Chemical potential formalism for polymer entropic forces

The entropic force is one of the most elusive interactions in macromolecules. It is clearly defined theoretically, but practically difficult to evaluate, thus restricting our knowledge of it often to a qualitative level. Here, we propose a formula for entropic force, f = f-o\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hskip0.3pt}^{\rm -{\hskip-4.3pt}o}$$\end{document} + T ln(α), for confined polymers and demonstrate its mathematical equivalence to the widely used chemical potential formula for solutions, μ = μ-o+RT̃ln(a)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\rm -{\hskip-4.3pt}o} +R\tilde T\,{\mathrm{ln}}(a)$$\end{document}. A systematic analysis based on this formalism clarifies the force magnitudes obtained in several recent experiments on polymers and granular chains and elucidates the common force scales for polymers studied in nanoscience and biological systems. This work provides a practical tool for instantaneously evaluating the entropic forces in polymer science and indicates the possibility of using a reference-based strategy to tackle general entropic problems beyond chemical solutions. Quantifying the entropic effects of confined polymers in biological environments or single molecule experiments is a challenging task. Using the same strategy as the chemical potential of solutions, the authors derive a simple entropic force formula, which largely reduces that difficulty and clarifies several recent experiments in confined polymers.

A wire threaded through a hole might be one of the simplest models one can imagine in physics, which, captures the core scenario in numerous polymer systems in laboratories, industrial applications, and real biological environments. This model has been studied intensively over decades, for polymer translocation and confined polymers in microfluidics, nanotechnology, and single-molecule experiments [1][2][3][4] . Although entropic effects are significant for systems scaled down to that size and were already formulated in the earlier theories in this field [5][6][7] , the estimation of their strength is challenging, because analytically or numerically solving partition functions is difficult 8,9 . Thus, qualitative judgments on entropic effects are more frequently reported than quantitative characterizations.
In contrast to these direct approaches, here we derive an entropic force formula for confined polymers and demonstrate its correspondence to the well-known chemical potential formula for solutions ( Fig. 1) 10 . The advantage of the latter formula is that one can calculate the chemical potential of a desired solution by summing an easily calculable correction term with a known reference value. In the same spirit, the convenience of the derived force formula is that it can be readily calculated based on the knowledge of the entropic force of a known reference system. The formalism is elucidated in several numerical experiments, including a polymer pulled by an optical tweezer into a twodimensional (2D) channel and a polymer straddling a 2D channel undergoing a tug-of-war. Different types of entropic forces in those systems are calculated by the Jarzynski equality (JE) 11 and a recursion formula (RF) for counting the exact configuration numbers. The highly coincident force magnitudes extracted from these two independent approaches justify the accuracy of the force formula. On the basis of this consistency, the analysis is extended to three-dimensional (3D) circular tubes and slits. The estimated force magnitudes are very close to the experimentally observed ones in 2D strips 12 , pillar arrays 13 , and 3D nanoslits 14 , and follow the same scaling behavior observed in granular chains 15 . More general studies on several polymers threaded through various natural and artificial pores unravel a common force range between femto-and piconewton and indicate a special ratio between the forces of different confinements. The proposed force formula reduces the complex evaluation of partition functions to a simple textbook-level calculation. This allows us to estimate the entropic effects on polymers algebraically by hand without resorting to elaborate computational efforts and perform a systematic analysis on the force scales for a broad class of confined-polymer systems.

Results
Configurational entropies. Let us consider a polymer partly confined in a channel. Suppose the polymer segment inside (outside) the channel is coarse-grained as a chain of basic unit (BU) C I (C O ), where C I and C O do not need to be the same (Fig. 2a). For instance, C I could be a rod in the Odijk regime and C O could be a Kuhn segment in a free space, or C I and C O could have the same shape but different numbers of monomer microstates inside them. The chain can be a non-self-avoiding chain (NSC) or a self-avoiding chain (SC). The channel is assumed to be narrow with respect to C I . If C I and C O are identical beads of size l in a bead-spring model (Fig. 2b, c), let n I (n O ) be the number of beads inside (outside) the channel and Ω n I Ω n O denote the configuration number of the chain segment of C I (C O ) inside (outside) the channel. Then the configurational entropy of the whole chain is S ¼ k B ln Ω n I Ω n O , with k B being the Boltzmann constant.
