Knots in Polymer Molecules Under Poiseuille Flow

Abstract

Knots are ubiquitous in polymers and biological macromolecules such as DNA and proteins, yet their behavior and functionality are still not sufficiently explored. Here we investigate the impact of Poiseuille flow on simple knots in flexible polymers placed in a quasi-rectangular micro-channel by systematically varying the flow strength for different chain lengths. Hydrodynamic interactions are accounted for by means of Multi-Particle Collision Dynamics (MPCD). We find that initially loosely localized knots in polymer coils typically tighten under shear to several segments beyond a certain body force threshold. At higher shear rates, intermittent transition from chain stretching to tumbling is observed which correlates with strong fluctuations in the knot size. Somewhat unexpectedly, our results indicate that the influence of channel width on tightening steadily increases with growing width even at equal mean shear rate $\overline{\dot{\gamma }}$.

1. Introduction

The biological role of knots in living matter, in particular their impact on enzymes and the kinetic stability of proteins [1] is of significant importance. Yet despite intensive studies in recent years, it is still not well understood. Knots are ubiquitous in polymers [2] and biological molecules [3], such as DNA [4,5,6,7,8] and proteins [9,10,11,12,13,14,15,16]. Knots become exponentially more common as the polymer length [17,18,19,20] or compactness increases [21,22]. Though formally defined in closed rings [23], open knots can be identified using suitable closures [24,25]. Knots tie and untie spontaneously via the polymer ends. In a polymer melt, for example, this occurs by the slithering motion of the polymer chains [26,27,28]. Their formation is often promoted by chain stiffness [29] or confinement. External forces influence knots differently depending on how the force is applied. Knots can either tighten [30] or untie [31] when pulling forces, for example, those applied in single-molecule experiments [32], are applied. However, the impact of common forces like shear (in its variety) on knots still poses a number of open questions, most notably, on whether knots generally rather tighten or untie subject to a particular type of flow.

Prior studies have shown that single ring polymers under simple (Couette) shear [33] exhibit a tumbling motion, which leads to a broad distribution of knot sizes, rather than fully tightened knots at all times. For linear chains an investigation on dynamics and topology of knots in steady shear flow [34] revealed repeated unknotting–knotting transitions in flexible chains. This is in contrast to Brownian dynamics simulations on the behavior of knotted chains in elongational flow [35], which have concluded earlier that knots untie. Other studies have driven knots by exposing knotted polymers to AC/DC fields [36]. In recent years, DNA knots stretched by elongational fields have been studied experimentally, where particular interests are the relaxation [37], dynamics [38], attraction [39], and untying [40] of the knots. Very recently, we demonstrated [41] that knots present in a linear polymer always tighten when placed in a Couette flow. In contrast to knots, the impact of shear on the conformational and dynamic evolution of polymer chains themselves has been the subject of intensive investigations for several decades now [42,43,44,45,46,47] and is currently much better understood.

While the influence of shear and elongational flows on knotted polymer chains has been studied previously, the specific case of Poiseuille flow presents a fundamentally distinct scenario that merits independent investigation. Unlike steady shear flow, which imposes a uniform velocity gradient, Poiseuille flow introduces spatial heterogeneity in the shear rate across the channel, leading to non-trivial polymer conformational dynamics. Prior studies have demonstrated that such flow fields induce local structural and orientational features that are not captured by linear shear flow [cf. Gompper et al. [48]]. Moreover, elongational flows—although also involving deformation—tend to untie rather than contract knots, highlighting the sensitive dependence of knot behavior on the flow details [35,37,40]. Given the absence of systematic studies on the impact of Poiseuille flow on intra-chain knots, and considering the relevance of such flow in microfluidic and biological environments, the present work addresses a critical gap in the literature and provides novel insights into the interplay between knot topology and non-uniform flow fields and gradients.

