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https://doi.org/10.1038/s42005-019-0118-8 OPEN Chemical potential formalism for polymer entropic forces

Hong-Qing Xie1 & Cheng-Hung Chang1 1234567890():,; 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 = fÀo + T ln(α), for confined polymers and demonstrate its mathematical equivalence to the widely used che- mical potential formula for solutions, μ = μÀo þ RT~ lnðaÞ. 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.

1 Institute of Physics, National Chiao Tung University, Hsinchu 300, Taiwan. Correspondence and requests for materials should be addressed to C.-H.C. (email: [email protected])

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wire threaded through a hole might be one of the sim- these two independent approaches justify the accuracy of the Aplest models one can imagine in physics, which, captures force formula. On the basis of this consistency, the analysis is the core scenario in numerous polymer systems in extended to three-dimensional (3D) circular tubes and slits. The laboratories, industrial applications, and real biological environ- estimated force magnitudes are very close to the experimentally ments. This model has been studied intensively over decades, for observed ones in 2D strips12, pillar arrays13, and 3D nanoslits14, polymer translocation and confined polymers in microfluidics, and follow the same scaling behavior observed in granular nanotechnology, and single-molecule experiments1–4. Although chains15. More general studies on several polymers threaded entropic effects are significant for systems scaled down to that size through various natural and artificial pores unravel a common and were already formulated in the earlier theories in this field5–7, force range between femto- and piconewton and indicate a special the estimation of their strength is challenging, because analyti- ratio between the forces of different confinements. The proposed cally or numerically solving partition functions is difficult8,9. force formula reduces the complex evaluation of partition func- Thus, qualitative judgments on entropic effects are more fre- tions to a simple textbook-level calculation. This allows us to quently reported than quantitative characterizations. estimate the entropic effects on polymers algebraically by hand In contrast to these direct approaches, here we derive an without resorting to elaborate computational efforts and perform entropic force formula for confined polymers and demonstrate its a systematic analysis on the force scales for a broad class of correspondence to the well-known chemical potential formula for confined-polymer systems. solutions (Fig. 1)10. The advantage of the latter formula is that one can calculate the chemical potential of a desired solution by Results summing an easily calculable correction term with a known Configurational entropies. Let us consider a polymer partly reference value. In the same spirit, the convenience of the derived confined in a channel. Suppose the polymer segment inside force formula is that it can be readily calculated based on the (outside) the channel is coarse-grained as a chain of basic unit knowledge of the entropic force of a known reference system. The (BU) CI (CO), where CI and CO do not need to be the same formalism is elucidated in several numerical experiments, (Fig. 2a). For instance, CI could be a rod in the Odijk regime and including a polymer pulled by an optical tweezer into a two- CO could be a Kuhn segment in a free space, or CI and CO could dimensional (2D) channel and a polymer straddling a 2D channel have the same shape but different numbers of monomer micro- undergoing a tug-of-war. Different types of entropic forces in states inside them. The chain can be a non-self-avoiding chain those systems are calculated by the Jarzynski equality (JE)11 and a (NSC) or a self-avoiding chain (SC). The channel is assumed to fi recursion formula (RF) for counting the exact con guration be narrow with respect to CI.IfCI and CO are identical beads of numbers. The highly coincident force magnitudes extracted from size l in a bead-spring model (Fig. 2b, c), let nI (nO) be the number of beads inside (outside) the channel and Ω Ω nI nO fi denote the con guration number of the chain segment of CI (CO)     = 1 f = f + T ln ( ), = inside (outside) the channel. Then the configurational entropy of Granular chain ¼ Ω Ω the whole chain is S kB ln n n , with kB being the Boltz- f f  I O Inter-unit factor mann constant. Intra-unit factor  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  =  + RT˜ ln(a), a = r c r = 1 ~ Dilute solution shifted distance out of the channel, where T is the Kelvin tem-   Concentration c perature.ÀÁ If thehi distance isl, this force is given by ~ ~ f ¼ k T=l ln Ω À Ω þ = Ω Ω . Taking the dimen- Activity coefficient r R B nI 1 nO 1 nI nO  ~=~ sionless temperature T T Troom with respect to the room ~ ¼ temperature Troom 298 K, it yields a dimensionless entropic Fig. 1 A correspondence between polymer physics and physical chemistry.  ~ =ϵ force fR fRl ,or Owing to a correspondence between the chemical potentials for solutions and the entropic forces for partly confined polymers, solving these two ! ! Ω À Ω þ Φ problems is mathematically equivalent. The chemical potential formula for a ¼ nI 1 nO 1 ¼ nO ; ð Þ Ào ~ fR T ln T ln 1 solution of activity a = rb is μ = μ + RT lnðaÞ, with the standard chemical Ω Ω Φ À nI nO nI 1 potential μÀo, gas constant R, Kelvin temperature T~, activity coefficient r, and b concentration of the solution. With this expression one can calculate the  μ of a desired solution by summing an easily calculable correction term ~ ~ Ào with ϵ  k T . Here, Φ  Ω þ =Ω Φ  Ω þ =Ω RT lnðaÞ with a known reference value μ . The entropic force formula for a B room nI nI 1 nI nO nO 1 nO polymer partly confined in a channel is f = fÀo + T ln(α), with a reference is the excess configuration number caused by an added bead force fÀo and a dimensionless temperature T. Here, the “activity” α = γβ inside (outside) the channel. As Φ outside the channel is always nO depends on the channel width and stiffness, as well as the polymer Φ larger than n À1 inside the channel, we have fR > 0. The positive “ ” β I persistence length and excluded volume. Therein, the concentration is sign indicates that the force points to the same direction as an fi decided by the con guration number of the basic unit to coarse-grain the outward-moving bead. If the channel is so narrow that only one polymer, such as the commonly adopted beads, rods, or blobs, whereas the straight configuration is allowed inside it, the force will degen- “activity coefficient” γ reflects the change in internal freedom in that basic erate to f ¼ T ln Φ , because Ω À ¼ Ω ¼ 1 implies unit when it enters or leaves the channel. If all the basic units along the R nO nI 1 nI Φ ¼ polymer chain are identical in shape and internal freedom, such as those in n À1 1. γ = α = β I ′ a granular chain, then 1 and , which corresponds to the ideal case Let fR be the force of a second chain of the same form as of dilute solutions with r = 1 and a = c in μ Eq. (1), with Φ , Φ , and l there replaced by Φ′, Φ′, and l′. For a nO nI nO nI

