Folding dynamics of tethered giant DNA under strong flowTakuya Saito, Takahiro Sakaue, Daiji Kaneko, Masao Washizu, and Hidehiro Oana Citation: J. Chem. Phys. 135, 154901 (2011); doi: 10.1063/1.3652957 View online: http://dx.doi.org/10.1063/1.3652957 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v135/i15 Published by the American Institute of Physics. Related ArticlesFolding dynamics of tethered giant DNA under strong flow JCP: BioChem. Phys. 5, 10B614 (2011) Rigorous coarse-graining for the dynamics of linear systems with applications to relaxation dynamics in proteins JCP: BioChem. Phys. 5, 08B605 (2011) Rigorous coarse-graining for the dynamics of linear systems with applications to relaxation dynamics in proteins J. Chem. Phys. 135, 054107 (2011) Comparison of two adaptive temperature-based replica exchange methods applied to a sharp phase transition ofprotein unfolding-folding JCP: BioChem. Phys. 5, 06B614 (2011) Comparison of two adaptive temperature-based replica exchange methods applied to a sharp phase transition ofprotein unfolding-folding J. Chem. Phys. 134, 244111 (2011) Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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THE JOURNAL OF CHEMICAL PHYSICS 135, 154901 (2011)

Folding dynamics of tethered giant DNA under strong flowTakuya Saito,1,a) Takahiro Sakaue,2,3 Daiji Kaneko,1 Masao Washizu,1,4,5

and Hidehiro Oana1,5,b)

1Department of Mechanical Engineering, The University of Tokyo, Tokyo 113-8656, Japan2Department of Physics, Kyushu University 33, Fukuoka 812-8581, Japan3JST, PRESTO, Japan4Department of Bioengineering, The University of Tokyo, Tokyo 113-8656, Japan5JST, CREST, Japan

(Received 17 May 2011; accepted 28 September 2011; published online 20 October 2011)

Using a microfluidic device, we investigate the folding dynamics of individual linear long DNA,whose one end is tethered under a strong flow in the presence of a condensing agent. Direct obser-vations of the folding process of DNA molecules reveal a characteristic dynamics with pronouncednon-monotonic velocity of the folded part at the free end against the flow. We discuss this unique dy-namics in relation to the inhomogeneous spatial fluctuation and the structure change at the multipleorder levels along the stretched DNA, which is induced by the increasing tension due to the build-upof the hydrodynamic drag force. © 2011 American Institute of Physics. [doi:10.1063/1.3652957]

I. INTRODUCTION

Genomic DNAs are very long biomacromolecules and asfor the eukaryote, their length range generally from mm tocm. Nonetheless, they are neatly folded in a narrow cellularspace and function properly with dynamical conformationalchanges. The formation of a higher-order structure by DNAfolding is thus expected to be essential for biological func-tions such as replication and transcription. The equilibriumaspects of DNA folding have been extensively studied. Oneaccomplishment is the fact that DNA folding has been clar-ified to be a discrete transition from a fluctuating coil to afolded compact state at a single chain level upon the additionof condensing agents.1

As compared to the transition manner between stablestructures involved in DNA folding, its dynamics have notyet been well understood. To obtain further insights, foldingprocess of individual DNA molecules has been observed uti-lizing stretched DNA obtained by tethering its one end un-der a flow of condensing agent’s solution. Previous studies onDNA chains shorter than ca. 100 μm have shown that the fold-ing rate is roughly constant during folding, i.e., the stretchedlength from the tethered point to the free end of the chainroughly monotonically shrinks in this process.2–5 It is stressedthat this situation is qualitatively different from the foldingtransition at equilibrium state in various aspects. Among oth-ers, the stretching effect caused by the flow is notable. Dueto the building-up of the hydrodynamic drag force along thechain, the degree of segmental spatial fluctuations should de-crease in the upstream direction.6 Thus, if we employ consid-erably longer chains, the inhomogeneous effect of segmentalspatial fluctuations should be significant.

a)Present Address: Department of Physics, Kyushu University 33, Fukuoka812-8581, Japan.

b)Author to whom correspondence should be addressed. Electronic mail:oana@mech.t.u-tokyo.ac.jp.

