074 (2017)

PHYSICAL REVIEW LETTERS 120 188003 (2018)

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While pure affine deformation is not expected its predictedbehavior sets an upper bound on the knot trajectoryConversely if there is no bias in the knotrsquos motion itcan be modeled as a particle undergoing one-dimensionalBrownian motion as in the work of Bao et al [16] andNarsimhan et al [14] In this case the population-averagedtrajectories will have a root-mean-square displacement thatgrows as the one-half power of time

κethtTHORN frac14 κeth0THORN thornffiffiffiffiffiffi

Dtp

frac14 κeth0THORN thornffiffiffiffiffiffiffiffiffiffi

2Dϵ

_ϵ

r

eth4THORN

As each knotrsquos position trends towards the chain end theposition of the knot diverges with time in a mannerinconsistent with and faster than Brownian motion (thenondimensionalization of the Brownian prediction with theknotrsquos hypothetical diffusivity of 05 μm2=s is discussed inthe SM [19]) The knot trajectories though driven towardsthe chain ends are considerably slower than the affine

prediction by a factor of 63 This presents a picture of knotmotion where the knots tend to follow the lines of theelongational field but in which motion is significantlyhampered by intrachain friction within the knotSimulations of simple knot topologies (52 and 63) [15]show that the knots are driven faster than what is typicallyobserved experimentally but still significantly slower thanaffine deformation These results describe the process bywhich the knot moves towards the chain end although ouranalysis halts before the knot reaches the end When thishappens the molecule often contracts as the knot swells inthe lower tension region near the end before spontaneouslyrestretching as the knot unties a process described ingreater detail in our recent publication [12]The picture of knot mobility is more complex than merely

following field lines A stronger field does not hasten theconvection of the knots but instead does the opposite themotion of knots can be halted or slowed by suddenlyincreasing the applied strain rate We performed an assayin which knotted molecules were stretched at Wi frac14 11 andas the knots approached the chain end the applied voltagewas suddenly increased (Fig 3) with an average post-increase strain rate of Wi frac14 19 Prior to the increase therewas a strong correlation between knot position and time(with a mean Pearson coefficient r frac14 09) but afterwards thecorrelation was weak (r frac14 minus035) and the mean-squareknot displacement at a given time lag over the measuredperiod dropped by a median factor of 3We observed that theendward motion of knots could be halted by a step change inextension suggesting that the decrease in knot mobility has a

FIG 2 (a) A kymograph of a knotted molecule stretched atWi frac14 16 The white streaks are from other molecules passingthrough the field of view (b) Averaged trajectory of translatingknots along an ensemble of molecules as a function of accumu-lated strain with individual trajectories underlaid Green trajec-tories occurred at Wifrac14 11ndash12 blue trajectories at Wifrac14 12ndash14and red trajectories at Wifrac14 14ndash17 The error bars representstandard error over a population average The nearly verticaldotted line represents affine motion [Eq (3)] and the curved linerepresents the hypothetical ensemble trajectory assuming a one-dimensional (1D) random walk with a diffusivity of 05 μm2=s[16] [Eq (4)] The thick blue and green curves representsimulations of 52 and 63 knots [15]

FIG 3 Time traces of knots moving towards the ends of thechain at Wi frac14 11 (gray) and halting as the strain rate is increasedto Wi frac14 19 (red) (Inset) An example of a molecule with twoknots with the larger one moving towards the end and stoppingas the molecule is stretched (corresponding to the bold curves)and the smaller one moving about the center of the molecule untilhalting when the local tension is increased

PHYSICAL REVIEW LETTERS 120 188003 (2018)

188003-3

stronger effect than the increased convective drive If theknot is sufficiently close to the end however its motioncannot be stoppedTo quantify the relationship between mobility and Wi

we performed an assay [Fig 4(a)] in which knottedmolecules were initially stretched at a high voltage for aminute then the strain rate was decreased every minuteuntil the knot untied or the molecule relaxed to its coiledstate The range of fields used was between Wi frac14 09 andWi frac14 29 corresponding to estimated tensions between 01and 05 pN at the centers of the molecules (see the SM[19]) but lower towards the ends Although the knotdiffusivity is a natural quantity with which to characterizethe knotrsquos motion the MSD trajectories did not displaydiffusive behavior but rather were characterized by sub-diffusion at short times and superdiffusion at long timesInstead we choose a fixed time interval of 5 s (long enoughto avoid tracking noise [26] but short enough to ensuresufficient statistics) and measure the knot MSD (described