Entropic recoiling forces. The entropic recoiling force describes the tendency of a chain to escape from a channel (Fig. 2b). It is the negative of the change in the entropic free energy, ÀTS, per shifted distance out of the channel, whereT is the Kelvin temperature. If the distance is l, this force is given bỹ Taking the dimensionless temperature T T=T room with respect to the room temperatureT room ¼ 298 K, it yields a dimensionless entropic force f R f R l=ϵ, or with ϵ k BTroom . Here, is the excess configuration number caused by an added bead inside (outside) the channel. As Φ n O outside the channel is always larger than Φ n I À1 inside the channel, we have f R > 0. The positive sign indicates that the force points to the same direction as an outward-moving bead. If the channel is so narrow that only one straight configuration is allowed inside it, the force will degenerate to Let f R ′ be the force of a second chain of the same form as Eq. (1), with Φ n O , Φ n I , and l there replaced by Φ n O ′, Φ n I ′, and l′. For a Owing to a correspondence between the chemical potentials for solutions and the entropic forces for partly confined polymers, solving these two problems is mathematically equivalent. The chemical potential formula for a solution of activity a = rb is μ = μ À o + RT lnðaÞ, with the standard chemical potential μ À o , gas constant R, Kelvin temperatureT, activity coefficient r, and concentration b of the solution. With this expression one can calculate the μ of a desired solution by summing an easily calculable correction term RT lnðaÞ with a known reference value μ À o . The entropic force formula for a polymer partly confined in a channel is f = f À o + T ln(α), with a reference force f À o and a dimensionless temperature T. Here, the "activity" α = γβ depends on the channel width and stiffness, as well as the polymer persistence length and excluded volume. Therein, the "concentration" β is decided by the configuration number of the basic unit to coarse-grain the polymer, such as the commonly adopted beads, rods, or blobs, whereas the "activity coefficient" γ reflects the change in internal freedom in that basic unit when it enters or leaves the channel. If all the basic units along the polymer chain are identical in shape and internal freedom, such as those in a granular chain, then γ = 1 and α = β, which corresponds to the ideal case of dilute solutions with r = 1 and a = c in μ Generally, the bead sizes l for f R and l′ for f R ′ in Eq. (2) are not the same. Therefore, even when the values of f R and f R ′ of the two systems are identical, the real forcesf R ¼ f R ϵ=l andf R ′ ¼ f R ′ϵ=l′ may be different.
If C I and C O do not have identical shapes or internal freedom, such as those in Fig. 2a, letΦ n IΦ n O be the excess "microstate" number caused by an added BU C I inside (C O outside) the channel. The "microstates" here include the allowed configurations of C I and C O considered in Eq. (1) and the internal freedom inside each individual BU, which are generally hybridized and cannot be decoupled. When a chain segment of length l moves into (out of) a channel, suppose n i (n o ) BUs of C I (C O ) appear inside (outside) it. Then, the total excess microstate number of n i (n o ) added BUs would beΦ n I À1 n iΦ n O n o , for which the entropic recoiling force in Eq. (1) is generalized to is the excess configuration of C I (C O ) defined in Eq. (1), while g n I and g n O as well as Ψ n I g n I þ1 =g n I and Ψ n O g n O þ1 =g n O are correction terms reflecting the difference between the shapes, or internal freedoms, of C I and C O . Notice that n O and n o as well as n I and n i in Eq. (3) have different meanings.
Two forces of the same form as Eq.
That is,α R is related to the α R in Eq. (2) byα R ¼ γα R with γ G α R =α R . Therefore, for general C I and C O , Eq. (2) is extended to the form with β = α R . Specifically, if the microstates of a polymer can be decomposed as those inside and outside C I (C O ), then g n I g n O is exactly the number of microstates inside C I (C O ). This decomposition is similar to that in the de Gennes blob, where the Flory statistics inside a blob is independent of the chain properties outside the blob. If C I and C O are identical, then n i ¼ n o ¼ g n I ¼ g n O ¼ 1 for all n I and n O . In this case, γ = 1 and α is reduced to the β = α R of an "ideal" confined chain in Eq. (2). Therefore, α, β, and γ play the same role as the activity a = rc, the concentration c, and the activity coefficient r, respectively, in the chemical potential formula μ = μ À o + T ln(a) of a non-ideal solution. Notice that the mathematical equivalence between Eq. (5) and the chemical potential formula for solutions does not mean the former is aimed at understanding polymer solutions.
