Thermodynamics and kinetics of DNA nanotube polymerization

Thermodynamics and kinetics of DNA nanotube

polymerization from single-filament measurements

Journal: Chemical Science

Manuscript ID: SC-EDG-12-2013-053331.R2

Article Type: Edge Article

Date Submitted by the Author: 29-Nov-2014

Complete List of Authors: Hariadi, Rizal; University of Michigan, Cell and Developmental Biology Yurke, Bernard; Boise State University, Materials Science and Engineering Department Winfree, Erik; California Institute of Technology, Computer Science, Computation and Neural Systems, Bioengineering

Chemical Science

Thermodynamics and kinetics of DNA nanotube polymerization

from single-filament measurements

Rizal F. Hariadi,1 Bernard Yurke,2 and Erik Winfree3

1Applied Physics, California Institute of Technology, Pasadena, California 91125, USA

present address: Cell and Developmental Biology,

University of Michigan, Ann Arbor, Michigan 48105, USA2Materials Science and Engineering Department and Electrical and Computer Engineering Department,

Boise State University, Boise, Idaho 83725, USA3Bioengineering, California Institute of Technology, Pasadena, California 91125, USA

(Dated: January 19, 2015)

DNA nanotubes provide a programmable architecture for molecular self-assembly and can serve asmodel systems for one-dimensional biomolecular assemblies. While a variety of DNA nanotubes havebeen synthesized and employed as models for natural biopolymers, an extensive investigation of DNAnanotube kinetics and thermodynamics has been lacking. Using total internal reflection microscopy,DNA nanotube polymerization was monitored in real time at the single filament level over a widerange of free monomer concentrations and temperatures. The measured polymerization rates weresubjected to a global nonlinear fit based on polymerization theory in order to simultaneously extractkinetic and thermodynamic parameters. For the DNA nanotubes used in this study, the associationrate constant is (5.99 ± 0.15)×105 /M/sec, the enthalpy is 87.9 ± 2.0 kcal/mol, and the entropyis 0.252 ± 0.006 kcal/mol/K. The qualitative and quantitative similarities between the kinetics ofDNA nanotubes, actin filaments, and microtubules polymerization highlight the prospect of buildingcomplex dynamic systems from DNA molecules inspired by biological architecture.

INTRODUCTION

The design and construction of collective behaviors outof rigorously-characterized molecular elements that rivalcellular systems is a challenge at the interface of biology,chemistry, physics, and computer science. Molecular bi-ology provides many proofs of principle for how chemistrycan be used to self-organize matter [5]. As an exam-ple, the cytoskeleton is a system of intracellular biopoly-mers that evaluates its environment to assemble and dis-assemble at the right time and the right place withincells. Interactions between the cytoskeleton, molecularmotors, and signaling proteins give rise to self-organizedintracellular structure [33] and motility [36], direct thegrowth of tissues [35], and guide the movement of organ-isms [48, 54].

DNA nanotubes [10, 45] have been proposed as apromising candidate material for de novo engineering ofan artificial cytoskeleton [22]. A successful demonstra-tion of an artificial cytoskeleton will recapitulate struc-tural, dynamic, force generation, and assembly controlaspects of the biological cytoskeleton. Toward that goal,we must understand the design principles of dynamictubular architectures and develop an accurate physicalmodel of how monomers can assemble and disassembletubular structures as they respond to information in theenvironment.

In structural DNA nanotechnology, synthetic oligonu-cleotides can be engineered to form a small DNA com-plex, called a DNA tile, that can polymerize to formlarger structures using the specificity of Watson-Crick hy-bridization [34, 44, 45, 51, 53, 65, 67, 76]. In this paper,

we use DNA tile and DNA monomer interchangeably.DNA nanotubes provide a simple example of how a longone-dimensional crystalline structure can arise from theinteraction between DNA tiles. Fig. 1 shows a DNA tilethat possesses 4 short single-stranded regions, known assticky ends, that serve as binding domains. The stickyend arrangement, in addition to the constraint providedby the biophysical properties of the DNA double helix,directs the interaction of DNA tiles to form tubular DNAstructures with a range of circumferences whose distribu-tion is determined by the thermodynamics and the kinet-ics of the DNA nanotube assembly process.

The first challenge for de novo construction of an artifi-cial cytoskeleton is constructing a long and rigid polymerout of artificial non-covalently-bound subunits. DNAnanotubes satisfy the length and rigidity criteria. Struc-turally, the DNA nanotubes used in this work are coop-erative polymers that are multiple monomers wide. Thecooperativity has two important consequences. First, thetubular organization of DNA tiles along the longitudinalaxis of a DNA nanotube gives rise to a long persistencelength, ξtubep ∼ 20 µm [41, 45], which is comparable to

the measured persistence length of actin filaments, ξactinp

= 6–25 µm [18, 29, 43]. Second, formation of coopera-tive polymers at reaction conditions where spontaneousnucleation is rare, gives rise to long polymers. The meanlength of the DNA nanotubes used in this study is on theorder of 5 µm for standard assembly conditions. EachDNA tile added to the tip of a growing nanotube inter-acts with two neighbors, whereas most of the collisionsbetween DNA tiles in solution result in contact with onlyone neighbor. As a result, there is a high kinetic barrier

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experimentally in detail. The rigorous testing of DNAnanotube polymerization theory requires assays that candetermine not only the concentration of free DNA tilemonomers, but also the number of nanotubes at anygiven time, and the rate of growth, without experimen-tally confounding effects, such as excessive spontaneousnucleation or the presence of tube bundles. Early studiesof DNA ribbons [51] used bulk UV absorbance data, incombination with static atomic force microscopy (AFM)assays, to measure the concentration of DNA tiles freein solution, and thereby to infer the kinetics of incorpo-ration into ribbon assemblies. Interpreting bulk data iscomplicated, because polymer growth kinetics dependsnot only on free monomer concentration, but also onthe size distribution of supramolecular assemblies and onthe number of such assemblies. This information can-not be accurately measured in bulk UV absorbance as-says and must be inferred indirectly, thus, introducinglarge uncertainties into the analysis. Recently, Evans et

al. used a single-molecule AFM movie to validate someof the kTAM assumptions for polymerization on micasurface [11]. Despite their rigorous analysis, the inter-action between DNA tiles and mica surfaces complicatedtheir measurements and limited their ability to determinequantitative measurements of the rate constants and freemonomer concentrations.In this work, we adopted the standard assay in biopoly-

mer research, namely time-lapse light microscopy [23, 55,56]. The power of single-filament cinematography hasenabled the continuous observation of non-equilibriumpolymers. To minimize background fluorescence fromthe sea of unlabeled monomers in solution, fluorescentpolymers must be excited either with an evanescentwave by total internal reflection fluorescence (TIRF) mi-croscopy [1, 17, 31] or by confocal illumination [26].Here, we report the application of TIRF microscopy to

the study of the polymerization of self-assembled DNAstructures. From a set of polymerization movies at a widerange of tile concentrations and reaction temperatures,we were able to measure both the kinetic and thermody-namic parameters of DNA nanotube assembly. The ex-perimental results are consistent with the common coremodel for nanotube polymerization; further, they pro-vide reliable direct measurements of key thermodynamicand kinetic parameters for DNA tile self-assembly andthus help resolve inconstancies among values that werepreviously measured by indirect means.