in greater detail in the SM [19]) over that interval and divide

by the time to characterize the knotrsquos mobility Figure 4(b)shows the knot mobility as a function of Wi for severalmolecules as well as the population average The mobilityof the knot decreases with an increasing strain rate It variesby more than an order of magnitude as a function of strainrate while the chain extension varies by less than a factorof 2 While there is variation in the mobility with respectto the position of the knot the observed variation inposition-dependent mobility is smaller than the strain-ratedependence and the variation in knot position throughoutthe step-down assay is small (see the SM [19]) Comparisonbetween individual knot trajectories is difficult due to boththe variation in their initial position and the diversesampling of topologies but overall populationwide trendsare observedTwo computational papers from our group [1415]

provide insight on the motion of knots under varyingstrain-rate conditions An earlier study by Renner andDoyle [15] predicted that knots would undergo convectivemotion beyond a critical length scale near the center of themolecule and indeed such convection was observed inour experiments However these simulations used a semi-flexible bead-spring model which was unable to suffi-ciently probe the role of intrachain friction because ofpathological chain crossings at high tensions A subsequentstudy by Narsimhan et al [14] overcame these difficultiesby using a bead-rod parametrization and by studyingflexible as well as semiflexible chains and it found thatfriction arising due to molecular corrugation (eg corre-sponding to the grooves along the double helix) can jamthe motion of knots as tension increases The data presentedin Fig 3(b) (of Ref [14]) exemplify this showing thatalthough the trend of knot motion is convective sufficientinternal friction (likely occurring when the strands in theknot are bent at subpersistence length scales) can overcomethe convective forces and halt the motion of the knot Thereis a small dynamic range of strain rates that can be appliedsuch that the molecule does not relax to the coiled state butthat the knot is noticeably mobile over minute-long time-scales roughly in the range Wifrac14 09ndash15Our results contrast with those of Bao et al [16] who

presented knot diffusivity data for molecules stretched attensions between 011 and 033 pN as determined by theirreported fractional extensions [27] (corresponding to cen-tral chain tensions found at Wi asymp 1 to 2 in our work see theSM [19]) and additionally state that they found no tensiondependence between 01 and 2 pN Because they sampledthe diffusivity of typically three to four molecules pertopology and averaged over tensions spanning a factor of 3they may not have had sufficient statistics to excludetension dependence Furthermore it is difficult to directlycompare our results to those of Bao et al due to the bufferused in their Letter (an overlapping polyethylene glycolsolution) the simpler knots they study (seven crossings andfewer) and the difference between uniform tension and the

FIG 4 (a) Kymograph of a molecule held at varying levels ofthe applied field (Wifrac14 09ndash25 5ndash30 V) for a minute eachshowing greater mobility at lower Wi Note that time runs right toleft in this image The scale bar is 10 μm (b) Relationshipbetween the measured effective knot mobility and the Weissen-berg number Each gray-scale trace represents a different mol-ecule and the red represents the population average The verticallines correspond to the high and lowWi in Fig 3 and the gray barcorresponds to the range observed in Fig 2(b) (Inset) The samedata with a logarithmic y axis

PHYSICAL REVIEW LETTERS 120 188003 (2018)

188003-4

quadratic tension in an elongational field [28] Assumingthat Bao et al did collect significant but unreportedevidence that knot motion is tension independent we positthat the experiments of Bao et al failed to see tensiondependence because they took place in a regime where thestrands in the knot are insufficiently close for corrugation-based friction to be relevant [29] According to the analysisof Bao et al the strands of DNA within the knot areseparated by 26 nm a factor of 10 greater than the DNAcorrugation length scale By comparison our experimentalbuffer gives DNA an effective width of roughly 105 nm[30] with 85 nm between the strands less than a factor of3 above the corrugation length scaleIn summary we examined the motion of knots in DNA

molecules stretched in elongational fields We observedthat knots moved noticeably across the molecule onminute-long timescales until they reached the end anduntied The motion of the knots was driven towards theends of the chain the knots reaching the end faster thanwould be expected from simple one-dimensional diffusionThe motion of the knots can be halted by increasing thestrain rate to induce greater intramolecular friction withinthe knot We measured the mobility of knots with respect tostrain rate and found that knots tend to slow with anincreasing Wi Our findings suggest a systematic way ofremoving unwanted knots from molecules This method ofremoving knots by applying an elongational field maysuggest a modification of nanochannel mapping assays [7]to include a cross flow [31] to remove knots

