The polymer physics of single DNA confined in nanochannels

Advances in Colloid and Interface Science 232 (2016) 80–100

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Advances in Colloid and Interface Science

j ourna l homepage: www.e lsev ie r .com/ locate /c i s

Historical perspective

The polymer physics of single DNA confined in nanochannels

Liang Dai a, C. Benjamin Renner b, Patrick S. Doyle a,b,⁎a BioSystems and Micromechanics (BioSyM) IRG, Singapore-MIT Alliance for Research and Technology (SMART) Centre, 138602, Singaporeb Department of Chemical Engineering, Massachusetts Institute of Technology (MIT), Cambridge, MA 02139, United States

⁎ Corresponding author.E-mail address: [email protected] (P.S. Doyle).

http://dx.doi.org/10.1016/j.cis.2015.12.0020001-8686/© 2015 Elsevier B.V. All rights reserved.

a b s t r a c t
a r t i c l e i n f o
Available online 23 December 2015
In recent years, applications and experimental studies of DNA in nanochannels have stimulated the investigationof the polymer physics of DNA in confinement. Recent advances in the physics of confined polymers, using DNAas amodel polymer, havemoved beyond the classic Odijk theory for the strong confinement, and the classic blobtheory for the weak confinement. In this review, we present the current understanding of the behaviors ofconfined polymers while briefly reviewing classic theories. Three aspects of confined DNA are presented: static,dynamic, and topological properties. The relevant simulationmethods are also summarized. In addition, compar-isons of confined DNAwith DNA under tension and DNA in semidilute solution are made to emphasize universalbehaviors. Finally, an outlook of the possible future research for confined DNA is given.
© 2015 Elsevier B.V. All rights reserved.

Keywords:Polymer physicsDNANanochannelConfinementMonte Carlo simulation

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812. DNA model and simulation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

2.1. Touching-bead model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822.2. Freely-jointed rod model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832.3. Other polymer models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842.4. Metropolis Monte Carlo simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842.5. Simplified PERM simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842.6. Comparison of simulation methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852.7. Other simulation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3. Polymer physics of confined DNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.1. Static properties of confined DNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.1.1. Classic de Gennes regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.1.2. Extended de Gennes regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.1.3. Classic Odijk regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883.1.4. Backfolded Odijk regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.2. Dynamic properties of confined DNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923.2.1. Dynamics of a single polymer in bulk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923.2.2. Hydrodynamic interaction in free space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923.2.3. Hydrodynamic interaction in confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.2.4. Diffusivity of confined DNA in slits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.2.5. Relaxation of confined DNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.3. Topological properties of confined DNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.3.1. DNA knots in free space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.3.2. DNA knots in nanochannels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4. Comparison of confined polymers and polymer under other conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.1. Polymers in confinement versus polymers under tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.2. Polymers in confinement versus polymers in semidilute solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

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5. Summary and outlook. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

1. Introduction

In recent years, experiments on DNA in nanochannels [1–7] andtheir relevant applications [8–14] have stimulated the systematicstudy of the physics of confined DNA molecules. These experimentshave beenmade possible due to concurrent advances in nanofabricationtechniques [15–27]. Beyond practical applications, DNA has often beenused as a model semiflexible polymer for single-molecule experimentsfor the purpose of exploring general polymer physics under confine-ment [28]. Single DNA molecules with well-defined length can be pre-pared by the molecular biology techniques, and visualization of singleDNA molecules is convenient with the aid of various fluorescencedyes. Two examples [2,29] of the visualization of single DNA moleculesin two types of confinement: tube-like channels [2], and the slit-likechannels [29] are shown in Fig. 1.

One promising application of confining DNA in nanochannels is ge-nomemapping [9,30]. The basic premise, shown in Fig 2, involves label-ing specific sequence motifs in DNA molecules by fluorescence dyes,confining and stretching thesemolecules in nanochannels, and then in-ferring the number of base pairs between sequencemotifs bymeasuringthe distances or the fluorescence intensities between motifs. Thesequence motif map can be used directly or facilitate the assembly ofshort DNA sequences [30]. Many experiments [14,24,31] and simula-tions [32] have been performed towards the development of this tech-nological platform. Confining DNA in nanochannels has been alsoapplied for other applications, such as DNA sorting [26,33,34], DNAdenaturation mapping [11,35], recognizing barcoded DNA [12], andstudying DNA-protein interactions [36].

From the viewpoint of polymer physics, confinement is a type ofperturbation to polymer systems, providing many fundamental ques-tions to be answered. Intuitively, confining a DNA molecule within ananochannel will elongate the DNA and slow down its dynamics. Theo-retical studies are motivated to obtain more quantitative relationshipsbetween the size of channels and resultant DNA physical properties,often expressed as scaling relationships. The dependence of DNA

Fig. 1. Schematic illustration of single DNA molecules in square/rectangular channels andslit-like channels. The left-bottom image shows experimental results [2] of the averagedintensity of λ-DNA in the 30 nm × 40 nm, 60 nm × 80 nm, 80 nm × 80 nm,140 nm × 130 nm, 230 nm × 150 nm, 300 nm × 440 nm, 440 nm × 440 nm channels(left to right). The right-bottom image shows experimental results of 2λ-DNA in a545 nm tall slit-like channel [29].

Fig. 2. (Top) Illustration of the steps in genome mapping. Adapted from Wang et al. [32]with permission. (Middle) examples of confined λ-DNA with sites labelled. The DNAmolecules are coated with cationic-neutral diblock polypeptides to increase stretchingand are confined in 150 × 250 nm2 channels. The scale bar is 5 μm. Adapted from Zhanget al. [91] with permission. (Bottom) more examples of confined DNA with siteslabelled. Image of a single field of view 73 × 73 μm [2]. Adapted from Lam et al. [9] withpermission.

82 L. Dai et al. / Advances in Colloid and Interface Science 232 (2016) 80–100

extension on channel size has beenmeasured in experiments [2,37–41]and simulations [42–47], and compared with the predictions by Odijktheory [48] and de Gennes theory [49]. These studies have led to the de-velopment of theories beyond the classic Odijk and de Gennes theories,including the theory for the extended de Gennes regime [50] and thebackfolded Odijk regime [51–53]. The ionic strength dependence ofDNA extension in nanochannels has also been investigated by experi-ments [4,6,39]. The effects of crowding agents on the conformations ofDNA were found to be altered by nanochannels using experiments[54,55] and simulations [56,57]. The force–extension relationship[58–61] and fluctuation in extensions [62–65] in confinement havebeen also explored. While most experiments involve linear DNA, circu-lar DNA molecules have been used in experiments [66,67] and simula-tions [68,69] as well. In addition to static properties, the dynamicproperties of confined DNA, such as diffusion [29,38,40,70,71], rotation[72], and relaxation [70,73–75], have been extensively studied by ex-periments and simulations and compared to the predictions by theories.The discrepancy between experimental results and theoretical predic-tions leads to amodified blob theory for DNA diffusivity in confinement.Confinementwithin nanochannels has been shown to collapseDNA [76,77], lead to segregation of DNA molecules [78–81], and affect the topo-logical properties of DNA, such as knotting probability, knot size, andknot lifetimes. [82–90]

Several review papers about DNA in nanochannels have been pub-lished in recent years, covering the polymer physics of confined DNA[8,92–94], experimental aspects [8,13,92,94,95], and relevant applica-tions [8,13,92,95]. However, recent advances beyond the classic Odijkand de Gennes have yet to be similarly summarized. In this review,we intend to combine these recent advances with the classic theoriesand present a complete theoretical understanding of the behaviour ofDNA under all degrees of confinement. Furthermore, in this review,we draw striking comparisons between polymers in confinement andpolymers in other situations, including polymers under tension, insemidilute solutions of polymers, and polymers with topological con-straints (knots). Such similarities are sometimes underappreciated inthe polymer community, yet the similar physics which are present canbe quite instructive. In particular, the knotting problem was found tobe essentially similar to the confinement problem [96].

This review starts with simulation methods and techniques fre-quently applied in the studies of confined DNA, and then moves to thestatic, dynamic and topological properties of confined DNA. While wefocus on presenting recent advances in this topic, we also briefly de-scribe classic theories for completeness. Finally, the similarities betweenconfined DNA and DNA in other situations are discussed.

2. DNA model and simulation method

While a myriad of polymer models and the simulation algorithmsexist, we will focus our attention on those which have been used toinvestigate confined DNA.

To simplify the problem of DNA in nanochannels while retaining theessential physics of the system, theoretical modelling often treats DNAas a homogenous semiflexible chain. Furthermore, the interactionsbetweenDNA segments, including the electrostatic interactions, are fre-quently expressed as purely hard-core repulsions [47] with an effectivechain width [97] or other similar repulsive potentials [42,45,75]. Thecontinuous chain needs to be discretized for simulations, and differentdiscretizations result in different levels of coarse-graining.

Using the simplified model for a DNA chain in a nanochannel, thesystem can be completely described by four parameters:

(i) the persistence length Lp;(ii) the effective chain width w;(iii) the contour length L;(iv) the channel size D for a tube-like channel or the slit heightH for a

slit-like channel.

There is a good description of these four length parameters for con-fined DNA in a review paper by Reisner et al. [92], and so here we onlybriefly introduce these parameters. Realistic values for these four pa-rameters are as follows. The persistence length depends on the ionicstrength of the experimental buffer and other factors, while the typicalvalue is 50 nm [98,99]. The definition of persistence length and howpersistence length is related to the bending stiffness and orientationalcorrelation can be found in standard polymer physics textbooks [100,101]. An effective chainwidth,w, is used to take account of the hardcoreand the electrostatic repulsions between DNA segments, ranging from10 nm to 3 nm for ionic strengths from 10 mM to 1 M [97]. The DNAmolecules used in experiments [2,4,38] are usually λ-DNA or T4-DNA,with contour lengths of approximately 16 μm and 56 μm, respectively.These contour lengths will be increased by florescence labeling [99]. Inexperiments, the confining dimension of the tube-like channel or slit-like channel typically ranges from 30 to 500 nm [2,4,38]. From theview point of theory, the effective channel size

D ¼ Dreal−δ ð1Þ

is more relevant than the real channel size Dreal because the centers ofmonomers are allowed to move in a cross-section of size (Dreal-δ) rath-er than Dreal. The DNA-wall depletion width δ is due to the hardcorerepulsion between DNA and walls, and also due to the electrostatic re-pulsion between DNA and walls in the case the wall is negativelycharged. Reisner et al. [92] discussed this depletion width δ in their re-view paper. In the simulations of a chain with only hardcore repulsions,the depletion width is simply the chain width, δ=w. Similarly for theslit-like channel, the effective slit height is H=Hreal-w.

It bears mentioning that obtaining converged simulation results isnot an easy task. Very long chains are required so that end effects arenegligible and the simulation results can be comparedwith the theoret-ical predictions for infinitely long chains. In order to obtain simulationresults for these long chains, it is critical to employ efficient algorithmsand select or develop polymer models that are coarse-grained at a levelthat balances precision with computational expense. After briefly de-scribing such models and algorithms, we also discuss their advantagesand shortcomings.

