As shown in figure 1(A), the simulation setup, which was composed of two macrospheres and a semi-flexible chain accommodating knots with different types, was confined in a cuboid nanochannel. This semi-flexible polymer was modelled by a coarse-grained bead-spring model, which was composed of L monomers of diameter σ and mass m (σ and m are set as the unit of length and mass, respectively), and the contour length of the polymer used in current simulations is 500σ. Its two ends were tethered to two macrospheres, and the diameter of the two tethered macrospheres should be larger than the knot size to block the unraveling of the knot.

The interaction between two adjacent monomers in the semi-flexible polymer can be described by the finite-extensible nonlinear elastic (FENE) potential as followsIt is a harmonic potential with a bond potential of ${k}_{\mathrm{FENE}}\,=\tfrac{30{k}_{{\rm{B}}}T}{{\sigma }^{2}}$ and a maximum degree of stretching of R₀ = 1.5σ. These parameters were taken from the standard polymer model of Kremer and Grest [26], and ${r}_{i,i-1}=| {\vec{r}}_{i}-{\vec{r}}_{i-1}| $ is the distance between any pairs of monomers. At the same time, the bending potential between two adjacent monomers can be introduced as followswhere ${\theta }_{i}=-\arccos \left(\tfrac{{\vec{b}}_{i-1}\cdot {\vec{b}}_{i}}{| {\vec{b}}_{i-1}| | {\vec{b}}_{i}| }\right)$ denotes the angle between two adjacent bond vectors ${\vec{b}}_{i-1}={\vec{r}}_{i}-{\vec{r}}_{i-1}$ and ${\vec{b}}_{i}={\vec{r}}_{i+1}-{\vec{r}}_{i}$, and k refers to the bending stiffness. The equation l_p = κσ that correlates the persistence length l_p with the bond stiffness was applied to map the coarse-grained bead-spring model to dsDNA, and the parameters of hydrated dsDNA were adopted as σ = 1.0 nm and κ = 50k_BT.

In addition, the spatial interactions between any two monomers i and j were given by a short range truncated and shifted Lennard–Jones (LJ) potential U_LJ of strength ε with a cutoff distance ${r}_{c}={2}^{\tfrac{1}{6}}\sigma $ [27] where ϵ = k_BT is the unit of energy, and k_B and T denote the Boltzmann constant and the temperature, respectively.

To identify the impact of confinement on the diffusion of the knot along the semi-flexible polymer under tension, the interactions between monomers and the nanochannel were described as follows:where d_i is the orthogonal distance between the ith monomer and the nanochannel. Finally, the function ϑ(x) in equations (3) and (4) is the Heaviside function, can be defined as

During the dynamics, the macrosphere center and the chain ends were constrained to move along the x-axis [28]. A constant force f ($f=\tfrac{{f}_{0}\sigma }{\varepsilon }=\tfrac{{f}_{0}\sigma }{{k}_{{\rm{B}}}T}$, in which ${f}_{0}=\tfrac{\varepsilon }{\sigma }$) was also applied to the semi-flexible chain via two macrospheres, and the corresponding stretching potential was defined as ${U}_{\mathrm{stretch}}=-\vec{f}\cdot {\sum }_{i=1}^{L}{\vec{b}}_{i}=-{fX}$. The x-axis coincides with the direction of the force f, and X is the projection of ${\sum }_{i=1}^{L}{\vec{b}}_{i}$ on the x-axis. The adimensional, reduced force f is typically varied in the range from 0.5 to 15 to characterize the mechanical tensile response of the knots under small, as well as large tensions. The model can explore a wide variety of polymer molecules with different values of l_p ranging from 1σ to 20σ, which correspond to polysaccharides and double-stranded DNA, and all of the simulations were performed at a constant temperature ($T=1\tfrac{\varepsilon }{{k}_{{\rm{B}}}}$). Since the macrospheres were free to rotate, and therefore no torsional stresses were generated in the simulations. The dynamics of the chain is generated by solving the Langevin equation as followswhere U_i is the total potential of the ith monomer, including all the potentials mentioned above, and ξ₀ is the friction coefficient for each monomer and its value was set as ${\xi }_{0}=2\tfrac{m}{{\tau }_{{\rm{LJ}}}}=2{\left(\tfrac{{\sigma }^{2}}{m\varepsilon }\right)}^{-\tfrac{1}{2}}$, where ${\tau }_{{\rm{LJ}}}=\sigma \sqrt{\tfrac{m}{{k}_{{\rm{B}}}T}}$ is the Lennard–Jones time. The Stokes friction equation for a spherical monomer is given by the following expression: ξ₀ = 3πη_solσ, in which η_sol is the viscosity of the solution. For a specified water viscosity η_sol = 1cP, T = 300 K, and σ = 1.0 nm, the characteristic Lennard–Jones time τ_LJ are related to $\tfrac{6\pi {\eta }_{{\rm{sol}}}{\sigma }^{3}}{\varepsilon }\approx 1.14\,{\rm{ns}}$. In addition, $\vec{R}(t)$ is a stochastic force term that satisfies the fluctuation-dissipation theorem. Finally, the dynamical evolution of the system is numerically integrated with the LAMMPS package with an integration time step of 0.005τ_ij ∼ 5.7 ps, and the total simulation time is about 10⁸τ_LJ.