In this paper, we investigate the folding process of longDNA (more than 200 μm in length) with one end tethered un-der a strong flow in the presence of a condensing agent. In thiscase, it is noteworthy that the describable inhomogeneity iscaused by the deformation of segments (variant from normalB-form on secondary structure) due to the overstretching ofthe double helical backbone7, 8 as well as spatial fluctuationsof invariant segments. Then, the hierarchical nature of theDNA molecule becomes explicit, and there may arise a cou-pling between transitions in different length scales under thisflow condition. In principle, clarifying these aspects shouldbe important for genome technology from the viewpoint ofunderstanding living matter, for which the reliable controlof hierarchical structures involving multiple order scales isrequired.

II. MATERIALS AND METHODS

In our experiment, we used large DNA molecules whichwere obtained from an agarose gel block (Bio-Rad Labora-tories, Inc.) in which yeast chromosomal DNAs (S. pombe)are embedded as size markers of linear DNAs with lengthsof 1.2, 1.6, and 1.9 mm. By melting a piece of the gel block(ca. 10 mm3) for 30 min at 80 ◦C in 1 ml of 2 M sodiumchloride solution, a solution of chromosomal DNA was ob-tained. With this procedure, we obtained a dilute DNA solu-tion which contains DNA molecules longer than 200 μm al-though large DNA molecules in random coil state are easyto be fragmented during pipette-base handling due to hy-drodynamic shear force. The direct observation of the singlechain folding was performed using a microfluidic channel de-vice fabricated by standard soft lithography with PDMS. Asschematically depicted in Fig. 1, this main channel (0.5 mmin width) was equipped with a micro-pillar array to hook andstretch linear DNA chains. The hooked DNA can be regardedas a chain with one end tethered at the pillar. In addition, thesepillars stand the stall force produced by the DNA chain. The

0021-9606/2011/135(15)/154901/5/$30.00 © 2011 American Institute of Physics135, 154901-1

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154901-2 Saito et al. J. Chem. Phys. 135, 154901 (2011)

Top view

m icro pillar region

2,0

00

μm

22 mm

0.5 mm

Cover slip

PDM S

Micro pillarIn lets O u tlet

2000 μm

20 μm

90 μ

m 60 μ

m

25 μm

Cross-sectional view

FIG. 1. The schematic representation of the micro-flow channel on the in-verted microscope for the observation of DNA folding process. The in-set represents arrangement of micro-pillars for the immobilization of DNAmolecules. (Reproduced and modified from Ref. 9, T. Saito et al., 2008 Inter-national Symposium on Micro-NanoMechatronics and Human Science, withpermission from c© [2008] IEEE.)

inlets and outlet were on the upstream and downstream sidesof the pillars, respectively. Each reagent was fed from dif-ferent inlets, which were connected with sample solutionsthrough silicone tubes, respectively. The flow of each solutionwas independently controlled by the microfluidics flow con-trol system (MFCS; FLUIGENT). First, the genomic DNAsolution was fed, hooked to the pillar by chance, and stretchedby a flow. Then, to rinse off sodium chloride and unhookedDNA from the main channel, purified water (by Milli-Q Gra-dient; Millipore) was introduced. After the immobilization ofa single DNA molecules on the micro pillars, i.e., obtain-ing tethered DNA molecules, the condensing agent (1 mMspermidine (trivalent cation)) was fed, and we observed thedynamics of DNA folding under inverted fluorescence mi-croscopy (IX-71; Olympus Corp.). In all solutions, 1 μM YO-PRO-1 as a fluorescent dye for DNA and 1 mM dithiothre-itol as a deoxidizer were dissolved. The fluorescence imagewas acquired using a high-sensitivity video camera (EB-CCDcamera, C7190-43; Hamamatsu Photonics K.K.).