2.1. Touching-bead model

The simplest model to capture bending and excluded volume (EV)interactions is the touching-beadmodel [45,47]. This model can be con-sidered as the continuous version of the lattice model for polymers be-cause the bond length is fixed, yet the angles between successive bondssample a continuous distribution. In thismodel (Fig. 3), the chain is rep-resented as a string of Nbead beads connected by (Nbead-1) inextensiblebonds of length lB, corresponding to a contour length L=(Nbead-1)lBThe bond length equals is equal to the bead diameter, also called thechain width w, and for this reason the touching-bead model is named.Beyond connectivity, only the bending energy and hardcore repulsionare considered. The bending energy between two adjacent bonds isused to reproduce the persistence length:

Ebendi;iþ1 θi;iþ1� � ¼ 1

2LplB

θ2i;iþ1; ð2Þ

where θi , i+1 is the bending angle between the bonds i and i+1. Hard-core repulsions are applied for every pair of beads and for every beadand thewalls of the confining geometry. Hardcore repulsions are imple-mented in a way that the interaction energy is infinitely large and theconformation is rejected if the distance between two beads is lessthan the bead diameter and if the distance from a bead to channelwall is less than the radius of the bead.

It is worth pointing out that discretizing a continuous wormlikechain with a bond length lB as well as using Eq. (2) to reproduce the

Fig. 3. Schematic illustration of three types ofmoves in aMonte Carlo simulation to samplethe chain conformations based on the touching-bead model.

Fig. 4. (a) Discretization of a continuous wormlike chain model. (b) The ratio of the actualcontour length in a bond to the preset bond length as a function of the relative bondlength. (c) The ratio of the actual persistence length to the preset persistence length as afunction of the relative bond length.


persistence length can precisely describe thewormlike chain onlywhenlB≪Lp. As the bond length lB increases to be comparable to Lp, thediscretization model leads to noticeable errors in representing the con-tour length and the persistence length, as shown in Fig. 4(a). Strictlyspeaking, each bond does not represent a bond with a contour lengthlB, but with an actual contour length lBactual following the equation[101,102]:

lactualB ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2L2p lB=Lp−1þ exp −lB=Lp

� �� q: ð3Þ

The ratio of the actual bond length to the preset bond length is plot-ted in Fig. 4(b).

The actual persistence length Lpactual determined by the bending en-ergy in Eq. (2) is also different from the preset persistence length Lp. Fol-lowing the standard definition, the persistence length is determined bythe correlation length in the orientational correlation along the chain.Then, the actual persistence length can be calculated through the aver-age bending angle over the separation of lB:

cos θ lBð Þh i ¼

Zexp −Ebend=kTð Þ sin θ cos θdθZ

exp −Ebend=kTð Þ sin θdθ¼ exp −

lBLactualp

!: ð4Þ

Recall that Ebend is related to the preset persistence length throughEbend=(1/2)(Lp/lB)θ2. The ratio of the actual persistence length to thepreset persistence length is plotted in Fig. 4(c). In a typical simulationwith lB=Lp/10, the actual bond length and the actual persistence lengthdeviate from the preset values by 0.84% and 0.85%, respectively.

2.2. Freely-jointed rod model

For very thin chains, i.e. w/Lp≪1, the touching-bead model is notcomputationally efficient because every persistence length needs to berepresented by many beads. In addition to the number of beads, thelarge bending stiffness κbend=Lp/lB=Lp/w as shown in Eq. (2) leadsto a small acceptance ratio when performing reptation or crankshaftmoves. To overcome the difficulties of the touching-bead model forthin chains, the freely jointed rod model is often employed, depictedin Fig 5(a). In this model, the chain consists of a series of rods withoutbending energy. Hardcore repulsions are the only interactions thatexist between rods. The problem of identifying rod−rod overlaps isequivalent to satisfying three conditions, illustrated in Fig. 5b. First,the distance between axes of two rods must be less than the chainwidth. Second, we calculate the vector v!12 that is normal to both rodaxes v!1 and v!2. We then determine the plane where v!1 and v!12 arelying, as shown in Fig. 5b. Both ends of the blue rod (with axis v!2), Aand B, must be located on the different sides of the plane if rods 1 and2 intersect. Third and finally, both ends of the red rod (with axis v!1),

Fig. 5. (a) Schematic of the freely-jointed rod model. (b) Illustration of how rod−rodoverlaps are identified. Adapted from Dai et al. [103] with permission.

Fig. 6. Flow chart for the simplified PERM simulation algorithm [50].


C and D, must be located on the different sides of the plane spanned byv!2 and v!12. For two infinitely long rods, only thefirst condition is need-ed to identify overlaps. The second and third conditions are necessaryfor finite length rods. This algorithm for the excluded volume interac-tion between rods is fast, but is so at the expense of missing some rareoverlap situations.

A key limitation to the freely-jointed rod model is that the bendingwithin a rod (Kuhn length) is ignored. As a result, the freely-jointedrod model is not suitable to study chains in the presence of tight exter-nal confinement, D≲4Lk since bending within a Kuhn length becomesimportant in this regime.

2.3. Other polymer models

Polymermodels other than the above twomodels have been used tostudy confined chains. For example, DNA has been modeled as a stringof rods with bending energy [83,104], a hybrid between the touching-bead model and freely-jointed rod model. In these models, the contourlength is inextensible. The discretized wormlike chain model [43,44]has also been used to simulate confined DNA, in which the bondconnecting two beads is described by a finitely extensible, nonlinearelastic potential (FENE). In the bead-springmodel, each bead representsa subchain larger a Kuhn segment, and the spring between beads is usedto capture the conformational entropy inside a subchain. Such bead-spring models are more coarse-grained than previously describedmodels, allowing for faster simulation of confined DNA [41,55,73,105–107].

As long as the approximations in the modeling are negligible for aparticular parameter set, the simulation results should be independentof the model selected. Usually, simulation results of a newly developedmodel are benchmarked through comparisons with known theoreticalresults, such as the distribution of end-to-end distance of ideal chains,the Marko–Siggia equation [108] for the force–extension relationship[105], or through the comparisons with the results from establishedmodels. For example, the freely-jointed rod model has beenbenchmarked [50] against the touching-bead model for chains in achannel with DN4Lk≡8Lp.

2.4. Metropolis Monte Carlo simulation

In a typical Metropolis Monte Carlo simulation of a polymer chain, arandom conformation is initially generated, and the conformationalphase space of the polymer is explored by successive reptation, pivotand crankshaft moves [45,47,109,110]. Fig. 3 illustrates these threetypes of moves. Both the pivot and crankshaft moves update the poly-mer conformation globally. These moves are more efficient in samplingconformations than local moves, typically used in Brownian dynamicsimulations [111] and some Monte Carlo simulations [42–44]. The rea-son for the greater sampling efficiency of global moves is straightfor-ward. The computational cost of moving N beads in a step (a globalmove) is roughly N times that of moving one bead in a step (a localmove). Achieving a similar change in the conformation by local movesalone requiresmuchmore than N steps. As a result, global moves are ef-ficient in sampling conformations. Owing to simplicity and computa-tional efficiency, these Monte Carlo sampling moves are often used inlarge computational efforts, such as the precise determination of theFlory exponent [112,113].

2.5. Simplified PERM simulation

Another popular and efficient technique for sampling polymer con-formations is known as PERM (Pruned-Enriched Rosenbluth Method).In PERM, polymer conformations are generated by chain growthstarting from one bead, and adding beads one by one until the desiredlength is achieved. The original PERM algorithm was developed by

Grassberger [114] for the lattice model and has been applied to studyconfined polymers [115,116]. To the best of our knowledge, theDorfman group was the first to apply the PERM simulation with anoff-lattice model to study confined polymers [46,53,117,118]. Othergroups have also applied PERM simulation with an off-lattice model tostudy confined polymers [50,84,116]. A recent review of the PERM sim-ulation of polymers can be found in Ref [119].

The basic idea of PERM is that some polymer conformations are du-plicated (enriched) or deleted (pruned) during the chain growth inorder to increase the sampling efficiency. The statistical weight of eachchain is adjusted in order to compensate for the enrichment or deletionof chains. To perform the duplication and deletion in a proper manner,parameter optimization and an initial guess of free energy are neededin the original PERM simulation [114].

Dai et al. [50] simplified the PERM algorithm so that no parameteroptimization is required. Only the simplified version is illustrated anddescribed here, while the original PERM algorithm can be found in Ref[114]. Fig. 6 illustrates the simplified PERM algorithm with thetouching-beadmodel. The chains are grown in a batch of Nc chains rath-er than individually. In the step of adding the i-th bead, the bead isadded such that the bending angle θ formed by (i-2)-th, (i-1)-th, andi-th beads follows a Boltzmann distribution exp[-(Lp/lB)θ2/2 ]. Supposethat Nd chains die after adding i-th due to the hardcore repulsion.Then, we randomly select Nd chains from (Nc-Nd) surviving chainsand duplicate them. The statistical weight of every chain is reduced bya factor (Nc-Nd)/Nc to account for this duplication. In such a way, thenumber of chain remains unchanged during the chain growth. Eventu-ally, Nc chain conformations, all with the same weight WL, are used tocalculate the quantities of interest.

The simplified version of PERM has some limitations. First, it re-quires large computer memory. In the simplified version of PERM,many chains must grow simultaneously in a batch, and the number ofchains in a batch should be as large as possible to minimize the statisti-cal errors. As a result, the computer memory may be a bottleneck to re-duce the statistical error. For example, in our PERMsimulation of a chainwith 10,000 monomers, the number of chains in a batch is set as 3000,


which is themaximumnumber allowed by our computermemory. Sec-ond, the simplified version of PERM is only suitable for chains withoutsoft pairwise interactions, i.e. only hardcore repulsions are allowed, be-cause for these chains, every surviving chain conformation (no overlap)has the same Boltzmannweight. Note that the contribution of the bend-ing energy to the Boltzmannweight has already been considered duringthe biased addition of a monomer to a chain.

Fig. 7. (a) The normalized density profile of an infinitely long wormlike chain in a slit ofwidth W calculated from the field-based method. Circles, squares, diamonds, andtriangles correspond to slit widths W/a= 4, 2, 1, and 0.25. The solid curve correspondsto the exact result for an infinitely wide slit. (b) The normalized confinement freeenergy as a function of the normalized slit width calculated from the field-based

2.6. Comparison of simulation methods

Next, we will discuss the relative advantages and disadvantages ofthe previously described simulation methods. First, PERM simulationgives the free energy directly, while Monte Carlo simulation does not.Note thatMonte Carlo simulation is also able to calculate the free energyusing techniques such as umbrella sampling [120,121] and thermody-namic integration [122], but these techniques will increase computa-tional cost. As a result, PERM simulation is preferred when theprimary simulation objective is the calculation of free energies. Second,Monte Carlo simulation becomes computationally inefficient for thinchains (w/Lp≪1)with the touching beadmodel because the acceptanceratio after reptation and crankshaft moves is low due to the large bend-ing energy coefficient Lp/lB as shown in Eq. (2). As a result, in the simu-lation of thin chains, PERM simulation is preferred (or the freely-jointedrod model). Third, PERM simulation becomes computational inefficientfor dense systems because chain growth is more likely to terminate dueto hardcore repulsions. Because single chains in good solvent are nottypically dense, this issue does not occur for PERM simulation of singlechains. However, in crowded systems, such as the mixture of DNA anddepletants, PERM simulation is expected to be ineffective.

method. The unit in length is the Kuhn length a≡2Lp. Reprinted from Chen and Sullivan[123] with permission.