MATERIALS AND METHODS

Total Internal Reflection Fluorescence microscope

Optics – The polymerization movies were acquiredwith a custom-built prism-based total internal reflec-tion fluorescence (TIRF) upright microscope (Fig. 2). A

solid-state green laser (GCL-025, 25 mW, CrystaLaser)equipped with an adjustable power supply (CL2005,CrystaLaser) provides the 532 nm excitation light. Thebeam was filtered with a Z532/10× laser filter (Chroma).The filtered excitation beam passed through a quarter-wave plate (Thorlabs) to produce a circularly polar-ized beam, which effectively has uniform polarization tocounter the orientation-dependent fluorescence of Cy3.A shutter is used to block the light path between snap-shots, to reduce photobleaching. Two mirrors (Thorlabs,not pictured in the drawing) were used to guide the illu-mination beam to the field of view below the objective.Another mirror (Thorlabs) and a 15 cm focusing lens(CVI Melles Griot) steered the excitation beam onto aSuprasil 1 right-angle prism (CVI Melles Griot) at ap-proximately 0◦ from the horizon to produce a weaklyfocused illumination spot. We calculated that the inci-dent angle between the incoming laser and the normalvector of the microscope slide is sufficiently larger thanthe critical angle for an evanescent wave to occur at theinterface between glass and liquid, where the sample andfocal plane of the objective are located.

In our experiment, DNA tiles and DNA nanotubesresided inside a rectangular glass capillary tube that wassituated between the prism and objective as shown inFig. 2. The emitted photons were captured by a 60× 1.2NA water immersion objective (Nikon) and focused tothe electron multiplier CCD camera (C9100-02, Hama-matsu) by a 20 cm tube lens (double achromat, CVIMelles Griot). We kept the laser power at below 10 mW,and closed the shutter when not imaging, to minimizephotobleaching. The combination of bright samples, lowbackground, and efficient light collection produces im-ages with high signal-to-noise ratio. Analysis of signal(image intensity of identified tubes) and noise (back-ground near the tubes) was performed on the kymo-graphs of Fig. S3, yielding µsignal = 510.4, σsignal =192.1, µnoise = 37.4, σnoise = 31.5, for a per-pixel signal-to-noise ratio of SNR = µsignal/σnoise = 16.2 and a sig-nal coefficient of variation of cv = σsignal/µsignal = 0.38.

Autofocus – The autofocus and temperature controlfeatures of our microscope were central in automatingthe data acquisition. A rotary motor (Z-drive, ASI) wasmechanically coupled to the translation stage of the ob-jective turret to control the vertical position of the ob-jective via computer. We used an autofocus module inthe µManager software [9, 49] to find and maintain thebest-focus-position of the objective based on the imagesharpness. A focused image has a higher sharpness thanan out-of-focus image. The DNA nanotube images weresufficient for finding the best-focus position without theneed for fiduciary beads. The autofocus method was ro-bust for long time-lapse imaging. The µManager plug-in used the image sharpness function as feedback to theautofocus routine. We set the µManager to run the aut-ofocus step either every 30 or 60 seconds to minimize



5

Name Sequence

NB-1 5′-CTCTGA-CTACCGCACCAGAATCTCGG-3′

NB-2 5′-AATTCC-CCGAGATTCTGGACGCCATAAGATAGCACCTCGACTCATTTGCCTGCGGTAG-3′

NB-3 5′-TCAGAG-GGTACAGTAGCCTGCTATCTTATGGCGTGGCAAATGAGTCGAGGACGGATCG-3′

NB-3-Cy3 5′-Cy3-TT-TCAGAG-GGTACAGTAGCCTGCTATCTTATGGCGTGGCAAATGAGTCGAGGACGGATCG-3′

NB-3-Cy5 5′-Cy5-TT-TCAGAG-GGTACAGTAGCCTGCTATCTTATGGCGTGGCAAATGAGTCGAGGACGGATCG-3′

NB-4 5′-GGAATT-CGATCCGTGGCTACTGTACC-3′

TABLE I: DNA sequences for a single-monomer-type DNA nanotube. – For the fluorophore-labeled strands, we inserted twoadditional T’s between the fluorophore and NB-3 sequence as a spacer to minimize any potential side effect of having the Cy3or Cy5 fluorophore at the end of a sticky end.

timization. We used our custom MATLAB code to de-sign the sequence for the extension of the arms and newpairs of sticky ends based on spurious binding minimiza-tion [63].

DNA stock solution – Each DNA strand (synthesizedby IDT DNA Technologies, Inc.) was resuspended sepa-rately and stored in purified water at a 10 µM stock con-centration. To expedite the subsequent sample prepara-tion step, we typically store our tile as an annealed DNAnanotube stock solution in a 4 ◦C refrigerator, and useit within 1 week after annealing. The stock of DNA nan-otube was made by mixing the four DNA strands at afinal equimolar concentration of 1.5 µM each in a bufferconsisting of 1×TAE [40 mM Tris-acetate and 1 mMEDTA (Ethylenediaminetetraacetic acid)] with 12.5 mMMg-acetate and then annealing from 90 ◦C to 20 ◦C at1 ◦C/min. In retrospect, we consider this annealing stepto be unnecessary because of another annealing step inthe preparation of the supersaturated DNA tile solution.

Pre-formed DNA nuclei with fiduciary markers – Thesimultaneous polymerization measurement of both DNAnanotube ends requires fiduciary markers. To create fidu-ciary markers, we pre-formed DNA nanotubes with ran-dom banding patterns to be used as nuclei. The bandingpattern along the DNA nanotubes established fiduciarycoordinates that enabled separate kinetic measurement ofboth ends of each DNA nanotube. The DNA nanotubeswith fiduciary markers were prepared, as discussed be-low, from Cy3- and {Cy3, Cy5}-labeled nanotubes, whichwere called bright and dim bands, respectively. All of thetiles in the bright nanotubes were labeled with Cy3. Forthe separately prepared dim nanotubes, only 33% of thetiles were labeled with Cy3 and the remaining 67% werelabeled with Cy5. Instead of using an unlabeled tile, wechose Cy5-labeled tiles to decrease the brightness of thetube fluorescence in the Cy3 channel. The aim was tominimize the physical difference between the Cy5-DNAand Cy3-DNA tiles so that both colors of tubes will besimilar in terms of melting temperatures, kinetics, andother aspects relevant to the common core model of nan-otube polymerization.