Entropic drift forces. The above formalism also applies to other entropic forces. For polymer tug-of-war in Fig. 2c, the chain straddling the channel is unstable and will gradually move out of a channel either rightward or leftward. The entropic drift force represents the tendency of the drift of the whole chain from the right to the left space. Here, n L (n R ) denotes the number of beads in the left (right) space and Φ n L Ω n L þ1 =Ω n L Φ n R Ω n R þ1 =Ω n R stands for the excess configuration number caused by an added bead in the left (right) space. Φ n L and Φ n R are the same function as Φ n O in Eq. (1). Since the configuration of an outside segment with fewer beads is more constrained by the halfplane boundary, such segment cannot have a larger excess number, which implies Φ n−1 ≤ Φ n for n ∈ {n O , n L , n R } (see a discrete example in Fig. 3). Hence, Fig The entropic recoiling force f R for a chain escaping from a two-dimensional (2D) channel is counterbalanced by the drag force f drag of an optical tweezer. n I (n O ) is the number of basic unit C I inside (C O outside) the channel. c The entropic drift force f D and tension force f T for polymer tug-of-war. n L (n R ) is the number of basic unit C O in the left (right) space. d A bead (blue circle) of size l is in a channel (black) of width w = l + d, where the lateral free space d is the distance the bead can freely move without touching the boundary. The potential well (blue curve) above the channel describes the bead-wall interaction versus the location of the bead center. The well boundary V(η) = κη 2 /2 is a function of the penetration depth η and stiffness κ. The probability density of finding the bead in the lateral direction of the channel (green area) has the same area as a uniform probability density of an effective width d + d e (magenta rectangle). The bead confined by the blue soft potential is analogous to being restricted to a hard potential of an effective width d + d e which the chain tends to move leftward (rightward). For n L = n R − 1, one has Φ n L ¼ Φ n R À1 and f D = 0, for which the chain movement does not have a biased direction. The drift force in Eq. (6) can be decomposed as a difference f D ¼ f R;ðn I ;n L Þ À f R;ðn I ;n R À1Þ between a leftward recoiling force, f R;ðn I ;n L Þ , and a rightward recoiling force, being a generalized expression of Eq. (1). Interestingly, the statistical forces f D and f R based on counting configurations also comply with the above subtraction rule as normal non-statistical forces in mechanics. Two drift forces of the same form as Eq. (6) are related as Eq. (2) with α R there replaced by α D ≡ α l /α r , where α l Φ n L =Φ n L ′ and α r Φ n R À1 =Φ n R À1 ′ are the same function as the α o below Eq. (2).
Entropic tension forces. The entropic tension force in Fig. 2c characterizes the tendency of polymer stretching inside a channel.
The simplest version of that force describes the tendency of moving one bead from the channel into the left or the right space, Since Φ n L ; Φ n R >Φ n I À1 , f T is always positive, indicative of a spontaneous trend of stretching, rather than compression, in that system. This trend will eventually be counterbalanced by the increased free energy during stretching caused by other mechanisms, such as bead deformation. Two tension forces of the same form as Eq. (7) can be related as Eq. (2), with α R there replaced by and l = l′ is not required, as in Eq. (2). For large n L and n R , Φ n L , Φ n R , Φ n L ′ , and Φ n R ′ are approximately the same as Φ n O at large n O , It is related to another tension force of the same form again by Eq. (5) with a slightly different α. For those C I and C O , the drift force f D in Eq. (6) does not need to be generalized, because C I is absent in that equation and C O in the left and right spaces are identical. The upper and lower limits of the above forces for NSCs and SCs in various confinements are derived in Supplementary Note 1 and summarized in Table 1), which serve as good candidates for the values of f À o in Eq. (5).
Optical tweezer experiments. To illustrate the above formulas and the analytical results in Table 1, let us carry out the following numerical experiments, with the Hamiltonian described in the Methods section. First, suppose an NSC is pulled into a 2D channel of width w by the drag force f drag of an optical tweezer, which moves slowly rightward with a constant speed (Fig. 2b, d).