2. Model

We adopt a coarse-grained approach to describe the segments, modeling a polymer chain as a bead–spring system, where N effective spherical monomeric units are connected by appropriate effective potentials. Any two beads $(i,j)$ in the system, separated by a distance $r=|{\overrightarrow{r}}_i-{\overrightarrow{r}}_j|$, interact through the purely repulsive Weeks–Chandler–Andersen (WCA) potential:

$$U^{WCA}\left(r\right)=\hspace{1em}4ϵ\left[{\left({\displaystyle \frac{\sigma }{r}}\right)}^{12}-{\left({\displaystyle \frac{\sigma }{r}}\right)}^6+{\displaystyle \frac{1}{4}}\right],\hspace{1em}\mathrm{if}\hspace{1em}r\le r_c=2^{1/6}\sigma ;\hspace{1em}0\hspace{1em}\mathrm{if}\hspace{1em}r>2^{1/6}\sigma .$$

(1)

$ϵ$ determines the interaction strength in units of thermal energy $k_{B}T$, while $\sigma$ defines the characteristic interaction range. We use reduced units where $ϵ=1$ and $\sigma =1$. Consecutive monomers along the polymer chain are connected via the finitely extensible nonlinear elastic (FENE) potential [49],

$$U^{\mathrm{FENE}}\left(r\right)=\hspace{1em}-0.5k{r}_0^{2}ln\left[1-{\left({\displaystyle \frac{r}{r_0}}\right)}^2\right]\hspace{1em}\mathrm{if}\hspace{1em}r<r_0;\hspace{1em}\infty \hspace{1em}\mathrm{if}\hspace{1em}r\ge r_0.$$

(2)

with standard parameter values $r_0=1.5\sigma$ and $k=30ϵ/{\sigma }^2$. Equations (1) and (2) establish a well-defined bond length of ${\ell }_b\approx 0.97\sigma$, with fluctuations limited to approximately $0.03\sigma$, independent of chain stiffness. In this work, we examine fully flexible polymer chains consisting of $N=512$ or $N=1024$ athermal segments, maintaining a constant temperature of $T=1ϵ/k_B$ across all simulations.

The polymer chains are confined to a slit environment, bounded by smooth repulsive walls that extend along the X-axis and are normal to the Z-axis. The simulation box dimensions are set to $L_x\times L_y\times L_z=200\times 80\times 80$ for chains with $N=512$ and $200\times 160\times 160$ for those with $N=1024$. At equilibrium in the absence of shear, the polymer’s mean radius of gyration, $\langle R_g^2\rangle =N^{-2}{\sum }_{i=1,j>i}^N\langle {(\overrightarrow{r_i}-\overrightarrow{r_j})}^2\rangle$, is measured as $\langle R_{g0}(N=512)\rangle =18.85$ and $\langle R_{g0}(N=1024)\rangle =32.91$. To integrate the equations of motion, we employ the NVE ensemble for the chain segments while temperature equilibration is incorporated in the MPCD method by an NVT ensemble, with a time step of $d{t}_{MD}=0.0003{\tau }_{MD}$ for $N=256,512$, and $0.00025{\tau }_{MD}$ for $N=1024$. Here, the fundamental molecular dynamics time unit is given by ${\tau }_{MD}=\sqrt{m{\sigma }^2/ϵ}$.

In the Multi-Particle Collision Dynamics (MPCD) method, solvent particles are treated as ideal point particles with unit mass $m=1$ and follow alternating streaming and collision steps [50,51]. During the streaming phase, solvent particles move ballistically over a time interval of $d{t}_{\mathrm{MPCD}}=0.0025{\tau }_{MD}$. In the collision phase, particles are sorted into cubic cells of volume ${\sigma }^3$ (with a numeric density of $\rho =5{\sigma }^{-3}$ per cell), where they undergo stochastic momentum exchanges while conserving both local and global momentum. Collisions occur at intervals of $10\times d{t}_{MPCD}$.

We utilize the HOOMD-Blue $v2.9.4$ software package [52] on GPUs, incorporating an Andersen thermostat collision scheme [53] to regulate temperature and bounce-back collisions at the confining walls to ensure no-slip boundary conditions. Before each collision phase, cells are displaced randomly by a three-dimensional vector with components uniformly drawn from $[-\sigma /2,\sigma /2]$, and the velocities of MPCD particles are rotated by an angle of $\alpha ={130}^{\circ }$ relative to the center-of-mass velocity, ensuring Galilean invariance [54].