The DNA nanotube nuclei were prepared as follows:First, we annealed bright and dim DNA nanotubes sep-

arately at a tile concentration of 1.0 µM from 90 ◦Cdown to 50 ◦C at 1 ◦C/min and from 50 ◦C to 20 ◦Cat 0.1 ◦C/min. This annealing protocol produces DNAnanotubes with mean length on the order of 5 µm. Onthe same day, equal volumes of 1 µM bright and dimDNA nanotubes were fragmented into shorter nanotubesby subjecting the DNA nanotube mix to a high elonga-tional fluid flow within a 20 µm×20 µm constriction ina microfluidic chip [21] at a 150 µL/min volumetric flowrate. The elongational flow near the constriction wassufficient to induce significant tension and induce DNAnanotube scission. The fragments had a mean size onthe order of 1 µm. Subsequently, the stochastic end-to-end joining between fragmented bright and dim DNAnanotubes produced hybrid DNA nanotubes with ran-dom banding patterns [10, 45] that were later used asfiducial markers for the construction of kymographs.

The bright and dim segments are visible in the mi-croscopy images (the left panels of Figs. 3 and 4) and aremore obvious in the kymograph (the right top panel ofFigs. 3 and 4). As expected, the position of bright anddim segments did not move relative to each other duringthe course of data acquisition, which justified the choiceof band positions along the DNA nanotubes to act asbonafide fiduciary markers.

Supersaturated DNA tile solution – For the polymer-ization assay, the supersaturated DNA tile solution wasprepared by annealing 10 µL of Cy3-labeled DNA tilemix at 15⁄8×of the desired DNA tile concentration in1×TAE/Mg++ from 90 ◦C to 50 ◦C. The slow anneal-ing was halted at 50 ◦C in order to prevent spontaneousnucleation of nanotubes; for our experimental concentra-tions and time scales, DNA nanotube nucleation is notnoticeable at temperatures above 40 ◦C. At 50 ◦C, 5 µLof 0.9% (w/v) methylcellulose (previously kept at 50 ◦C)was added to the 10 µL supersaturated DNA tiles. Notethat the formation temperature depends on the tile con-centration and on the annealing speed. By the end of theexperiment, we determined that the observed formationtemperature range is 35.2−38.3 ◦C, which corresponds tothe temperature where spontaneous nucleation was ob-served in 100 nM and 500 nM samples after ∼5 minutesof imaging time, respectively. Thus, 50 ◦C incubation


6

is above the formation temperature in any free monomerconcentration used in this work. The sample temperaturewas then lowered to 45 ◦C. 2 µL of pre-formed bandednuclei were added at 45 ◦C and immediately the mixwas gently injected into the capillary tube, which was al-ready at the specified reaction temperature between thetemperature-controlled prism and the objective. Bothends of the capillary chamber were immediately sealedwith Vaseline to prevent evaporation.Crowding agent confines the nanotubes close to the

glass surface – We included 0.3%(w/v) methylcellulose(viscosity 4,000 cP at 2% in H2O at 20 ◦C, purchasedfrom Sigma-Aldrich M0512) as a crowding agent to con-fine DNA nanotubes near the bottom (as well as topand side) of the glass surface where the focal plane andevanescence field were positioned. In a crowded environ-ment, the entropy of the system is maximized when all ofthe long structures are pushed close to another surface,such as capillary tube walls. This entropic confinementdid not hinder the mobility of confined DNA nanotubeswithin their confinement space (middle columns of Figs. 3and 4). This behavior is in accord with previous obser-vations of confined biopolymers in the presence of crowd-ing agents [19, 59]. The side effect of this confinementstrategy is that the same entropic force also favors con-fining DNA nanotubes to other surfaces, including thesurfaces of other DNA nanotubes. Consequently, at highDNA nanotube densities, DNA nanotubes were observedto exhibit side-to-side joining (Movies S1 and S2) andlateral aggregation. The increasing intensity of tubes inimages corresponds to the lateral “bundling” of multipleDNA nanotubes or side-to-side joining (observed directlyin Movies S1 and S2).

Data acquisition

Since polymerization is temperature sensitive, we paidclose attention to the temperature of our sample and min-imized exposure to the room temperature.Before the injection of DNA monomers, the empty

sample chamber was mounted onto the heated prism andunder the heated objective and immersion water to bringthe sample chamber to the desired steady state temper-ature. Skipping this step will result in a sample chamberthat is initially at room temperature, which would causeDNA nanotubes to nucleate very rapidly. In addition,our autofocus did not work well when the temperature ofthe sample, prism, and objective changed rapidly, suchas in the initial heating step of our method of tempera-ture control. The chamber was left empty at the desiredsteady state reaction temperature until the polymeriza-tion mix was ready.In contrast to adding a liquid sample to a filled cham-

ber, injecting a sample into an empty capillary chamberresults in a known initial sample concentration. Previ-

ously, studies that used a similar sample chamber wouldflush the filled chamber with at least twice the cham-ber volume to ensure that the reaction conditions heldduring measurements. Because the fluid flow approacheszero near the channel walls, it is difficult to produce sam-ples with known concentration using that method. Thesecond advantage of starting with an empty chamber isfast injection time. Due to stronger capillary action, in-jecting the sample into an empty chamber requires lesstime than infusing a filled chamber with sample. The fastinjection may also be important in minimizing thermalcontact between the heated liquid of DNA tiles, DNAnanotubes, and the ambient room temperature. How-ever, an empty chamber also possesses an intrinsic prob-lem; the fast injection flow of DNA nanotubes, especiallyat high temperatures, induces DNA nanotube scission.We minimized the scission problem by adding the sam-ple gently at the opening of the empty chamber. Theinjection time was approximately 5 sec for . 6 µL sam-ple. Quantifying how much scission occurs with our in-jection protocol is not necessary since the polymerizationrate measurements should be independent of the initialamount of fragmentation.