The work done by f drag is converted to the entropic recoiling force f ðNSCÞ R via the JE (Methods section), as depicted by the magenta dashed curve in Fig. 3a. This curve shows that for n O > 10, f ðNSCÞ R has almost saturated at its maximum value f ðNSCÞ R on a plateau (blue solid line), which is independent of the tail length n O l outside the channel. In contrast to other studies in experiments 13 and simulations 16,17 , this tail length independent force is, to our   knowledge, first time determined by the JE. For n O < 10, the curve falls monotonically and drastically with a decreasing n O to its minimum. The sharp kink around n O = 5 reveals that after most beads have been pulled into the channel, the rest beads in the tail will surrender and give up to resist, which behaves like a first order phase transition 16 . In Fig. 3a, the maximum force f ðNSCÞ R is also calculated by the RF (blue dotted line) (Methods section), which is slightly larger than the magenta dashed curve calculated via the JE.
To inspect the cause of this slight difference, we raise the temperature from T = 1 (T ¼ 298 K) up to a physically extreme value T = 8 (T ¼ 2384 K) in Fig. 3b. At that T, the f ðNSCÞ R evaluated by the RF (blue dotted line) has deviated 10% from that calculated by the JE (blue circles). However, taking into account the penetration effect by enlarging the channel width w = l + d in the RF approach to w e = l + d + d e with d e being the mean penetration depth in Eq. (8) in the Methods section, the blue dotted line is reduced to the blue solid line in the same plot. The latter passes precisely through all blue circles evaluated by the JE, which justifies the crucial role of the penetration effect and the accuracy of formula (8). Subtracting this effect, the plateau of the magenta dashed curve in Fig. 3a is shifted to the blue dotted line, which is rather close to the magnitude f ðNSCÞ R ¼ T lnð2πl=dÞ predicted in Table 1. Although that analytical magnitude is derived under the strict assumption of n O ) 1, the simulation in Fig. 3a shows its correctness even for n O > 10.
If the chain is an SC, the maximum values f ðSCÞ R of the recoiling force f ðSCÞ R of that chain are calculated by the JE at different T and depicted by the red squares in Fig. 3b. To determine those values from an NSC by Eq. (2), one needs to evaluate α o , which is the α o at n O ) 1. For a 2D half-lattice, our Monte Carlo simulation (Fig. 3c) shows that α o has almost saturated at its minimum α o % 0:66 after n O = 10, which is slightly smaller than that, b α R % 0:78, observed in a 3D half-lattice 18 Fig. 3b is converted to f ðSCÞ R (red dashed line), which is the RF calculated force for the SC without the penetration effect. Taking into account that effect by enlarging d to d + d e , the red dashed line is further shifted down to the red dash-dotted line. The latter is very close to the JE calculated red squares, which confirms the specific version of formula (5) written in Fig. 3b.
In Fig. 4a, the maximum recoiling force f ðNSCÞ R for a channel of width w = l + d at different d is calculated by the RF and depicted by the blue solid curve. It deviates from the force evaluated by the JE when the channel has stiffness κ ¼ 10 3 ϵ (diamonds) and 10 4 ϵ (triangles). However, after considering the penetration effect by replacing d by d + d e as before for κ=ϵ ¼ 10 3 κ=ϵ ¼ 10 4 ð Þ , the blue solid curve is shifted to the red dash-dotted (dashed) curve, which precisely passes through the diamonds (triangles) and again justifies the significance of formula (8). All force magnitudes on the blue solid curve agree very well with the analytical results in Table 1, which validates those results at least up to d = 0.6l, far beyond the ideal condition d ( l. In Fig. 4b, the stiffness dependent f ðNSCÞ R calculated by the JE at two different T (squares and circles) coincide precisely with those evaluated by the RF (dashed and solid curves). Both curves show a kink around κ=ϵ ¼ 10 3 , indicative of a separation between the stiffness effect dominant regime κ=ϵ<10 3 ð Þ and the width effect dominant regime κ=ϵ>10 3 ð Þfor the force. A similar stiffness analysis on a linear channel potential can be found in ref. 19 . In Fig. 4c, the potentials of three different stiffnesses are plotted. The force magnitudes calculated in Fig. 4a, b confirm the specific version of formula (5) written in Fig. 4a.