2.7. Other simulation methods

Dynamic simulations have also been used to investigate confinedpolymers [41,73]. While dynamic simulations provide informationsuch as diffusion and relaxation, they are usually not as efficient asMonte Carlo simulations in terms of sampling conformations. In addi-tion to sampling conformations in simulations, other numericalmethods were also applied to investigate confined polymers. Chenand co-workers applied field-based methods to calculate the segmentdensity (Fig. 7a) and the confinement free energy (Fig. 7b) in a slit[123] or a tube [124]. The basic idea of the field-basedmethods is as fol-lows. The probability of segments of wormlike chains in space satisfies apartial differential equation [125], analogous to Feynman path integralsin quantum mechanics. Solving the eigenfunctions and eigenvalues ofthis partial differential equation yields the distribution of segments inspace and the confinement free energy. The field-basedmethods are su-perior to the method of sampling conformations in the sense that theygive exact results for infinitely long chains, ignoring the usually smallnumerical errors in solving partial differential equations. However, thefield-basedmethods are usually applied towormlike chainswithout ex-cluded volume interactions [123,124], due to the difficulty caused by in-cluding these interactions.

Another effective numerical method to investigate confined poly-mers is the randomly-accelerated-particle method developed byBurkhardt, Gompper and coworkers [126–128]. The premise of thismethod is that each configuration of a strongly confined polymer isinterpreted as the position of a randomly accelerated particle in two di-mensions. This method yields the prefactors in the scaling behaviors ofthe free energy, the extension and the fluctuation for D→0, and theseprefactors are widely adopted in polymer community. This method isonly applicable in very strong confinement, because the interpretationof a confined polymer as a randomly accelerated particle in two dimen-sions assumes the vectors of segments are nearly parallel with thechannel walls.

3. Polymer physics of confined DNA

We now turn our attention to the behaviors of DNA in confinement.As we describe the current physical understanding and the theoreticalpredictions of confined polymers, simulation results are presented tovalidate theoretical predictions. In addition, the experimental resultsof extension, fluctuation, diffusion and relaxation of confined DNA arecompared to simulations and theory. Bear in mind that experimentalsystems of DNA in nanochannels involve many unwanted factors thatblur the physical picture of confined chains, such as the interaction be-tween DNA and channel walls [129] and the effect of fluorescence dyeson DNAphysical properties [99]. Furthermore, DNAmolecules in exper-iments are often too short to follow the long chain behavior [130].

3.1. Static properties of confined DNA

Wewill begin by first discussing how various scaling regimes for thestatic properties of confinedDNA (Fig. 8) are created by the competitionof the three interactions: bending energy, excluded volume (EV) inter-action, and confinement.Wewill note that confinement has two effects.The first (and most obvious) is to restrict the chain to reside within theconfining geometry. A secondary effect is that the segments of the chainnear the confining walls become aligned with the channel walls.

In the classic Odijk regime, the effect of the alignment of segmentsby channel walls dominates, and all segments are strongly aligned andare unidirectional. When the bending energy and EV interactions arenot strong enough to prohibit backfolding [52,53,131], the chain entersthe backfolded Odijk regime. In the backfolded Odijk regime, all seg-ments are also aligned, but the segments are bidirectional due tobackfolding. In the partial alignment (transition) regime, the segmentsclose to walls are aligned, while segments further from the walls arerandomly orientated. In this regime, the extension is significantly affect-ed by the effect of segment alignment because the fraction of aligned

Fig. 8. Illustration of different regimes experienced by a long polymer in a tube-likechannel when varying the channel size D and the chain width w. The black curvesrepresent polymer chains, red circles represent thermal blobs, and blue circles representthe self-avoiding blobs. The dashed lines represent the boundaries between regimes.Note that the transition in the middle of the diagram has no distinct scaling, and soshould be considered as a crossover from the Odijk regimes to the de Gennes regimes.


segments is substantial. As the channel becomes wider, the fraction ofaligned segments decreases, and the chain gradually enters the coiledregime [46], where all segments can be considered as randomly orien-tated locally. In this sense, the partial alignment regime can be consid-ered as a transition regime. In the coiled regime, the alignment effectis not significant locally, but it is significant on length scales largerthan the channel size D. The coiled regime is separated into the classicand the extended deGennes regimes due to the competition of EV inter-actions and the thermal energy.

It is important to note that Odijk [51] first described how the compe-tition of bending energy, excluded volume interaction and the channel-confinement leads to various scaling regimes. The understanding ofthese regimes was greatly deepened by the following studies, in partic-ular, with the aid of simulations.

Next,we describe the typical theoretical treatment for bending ener-gy and excluded volume interactions. In theory, the bending energy isusually considered through the elastic entropy:

Fentropy≈R2

R20

þ R20

R2

R0≈ffiffiffiffiffiffiffiffiLLp

q;

ð5Þ

where R is the size of the polymer in three-dimensions. The elastic en-tropy tends to pull DNA size towards

ffiffiffiffiffiffiffiffiLLp

p, which is the most probable

size of the polymer, given by the statistics of a randomwalk. For a swol-len chain in a good solvent, the equilibrium size is larger than R0, andthen the second term R0

2/R2b1 is usually ignored.The second interaction is the excluded volume interaction. The ex-

cluded volume refers to the volume that can be not accessed by othersegments due to the occupancy of a segment. For a pair of persistence-length rods, the mutual excluded volume [92,101] is approximatelyVLp≈Lp2w. The contribution of the excluded volume interaction to freeenergy is as follows. The number of persistence-length segments isNp=L/Lp, and the number density is c=Np/V, where V is the volumeoccupied by the polymer conformation. These segments produce an en-vironment like a real gas, increasing the free energy of each segment bycVLp relative to the ideal-gas case. Then, the total free energy increase ofall segments is NpcVLp or the simplified form:

Fev≈L2wV

: ð6Þ

The confinement reshapes the chain so that one has V≈D2L|| intubes, where L|| is the extension of the chain along the tube, orV≈HR||2 in slits, where R|| is the in-plane radius.

For a chain in free space, the Flory-type free energy is built uponEqs. (5) and (6):

Fbulk ≈R2

LLpþ L2w

R3 : ð7Þ

Minimization of Fbulkwith respect to R yields the Flory scaling for thepolymer size

Rbulk ≈ L3=5L1=5p w1=5: ð8Þ

As pointed by de Gennes [132], Eq. (8) yields the Flory exponent3/5 close to the precise value [112,113] 0.5876 but leads to a wrongprediction of the fluctuation around the equilibrium size σ2

bulk≈�∂2 Fbulk∂R2 jR¼Rbulk

�−1≈ LLp.

The proper fluctuation [132] is

σ2bulk ≈ R2

bulk ≈ L6=5L2=5p w2=5 ð9Þ

de Gennes [132] explained thatwhile the scalings in Fentropy≈R2/R02 and

Fev≈L2w/R3 are not precisely correct, they lead to a scaling of Rbulk closeto the precise value due to cancellation of errors. The calculation of thefluctuation in size fails to benefit from a cancellation of errors.

A similar situation occurs when the Flory-type free energy is appliedto chains in sufficiently wide channels. It predicts the correct scaling inextension but the wrong scaling in fluctuation [133]. The details behindthis discrepancy will be presented in the following subsections.

In the following sub-sections, we will describe the physical picturesand predicted scaling behaviors in various regimes, as well as presentcomparisons of these predicted scalings with simulations and experi-ments. Note that we only review the results for tube-like channelshere, while the case of slit-like channel is similar [47,118]. In addition,we mainly review the results for the infinitely long chains in confine-ment. For chains of finite lengths, more scaling regimes will appear[53,118].

3.1.1. Classic de Gennes regimeIn the classic de Gennes regime, the chain conformation is described

by the blob model [49] (Fig. 8). The blob size equals the channel size D.Inside each blob, Flory scaling is applied to calculate the contour lengthinside a blob Lblob=D5/3Lp-1/3w-1/3. Then, the number of blobs is Nblob=L/Lblob=LD-5/3Lp1/3w1/3. The extension is obtained as:

LdeGjj ¼ NblobD≈ 1:176LD−2=3L1=3p w1=3: ð10Þ

The prefactor 1.176 was determined with simulations by Wernerand Mehlig [134], while Dai et al. [50] obtained a prefactor 1.05. Theconfinement free energy simply equals the number of blobs:

FdeGconf¼ Nblob ≈ 5:04LD−5=3L1=3p w1=3: ð11Þ

Again, the prefactor 5.04 was determined by simulations [50].The fluctuation in extension may be derived by the blob model as

well. The fluctuation in size of each blob is proportional to the blobsize σblob

2 ≈Rblob2 ≈D2. So the total fluctuation is determined by

σ2deG ¼ Nblobσ2

blob≈ 0:16LD1=3L1=3p w1=3: ð12Þ

The prefactor 0.16 was determined by simulations [103]. The effec-tive spring constant is

kdeGspr ≈ kBT=σ2deG≈kBTL

−1D−1=3L−1=3p w1=3: ð13Þ

Fig. 10. Extension as a function of the square channel size for three different chain widths.The symbols are from simulations [50] using the touching-beadmodel with 2×104 beads.The three dashed lines on the right-hand side are plotted based on Eq. (10). The green lineon the left-hand side is plotted based on Eq. (23) with prefactor 0.1827.


In addition to counting the number of blobs, the extension and con-finement free energy has also been derived by another approach. Junet al. [133] pointed out that the Flory-type free energy by Eq. (4) andEq. (5) is wrong. They derived the free energy in the classic de Gennesregime by considering a 1-D random walk of blobs [133]:

FdeGconf ≈L2jj

L=Lblobð ÞD2 þD L=Lblobð Þ2

Ljj: ð14Þ

The first term describes the elastic entropy for a 1-D random walkwith a step size D, while the second term describes the EV interactions.Eq. (14) yields the same scaling of extensionwith Eq. (10) and yields thesame scaling of free energy with Eq. (11). The effective spring constantin Eq. (13) can be reproduced by ∂2FconfdeG/∂L||2.

