DNA confined in nanochannels: Hairpin tightening by entropic depletion
Theo Odijka兲
Complex Fluids Theory, Kluyver Laboratory of Biotechnology, Delft University of Technology, Julianalaan 67, 2628 BC Delft, The Netherlands
共Received 25 September 2006; accepted 26 October 2006; published online 27 November 2006兲 A theory is presented of the elongation of double-stranded DNA confined in a nanochannel based on a study of the formation of hairpins. A hairpin becomes constrained as it approaches the wall of a channel which leads to an entropic force causing the hairpin to tighten. The DNA in the hairpin remains double-stranded. The free energy of the hairpin is significantly larger than what one would expect if this entropic effect were unimportant. As a result, the distance between hairpins or the global persistence length is often tens of micrometer long and may even reach millimeter sizes for 10 nm thin channels. The hairpin shape and size and the DNA elongation are computed for nanoslits and circular and square nanochannels. A comparison with experiment is given. © 2006 American
Institute of Physics.关DOI:10.1063/1.2400227兴
I. INTRODUCTION
There has been a long-standing interest in the effects of confining polymer chains in tubes and slits, for instance, in relation to the field of size exclusion chromatography.1–8 Re-cent advances in chip lithography have led to the fabrication of well-defined nanochannels9,10 so that issues in polymer physics may now be addressed which were elusive in the past.11–19At the same time, stiff polymers such as DNA and actin can now be well aligned which is of considerable in-terest in the biosciences.20–22
In order to understand the mechanism of chain elonga-tion, it is crucial to investigate the alignment as a function of the size of the channel cross section and this has been carried out recently by Reisner et al.13The channel width was of the order of the DNA persistence length. They focused espe-cially on the relaxation times of monodisperse double-stranded DNA fluctuating in square and rectangular nanochannels which they compared with estimates inferred from theory.4,7 The pictures of fluorescently labeled DNA they show are, however, quite remarkable with respect to their extreme degree of elongation, which has not been com-mented on before. Let us consider, in fact, a naive estimate of the distance g between two consecutive hairpins of the nanoconfined DNA which would be something like
g =r exp共U0/kBT兲. 共1兲
The radius of a hairpin, which for the moment is supposed to be semicircular, is r and its bending energy is U0. The
tem-perature is T and kBis Boltzmann’s constant. A form similar to Eq. 共1兲 arises in the theory of nematically confined stiff chains23–33and will be rederived below. Note that the B-helix is assumed to remain intact. Thus, in the experiment of Reisner et al. in the case of DNA confined within a 60 ⫻80 nm2channel for instance, one would expect the hairpin
to extend to its maximum diameter of 100 nm 共equal to the diagonal兲. This leads to g⯝1m from Eq. 共1兲. This is the
mere step length of the effectively one-dimensional random walk, yet the T2 DNA having a contour length of 63m is almost fully elongated.13
I argue here that there is an important effect missing in Eq. 共1兲, namely, entropic depletion of the DNA hairpin by the wall of the nanochannel. As the hairpin approaches the wall, it loses orientational and translational degrees of free-dom. The resulting entropic repulsion forces the hairpin to-ward the axis of the channel so the hairpin is tightened up. Equation共1兲seriously underestimates the elongation for this reason. An analytical solution of the relevant Fokker-Planck equation for a stiff chain trapped in a pore, which addresses all fluctuations, is well known to be difficult, although nu-merical analyses have been performed.34–40 My emphasis will be on an analysis of this phenomenon in the mechanical limit for the hairpin curve. The hairpin will be assumed to be a two-dimensional curve within a plane aligned along the long axis of the nanochannel. The curve is free from thermal undulations and its shape is determined by the bending en-ergy U. The hairpin chord, which is the vector distance be-tween the two ends, is perpendicular to this axis but the shape of the curve is not semicircular and needs to be deter-mined. The chord of length 2r has to be fitted into the cross section of the channel. This is a geometrical problem in two dimensions. Clearly, the confinement of the chord entails a loss in both orientational and translational degrees of free-dom, giving rise to a free energy Fconf of entropic origin.