III. RESULTS AND DISCUSSION

Figure 2 shows a time series of fluorescence images ofDNA folding induced by the addition of the condensing agent,spermidine. In this article, we set the term “stretched length”,which is the distance from tethered point to the downstreamfree end of the DNA chain. Typically, before folding, thestretched length is greater than the width of the field of viewdue to the optical setup (ca. 200 μm), and the free down-stream end is not observed in the flame. Upon the additionof condensing agents, the stretched length starts decreasing.Shortly after the condensing agent is introduced, the down-stream free end enters the visual field owing to the shrinkageof the stretched length. Note that the free downstream endwas observed as the first bright spot that corresponds to thefolded part. Here, we set this time as t = 0. This downstreamfolded part (#1) moves toward the tethered point at the pillarwith time. In many samples, plural bright folded parts (#2, #3,. . . ) were newly observed along the stretched chain, especially

0 s

0.17 s

1.0 s

10.0 s

59.0 s

60.0 s

60.4 s

60.8 s

61.2 s

61.6 s

61.8 s

62.0 s 100 μmt (

s)

Flow direction

#1#3#4

#1

#2

Tethered point at micro pillar

DNA

FIG. 2. The typical folding process of long DNA at a flow velocity of ca.300 μm/s. The flow is directed rightward. The bright spots indicate foldedparts, which are labeled by #1 ∼ #4 from the downstream end. (Reproducedand modified from Ref. 9, T. Saito et al., 2008 International Symposiumon Micro-NanoMechatronics and Human Science, with permission from c©[2008] IEEE.)

near the downstream free end through the folding process, asseen after 60 s. These move against the flow while fusing withthe downstream end or other folded parts.

To clearly characterize the folding dynamics using thesample shown in Fig. 2, we established the time evolutionof the stretched length from the tethered point to the foldedpart at the downstream end, together with the positions ofother folded parts, as shown in Fig. 3. Hereafter, the veloc-ity at the downstream end is called the folding velocity, andit corresponds to the shrinkage rate of the stretched length.Figure 3(a) shows the overall folding process. This resultindicates that folding proceeds through three stages: [I] thefolded part at the free end rapidly moves toward the teth-ered point, but slows down at a length of around 170 μm;[II] then, the folding almost plateaus in around a dozen sec-onds; and [III] finally, the chain suddenly and rapidly foldscompletely within a few seconds. This non-monotonic fold-ing process was observed under flow velocity of more thanca. 200 μm/s (we examined up to ca. 600 μm/s). On the otherhand, for the relatively shorter DNA(<200 μm), the plateaustage was not observed during the folding under flow velocityof less than ca. 300 μm/s (not shown). This rather monotonicfolding for the shorter DNA under the weak flow is in agree-ment with previous reports.2–5 Figures 3(b) and 3(c) showmagnified views of stages [I] and [III], respectively. Also,the plural folded parts have been often observed. Similarly,

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154901-3 DNA folding under flow J. Chem. Phys. 135, 154901 (2011)

I II III

I III

II

200

150

100

50

0302010 500 40 60 70t (s)

180

170

t (s)0 1 2 3

200

100

059 60 61 62

t (s)

(a)

(b) (c)

Str

etch

ed le

ng

th L

(μm

)

Str

etch

ed le

ng

th

L (

μm)

FIG. 3. Time evolution of the length from the tethered point to the foldedparts along the folding. Data points are obtained from the sample shownin Fig. 2. (©), (�), (�), and (�) correspond to folded parts, #1 ∼ #4,from the downstream end, respectively. (a) Overall process. Magnified views:(b) [I] first stage and (c) [III] last stage. Red lines serve as a visual guide forthe stretched length. (Reproduced and modified from Ref. 9, T. Saito et al.,2008 International Symposium on Micro-NanoMechatronics and Human Sci-ence, with permission from c© [2008] IEEE.)

this phenomenon has been observed in other experiments byusing shorter DNA chain.2 Note that, in our experiment, theplural folded parts show the explicit tendency to emerge nearthe downstream end, which should reflect the great differenceof the spatial fluctuation along the chain in relation to the nu-cleus formation. The non-monotonic spatial fluctuation alongthe chain will be discussed.