The scaling of extension L||~D-2/3 in the classic de Gennes regimewas first validated by the ground-breaking Metropolis simulations ofDorfman group [45] as shown in Fig. 9(top). Due to the limitation ofthe chain length in simulations at that time, the classic de Gennes re-gime (combined with the extended de Gennes regime) spans a narrowrange of the channelwidth. Good agreementwas also obtained betweenthe simulation results by Wang et al. [45] and the experimental resultsby Reisner et al. [2] after adjusting the effective DNA width in experi-ments as shown in Fig. 9(bottom). Later, PERM simulations [46,50]with an order of magnitude longer chains observed the predicted scal-ing of extension in wider ranges of the channel size. One example ofthe simulation results is presented in Fig. 10. The simulation data inFig. 10 agree with Eq. (10) for D≳4Lp. For D≲4Lp, Eq. (10) underesti-mates the extension because the blob model ignores the effect of seg-ment alignment close to channel walls, discussed in the beginning ofSection 3.1.

The scaling of fluctuation in Eq. (12) was validated by Metropolissimulations [103] and will be shown in Fig. 35, while more precise re-sults could be obtained if the PERM algorithm is applied.

Fig. 9. (Top) extension of the touching bead model (w = 4.6 nm, Lp = 53 nm) in squarenanochannels. Deff is the effective channel size Din Eq. (1). Symbols: simulations. Thickersolid line: theoretical prediction of Odijk’s regime. Thinner solid line: power-law fit to thedata for channel widths ranging from60 to 120 nm. Dash-dotted line: power-lawfit to thedata for channel widths ranging from 120 to 200 nm. (Bottom) Comparison of simulation(w = 4.6 nm, Lp = 53 nm) and experimental data [2]. Three widths are used to calculateDeff/(wLp)1/2 for the experimental data. Adapted fromWang et al. [45] with permission.

The scaling of the confinement free energy in Eq. (11) was con-firmed by simulations of Dorfman group [46] in Fig. 12. For a relativelythick chain with w=0.2Lp (purple asterisks), the de Gennes scalingF~D-5/3 is reachedwhen D≳10Lp. The de Gennes scaling of confinementfree energy was also observed in the simulations by Dai et al. [50] asshown in Fig. 12. The confinement free energy follows Eq. (11) inwide channels (right-half of Fig. 12b). Note that the simulation resultsin Fig. 12 are based on freely-jointed rod model. As a result, the persis-tence length is replaced by the Kuhn length Lk≡2Lp.

3.1.2. Extended de Gennes regimeThere was a debate about the existence of the extended de Gennes

regime. This regime was first investigated by Brochard-Wyart andRaphael [135] in 1990, and the confinement free energy was proposedto be F≈LD-4/3Lp-1/3w2/3. Later, the same result was reached by Reisneret al. [92] using a Flory-type free energy approach. More recently, Treeet al. found that the prediction F≈LD-4/3Lp-1/3w2/3 was inconsistentwith the confinement free energy calculated from simulations, whichled them to doubt the existence of the extended de Gennes regime[46]. In 2014, Dai et al. [50] pointed out that Tree et al. [46] interpretedthe simulation results in an improper manner, and, Dai et al. confirmedthe existence of the extended de Gennes regime with their simulationresults [50].

The physical picture underpinning the extended de Gennes regimeis as follows. One assumption in the classic blobmodel is that the repul-sion between spherical blobs is strong enough to segregate blobs.Whenthe repulsion between spherical blobs is not strong enough to segregateblobs and D≫Lp, the chain enters the extended de Gennes regime. Theboundary between the classic and extended de Gennes regime can beestimated [45,51] by setting the repulsion between two sphericalblobs as kBT, i.e. Frep≈Lblob2 w/D3=D1/3Lp-2/3w1/3=1. The extended deGennes regime corresponds to D≲Lp2/w. This regime can be describedby replacing the spherical blob with an anisomeric blob. In this model[45,51], the shape of a blob is not a sphere but a cylinder, and the lengthof each blob is increased from D to

Rexblob ≈D2=3L2=3p w−1=3 ð15Þ

such that the EV interaction between two blobs equals kBT [45,51]. Insidethe anisometric blob, the ideal chain scaling is applied: Lblobex ≈(Rblobex )2/Lp=D4/3Lp1/3w-2/3. Accordingly, the number of blobs is

Nexblob≈ LD−4=3L−1=3

p w2=3: ð16Þ

Then, the extension for the extendeddeGennes regime follows [45,51]

Lexjj ¼ NexblobR

exblob≈1:176LD−2=3L1=3p w1=3: ð17Þ


The extension in Eq. (17) has the same scaling lawwith the classic deGennes regime Eq. (10), and that is the reason why this regime is calledas “extended de Gennes regime”.

The anisometric model can also be used to derive the confinementfree energy. As shown in Fig. 13, the channel walls not only restrictthe arrangement of these anisometric blobs, but they also compresseach anisometric blob. The compression is due to the fact that ananisometric blob would restore to an isotropic shape in the absence ofthe channel walls. The free energy contributions from the alignmentand compression of blobs are captured by the first and second term inthe following equation

Fexconf ¼ 2:4LD−4=3L−1=3w2=3 þ 2π2=3� �

LLpD−2: ð18Þ

The first prefactor 2.4 has been determined by the results of simula-tions [50]. The second prefactor 2π2/3 is the theoretical value [123,136].It can be seen that the second term is the leading term because the ratioof the first term to the second term is (Dw/Lp2)2/3, which is vanishinglysmall for D≪Lp2/w. The second term, as a leading term, has beenneglected in a number of previous studies [46,59,92,135].

The fluctuation in extension in the extended de Gennes regime canbe derived by the anisometric blob model through the fluctuation ofeach anisometric blob [45,51] [103]:

σ2ex ≈Nex

blob Rexblob

� �2≈ 0:264LLp: ð19Þ

The prefactor 0.264 was determined with simulations by Wernerand Mehlig [134], while Dai et al [50] obtained a prefactor 0.28. The in-dependence of fluctuationwith respect to the channel size was recentlyobserved in experiments of DNA in nanochannels [65]. Accordingly, theeffective spring constant is [45,51]

kexspr≈kBTL−1L−1

p : ð20Þ

In addition to the anisometric blobmodel, the Flory-type free energycan be also used to describe the extended de Gennes regime:

Fexconf≈LLpD2 þ L2jj

LLpþ L2w

D2Ljj: ð21Þ

The application of the above equation in the extended de Gennes re-gime can be justified in the followingmanner. The first two terms corre-spond to the free energy of an ideal chain. If the chain width graduallyincreases from zero, EV interactions would also gradually increasefrom zero. When the EV interactions are weak, they can be consideredas a weak perturbation to the free energy of an ideal chain and can becaptured by the third term in Eq. (21). Therefore, Eq. (21) is valid inthe case of weak EV interaction, i.e., the extended de Gennes regime.Eq (21) can be used to calculate the extension, the fluctuation, and theconfinement free energy.

The predictions of extension, fluctuation, and confinement free en-ergy in the extended de Gennes regime have been validated by simula-tion results [50,103]. Fig. 12 shows that the scaling of extension remainsunchanged, while the scaling of confinement free energy graduallyvaries from ~D-2 to ~D-5/3 during the crossover from the extended deGennes regime Dw/Lp2≪1 to the classic de Gennes regime Dw/Lp2≫1.The boundary between these two de Gennes regimes was determinedas Dex⁎≈2Lp2/w by the crossing point of two scalings. We highlight thatusing typical parameters for DNA Lp=50 nm andw=5 nm, the criticalchannel size is Dex⁎=1000 nm, which is larger than the most channelsizes in DNA confinement experiments. As a result, the confined DNAin most experiments does not enter the classic de Gennes regime. Tospecifically validate the first term in Eq. (18), the confinement free ener-gy of the ideal chain, the second term in Eq. (18) is subtracted from thetotal confinement free energy and the results are plotted in Fig. 12(c).

The independence of fluctuation with respect to the channel size inextended de Gennes regime was observed in simulations [103,137] aswell as in experiments as shown in Fig. 14. The simulation results offluctuation will be shown in Fig. 35.

Recently, the extended de Gennes regime was also studied throughthe mapping of a confined polymer in this regime to one-dimensionalweakly-self-avoided random-walk problem, which has analytic solu-tions [134]. Using themapping,Werner andMehlig successfully derivedthe scaling of the extension in agree with the ones by Flory theory andthe anisometric blob model. Furthermore, such mapping gives the ex-tension and the fluctuation with precise prefactors [134].

3.1.3. Classic Odijk regimeIn the classic Odijk regime, the conformation of the DNAmolecule is

described by the deflection model [48] (Fig. 15), where the chain is fre-quently deflected by channel walls. The extension and confinement freeenergy in the Odijk regime are derived as follows [48]. According to theexponential decay of orientation correlation in the semiflexible chain,one has ⟨cos θ⟩=exp(-λ/Lp), where θ is the angle formed by the chan-nel axis and the segment between two deflections, and λ is the averagecontour length between two deflections (Fig. 15). Because θ is a smallquantity, the above equation can be approximated as 1-θ2/2≈1-λ/Lp.Applying θ≈ sinθ≈D/λ, we obtain the Odijk scaling

λ≈D2=3L1=3p : ð22Þ

Accordingly, the relative extension follows

Ljj=L ¼ cosθh i≈1−αodijkjj D=Lp

� �2=3: ð23Þ

The prefactor α||odijk has been precisely determined [127,138] for

channels with square cross-section α||odijk=0.1827±0.0001 and

tubes with circular cross-section α||odijk=0.1701±0.0001. Recall that

these prefactors were obtained from the randomly-accelerated-particle method mentioned in Section 2.7. Since every deflection re-stricts the chain conformation and contributes a confinement free ener-gy on the order of kBT, the confinement free energy is proportional tothe number of deflections:

Fconf ≈ L=λ≈αodijkF LD−2=3L−1=3

p : ð24Þ

The prefactor αFodijk has been precisely determined [127,138] for

channels with square cross-section αFodijk=2.2072 and tubes with

circular cross-section αFodijk=2.3565.

The fluctuation around the equilibrium extension can be alsoestimated from the deflection model. It can be considered that thedeflection angle fluctuates between 0 and θ≈D/λ. Then, the extensionof each deflection segment fluctuates between λ and λcosθ, with theamplitude of fluctuation σλ≈λ(1− cosθ)≈ D2/(2λ). The fluctuationsof (L/λ) deflection segments are independent of each other, andhence, the fluctuation of total extension [48,92,138] is

σ2odijk≈

Lλσ2

λ≈αodijkδ LD2=Lp: ð25Þ

Again, the prefactor has been precisely determined [127,138] by therandomly-accelerated-particle method for channels with square cross-section αδ

odijk=0.0096±0.0002 and tubes with circular cross-sectionαδodijk=0.0150±0.0002.The effective spring constant for the force-extension relationship is

related to the fluctuation through kspr(δR)2≈ kBT because the fluctua-tion in the free energy is on the order of kBT. As a result, one has

kodijkspr ≈ kBTL−1D−2Lp: ð26Þ

Fig. 11. The normalized confinement free energy as a function of the normalized channelsize. The dashed line is from an empirical equation proposed by Tree et al. [46] to coverthe confinement free energy of a long ideal chain from strong to weak confinement. Thesymbols are from PERM simulations for w/Lp=0.002 (open upward triangle), 0.005(open downward triangle), 0.01 (plusses), 0.02 (open diamond), 0.05 (crosses), 0.1 (openpentagons, DNA) and 0.2 (asterisks). The Gauss-de Gennes regime was proposed whenthe understanding of the extended de Gennes regime was not clear, and it corresponds tothe extended de Gennes regime. Adapted from Tree et al. [46] with permission.