Tug-of-war experiments. In Fig. 5a, the drift forces f ðNSCÞ D of an NSC of N′ beads straddling a channel are calculated by the RF, where the translocation ratio ξ ≡ n L /N, with N = N′ − n I , is the fraction of the chain segment in the left space. The left (right) ends of the force curves of different N approach their minimum, −T ln 2, (maximum, T ln 2,) as derived in Table 1. These force curves are insensitive to the penetration depth and channel size, in consistent with the absence of Ω n I and Φ n I in Eq. (6). Integrating these curves leads to the free energy profiles F(ξ) in Fig. 5b. The barriers of F(ξ) can be used to calculate Krammers' escape rate for polymer translocation, if the process is quasiequilibrium and ergodicity is preserved 20 . The curves in Fig. 5b will converge to an invariant shape at large N. In Fig. 5c, the tension forces f ðNSCÞ T of different N start from some initial values close to T ln(3πl/d) at ξ = 0, increase monotonically to their maxima below T ln(4πl/d), and then fall back to their initial values when ξ = 1. For larger (smaller) N, the maxima (minima) of the forces will approach T ln(4πl/d) (T ln(2πl/d)). All these numerical observations agree with the analytical bounds of f ðNSCÞ T predicted in Table 1 and Supplementary Note 1. Let Δ n be the difference between the entropic free energies of a chain at n L = n and N/2, where the latter is the position of the maximum of the F(ξ) in Fig. 5b. The value of Δ n reveals the free energy barrier for a chain translocation starting from n L = n. According to a scaling argument, the entropic free energy of a chain of n L = n is F n =k BT = ð1 À rÞ lnðnÞ þ ð1 À rÞ lnðN À nÞ when n and N À n ) 1 21,22 , where r = 1/2 for NSCs. It implies Δ n = F N=2 À F n = k BT À Á = ð1=2Þln ðN=2Þ 2 =ðnðN À nÞÞ Â Ã for an NSC. In Fig. 5d, the Δ n calculated by that scaling argument are found to overestimate those measured from the F(ξ) in Fig. 5b. The discrepancy is especially significant in Δ 1 between the solid curve and circles. This is ascribed to the overestimated difficulty of pulling the first few beads into the left space by the scaling formula. It is in echo with the finding in Fig. 3a that f ðNSCÞ R is weaker than expected for chains with few beads outside the channel.
. Despite that discrepancy, these two forces are of the same order.
More generally, if a single-stranded DNA (ssDNA) of l p ≈ 0.75 nm 23 , partly confined in a 2D nanochannel or a 3D circular nanotube of width 1.6 nm and κ = ∞ pN nm −1 , is modeled by an SC comprising beads of three nucleotide bases 24 , one obtains l = l K ≈ 1.5 nm and d ≈ 0.1 nm. This yieldsf ðSCÞ R % 11:32 pN for the nanochannel and 21.76 pN for the nanotube (Supplementary Note 2), which are even larger than the drag force, 5~6 pN, of a myosin. The increase in the force magnitude from dsDNA to ssDNA is a consequence of the shortened Kuhn length and the narrowed channel width. Owing to the physical limits on these two lengths, 20 pN seems to be close to the maximum entropic force one can observe in nature for a recoiling polymer. For the examples of soft channels, consider a 120-nt ssDNA oligomer 25 threaded through an α-hemolysin, which is 10 nm long and has an inner size varying between 1.4 and 4.1 nm 1 . If this system is modeled by an NSC of 40 Kuhn segments of l ≈ 1.5 nm 24 and threaded through a circular tube of the same width as l, with a of a non-self-avoiding chain of N = 4, 20, and 100 are calculated by the recursion formula at different ξ, and fitted by the black dash-dotted, red dashed, and blue solid curves, respectively. When the chain is pulled leftward to increase its n L starting from n L = 0, it will be resisted by a rightward force, f ðNSCÞ D ðξÞ<0, when ξ < 0.5, but assisted by a leftward force, f ðNSCÞ D ðξÞ>0, when ξ > 0.5. b The free energy profiles F(ξ) are integrated from the curves in a, where the two ends of those profiles are set to zero. The integration is carried out along the real length n L l, and then reexpressed as a function of ξ. Therefore, F(ξ) rises with N, although the areas bounded by the curves f ðNSCÞ D ðξÞ in Fig. 5a decrease with N. c The tension forces f ðNSCÞ T calculated by the recursion formula are fitted by the black dash-dotted, red dashed, and blue solid curves for N = 4, 20, and 100, respectively. d The free-energy barrier Δ n measured from the energy profiles in Fig. 5b gives the blue circles, red squares, and black triangles for n = 1, 2, and 3, respectively, where n I ) 1. The Δ n calculated by the scaling formula in the main text are the blue solid, red dashed, and black dash-dotted curves, respectively hypothesized effective stiffness κ = 1~10 2 pN nm −1 , then the maximum tension force isf T % 1:47 $ 14:12 pN (Supplementary Note 2). This analysis provides an instantaneous check for the consistency between the hypothesized d, κ, and forces.