This decreases as the hairpin tightens up so it balances the bending energy U at an optimum radius r. Accordingly, the free energy of the hairpin in the mechanical limit共index mc兲 is
F ¯
mc= U¯ 共r¯兲 + F¯conf共r¯兲, 共2兲
where a bar denotes the state of minimum free energy. My main objective will be to compute this quantity. It is convenient to express the scale g as
g =␣¯ expr 共Ftot/kBT兲, 共3兲
a兲Electronic mail: odijktcf@orange.nl
Ftot= F¯mc共r¯兲 + H. 共4兲
Here,␣is a constant and H is a sum of a variety of fluctua-tional free energies beyond the mechanical limit as it is de-fined here, which I will analyze qualitatively.
The looping of DNA occurs widely in biology and tightly curved DNA with both ends constrained has been studied often theoretically 共for recent work, see, e.g., Refs. 41–46兲. The mathematical problem dealt with here is differ-ent because there is an additional restriction imposed by the nanoconfinement of the hairpin curve. In addition, since the distance between hairpins is dominated by an exponential in Eq.共3兲, an accurate analysis of the shape of the curve and its bending energy is warranted.
The paper is organized as follows. In the next section, I compute the optimum shape of the hairpin and the minimum bending energy. In Sec. III, the free energy of confinement is evaluated for three types of cross section: circular, square, and rectangular. The distance between hairpins or the global persistence length is calculated in Sec. IV. In the last section, a discussion will be given of the resulting free energies, hair-pin sizes, and DNA elongations in comparison with experi-ment. Therein, the terms neglected in the mechanical limit will also be remarked upon.
II. HAIRPIN SHAPE
The hairpin of double-stranded DNA has a length l which is to be determined while its end-to-end distance is constrained to be 2r. It is described by the tangential unit vector u共s兲 at contour position s 共see Fig. 1兲. The hairpin curve is assumed to be symmetric about the y axis which is parallel to the central axis of the nanochannel. The angle between u共s兲 and the x axis is−共s兲 so that the unit vector may be written in terms of the coordinate unit vectors u = −excos+ eysin. The boundary conditions are assumed to be共0兲=12 and共l/2兲=0.
I first compute the shape of the curve by minimizing the bending energy of the hairpin of length l,
U = PkBT
冕
0 l/2 ds冉
d ds冊
2 , 共5兲subject to the constraint
冕
0 l/2 dsu · ex= −冕
0 l/2 ds cos= − r. 共6兲In Eq.共5兲, P is the persistence length and d/ ds equals the inverse radius of curvature. Minimizing Eq.共5兲 with respect to共s兲 leads to
d2 ds2 =
␥
2l2sin, 共7兲
where the Lagrange multiplier ␥ may be positive, zero, or negative. Yamakawa and Stockmayer47have previously ana-lyzed this equation in their theory of the excluded-volume effect of stiff chains, but our treatment differs from theirs because the constraint is nonzero here 关Eq.共6兲兴. Integration of Eq.共7兲yields
l2
冉
dw ds冊
2
= f −␥cos, 共8兲
where the constant f must be positive.
In the remainder of the analysis we need not compute 共s兲 explicitly. The bending energy is as yet given in terms of three unknown parameters
U PkBT = f 2l− ␥r l2. 共9兲
We ultimately wish to minimize this with respect to l subject to Eq. 共6兲. But it turns out that it is possible to do this ana-lytically via the dummy variable B defined below.
Actually, there is another restriction on Eq.共8兲: the chain cannot cross the wall of the channel. This is difficult to take into account mathematically because it is a local constraint defined at every point of the contour s. Nevertheless, on physical grounds, one expects deviations of reverse curva-ture away from a typical hairpin configuration to be forbid-den in view of the increase in elastic energy associated with such bending. Accordingly, I assume that the simpler con-straint d/ ds⬍0 holds.