This research was supported by the National ResearchFoundation Singapore through the Singapore MIT Alliancefor Research and Technologyrsquos research program inBioSystems and Micromechanics and the NationalScience Foundation (Grant No CBET-1602406) A R Kis partially funded by a NSERC Postdoctoral FellowshipWe thank C Benjamin Renner and Vivek Narsimhan forthe fruitful discussion

pdoylemitedudaggerORCID httporcidorg0000-0002-1581-6956DaggerORCID httporcidorg0000-0001-8399-5995sectORCID httporcidorg0000-0003-2147-9172

[1] D Sumners and S G Whittington Knots in self-avoidingwalks J Phys A 21 1689 (1988)

[2] L F Liu L Perkocha R Calendar and J C Wang KnottedDNA from bacteriophage capsids Proc Natl Acad SciUSA 78 5498 (1981)

[3] J M Sogo A Stasiak M L Martınez-Robles D BKrimer P Hernaacutendez and J B Schvartzman Formationof knots in partially replicated DNAmolecules J Mol Biol286 637 (1999)

[4] V Lopez M-L Martiacutenez-Robles P Hernandez D BKrimer and J B Schvartzman Topo IV is the topoisomer-ase that knots and unknots sister duplexes during DNAreplication Nucleic Acids Res 40 3563 (2012)

[5] C Benjamin Renner and P S Doyle Stretching self-entangled DNA molecules in elongational fields SoftMatter 11 3105 (2015)

[6] A Y Grosberg and Y Rabin Metastable Tight Knots in aWormlike Polymer Phys Rev Lett 99 217801 (2007)

[7] J G Reifenberger K D Dorfman and HCao Topologicalevents in single molecules of E coli DNA confined innanochannels Analyst 140 4887 (2015)

[8] V Narsimhan C Benjamin Renner and P S DoyleTranslocation dynamics of knotted polymers under a con-stant or periodic external field Soft Matter 12 5041 (2016)

[9] While any open string such as a linear molecule ismathematically unknotted and the topology of knots inthe center of a chain can be discussed if the ends are treatedas being connected ldquoat infinityrdquo

[10] L Huang and D E Makarov Langevin dynamics simu-lations of the diffusion of molecular knots in tensionedpolymer chains J Phys Chem A 111 10338 (2007)

[11] R Matthews A A Louis and J M Yeomans Effect oftopology on dynamics of knots in polymers under tensionEurophys Lett 89 20001 (2010)

[12] V Narsimhan A R Klotz and P S Doyle Steady-state andtransient behavior of knotted chains in extensional fieldsACS Macro Lett 6 1285 (2017)

[13] L Dai C Benjamin Renner and P S Doyle Metastabletight knots in semiflexible chains Macromolecules 47 6135(2014)

[14] V Narsimhan C Benjamin Renner and P S DoyleJamming of knots along a tensioned chain ACS MacroLett 5 123 (2016)

[15] C Benjamin Renner and P S Doyle Untying knotted DNAwith elongational flows ACS Macro Lett 3 963 (2014)

[16] X R Bao H J Lee and S R Quake Behavior of ComplexKnots in Single DNA Molecules Phys Rev Lett 91265506 (2003)

[17] R Metzler W Reisner R Riehn R Austin J O Tegenfeldtand I M Sokolov Diffusion mechanisms of localised knotsalong a polymer Europhys Lett 76 696 (2006)

[18] C Plesa D Verschueren S Pud J van der Torre J WRuitenberg M J Witteveen M P Jonsson A Y GrosbergY Rabin and C Dekker Direct observation of DNA knotsusing a solid-state nanopore Nat Nanotechnol 11 1093(2016)

[19] See Supplemental Material at httplinkapsorgsupplemental101103PhysRevLett120188003 for cali-bration of the Weissenberg number additional informationabout the diffusive behavior of the knots and a discussion ofintrachain tension in elongational fields

[20] J Tang and P S Doyle Electrophoretic stretching of DNAmolecules using microscale T junctions Appl Phys Lett90 224103 (2007)

[21] B Kundukad J Yan and P S Doyle Effect of YOYO-1 onthe mechanical properties of DNA Soft Matter 10 9721(2014)