We identified three instances in our protocol in whichthe polymerization mix was exposed to ambient environ-ment. First, we pipetted 5 µL methylcellulose to a su-persaturated tile solution with a pipette tip that was atroom temperature. Second, after we incubated the solu-tion of supersaturated tiles and methylcellulose at 50 ◦C,we took the sample out from the temperature cycler andmixed it for ≈5 sec at room temperature. Third, we in-jected the supersaturated tiles, methylcellulose, and pre-formed DNA nuclei at the opening of the heated glasscapillary chamber with a pipette tip that was not heated.To minimize potential problems, such as the rapid nucle-ation of DNA nanotubes from supersaturated DNA tilesbefore the sample was injected to the heated glass capil-lary chamber, we performed these three steps as rapidlyas we could. The typical execution time for these stepswas 5 sec and no longer than 10 sec. In almost all cases,the fast sample handling seemed to be sufficient to avoidspontaneous nucleation before imaging, with the excep-tion of a polymerization assay at 600 nM and 41.4 ◦C.We therefore did not use data taken at 600 nM.

DNA nanotube imaging – Our prism-based TIRF mi-croscope, equipped with temperature control and auto-mated focusing, monitored the dynamics of the DNAnanotubes that were confined close to the glass surface(Figs. 3 and 4) by imaging with a 100 ms exposure timeand 2–4 frame/min acquisition rate. The signal-to-noiseratio was very high, even in the presence of a high con-centration of Cy3-labeled free monomers in solution. Forall of the nanotubes that were analyzed, we did notencounter any pausing of polymerization in any of ourmovies, which provides evidence that the untreated glasssurface is not too sticky. The majority of DNA tiles were


7

in the free monomer state. The typical total concen-tration of DNA tiles in pre-formed DNA nuclei was ap-proximately 7 nM, which is more than 10× smaller thanthe most dilute free monomer concentration in our assay(100 nM). Even after 2 hours of imaging, we typicallyobserved a difference of less than a factor of 2 in contourlength for all DNA nanotubes, which corresponded to asmall DNA tile concentration change.Under reaction conditions where spontaneous nucle-

ation was hardly observable, DNA nanotube polymer-ization was followed for at least 1 hour and no longerthan 2 hours. Much to our surprise, our imaging proto-col did not require an oxygen scavenger buffer to achieveand maintain a high signal-to-noise ratio for the full du-ration of time-lapse imaging. At an acquisition rate oftypically 4 frames/min, we usually acquired enough datapoints in less than 30 minutes. If significant sponta-neous nucleation was observed, we terminated the dataacquisition after ∼5 minutes because the newly formednuclei rapidly obscured the visibility of the pre-formednuclei. Moreover, the new nuclei can also end-to-endjoin to a growing DNA nanotube end, which would havemade our polymerization rate measurements unreliable.Thus, spontaneous nucleation limited the range of tem-peratures and concentrations for which we could obtainaccurate rate measurements. The range of hysteresis ob-served in UV absorbance annealing and melting curvesat 200 nM (Fig. S1) predict a similar range of tempera-tures for which the DNA nanotubes can be held out ofequilibrium during the time required for a typical movie.

Data analysis

The polymerization rate discussed here refers to theelongation or shortening of DNA nanotubes (and thus ismeasured in layers/sec) as opposed to the rates of asso-ciation and dissociation of a single DNA tile to a sin-gle binding site. (The polymerization rate constant konis roughly half the association rate constant for a singlesite, ksiteon , under usual growth conditions where there areon average m

2available binding sites in a nanotube that

is m tiles in circumference. For the same reason, the de-polymerization rate constant koff is roughly half the dis-sociation rate constant for a single site, ksiteoff,2. This dis-tinction is discussed more in the Discussion section andin the Supplementary Information analysis of Hypothe-sis 2, pages S8–S11.) A layer is defined as a ring of mtiles for a m-tile-wide nanotube; it is estimated to be thelength of 42 bp’s or 14 nm (accounting for the expectedcurvature of the helix axis [2]). In our analysis, the num-ber of layers in a nanotube is calculated by dividing itslength by 14 nm. The polymerization rate was measuredusing two methods: (1) kymographs [32] or (2) lengthmeasurements taken at two frames with a sufficient timedifference. The kymograph allows separate measurement

of both nanotube ends at the cost of time to constructa kymograph. Obtaining the polymerization rate fromthe nanotube length at two data points is fast but canonly measure the net polymerization rate of a nanotubeend. Although alignment of fiducial markers is reliablein multi-frame kymographs, image shot noise precludesreliable alignment in two-frame measurements and anyasymmetry in the polymerization rate at the nanotubeends will be unobservable.

In the first method, we applied an ImageJ [49] plugindeveloped by Kuhn and Pollard [32] to construct kymo-graphs from a series of DNA nanotube images. We usedtheir image analysis routine to convert a rough hand traceof each DNA nanotube to a refined trace of the nanotubeby snapping each pixel along the trace to the DNA nan-otube axis. The intensity along the refined traces wasused to construct equivalent straightened images of thecurvilinear DNA nanotubes. The straightened images ofthe same nanotube at different time points were alignedand stacked into a kymograph. We wrote Mathemat-ica (Wolfram Research) code that shifted the longitudi-nal offset between straightened DNA nanotubes until thesum of the correlations between straightened images ina kymograph was maximized, i.e., the banding patternswere vertically aligned. The longitudinal position of bothnanotube ends in a kymograph was detected by setting achosen threshold for both DNA nanotube ends, typicallyless than the half maximum value of any given straight-ened images. For each nanotube end, we performed alinear fit through the coordinates of nanotube ends inthe aligned image to measure the polymerization rate(Figs. 3, 4, and S3). The curve fitting was computing inMathematica using the LinearModelFit function. Out-liers were detected by calculating the Euclidean distancebetween the measurement and the estimates. For qualitycontrol, data points that are > 5 pixels (649 nm) awayfrom the estimates were excluded from the curve fitting.Error bars represent standard errors (1σ).

In the second technique, we simply calculated the netpolymerization rate from the ratio of the length changebetween two frames and the time interval between theframes. Because the kymograph integrates over mul-tiple frames, its standard error is likely to be smaller.However, the simpler technique is far quicker than con-structing a kymograph for each DNA nanotube, and, bybypassing the alignment process, we could measure therates from DNA nanotubes that did not have multiplebands.

The accuracy of our measurements was not limitedby the per-pixel signal-to-noise of microscopy imag-ing (µsignal/σnoise = 16.2). Rather, variability ofpixel brightness within the straightened nanotube im-ages (cv = σsignal/µsignal = 0.38, which is presumablyattributable to shot noise, fluorophore blinking, and pro-cessing to straighten nanotube images) limited the ac-curacy of locating nanotube ends and aligning frames in


8

kymographs. Presuming that each nanotube end is local-ized to within 1 or at most 2 pixels, we expect the lengthmeasurements to have an error of less than 0.52 µm(given pixel size 130 nm). The empirical localization er-rors (per-frame root-mean-square residuals from the lin-ear fits in the kymographs of Figs. 3F, 4F, and S3) werecomparable (σL = 0.34 µm).