Validation of the extension in Eq. (23) and the fluctuation in Eq. (25)by simulation results is shown in Fig. 16. The extensions in simulationsagree with Eq. (23) for D/Lp≲0.4. The fluctuations in simulations followEq. (25) for D/Lp≲0.3. As shown in Fig. 11, the confinement free energyfollows Eq. (24) for D/Lp≲0.5.

Fig. 12. Results from freely-jointed rod simulations of 104 rods, i.e., L/Lk=104.(a) Normalized extension as a function of normalized size of a square channel(b) Normalized confinement free energy as a function of normalized channel size.(c) The difference in confinement free energy between a real chain and an ideal chain.The crossover between the extended and classic de Gennes regimes is estimated asDex

⁎≈0.56Lk2/w≈2Lp2/w. Adapted from Dai et al. [50] with permission.

The experimental observation of the deflection conformation in theOdijk regime was not easy for DNA because the deflection length is onthe order of or less than the persistence length Lp≈50 50 nm. Asshown in Fig. 17, such deflection conformations can be observed exper-imentally [139] in the case of F-actin in microchannels, because F-actinhas a persistence length on the order of a micron.

3.1.4. Backfolded Odijk regimeDue to backfolding, as illustrated in Fig. 18, the extension atD≈ Lp is

shorter than the prediction by the deflection model. The simulations ofDNA in slits demonstrate that the projected contour length of a longchain on the slit wall follows Eq [23] for H≲2Lp [47]. Recall that thereis not backfolding for chains in slits. Such a result [47] suggests that ifthe backfolding does not occur in tube-like channels, the deflectionmodel should be applicable to calculate the extension for D≲2Lp.Hence, efforts have been made to add the effect of backfolding intothe deflection models [51–53,140].

Backfolding leads to the formation of S-loop domains. The formationof S-loop domain costs bothbending energy and excluded volume inter-action energy. The bending energy is mainly due to the two haipins[140] at both ends of S-loop domain, while the EV interaction is due tohigh segment density in S-loop domains.

Two approaches have been used to quantify backfolding. The firstapproach was developed by Odijk [140] and recently examined and re-fined by simulations [53]. In this approach, the chain in the backfoldedOdijk regime is viewed as 1-D random walk in the chain. In analogy tothe Flory theory for the 3-D randomwalk, two parameters are calculat-ed: the step size of randomwalk and the excluded volume interactions.

The step size of this randomwalk is defined as the global persistencelength Lg, which is approximately the average contour length betweentwo occurrences of backfolding in the absence of EV interactions [51].The global persistence length should follow Lg≈ Lbackexp(+Fback),where Lback is the average contour length inside a backfold, and Fbackis the free energy cost of backfolding, mainly attributed to bending en-ergy. Odijk [140] derived the expression of Lg:

Lg ¼ αr exp F� �

r ¼ Lp=6� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

E2m þ 6ffiffiffi2

pEm D=Lp� �r

−Em

;

F ¼ Em Lp=r� �

−3 ln D−rffiffiffi2

p� �=D

h i− ln 8= 3πð Þ½ �;

ð27Þ

where α≈3.3082 and Em≈1.5071. The global persistence length wasnumerically calculated from simulations of ideal chains by Muralidharet al. as shown in Fig. 19. Eq. (27) systematically overestimates theglobal persistence length, and so Muralidhar et al. [53] modified theexpression of Lg :

Lg ¼ αr exp F−4:91� �

; ð28Þ

in order to match simulation results.

Fig. 13. Illustration of the anisometric blobmodel for the chain in the extended de Gennesregime.

Fig. 14. Semilog plot of the fluctuation in extension δX as a function of effective channelsize Deff for rectangular channels with depth D1 = 300 nm and widths D2 ranging from350 to 750 nm. The experimental mean extension variance (blue □) is the mean of 29molecules. PERM simulations in the rectangular channels of dimension D1 = 300 nmand D2 = 350 to 750 nm for a contour length L = 70.2 μm are included for rectangularchannels (black ○) and equivalent square channels (red △) of size Deff+w. The blackline corresponds to Eq. (19) with the prefactor 0.264. Adapted from Gupta et al. [65]with permission.

Fig. 16. (Top) the deviation of the normalized extension from unity as a function of thenormalized channel size from simulations. The Lp/w values are 17.2 (green), 35.03(gold), 71.35 (orange), 145.32(red), 295.96 (black), 602.74 (gray), 1227.54 (dark-green), and 2500 (brown). The dashed line is Eq. (23). (Bottom) the normalizedfluctuation in extension as a function of the channel size from simulations. The dashedline is from Eq. (25) with the prefactor determined by Burkhardt et al. [138] Adaptedfrom Muralidhar et al. [53] with permission.


The excluded volume interactions are calculated as follows. For apair of segments of deflection length λ, after considering the orienta-tional correlation, the excluded volume interaction is

νλ≈λ2w D=Lp� �1=3

: ð29Þ

After knowing the step size Lg and the EV interaction, the free energyof a chain in this regime is

F ¼ Lλþ L2jjLLg

þ N2λνλ

LjjD2 : ð30Þ

The first tem is the leading-order term as presented in Eq. (24). Thesecond and third terms are the elastic entropy and the EV interaction,respectively. Minimization of F with respect to L|| yields

Ljj≈ Lξ1=3g ; ð31Þ

where ξg is a dimensionless parameter that describes the ratio of thevolume caused by the excluded volume interactions relative to the vol-ume available for interactions:

ξg ¼ n2λνex

LgD2 ¼ Lgw

D5=3L1=3p

: ð32Þ

Here, nλ= Lg/λ is the number of deflection length in a global persis-tence length. In the case of ξgN1, the EV interactions are strong enoughsuch that backfolding is unlikely to occur, and the chain enters theclassic Odijk regime. In the case of ξgb1, backfolds can occur, and theextension follows the scaling in Eq (31). Accordingly, the boundary

Fig. 15. Illustration of a semiflexible chain in strong confinement. The average contourlength between two deflections is denoted as λ, and the angle formed by the segmentand the channel wall is denoted as θ.

between the classic and the backfolded Odijk regime is specified byξg = 1, corresponding to

D�back ¼ L3=5g w3=5L−1=5

p : ð33Þ

The scaling of the extension with respect to the dimensionlessparameter ξg was confirmed by simulation results [53] in Fig. 20.

The second approach was proposed by Dai et al. [52] In this ap-proach, the chain is considered as a series of units of contour lengthπD, and these units can assume one of two states: the deflection stateor the S-loop state, illustrated in Fig. 21. Using this two-state model,analytical calculation becomes feasible based on the Bragg–Zimmcooperativitymodel [141] or the Isingmodel. The two input parametersfor the cooperativitymodel are: the excess free energy Fs of a unit in theS-loop state with respect to the deflection state and the free energy cost

Fig. 17. Fluorescence video microscopy images of F-actin fluctuations in microchannels:from left to right, the channels are 20, 10, 5, and 3 μm wide, respectively, and 1 μmdeep. Top and bottom images were taken in arbitrary time. Scale bar indicates 10 μm.Adapted from Choi et al. [139] with permission.

Fig. 18. Illustration of backfolding of a chain in confinement. An S-loop domain is createdby two occurrences of backfolding.

Fig. 20. The extension as a function of the dimensionless parameter ξ shown in Eq (32).Adapted from Muralidhar et al. [53] with permission.


2Fu to create an S-loop domain. Here, Fu is the free energy cost of hairpinformation, similar to Eq. (27). The expression of Fs is proposed as

FS ¼ a36w

D2=3L1=3p

ð34Þ

to take account of the excluded volume interactions between partiallyaligned segments of length πD. Here, a3 is a prefactor, which is deter-mined as a3≈1.08 by simulations for w/Lp=0.1. The expression of Fuis proposed as

Fu ¼ a1πLp=D−a2; ð35Þ

where the first term captures the bending energy in the hairpin, and thesecond term is an entropic contribution. The two prefactors a1≈1.2 anda2≈1.7 are determined by simulations for w/Lp=0.1. For other chainwidths, the values of a1 and a2 are slightly varied.

Based on Fs and Fu, the propagation and nucleation parameters aredefined as

s ¼ exp −FSð Þu ¼ exp −2Fuð Þ: ð36Þ

In the ground state dominance, the cooperativity model gives theanalytic solutions for the average number of S-loop domains per unitcontour length fs, and the the average contour length stored in an S-loop domain:

fs ¼ u2πD 1−uð Þ

1þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−sð Þ2 þ 4su

q −1

264

375

LS ¼ πD 1þ 12u

s−1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−sð Þ2 þ 4su

q� � :

ð37Þ

After knowing the frequency and the size of S-loop domains, theextension in the backfolded Odijk regime follows:

Lbackjj ¼ Lodijkjj 1−2fSLS=3−πDfS=3ð Þ: ð38Þ

Fig. 19. The global persistence length as a function of the normalized channel size. Thesymbols are from PERM simulations. The dashed and solid lines are from Odijk theorywithout and with rescaling. Adapted from Muralidhar et al. [53] with permission.

In the above equation, the term −2fsLs/3 is added because the seg-ments in the S-loop state contribute 1/3 of the extension respective tothe deflection state. The term −πDfs/3 is added because the segmentsin hairpin contribute no net extension. Note that the segments in hair-pins are considered as the S-loop state (but at junctions) to make thetwo-state model applicable.

Fig. 22 shows the comparison of Eq. (38) with simulation results[52]. The green line calculated by Eq. (38) overestimates the extensionbecause the formation of C-loops at both ends of the chain is ignored.After considering the reduction of extension by C-loops, the simulationresults in the range D≲2Lp can be explained by theory.

Both approaches are based on the Odijk (deflection) theory and con-sider the effect of backfolding on the extension. While these two ap-proaches capture some features of backfolding, they have limitations.

In the first approach (Flory-type theory), the chain is considered as a1-D randomwalkwith a step size Lg, based onwhich the elastic entropyis calculated. The excluded volume interaction (the third term inEq. (30)) is considered through a mean-field approximation, whichmay only be validwhen the excluded volume interaction can be consid-ered as a weak perturbation, just like the case of the extended deGennes regime (Eq. (21)). In this sense, this Flory-type approach shouldwork better as the chain width decreases and work best for w→0. Onthe other hand, it is expected that as the excluded volume interactionbecomes stronger, the S-loop domain become less frequent and shorter.Intuitively, a configuration containing rare and short S-loop domainswith the domain extension Ls≪ Lg is very different from 1-D randomwalk with a step size Lg and not suitable to be described by this Flory-type approach. Such configuration should occur when the dimension-less parameter ξg is around one in Fig. 20.