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 n n Φ~ i Φ~ o a Φ 2C 3C Φ (no) added BUs would be À , for which the nO O I nI nI 1 nO entropic recoiling force in Eq. (1) is generalized to 0 1 ~ no Ψn Ψ Φ O C CI nI B nO C O f ¼ T ln@A; ð3Þ R ~ ni Φ À nI 1 b Optical d tweezer  d + de Φ~  Ω = Ω = Ψ Φ Φ~  with n gn þ1 n þ1 gn n n n and n gn þ1 nO nI I I I I I I I O O Ω þ = g Ω = Ψ Φ . Therein, Φ Φ is the excess nO 1 nO nO nO nO nI nO configuration of C (C )defined in Eq. (1), while g and g as h I O nI nO well as Ψ  g þ =g and Ψ  g þ =g are correction terms f f nI nI 1 nI nO nO 1 nO R drag fl d re ecting the difference between the shapes, or internal freedoms, of CI and CO. Notice that nO and no as well as nI and ni in Eq. (3) have different meanings. c fD n R Two forces of the same form as Eq. (3) are related as Eq. (2) α α~ ¼ α n with R there replaced by R G R where n I L n n n n l g′ À i g o Φ′ À i Φ o  nI 1 nO α  nI 1 nO : ð Þ G n n and R n n 4 g À i g′ o Φ À i Φ′ o nI 1 nO nI 1 nO l + d fT α~ α α~ ¼ γα That is, R is related to the R in Eq. (2)by R R with γ  α =α G R R. Therefore, for general CI and CO, Eq. (2) is extended Fig. 2 Three kinds of entropic forces for partly confined polymers. a A chain to the form fi C partly con ned by a channel, for which three basic units of I (red) appear À f ¼ f o þ T lnðÞα ; with α ¼ γβ; ð5Þ inside the channel when two basic units of CO (blue) move into the channel. Φn Φn is the excess configuration number inside (outside) the channel I O β = α fi in Eq. (1), whereas Ψn and Ψn reflect the changes of internal freedoms in with R. Speci cally, if the microstates of a polymer can be I O basic units in Eq. (3). b The entropic recoiling force f for a chain escaping R decomposed as those inside and outside CI (CO), then gn gn is from a two-dimensional (2D) channel is counterbalanced by the drag force I O exactly the number of microstates inside C (C ). This f of an optical tweezer. n (n ) is the number of basic unit C inside (C I O drag I O I O decomposition is similar to that in the de Gennes blob, where outside) the channel. c The entropic drift force fD and tension force fT for n n C the Flory statistics inside a blob is independent of the chain polymer tug-of-war. L ( R) is the number of basic unit O in the left (right) ¼ l w = l properties outside the blob. If CI and CO are identical, then ni space. d A bead (blue circle) of size is in a channel (black) of width ¼ ¼ ¼ γ = α + d, where the lateral free space d is the distance the bead can freely move no gn gn 1 for all nI and nO. In this case, 1 and is I O β = α “ ” fi without touching the boundary. The potential well (blue curve) above the reduced to the R of an ideal con ned chain in Eq. (2). channel describes the bead-wall interaction versus the location of the bead Therefore, α, β, and γ play the same role as the activity a = rc, the center. The well boundary V(η) = κη2/2 is a function of the penetration concentration c, and the activity coefficient r, respectively, in the Ào depth η and stiffness κ. The probability density of finding the bead in the chemical potential formula μ = μ + T ln(a) of a non-ideal lateral direction of the channel (green area) has the same area as a uniform solution. Notice that the mathematical equivalence between probability density of an effective width d + de (magenta rectangle). The Eq. (5) and the chemical potential formula for solutions does bead confined by the blue soft potential is analogous to being restricted to a not mean the former is aimed at understanding polymer hard potential of an effective width d + de solutions. ′ given (nI, nO), fR is related to fR by Entropic drift forces. The above formalism also applies to other ! entropic forces. For polymer tug-of-war in Fig. 2c, the chain Φ′ À Φ straddling the channel is unstable and will gradually move out of ¼ ′ þ nI 1 nO ¼ ′ þ ðÞα ; ð Þ fR fR T ln fR T ln R 2 Φ À Φ′ a channel either rightward or leftward. The entropic drift force nI 1 nO ! ! Ω À Ω þ Φ f ¼ T ln nR 1 nL 1 ¼ T ln nL ð6Þ with α ≡ α α , where α  Φ′ À =Φ À and α  Φ =Φ′ . D Ω Ω Φ R i o i nI 1 nI 1 o nO nO À ′ ′ nR nL nR 1 Generally, the bead sizes l for fR and l for fR in Eq. (2) are not ′ the same. Therefore, even when the values of fR and fR of the two represents the tendency of the drift of the whole chain from the ~ ¼ ϵ= ~′ ¼ ′ϵ= ′ systems are identical, the real forces fR fR l and fR fR l right to the left space. Here, nL (nR) denotes the number of beads may be different. in the left (right) space and Φ  Ω þ =Ω  nL nL 1 nL If CI and CO do not have identical shapes or internal freedom, Φ  Ω =Ω fi ~ ~ n n þ1 n stands for the excess con guration number such as those in Fig. 2a, let Φ Φ be the excess “microstate” R R R nI nO caused by an added bead in the left (right) space. Φ and Φ are number caused by an added BU C inside (C outside) the nL nR I O the same function as Φ in Eq. (1). Since the configuration of an channel. The “microstates” here include the allowed configura- nO tions of C and C considered in Eq. (1) and the internal freedom outside segment with fewer beads is more constrained by the half- I O plane boundary, such segment cannot have a larger excess inside each individual BU, which are generally hybridized and Φ ≤ Φ ∈ number, which implies n−1 n for n {nO, nL, nR} (see a cannot be decoupled. When a chain segment of length l moves − − discrete example in Fig. 3). Hence, nL > nR 1(nL < nR 1) in into (out of) a channel, suppose ni (no) BUs of CI (CO) appear inside (outside) it. Then, the total excess microstate number of n Eq. (6) implies Φ >Φ À Φ <Φ À and f >0(f < 0), for i nL nR 1 nL nR 1 D D