The right hand side of Eq.共8兲is now rewritten in terms of the new angular variablev =共−兲/2 with boundary val-uesv共0兲=/ 4 and v共l/2兲=/ 2,
ld
ds = −共f +␥兲
1/2共1 − B sin2v兲1/2. 共10兲
The constant B⬍1 is defined by
B⬅ 2␥
f +␥. 共11兲
Note that B may be positive or negative though the combi-nation f +␥ cannot be negative. Next, the objective is to the express the bending energy given by Eq.共9兲 solely in terms of B. First, I integrate Eq.共10兲over the entire contour from 0 to l / 2. This leads to
共f +␥兲1/2= 4 共12兲
in terms of the elliptic integral FIG. 1. Half of a hairpin curve in a two-dimensional Cartesian coordinate
⬅
冕
/4
/2 dv
共1 − B sin2v兲1/2. 共13兲
The constraint Eq.共6兲is rewritten as
r = −
冕
0/2 d
冉
dsd
冊
cos. 共14兲The derivative is now substituted from Eq.共10兲and cosis expressed in terms of v. The scaled contour length ␣⬅l/r
thus becomes a function explicitly dependent on B only, ␣−1B=
冉
1 −12B
冊
−, 共15兲whereis another elliptic integral ⬅
冕
/4 /2
dv共1 − B sin2v兲1/2. 共16兲
The constants f and␥can also be written in terms of B with the help of Eqs.共11兲and共12兲,
f = 82共2 − B兲, 共17兲
␥= 8B2. 共18兲
We now need to derive an expression for the bending energy which may be differentiated with respect to B as expediently as is possible. Equations 共17兲 and 共18兲 are first substituted into Eq. 共9兲 and then B is eliminated with the help of Eq. 共15兲, E⬅ Ur PkBT =8 2 ␣
冉
1 − 1 2B − B␣ −1冊
=8 ␣ . 共19兲This is still too complicated because it contains␣. Hence, I eliminate␣by inserting Eq.共10兲into Eq.共14兲and using Eq. 共12兲. Equation共19兲then reduces to the form
E共B兲 = 4 共20兲
in terms of the integral ⬅
冕
/4 /2
dv g共v兲
共1 − B sin2v兲1/2 共21兲
and the function
g共v兲 ⬅ 2 sin2v − 1. 共22兲
The advantage of writing the energy as the productis that the numerator and denominator are identical in the respective integrands so that one may expect cancellation of terms after differentiation 共apart from the factor g which, not inconve-niently, increases monotonically withv兲. In fact, the
deriva-tive of Eq. 共20兲with respect to B introduces sin2v into the
integrands which is then conveniently rewritten in terms of 共1−B sin2v兲. Terms of the kindthen cancel and we
ob-tain 1 2 dE dB= B −1
冋
冕
/4 /2 dv g共v兲 共1 − B sin2v兲3/2−册
. 共23兲This form is reexpressed as a two-dimensional integral with the help of Eqs. 共13兲, 共16兲, and 共21兲. After rearrangement, this yields 1 2 dE dB=
冕
/4 /2冕
/4 /2 dudv g共u兲共sin 2u − sin2v兲 共1 − B sin2u兲3/2共1 − B sin2v兲1/2. 共24兲 Next, one notes that the function g共u兲/共1−B sin2u兲 increasesmonotonically with u for all B, whatever the sign. The re-maining factor in the integrand is antisymmetric in u andv.
Hence, the derivative must be positive for all B.
In conclusion, the bending energy is a minimum as
B→−⬁. Asymptotically, we have from Eqs.共13兲and共16兲
⬃ C1共− B兲−1/2, 共25兲 with C1= − ln tan 8 = 0.881 374, ⬃ C2共− B兲1/2, 共26兲 with C2=1 2
冑2.
Equation共15兲yields for the optimum length of the hairpin,
l/r =␣=
冉
C2 C1 −1 2冊
−1 = 3.3082. 共27兲The minimum elastic energy of the hairpin according to Eq. 共19兲is
Umr
PkBT
= Em= 4共1 − C1C2兲 = 1.5071. 共28兲
I have integrated Eqs.共15兲and共19兲numerically in order to assess the nature of the elastic “well” 共see TableI兲. The optimum curve is very close to that of a semicircle 共B=0; TABLE I. Scaled length and scaled energy E of the hairpin as a function of
B. At B = 0 the hairpin is a semicircle so␣=and E =/ 2.
␣=兲. The bending energy only starts to increase markedly as B approaches unity. The hairpin is almost a straight line in that case, with highly rounded ends.
III. ENTROPY OF DEPLETION
In the Introduction it was argued that it is plausible to introduce a mechanical limit for the state of the hairpin which is aligned along the centerline of the nanochannel. Then, we need to address two two-dimensional problems: the shape of the hairpin treated above and how the chord of the hairpin can be fitted into the cross section of the channel. In the latter case, we may speak of an entropy of depletion since orientational and translational degrees of the chain are cut off as the hairpin is formed and restricted by the channel walls. In this section I give approximate expressions for the deple-tion entropy for three characteristic types of nanochannels: circular, square, and rectangular.