In the second approach, only the occurrence of S-loop is added to theclassic deflectionmodel. However, structures more complicated than S-loop can occur. In particular, for thin chains, more than three strandsmay share the same location in the channel, and such structures arenot captured in the S-loop model.

In the backfolded Odijk regime, despite lacking a simple scaling rela-tionship between extension and the channel size, distinct scaling rela-tionships exist as shown in Eq. (31). In this sense, the backfoldedOdijk regime is a real scaling regime. On the other hand, the transitionzone (the middle part) in Fig. 8 has no clear scaling relationship so far

Fig. 21.A two-statemodel is applied to the chain in the backfoldedOdijk regime. The chainis viewed as a series of units of contour length πD.

Fig. 22. The chain extension in a square channel as a function the channel size. In thesimulations, the contour length is L=160Lp. Adapted from Dai et al. [52] with permission.

Fig. 24. Diffusivity as a function of slit height. The symbols are from experiments [29] forboth λ-DNA and 1/2 λ-DNA. The two lines are calculated from Eq. (58) using theprefactor c2=1.68. The triangle indicates the de Gennes scaling of 2/3. Adapted fromDai et al. [152] with permission.


and cannot be considered as a scaling regime but just a crossover fromthe backfolded Odijk regime to the extended de Gennes regime.

3.2. Dynamic properties of confined DNA

3.2.1. Dynamics of a single polymer in bulkWe next move beyond static properties to discuss dynamics of DNA

in confinement. We begin by first recalling basic arguments for dynam-ics of DNA in the absence of confinement. The diffusion of DNA dependson the friction coefficient ζ through the Einstein relation

Ddiff¼ kBT=ζ: ð39Þ

The relaxation time can be approximated as the time required forthe DNA to move a distance of its equilibrium size R or its fluctuationin size δR. Recall that δR is proportional to R in bulk [132].

τ≈R2=Ddiff � R2ζ: ð40Þ

The relaxation time can be also derived by the dumbbell model,where the chain is simplified as two massless beads connected by aspring. The spring force Fspr≈ ksprR is balanced by the drag force Fdrag≈

ζ dRdt. Solving the equation Fspr+ Fdrag=0 yields

R tð Þ≈ R 0ð Þ exp −t=τð Þ ð41Þ

with

τ≈ ζ=kspr: ð42Þ

Fig. 23.Normalized DNA diffusivity as a function of slit height. The filled circles are locatedin 2LpbHbRg,bulk. Adapted from Dai et al. [152] with permission.

Eq. (42) is equivalent to Eq. (40) after applying kspr≈ kBT/δR2 andδR≈R. Eq. (41) suggests a perturbation to the equilibrium chain sizewill decay exponentially with a relaxation time τ.

The friction coefficient ζ is needed to calculate Ddiff and τ. In theRouse model [142], the drag force of the chain is assumed as the sumof drag force of each monomer, i.e. the freely-draining condition:

ζRouse ¼ Nmζm; ð43Þ

where ζm is the friction coefficient of a monomer.In the Zimmmodel [143], the chain is assumed to move like a solid

object and the drag force is assumed to proportional to the radius ofchain conformation, i.e. the no-draining condition:

ζZimm≈ ηR; ð44Þ

where η is the viscosity of solvent.

3.2.2. Hydrodynamic interaction in free spaceThe Rouse model and Zimm model consider two extreme cases for

hydrodynamic interactions within a single DNA molecule. In the morerigorous calculations of dynamic properties, more detailed hydrody-namic interactions (HI) should be considered. Here, HI refers to theforce experienced by one particle due to the flow field generated bythe motion of another particle. The relationship between the force andthe flow field is often calculated based on the Stokeslet approximation.

Fig. 25. Relaxation time obtained from simulations and compared to the experimentaldata of Reisner et al. [2] Adapted from Tree et al. [137] with permission.

Fig. 26. (a) Illustration of a trefoil knot on an linear chain. The red portion represents theknot core, while the blue portion is not involved in knot formation. The knot core (red)with a contour length of Lknot is confined in a virtual tube (gray) with a diameter ofDvirtual. (b) A chain is confined in a straight tube with diameter Dvirtual and lengthLknot≈pDvirtual. Adapted from Dai et al. [173] with permission.

Fig. 28. The most probable size of a trefoil knot as a function of the rescaled chain width.The solid line is calculated from Eq. (69) with k1=17.06, k2=1.86 and p=16. Adaptedfrom Dai et al. [173] with permission.


In this approximation, a point force F!

generates a flow field v!ð r! Þ infree space as:

v! r!� �

¼ F!�Ω; ð45Þ

where the Oseen tensor is

Ω r!� �

¼ 18πμ

1rj j þ

r! r!rj j3

!: ð46Þ

This approximation is based on the fact that the far-field flow gener-ated by a force applied on a sphere is independent of the size of sphere.For simplicity, the Oseen tensor is averaged over all orientations:

Ω rð Þh i ¼ 16πηr

: ð47Þ

Fig. 27. (a) Probability of a wormlike chain containing a trefoil knot with L=400Lp.(b) Potential of mean force as a function of knot size. The line of best fit is shown in red:y=17.06x−1+1.86x1/3−5.69. Adapted from Dai et al. [173]

The pre-averaged Oseen tensor can be applied to calculate the diffu-sivity using the Kirkwood approximation [144]:

Ddiff ≈kBT

N2m

XNm

i¼1

XNm

j¼1; j≠iΩ r!i− r!j

� �D E; ð48Þ

where r!i is the position of i-th monomer, and Nm is the number ofmonomers. The above equation can be understood in the followingmanner. A force F is uniformly distributed among monomers such thateach monomer experiences a force F/Nm. Each point force generates aflow field in the space, and these flow fields are linearly superimposedto each other. The velocity of the chain is approximated as the averagevelocity of monomers. Eq. (48) can be adapted for a continuous chain

Ddiff≈kBTL

ZΩ rð Þh ig rð Þ4πr2dr; ð49Þ

where g(r)≡dL/(4πr2dr) is the pair correlation of monomers, and dLcorresponds to the contour length of the sub-chain within the sphericalshell of thickness dr at distance r. The contribution of long-range HI todiffusivity can be inferred from scaling analysis. Considering thatΩ(r) scales as r−1, the contribution of long-range HI to diffusivitywould be vanishing small if the scaling of g(r)4πr2~rβ has an exponentβb0. The exponent β is 1 for ideal chains and 2/3 for real chains. Hence,the long-range HI play an important role in the diffusivity of single longpolymers. In this sense, the Zimmmodel ismore suitable than the Rousemodel to describe the diffusivity of dilute long polymer solutions, whichhas been validated by experiments [145].

Note that the Kirkwood approximation is based on a pre-averagedOseen tensor and will lead to some errors. Recently, Jain and Dorfmanfound that the Kirkwood diffusivity is always higher than the one calcu-lated by Brownian dynamic simulations with HI, and the maximumerror caused by Kirkwood approximation is about 2% for their simula-tion of confined DNA in the de Gennes regime [146].

Fig. 29. (a) A small knot in a channel (b) A large knot in channel. Adapted from Dai et al.[84] with permission.

Fig. 30. DNA knotting probability as a function of the slit height for DNA with Lp=50 nmand w=2.5 nm. Reproduced from Micheletti and Orlandini [104] with permission.

Fig. 32. The most probable size of a trefoil knot as a function of the channel size. The solidline corresponds to the minimization of free energy in Eq. (70) with respect to Lknot usingnumerical coefficients α=0.1 and β=0.02. The contour length is fixed as L=400Lp, andthe chain width is fixed as w=0.4Lp. Adapted from Dai et al. [84] with permission.


3.2.3. Hydrodynamic interaction in confinementHydrodynamic interactions are altered in confined geometries due

to the presence of the channel walls [147,148]. A comprehensive reviewof HI in various confined geometries can be found in Ref [148]. In squarechannels ofwidthD, hydrodynamic interactions decay exponentially forrND [149]. Accordingly, HI can be considered as screened for r ≫ D . Inthe calculation of the diffusivity, the HI with r≥D can be ignored.

In slit-like channels of heightH, hydrodynamic interactions decay al-gebraically as Ω~r−2 for r ≫ H [150,151]. Whether the scaling for thedecay of hydrodynamic interactions results in the screening of long-range HI depends on the spatial distribution of monomers. RewritingEq. (49) for slits with r ≫ H (quasi-2D)

Ddiff≈kBTL

Zg2D rð Þ Ω rð Þh i2πrdr: ð50Þ

Here, g2D(r) is the pair correlation of monomers in two dimensions.Balducci et al. [29] revealed that HI is local (screened) for a real chainconsidering g(r)r~r1/3, and HI is non-local for an ideal chain consideringg(r)r~r. In experiments of confined DNA, DNA usually is seen to followreal-chain behavior. Therefore, the long-range HI of DNA in both tube-like and slit-like channels can be considered as screened.

3.2.4. Diffusivity of confined DNA in slitsIn the classic de Gennes regime, the diffusivity can be derived using

the blob model. The chain is viewed as a series of blobs, and the frictionof each blob is proportional to the blob size. After ignoring the HI be-tween blobs, the diffusivity can be derived as

DdeGdiff ≈ c2D

2=3L−1=3p w−1=3D0

diff ; ð51Þ

Fig. 31. Probability distributions of the sizes of trefoil knots for different confining channelwidths. For all curves, the contour length isfixed as L=400Lp, and the chainwidth isfixedatw=0.4Lp. The inset shows the total probability of trefoil knots as a function of channelsize. The dashed line in the inset shows the probability of trefoil knots in bulk. Adaptedfrom Dai et al. [173] with permission.

where

D0dif f ≡ kBT= 6πηLð Þ; ð52Þ

and c2 is a prefactor.The above result can be reproduced by Eq. (49). Considering that HI

is screened, and defining a dimensionless pair correlation function

h rð Þ ≡ 4πr2g rð Þ; ð53Þ

Eq. (49) becomes

Ddiff≈kBTL

Z D=2

0Ω rð Þh ih rð Þdr: ð54Þ

With the dimensionless pair correlation inside a blob derived fromthe Flory scaling

h rð Þ ¼ c1r2=3L−1=3p w−1=3; ð55Þ

Fig. 33. Sliding motion of a localized knot along a DNA chain confined in a channel. Theknot appears a bright spot on fluorescence labelled DNA, and disappears at one end ofDNA. Adapted from Metzler et al. [163] with permission.

Fig. 34. The chain extension in a square channel as a function the channel size. In thesimulations, the contour length is L=160Lp. Adapted from Dai et al. [103] withpermission.