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a c 1 Table 1 The upper and lower bounds of various entropic forces 5 0.9 4.5 o

 0.8 (NSC) R 4 f Force range Strip (2D) Tube (3D) Slit (3D) 3.5 0.7 "# 2 3 "# ÀÁ l2 ÀÁ 3 0.6 ðNSCÞ 2πl Tln 16 2lq f Tln d2 Tln 020400 10 20 R ÀÁd 4 5 ÀÁd ðNSCÞ πl lq Tln 8l2 nO nO f d Tln Tln d "#R d2 ðNSCÞ   f T ln 2 T ln 2 T ln 2 b 30 D fðNSCÞ ÀT ln 2 ÀT ln 2 ÀT ln 2 D 2 3 R "# "# ÀÁ 2 ÀÁ

f l 25 ðNSCÞ 4πl Tln 32 4lq f Tln d2 Tln T ÀÁd 4 5 ÀÁd (SC) (NSC) ðNSCÞ 2πl l2 2lq  f Tln d Tln 16 Tln 20 f R = f R + T ln( o) T d2 d The maxima (minima) of the entropic recoiling, drift, and tension forces of a non-self-avoiding 15 chain are denoted by the upper (lower) components in the column vectors, where d >1/ρ, with ρ the linear density of the bead configurations (Supplementary Note 1). The parameter q ∈ [1, ∞) characterizes the angle 2π/q a bead in a stretched chain segment inside a quasi-2D slit can 10 rotate. Because the force f(NSC) of the non-self-avoiding chain is smaller than the force f(SC) of ðNSCÞ the self-avoiding chain, fðSCÞ fðNSCÞ , the maximum f of f(NSC) is an upper bound of the ðSCÞ < f f(SC) 5 maximum of Entropic recoiling force 0 12345678The simplest version of that force describes the tendency of moving one bead from the channel into the left or the right space, Temperature T  Ω þ Ω À Ω þ Ω Ω À Ω þ nL 1 nI 1 nR nL nI 1 nR 1 Fig. 3 Relations between the entropic recoiling forces of different kinds of f ¼ T ln Ω Ω Ω T n n n polymer. a If a non-self-avoiding chain is pulled into a channel by an optical L I R ð7Þ ð Þ Φ þ Φ f NSC ¼ nL nR : tweezer (as shown in the inset of b), the entropic recoiling forces R T ln Φ n À1 calculated by the Jarzynski equality at d = 0.05l, κ=ϵ ¼ 104, T = 1, and I n different bead numbers outside the channel, O, are depicted by the green Φ ; Φ Φ Since > À , fT is always positive, indicative of a points. Increasing the number of systems in the ensemble of the Jarzynski nL nR nI 1 equality approach, the green points will become more concentrated onto spontaneous trend of stretching, rather than compression, in that the magenta dashed curve. The curve saturates at the maximum magnitude system. This trend will eventually be counterbalanced by the ð Þ ð Þ f NSC when n > 10. The blue dotted (solid) line denotes the force f NSC increased free energy during stretching caused by other R O R mechanisms, such as bead deformation. Two tension forces of the calculated by the recursion formula, without (with) the penetration effect, α same form as Eq. (7) can be related as Eq. (2), withR there for which the lateral free space in Fig. 2disd (is d + de). b For d= 0.2l at ð Þ ð Þ different temperatures, the circles (squares) are the force f NSC f SC of replaced by α ≡ α α , where α  Φ þ Φ = Φ′ þ Φ′ R R T i lr lr nL nR nL nR the non-self-avoiding chain (self-avoiding chain) measured via the = ′ Φ and l l is not required, as in Eq. (2). For large nL and nR, , fðNSCÞ fðSCÞ nL Jarzynski equality. The dotted (dashed) line denotes the R R Φ Φ′ Φ′ Φ , , and are approximately the same as at large nO, calculated by the recursion formula without the penetration effect, while nR nL nR  nO ð Þ ð Þ the solid (dash-dotted) line represents the f NSC f SC calculated by the which implies f  T ln 2Φ =Φ and α  α α ¼ α . R R T nO nI T i o R recursion formula with that effect, which agrees with the circles (squares). If CI and CO are different,hi in analogy to Eq. (3),hi the generalized ð Þ ð Þ n n n The forces f NSC and f SC are related by a specific version of formula (5) ~ o ~ o ~ i R R tension force is f = T ln Φ þ Φ = Φ À .It T nL nR nI 1 written in the plot. c The function αo (blue curve) in Eq. (2) converges to α is related to another tension force of the same form again by Eq. 0.66 (red dashed line). It has a zigzag shape, because o behaves α (5) with a slightly different . For those CI and CO, the drift force differently at even and odd nO in a lattice space. The error bars in the Jarzynski equality approach vary with the selected ensemble size and are, fD in Eq. (6) does not need to be generalized, because CI is absent therefore, not emphasized in b in that equation and CO 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 which the chain tends to move leftward (rightward). For nL nR 1, Φ ¼ Φ = summarized in Table 1), which serve as good candidates for the one has À and fD 0, for which the chain movement À nL nR 1 values of f o in Eq. (5). does not have a biased direction. The drift force in Eq. (6) can be decomposed as a difference f ¼ f ;ð ; Þ À f ;ð ; À Þ between a D R nI nL R nI nR 1 Optical tweezer experiments. To illustrate the above formulas leftward recoiling force, f ;ð ; Þ, and a rightward recoiling force, R nI nL ÂÃÀÁ and the analytical results in Table 1, let us carry out the following f ;ð ; À Þ, with f ;ð ; ′Þ  T ln Ω À Ω ′þ =ðÞΩ Ω ′ being a R nI nR 1 R n n n 1 n 1 n n numerical experiments, with the Hamiltonian described in the generalized expression of Eq. (1). Interestingly, the statistical Methods section. First, suppose an NSC is pulled into a 2D fi forces fD and fR based on counting con gurations also comply channel of width w by the drag force fdrag of an optical tweezer, with the above subtraction rule as normal non-statistical forces in which moves slowly rightward with a constant speed (Fig. 2b, d). mechanics. Two drift forces of the same form as Eq. (6) are The work done by fdrag is converted to the entropic recoiling force α α ≡ α α ð Þ related as Eq. (2) with R there replaced by D l/ r, where NSC fR via the JE (Methods section), as depicted by the magenta α  Φ =Φ′ and α  Φ À =Φ′ À are the same function as the ð Þ l nL nL r nR 1 nR 1 NSC α dashed curve in Fig. 3a. This curve shows that for nO > 10, fR o below Eq. (2). ðNSCÞ has almost saturated at its maximum value fR on a plateau (blue solid line), which is independent of the tail length nOl Entropic tension forces. The entropic tension force in Fig. 2c outside the channel. In contrast to other studies in experiments13 characterizes the tendency of polymer stretching inside a channel. and simulations16,17, this tail length independent force is, to our