A. Circular cross section
The chord of the hairpin is a segment of length 2r en-closed within a circle of radius a 共see Fig. 2兲. The transla-tional degrees of freedom of the segment are assessed by how we may position point P in the circle which is at a distance t from the origin O. The orientational degrees of freedom are expressed by the angle . In this section, all lengths will be scaled by a for convenience. Note that we require t艋2r−1; otherwise the segment does not fit into the circle. In addition, if t⬎1−2r, the segment cannot rotate freely and the maximum anglemis given by
1 = 4r2+ t2− 4rt cosm. 共29兲
It is in this regime that an approximation to the entropy is derived. The fraction of realizable states is written as
Ic=共22兲−14
冕
2r−1 1 dtt冕
0 m d. 共30兲This is simply a sum over all degrees of freedom noting that m=m共t兲. It is normalized by a factor 22applicable to the state if the segment were to orient and translate freely within the circle.
The bending forces of the previous section tend to open up the hairpin toward the channel wall. Therefore, let us assume r and t are close to unity and introduce the small quantities ␦ and :␦= 1 − r and=1−t 共0⬍⬍2␦兲. In that
case, the angle m is also small and we may write cosm ⯝1−1
2m2. From Eq.共29兲we have to the leading order
m
2
= 2␦−⑀. 共31兲
Performing the integrations in Eq.共30兲, we get the entropy of depletion Sc= kBln Ic= 3 2kBln␦+ kBln
冉
8.21/2 3冊
. 共32兲Although this expression is formally valid in the limit␦Ⰶ1, it turns out to be a very good approximation for all r, even when the segment is virtually free to rotate. As r = 1 −␦→0,
Ic⯝1.200 is close to unity which is the exact value. Further-more, Ic divided by the integral computed numerically is a monotone increasing function of␦. The leading approxima-tion to the integrand tm共t兲 in Eq.共30兲, namely, amwithm given by Eq. 共31兲, is also very near the numerical value of
tm共t兲 for all r⬎共1−t兲/2.
B. Square cross section
Trying to fit a line segment into a square is different from the case discussed above. If the side of the square has a length A and the length of the segment is restricted to the regime A⬍2r⬍A
冑2, then the segment cannot rotate through
360° 共see Fig. 3兲. One of its ends 共denoted by point P兲 is constrained to lie inside the region that is almost like a tri-angle in Fig.3. Its boundary in terms of the Cartesian coor-dinates xband ybis given byxb2+ yb2= 4r2. 共33兲
Again, I attempt to approximate the entropy of depletion by a suitable limiting form in this case, when the segment is almost as long as the diagonal of the square. For conve-nience, all lengths are scaled by A in the rest of this section. Thus the small quantities ␦s, x, and y are introduced: ␦s = 1 − r冑2,x= 1 − x, andy= 1 − y in terms of the coordinates x and y of point P. To the leading order, we then have from Eq. 共33兲the requirement
x+y⬍ 2␦s. 共34兲
The orientational degree of freedom is expressed by the angle which is bounded by the angles1 and2共see Fig.
3兲. They are given by FIG. 2. Line segment共i.e., chord of the hairpin兲 enclosed in a circular cross
section.
cos1= y 2r, 共35兲 sin2= x 2r. 共36兲
Since the segment approaches a diagonal in size, it is convenient to introduce the small angles1 and2 defined by 1=
1
4−1 and 2=2− 1
4, respectively. The left and
right hand sides of Eqs. 共35兲 and 共36兲 are now expanded in all of the small variables which yields 1=␦s−x and
2=␦s−yto the leading order. Hence, the range of the ori-entational degree of freedom is
2−1= 2␦s−x−y. 共37兲
Accordingly, the fraction of realizable states is, again, a sum over all degrees of freedom,
Is= 2
冕
triangle冕
dxdy共2−1兲 ⯝ 2 冕
0 2␦s dx冕
0 2␦s−x dy共2␦s−x−y兲 = 8␦s3 3, 共38兲 and the entropy of depletion is expressed asSs/kB= ln Is= 3 ln␦s+ ln
冉
83
冊
. 共39兲The number of corners is 4 which has to be included in the normalization共because of fluctuations, the chord of the hair-pin can, of course, sample the entire square cross section of the nanochannel兲. It would appear that Eq. 共38兲 is a very good approximation to the exact Is because Eq. 共38兲equals 0.8488 at␦s= 1 which is quite close to unity. However, the numerically computed Isis more than twice that predicted by Eq. 共38兲 at r = 0.5 so the approximation must be viewed as fairly accurate at best for all r⬍21
冑2A.