Fig. 36. Diagram of regimes for polymer solutions. The y-axis is the chain width or thetemperature. Adapted from Schaefer et al. [176] and Dai et al. [50] with permission


the diffusivity in Eq. (51) is reproduced. Eq. (54) can be adapted for slitconfinement with the replacement of D with H:

Ddiff≈kBTL

Z H=2

0Ω rð Þh ih rð Þdr: ð56Þ

Recently, Dai et al. [152] revisited the blob model for the diffusivityof confinedDNAbecause all experiments of DNA in slits observed a scal-ing Ddiff~Hβ with the apparent exponent β less than 2/3 predicted bythe blob model. Dai et al. proposed that the pair correlation in Eq. (55)

Fig. 35. Extension (a) and fluctuation (b) as a function of channel size in the simulations ofpolymers in cylindrical confinement using different chain widths. (c) and (d) are for thesimulations of polymers under stretching force. Adapted from Dai et al. [103] withpermission.

does not capture the rod-like property at the short length scale.Hence, the following pair correlation was proposed:

h rð Þ ¼ 2 r b Lp=2c1r2=3L

−1=3p w−1=3 r ≥ Lp=2

ð57Þ

With this modified pair correlation, the diffusivity becomes

Ddiff¼ DdeGdiff þ Ddiff2−Ddiff3; ð58Þ

where

Ddiff2≈2 ln Lp=a� �

D0dif f ; ð59Þ

Ddiff3 ¼ c2 Lp=w� �1=3D0

diff : ð60Þ

In Eq. (59), a is the hydrodynamic radius of the chain, which is need-ed to remove the singularity in the integral of Eq. (56). Eq. (59) approx-imately corresponds to the diffusivity of a randomly oriented rod withthe length Lp and the radius a. [153–155] Since (Ddiff2−Ddiff3) is inde-pendent of H, i.e. scaling as H0, adding it into Ddiff

deG~H2/3 leads to an ap-parent exponent less than 2/3. The apparent exponent approaches 2/3with the increasing H.

The diffusivity in Eq. (58) has been used to explain simulation [152](Fig. 23) and experimental results [29] (Fig. 24) of confined DNA. Thesimulation results correspond to the Kirkwooddiffusivities of chain con-formations sampled in Monte Carlo simulations [152]. The theoreticalpredictions agree with simulation and experimental results using theprefactors c1=2.8 and c2=1.68. Note that the prefactor c1, defined inEq. (57), is determined from the simulations results of the contourlength in blob Lblob versus the slit height. Only the prefactor c2 is usedto fit the theoretical prediction of diffusivity with the simulation result.

It is worth mentioning that the modification of the short-scale paircorrelation in Eq. (57) should also apply to the static properties, suchas the extension as a function of the channel size. The modificationhasminor effect on statistic properties, but a significant effect on the dy-namics properties because the Oseen tensorΩ(r)~ r−1 makes dynamicproperties much more sensitive on the short-scale properties.

Other explanations have been presented regarding the discrepancybetween blob theory and experiments in the scaling exponent of DNAdiffusivity in slits. Lin et al. suggested this discrepancy is caused by par-tial hydrodynamic screening of DNA in slits [156]. As shown by theabove calculation, the discrepancy is reconciled without consideringpartial hydrodynamic screening. Jendrejack et al. [107] observed thatthe scaling exponent of diffusivity is less than 2/3 in their simulation re-sults, and they attributed the discrepancy between their simulation re-sults and the classic blob theory to polymer-wall interactions, which are


not fully represented in the bead-spring model. However, as discussedby Muralidhar and Dorfman [157], the high-resolution touching beadmodel yields similar results, suggesting that the polymer-wall interac-tions are unlikely to be the major reason for the discrepancy.

Next, we review the results for the diffusivities in other regimes. Themodified blob theory [152] for diffusivity was developed before a clearunderstanding [50] of the extended de Gennes regime. Later,Muralidhar and Dorfman [157] systematically investigated the diffusiv-ity of polymers in square channels in different regimes by varying thechannel size and the relative chain width w/Lp. It was found that themodified blob theory correctly predicts the diffusivity in the extendedde Gennes regime using simulation results of thin chains. It was alsofound that the classic blob theory is sufficient to describe the diffusivityin the classic de Gennes regime using simulation results of thick chains.It is easy to see that for thick chainswithw/Lp and a/Lp approaching 1, thecorrection term (Ddiff2−Ddiff3) in themodified blob theory becomes van-ishingly small. For a flexible chain, the correction to the classic blob theo-ry as shown by Eq. (58) becomes unnecessary. It would be interesting tosimulate a sufficiently long and thin chain in the classic de Gennes regimeand examine whether the correction term (Ddiff2−Ddiff3) is needed.

In the Odijk regime, a deflection segment can be approximated as arod because it is much shorter than persistence length. For a rod, thefriction coefficient for themotion along its axis in the free space [158] is

ζrodjj ¼ 2πηlrodln lrod=að Þ ; ð61Þ

where lrod is the length of rod, and a is the hydrodynamic diameter. Thechain is viewed as (L/D) rods with length D, considering that HI for rNDis screened and the HI for rbD is assumed to be unaffected by channels.Then, the diffusivity in the Odijk regime is obtained as [92,159]

Dodijkdiff ≈ kBT

ln D=að Þ2πηL

: ð62Þ

The predicted diffusivity for the Odijk regime has been confirmed bysimulation results [157,159].

In the backfolded Odijk regime and the transition regime, Tree et al.[159] proposed that the number of segments in the screening volume isnot high enough for substantial HI, and hence the diffusivity follows theRouse-like scaling:

Dtrandiff � kBT

ηL: ð63Þ

The predicted diffusivity for the backfolded Odijk regime and thetransition regime was also confirmed by simulation results [157,159].It is intriguing that the extension in the backfolded Odijk regime andtransition regime is very sensitive to the channel size, while the diffusiv-ity is insensitive to the channel size [159].

3.2.5. Relaxation of confined DNAAs presented in the beginning of Section 3.2, the relaxation time de-

pends on the diffusivity and the spring constant in the force-extensionrelationship through τ≈kBT/(ksprDdiff). The effective spring constantsin various regimes have been shown in Eqs. (13), (20), and (26). Com-bining the diffusivities in Eq. (51), (58), and (62), the relaxation timesfor polymers confined in tube-like channel are

τodijk ≈2πηkBT

L2D2

Lp ln D=að Þ ð64Þ

in the Odijk regime,

τdeG ≈6πηkBT

L2=3p w2=3L2

D1=3 : ð65Þ

in the classic de Gennes regime, and

τex≈6πηkBT

L2Lp

D2=3L−1=3w−1=3 þ 2 ln Lp=a� �

−c2L1=3p w−1=3

h i ð66Þ

in the extended classic de Gennes regime. Eqs. (64) and (65) are identi-cal with the ones in the review paper by Reisner et al. [92], but Eq. (66)is different because the diffusivity is the extended de Gennes regime ismodified in Eq. (58).

In the extreme caseD→0, the relaxation time in Eq. (64) approacheszero because the reduction in fluctuation σodijk

2 ~D2 overwhelms the re-duction in the diffusivity Ddiff

odijk~ ln(D/ahydro). In the other extreme caseD/Lp→+∞, the relaxation time in Eq. (65) also approaches zero becausethe increase of diffusivity Ddiff~D2/3 overwhelms the increases in thefluctuation σ2~D1/3. Considering the zero relaxation time in both limitsof channel size, the finite relaxation time in the between, it is expectedthat there is non-monotonic dependence of relaxation time on thechannel size, which was observed in experiments [2]. Tree et al. [137]used simulation results of the diffusivity and spring constant of DNAto explain thenon-monotonic behavior and the results are in agreementwith the experimental data by Reisner et al. [2] as shown in Fig. 25. Be-cause no analytic expression exists for the diffusivity and the fluctuationin the crossover from strong to weak confinement, Tree et al. [137] ex-tracted those data from simulations, and used them in the calculationof relaxation time.

In the above paragraphs, we discuss the relaxation around the equi-librium size. The relaxation of highly-stretched DNA in nanoslits wasalso investigated by experiments [74] and simulations [73]. Two distinctrelaxation regimeswere observedwith different relaxation times. In thefirst relaxation regime, highly-stretched DNA is relaxed to a line ofblobs. In the second relaxation regime, the blobs are relaxed in two-dimension.

3.3. Topological properties of confined DNA

DNA can form knots in free space [160–162] and confinement [31,163]. The DNA knots in free space [162,164] and in nanochannels [31,163] have been directly observed in experiments. Simulation studieshave been performed to investigate the effects of confinement on knot-ting probability [82,89,104,165–167], the knot size [168,169], the knottype [83,88,170] and the knotting/unknotting kinetics [90,171].Orlandini and Micheletti and co-workers performed numerouspioneering simulation studies of knots. Two review papers by OrlandiniandWhittington [172], and Micheletti et al. [93] have covered the basicknowledge of knots - polymer physics, relevant simulation techniquesand simulation results of knots in free space and confinement. In thissection, we focus on describing the recent studies of DNA knots usinga theory proposed by Grosberg and Rabin [96] due to two reasons.First, as can be seen below, the topological constraint by knots is funda-mentally similar to the spatial confinement, which is the theme of thisreview. Second, this theory can be extended to understand the sizeand probability of knots in confined DNA.

Strictly speaking, a knot is only strictly defined on a closed loop (cir-cular chain). However, for linear chains, a tight knot localized in a smallportion of a linear chain can be easily identified visually and the applica-tion of chain closure schemes allow for it to be located algorithmically.The knots of linear chains are reviewed in the following sub-sections.

Recently, free energy analysis has been applied to study the knottingprobability. As pointed by Grosberg and Rabin [96], a polymer knotshares a similar free energy to confined DNA because the polymerchain inside the knot core is essentially confined by a virtual tube(Fig. 26). The free energy analysis of Grosberg and Rabin predictsknots on wormlike chains will spontaneously shrink to a tightenedsize. This prediction was subsequently validated by simulations of Daiet al. [173] Later, Dai et al. extended the Grosberg-Rabin theory to


semiflexible chains with EV interaction [173], flexible chains [174], andchains in confinement [84]. The relevant theories and simulation resultsare summarized in this section.

3.3.1. DNA knots in free spaceGrosberg and Rabin [96] proposed that the sub-chain in the knot

core is confined by a virtual tube formed by sub-chain itself (Fig. 26a).The size and shape of the virtual tube can be estimated through thetight knot formed by pulling both ends of a rope with diameter Dvirtual.The length of virtual tube is related to the diameter of virtual tube bya parameter:

p ≡ Lknot=Dvirtual; ð67Þ

The confinement of the virtual tube leads to a confinement free en-ergy Fconf≈LknotDvirtual

−2/3 Lp−1/3, using the Odijk scaling in Eq. (24), where

Lknot is the contour length in the knot core. It will be shown later thatthe confinement by the virtual tube is strong such that the Odijk scalingis applicable. In addition to the confinement free energy, the subchainwithin the knot has a bending energy contribution scaling asFbend≈LknotRknot

−2 , where Rknot is the size of knot. For a tight knot, thequantities Lknot, Rknot, and Dvirtual are proportional to each other, andhence, both Rknot and Dvirtual can be replaced by Lknot after ignoring nu-merical coefficients. Eventually, Grosberg and Rabin [96] obtained thefree energy for the knot formation

Fwlc ¼ k1 Lknot=Lp� �−1

þ k2 Lknot=Lp� �1=3

; ð68Þ

where k1 and k2 are prefactors that take account of all numerical coeffi-cients ignored in the derivation. The first and second terms in the aboveequation tend to swell and shrink the knot size, respectively. The com-petition of these terms leads to a local minimum of free energy, whichcorresponds to a metastable knot.