Discussion
The experimental measurement of polymer entropic forces is challenging. The currently reported "measured" force magnitudes are all indirectly inferred from certain force-velocity or forceextension relations. The observed velocity and extension can be affected by several factors, including interfacial friction, Debye length, osmotic force, dye and salt concentrations, and nature of solvent. Even for the more fundamental polymer characteristics, such as persistence length or width of DNA and RNA, the experimentally measured values are often not unique. Thus, some experimentally measured entropic force was considered an "effective" one 13 . With these technical difficulties and system varieties, a coincidence close to 100% between a theoretically predicted force and an experimentally measured one could be accidental, when these forces are as small as a few piconewtons. However, if coincidences of the same order occur concurrently in several systems, as in the above-mentioned examples, the actual force magnitudes are probably not much different from the experimentally and theoretically determined ones.
The formalism proposed in this work overcomes the problem of directly solving partition functions and provides a less-timeconsuming reference-based algorithm to assess entropic contributions. This approach is especially simple when a polymer system can be modeled by an ideal confined chain with γ = 1. Beyond that regime, such as in the de Gennes or Odijk regime, the forces obtained from Eqs. (3) and (5) are as complex as those extracted by other half-analytical approaches (Supplementary Note 3). If other specific interactions, such as the angular and dihedral force-fields, are not coarse-grained in the BUs of a chain model, their entropic contribution has to be derived case by case, as reflected in the widened channel width in Eq. (8), or empirically measured, as the α o in Fig. 3c. Nevertheless, the forces of a nearby system with γ = 1, calculated by Eq. (5), can readily provide an upper or a lower bound for the desired systems with γ ≠ 1, as the pillar array discussed above. To sharpen these force bounds, the current formalism further serves as a proper framework, for instance, for selecting more sophisticated UBs C I and C O or for performing perturbation approaches on Eq. (5) around γ = 1.

Methods
Hamiltonian. For a chain of N identical beads in the bead-spring model in Fig. 2b, let m and l be the mass and the size of a bead, respectively, whereas x i and _ x i be the position and the velocity of the ith bead, respectively. The Hamiltonian of the chain is , the excluded volume interaction between non-consecutive beads , and the interaction between the last bead and the optical tweezer E O = k O |x N − z| 2 /2. Here, the natural length of the spring between two consecutive beads is the same as the bead size l. The value σ ij = 1 for |x i − x j | ≤ l and 0 for |x i − x j | > l. The position z denotes the location of the optical tweezer. The potential Vðη i Þ ¼ κη 2 i =2 when the ith bead penetrates a distance η i into a channel wall (Fig. 2d). Therein, k C , k V , k O , and κ are the force constants in E C , E V , E O , and E W , respectively. In this system, there is no facial friction and the diffusion coefficient is position independent. For an SC, the bending angle formed by three consecutive beads lies within [0, 2π/3].
Penetration effects. Since the channel is narrow, the slight lateral motion of a bead is less affected by its motion along the channel axis. Therefore, the probability density of finding a bead at some lateral position y follows the Boltzmann distribution e ÀUðyÞ=k BT , which is depicted by the green area in the upper part of Fig. 2d.
Here, the potential well U(y) consists of a zero-potential area flanked by two walls of the form of the above V(η). If that distribution has the same area as an effective uniform distribution (magenta rectangle) of the same height h and of a larger width d + d e , the equality in area implies 2 Þ . In analogy, for a 3D circular tube, the equality in volume leads to R 1 0 2πðη þ d=2Þh exp Àκη 2 = 2k BT À Á À Á dη + hðd=2Þ 2 π = hðd=2 þ d e =2Þ 2 π. Therein, d e is the mean penetration depth into two strip walls in Fig. 2b, c or into the circular tube wall, which is given by for 2D strips; ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi d 2 þ B p À d; for 3D circular tubes; ( ð8Þ with ϵ defined in Eq. (1) and B 8ϵT=κ þ 2d ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2πϵT=κ p . Although the above soft potential U can generally be interpreted as a soft channel or a soft bead, the latter requires a more subtle model with a varying bead shape during bead-wall collision, which is not considered here.