Statistical analysis – Curve fitting was computed inMathematica (Wolfram Research) using the commandNonlinearModelFit. For global fits, we subjected all ofour data (N = 347 nanotubes) to Eqs. (1) and (2) forplots shown in Figs. 5, 6 and S2. For local fits (Fig. 6and Tables S1 and S2), each curve fitting was limited tothe data obtained at the indicated monomer concentra-tion (Table S1) or temperature (Table S2). In the ∆G◦

37

analysis in the discussion section, the ∆H term in Eq. 2was substituted by ∆H◦ = ∆G◦

37−(273.15+37 K)×∆S◦.In all curve fitting, all data points were weighted equally.The fitting routine minimizes the sum of the squareddifferences between the model’s predicted rate and theexperimentally-measured rate. The results from theglobal and local fits are reported as mean ± standarderror (1σ) for each parameter estimate.

RESULTS

Polymerization rate measurements

At the critical monomer concentration [tile]crit, thetile attachment rate and the tile detachment rate areequal. As a result, the length of DNA nanotube i fluc-tuates around its length Li. At a constant tile concen-tration away from [tile]crit, DNA nanotubes either elon-gate or shrink at a constant rate. The resolution of themicroscopy assay was diffraction limited at ∼250 nm or∼18× the length of a DNA tile (∼14 nm; Fig. 1). Ourimaging optics produced movies that were sufficient toaccurately track both ends of individual DNA nanotubes.However, the optics were insufficient to discriminate theprecise tile arrangement at nanotube ends and could notdetect individual tile attachment and detachment events.The polymerization rate was measured from time-lapse

images (Figs. 3 and 4) by (1) constructing kymographsand (2) measuring DNA nanotube lengths at two timepoints. We address the results and merits of both ap-proaches below.DNA nanotubes depolymerize at a steady rate at low

free tile concentration ([tile] < [tile]crit) – To measurethe rate at which monomers dissociated from DNA nan-otubes koff , we diluted 1 µM of DNA tiles (as pre-formedDNA nanotubes) at room temperature by a dilution fac-tor of 143 in imaging buffer [1×TAE/Mg++, 0.3% (w/v)methylcellulose]. In the depolymerization experimentswith non-zero free DNA tile concentrations, we addedCy3-labeled DNA tiles to the pre-formed DNA nanotube

nuclei.Fig. 3 shows the steady-state shortening of DNA nan-

otubes at 38.3 ◦C at [tile] = 0. Since the criticalmonomer concentration is always positive, DNA nan-otubes are expected to depolymerize at [tile] = 0. Thefree tile concentration will increase as depolymerizationincreases, but this is limited by the total concentration ofmonomers within the pre-formed DNA nanotube nuclei,which was just 7 nM. As this limit won’t be reached un-less all nanotubes completely depolymerize, and as theconcentration interval in our experiments was substan-tially larger at 100 nM, we considered deviations fromthe nominal free tile concentration to be negligible.DNA nanotubes elongate at high free tile concentra-

tion ([tile] > [tile]crit) – To measure the second-order for-ward rate constant kon with which DNA tiles associatedto DNA nanotube ends, we assayed the DNA nanotubepolymerization at multiple DNA tile concentrations withintervals of 100 nM and at multiple temperatures rang-ing from 28.9 to 41.3 ◦C. An example of DNA nanotubeelongation is shown in Fig. 4 for the case of a tile concen-tration of 400 nM and a temperature of 38.3 ◦C. In theseexperiments, the net elongation results from the excessof tile association events over tile dissociation events.As in the depolymerization case, changes in the free tile

concentration during the experiment can be considerednegligible. Since reaction conditions were chosen suchthat nanotubes did not double in length during the ex-periment, and such that spontaneous nucleation was veryrare, the decrease in free tile concentration can again bebounded by the total tile concentration within the pre-formed nuclei, i.e., 7 nM, which is less than 10% of thesmallest non-zero concentration in our experiments.DNA nanotubes polymerize at steady rates – The lin-

ear fits of DNA nanotube end positions in Figs. 3 and 4show that both polymerization and depolymerization ofDNA nanotubes proceeded at steady rates. Surprisingly,the kymographs of DNA nanotube polymerization revealthe relatively high prevalence of apparently asymmetricpolymerization rates between the two ends of an individ-ual DNA nanotube. This phenomenon, and its statisticalsignificance, is considered further in the Discussion.

Kinetic and thermodynamic analysis of DNAnanotube polymerization

To test the common core nanotube polymerizationmodel, we measured the polymerization rates of 347 DNAnanotubes within a 0−500 nM concentration range anda 28.9−41.3 ◦C temperature range. Having establishedconfidence in the steady polymerization rate, averagepolymerization rates were measured by comparing thenanotube lengths at two time points determined to besufficiently far apart (∆t >10 mins). We excluded DNAnanotubes that had undergone spontaneous scission, end-




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FIG. 5: Dependence of DNA nanotube polymerization rates on free tile concentration for several reaction temperatures. Asexpected, the polymerization rate was faster at lower temperatures and higher free monomer concentrations. The gray-shadedregion represents the parameter space where we observed spontaneous nucleation and end-to-end joining, which invalidatesmeasurements due to side-to-side joining between pre-formed nuclei and the newly nucleated nanotubes. Furthermore, theside-to-side joining obscured the time evolution of individual DNA nanotubes. As a consequence, the movies in the shadedparameter space were not analyzed. The fitting line is the global linear fit (Eq. 1). The numbers on the top horizontal axis arethe inferred critical monomer concentrations (in nM), which were calculated by setting Eq. 1 to zero at given temperature T .The data at a given temperature and at different monomer concentrations was fitted separately (not shown), and the fittingresults are presented in Table S2.

rate constant kon to model the concentration dependenceof the net nanotube elongation rate (in layers per second)and the depolymerization rate constant koff to modelthe net rate of shrinking (also in layers per second), thekTAM models the growth and shrinking process at thelevel of individual tile addition and removal steps. In thekTAM, the association between a free tile and a bindingsite is assumed to be a reaction with forward rate

rf = ksiteon [tile], (4)

where ksiteon is the second-order association rate constantfor an individual tile to an individual site. The reversereaction rate depends on the stability of the binding andis modeled to be

rr,b = ksiteoff,b = ksiteon e−b∆G◦

se/RT+α × u0, (5)

where b is the number of sticky end bonds, ∆G◦

se > 0is the standard free energy for breaking a single stickyend bond at standard concentration u0 = 1 M, and αRTis the initiation energy for double-stranded DNA forma-