[22] J Tang N Du and P S Doyle Compression and self-entanglement of single DNAmolecules under uniform electricfield Proc Natl Acad Sci USA 108 16153 (2011)

[23] A R Klotz V Narsimhan B W Soh and P S DoyleDynamics of DNA knots during chain relaxation Macro-molecules 50 4074 (2017)

PHYSICAL REVIEW LETTERS 120 188003 (2018)

188003-5

stronger effect than the increased convective drive If theknot is sufficiently close to the end however its motioncannot be stoppedTo quantify the relationship between mobility and Wi

we performed an assay [Fig 4(a)] in which knottedmolecules were initially stretched at a high voltage for aminute then the strain rate was decreased every minuteuntil the knot untied or the molecule relaxed to its coiledstate The range of fields used was between Wi frac14 09 andWi frac14 29 corresponding to estimated tensions between 01and 05 pN at the centers of the molecules (see the SM[19]) but lower towards the ends Although the knotdiffusivity is a natural quantity with which to characterizethe knotrsquos motion the MSD trajectories did not displaydiffusive behavior but rather were characterized by sub-diffusion at short times and superdiffusion at long timesInstead we choose a fixed time interval of 5 s (long enoughto avoid tracking noise [26] but short enough to ensuresufficient statistics) and measure the knot MSD (described

in greater detail in the SM [19]) over that interval and divide

by the time to characterize the knotrsquos mobility Figure 4(b)shows the knot mobility as a function of Wi for severalmolecules as well as the population average The mobilityof the knot decreases with an increasing strain rate It variesby more than an order of magnitude as a function of strainrate while the chain extension varies by less than a factorof 2 While there is variation in the mobility with respectto the position of the knot the observed variation inposition-dependent mobility is smaller than the strain-ratedependence and the variation in knot position throughoutthe step-down assay is small (see the SM [19]) Comparisonbetween individual knot trajectories is difficult due to boththe variation in their initial position and the diversesampling of topologies but overall populationwide trendsare observedTwo computational papers from our group [1415]

provide insight on the motion of knots under varyingstrain-rate conditions An earlier study by Renner andDoyle [15] predicted that knots would undergo convectivemotion beyond a critical length scale near the center of themolecule and indeed such convection was observed inour experiments However these simulations used a semi-flexible bead-spring model which was unable to suffi-ciently probe the role of intrachain friction because ofpathological chain crossings at high tensions A subsequentstudy by Narsimhan et al [14] overcame these difficultiesby using a bead-rod parametrization and by studyingflexible as well as semiflexible chains and it found thatfriction arising due to molecular corrugation (eg corre-sponding to the grooves along the double helix) can jamthe motion of knots as tension increases The data presentedin Fig 3(b) (of Ref [14]) exemplify this showing thatalthough the trend of knot motion is convective sufficientinternal friction (likely occurring when the strands in theknot are bent at subpersistence length scales) can overcomethe convective forces and halt the motion of the knot Thereis a small dynamic range of strain rates that can be appliedsuch that the molecule does not relax to the coiled state butthat the knot is noticeably mobile over minute-long time-scales roughly in the range Wifrac14 09ndash15Our results contrast with those of Bao et al [16] who

presented knot diffusivity data for molecules stretched attensions between 011 and 033 pN as determined by theirreported fractional extensions [27] (corresponding to cen-tral chain tensions found at Wi asymp 1 to 2 in our work see theSM [19]) and additionally state that they found no tensiondependence between 01 and 2 pN Because they sampledthe diffusivity of typically three to four molecules pertopology and averaged over tensions spanning a factor of 3they may not have had sufficient statistics to excludetension dependence Furthermore it is difficult to directlycompare our results to those of Bao et al due to the bufferused in their Letter (an overlapping polyethylene glycolsolution) the simpler knots they study (seven crossings andfewer) and the difference between uniform tension and the

FIG 4 (a) Kymograph of a molecule held at varying levels ofthe applied field (Wifrac14 09ndash25 5ndash30 V) for a minute eachshowing greater mobility at lower Wi Note that time runs right toleft in this image The scale bar is 10 μm (b) Relationshipbetween the measured effective knot mobility and the Weissen-berg number Each gray-scale trace represents a different mol-ecule and the red represents the population average The verticallines correspond to the high and lowWi in Fig 3 and the gray barcorresponds to the range observed in Fig 2(b) (Inset) The samedata with a logarithmic y axis