In the Poiseuille flow setup, a uniform body force with magnitude f is exerted on the solvent in the X-direction, driving the fluid. When polymer concentration is low, the solvent behaves as a Newtonian fluid with viscosity $\eta$, generating the well-known characteristic parabolic velocity distribution:

$$v_x\left(z\right)={\displaystyle \frac{\rho f}{2\eta }}\left({\displaystyle \frac{L_z^2}{4}}-z^2\right).$$

(3)

The flow intensity is characterized by the mean shear rate $\overline{\dot{\gamma }}=fH/2\nu$ [48,55], representing the shear rate averaged across the channel. Here, $H=L_z/2$ denotes the channel half-width, while $\nu =\eta /\rho$ corresponds to the kinematic viscosity. Since the kinematic viscosity is constant, we characterize the mean shear rate by the value of $\overline{\dot{\gamma }}·\nu$ later in this study.

To express our results in hydrodynamic terms, we define the Weissenberg number as $\mathrm{Wi}=\dot{\gamma }{\tau }_0$, where ${\tau }_0$ represents the longest relaxation time of the polymer chain. This relaxation time is determined by examining the exponential decay of the auto-correlation function:

$$C\left(t\right)={\displaystyle \frac{\langle \overrightarrow{R}(t_0+t)\overrightarrow{R}\left(t_0\right)\rangle }{\langle \overrightarrow{R}\left(t_0\right)\overrightarrow{R}\left(t_0\right)\rangle }}=exp(-t/{\tau }_0),$$

(4)

where $\overrightarrow{R}\left(t\right)$ is the polymer’s end-to-end vector (shifted by its mean), and $t_0$ is a reference time. For a coil of length $N=512$, we find ${\tau }_0=31,892{\tau }_{MD}$, while for $N=1024$, ${\tau }_0=214,285.5{\tau }_{MD}$. Our model for a Poiseuille flow [56] yields a value of $\eta /\rho =2.83k_{B}T{\tau }_{MD}$ for the kinematic viscosity, where $\eta$ is the dynamics viscosity of the fluid. Thus, one obtains the Reynolds number $Re=v\rho L_z/\eta \approx 1.7$ for a velocity v at body force $f=0.004.$

In the swollen coil state, the likelihood of encountering a knot in these chain lengths is generally around 1% or lower [22]. Knots in this regime tend to be weakly localized, occupying only a portion of the polymer and undergoing continuous fluctuations in both size and position along the contour.

Throughout simulations, we track the position and size of a pre-existing knot, which is initially introduced as a trefoil knot selected from equilibrium configurations. Knot analysis is conducted using the KymoKnot package [57], which identifies the knot type, location, and size by computing the Alexander polynomial [23] with a minimally interfering closure. Knot size determination follows a bottom-up approach, progressively incorporating monomers until a knotted region consistent with the overall chain topology is identified. The minimally interfering closure either connects the chain termini directly or joins them through the shortest distance to the convex hull, depending on whether the direct distance between the ends is shorter than the summed minimal distances of the termini to the convex hull of the polymer. Alternative methods for detecting knots in open chains without prior closure have also been developed [58,59].

3. Results

3.1. Effect of Shear on Knot Size in a Poiseuille Flow

We show in Figure 1a the temporal changes in knot size $K_n\left(t\right)$ over a longer period of time in a channel at zero body force f demonstrating that in the absence of shear $K_n$ undergoes moderate fluctuations of $\pm 21\%$ around its mean size (here $\langle K_n\left(t\right)\rangle =116.35$). Although the knot position undergoes a certain diffusion along the chain contour, (e.g., the center of the trefoil knot $K_n\text{-}\mathrm{mid}$ diffuses over ≈60 beads from its initial position in the polymer chain over the course of the simulation in Figure 1a), its location is found to change little during the whole time interval, similar to the radius of gyration, $R_g\left(t\right)$, of the chain. When a constant body force f is applied to the fluid particles, the polymer chain starts moving in the resulting flow. Beyond a certain threshold value $f^{cr}$, the knot undergoes a process of tightening at a rate which is found to increase linearly with the growing force f, shown in Figure 1b. Apparently, the tightening rate ${\dot{K}}_n=d{K}_n\left(t\right)/dt$ (determined from the slope of a linear fit to $K_n\left(t\right)$ during the tightening process, more details in Supplementary Section S1) of the long chain, $N=1024$, is significantly larger due to the stronger local shear acting on its building units. Figure 2 shows a series of snapshots of a knot, taken at different moments after the onset of flow, clearly demonstrating the gradual tightening of the knot down to only a few remaining beads. Note that in this study we employ only weak to moderate forces since rather strong body forces f may lead to knot removal by intersegmental passage. While this results in knot elimination similar to the action of type II topoisomerase in DNA [60], the underlying mechanisms are fundamentally different. The Supplementary Video provides a visualization of a possible stretched conformation, where the chain ends fall into regions in the vicinity of the walls where the flow is slower.