C. Slitlike cross section
A simple approximation for the entropy of depletion can be given in the case of a nanoslit of large aspect ratio, i.e., when the cross section is a rectangle of width A very much larger than height D 关see Fig. 4共a兲兴. We suppose D is so small that we may writem⯝x/r for the maximum anglem when the hairpin chord just touches one of the walls关see Fig. 4共b兲兴. Ultimately, this implies that the hairpin must be suffi-ciently stiff共PⲏD, say兲. If the chord or line segment is at
a distance x from the wall, the contribution to the
orienta-tional entropy is propororienta-tional to ln共2x/r兲 owing to the
an-gular restriction 共m would be/ 2 if the segment were to orient freely兲. Hence, the total orientational entropy is given by Sor/kB= 2 D
冕
0 D/2 dx ln共2x/r兲 = ln冉
D r冊
− 1. 共40兲The segment may also translate along the longitudinal axis of length A implying a fraction of realizable states equal to about 共A−2r兲/A. The total entropy of depletion thus be-comes Sslit/kB= ln
冋
冉
A − 2r A冊
D r册
− 1. 共41兲This is only valid at small enough D. For this reason a smooth crossover from Eq.共41兲to Eq.共39兲does not exist as
A→D and r
冑2
→A.IV. GLOBAL PERSISTENCE LENGTH
The distance between hairpins for a chain in a uniaxially ordered matrix scales as the global persistence length which is given in terms of a fluctuation theorem involving the chain susceptibility.23,27,32This is a rigorous way of deriving g but is only useful if one has a precise analytical theory for the segmental distribution inside the nanochannel, which is lack-ing at present. I have yet to establish Eqs.共1兲and共2兲 and in the mechanical limit the following physical argument is well known.24,27The hairpins of length␣¯ may be viewed as de-r
fects in a one-dimensional chain model of contour length L. There are L / g defects of energy F¯mc= F¯mc共r¯兲 and
concentra-tion␣¯ / g. Therefore, the free energy of the chain becomesr FL kBT =L g ln
冉
␣¯r g冊
− L g +冉
L g冊
F ¯ mc kBT , 共42兲where the first two terms derive from the ideal entropy of the defect gas. Minimization of the free energy with respect to g leads to
gmc=␣¯ expr 共F¯mc/kBT兲. 共43兲 The free energy of confinement or depletion Fconf= −TS has
been discussed in the previous section. If we set S = 0 and
U = U0 and we assume the bend is semicircular, we regain
Eq. 共1兲. Any approximations incurred in the previous analy-ses can be accounted for by adding an energy H to F¯mc. The
global persistence length will now be evaluated for the vari-ous nanochannels.
A. Circular nanochannel
The bending energy of a hairpin is given by Eqs.共19兲 and共28兲, and Eq.共32兲expresses the free energy of depletion for the hairpin enclosed in the circular nanochannel. Upon minimization, the total free energy of the hairpin in the me-chanical limit,
Fmcc 共r兲 kBT =EmP r − 3 2ln
冉
a − r a冊
− ln冉
8.21/2 3冊
, 共44兲leads to an optimum size of the hairpin,
r ¯ =1
3关共Em
2
P2+ 6EmPa兲1/2− EmP兴. 共45兲 Using Eqs.共43兲–共45兲, I have tabulated r¯ and g for DNA with
a typical persistence length of 50 nm 共see Table II兲. These results will be discussed in the next section.
B. Square nanochannel
The entropy of depletion is now given by Eq. 共39兲 so that the free energy of a hairpin in a square nanochannel is
Fmcs 共r兲 kBT =EmP r − 3 ln
冉
A − r冑2
A冊
− ln冉
8 3冊
. 共46兲Minimization of Fmcs with respect to r gives
r ¯ =1
6关共Em
2P2+ 6冑2E
mAP兲1/2− EmP兴. 共47兲 The optimum half length of the chord and distance between hairpins are displayed in Table III for DNA which has a persistence length of 57.5 nm. These predictions will be compared with the experiments of Reisner et al.13
C. Nanoslit
Equation共41兲leads to the free energy of a hairpin in a nanoslit, Fmcslit共r兲 kBT =EmP r − ln
冋
冉
A − 2r A冊
D r册
+ 1. 共48兲Minimization yields an optimum size
r
¯ = EmPA A + 2EmP
. 共49兲
Note that this expression does not depend on D. In this case, it is possible to present a convenient formula for the distance between hairpins
g =␣EmPr¯ D exp
冉
2A + 2EmP
A
冊
. 共50兲In a recent experiment,22the dimensions of the nanoslit were kept fixed but the persistence length was varied by changing the ionic strength. In TableIV, g is presented as a function of
P for a 100⫻1000 nm2nanoslit.