12

30.5

32.1

33.6

35.2

36.7

38.3

39.8

41.4

7 105

0

93

191

428

901

Temperature [˚C]

Dis

soci

ati

on

co

nst

an

t

Kd [nM

]

Po

lym

eri

zati

on

ra

te c

on

sta

nt

kon [

×1

05 /

M/s

ec]

0

5

6

7 A

B

FIG. 6: (A) The polymerization rate constant, kon, as in-ferred from independent “local” linear fits at each temper-ature, is relatively constant. (B) The inferred equilibriumdissociation constant, Kd, for DNA nanotube polymerizationgrows exponentially with the temperature. The dissociationconstant was calculated by taking the ratio of inferred koffand kon parameters from the local fits. The solid line wascomputed by employing the ∆H◦ and ∆S◦ parameters fromthe global data fit and Eq. 2.

tion with α ∼ ln(20) in solution [64]. The standardfree energy ∆G◦

se can be further expressed as ∆G◦

se =∆H◦

se − T∆S◦

se. In the kTAM, due to the weak bondstrength of one sticky end interaction, a tile that bindswith one sticky end will quickly disassociate from thenanotube end, as illustrated by the large arrow in theleft panel of Fig. 7. In DNA nanotube polymeriza-tion, configurations where an incoming tile can bind with3 or 4 bonds can be neglected because a DNA nan-otube end is highly unlikely to contain any tile arrange-ment allowing for a tile to bind with 3 or 4 sticky ends.For the quantitative analysis, we ignored 1, 3, and 4sticky end interactions and assigned the inferred ∆G◦

as the free energy of an interaction with two sticky-ends∆G◦ = 2∆G◦

se − αRT , ∆G◦ > 0 (Fig. 7 right).

Under the assumption that tiles attach or detach onlywhen bound by exactly 2 sticky ends, we derive (Sup-plementary Information, pages S8–S11) that steady-stategrowth (or shrinkage) results in on average m

2potential

attachment sites and on average m2potential detachment

sites, for nanotubes of large enough circumference m.Since m tiles must attach (or detach) to grow (or shrink)by one layer, we have kon = 1

2ksiteon and koff = 1

2ksiteoff,2.

Thus the thermodynamic and kinetic parameters of thekTAM can be put in exact correspondence with those ofthe common core model.

Symmetrical or asymmetrical polymerization?

Many kymographs exhibited asymmetrical growthrates in the two ends (Fig. S3). Asymmetrical growthcould be expected for several reasons. For example,one end of the nanotube presents sticky ends 1 and 2*,while the other end presents sticky ends 1* (which isfluorophore-labeled) and 2. Thus, although the ther-modynamics of adding tiles must be the same for bothends, the kinetics could be different. Alternatively, evenif constant-diameter nanotubes have identical kineticson both ends, growth rates could differ in nanotubesthat were formed by joining events [10] between precur-sor nanotubes of two distinct diameters. Under our an-nealing protocol for creating nuclei, nanotube diametersvaried between 5 and 11 tiles in circumference (Figs. 8and S4). Both kinetic factors related to tile assemblypathways, as well as thermodynamic factors related tonanotube strain [45], could differ for tubes of differentcircumferences, although the latter factors are likely tobe more significant (Figs. S5 and S6). However, statis-tical analysis suggests that some (or all) of the experi-mentally observed asymmetry can be attributed to noisein the rate measurements (Fig. S7), and thus none ofthese potential effects are likely to factor largely in ourexperiments. Using DNA origami seeds [3, 38] that al-low growth from only one side, rather than pre-formednanotube nuclei that can grow on both ends, would inprinciple allow this issue to be resolved.

Comparison with previously reported kinetic andthermodynamic parameters

Kinetic parameters

We measured kon to be (5.99±0.15) ×105 /M/secfor the polymerization model. As noted earlier, notall parts of growing/shrinking nanotubes contribute tothe polymerization/depolymerization. Only sites with 2sticky ends contribute to the process (Figs. 7 and S5–S6). Consequently, the inferred association rate con-stant for a single DNA tile binding to an available siteat the end of a DNA nanotube was expected to beksiteon ≈ 2 × kon ≈ 1.2 × 106 /M/sec. As an oversimpli-fication, we initially compare the association rate con-stant for a tile attaching to a site by two length-sixsticky ends, ksiteon , to the second order association ratek2 for a 12 nucleotide DNA strand hybridizing to its per-fect complement. Wetmur and Davidson [61] determinedk2 ≈ 3.5 × 105 ×

√L = 1.2 × 106 /M/sec for L = 12;

Morrison and Stols [39] measure k2 ≈ 107 /M/sec for a10-mer; while Zhang and Winfree [75] obtained the rangeof forward rate to be (1−6)×106 /M/sec for toeholds ofvarious sequences and lengths up to 15 nucleotides. Thus,to first approximation, tile association rate constants ap-


13

for Students

konkon koff,1 koff,2site sitesite site

FIG. 7: In the kinetic Tile Assembly Model [64], the rate of a free tile attachment to an available site in DNA nanotubeend ksite

on is independent of the number of sticky ends in the potential binding site. To satisfy detailed balance, the reverserates depend on the number of available sticky ends. Hence, a DNA tile that only binds with one sticky end (left panel) willdissociate from a DNA nanotube faster than DNA tile with 2 bonds (right panel). The configuration of DNA nanotube endsand the position of dark tiles are different between left and right panels. Here, the highlighted attachment sites in the left andright panels provide 1 and 2 sticky ends, respectively. The attachment sites are illustrated as darker colored tiles. The stickyends, illustrated as short green or orange tubes, are complementary when the colors match. The faster rate is indicated by thelarger arrow of ksite

off,1 than ksiteoff,2.

0

0.1

0.2

0.3

0.4

0

mode = 7 DNA tiles

2 4 6 8 10 12 14

No

rma

lize

d f

req

ue

ncy

Nanotube circumference [DNA tiles]

A B

20 nm 20 nm

FIG. 8: Our annealing protocol produces DNA nanotube nu-clei that are 5–11 tiles in circumference, with 7-tile-wide DNAnanotubes as the most prevalent. Images of individual openedDNA nanotubes for constructing this histogram are presentedin the (Fig. S4). Heterogeneous circumferences were also ob-served in the in-vitro self-assembly of other tubular structures,such as protein microtubule [8, 57, 58]. The error bars are thestandard deviation for each bin calculated using a bootstrap-ping method. (Insets) Representative images of opened DNAnanotubes with circumference of 5 (A) and 11 (B) DNA tiles.

pear to be comparable with oligonucleotide hybridization

rate constants.