The variation of the knot size $K_n\left(t\right)$ in polymer chains with length $N=512$ and $N=1024$ with elapsed time is shown in Figure 3. While for $N=512$ and $f=5\times {10}^{-5}$ the knot size stays largely constant, Figure 3a, for $f\ge {10}^{-4}$ one observes a steady decline of $K_n$, which becomes faster with growing f along with intermittent short-lived fluctuations. The same qualitative observation is made for $N=1024$, although tightening occurs at lower body forces due to the larger shear on the larger chain. Note that occasional sudden changes (spikes) in the knot size are caused by the knot detection scheme, which can measure significantly different knot lengths between frames due to the closure scheme necessary for open chains (e.g., the minimally interfering closure of KymoKnot might use its bridging closure in one frame and the outer closure in the next; more details in Ref. [57]).

3.2. Cross-Correlation Between Chain Tumbling and Knot Size

Generally, the type of flow strongly affects the conformational response of the polymer molecule. Similar to the case of simple shear [41,62], one observes large conformational fluctuations in pressure-driven flow with a Poiseuille profile, whereby the polymer chain stretches and recoils [48]. In addition to tumbling dynamics, however, the transverse migration of polymers toward the center line of the channel is observed [55].

Our observations reveal an immediate relation between the appearance of recoil conformations and the ensuing abrupt changes in knot size. As shown in Figure 4a, the size distributions of trefoil knots for two different forces f differ markedly, revealing considerably larger knots for the weaker body force $f=0.00009$. Moreover, as suggested by the inset to Figure 4a, knots close to the minimum or maximum x-position relative to the chain (or ‘outer’ knots) are generally larger than those with central (or ‘inner’) relative x-positions if $f\gg f^{cr}$ (here, the relative x-position refers to the range of x-positions between the monomer with the smallest x-coordinate and the monomer with the largest x-position). This is due to the larger elongational forces acting on the center of the chain. The long tail of the overall trefoil length distribution at $f=0.00165$ is thus shown to be predominantly caused by knots with outer relative x-positions, while the occurrences of very tight trefoil knots are predominantly caused by knots with inner relative x-position. This point is further illustrated by Figure 4b: When the knot positions (snapshots, bottom of the panel) are far from the relative termini $x_{\mathrm{rel}}=0$ or $x_{\mathrm{rel}}=100\%$ then the knot size is small (cf. inset in Figure 4a), and the knot position is constant, indicated by the fairly thin and constant kymograph bands (spanning from knot start to knot end positions) in the top of the panel in Figure 4b. Conversely, if the knots lie close to the relative end points of the chain, then the knot becomes larger and the knot center can move. Since the knot thus alternates between tightened and loose states as the polymer tumbles, the knot is “breathing” [63]. This correlation between tumbling and knot breathing is qualitatively similar to the behavior we previously observed for knotted polymers in Couette flow [41]. We performed additional simulations of a polymer with $N=256$ monomers and analyzed further configurational observables and their correlations to tumbling and knot breathing in Supplementary Section S2.