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fi knowledge, rst time determined by the JE. For nO < 10, the curve a falls monotonically and drastically with a decreasing nO to its 7 = b c 2 minimum. The sharp kink around nO 5 reveals that after most / = 10 5 1 beads have been pulled into the channel, the rest beads in the tail 3.5 will surrender and give up to resist, which behaves like a first ðNSCÞ 6 16 R 3 order phase transition . In Fig. 3a, the maximum force fR is f also calculated by the RF (blue dotted line) (Methods section), d which is slightly larger than the magenta dashed curve calculated 2.5 0 5000 10,000 5 via the JE.  To inspect the cause of this slight difference, we raise the Channel stiffness / temperature from T = 1(T~ ¼ 298 K) up to a physically extreme ~ ðNSCÞ value T = 8(T ¼ 2384 K) in Fig. 3b. At that T, the f 4 R (w) (w′)  evaluated by the RF (blue dotted line) has deviated 10% from that f R = f R + T ln( i) calculated by the JE (blue circles). However, taking into account Entropic recoiling force the penetration effect by enlarging the channel width w = l + d in = + + 3 the RF approach to we l d de with de 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, 2 which justifies the crucial role of the penetration effect and the 0 0.1 0.2 0.3 0.4 0.5 0.6 accuracy of formula (8). Subtracting this effect, the plateau of the Channel free space d (in units of l ) magenta dashed curve in Fig. 3a is shifted to the blue dotted line, Fig. 4 Relations between the entropic recoiling forces of different channel ð Þ ð Þ  NSC ¼ ð π = Þ sizes and stiffnesses. a The recoiling forces f NSC of a non-self-avoiding which is rather close to the magnitude fR T ln 2 l d R predicted in Table 1. Although that analytical magnitude is chain for κ=ϵ ¼ 103 and 104 at T = 1 and different d measured by the  Jarzynski equality approach are depicted by the diamonds and triangles, derived under the strict assumption of nO 1, the simulation in respectively. The same force for a hard-wall channel (κ = ∞) calculated by Fig. 3a shows its correctness even for nO > 10.ð Þ  SC the recursion formula is depicted by the blue solid curve. The divergence of If the chain is an SC, the maximum values fR of the recoiling ð Þ d force f SC of that chain are calculated by the JE at different T and this curve is bounded by a maximum value when falls into the small area R ρ depicted by the red squares in Fig. 3b. To determine those values (0, 1/ ), which is the degenerate force explained below Eq. (1). b The fðNSCÞ α α squares (circles) are the R measured by the Jarzynski equality from an NSC by Eq. (2), one needs to evaluate o, which is the o  approach, at d = 0.2l, T = 1(T = 1.1), and different stiffnesses κ scaled by ϵ. at nO 1. For a 2D half-lattice, our Monte Carlo simulation (Fig. 3c) shows that α has almost saturated at its minimum The dashed (solid) curve denotes those forces calculated by the recursion o w = l + d + d l α  : = formula for the effective channel width e e, in units of . c The o 0 66 after nO 10, which is slightly smaller than that, bα  : 18 ′ potentials of three different stiffnesses studied in b are plotted on the same R 0 78, observed in a 3D half-lattice .IffR and fR in formula length scale. The error bars in the Jarzynski equality approach vary with the ðSCÞ ðNSCÞ (2) represent the fR of an SC and the fR of an NSC, selected ensemble size and are, therefore, not emphasized in a, b fi α respectively, partly con ned in the same channel, R there should α ¼ α α  : α ¼ be equal to R i o 0 66, where i 1 due to identical ðNSCÞ 19 channels (Supplementary Note 1). Using this formula, fR linear channel