How can we describe the non-monotonic folding processof the tethered long DNA under strong flow? To begin with,let us describe the DNA as “an inextensible worm-like chain”(inextensible WLC), the contour length of which is invariant.As shown in Fig. 4(a), its one end is fixed at the origin x = 0.Suppose that the DNA is folded against a uniform flow withvelocity Vs where single folded part at the downstream end isconsidered for simplicity. This WLC polymer consists of M0

segments, which are labeled from the tethered point (n = 0).M(t)-th segment is the boundary between the folded and un-folded parts at time t, and the unfolded and folded segmentsare assumed to be in the range 0 ≤ n ≤ M(t) and M(t) ≤ n≤ M0, respectively. The distance from the origin to n-th seg-ment is given by

x[n] =n∑

k=1

l[k], (1)

where l[k] is the projected length of the k-th segment to theflow direction. Here we obtain the stretched length of the un-folded DNA as L = x[M(t)] (= ∑M(t)

k=1 l[k]). Note that the arclength of each segment is the Kuhn length b0, and the totalcontour length is given by Lc = b0M0.

The tension at n-th segment is given by the building-up of the hydrodynamic force from downstream end to

(c)

0

Vs Flow velocity:

Tethered pointL x

(a)

Folded part

Inextensible WLC

~100 nm

(b)

0 L

b0

l(x)

b0

l(x)

0x

t / τ0

200010000

L/L

0

1

0.5

0

εb0/k

BT:

4.05.06.0

FIG. 4. (a) Schematic representation of long DNA conformation under flow.x-axis indicates the direction of flow. (b) Plot of the projected length persegment l(x) in the case of the “inextensible WLC” model under steady strongflow. Note that the arc length per segment is constant b0. (c) Theoretical timeevolution of the stretched length L for the inextensible WLC.

n-th segment.10 Assuming that the extension of each seg-ment could be characterized by the uniform tension-stretchingrelation,11, 12 we find the local force balance as follows:

2πη(M(t) − n)b0Vs � kBT

2b0(1 − l[n]/b0)2, (2)

where kBT is the thermal energy, and η is the viscosity of thesolvent, and the left-hand side corresponds to the tension ofn-th segment. Note that the drag force acting on thefolded part is comparable with that for the single unfoldedsegment.13 This balance implies that an upstream transversefluctuation decreases because of the increasing tension due tothe build-up of the drag force along the stretched chain.10 Wedemonstrate the projected length per segment l(x) in the caseof the inextensible WLC under the steady flow Vs as shownin Figure 4(b). The projected length per segment should prac-tically approach its arc length b0 as x goes from downstreamend to 0.

The free energy of the folding chain may be written as F= −ε b0[M0 − M(t)], where a phenomenological parameter

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154901-4 Saito et al. J. Chem. Phys. 135, 154901 (2011)

ε( > 0) represents a free energy gain upon folding per length.Balancing the rate of free energy change with the dissipationrate associated with the folding nucleus dF/dt = −dQ/dt, weobtain

εb0dM(t)

dt= �

(Vs − dL

dt

)dL

dt. (3)

where it is assumed that the free energy change in the foldedpart is dissipated owing to the drag force against the foldedpart, and � is the relevant drag coefficient. Introducing con-tinuous variable s( ≡ b0k), we replace the summation with theintegral in Eq. (1), and then putting Eq. (2) into the resultantintegral leads to

L � b0

[M −

√2Mb0

τ0Vs

], (4)

where τ0 = 2πηb30/kBT . Its derivative with respect to t is

dL

dt� b0

[1 −

√b0

2Mτ0Vs

]dM

dt. (5)

The combination of this result with Eq. (3) gives the followingtime evolution for M(t):

dM

dt�

⎡⎢⎣Vs

b0− εb0/kBT

cτ0

[1 −

√b0

2Mτ0Vs

]⎤⎥⎦

/ [1 −

√b0

2Mτ0Vs

].

