# Knot Factories with Helical Geometry Enhance Knotting and Induce Handedness to Knots

^{*}

## Abstract

**:**

## 1. Introduction

_{g}, and chain extension, R, are greatly affected if polymeric chains are constrained by confinement in narrow spaces [12,13]. The effect of confinement arises from competition between geometric measures of confining spaces and metrics of a polymer chain in unperturbed state. In the case of polymers, the effect of confinement is expressed as confinement strength given as a ratio of polymer stiffness, P, and a geometric characteristic of confinement such as width or diameter of confining channels, D. Here, several regimes of confinement strength on the scaling of polymer metrics have been described. For very large, basically, open spaces characterized by D >> P, scaling of polymer properties such as the gyration radii followed de Gennes regime [14], which describes polymer as a succession of blobs of a diameter D. In such an extremely weak confinement, the metric properties of polymer chains are unconstrained and the same as in the case of unperturbed polymers in the bulk. On the other hand, if the geometric dimensions of confining spaces D are much smaller than the characteristic dimension of polymer P, D << P, the scaling properties of polymers are described by Odijk’s deflective regime where the polymer segments are stretched as the confinement prevents the polymer chain from folding on itself [15]. In between these two boundary regimes, where the ratio between polymer rigidity and geometric dimensions of confinement D/P~1, existence of multiple intermediate and transition regions have been predicted [16], such as extended de Gennes regime or back-folding Odijk’s regimes [17,18]. Polymers in a confined state can be found in nano-technological applications such as polymer nano-composites, where the polymers interact with structurally rigid nano-particles of a filler, or polymers are squeezed between layered nano-clays, but the confined systems include also polymer blends with large difference of T

_{g}[19,20,21]. In the case of DNA, that is a bio-polymer and vital molecule for life whose production greatly exceeds man-made polymers [22], the DNA is found in confinement naturally when it is tightly packed in cells nuclei or viral capsids. In addition, in the case of the DNA, the confinement is encountered in nanotechnological devices used in single molecular experiments to study DNA, where the confinement is used to keep DNA stretched and exposed to sensors [5,23].

_{0}<< 1) and weak (R/R

_{0}> 0.5) compression regimes characterized by the ratio of the chain extension R in compressed and unperturbed states, R

_{0}[38]. The MC simulations were used to study force displacement relations with conformational and free energy changes during compression of linear [32,33] and ring polymers [34] and indicated different energy costs of confinement strength and external force on polymer extension with potential implications for experiments in nanofluidic chips. Computer simulations of compressed confined polymer chains employing different mechanisms of compression performed whether by pulling the distant parts of polymer chain against each other [35], piston compression [37,39] or compression by flux of media [31,40] repeatedly reported a new topological phenomenon of chiral helical structures formation. Extensive computer simulations of compression allowed exploring parameter space and revealed complex topological behaviour of emerging self-entanglements with a promise of producing knots with a desired topology by controlling compression “waiting time” and applied compressive force [36].

_{H}= 0.03 R

_{ch}. Therefore, in the current work we also explore the aspect of different application of the helical channels, i.e., whether and at which extent of compression in helical channels similar to the experimental setup of knot factories can produce knots with given chirality induced by handedness of the helical channel.

## 2. Materials and Methods

#### 2.1. The Model of DNA

_{s}(r) = k

_{s}(r − r

_{0})

^{2}, where the constant k

_{s}represents penalty against stretching of the bond and r

_{0}is an equilibrium distance that is set to 1 σ. We limited the stretching of the bonds in order to prevent artificial passages of the discretized DNA strands at high compressive forces by using k

_{s}= 80ε

_{0}, where ε

_{0}= k

_{B}T. The second bonded interaction is used to introduce bending stiffness of the DNA molecules. The bending stiffness was modelled by harmonic interaction U

_{b}(θ) = k

_{b}(θ − θ

_{0})