Figure 4c illustrates the process of tumbling (or recoiling) and how the knot size and relative x-position evolves over a typical recoiling process. The snapshots correspond to the trajectory shown in red in panel (b). When a chain is in a stretched state, then the monomer positions near the ends of the chains fluctuate more than central monomers due to the elongational forces. Knots tend to be tightened in such configurations, as the shear at the knot center position is likely to be large enough to tighten the knot. As the chain starts to recoil, the knot’s relative x-position can reach an extreme value $x_{\mathrm{rel}}=0$ or $x_{\mathrm{rel}}=100\%$, where the local elongational forces weaken such that the knot center position is no longer stationary and the knot sizes increase. Note that the change in knot center position is small compared to the change in $x_{\mathrm{rel}}$ over time, indicating that the change in $x_{\mathrm{rel}}$ is dominated by the recoiling, not by the diffusion of the knot along the chain. As the recoiling process continues, the knot center position eventually reaches the opposite extreme $x_{\mathrm{rel}}$ position before the chain stretches again, likely leaving the knot in a more central relative x-position, thus fixing and tightening the knot again.

3.3. Effect of the Channel Width on Knot Tightening

Given the established result that Poiseuille flow generally leads to knot tightening, an important aspect of this process concerns the influence of channel width $2H$ on the tightening rate ${\dot{K}}_n\left(t\right)$. According to Equation (3), the maximal velocity is $v_x^{max}(z=0)\propto f{L}_z^2$, so one would trivially expect ${\dot{K}}_n\left(t\right)$ to rise rapidly with increasing channel width $L_z$ at fixed f. Less obvious is the influence of channel width on the tightening rate ${\dot{K}}_n\left(t\right)$ in channels with different width at equal mean shear rate $\overline{\dot{\gamma }}$.

In Figure 5, we show three different aspects characterizing the impact of channel width $L_z$ on polymer chains in a Poiseuille flow that unambiguously demonstrate the increase in efficiency of knot tightening with growing channel width even in the case of equal mean shear rate $\overline{\dot{\gamma }}$. The efficiency of knot tightening, cf. Figure 5a, is found to increase steadily with the channel width increase as also does the degree of chain stretching (i.e., the effective shear-induced pulling on the knot), cf. Figure 5b, suggested by the growth of the two largest eigenvalues ${\lambda }_1,{\lambda }_2$ of the $R_g^2$-tensor. Since ${\lambda }_1$ describes the strongest stretching of the macromolecule along the flow, with growing ${\lambda }_1$, the chain segments of the macromolecule experience increasing difference in local flow velocity; therefore, any increase in ${\lambda }_1$ leads to a stronger shear-induced pulling force on the knot and and thus to increased tightening. The typical correlation time ${\tau }_0$ of the end-to-end vector, ${\overrightarrow{R}}_e$, becomes also progressively shorter with growing channel width $L_z$ at equal $\overline{\dot{\gamma }}$, see Figure 5c, indicating a faster renewal of the driven chain conformation with growing $L_z$. In Figure 5d, we compare the velocity profiles of the MPCD fluid particles in the narrow, $H/R_{g0}=0.53$, and medium-wide channel, $H/R_{g0}=0.85$, demonstrating nearly perfect parabolic profiles of the flow. For medium-wide $H/R_{g0}=0.85$, $f=0.00103125$ corresponds to a mean shear $\overline{\dot{\gamma }}$ equal to that for $f=0.00165$ in the narrow $H/R_{g0}=0.53$ channel, while $f=0.00165$ shows the effect of equal body force f for the two different half-widths H.

Based on the established velocity profiles of Poiseuille flow, one may estimate the effective shear acting on the polymer at equal mean shear $\overline{\dot{\gamma }}$ by taking into account the distribution of chain segments exposed to local shear $\dot{\gamma }\left(z\right)$ (or $\dot{\gamma }(z/H)$) across the channel averaged over particular time intervals ${\overline{P}}_{mon}(z/H)$, Figure 6. We obtain the effective shear rate acting on the monomers by calculating the expectation value of the shear rate $|\dot{\gamma }(z/H)|$ with respect to the monomer distribution ${\overline{P}}_{mon}(z/H)$ across the channel averaged over a specific time interval of the simulation ${\dot{\gamma }}_{mon}={\int }_{-1}^1{\overline{P}}_{mon}(z/H)\left|\dot{\gamma }(z/H)\right|d(z/H)$. As indicated by the insets to Figure 6a, the effective shear acting on the respective chain segments during the first $10\%$ of the simulation significantly increases with growing channel width $H/R_{g0}$ notwithstanding the equal mean shear rate $\overline{\dot{\gamma }}$. The chains prefer to move towards the channel walls during the beginning of the simulations as can be seen by the more asymmetric monomer distribution ${\overline{P}}_{mon}(z/H)$ in panel (a). This leads to stronger shear experienced by the monomers ${\dot{\gamma }}_{mon}$ in wider channels since they can reach higher shear regions closer to the wall in units of $z/H$ more easily. At later times, the Poiseuille flow progressively drives the chain towards $z=0$, Figure 6b, in line with previous observations [55].