PHYSICAL REVIEW LETTERS 120 188003 (2018)

188003-4

quadratic tension in an elongational field [28] Assumingthat Bao et al did collect significant but unreportedevidence that knot motion is tension independent we positthat the experiments of Bao et al failed to see tensiondependence because they took place in a regime where thestrands in the knot are insufficiently close for corrugation-based friction to be relevant [29] According to the analysisof Bao et al the strands of DNA within the knot areseparated by 26 nm a factor of 10 greater than the DNAcorrugation length scale By comparison our experimentalbuffer gives DNA an effective width of roughly 105 nm[30] with 85 nm between the strands less than a factor of3 above the corrugation length scaleIn summary we examined the motion of knots in DNA

molecules stretched in elongational fields We observedthat knots moved noticeably across the molecule onminute-long timescales until they reached the end anduntied The motion of the knots was driven towards theends of the chain the knots reaching the end faster thanwould be expected from simple one-dimensional diffusionThe motion of the knots can be halted by increasing thestrain rate to induce greater intramolecular friction withinthe knot We measured the mobility of knots with respect tostrain rate and found that knots tend to slow with anincreasing Wi Our findings suggest a systematic way ofremoving unwanted knots from molecules This method ofremoving knots by applying an elongational field maysuggest a modification of nanochannel mapping assays [7]to include a cross flow [31] to remove knots

This research was supported by the National ResearchFoundation Singapore through the Singapore MIT Alliancefor Research and Technologyrsquos research program inBioSystems and Micromechanics and the NationalScience Foundation (Grant No CBET-1602406) A R Kis partially funded by a NSERC Postdoctoral FellowshipWe thank C Benjamin Renner and Vivek Narsimhan forthe fruitful discussion

pdoylemitedudaggerORCID httporcidorg0000-0002-1581-6956DaggerORCID httporcidorg0000-0001-8399-5995sectORCID httporcidorg0000-0003-2147-9172

[1] D Sumners and S G Whittington Knots in self-avoidingwalks J Phys A 21 1689 (1988)

[2] L F Liu L Perkocha R Calendar and J C Wang KnottedDNA from bacteriophage capsids Proc Natl Acad SciUSA 78 5498 (1981)

[3] J M Sogo A Stasiak M L Martınez-Robles D BKrimer P Hernaacutendez and J B Schvartzman Formationof knots in partially replicated DNAmolecules J Mol Biol286 637 (1999)

[4] V Lopez M-L Martiacutenez-Robles P Hernandez D BKrimer and J B Schvartzman Topo IV is the topoisomer-ase that knots and unknots sister duplexes during DNAreplication Nucleic Acids Res 40 3563 (2012)

[5] C Benjamin Renner and P S Doyle Stretching self-entangled DNA molecules in elongational fields SoftMatter 11 3105 (2015)

[6] A Y Grosberg and Y Rabin Metastable Tight Knots in aWormlike Polymer Phys Rev Lett 99 217801 (2007)

[7] J G Reifenberger K D Dorfman and HCao Topologicalevents in single molecules of E coli DNA confined innanochannels Analyst 140 4887 (2015)

[8] V Narsimhan C Benjamin Renner and P S DoyleTranslocation dynamics of knotted polymers under a con-stant or periodic external field Soft Matter 12 5041 (2016)

[9] While any open string such as a linear molecule ismathematically unknotted and the topology of knots inthe center of a chain can be discussed if the ends are treatedas being connected ldquoat infinityrdquo

[10] L Huang and D E Makarov Langevin dynamics simu-lations of the diffusion of molecular knots in tensionedpolymer chains J Phys Chem A 111 10338 (2007)

[11] R Matthews A A Louis and J M Yeomans Effect oftopology on dynamics of knots in polymers under tensionEurophys Lett 89 20001 (2010)

[12] V Narsimhan A R Klotz and P S Doyle Steady-state andtransient behavior of knotted chains in extensional fieldsACS Macro Lett 6 1285 (2017)

[13] L Dai C Benjamin Renner and P S Doyle Metastabletight knots in semiflexible chains Macromolecules 47 6135(2014)

[14] V Narsimhan C Benjamin Renner and P S DoyleJamming of knots along a tensioned chain ACS MacroLett 5 123 (2016)

[15] C Benjamin Renner and P S Doyle Untying knotted DNAwith elongational flows ACS Macro Lett 3 963 (2014)