potential can be found in ref. . In Fig. 4c, the ðSCÞ potentials of three different stiffnesses are plotted. The force (blue dotted line) in Fig. 3b is converted to fR (red dashed line), fi fi which is the RF calculated force for the SC without the magnitudes calculated in Fig. 4a, b con rm the speci c version of penetration effect. Taking into account that effect by enlarging formula (5) written in Fig. 4a. + d to d de, the red dashed line is further shifted down to the red dash-dotted line. The latter is very close to the JE calculated red ðNSCÞ squares, which confirms the specific version of formula (5) Tug-of-war experiments. In Fig. 5a, the drift forces fD of an ′ written in Fig. 3b. NSC of N beads straddling a channel are calculated by the RF, ðNSCÞ ξ ≡ = ′ −  where the translocation ratio nL/N, with N N nI, is the In Fig. 4a, the maximum recoiling force fR for a channel of width w = l + d at different d is calculated by the RF and depicted fraction of the chain segment in the left space. The left (right) by the blue solid curve. It deviates from the force evaluated by the ends of the force curves of different N approach their minimum, − JE when the channel has stiffness κ ¼ 103ϵ (diamonds) and 104ϵ T ln 2, (maximum, T ln 2,) as derived in Table 1. These force (triangles). However, after considering the penetration effect by curves are insensitive to the penetration depth and channel size, + κ=ϵ ¼ 3 ðÞκ=ϵ ¼ 4 in consistent with the absence of Ω and Φ in Eq. (6). Inte- replacing d by d de as before for 10 10 , the nI nI blue solid curve is shifted to the red dash-dotted (dashed) curve, grating these curves leads to the free energy profiles F(ξ)in which precisely passes through the diamonds (triangles) and Fig. 5b. The barriers of F(ξ) can be used to calculate Krammers' again justifies the significance of formula (8). All force escape rate for polymer translocation, if the process is quasi- magnitudes on the blue solid curve agree very well with the equilibrium and ergodicity is preserved20. The curves in Fig. 5b analytical results in Table 1, which validates those results at least will converge to an invariant shape at large N. In Fig. 5c, the =  ðNSCÞ up to d 0.6l, far beyond the ideal condition d l. In Fig. 4b, tension forces fT of different N start from some initial values ðNSCÞ close to T ln(3πl/d)atξ = 0, increase monotonically to their the stiffness dependent fR calculated by the JE at two different T (squares and circles) coincide precisely with those evaluated by maxima below T ln(4πl/d), and then fall back to their initial the RF (dashed and solid curves). Both curves show a kink around values when ξ = 1. For larger (smaller) N, the maxima (minima) π π κ=ϵ ¼ 103, indicative of a separation between the stiffness effect of the forces will approach T ln(4 l/d)(T ln(2 l/d)). All these ðÞκ=ϵ 3 ðNSCÞ dominant regime <10 and the width effect dominant numerical observations agree with the analytical bounds of fT regime ðÞκ=ϵ>103 for the force. A similar stiffness analysis on a predicted in Table 1 and Supplementary Note 1.

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5 l = l ≈ 2l ≈ 106 nm. According to Eq. (5) (Supplementary D K p 

f 0.6 a ð Þ ð Þ ð Þ ^ SC ; ^ SC ; ^ SC ≈ Note 2), the maximum real forces are fR fD fT 4.5 ð : ; : ; : Þ 0 0 18 0 01 0 21 pN for the strip, (0.35, 0.02, 0.38) pN for the tube, (0.16, 0.02, 0.19) pN for the slit when nI is so large that an 4 π d added bead in the slit can freely rotate 2 , and (0.28, 0.02, 0.31) Entropic force –0.6 n pN if nI is so small during strong stretching that an added bead Δ 3.5 can only rotate a small angle around 2π/q = π/16 for q = 32.