The Grosberg-Rabin theory was modified by Dai et al. [173] for realchainswithfinite thicknesses. For a chainwith aneffectivewidthwcon-fined in a tubewith a diameter Dvirtual, the effective diameter of the con-fining tube becomes (Dvirtual−w) due to the repulsion between thechain and tube walls. As a result, the confinement free energy becomesFconf≈Lknot(Dvirtual−w)−2/3Lp−1/3. Eventually, Eq. (68) becomes

Freal ¼ k1 Lknot=Lp� �−1

þ k2Lknot Lknot−pwð Þ−2=3L−1=3p : ð69Þ

It is easy to see that when w=0, Eq. (69) returns to Eq. (68).Fig. 27 shows simulation results [173] for the size distribution of tre-

foil knots inwormlike chainswithw=0. The peak value corresponds toa metastable knot with contour length of 12Lp. The radius of gyration ofthemetastable knot is only 1.3Lp, corresponding to a tight knot. The sizedistributionwas converted to the free energy through F=− log(Pknot).The fit of Eq. (68) to the free energy yields k1≈17.06 and k2≈1.86.

Fig. 28 shows simulation results [173] for themetastable knot size asa function of the chainwidth. Note that the effective chainwidth of DNAcan be controlled by the ionic strength of the buffer [97]. Themetastableknot becomes larger with the increasing chain width. The prediction ofEq. (69) was used to fit the simulation results, yielding p≈16.

3.3.2. DNA knots in nanochannelsDai et al. [84] further extended the Grosberg-Rabin theory to DNA

knots in nanochannels, based on the fact that the segments in knots ex-perience the different confinement free energy from the ones inunknotted portions. If the knotted segments experience less confine-ment free energy, then the knot formation should be favored in confine-ment in order to reduce the overall free energy. In addition, the knots ofdifferent sizes should experience different confinement free energy, andhence, the size distribution is expected to be reshaped by the

confinement. Following this idea, Dai et al. [84] wrote the free energycost of knot formation in a nanochannel as

Fchannelknot Lknotð Þ ¼ Fbulkknot Lknotð Þ þ Fexcess; ð70Þ

Fexcess Lknotð Þ ≡ Fwallknot Lknotð Þ−Fwall

unknot Lknotð Þ; ð71Þ

where Fknotwall (Lknot) corresponds to the free energy cost of confininga knot while the contour length of the knot is maintained as Lknot, andFunknotwall (Lknot) corresponds to the confinement free energy of anunknotted chain with contour length Lknot. The term Fknotbulk(Lknot) followsthe expression in Eq. (69).

The excess free energy is calculated as follows. The confinement freeenergy of unknotted segments Fwall

unknot(Lknot) was approximated byEq. (11):

Fwallunknot ¼ 5:0D−5=3w1=3L1=3p Lknot; ð72Þ

because Dai et al. focused on the confinement strength in the classicde Gennes regime. The confinement free energy of knotted segmentsFknotwall (Lknot) depends on the size of the knot relative to that of the chan-nel. When the knot size is much smaller than the channel size (“small”knot regime in Fig. 29a), the knot is weakly deformed and can be con-sidered as a ball with an effective diameter bknot. The ball size may beapproximated as bknot=αLknot, where α is a numerical coefficient lessthan 1. The confinement free energy of this ball was approximated asthe one for a bead on a flexible chain of identical balls [133]:

Fwallknot¼ βL5=3knot D−αLknotð Þ−5=3: ð73Þ

When the knot size is much larger than the channel size (“large”knot regime in Fig. 29b), different portions of the knot were consideredto be confined in sub-channels with channel size less than D, whichwasdiscussed by Nakajima and Sakaue [168]. In the “large” knot regime,knotted segments experience a larger confinement free energy penaltythan outside the knot. As a result, “large” knots are rare and are not con-sidered in the calculation of the metastable knot size in channels.

Fig. 30 shows the total knotting probability of DNA as a function ofthe slit heightwith Lp=50nmandw=2.5 nm. The total knotting prob-ability exhibits a non-monotonic dependence on the slit height. Suchnon-monotonic behavior is due to the competition of the confinementfree energy density in the knotted portion Eq. (73) and the unknottedportion Eq. (72).

Fig. 31 shows the size distributions of trefoil knots in channels fromsimulations. The peak locations, corresponding to the metastable knotsizes, are plotted in Fig. 32. The dependence of the metastable knotsize on the channel size can be explained by the theoretical predictionusing Eqs. (70–73) with numerical coefficients α=0.1 and β=0.02.

The knots in confinedDNAhave been observed in experiments [163]as shown in Fig. 33. The channel confinement stretches DNA, and hencea knot appears a bright spot diffusing DNA and disappears at one end ofDNA.

4. Comparison of confined polymers and polymer under otherconditions

Polymers in confinement are similar to polymers under tension andpolymers in the semidilute solution because the blob model can be ap-plied in all of these three cases. For polymers in confinement, the sepa-ration of classic/extended de Gennes regime occurs because the real-chain scalingwill be replaced by the ideal-chain behavior at short lengthscales when the EV interactions are weaker than the thermal energy.Similar separations of regimes also occur for polymers under tensionand polymer in semidilute solutions, which are presented below.


4.1. Polymers in confinement versus polymers under tension

It is well known that a polymer under tension shares the great sim-ilarity with the polymer in confinement because the force f applied tothe end of a polymer introduces a characteristic length ξ=kBT/f thatis similar to the channel size D [175]. Fig. 34 shows the comparison ofregimes of polymers in confinement versus polymer under tension[103]. In the classic Pincus regime, the scalings of extension L|| ~ f2/3

and fluctuation σ2~ f−1/3 are similar with the scaling L|| ~D−2/3 andσ2~D1/3 in the classic de Gennes regime. However, in the extendedPincus regime, the scaling of extension L||~f1 is different from the scal-ing L|| ~D−2/3 in the extended de Gennes regime. The fluctuation inthe extended Pincus regime σ2~f0 is still similar with the scalingσ2~D0 in the extended de Gennes regime. These scalings are confirmedby simulation results in Fig 23. The difference in scaling of extension be-tween polymers in confinement and under tension is caused by thedriving force to separate the blobs [103]. For polymers in confinement,the blobs are separated by excluded volume interactions. If the blob issmaller than the thermal blob, EV interactions are not strong enoughto separate the spherical blobs. For polymers under tension, the blobsare separated by tensile forces. The backfolding of blobs costs energyof kBT and hence is prohibited. In this case, even if the blob is smallerthan a thermal blob, the tensile force can still separate the blobs.

4.2. Polymers in confinement versus polymers in semidilute solution

Polymers in confinement also share great similarities with polymersin semidilute solutions because both cases can be explained by the blobmodel. Recently, Dai et al. revealed the similarity between the extendedde Gennes regime for a confined polymer and the semidilute marginalcondition for polymer solutions. The semidilute marginal conditionwas proposed and studied by Schaefer et al. [176] and Birshtein [177],which is shown in Fig. 36. Note that the y-axis in the diagram bySchaefer et al. is the normalized temperature (T−Tθ)/T, where Tθ isthe θ temperature. The effective chain width is determined by the effec-tive excluded volume interaction, which is determined by the tempera-ture. The conceptual picture behind the semidilute marginal conditionis as follows. Applying the classic blob model, a chain in the semidilutesolution is viewed as a string of blobs. As in the extended de Gennes re-gime, intra-blob conformations should follow ideal-chain scaling ratherthan real-chain scalingwhen the EV interactions inside a blob areweak-er than kBT. This situation was investigated and termed as semidilutemarginal condition by Schaefer et al. [176] The boundary for the mar-

ginal condition was determined as the critical concentration ~ρ≈wL−4k

of Kuhn segments. A new characteristic length ξ≈w−1/2c−1/2wasalso determined in the monomer−monomer pair-correlationg(r)~(1/r)exp(−r/ξ). The critical concentration in the marginal con-dition shares the same expression with the one in the extended deGennes regime c=(L/Lk)/(L||D2)≈wLk−4. The characteristic length inthe marginal condition is similar to the length of anisometric blob.

5. Summary and outlook

The competition of three interactions, bending (elastic entropy), ex-cluded volume, and confinement, leads to distinct chain behaviors invarious regimes of a confined polymer molecule. The competitions ofthree interactions can be also as the competition of three length scales:the persistence length Lp, the chain width w, and the channel size D. Inthe case of DNA, the chain width w is usually much less than Lp and D.Accordingly, the parameter space is often separated to D≪Lp, D≈Lp,and D≫Lp. For the two extreme cases, D≪Lp and D≫Lp, theoreticaltreatments are relatively simple, and classic models give predictionsthat are validated by simulations and experiments in the last decade.A notable recent development is a deeper understanding of the extend-ed de Gennes regime and the backfolded Odijk regime.

An important interaction underappreciated in this review is that ofelectrostatics [178]. Accordingly, the corresponding length scale, theDebye length, is underappreciated. In this review, long-range electro-static interactions are approximated by a short-range hardcore repul-sion with an effective chain width. The introduction of this new lengthscale, Debye length, should lead to more regimes in the diagramshown in Fig. 8. Experiments of stretching DNA under different ionicstrengths have revealed rich phenomenon due to the competition ofDebye length with other length scales [179,180]. In addition, DNA–DNA interactions can be complex and even attractive in some cases[121,181].

Compared to equilibrium properties of confined DNA, the theoriesfor of non-equilibrium properties of confined DNA, for example, thecoil-globule transition, are less developed. Several experiments discov-ered that DNA compaction is much easier and much faster in confine-ment. For example, the coil-globule transition or the compaction ofDNA in channels induced by neutral crowding agents or charged ligandshas been investigated bymany recent experiments [54,95,182] and sim-ulations [55,183], all of which reveal that compaction is greatly facilitat-ed by confinement. To understand the effect of confinement on thecompaction of DNA in confinement, explicit consideration of electro-static interactions should be required.

Exploring the knots in confined DNA by experiment is feasible fromthe technical point of view, and much of work can be done to deepenour understanding of knots aswell as the effect of confinement on poly-mer. It is fortunate that the experimental observation of a knot in DNAshould be easier in confinement than in bulk, because the channel-confinement stretches DNA and the knot appears as a bright spot anddiffuses along DNA until untied at ends [163]. Systematic experimentscould be performed to measure the diffusion coefficient of a knot as afunction of the channel size, the effective width and persistence lengthof DNA, and these results can be comparedwith simulations and theory.The size of knots can be also inferred from the total intensity in thebright spot.

Acknowledgment

This research was supported by the National Research FoundationSingapore through the Singapore MIT Alliance for Research andTechnology's research program in BioSystems and Micromechanics,the National Science Foundation CBET-1335938.

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The polymer physics of single DNA confined in nanochannels