From a mathematically rigorous point of view, knots are defined only for closed chains. For our analysis, it is crucial to extend the knot notion to open chains by introducing appropriate schemes for joining the ends of the open chain by an arc and turning it into a closed ring. After closure, the physical knotted state of an open chain can be established by using the Alexander determinants [29], and the knotted portion of a chain is finally determined as the shortest chain segment that maintains the topology of the whole chain. The shortest chain segment is measured and identified based on a bottom-up searching scheme, which consists of three steps: (i) compute the center of mass of the portion, (ii) construct two segments starting from the ends of the portion and going far away from the center of mass and (iii) connect the termini of these segments at ‘infinity’. Knots are designated by C_k, where C refers to the minimal number of self-crossings displayed when the knot is projected into a plane, and k denotes a cardinal index that is used to distinguish between topologically different knots with the same C [30]. In the current study, the Python package was employed to generate different knots. As shown in figure 1(A), a typical complex knot 5₁ is accommodated in a polymer under tension f = 0.5, and the other simple knots 3₁, 4₁, 5₁, and 5₂ are successively obtained, as demonstrated by figure 1(B).

The diffusion of a knot along a polymer can be monitored by a discrete monomer index n, known as the diffusion coordinate. As indicated by the conformation with a 3₁ knot in figure 1(B), the boundaries of the knot region are defined as n_l and n_r, and the knot position along the chain is then defined as n = (n_l + n_r)/2. The knot diffusion coefficient can be identified by the following relationship $D=\langle {\left[n(\bigtriangleup t)-n(0)\right]}^{2}\rangle /(2\,\bigtriangleup t)$, where the square of the knot displacement ${\left[n(\bigtriangleup t)-n(0)\right]}^{2}$ is averaged over a series of short-time simulations, with the knot initially located at n(0) [25]. In the current simulations, the unraveling of the knot from the polymer is prohibited by tethering its ends to two macrospheres. Therefore, the diffusion coefficient in the low-force limit can be extracted from the current simulations [28]. According to this definition, the tension dependence of the diffusion coefficients of a knot with different bending stiffness k under different spatial constraints can be identified, and those of the 3₁ knot are demonstrated in figures 2(A)–(D), in which the knot diffusion coefficients and tension f are scaled by σ²/τ and ϵ/σ, respectively. Furthermore, the dependence of the diffusion coefficients of a 3₁ knot along a polymer with the bending stiffness of κ = 40k_BT on the spatial constraint is outlined in figure 2(D), in which the curves for the space without constraints or weak constraints have two turnover behaviours, and the first turnover behaviour becomes totally disappeared in the space with a strong constraint of 600 × 6 × 6.

In the space without constraints and with weak constraints shown in figures 2(A) and (B), the diffusion coefficient of the 3₁ knot as a function of the tension f at relatively large bending stiffness κ > 3k_BT exhibits a non-monotonic pattern. Their corresponding curves have two turnover behaviours and are divided into three regimes, namely, the regime before the first turnover behaviour, the regime between the two turnover behaviours and the regime after the second turnover behaviour. Their corresponding knot diffusion coefficient is first reduced, and then increased, and finally decreased as the applied tension f became bigger. Those at relatively small bending stiffness κ ≤ 3k_BT monotonically decreased with the increasing tension f.

In the regime before the first turnover behaviour, the diffusion coefficient of the 3₁ knot reduced with increasing tension, which can be reasonably argued based on the knot-region breathing. Particularly, this effect involves the exchange of the stored length locally with the vicinal chain and is expected to dominate in large knots. The increasing tension f gradually inhibits the knot-region breathing by reducing the knot size, consequently resulting in slower diffusion. This pattern is captured in figures 3(A)–(C), in which the relatively large knot in figure 3(A) is reduced to the relatively small one in figure 3(C) by increasing the tension from 0.5 to 2. When the knot conformation further developed into the loose knot corresponding to the regime between the two turnover behaviours (figures 3(D)–(F)), the 3₁ knot self-reptated along the polymer under tension and its diffusion coefficient D increased with the increasing tension. This effect could be interpreted considering that the knot diffusion is accomplished via a concerted motion of the loose knot. Hence, the total friction drag force that acts on the knot is proportional to the number N of monomers within the knot multiplied by the friction coefficient ξ₀ per single monomer. Thus the friction coefficient ξ can be approximated as Nξ₀, and the Einstein relation D = k_BT/ξ for a knot can be re-expressed as D ≃ k_BT/Nξ₀, The increasing tension could also effectively accelerate the knot diffusion by reducing the knot size N.