F b 12 2 In an experiment , a dsDNA recoiling from a 90-nm-wide and 3 100-nm-deep channel is estimated to encounter a force ^ðexpÞ ^ðSCÞ 1 f  0:22 pN. It is slightly larger than the value f  0:18 2.5 R R pN for a strip of a slightly greater width 110 nm calculated above.

Free energy 13 0 In an experiment , a T2 phage dsDNA recoils from an array of 2 nanopillars 35 nm in diameter with a center-to-center spacing of Free energy barriers

T c f 4.2 160 nm. The experimentally estimated recoiling force 1.5 ^ðexpÞ  : fR 0 04 pN in that array was bounded by the force ð Þ 4 ^ SC  : 1 fR 0 12 pN in a 2D channel of the same width estimated by formula (5) (Supplementary Note 2). This upper bound is 3.8 reasonable because the confinement of an array is much weaker Entropic force 0.5 15 0 0.5 1 20 40 60 80 100 than that of a channel of the same size. In an experiment ,a Translocation ratio  Bead number N granular chain above a vibrating platform escapes from a channel. The recorded recoiling force follows exactly the same relation fR Fig. 5 Entropic drift and tension forces at different chain lengths and ~ −ln(w − l) as in Fig. 4a. In another experiment14, when a fðNSCÞ translocation ratios. a The drift forces D of a non-self-avoiding chain of dsDNA straddling a nanoslit is entropically pulled out of the slit, N = 4, 20, and 100 are calculated by the recursion formula at different ξ, it undergoes (i) a tug-of-war scenario, followed by (ii) a recoiling- fi ~ðexpÞ and tted by the black dash-dotted, red dashed, and blue solid curves, retraction scenario. The force, f , in (i) is similar to the above n ðiÞ respectively. When the chain is pulled leftward to increase its L starting ð Þ ðNSCÞ ~ SC from nL = 0, it will be resisted by a rightward force, f ðξÞ<0, when ξ < fT , which is counterbalanced by the entropic elasticity of the ð Þ D ð Þ 0.5, but assisted by a leftward force, f NSC ðξÞ 0, when ξ > 0.5. b The free ~ exp D > stretched dsDNA strand inside the nanoslit. The force, fð Þ ,in fi F ξ ii energy pro les ( ) are integrated from the curves in a, where the two ends ~ðSCÞ of those profiles are set to zero. The integration is carried out along the real (ii) is closer to our fR , which are dragged by the friction force and the optical tweezer (Fig. 2b), respectively. For a slit 110 nm in length nLl, and then reexpressed as a function of ξ. Therefore, F(ξ) rises  N fðNSCÞðξÞ ^ðexpÞ ^ðexpÞ with , although the areas bounded by the curves in Fig. 5a height, the forces extracted from the experiment, fð Þ ; fð Þ ≈ ð Þ D i ii decrease with N. c The tension forces f NSC calculated by the recursion T ð : ; : Þ ^ðSCÞ; ^ðSCÞ ≈ ð : ; : Þ formula are fitted by the black dash-dotted, red dashed, and blue solid 0 17 0 30 pN, are not far from fT fR 0 31 0 28 pN curves for N = 4, 20, and 100, respectively. d The free-energy barrier Δn of a tightened chain estimated in the previous paragraph. The measured from the energy profiles in Fig. 5b gives the blue circles, red ^ðexpÞ ^ðSCÞ relation fð Þ  f indicates that the allowed rotational angle squares, and black triangles for n = 1, 2, and 3, respectively, where n  1. ii R I ^ðexpÞ ^ðSCÞ Δ π The n calculated by the scaling formula in the main text are the blue solid, for beads inside the slit is around /16. The ordering fðiÞ < fT red dashed, and black dash-dotted curves, respectively suggests the existence of a non-configurational-entropy effect in ^ðexpÞ f . Indeed, in addition to increasing the configurational Δ ðiÞ Let n be the difference between the entropic free energies of a = entropy, strong chain stretching may also decrease the microstate chain at nL n and N/2, where the latter is the position of the ξ Δ number inside a bead or deform the bead. The subsequent maximum of the F( ) in Fig. 5b. The value of n reveals the free ð Þ ð Þ = ^ exp ^ SC energy barrier for a chain translocation starting from nL n. increase of free energy will weaken fðiÞ , but not fT , leading to ^ðexpÞ ^ðSCÞ According to a scaling argument, the entropic free energy of a f < f . Despite that discrepancy, these two forces are of the = = ~ = ð À Þ ð Þþð À Þ ð À Þ ðiÞ T chain of nL n is Fn kBT 1 r ln n 1 r ln N n same order. whenn and N À n  121,22, where r = 1/2 for NSCs. It implies ≈ ÀÁ ÂÃMore generally, if a single-stranded DNA (ssDNA) of lp Δ = À = ~ = ð = Þ ð = Þ2=ð ð À ÞÞ 23 n FN=2 Fn kBT 1 2 ln N 2 n N n for an 0.75 nm , partly confined in a 2D nanochannel or a 3D circular κ = ∞ −1 NSC. In Fig. 5d, the Δ calculated by that scaling argument are nanotube of width 1.6 nm and pN nm , is modeled by an n 24 = found to overestimate those measured from the F(ξ) in Fig. 5b. SC comprising beads of three nucleotide bases , one obtains l fi Δ ≈ ≈ ^ðSCÞ  : The discrepancy is especially signi cant in 1 between the solid lK 1.5 nm and d 0.1 nm. This yields fR 11 32 pN for the curve and circles. This is ascribed to the overestimated difficulty nanochannel and 21.76 pN for the nanotube (Supplementary of pulling the first few beads into the left space by the scaling Note 2), which are even larger than the drag force, 5 ~ 6 pN, of a fi ðNSCÞ myosin. The increase in the force magnitude from dsDNA to formula. It is in echo with the nding in Fig. 3a that fR is weaker than expected for chains with few beads outside the ssDNA is a consequence of the shortened Kuhn length and the channel. 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 Applications. Suppose we have a double-stranded DNA examples of soft channels, consider a 120-nt ssDNA oligomer25 ≈ fi (dsDNA) of persistence length lp 53 nm partly con ned in a 2D threaded through an α-hemolysin, which is 10 nm long and has strip (Fig. 2b, c), a 3D circular tube, or a 3D slit of size an inner size varying between 1.4 and 4.1 nm1. If this system is − ~ w = 110 nm and stiffness κ = ∞ pN nm 1 at T ¼ 298 K. If modeled by an NSC of 40 Kuhn segments of l ≈ 1.5 nm24 and this system is modeled by an SC of beads of Kuhn length lK, then threaded through a circular tube of the same width as l, with a