[16] X R Bao H J Lee and S R Quake Behavior of ComplexKnots in Single DNA Molecules Phys Rev Lett 91265506 (2003)

[17] R Metzler W Reisner R Riehn R Austin J O Tegenfeldtand I M Sokolov Diffusion mechanisms of localised knotsalong a polymer Europhys Lett 76 696 (2006)

[18] C Plesa D Verschueren S Pud J van der Torre J WRuitenberg M J Witteveen M P Jonsson A Y GrosbergY Rabin and C Dekker Direct observation of DNA knotsusing a solid-state nanopore Nat Nanotechnol 11 1093(2016)

[19] See Supplemental Material at httplinkapsorgsupplemental101103PhysRevLett120188003 for cali-bration of the Weissenberg number additional informationabout the diffusive behavior of the knots and a discussion ofintrachain tension in elongational fields

[20] J Tang and P S Doyle Electrophoretic stretching of DNAmolecules using microscale T junctions Appl Phys Lett90 224103 (2007)

[21] B Kundukad J Yan and P S Doyle Effect of YOYO-1 onthe mechanical properties of DNA Soft Matter 10 9721(2014)

[22] J Tang N Du and P S Doyle Compression and self-entanglement of single DNAmolecules under uniform electricfield Proc Natl Acad Sci USA 108 16153 (2011)

[23] A R Klotz V Narsimhan B W Soh and P S DoyleDynamics of DNA knots during chain relaxation Macro-molecules 50 4074 (2017)

PHYSICAL REVIEW LETTERS 120 188003 (2018)

188003-5

quadratic tension in an elongational field [28] Assumingthat Bao et al did collect significant but unreportedevidence that knot motion is tension independent we positthat the experiments of Bao et al failed to see tensiondependence because they took place in a regime where thestrands in the knot are insufficiently close for corrugation-based friction to be relevant [29] According to the analysisof Bao et al the strands of DNA within the knot areseparated by 26 nm a factor of 10 greater than the DNAcorrugation length scale By comparison our experimentalbuffer gives DNA an effective width of roughly 105 nm[30] with 85 nm between the strands less than a factor of3 above the corrugation length scaleIn summary we examined the motion of knots in DNA

molecules stretched in elongational fields We observedthat knots moved noticeably across the molecule onminute-long timescales until they reached the end anduntied The motion of the knots was driven towards theends of the chain the knots reaching the end faster thanwould be expected from simple one-dimensional diffusionThe motion of the knots can be halted by increasing thestrain rate to induce greater intramolecular friction withinthe knot We measured the mobility of knots with respect tostrain rate and found that knots tend to slow with anincreasing Wi Our findings suggest a systematic way ofremoving unwanted knots from molecules This method ofremoving knots by applying an elongational field maysuggest a modification of nanochannel mapping assays [7]to include a cross flow [31] to remove knots

This research was supported by the National ResearchFoundation Singapore through the Singapore MIT Alliancefor Research and Technologyrsquos research program inBioSystems and Micromechanics and the NationalScience Foundation (Grant No CBET-1602406) A R Kis partially funded by a NSERC Postdoctoral FellowshipWe thank C Benjamin Renner and Vivek Narsimhan forthe fruitful discussion

pdoylemitedudaggerORCID httporcidorg0000-0002-1581-6956DaggerORCID httporcidorg0000-0001-8399-5995sectORCID httporcidorg0000-0003-2147-9172

[1] D Sumners and S G Whittington Knots in self-avoidingwalks J Phys A 21 1689 (1988)

[2] L F Liu L Perkocha R Calendar and J C Wang KnottedDNA from bacteriophage capsids Proc Natl Acad SciUSA 78 5498 (1981)

[3] J M Sogo A Stasiak M L Martınez-Robles D BKrimer P Hernaacutendez and J B Schvartzman Formationof knots in partially replicated DNAmolecules J Mol Biol286 637 (1999)

[4] V Lopez M-L Martiacutenez-Robles P Hernandez D BKrimer and J B Schvartzman Topo IV is the topoisomer-ase that knots and unknots sister duplexes during DNAreplication Nucleic Acids Res 40 3563 (2012)

[5] C Benjamin Renner and P S Doyle Stretching self-entangled DNA molecules in elongational fields SoftMatter 11 3105 (2015)

[6] A Y Grosberg and Y Rabin Metastable Tight Knots in aWormlike Polymer Phys Rev Lett 99 217801 (2007)