When the loose knot was switched to the tight knot corresponding to the regime after the second turnover behaviour, the intrachain interactions within the tight knot enhanced the ‘bumpiness’ of the energy landscape of the knot, which is the microscopic origin of internal friction [31–34]. The application of a higher tension f induced a rougher energy landscape and consequently a slower diffusion. This assumption on the internal friction provided valuable insights for understanding the monotonic decrease of the diffusion coefficient of the 3₁ knot with relatively small bending stiffness κ ≤ 3k_BT as a function of the increasing tension f in the space without spatial constraints. As indicated by figures 3(H)–(J), because of the relatively small bending stiffness, the conformation of 3₁ knot under tension is directly developed into the tight knot. For this reason, the diffusion coefficient was gradually reduced by the internal friction.

In figure 2(C) with strong constraints of 600 × 6 × 6, the curves are divided into two regimes by the second turnover behaviour, namely, the regime before the second turnover behaviour, and the regime after the second turnover behaviour. In the regime before the second turnover behaviour, the change in the diffusion coefficient with the increasing tension could be ascribed to a similar assumption to that corresponding to the regime between the two turnover behaviours in the space without constraints or weak constraints (figures 2(A) and (B)). However, the conformation transition of the 3₁ under the strong constraints of 600 × 6 × 6 is obviously different. As demonstrated by the typical conformations with large bending stiffness κ = 50k_BT in figures 4(A)–(F), the 3₁ knot was compressed into the back-folding state (figure 4(A)) by the strong constraints. This effect resulted in the disappearance of the regime before the first turnover behaviour indicated in the space without constraints, namely, the knot-region breathing is inhibited by the strong constraints. The 3₁ knot in the back-folding state and its successive loose knot (figures 4(A)–(F)) self-reptate along the polymer, and their diffusion is accelerated by reducing the number of monomers within this knot. As a result, its corresponding diffusion coefficients are increased. In the regime after the second turnover behaviour, the diffusion coefficient decreased with the increasing tension, which can be also reasonably understood by the internal friction in a tight knot. As indicated by figures 4(E)–(F), by increasing tension f more than 10, the 3₁ knot switched from the transition state (figure 4(E)) to the tight knot (figure 4(F)). On top of that, the internal friction between the monomers within a knot, rather than the viscous friction due to the solvent, became gradually dominant over the knot diffusion, which induced the gradual reduction in the diffusion coefficients of the 3₁ knot in the transition state and its following tight knot. The implementation of a higher tension f reduced the diffusion of the knot, and even the 3₁ knot was almost jammed in the polymer, which was observed in current simulations and recent single-molecule experiments [9].

The conformation transitions of the 3₁ knot with relatively small bending rigidity κ ≤ 3k_BT in space with strong constraints of 600 × 6 × 6 were different from those with relatively large bending rigidity κ > 3k_BT. As indicated by figures 4(G)–(L), the typical knot conformations with bending stiffness κ = 1k_BT are directly developed into a loose knot by passing over the back-folding state, and then switched to the tight knot. Its corresponding diffusion coefficients first increased, and then decreased with the increasing tension f.

The diffusion coefficients for the knots with types of 3₁, 4₁, 5₁, and 5₂ and different values of κ, f under different spatial constraints were systematically computed. The results for the bending stiffness κ = 50k_BT are outlined in figure 5, where the effective friction coefficients ξ = k_BT/D of the knots of type 3₁, 4₁, 5₁ and 5₂ as a function of the knot size N at a fixed tension were well fitted by the relationship ξ ∝ N, where the ratio ξ/N represents the friction coefficient ξ₀ of a single monomer [9, 25]. Moreover, some physical information enclosed in the knot diffusion can be drawn from the comparison among figures 5(A)–(D). More specifically, the effective friction coefficients of knots of type 3₁, 4₁, 5₁ and 5₂ as a function of the knot size N at relatively large tension f > 3 sharply increased with the knot complexity, which is not obviously dependent on the spatial constraints. By contrast, the effective friction coefficient ξ of knots with 3₁, 4₁, 5₁ and 5₂ types as a function of the knot size N at relatively small tension changing with the knot complexity is remarkably dependent on the spatial constraints. Take the relatively small tension f = 0.5 for example, in the space without constraints, the effective friction coefficients with 3₁, 4₁, 5₁ and 5₂ types as a function of the knot size N slightly increased with the knot complexity. With the gradual shrinking of the spatial constraints from 600 × 10 × 10 to 600 × 8 × 8, and then to 600 × 6 × 6, their corresponding effective friction coefficients of knots with types of 3₁, 4₁, 5₁, and 5₂ as a function of the knot size N were obviously reduced, then slightly decreased, and finally sharply increased with the knot complexity. All these effects can be attributed to knot-region breathing and internal friction.