V. DISCUSSION
The first striking conclusion is that the distance between hairpins is considerably larger than the usual persistence length of DNA. This is true even when P is smaller than the typical width of the channel, irrespective of the type 共see Tables II–IV兲. In the larger channels, the effect of entropic depletion is clearly seen for the hairpins are tightened up significantly共compare r with a or A兲. At the other extreme, the global persistence length may reach the scale of a milli-meter which means that it may prove feasible to align bac-terial chromosomes in nanochannels completely. Entropic depletion is an important factor in the elongation.
The large values of g are intimately related to the small size of the orientational fluctuations of the aligned 共nonhair-pinlike兲 sections of the chain with respect to the channel axis. This relation has been discussed at length for a worm-like chain in a nematic.32Here, the mathematics of a chain in a channel involves both translational and orientational de-grees of freedom so is significantly more complicated. Nev-ertheless, the scaling estimate7 for the angular fluctuations when amended by a small numerical coefficient关see Eq.共53兲 below兴 qualitatively corroborates the large values of g pro-posed here. A further implication is that the crossover to the scaling regime applicable in the limit of flexible chains4 is not trivial. The transition is delayed because g is still very large at P⯝channel width. The crossover must occur at g ⯝ P, but, as I argue below, the classical limit is no longer valid then.
It is interesting to note that the depletion effect is much stronger when the nanochannel is square than when it is cir-cular 关compare the coefficients 1.5 and 3 in Eqs. 共44兲 and 共46兲兴. As a hairpin retracts from the edge of a square channel, TABLE II. Half length of the hairpin chord r and global persistence length g for DNA confined in a circular nanochannel of radius a. The DNA persistence length is 50 nm.
a共nm兲 10 15 20 25 30 50 70 100 150
r共nm兲 8.6 12.1 15.3 18.3 21.1 31 39 50 65
g共m兲 2900 199 51 22 13 4.1 2.5 1.8 1.3
TABLE III. Half length of the hairpin chord r and global persistence length g for DNA confined in a square nanochannel with side equal to A. The DNA persistence length is 57.5 nm. These results pertain to the experi-ments of Reisner et al.共Ref.13兲.
A共nm兲 35 70 80 135 190 370 440
r共nm兲 15.9 26.0 28.5 40 50 74 82
its freedom to move increases rapidly. This is much less so in the case the channel is circular for the curvature is directed smoothly inward then.
The entropic depletion by a nanoslit given by Eq.共41兲is quite peculiar. It leads to an expression for the optimum size 关Eq. 共49兲兴 which is independent of D. Moreover, for highly elongated slits, the half length of the hairpin chord never increases much beyond the persistence length. However, the expressions derived in Sec. III C and IV C must be regarded as tentative. If Eq.共48兲is expanded about the minimum state to second order in r, the root-mean-square fluctuation in the hairpin chord is computed to be具␦r2典1/2⯝A2P /共A+2E
mP兲2. For large enough width A, this becomes of the order of the radius r¯ itself as given by Eq.共49兲. Therefore, the computa-tions based on a mechanical limit where fluctuacomputa-tions are pre-supposed to be minor can only be approximate. The relative enhancement of fluctuations as A increases is in accord with the fact that, ultimately, a semiflexible chain confined in a very thin slit of infinite dimensions must behave as a two-dimensional chain of persistence length 2P. Evidently, a full statistical mechanical analysis is needed to investigate the transition from a one-dimensional to a two-dimensional ran-dom walk for a chain confined in a nanoslit as a function of
A. Meanwhile, Eq. 共50兲 predicts that DNA is highly elon-gated in nanoslits of dimensions presented in Table IV, in agreement with the experimental work of Jo et al.22Yet, at the same time, Eq.共50兲seems to overestimate the elongation in other experiments.55 It is clear that a number of issues need to be resolved for DNA confined in nanoslits.