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− hypothesized effective stiffness κ = 1~102 pN nm 1, then the Here, the potential well U(y) consists of a zero-potential area flanked by two walls ^ η maximum tension force is f  1:47  14:12 pN (Supplementary of the form of the above V( ). If that distribution has the same area as an effective T uniform distribution (magenta rectangle) of the same height h and of a larger width R 1 ÀÁÀÁ Note 2). This analysis provides an instantaneous check for the d + d , the equality in area implies 2 hexp Àκη2= 2k T~ dη þ hd = hdðÞþ d . κ e 0 B e consistency between the hypothesized d, , and forces. RIn analogy, for a 3DÀÁ circularÀÁ tube, the equality in volume leads to 1 πðη þ = Þ Àκη2= ~ η + ð = Þ2π = ð = þ = Þ2π 0 2 d 2 h exp 2kBT d h d 2 h d 2 de 2 . Therein, Discussion de is the mean penetration depth into two strip walls in Fig. 2b, c or into the circular tube wall, which is given by The experimental measurement of polymer entropic forces is challenging. The currently reported “measured” force magnitudes are all indirectly inferred from certain force-velocity or force- ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2πTϵ=κ; for 2D strips; ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ extension relations. The observed velocity and extension can be de 8 2 þ À ; ; affected by several factors, including interfacial friction, Debye d B d for 3D circular tubes length, osmotic force, dye and salt concentrations, and nature of solvent. Even for the more fundamental polymer characteristics, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi such as persistence length or width of DNA and RNA, the with ϵ defined in Eq. (1) and B  8ϵT=κ þ 2d 2πϵT=κ. Although the above soft experimentally measured values are often not unique. Thus, some potential U can generally be interpreted as a soft channel or a soft bead, the latter experimentally measured entropic force was considered an requires a more subtle model with a varying bead shape during bead-wall collision, “effective” one13. With these technical difficulties and system which is not considered here. varieties, a coincidence close to 100% between a theoretically predicted force and an experimentally measured one could be Langevin dynamics. With the above Hamiltonian H, the ith bead follows a Lan- gevin equation mx€ ¼À∇ H À Ξx_ þ ζ , where ∇ H is the derivative of H with i xi i i xi accidental, when these forces are as small as a few piconewtons. Ξ fi ζ 24 respect to xi, denotes the friction coef cient, and i stands for a random force . However, if coincidences of the same order occur concurrently in That force is GaussianDE and white and satisfies the fluctuation-dissipation theorem, several systems, as in the above-mentioned examples, the actual 〈ζ 〉 = ζ ð ÞζTð ′Þ = Ξ ~δð À ′Þδð À Þ with i(t) 0 and i t j t 4 kBT t t i j , where the superscript force magnitudes are probably not much different from the T represents the transpose. Since the strength of the above random force changes experimentally and theoretically determined ones. with temperature, the diffusion of beads is temperature dependent as well. To study The formalism proposed in this work overcomes the problem general real systems, the above equation can be reduced to a dimensionless one, ~ ~€ ¼À∇ ~ À Ξ~~_ þ ~ζ ~ ¼ = ~ ≡ of directly solving partition functions and provides a less-time- mxi x~H xi i, with the dimensionless quantities: m m M, x x/l, ~  =τ ~  ðÞτ2=ðÞ2 ~  ðÞτ2= ~  ðÞτ2= ~κ  ðÞτ2= κ consuming reference-based algorithm to assess entropic con- t t , H Ml H, kC M kC, kV M kV, M , ~ tributions. This approach is especially simple when a polymer Ξ~ ðτ=MÞΞ, and ζ  ðÞτ2=ðMlÞ ζ in units of some convenient mass M and time τ24 system can be modeled by an ideal confined chain with γ = 1. . 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 Beyond that regime, such as in the de Gennes or Odijk regime, simple parameter values for the original Langevin equation. In this study, we take ð ; ; Ξ; ; ; ; κ=ϵÞ = ð ; ; ; 4; 4; 4;  4Þ the forces obtained from Eqs. (3) and (5) are as complex as those l m kC kV kO 1 1 1 10 10 10 0 10 , where units are not extracted by other half-analytical approaches (Supplementary specified. Note 3). If other specific interactions, such as the angular and ~ dihedral force-fields, are not coarse-grained in the BUs of a chain Jarzynski equality. Let W be the work done by fdrag to pull a bead of length l into ~ model, their entropic contribution has to be derived case by case, the channel in Fig. 2b. Owing to fluctuations, the W in different pulling realizations 11 as reflected in the widened channel width in Eq. (8), or empiri- will be different, which form a work distribution. According toDE the JE , the free ~ ~ ÀΔ~= ~ À ~ = ~ α energy change, ΔF, per shifted l is related to W by e F kB T ¼ e W kB T , where cally measured, as the o in Fig. 3c. Nevertheless, the forces of a nearby system with γ = 1, calculated by Eq. (5), can readily the ensemble mean 〈⋅〉 is averaged over the whole work distribution. Therefore, the provide an upper or a lower bound for the desired systems with mean force toÀÁ counterbalanceDEfdrag is the free energy change per shifted bead ~ ~ ~ À ~ = ~ f ¼ÀΔF=l ¼ k T=l ln e W kB T , which has a dimensionless version γ ≠ 1, as the pillar array discussed above. To sharpen these force ÀÁB ~ À = bounds, the current formalism further serves as a proper frame- f ¼ fl=ϵ ¼ T ln e W T , with the dimensionless work W ¼ W~ =ϵ, where ϵ is defined in Eq. (1). The f measured in our numerical recoiling experiments should work, for instance, for selecting more sophisticated UBs CI and ð Þ be the f NSC derived in Eq. (1) if the change of internal energy is negligible during CO or for performing perturbation approaches on Eq. (5) around R γ = 1. pulling, which occurs when the channel has a hard-wall or its soft wall is replaced by an effective hard-wall.