^{2}, where k

_{b}represents penalty against bending of the DNA chain and θ

_{0}is the equilibrium angle set to θ

_{0}= π. The constant k

_{b}is set to 20 σ/ε

_{0}what corresponds to the experimentally common value of persistence length P = 50 nm [46,47]. The volume of the polymer chain is modelled by the non-bonded excluded volume interaction in the form of fully repulsive truncated and shifted Lennard-Jones potential U

_{ex}(r) = 4ε

_{0}[(σ/r)

^{12}− (σ/r)

^{6}+ 0.25], if r < 2

^{1/6}σ and U

_{ex}(r) = 0 otherwise. The DNA in all simulations was considered nicked and torsionally relaxed. The torsional stiffness can be induced by using some of our previously developed models [48,49,50] for the price of increased computational costs. However, for small DNA chains below 1 kb the torsional relaxation of linear chains is very fast, and in our previous work we have also shown that the torsionally stiff molecule is able to quickly relax emerging supercoiling by effusion through chain ends, if the molecule is short enough (100 beads), so no differences on stereospecificity were observed as compared to simple model of DNA without torsional stiffness, (Figure S2 in [44]). Hence, our current model can represent DNA with 2 nicks 100 beads distant. All the simulations were performed by using Extensible Simulation Package for Research on Soft matter [51,52]. We performed Langevin molecular dynamics simulations with solving equations of motion m$\ddot{r}$ = −γm$\dot{r}$ − $\nabla $U(r) + R(t)(2ε

_{0}mγ)

^{0.5}, where r represents position vectors of the beads, $\nabla $U is the force acting on the bead calculated from bead’s interactions described above γm$\dot{r}=$ ξv is a damping constant in units of reciprocal time and the last term represents implicit random kicking force from surrounding media, where R(t) is a delta-correlated stationary Gaussian process. The equations of motion were integrated with the time step d τ = 0.01 τ. The physical dimension of the time unit [τ] = 74 ns [45]. After inserting polymer chain into the channel, we performed a pre-equilibration run of 10

^{9}MD integration steps equal to 10

^{7}τ, followed by the production run of the same duration, during which we collected 5000 samples for analyses, adding also 5 repeated production runs for each setting of confinement strength D/P and compression force F (see below). The length of our pre-equilibration and production runs is 10 times longer than used in the previous simulations works [35,36], but we did fewer repeated runs. We arranged our simulations in the way the compression uses longer “waiting time” instead of cycling compression/decompression cycles, based on the previous work that showed that an equilibrated entangled state was reached in one cycle when using long waiting times [36].

#### 2.2. The Model of Helical Nanochannels

_{0}(t) = k$t\widehat{i}$ + R

_{H}cos (ωt) $\widehat{j}$ + R

_{H}sin (ωt) $\widehat{k}$ [53], where t is a periodic parameter in radial space and ω gives a subtended angle as t increases and ω carries also information on the handedness of the helix whether it has plus, +ω, or minus, −ω, sign. Unlike cylinders, helical tubes are characterized by three additional parameters in addition to the diameter of the channel [53], D = 2 R

_{ch}: radius of the helix, R

_{H}, the pitch k denoting the distance between helical loops equal to d

_{H}= 2 πkσ and handedness of the helical tube, sign (ω), that can be negative or positive. The radius of the helix in all simulations with helical channels is R

_{H}= ⅓ R

_{ch}, based on our previous MD simulations that determined the strongest effect of channel helicity on chiral properties of knotted polymer for R

_{H}in the range between ⅓R

_{ch}to ½R

_{ch}[44]. Additionally, simulations with R

_{H}= 0 were performed in order to obtain trajectories for cylindrical geometry. Similarly, the setting of the pitch k = D/(2 π) was chosen based on our previous investigations of knotted polymers in helical channels [44]. We performed simulations with various ratios of confinement strength D/P = 2

