Dynamics in the Folding of Long DNA Chain under Strong Flow Takuya 1Saito1, Takahiro Sakaue2, Daiji Kaneko1, Masao Washizul34, Hidehiro Oana1'4 Department of Mechanical Engineering, The University of Tokyo, Tokyo 1138656, Japan, 2Fukui Institute for Fundamental Chemistry, Kyoto University, Kyoto 6068103 Japan, 3Department of Bioengineering, The University of Tokyo, Tokyo 1138656, Japan, 4JST, CREST, Japan
Abstract: This paper presents the study about the dynamics in the folding of long DNA under strong flow. Direct observation of the folding process of the tethered DNA molecule induced by adding a condensing agent is demonstrated using a microfluidic device, and this theoretical interpretation of the process is provided. From the direct observation, it is found that the folding dynamics typically exhibits characteristic three stages: (i) initial rapid compaction, (ii) slowingdown, (iii) sudden speedup. Such a notable feature pinpoints a nonequilibrium flow effect and can be interpreted by a simple theory, which takes account of the nonuniformity of the tethered chain stretched by flow.
1.eIntroductioAsare
GenomicDNAs arenegaivelychargedandextremely negaively charged and extremely long biomacromolecules. However, in nature, they are complexed with Histones (positively charged proteins) etc. and tightly folded into a small cellular space. Fur
thermore, they function properly together with dynamic change of their structure. Understanding of the dynamics of structure change at higherorder level would be of importance for various purposes, particularly, in biology, and many attempts are currently under potential development. In aqueous environments, long DNA molecule is dissolved, and behaves as the fluctuated flexibile chain in the length scale larger than ca. pm, despite of local stiffness. Long DNA molecules with the fluorescent labels enable us to examine the nature of individual DNA as an ideal physical model. In fact, this methodology reveals that DNA folding is discrete transition from the fluctuated coil to compact states at the level of a single chain upon the addition of condensing agents such as multivalent cations [1], which is consistent with that in theoretical prediction [24]. Furthermore, the fundamental nature of DNA has been recenly eamine by
tilizng te mico deice,
here
single DNA molecules are confined in narrow space [58], or deformed by hydrodynamic flow [9, 101. The charac
teristic average dimensions and relaxation process were investigated in these experiments. For DNA folding, to
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obtain the detailed spatial information along the chain, earlier works utilized hydrodynamic flow to stretch the DNA with the tethered one end, and observed the folding process, upon the addition of condensing agents [1113]. These studies report that the folding velocity is nearly constant, and becomes faster with increasing the cocentration of condensing agents. However, one may naturally wonder the effect of flow. In fact, the segments are much more nonuniformly distributed because ten
sion builds up along the tethered chain, than that of the coil without flow. DNA folding has been also discussed in the framework of coilglobule transition [24] developed in the system with uniform tension. Therefore, in this sense, DNA folding under flow is an open physical subject. Moreover, it is of importance not only for the physical interest, but for the understanding of the living matter, since, in nature, DNA is folded in a nonequilibrium environment, where flow should be induced by the molecular motor pulling, etc. However, this remarkable effect under flow should not clearly appear in past study because of the smaller buildup of tension in relatively shorter DNA [1113]. In this study, to clarify the effect of the induced nonuniformity by flow on DNA folding, we performed direct Top view 22mm mm OaE
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4. Discussion The previous experimental results report the
folded part
~LF
Trumpet (Flower)
[ III ]
.
(#2, #3,....) appeared in the middle of the chain in
many samples. To characterize the folding velocity clearly, on the basis of ]the tvsample of Fig. 2, we established the time evolution of the length from the tethered point to the downstream as in Fig. 3. This result shows that the folding goes through three stages: [I] the folded part at free end goes to approach the terthered point rapidly, but slow down it velocity around 170 pm length; [II] in middle stage, its velocity is almost plateaued in about dozen seconds; [III] in last stage, the chain is suddenly folded completely with rapid speed within about several seconds.
Tetheredpoint junction point
X4X4 L Stem LF
[
parts
f>.(Q) folded part
FIG. 4: Schematic representation of three stages of dynamics in
DNA folding under flow: [0] The steady state conformation of StemFlower (Flower (O < x < LF), Stem (LF < x < L)) under flow velocity Vs before adding the condensing agent, [I] in initial stage, the segments in Stem part (the full extension part) mainly decrease, [II] the folding velocity slow down drastically soon after Stem disappear, [III] in last stage, the folding velocity is suddenly rapid due to the decrease of the extension length. The Trumpet conformation corresponds to regimes [II] and [III]. Balls indicated by white arrows represent the folded parts, and the sequenced dashed circles along the chain in [0] represent coilings (blobs).