Methods Recursion formulas. A recursion formula can be used to count the configuration Hamiltonian. For a chain of N identical beads in the bead-spring model in Fig. 2b, number of a chain of n beads based on that of a chain of n − 1 beads. For a one- _ n fi let m and l be the mass and the size of a bead, respectively, whereas xi and xi be the dimensional NSC consisting of n rods of unit length, let Nx be the con guration position and the velocity of the ith bead, respectively. The HamiltonianP of the chain number when one end of the chain is anchored to the origin and the other end is = + + + + ¼ N _ 2= − + is H EK EC EV EO EW, with the kinetic energy E ¼ mx 2, the located at x between n and n, where x is an integer. It follows a simple P ÀÁ K i 1 i À À NÀ1 2 recursion relation Nn ¼ Nn 1 þ Nn 1 for x = −n, −n + 1, …, n − 1, n, where interaction between consecutive beads E ¼ ¼ k x À x þ À l =2, the x xÀ1 xþ1 C i 1 C i i 1 n ¼ fi excluded volume interaction between non-consecutive beads Nx 0 for |x|>n. The solutions of this equation are the binomial coef cients  n n n P 2 N ¼ C when x = −n, −n + 2, …, n − 2, n and N ¼ 0 elsewhere. With ¼ N À À σ = x ðnþxÞ=2 x P EV i;j¼1 ði>jÞ kV xi xj l ij 2, the bead-channel interaction n fi Ω ¼ n P ÀÁ those Nx , the total number of the chain con gurations of n rods is n x¼Àn ¼ N η n ¼ n fi = Ω = EW i¼1 V i , and the interaction between the last bead and the optical Nx 2 and the corresponding con gurational entropy is S kB ln n nkB ln 2. = − 2 tweezer EO kO|xN z| /2. Here, the natural length of the spring between two For a chain of n beads of sizeRl in a high-dimensional off-lattice space, the above RF σ = − ≤ n ¼ nÀ1δðÞjjÀ′ À ′ consecutive beads is the same as the bead size l. The value ij 1 for |xi xj| l and can be generalized to Nr Nr′R r r l dr , for which the total number of − fi Ω ¼ n 0 for |xi xj|>l. The position z denotes the location of the optical tweezer. The the chain con gurations, n Nr dr, is an integration over all end-to-end dis- ðη Þ¼κη2= η fi potential V i i 2 when the ith bead penetrates a distance i into a channel placement vectors r of a chain of n beads. To count the con gurations of a mac- κ fi fi wall (Fig. 2d). Therein, kC, kV, kO, and are the force constants in EC, EV, EO, and rostate of a chain, speci ed by some (nI, nO), in Fig. 2b, rst calculate the EW, respectively. In this system, there is no facial friction and the diffusion coef- fi Ω Ω con guration number of a chain segment of nI (nO) beads inside ficient is position independent. For an SC, the bending angle formed by three nI nO consecutive beads lies within [0, 2π/3]. (outside) the channel when one segment end is anchored to a point on the cross- section of the channel opening. Summing up the product Ω Ω over all different nI nO points on that cross-section yields the total configuration number needed for Penetration effects. Since the channel is narrow, the slight lateral motion of a calculating the entropy of the whole chain. Notice that although the RF is able to bead is less affected by its motion along the channel axis. Therefore, the probability count the chain configurations within a channel, it cannot reflect the stiffness effect density of finding a bead at some lateral position y follows the Boltzmann dis- of the channel, unless we use Eq. (8) to replace a soft channel by an effective wider À ð Þ= ~ tribution e U y kB T , which is depicted by the green area in the upper part of Fig. 2d. hard channel.

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