^{i}where i = −1, 0, 1, on the borders of strong, intermediate and weak confinement. In our previous MD simulations [44], we have used a helical tube that was modelled explicitly by constructing the walls of the channels using beads with diameter of 1 σ. Despite the positions of the explicit beads forming the walls of the channels being fixed during integrations (i.e., their equations of motions did not have to be solved), the pairwise excluded-volume interactions between the polymer and the walls still had to be computed, thus making the use of explicit walls computationally heavy. Because of the length of the intended simulation runs of 10

^{9}integration steps per each trajectory, and especially also radius of the channel R

_{ch}as free parameter, we developed an implicit model of the confining walls of helical channels. For this purpose, we needed to implement a function that computes the nearest distance of a bead to the helical wall provided along with its norm vector to be passed into the simulation engine of the simulation software ESPResSo. The developed implicit constraint uses a solution derived for solving mechanics of celestial bodies (Algorithm 1 in [54]), solving Kepler’s equation by Newton’s non-linear iterative method using an initial estimate for eccentric anomaly E (Equation (10) in [54]). The initial estimate made the MD simulations stable, converging and fast. Note, that in the case of a point laying outside of a helix, there could exist several valid solutions, however, we think this is not the case when a bead is contained inside a helical tube, 2 πk > R

_{ch}. We experienced that the implementation of the algorithm to model implicit helical confinement was fast and stable during hundreds of repeated runs with 3 × 10

^{11}distance calculations performed for each bead in every trajectory.

#### 2.3. Piston Compression

_{H}= 0σ) and helical (R

_{H}= ⅓R

_{ch}) channels (see Section 2.2), also with an impenetrable wall at the bottom of the channels. The piston bead had a very large radius of 100σ. We used a range of compression forces covering region from weak compression forces Fσ/ε

_{0}≤ 0, 0.1, 0.35, 0.5 and 1 to the strong compression Fσ/ε

_{0}≥ 2, 5, 10, 15 and 20 similarly to previous works [34,35,36].

#### 2.4. Topological Analyses

## 3. Results and Discussion

#### 3.1. DNA in Cylindrical versus Helical Confinement

_{H}= 0 in our implicit model for helical channels) in order to have access to all polymer metrics and properties that could show useful to demonstrate distinctive behaviour of the chain in channels with helical geometry. One of the most basic metrics of polymer chain is expressed by radius of gyration of the polymer and its components.

_{t}= C/L. Tortuosity increases the effective distance between two points on the helix, hence, if the polymer would localize mainly in the centre of the channels, following the channels curvature, its length would be shorter than the length of the polymer in cylindrical channel by factor of $f={[{(2\pi k)}^{2}+{R}_{\mathrm{H}}^{2}]}^{0.5}/2\pi k$ ~20% [44]. Since the polymer in the helical channels appears to be slightly more extended than in the cylindrical confinement, it demonstrates that the interplay between the chain stiffness and the helicity of the channel does not allow the polymer to freely explore loops of the helix, and hence the helical confinement in terms of the polymer extension appears to be slightly stronger than the cylindrical one. In such case, the polymer would stay in the inner section of the channel that would have in our setting R

_{H}= 0.3 R

_{ch}diameter of D

_{in}= 2/3 D. This is partially confirmed by the plot of the transversal component of the gyration radius, in Figure 1b. The transversal radius of gyration represents distribution of polymer into lateral sides of the channel [57]. Figure 1b shows, that the ${R}_{\perp}$ increases with the increasing diameter of the channel and decreasing the confinement strength D/P indicated on the plots by numbers D/P = 0.5, 1.0 and 2.0. At the same time, the running averages show that the distribution to the lateral sides of the channel is smaller for the helical channels. The average values of the lateral distribution by transversal gyration radius are not a sensitive quantity to fully capture the distribution of the chains across the cross section of the channels, and we will demonstrate this additionally later by calculating radial distributions of monomers, number densities on the surface and planar projections of the monomer distributions. The values of the transversal gyration radii however show that the difference between the values obtained in the helical channel and the cylinder differ less than by one third, hence the polymer does not reside exclusively in the inner section of the helical channel away from the helical grooves of the channel.