4. Methods for Experimental Verification

While there are many theoretical studies on knot topology and behavior in conjunction with DNA and proteins, there are currently very few experimental studies. Estimating the tightness of a knot in a polymer experimentally is challenging but feasible using techniques that probe conformational constraints, local density, and spatial distribution of the knot within the chain. Thus, there are several techniques to estimate the knot’s spatial extent and how confined it is in the polymer backbone:

Atomic Force Microscopy (AFM) or Cryo-Electron Microscopy (cryo-EM) whereby polymers (like DNA or proteins) are deposited on a surface and imaged with tight knots appearing as high-density regions with distinct crossings. Spatial resolution then allows visual estimation of knot diameter or area [30].

Optical or Magnetic Tweezers where a single polymer chain is stretched by its ends. When a knot is pulled, it slides and tightens while resistance to sliding indicates entropic tightening. Measured force–extension curves of tighter knots result in force plateaus or sharp increases. Also the distance the knot migrates before getting trapped at an endpoint is indicative [64].

Fluorescence Resonance Energy Transfer (FRET). Attaching donor and acceptor dyes near suspected knot sites leads to higher FRET efficiency, e.g., with shorter distances (i.e., tighter knots) [65].

Nanopore Translocation, when knotted polymers are electrophoretically driven through nanopores. Tight knots slow down translocation and may cause blockade spikes in ionic current. The depth and duration of current blockade is linked to knot compactness [8].

Gel Electrophoresis (for DNA) shows that knotted DNA migrates slower than unknotted DNA, while tighter knots have reduced electrophoretic mobility. Combined with enzyme treatment (e.g., topoisomerases), this method can infer knot position and size [66].

A possible verification of the present computational results by the aforementioned methods would extend and corroborate our observations.

5. Conclusions

Given the insufficient understanding at present regarding the behavior and functionality of knots in macromolecules and proteins, and the influence of external forces on knot properties, the main objective of the present study is the clarification of the impact of pressure-driven Poiseuille flow on existing knots in linear polymer chains. To this end we carried out extensive mesoscale molecular dynamics simulations taking into account the hydrodynamic interactions by means of the Multi-Particle Collision Dynamics (MPCD) method. As a main result of the studies performed for a variety of body forces f, channel widths, $L_z$, and chain lengths N, one should point out the revealed general trend to knot tightening which takes place beyond a certain critical body force $f^{cr}$ and intensifies with growing f.

In addition, we have found that thermal fluctuations play a dramatic role in the dynamics of confined polymer chains under flow, for example, leading to tumbling motion associated with large conformational changes. The observed strong cross-correlation between intermittent chain tumbling and ensuing knot size fluctuations has been demonstrated to depend essentially on the knot position relative to the chain, indicating diminishing knot size stability as the proximity of the knot to the most extreme relative x-positions (largest x-position of all monomers or smallest x-position of all monomers) increases.

An interesting and non-trivial effect of the interplay between confinement and hydrodynamic interactions is also the observed impact of the channel width on the knot tightening rate ${\dot{K}}_n$. For polymers in the crossover from weak to strong confinement, $H/R_{g0}$, the efficiency of knot tightening is found to diminish with increasing confinement even at equal mean shear rate $\overline{\dot{\gamma }}$. Using the probability distribution of chain segments across the channel in conjunction with the local shear rate of the flow acting on them, we have shown that this result is a consequence of the steadily increasing shear acting on the chain conformation during tightening as the channel width grows [8,30,64,65,66].