[7] J G Reifenberger K D Dorfman and HCao Topologicalevents in single molecules of E coli DNA confined innanochannels Analyst 140 4887 (2015)

[8] V Narsimhan C Benjamin Renner and P S DoyleTranslocation dynamics of knotted polymers under a con-stant or periodic external field Soft Matter 12 5041 (2016)

[9] While any open string such as a linear molecule ismathematically unknotted and the topology of knots inthe center of a chain can be discussed if the ends are treatedas being connected ldquoat infinityrdquo

[10] L Huang and D E Makarov Langevin dynamics simu-lations of the diffusion of molecular knots in tensionedpolymer chains J Phys Chem A 111 10338 (2007)

[11] R Matthews A A Louis and J M Yeomans Effect oftopology on dynamics of knots in polymers under tensionEurophys Lett 89 20001 (2010)

[12] V Narsimhan A R Klotz and P S Doyle Steady-state andtransient behavior of knotted chains in extensional fieldsACS Macro Lett 6 1285 (2017)

[13] L Dai C Benjamin Renner and P S Doyle Metastabletight knots in semiflexible chains Macromolecules 47 6135(2014)

[14] V Narsimhan C Benjamin Renner and P S DoyleJamming of knots along a tensioned chain ACS MacroLett 5 123 (2016)

[15] C Benjamin Renner and P S Doyle Untying knotted DNAwith elongational flows ACS Macro Lett 3 963 (2014)

[16] X R Bao H J Lee and S R Quake Behavior of ComplexKnots in Single DNA Molecules Phys Rev Lett 91265506 (2003)

[17] R Metzler W Reisner R Riehn R Austin J O Tegenfeldtand I M Sokolov Diffusion mechanisms of localised knotsalong a polymer Europhys Lett 76 696 (2006)

[18] C Plesa D Verschueren S Pud J van der Torre J WRuitenberg M J Witteveen M P Jonsson A Y GrosbergY Rabin and C Dekker Direct observation of DNA knotsusing a solid-state nanopore Nat Nanotechnol 11 1093(2016)

[19] See Supplemental Material at httplinkapsorgsupplemental101103PhysRevLett120188003 for cali-bration of the Weissenberg number additional informationabout the diffusive behavior of the knots and a discussion ofintrachain tension in elongational fields

[20] J Tang and P S Doyle Electrophoretic stretching of DNAmolecules using microscale T junctions Appl Phys Lett90 224103 (2007)

[21] B Kundukad J Yan and P S Doyle Effect of YOYO-1 onthe mechanical properties of DNA Soft Matter 10 9721(2014)

[22] J Tang N Du and P S Doyle Compression and self-entanglement of single DNAmolecules under uniform electricfield Proc Natl Acad Sci USA 108 16153 (2011)

[23] A R Klotz V Narsimhan B W Soh and P S DoyleDynamics of DNA knots during chain relaxation Macro-molecules 50 4074 (2017)

PHYSICAL REVIEW LETTERS 120 188003 (2018)

188003-5

[24] B W Soh V Narsimhan A R Klotz and P S DoyleKnots modify the coil-stretch transition in linear DNApolymers Soft Matter 14 1689 (2018)

[25] R G Larson Constitutive Equations for Polymer Melts andSolutions Butterworths Series in Chemical Engineering(Butterworth-Heinemann London 2013)

[26] D S Martin M B Forstner and J A Kaumls Apparentsubdiffusion inherent to single particle tracking BiophysJ 83 2109 (2002)

[27] J F Marko and E D Siggia Stretching DNA Macro-molecules 28 8759 (1995)

[28] J W Griffis E Protozanova D B Cameron and R HMeltzer High-throughput genome scanning in constanttension fluidic funnels Lab Chip 13 240 (2013)

[29] A Ward F Hilitski W Schwenger D WelchAW C Lau V Vitelli L Mahadevan and Z DogicSolid friction between soft filaments Nat Mater 14 583(2015)

[30] A R Klotz L Duong M Mamaev H W de HaanJ Z Y Chen and WW Reisner Measuring the confine-ment free energy and effective width of single polymerchains via single-molecule tetris Macromolecules 485028 (2015)

[31] J N Pedersen R Marie A Kristensen and HFlyvbjerg How to determine local stretching and tensionin a flow-stretched DNA molecule Phys Rev E 93042405 (2016)

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