**r**

_{i},

**r**

_{j}), i ≠ j and i,j $\in $ N. The span of the molecule can be obtained by using all cartesian coordinates x, y, z, sometimes denoted as R≡S (

**r**) or, alternatively, by using only the coordinate along the major axis of inertia of the channel, denoted as R≡S (x). In our work, we are using primarily the span defined as R≡S (x), as the data on compression with large forces and very large diameters inferred a possible chain size bias when the chain laid flat on the bottom of the channel, and the values of the maximum span obtained as S (

**r**) were distorted by showing non-monotonous behaviour with the compressive force. In Figure 1c, we show the extension of the DNA in terms of chain span R as a function of confinement strength D/P. The values are shown in black and orange colours for cylindrical and helical channels respectively, together with the statistical errors. Again, the pre-extension induced by helical and cylindrical channels results into very similar values, where the obtained difference is statistically reliable mainly in strong confinement D/P = 0.5. The values of the span obtained in the cylindrical channel is R = 266.7 ± 3.3 σ and for the helical channels R = 271.2 ± 2.6 σ, hence the difference between the values of the R is larger than is the calculated standard deviation error. The values in the region between D/P = 0.5 to 1.0 can be well approximated by theoretical prediction given by equation R = L [1 − A (D/P)

^{2/3}] (the dashed line in Figure 1c, where A = 0.1701 was obtained for a cylindrical channel [58]. The value of A seems to be slightly smaller for the helical geometry, but since we have only a single data point and the difference from predicted value is within the statistical error, we currently use this value as satisfactory for prediction of the polymer span also for the helical channels (with R

_{H}< R

_{ch}). In the region between D/P = 1.0 to 2.0, the dependence of the chain extension is approximated by a line corresponding to a power law fit that predicted the exponent R ≈ (D/P)

^{−1.02}(the dotted line, in Figure 1c [17]). The small differences arising from helical and cylindrical geometries are probably related to the setting of the pitch of the helical channels. The pitch of the helical channels, k, is set so that the distance between helical loops, d

_{H}, scales with the strength of confinement d

_{H}= ½P, P and 2 P respectively. The deflection length of a polymer in cylindrical channel of diameter D is λ = D

^{2/3}P

^{1/3}[16], that corresponds to 12.6, 20 and 31.7 σ in the investigated confinement strengths D/P = 0.5, 1.0 and 2.0. Since the deflection length is slightly larger λ > d

_{H}only for the regime of D/P = 0.5, this might be responsible for observing significant differences in the scaling of the polymer metrics between helical and cylindrical channels only in the strong confinement regime.

_{B}TlnP (R), while by differentiation one obtains force F = −dA (R)/dR that acts on the endpoints of the chain in attempt to restore unperturbed equilibrium properties. It has been shown, that this method originally developed to study elasticity of polymers, can be used to calculate the pre-stretching force applied by the confinement to the polymer chain [32]. The computed probability distributions indicate existence of a higher pre-stretching force in helical channels that will counteract also the external compressive forces. As an inlay in Figure 2, we show also representative snapshots of prevalent conformations of the polymer chain together with simplified impressions given by letters, as discussed above along with the discussion of Figure 1a.

_{ch}(0.5 σ)

^{2}[(n + 1)

^{2}− n

^{2}]. The distributions were normalized in the way that the obtained curves represent the probability of finding a monomer of the chain with length N in the radial distance from the centre of the channel, r/R

_{ch}. It is also important to note, that in the case of helical channels, in order to calculate the distance from the helical centre, we employed the same algorithm described in [54] that was implemented to calculate the distance of beads from walls during solving equations of motions in molecular dynamics simulations (see Section 2.2). The data in Figure 3a show that the radial distributions of monomers are mis-shaped as compared to channels with cylindrical geometry as a result of tri-axial symmetry breaking in the channels with helical geometry. The distributions in the helical channels shows slight increase of the monomer concentration in the middle of the channel (r/R