Our results may also be compared to previous investi-gations of the kinetics of tile assembly. In two separateworks with different types of DNA ribbon, Schulman andWinfree [51] as well as Fujibayashi and Murata [16] usedthe values suggested in the original kTAM paper [64] (orwithin 25% thereof, i.e. 0.8–1.0×106 /M/sec) for the tileassociation rate constant in their kTAM-variant simula-tions, yielding satisfactory (though indirect) agreementwith experimental data. In contrast, Chen et al ’s studyof “snaked tile” growth and facet nucleation [7] increasedksiteon to 17 × 106 /M/sec in order to match their exper-imental data; this is 14× faster than our measurement.However, given the sensitivity of their complex simula-tion to both ksiteon and to the various sticky-end free ener-gies, none of which were determined experimentally, wepresume that their inferred value for ksiteon is unreliable.The final comparison is to Jiang et al ’s careful study ofthe kinetics of dimer association for a variety of DX andDX-like tiles [27]. This careful study found that at 24 ◦C,dimers binding by two 5 nt sticky ends formed at approx-imately twice the rate as dimers binding by a single 10 ntsticky end (roughly 2.5×106 /M/sec vs 1.2×106 /M/sec).They also studied the temperature dependence of these


14

rates, down to 12 ◦C, from which we extrapolate a rateof 3.5× 106 /M/sec for association by two sticky ends at37 ◦C, which is within the range of temperatures in ourstudy. The roughly three-fold difference, compared toour results, could be attributed to sequence dependenceor induced strain due to the DNA nanotube context, asdiscussed in the thermodynamics section below.

Thermodynamic parameters

To assess the experimentally-measured thermody-namic parameters in our study (Table S1), we com-pared the enthalpy, the entropy, and the standard freeenergy of two sticky end interactions at 37 ◦C, to thetheoretical predictions and to values from previously re-ported studies of the free energy of DNA hybridization.The first comparison that we examine is with respect tothe kTAM [64]. In that model, the enthalpy of disas-sembly was estimated to be ∆H◦ = R × sb×(4000 K),where s is the number of base pairs in a sticky endand b is the number of sticky end bonds. For b=2and s=6, the simple expression yields the value of∆H◦ = 95 kcal/mol, which is within 10% of our mea-surement of ∆H◦ = 87.9±2.0 kcal/mol. For the en-tropy, using the kTAM and α=ln(20) [64], the theo-retical value was predicted to be ∆S◦ = R × (11sb +α) = 0.268 kcal/mol/K, which is within 10% of the mea-sured ∆S◦ = 0.252±0.006 kcal/mol/K in our experiment.A more insightful thermodynamic parameter to describethe polymerization process is the standard free energy.Using the predicted and experimentally-measured val-ues of ∆H◦ and ∆S◦, we calculated the standard freeenergy of two sticky end interactions at 37 ◦C to be∆G◦

37 = ∆H◦ − [(273 + 37) K]∆S◦ = 12.2 kcal/mol.In our experiment, the global data fit yielded the freeenergy ∆G◦

37 = 9.84±0.02 kcal/mol. The ∆G◦

37 from thekTAM is 3.8 RT higher than our measured ∆G◦

37.To our knowledge, the only published values for ther-

modynamic parameters of double crossover tile-basedDNA crystal structures in solution were obtained frombulk studies of DNA ribbons of designed widths [51].The ribbons were composed of multiple tiles with 5 bpsticky ends, which is shorter than the 6 bp sticky endsin our tiles. Schulman and Winfree extracted ∆H◦

and ∆S◦ from a series of UV absorbance data by van’tHoff analysis. They measured ∆H◦ = 102.4 kcal/moland ∆S◦ = 0.300 kcal/mol/K. To compare our mea-surements to these values, we multiplied these valuesby 6/5, which is the ratio of sticky end lengths of ourDNA nanotube and Schulman andWinfree’s ribbon. Thelinear extrapolation gives ∆H◦ = 122.9 kcal/mol and∆S◦ = 0.360 kcal/mol/K. Using these adjusted ∆H◦

and ∆S◦ values, the free energy was calculated to be∆G◦

37◦C = 11.3 kcal/mol, which is 2.4 RT higher thanour measured ∆G◦

37.

One class of thermodynamics measurements relevantto DNA nanotube polymerization is the dimerization re-actions of rigid DNA molecules, such as (1) quadruple-crossover (QX) molecules [40] and (2) double-crossover(DX) DNA tiles [30]. The QX molecule, in essence, is asheet of 4 parallel DNA helices. By attaching different 6bp sticky ends to a QX pair, the thermodynamic prop-erties of different configurations of sticky ends were sys-tematically studied. In their second study, the reactionconsisted of DNA tiles similar to our monomers in Fig.1, with shorter 5-bp sticky end pairs. The most relevantsubset of their experiments [30, 40] is the dimerization ofrigid QX and DX structures with 2 pairs of sticky endsthat are located adjacent to each other. In these vari-ants, the adjusted enthalpy was measured to be 105.1–122.4 kcal/mol. For the entropy of the reaction, theydetermined the values to be 0.301–0.352 kcal/mol/K. To-gether, the free energy of DNA tile dimerization wascomputed to be 11.8–13.3 kcal/mol, which is 3.2–5.6 RThigher than our measured free energy.

The relatively-low free energy in our measurement sug-gests that the sum of inter-molecular penalties betweenDNA monomers within the nanotubes, which includesthe inter molecular strain and electrostatic repulsion,is higher than the energy cost for DX dimers [30], QXdimers [40], and DNA ribbons [51]. If we consider Nang-reave et al.’s DX and QX molecules as simplified modelsof the association and disassociation of a DNA tile toa growing DNA nanotube, one consideration is that thedimerization process will yield structures with less strainthan in our DNA nanotubes. In a related system, strainin tile attachment to a DNA origami seed was inferredto account for a 2 kcal/mol (or 3.3 RT ) deviation [38].Moreover, the close proximity between DNA tiles alongthe circumference gives rise to higher electrostatic repul-sion than in DX dimers, QX dimers, and two-dimensionalDNA ribbons. In addition to the inter-molecular strain,the free energy is also dependent on the sticky end se-quences. Theoretical ∆G◦ at 37 ◦C for a random 12-bpDNA helix, using Santa Lucia’s nearest neighbor param-eters [47] excluding any stacking energies for the flankingnucleotides, was computed to be 9.9–20.7 kcal/mol with〈∆G◦〉 = 15.5±1.9 kcal/mol. Interestingly, the ∆G◦ ofthe sticky end sequences in DNA ribbons, QX, and DXwere calculated to be 0.4–2.4 kcal/mol or 0.7–4 RT higherthan the calculated ∆G◦ for our DNA nanotubes. To-gether, the sum of higher strain and weaker sticky end issufficient to explain the relatively low free energy valuein our experiments.