Reisner et al. have presented fluorescent images of DNA in square 共or almost square兲 nanochannels.13 The elongated length Rexpof the DNA represents the typical average span of the chain and is shown in Table V as a function of the nanochannel size in the case of T2 DNA. The span of a polymer chain is the smallest size of the box it can be
en-closed in, and in one dimension its average has been com-puted in the flexible limit 共LⰇg; see Refs.48–50兲,
S = 4
冉
gL 冊
1/2
. 共51兲
The average span is not known for the wormlike chain model but as g tends to L or becomes larger, the root-mean-square extension Reshould be close to S,
Re2= 2Lg − 2g2共1 − e−L/g兲. 共52兲
Besides the hairpins, orientational fluctuations also shorten the chain. Ifis the angle between its tangent and the axis of the channel, we have the following average:
具2典 = 0.340
冉
A P冊
2/3
. 共53兲
This is derived from Burkhardt’s expression51 for the distri-bution of a harmonically confined worm, suitably 共though approximately兲 rescaled to account for the hard wall repul-sion in a square nanochannel 共see Ref.22兲. In TableV, the effective span 共S具cos典 or Re具cos典兲 is compared with the experimental Rexp共Ref.13兲 where the cosine is computed to second order via Eq.共53兲. The agreement is quite good if one bears in mind that no adjustable parameters have been used. I finally discuss a variety of fluctuational terms— signified by H in Eq. 共4兲—that have been neglected in the mechanical limit as defined in the Introduction.
共1兲 Equation共44兲may be expanded to second order in or-der to evaluate fluctuations in the chord or size of a hairpin. The root-mean-square fluctuation is then 具␦r2典1/2⯝r¯2/ P, with r¯ given by Eq. 共45兲. Similar
con-siderations apply to square nanochannels but not to nanoslits共see above兲.
共2兲 The shape of a hairpin fluctuates at constant r. The length of the hairpin is largest when the bending energy is a minimum. Away from the minimum, the energy and length are shown in Table Ifrom which one esti-mates the fluctuation in l to be兩␦l兩⯝r2/ P.
共3兲 The hairpin may tilt away from the channel axis but one expects the tilt to be given by Eq.共53兲or a similar expression when the cross section is circular. Further-more, the hairpin is only slightly more eccentric than semicircular, at best, so the influence of tilting would appear to be negligible.
共4兲 The entropy of depletion has been computed only ap-proximately when the cross section of the nanochannel TABLE IV. Global persistence length g of DNA enclosed in a nanoslit as a
function of the persistence length P. The width A of the nanoslit is 1000 nm and its height D is 100 nm关values pertaining to the experiments of Jo et al. 共Ref.22兲兴. P共nm兲 g共m兲 100 18 150 42 200 79 250 32 300 204
TABLE V. Experimental elongations Rexpcompared with theoretical predictions Re具cos典 and S具cos典 in their
is square. In the case of DNA, this may suffice because any correction terms to Eq.共39兲will be sensitive to the fluctuations in r discussed above. However, a complete analysis may turn out to be useful for very stiff poly-mers such as actin.
共5兲 The hairpin curve is perturbed by thermal motion. The deformations are undulatory in nature and may be ex-pressed by⌬⯝r¯3/2/ P1/2.32,52,53
On the whole, we conclude that the mechanical limit breaks down as r approaches P. For the widest channels in TableIIandIII, the predictions for g must be considered to be tentative only, for this reason. Complete analyses of the global persistence length including fluctuations have been given for chains in the nematic state.27,33It would be impor-tant to have similar rigorous treatments in the case of nanochannels.
The DNA chain has been treated here as if it were un-charged. A sufficient condition for this to be reasonable is when the typical fluctuation具␦r2典1/2is larger than the Debye
screening length. This applies to the experiment by Reisner
et al.13 which has been discussed at length. At low ionic strength, one must include the interaction between a DNA hairpin and the nanochannel which is often negatively charged. In the rare case of the channel being uncharged, it would be important to deal with repulsive forces owing to image charges共the dielectric permittivity of the channel ma-terial is lower than that of water兲. It is emphasized that for tight bends, one needs to account for the self-interaction of the DNA which is pumped up by undulatory electrostatics.54
ACKNOWLEDGMENTS
I thank Peter Prinsen, Marc-Olivier Coppens, and David C. Schwartz for discussions and Derek Smith and David Wil-liams for correspondence.
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