_{ch}= 0). Then, the line crosses the distribution computed for cylindrical channels two times, suggesting that the monomers would concentrate on the surface of the inner ridges (threads) of the helical grooves. In addition, we have calculated an integral of the number density of monomers (${\phi}_{0}$) at the surface layer δ = ⅕ σ thick (inset of Figure 3a). By this approach, that we used also earlier on studies of entropic segregations, we have calculated the confinement free energy A

_{C}$=2\pi {L}_{\mathrm{ch}}\int {\phi}_{0}\left(r\right)dr$

_{,}for r $\in $ <R

_{ch}− δ; R

_{ch}> [29]. The calculation shows that despite smaller lateral distributions of monomers, indicated by transversal gyration radii and radial distributions also indicating increase of the monomer concentration in the middle of the helical channels, the distribution of the monomers in the cross section of the channel concentrates around inner ridges of the helical grooves (see also heatmaps in Section 3.2). The increased concentration of monomers in the inner channel with ⅔D diameter is also demonstrated by computing radial distribution function from the major axis of inertia of the channel (Figure 3b).

#### 3.2. DNA under Compression in Cylindrical and Helical Confinement

_{0}= 0.35, while it vanishes at Fσ/ε

_{0}= 0.2 in cylindrical channels.

_{0}= 0.5, the chain already forms double-backfolded “U” shaped hairpins and partially triple-folded structures (snapshots in Figure 2 and Figure 6). For the compressive forces Fσ/ε

_{0}= 0.5 we did not observe higher order folding in the strong confinement D/P = 0.5. For strong compressive forces above Fσ/ε

_{0}> 1, data computed in Figure 5a indicate collapse of the chain span, which is illustrated by the snapshots in Figure 6a. At very high compressive forces, Fσ/ε

_{0}= 20, the chain exhibits effect known as spooling encountered in DNA tightly packed in bacteriophages [62,63] and viral capsids [64]. The orientation of the spools in the narrow channels, D/P = 0.5, is longitudinal with the chain aligning with the direction of the main axis of the inertia of the channel while their orientation changes in larger channels and they wing around the main axis of the inertia of the channel (Figure 6a–c). In the helical channels, the spools are also distorted and skewed following the helical curvature of the channel. The monomer radial distribution function for helical and cylindrical channels obtained for D/P = 0.5 are compared in Figure 6d for 3 compressive forces Fσ/ε

_{0}= 0.1, 1 and 20. The comparison shows that with increasing compressive force the monomers shift towards the walls, while the monomer concentration in the middle of the channel decreases. This process is more prominent in cylindrical channels where in the case of the highest compressive force Fσ/ε

_{0}= 20 the monomers are expelled from the middle of the channel with the probability of finding the monomers close to the walls being higher than in the middle of the channel.

^{Y}with the exponent Y = −9/4 = −2.25 [38], shown by the dashed line. The concatenated fits over our computed data yielded a value of the exponent Y = 2.108. The deviation from the established value arose mainly from the plateau region at D/P = 0.5 that is not in the interval of values of D/P where the theoretical fit was originally designated.

_{0}≥ 5 and wider channels characterized by D/P = 1 and 2, the DNA chains are very much collapsed, so that the molecular span R is smaller than the radius of the channel. In relation with this observation, we expect a chain length bias can be encountered in the data. For example, in Figure 7, we show the number concentration of monomers on the surface of the channels computed as number of monomers in a thin layer of δ = ⅕ σ that corresponds to the confinement free energy [29]. We observe, that for strong confinement D/P = 0.5, where the R of the molecule does not drop below the diameter of the channel, the number concentration of the monomers obtained in helical channels is larger as compared to the cylindrical ones in the whole investigated range of the compressive forces, consistently with the observations made for uncompressed polymers discussed in the previous Section 3.1. On the other hand, in the case of larger channels with D/P ≥ 1, the concentration of monomers obtained for helical channels crosses the curve obtained in cylindrical channels (indicated by arrows). The computed concentration dependences cross at Fσ/ε

_{0}= 2 for D/P = 2 and at Fσ/ε

_{0}= 1 for channels with D/P = 2. The chain size bias is also related to the fact that the concentration of monomers on the bottom is not counted to evaluate effect of walls, and the cross section of helical channels has larger area than the cylindrical geometry.