(6)

where the coefficient c = �/2πηb0 is assumed to be a con-stant of order unity, which might be a reasonable approxi-mation from the observation that the hydrodynamic radiusof the compactly folded DNA is of the order of the Kuhnsegment.13 Figure 4(c) shows the theoretical time evolutionof the stretched length L(t) obtained from Eqs. (4), (6), whichwell captures the qualitative feature of the folding dynamicsin stage [III].

However, this analysis does not interpret a longer plateauafter initial rapid folding at stage [I]. Here, as the poten-tial mechanism, we discuss the secondary structure change,i.e., overstretching beyond its natural contour length in B-form. Caron’s group7 and Bustamante’s group8 independentlyfound that DNA shows abrupt overstretching when it is pulledbeyond the critical tension ∼70 pN. Caron’s group termed thisoverstretched form of the secondary structure as S-form. Inour experimental condition (Vs ∼ 100 − 600 μm/s), the dragforce per segment is more than 2πηb0Vs ∼ 0.1 − 1 pN.14

Therefore, as for the long tethered DNA (more than ∼200μm in contour length), its segmental number greater than 2× 103 should result in a large upstream tension ∼102 pN. Tak-ing account of this estimation, the upstream secondary struc-ture of DNA in our experiments should be remarkably de-formed and might become S-form as shown in Fig. 5(a) dueto the extreme building-up of the hydrodynamic drag forcefrom its downstream part.

To discuss how S-DNA is involved in the non-monotonicfolding process as shown in Fig. 3(a), we introduced an exten-sible WLC model, the contour length of which is extensible.In this model, when the tension is small, the DNA chain is

~100 nm

Folded part

S-DNA B-DNA

x

Extensible WLC

0

b0

l(x)

0 L

Secondary structure

(i) S-form DNA (ii) B-form DNA(S-DNA) (B-DNA)

(a)

(b)

xFlow

0 L

(c)

Thr

esho

ld le

ngth

Lc

Flow velocity Vs

Monotonic

Folding

Non-monotonic Folding

Vs*

FIG. 5. (a) Schematic representation of long tethered DNA under strong flowas “extensible WLC” model. The upstream segments are remarkably over-stretched accompanied with the secondary structure change (closeup (i)).This overstretched part (S-form) of the DNA is indicated by a red dashedrectangle. Other downstream part of the DNA takes normal B-form DNA(closeup (ii)) as the secondary structure, and the downstream end part isfolded. (b) Plot of the projected length per segment l(x), corresponding toFig. 5(a). The value l(x) at the upstream segment (S-DNA) is greater than itsarc length b0 in the absence of the tension. The red dashed rectangle indi-cates the overstretched part of the DNA. (c) Diagram separating the mono-tonic folding and non-monotonic folding regions in (Vs − M) plane with theboundary given by Eq. (7). From our experiments, the order of the criticalvelocity is V ∗

s ∼ 200 μ m/s in the present spermidine concentration.

almost inextensible with keeping its structure as B-form, andwhen the tension is larger than a certain threshold value, itbecomes extensible with changing its structure from B-formto S-form. Based on this model, we established the conjec-tured plot of the projected length per segment, l(x), to the flowdirection as shown in Fig. 5(b). Here, this plot is obtainedby utilizing the reported experimental results of “overstretch-ing” at uniform tension.7, 8 Figure 5(b) represents that in thedownstream part, l(x) approaches the arc interval b0 towardthe upstream with keeping its secondary structure in the B-form. Furthermore, it represents that in the upstream part ofthe tethered DNA, the abrupt secondary structure change, i.e.,overstretching occurs, and l(x) in the S-form becomes greaterthan b0 due to suffering a large tension which exceeds thecritical tension (∼70 pN7, 8).

While the upstream part moving against the flow mustpull the downstream part, the secondary structure changein the extremely overstretched part should provide a muchgreater gain in the free energy. Hence, during the early stage

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154901-5 DNA folding under flow J. Chem. Phys. 135, 154901 (2011)

in the folding, the overstretched upstream part could be re-tracted, leading to greater shrinkage of the stretched lengthtogether with free energy gain. However, with a decrease inthe unfolded segments, the upstream structure approaches thenormal B-form, and the gain in the free energy due to the sec-ondary structure change decreases. In this situation, the fold-ing velocity should decrease. This process should correspondto the period stage [I] in Fig. 3.