Langevin dynamics. With the above Hamiltonian H, the ith bead follows a Langevin equation m€ where ∇ x i H is the derivative of H with respect to x i , Ξ denotes the friction coefficient, and ζ i stands for a random force 24 . That force is Gaussian and white and satisfies the fluctuation-dissipation theorem, with 〈ζ i (t)〉 = 0 and ζ i ðtÞζ T j ðt′Þ D E = 4Ξk BT δðt À t′Þδði À jÞ, where the superscript T represents the transpose. Since the strength of the above random force changes with temperature, the diffusion of beads is temperature dependent as well. To study general real systems, the above equation can be reduced to a dimensionless one, ð Þ ζ in units of some convenient mass M and time τ 24 . However, to test the consistency between the force formula (5), the JE and RF approaches, and the analytical results in Table 1, it is sufficient to consider some simple parameter values for the original Langevin equation. In this study, we take ðl; m; Ξ; k C ; k V ; k O ; κ=ϵÞ = ð1; 1; 1; 10 4 ; 10 4 ; 10 4 ; 0 $ 10 4 Þ, where units are not specified.
Jarzynski equality. LetW be the work done by f drag to pull a bead of length l into the channel in Fig. 2b. Owing to fluctuations, theW in different pulling realizations will be different, which form a work distribution. According to the JE 11 , the free energy change, ΔF, per shifted l is related toW by e ÀΔF=k BT ¼ e ÀW=k BT D E , where the ensemble mean 〈⋅〉 is averaged over the whole work distribution. Therefore, the mean force to counterbalance f drag is the free energy change per shifted bead , which has a dimensionless version f ¼f l=ϵ ¼ T ln e ÀW=T À Á , with the dimensionless work W ¼W=ϵ, where ϵ is defined in Eq. (1). The f measured in our numerical recoiling experiments should be the f ðNSCÞ R derived in Eq. (1) if the change of internal energy is negligible during pulling, which occurs when the channel has a hard-wall or its soft wall is replaced by an effective hard-wall.
Recursion formulas. A recursion formula can be used to count the configuration number of a chain of n beads based on that of a chain of n − 1 beads. For a onedimensional NSC consisting of n rods of unit length, let N n x be the configuration number when one end of the chain is anchored to the origin and the other end is located at x between −n and + n, where x is an integer. It follows a simple recursion relation N n x ¼ N nÀ1 xÀ1 þ N nÀ1 xþ1 for x = −n, −n + 1, …, n − 1, n, where N n x ¼ 0 for |x| > n. The solutions of this equation are the binomial coefficients N n x ¼ C n ðnþxÞ=2 when x = −n, −n + 2, …, n − 2, n and N n x ¼ 0 elsewhere. With those N n x , the total number of the chain configurations of n rods is Ω n ¼ P n x¼Àn N n x ¼ 2 n and the corresponding configurational entropy is S = k B lnΩ n = nk B ln 2. For a chain of n beads of size l in a high-dimensional off-lattice space, the above RF can be generalized to N n r ¼ R N nÀ1 r′ δ r′ À r j jÀl ð Þ dr′, for which the total number of the chain configurations, Ω n ¼ R N n r dr, is an integration over all end-to-end displacement vectors r of a chain of n beads. To count the configurations of a macrostate of a chain, specified by some (n I , n O ), in Fig. 2b, first calculate the configuration number Ω n I Ω n O of a chain segment of n I (n O ) beads inside (outside) the channel when one segment end is anchored to a point on the crosssection of the channel opening. Summing up the product Ω n I Ω n O over all different points on that cross-section yields the total configuration number needed for calculating the entropy of the whole chain. Notice that although the RF is able to count the chain configurations within a channel, it cannot reflect the stiffness effect of the channel, unless we use Eq. (8) to replace a soft channel by an effective wider hard channel.

Data availability
All data generated or analyzed during this study are included in this published article (and its Supplementary Information file).