#### 3.3. Topology of DNA under Compression in Helical and Cylindrical Confinement

_{0}= 0.1, 0.5, 1.0, 5.0 and 20.0, and a dashed line showing the exponential decay of the orientational correlations in the bulk for reference, obtained as <cos θ> = exp (−s/P) [61]. The orientational correlations computed for very small compressive force Fσ/ε

_{0}= 0.1 show very similar behaviour as observed for uncompressed confined DNA, shown in Figure 4 and discussed in Section 3.1. The orientational correlations show three distinct regions given by separation of monomers along the chain s/σ. First, at s ≤ D, the correlations show an onset of exponential decay delimited by a shallow minimum, next evolving a plateau region in strong confinement and producing elastic stiffening at D/P = 2, and finally at s → L the orientational correlations quickly drop as a result of random motions of polymer ends.

_{1}and 3

_{1}m, and single achiral knot 4

_{1}, are shown, to more complex knots that could be identified and named according to the Rolfsen table [72]. These are shown with different shades of colour and separated by lines, together with the colour scale of k

_{n}in the associated legend, where the number indicates the number of crossings and index n denotes the knot type as shown in the Rolfsen table.

_{0}~0.1 no knots are detected, what is consistent with observation of extended “I” shaped conformation of chains, as discussed with Figure 1 in Section 3.1 and as shown also as snapshots in Figure 6a, and indicated also by orientational correlations functions in Figure 8a,d. As the compressive force increases Fσ/ε

_{0}~1, the chain becomes more readily folded with emergence of simpler knots with smaller number of crossings indicated by the computed knotting probabilities. At even higher compressive forces 1< Fσ/ε

_{0}< 5, one can observe that very complex knots with crossing number larger than 11 come more and more into play to represent topology of the chain. The probability of occurrence for the spectrum of simpler knots, with crossing number between 3 to 11, also widens. At very high compressive forces, Fσ/ε

_{0}> 5, the unknots (0

_{1}) are heavily suppressed, and the knotting probability is dominated by the occurrence of very complex knots. At the same time, also the probabilities of the spectrum of simpler knots are suppressed. The distribution of knotting probabilities in the cylindrical channels appears to be, however, much wider that that observed for helical channels, suggesting that the helical geometry of the channels enhances knotting as compared to cylindrical channel.

_{0}< 1, the probability of them existing in unknotted state is however smaller, than observed for strong confinement. This is can be related to the fact, that at weaker confinement smaller compressive forces are needed to induce folding, as shown in Figure 6b and discussed also with the orientational correlations and effect of elastic softening in Figure 8. At high compressive forces, Fσ/ε

_{0}> 5, the computed data start showing instability prominent especially for cylindrical channels. As we discussed earlier, this instability was foreseen as we became aware of the size bias of the chain due to the finite number of monomers and rapid increase of volume in wider channels explored (discussion to Figure 7). As the conformation of the polymer transforms to the spooled structure, the knotting probability associated with this transformation may drop. In the case of the helical channels, they still induce some asymmetry due to tri-axial symmetry breaking in the helical geometry, hence, the monotonous appearance of the knotting probability dependence is not so affected.

_{0}= 0.1. This is consistent with observation of spontaneous knot formation on DNA confined in nano-channels with diameter 100 nm [73,74]. At higher compressive forces, Fσ/ε

_{0}≥ 1, the knotting probabilities seem to stabilize to a plateau region, as the chains are already spooled at the bottom of the channel, as indicated by Figure 8c,f. The knotting probabilities in helical channels still exhibit formation of complex knots, with a crossing number larger than 11, that is not observed for cylindrical channels, probably as the helical channels still maintain some effect of asymmetry acting on the chain.

_{H}has more space available around it.