Photonics K.K.). 3. Results Figure 2 shows the fluorescence images in time evolution of the folding induced by the addtion of condensing agent after stretching by flow. The bright spots along the stretched chain correspond to folded parts, which are labeled by #1, #2,... from the downstream end in the case of multiple nucleation. Due to a limitation of the optical setup, the downstream free end is out of frame if the DNA is longer than ca. 200 pu m, and we observed the downstream folded part (#1) initially in any samples. Thus, we set the initial time as t=0 when it appears. During the folding, this end part (#1) proceeds to the tethered point at pillar, and the folding is completed. In the course of the folding process, newly created folded
120
nearly
the folding [1113]. However, our observed result shows three stages dynamics in the folding velocity. This difference between them should be attributed to the degree of nonuniformity in tension depending on the DNA length. This indicates that the folding velocity for shorter DNA should correspond to that at stage [III] for longer DNA, since the nonuniformity of tension along the chain decreases through the folding. In constant
velocity
on
fact, many theoretical studies under the assmption of the uniform tension, neglecting its nonuniformity [14, 15],
if the acceleration at [III] stage, or the folding velocity for the shorter DNA can be explained. To introduce the nonuniformity of tension for the
even
chain deformation under the flow, we utilize the scaling
theory developed by BrochardWyart et al. [16]. Then, the time evolution of the folding process is deduced from the balance between the free energy change and the dissipation heat. 41. Conformation under flow This deformed conformation is approximated as that in steady state, since the relaxation time of the chain is very fast compared with the folding velocity [17]. First, we overview the unfolded conformation under flow, according to BrochardWyart et al. [16]. Let us consider the polymer, composed of a sequence of m Kuhn segments, each of length b, under flow velocity Vs. The deformation behaviour by flow can be distinguished by the magnitude of the segmental number m, since the tension is accumulative along the chain from free end to the tethered point due to dragging force. In fact, the spatial distribution of upstream segments is narrower as the deformed shape.
(411) StemFlower regime For sufficient long chains, the upstream segments near the tethered point should be fully extended due to stronger tension as shown Fig. 4 [0]. This conformation is called "StemFlower" (Stem: full extension, Flower: the sequence of coiling). First, if we see the "Flower" part, the chain exhibits some coiling due to the entropic elasticity (thermal fluctuation). The dragging force builds up along the chain, thus, the spatial size of this coiling (blob size) ((x) depends on the position x, which is determined by the local force balance equation as follows:
kBT
~(X)
TSxVs.
(1)
where x is the distance measured from the free end, 7 the viscosity of solvent, kB is the Boltzmann constant, and T is the thermal temperature. The force balance for the "junction" point (boundary between Flower and Stem) is kBT/b ILFVS, where LF is the extension length in Flower part. Segments above Flower are also fully strethed, i.e., Stem conformation. Therefore, the extension length L is L
=
b (m  NF) + CFbNF,
(2)
k7zBT/rlb2V. is the segmental number in where NF Flower part, and LF sCFbNF (CF < 1is the numerical coefficient).
(412) Trumpet regime Next, we discuss the extension length L, if L < LF, i.e., m < NF. The conformation in this regime looks like "Trumpet", and the force balance equation (1) in Flower conformation is valid. To obtain the relation between segmental number m and the extension length L, we apply the coarsegrained deformation dx/dm(x) to ((x)/g(x) as follows:
dm(x) (g(x)'
(t1xV N VKkBT,
Lmin 2v'1
L
CFbNFYN)
(5)
in which, we use the continuous condition of the length at the boundary between StemFlower and Trumpet to determine the numerical coefficient. Now, we utilize these results for the conformation in the folding dynamics. In this model, the dragging force to the folded part is negligible, since its size is sufficiently smaller than the spatial size of the coiling ((x) (blob size). This means that the conformation of the unfolded chain does not depend on where and how many folded parts appear through the folding process. Therefore, the extension length L(t) is determined by the number of unfolded i.e., "active" segments m(t) at time t. Thus, the conformations obtained in steady state are available by the following replacement: V

L(t),
X) m X m(t):
NF NX 7(t), LF X LF(t)(_ cFbNF(t)), where
NF(t) = NFVS/ (v
(6)
 L(t)) is the segmental num
ber at junction point (boundary) from StemFlower to Trumpet through this folding dynamics. 42. Time evolution of the folding process Now we need the equation of motion, which is obtained by balancing the rate of free energy gain with the dissipation heat as follows: dF dQex
(7 dt dt where F is the free energy and dQex/dt is the excess dissipation heat, meaning that the dissipation heat to maintain the steady state (house keeping heat) is eliminated from the total dissipation. (3) dm(x) g9(x) 'The plausible form for the free energy change is
where m(x) is the number of segments from free end to x, and g(x) is the number of segments in the coiling (blob) of size ((x), i.e., ((x) bg(x)v. Here, employing Eq. (1), we have d dm(x)
Therefore, the length L in "Trumpet" regime is given by total number of segments m as follows:
(4
121
dF _ (m(0)m(t))(8) dt dt where e (> 0) represents the depth of free energy gain per segmental units upon the folding (chemical potential difference). On 4 the other hand, the number of active segment in the coiled part decreases, which results in the decrease of
the apparent chain length L(t) with the velocity of folding (retraction) L(t). The main excess dissipation arises from the retraction of the flower against the solvent flow. By noting the friction force is proportional to the relative velocity Vs  L(t), the dissipation can be evaluated as dQex dt

FHyd(t)
L(t).