Finally, another meaningful value to compute is themelting temperature Tm of DNA nanotubes. From sim-ple thermodynamics, the melting temperature in Kelvincan be calculated as

Tm =∆H◦

∆S◦ −R ln[tile]. (6)


15

Using the theoretical values from the kTAM [64], themelting temperature for a reaction with 100 nM, 200 nM,300 nM, 400 nM, and 500 nM free tile concentration iscalculated to be 43.5 ◦C, 44.9 ◦C, 45.7 ◦C, 46.4 ◦C, and46.9 ◦C, respectively. The calculated values are less than8 ◦C higher than the measured equilibrium temperaturein the polymerization rate vs. temperature plots (Fig.S2). Similarly, the discrepancy in Tm is likely becausethe kTAM number is derived from a simple model and ig-nores the inter-molecular strain and sequence dependenceof ∆H◦ and ∆S◦. Nonetheless, this relative agreementhints at the usefulness of the simple energetics model inthe kTAM for estimating thermodynamic values in DNAself-assembly.

Comparison with the polymerization rates of actinand microtubules

The kinetic Tile Assembly Model posseseses the samekinetic and thermodynamic features as the kinetic modelfor actin polymerization [31]. The forward rates of DNAnanotube, actin filament, and microtubule assemblies aremodeled as reactions that depend on the free monomerconcentration-dependent reaction. Actin filaments andmicrotubules are asymmetric polymers. The polymerends have different thermodynamic free energies and ki-netic rates. The association rate constant for an ATPbound actin monomer to attach to an actin filamenthas been measured at the single molecule level to be0.5×106 /M/sec and 7.4×106 /M/sec for the pointed andthe barbed end, respectively [31]. For microtubules, theassociation rate constant for α,β-tubulin bound GMP-CPP, an unhydrolyzable analog of GTP, to dock to amicrotubule at 37 ◦C has been measured by bulk assayto be 5.4×106 /M/sec [25]. The association rate constantkon values of actin and microtubules are comparable tothe measured kon in our assay. The monomer dissoci-ation rate for actin and microtubule polymerization de-pends on the bond strength. The dissociation rate of fuel-bound monomers, such as ATP-actin and GTP-tubulin,is slower than waste-bound monomers. The qualitativeand quantitative similarities between the DNA nanotubeand actin provide additional support for the DNA nan-otube as an attractive engineering material for de novo

creation of an artificial cytoskeleton.

Although both polymers have comparable kon values,typical polymerization of actin and microtubules is onthe order of 1 layer/sec or faster, compared to the 0.1layer/sec mean polymerization rate of DNA nanotubesreported here. Faster polymerization gives actin and mi-crotubules morphological flexibility. These biopolymerscan assemble structures when a cell needs them and sta-bilize them by capping proteins. The faster cytoskeletonpolymerization rate is a direct result of the higher freemonomer concentration in cellular milieu, which is on the

order of 1 µM. In our study, the relatively high sponta-neous nucleation rate in DNA nanotubes prevented usfrom performing polymerization assays at comparableconcentrations to those of the actin and microtubules.Hyman et al. [25] have shown that the coupling betweenpolymerization and stochastic GTP hydrolysis is respon-sible for the slow spontaneous nucleation rate of proteinmicrotubules. Docking of an α,β-tubulin monomer thatis bound to GTP on a growing microtubule, triggers thestochastic GTP hydrolysis reaction, which weakens thetubulin−microtubule binding and increases the dissocia-tion rate significantly. Inspired by this elegant solution, itwill be interesting to examine how to incorporate energyconsuming reactions into the interaction between DNAtiles and between DNA tiles and DNA nanotubes in or-der to achieve a higher nucleation barrier than the oneobserved in the existing passive DNA nanotube system,such as the one used in this work.

CONCLUDING REMARKS AND OUTLOOK

From single-filament movies, we were able to system-atically test a mathematical model of DNA self assem-bly while extracting both the kinetic and thermodynamicparameters of DNA nanotube polymerization. The poly-merization model depends on the tile concentrations andis sensitive to reaction temperature. To the best of ourknowledge, this experiment is the most accurate mea-surement of DNA tile-based self-assembly to date. Ourexperiment justifies the use of polymerization theory de-veloped for one-dimensional cooperative polymers, suchas microtubules and actin, to accurately model DNA nan-otube polymerization.Further, we expect that the common core model will be

suitable for many types of tile-based self-assembled DNAnanotubes – such as variations on DX tiles [45, 69], triple-crossover (TX) tiles [70], multi-crossover (MX) tiles [71]and even “4x4” tiles [72] – so long as tile formation iscleanly separated from tube formation, tiles are relativelyrigid, a regular lattice is formed, and similar tile assem-bly pathways arise. It is less clear whether the basicmodels studied here can be applied without modifica-tion to DNA nanotubes that self-assemble directly fromsingle-stranded DNA [66–68], because the increased po-tential for flexible interactions between unformed or par-tially formed oligonucleotides may introduce complica-tions. Synthetic RNA filaments [73, 74], which may formvia distinct self-assembly pathways, may also require dis-tinct modeling features.The most basic demonstration of non-equilibrium poly-

mer dynamics is steady elongation or shortening at aconstant monomer concentration that is far from thecritical DNA tile concentration. More elaborate non-equilibrium polymer behaviors can be envisioned, suchas coupling the DNA nanotube polymerization described


16

in this paper with an analog of nucleotide hydrolysis [22].It is conceivable that this biomimetic strategy could po-tentially recapitulate the more complex non-equilibriumcytoskeleton-based dynamics [24], such as treadmilling [6]and dynamic instability [37], where polymerization anddepolymerization co-exist at steady state without everreaching equilibrium. These novel dynamics can onlybe observed at the single-filament level, as demonstratedwith the TIRF assay reported here.

ACKNOWLEDGMENTS

We would like to acknowledge Rebecca Schulman,Damien Woods, Matthew Cook, Tosan Omabegho, HeunJin Lee, Ethan Garner, Michael Diehl, Zahid Yaqoob,Nadine Dabby, and Paul Rothemund for their helpfuldiscussions. The authors especially thank Rebecca Schul-man and Ann McEvoy for pointing our attention to glasscapillary chambers, and Matthew Cook for analysis of2D crystal growth front dynamics. The length mea-surements were made possible because of Jeffry Kuhn’sgenerosity in sharing his filament snapping and lengthmeasurement codes. This work was supported by NSFthrough the grants EMT-0622254, NIRT-0608889, CCF-0832824 (The Molecular Programming Project), andCCF-0855212.

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Thermodynamics and kinetics of DNA nanotube polymerization