_{0}= <0;5>, and in the case of the channels with D = 2 P, the data unaffected by chain bias reach only to Fσ/ε

_{0}= 1.

## 4. Conclusions

_{ch}, radius of the helix, R

_{H}, pitch of the channels, k, and their handedness distinguished by the sign. The investigated were channels with three different widths, in order to investigate also the effect of the confinement strength, D/P = 0.5, 1.0 and 2.0. The radius of the helix was chosen based on our previous work that showed the chiral properties were exhibited in maximum extent for R

_{H}= 0.3 R

_{ch}. And stereospecific effect vanished as R

_{H}$\to $ R

_{ch}. The pitch of the helical channels was chosen D/(2 π) in order to maintain a geometrical self-similarity of the channels upon scaling dimensions of the channels.

_{H}, scales with the confinement strengths 10, 20 and 40 σ. In these conditions, the deflection length in the channels is similar to the distance of the helical loops λ = D

^{2/3}P

^{1/3}= 12.6, 20 and 31.7. This is consistent with the main scaling differences between helical and cylindrical channels observed in the channels with D/P = 0.5.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**DNA confined in nano-channels without external compressive force. Data for cylindrical channels are shown in black and helical channels are in orange. The confinement strengths investigated are indicated by numbers D/P = 0.5, 1.0 and 2.0. Panels (

**a**,

**b**) show evolution of instantaneous values of gyration radius and transversal radius from the initial structure. The letters I, J, U, and W show impressions of DNA conformations. (

**c**) average values of polymer span as a function of confinement strength D/P. The dashed and dotted lines show theoretically predicted values.

**Figure 2.**Histograms showing probability distributions of DNA span, R, obtained from simulations of DNA confined in nano-channels without external compressive force. The histograms obtained for cylindrical and helical nano-channels are shown in black and orange colours respectively. The panel shows also snapshots of the DNA chain in a rainbow colour scale from simulations for typical conformations represented by simplified impressions using the letters I, J, U and W. The numbers indicate confinement strength indicated as the D/P ratio 0.5, 1.0 and 2.0.

**Figure 3.**Radial distributions of DNA monomers obtained from simulations of DNA confined in nano-channels without external compressive force and calculated (

**a**) from the centre of the nano-channel and (

**b**) from the major axis of inertia of the channels in normalized radial coordinates, r/R

_{ch}. The values computed for cylindrical and helical geometries of the channels are shown in black and orange colours respectively. The numbers indicate regime of the confinement strength expressed as the ratio D/P. The inlay on the panel (

**a**) shows also the dependence of the confinement free energy, A

_{C}, obtained as the integral of the monomer concentration on the surface of the channels [29].

**Figure 4.**The orientational correlation functions along the coarse-grained curvature s/σ of the DNA polymer as obtained from simulations of DNA confined in nano-channels without external compressive force. The values obtained for cylindrical and helical channels are distinguished by black and orange colours and the strength of confinement in terms of the ratio D/P is indicated by number 0.5, 1.0 and 2.0. The the dashed line corresponds to the decay of orientational correlations of unperturbed DNA in the bulk <cosθ> = exp (−s/P) [61].

**Figure 5.**The evolution of polymer metrics during compression of DNA in helical and cylindrical nano-channels. In the plots, the black lines correspond to cylindrical channels, blue lines and orange lines to helical channels with negative −ω and positive +ω handedness respectively. The regime in terms of confinement strength expressed as the ration D/P = 0.5, 1.0 and 2.0 is indicated by numbers along the computed values of the span. Panels (

**a**,

**b**) show average extension of the DNA, R, for weak Fσ/ε

_{0}≤ 2 and strong Fσ/ε

_{0}≥ 2 compressive forces respectively. (

**c**) Plot shows the extension of the DNA in log-log scale, while the dashed line corresponds to the theoretically predicted relation R ≈ F

^{Y}, where Y = −9/4.