In addition, dM/dt = 0 in the equation of time evolution(Eq. (6)) suggests the presence of the threshold in the un-folded contour length Lc(ε, Vs) (= b0Mc(ε, Vs)) with the givenattraction strength ε and flow velocity Vs, implying that DNAlonger than that threshold is not folded within the frameworkof the inextensible WLC model. From Eq. (6), the followingrelation is satisfied at the threshold condition;

εb0

kBT� τ0Vs

b0

(1 −

√b0

2Mτ0Vs

). (7)

In Fig. 5(c), we show a diagram separating the monotonicfolding and non-monotonic folding regions in (Vs − M) planewith the boundary curve given by Eq. (7). Below the flow ve-locity V ∗

s � εb20/(τ0kBT ), DNA chain shows the monotonic

folding irrespective of the chain length, and then the boundaryline of the threshold length Lc appears beyond that. Indeed, inour experiments, the non-monotonic folding was not observedfor the relatively shorter DNA (<200 μm) and slower flowvelocity (<ca. 300 μm/s), and the non-monotonic folding ap-peared for longer DNA (>200 μm) and faster flow velocity(>ca. 200 μm/s). These correspond to our theoretical trend.Given the simplicity in our theoretical model, this should bea rather reasonable agreement, which qualifies the presentargument.

What is the mechanism inducing the transition to stage[III]? The observed plateau time in experiments differs eachtime even under the same conditions, suggesting that the tran-sition to stage [III] in Fig. 3 is an activation process, althoughthe precise mechanism is not known.

Here, let us compare our experimental results with previ-ous ones.2–5 These experiments have been performed in a sim-ilar situation. The folding dynamics, however, differs consid-erably, i.e., previous works reported a nearly constant speedon folding, whereas our observed result indicates more in-volved dynamics with a highly non-monotonic folding veloc-ity. This difference between them should be attributable tothe initial chain length. In other words, the folding velocityfor DNA smaller than 100 μm should correspond to that atthe late stage [III] for longer DNA, because initially large ten-sion variation in long chain becomes less noticeable with thegrowth of the folded part.

Our scenario for three-stage-folding [I]–[III] is summa-rized as follows:

[I]: The upstream segments are initially overstretched withthe deformation of the secondary structure. The decreasein the unfolded downstream segments reduces the ten-sion at the upstream. Hence, the overstretched segmentsare retracted, accompanied by the secondary structurechange.

[II]: Then, the folding becomes stagnant because the free en-ergy gain by the overstretching is exhausted. A longplateau continues until the length of the unfolded partbecomes lesser than the threshold by some sort offluctuation.

[III]: Eventually, the rapid folding is caused, because the up-stream part of the tensed B-form DNA is slacked dueto the lowering of the tension accompanied with the nu-cleus growth.

IV. CONCLUDING REMARKS

In summary, we have studied the folding process of longDNA under a strong flow at a velocity ca. 300 μm/s. Directobservations revealed the following non-monotonic changein the folding velocity: [I] an initial fast folding; [II] subse-quent slowing down, leading to the long plateau; and [III]the final sudden acceleration and the completion of the fold-ing. We consider that this notable feature in the foldingdynamics should arise from the inhomogeneous conforma-tion change involving various order levels on the hierarchi-cal structure together with the non-uniform spatial fluctuationthat is inherent to a much longer tethered chain under a strongflow.

ACKNOWLEDGMENTS

T.S. and T.S. thank H. Nakanishi at Kyushu Universityfor useful discussions. This work was supported in part by anIndustrial Technology Research Grant Program from NEDO,Japan, and KAKENHI (No. 20034008, No. 20840027, No.21114507) from the MEXT, Japan. Photography masks werefabricated using the EB lithography apparatus available at theVLSI Design and Education Centre (VDEC), the Universityof Tokyo.

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