(9) t

L(t)
R(t) = LF(t).
Applying the steady state conformation (Eqs. (2) and (5)) to this balancing relation (Eqs. (7), (8) and (9)), we have the differential equation of time evolution as follows:
(421) StemFlower regime (NF(t) < m(t)) ____ L__±t
__) vL(~
V (1tc)L (10) +
(I Ct)N22
[II]
As shown in Fig. 4 [II], fully extended segments disappear, and the deformed conformation enters Trumpet regime. Equation (11) predicts that the velocity suddenly decreases at the onset of the trumpet regime to balance the free energy gain with dissipation heat, since the growth of the nucleus (increase of m(t)) inevitably causes the high retraction velocity, which requires large dissipation. This regime should correspond to the sudden slow down observed in experiment. In fact, numerical solutions of Eq. (11) show the extremely slow regime in appropriate parameters.
[III] Then, after some period of the slow folding process as in Fig. 4 [III], the extreme speedup follows, since the decrease of the stretched length leads to the smaller dragging force, indicating that the folded part must be fast to cause greater dissipation heat. The dominant balance in Eq. (1 1) leads to
n heslwfodig egm a sag [I.
Ct
(422) Trumpet regime (m(t) < NF(t))
hi df
where Lstat is the stationary length, or the length
"'22v1L(t)
1v L(t) ferential equation represents the extremely rapid folding at last stage, since log (LV(t)/L ) Xoo as v VtL(t) L(t) V0. This regime should correspond to the _1 C2L(t) (l(t)) v (lVsL(t)) vL observed rapid folding at last stage. LVSummarizing the above time evolution, these overall whee= L(BT) Note that V(t) 0, i.e., the steady state is realized if feature of the process resembles that observed in real in 0 both regimes. experiments, i.e., [I] the initial fast, [II] the very slow e= growth al stationary) end s the elastrapidefolding Here, we interpret the mechanism aboutt velocity on the basis of Eq. (10), (11). For simplicity, should correspond to StemFlower", Trumpet in near we examine it under an approximation of first order of m sier rad s ing t t"e.. as 1). Our scenario is it is difficult to observe L(t)/V.(<< follows: However, StemFlower and [I] The initial shape takes StemFlower as shown in Trumpet conformation directly. As a test for the above Fig. 4 [I]. In this regime, the length of fully ex mechanism, Eq. (11) predicts the stationary length, or tended part (stem) mainly decreases, due to the the length in the slow folding regime is estimated as upstream progress of "slacking". On the basis of the balance in Eq. (10), the estimation of time evof (l(t.)\L(tj) \\ (14 lution is Lstat 2 LF\YtLFJ) V: ) i (Ca)vl s ex where Lr is the junction time from StemFlower to TrumJtr
kBTT_ sL(t)
where Lo is the initial stretched length, and C is the numerical coefficient. This time evolution indicates that the folding velocity increases exponentially. In addition, this acceleration is attributed to the gradual decrease of Flower length.
122
pet. This folding velocity L(tA) is also faster for the greater initial length from Eq. (12), owing to the longer acceleration time in StemFlower regime. These mean that the stationary length Lstat is smaller with stronger attraction or greater initial length. Therefore, this hy
pothesis can be tested by controlling these parameters, and this is future work. Finally, we notice that intrinsically flexible chains may start collapsing just after the quench by the process of .inodaldecompositio .Incontrastthel t spinodal decomposition [18]. In contrast, the local stiffness may avoid it, and the coil state survives for a while even after the quench [19, 20]. This enables us to apply the Brochard's theory to the metastable coil. In this sense, the semiflexibility is an crucial factor for the observed phenomena. 5. Concluding Remarks In this article, we studied the dynamics in the folding of long DNA under strong flow at the velocity ca. 300 pm/s. Our experimental result shows that the folding velocity iS largely changed through this process as follows: d Iw an initial fast folding; [II] the subsequent slowing down;
[III] the last sudden acceleration and the completion of the folding. This notable feature in the folding dynamics should arise from the segmental nonuniformity of spatial distribution. We theoretically show the extreme sensitivity of the folding velocity on the degree of the fluctuation Or slacking" of theaupstream tetheringpartdofcthe hain which naturally leads to the observed dynamics of three stages.
Acknowledgements This work was supported in part by Industrial Technology Research Grant Program from NEDO of Japan and GrantinAid for Scientific Research on Priority Areas from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. Photography masks were fabricated using EB lithography apparatus of VLSI Design and Education Centre (VDEC), the University of Tokyo.
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