**Figure 6.**The compression of DNA in helical and cylindrical nano-channels. Panels (

**a**–

**c**) show planar projection heatmaps of distributions of DNA monomers across the channel together with snapshots for cylindrical channels and helical channels with negative and positive handedness. F’s with the arrows indicate the direction of increasing force (Fσ/ε

_{0}= 0.1, 0.5, 1, 5, 20); (

**d**) the panel shows the radial distribution function of the DNA monomers along the radial coordinate of the channel, for D/P = 0.5 shown with three lines Fσ/ε

_{0}= 0.1, 1 and 20; and Fσ/ε

_{0}= 0.1, 0.5, 1, 5, 20 for D/P ≥ 1.

**Figure 7.**The confinement free energy during compression of DNA in helical and cylindrical nano-channels. In the plots, the black lines correspond to cylindrical channels, blue lines and orange lines to helical channels with negative −ω and positive +ω handedness respectively. Graph shows confinement free energy A

_{C}, obtained as the integral of the number density of monomers on the surface of the channel in a layer δ = ⅕ σ thick. The regime of the confinement strength is indicated by the numbers along the lines in terms of the ratio D/P = 0.5, 1.0 and 2.0. The arrows indicate where the chain size bias is expected to take place in larger channels.

**Figure 8.**Orientational correlations as a function of the channel geometry, confinement strength and compressive force. Panels (

**a**–

**c**) show orientational correlations for cylindrical channels, and panels (

**d**–

**f**) show the orientational correlations obtained as average for right-handed and left-handed helical channels. The investigated range of confinement strengths D/P = 0.5, 1.0 and 2.0 is indicated by numbers in the upper right corner of the plots. The data are shown for compressive force Fσ/ε

_{0}= 0.1, 0.5, 1, 5 and 20 indicated by the color-scale legend. The dashed line corresponds to the orientational correlations decay in the bulk, given as <cosθ> = exp (−s/P), where s is the coordinate along the chain.

**Figure 9.**The topology of DNA chain and knots. (

**a**) the graph shows knotting probability in terms of knot types and shown in thermometer colour scale representing knots by their crossing number and indicated by the legend. The knotting probability is shown for different confinement strengths expressed as D/P = 0.5, 1.0 and 2.0 (rows) and different geometries of the channel (columns). The comparison is made for right-handed (+ω) helical channels and left-handed (−ω) channels against cylindrical geometry. The arrow indicates direction of increasing compressive force in the range of Fσ/ε

_{0}= <0;20>. (

**b**) the graph shows difference of knotting probability (filled area) in channels with different geometry for knots with complex topology and crossing number above 11 and different confinement strengths D/P indicated by numbers adjacent to the curves. Orange lines correspond to averaged values for helical channels and blue lines correspond to the values obtained in cylindrical channels. (

**c**) Average crossing number for different confinement strengths and geometries of the channel. Panels (

**d**–

**f**) show the dependence of writhe of the chain (lines) and writhe of the knotted portion (filled areas). The writhe of the chain and the knots obtained in the right-handed helical channels with positive handedness (+ω) is shown in blue and for the left-handed channels with negative handedness (−ω) is shown in orange. The writhe of the chain is investigated for three different confinement strengths indicated in terms of D/P = 0.5, 1.0 and 2.0 in the upper right corner of the plots.

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**MDPI and ACS Style**

Rusková, R.; Račko, D.
Knot Factories with Helical Geometry Enhance Knotting and Induce Handedness to Knots. *Polymers* **2022**, *14*, 4201.
https://doi.org/10.3390/polym14194201

**AMA Style**

Rusková R, Račko D.
Knot Factories with Helical Geometry Enhance Knotting and Induce Handedness to Knots. *Polymers*. 2022; 14(19):4201.
https://doi.org/10.3390/polym14194201

**Chicago/Turabian Style**

Rusková, Renáta, and Dušan Račko.
2022. "Knot Factories with Helical Geometry Enhance Knotting and Induce Handedness to Knots" *Polymers* 14, no. 19: 4201.
https://doi.org/10.3390/polym14194201