1 Introduction

Physical properties of biomacromolecules and biopolymers in complex microenvironments are usually influenced by external conditions such as geometrical confinements and applied forces ^[1] . Polymers in geometrical confinements that are smaller than their unconfined molecular sizes are of great significance in fields from polymeric liquid crystals to biological structures including cytoskeleton,stress fibres,nucleosomes,and viruses ^{[1,2,3,4,5,6,7,8,9]} . However,since the governing equations of these problems are very complex in mathematics,most theoretical studies are just based on scaling discussions,among which,two well-known scaling regimes on nanotubeconfined polymer chains were given by de Gennes ^[3] and Odijk ^[4,9,10] . The de Genne regime is for the case that the confinement size is larger than the chain’s persistence length,whose corresponding scaling theory regards that the chain forms a series of spherical blobs that are comparable with the tube diameter. At the other extreme,the Odijk regime is for the case that the confinement size is much smaller than the chain’s persistence length. In this case,the confined chains fluctuate back and forth by the tube-wall,where a typical length scale called the Odijk deflection length is identified. For the intermediate confinements,Odijk ^[10] presented a scaling analysis on the statistics of long DNA chains confined in both nanochannels and nanoslits and pointed out that more regimes are needed besides the two extremes originally described by de Gennes and Odijk. This conclusion has been confirmed by Wang et al. ^[11] and most recently by Hsu and Binder ^[12] using Monte Carlo simulations. They found that there exist more transition regimes to link the de Genne and Odijk regimes. For the semiflexible polymer chain simultaneously subject to both geometrical and tensile constraints,studies on the quantitative force-extension relations of the chain are rare. As one of few exceptions,Wang and Gao ^[13] analytically derived the end-to-end extension of a semiflexible polymer chain under strong confinements of cirlular tubes in the Odijk regime by modeling the confinement effect as a quadratic potential. Based on the so-called Blob theory ^[3] in de Genne’s regime,Dai and Doyle ^[14] investigated the similarities and differences between effects of nanoslit constraint and tensile load on polymers.

In spite of the above progresses,our understanding on the mechanics of confined semiflexible polymer chains is still far from complete. According to this view,in this study,we intend to furnish a theoretical framework based on Odijk’s theory of confined polymer to analytically derive the force-extension relation of the semiflexible chains under various confinements including circular tubes,rectangular tubes and slits,and use Brownian dynamics (BD) simulations based on the generalized bead-rod (GBR) model ^{[2,13,15,16,17]} to verify the corresponding theoretical predictions. 2 Confinement vs. effective force

We consider a wormlike chain (WLC) with the persistence length p and the contour length L (L p) subject to a large tensile force ƒ_s on its two ends. We denote the position,the tangential vectors,and their vertical components along the chain’s contour as r,u,r⊥,and u⊥,respectively. Wang and Gao ^[13] have derived the following relation:

When the chain is at the same time confined in a nanochannel,we assume that the effect of constraint can be effectively equivalent to a quadratic potential ^[13] ,,where the potential strength Ξ depends on the geometrically constraining conditions. Following this assumption,Wang and Gao ^[13] have further derived By comparing Eqs. (1) and (2),we can obtain an effective force f c induced by the geometrical constraint

In the Odijk regime,the confinement free energy of the chain can be given by

where a constant is omitted and λ is the so-called Odijk deflection length depending on the size of the confinement. Note that where θ = arccos(u(0) · u(λ )).

Considering small undulation,we have

where according to Eq. (4). Thus,the behavior of a strongly confined polymer chain under stretching can be described by that of an unconfined chain subject to the effective force equal to In other words,the confining effect of a nanochannel to a semiflexible polymer chain might be able to be viewed as the chain in a free solution subject to a stretching force. The strength of the force depends on the size and the shape of the confinement. Eventually, we can obtain the force-extension relation of a long WLC confined in nanochannels ^[13,18] where b is the bond length ^[13,18] , is the normalized average distance between the two ends of the chain in the nanochannel.

Figures 2(a)-2(c) show the schematic plot of the polymer chain,respectively,confined to a slit of height d_s ,a rectangular tube with height d₂ and width d₁ ,and a capped circular tube with length l close to L and diameter d_c . Based on the WLC model and our previous studies, we can derive the effective stretching force due to the confinements

For the slit confinement,c = 1.4,and For the rectangular tube,c = 1.10 ^[19,20] ,and For the capped circular tube,c = 2.36. We note that,the pre-factor c appearing in Eqs. (8)-(10) is a universal dimensionless parameter ^[19,20] ,which depends on the forms of confinements,but does not depend on the persistence length p and the confinement sizes d_s,d₁,d₂ ,and d_c .

Numerical methods such as the BD simulation are viable to the above theoretical predictions,although efficiently realizing strong geometrical confinement can be challenging even for simulations. In this study,based on the powerful GBR model ^[2,15,16,17] ,BD simulations have been intensively performed for the chains confined in various nanochannels. In all the simulations,the chains are initially set in a straight configuration. Various confinements as shown in Figs. 1(a)-1(c) and constant tensile forces are then used during the chains’ relaxation. We record the normalized end-to-end distance of the chain along the z -axis at each time increment, and more than 10 different trajectories with different random seeds are averaged to obtain hz i.

From Figs. 2(a)-2(c),we can see that the theoretical predictions agree well with the BD simulations. Thus,we can conclude that the effect of geometrical confinements on the force-extension relation of a polymer can be represented by an effective force applied to an unconfined polymer chain.