Coarse-grained models of double-stranded DNA based on experimentally determined knotting probabilities

Reactive and Functional Polymers 131 (2018) 243–250

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Coarse-grained models of double-stranded DNA based on experimentally determined knotting probabilities

T

⁎

Florian C. Riegera,b, , Peter Virnaua a b

Institute of Physics, Johannes Gutenberg University Mainz, Staudingerweg 9, D-55128 Mainz, Germany Graduate School Materials Science in Mainz, Staudingerweg 9, D-55128 Mainz, Germany

A R T I C LE I N FO

A B S T R A C T

Keywords: DNA modelling DNA knots Monte Carlo simulation Model fitting Kratky-Porod wormlike chain model

To accurately model double-stranded DNA in a manner that is computationally efficient, coarse-grained models of DNA are introduced, where model parameters are selected by fitting the spectrum of observable DNA knots: We develop a general method to fit free parameters of coarse-grained chain models by comparing experimentally obtained knotting probabilities of short DNA chains to knotting probabilities that are computed in Monte Carlo simulations, resulting in coarse-grained DNA models which are tailored to reflect DNA topology in the best possible way. The method is exemplified by fitting ideal chain models as well as a bead-spring model with excluded volume interactions, to model double-stranded DNA for physiological as well as for high salt concentrations. The resulting coarse-grained DNA models predict the correct persistence length and effective diameter of double-stranded DNA, and can in principle be used for dynamical investigations using Molecular Dynamics. Our modelling ansatz thus provides a blueprint for building coarse-grained models of polymers, which are solely based on knotting spectra.

1. Introduction Mathematical polymer models, designed for large-scale computational studies of double-stranded DNA (dsDNA), are coarsened descriptions of the DNA molecule, tailored to model its most important physical properties. With time and growing computational power, DNA models which include more details of DNA structure have become tractable, meeting the demands of more sophisticated theoretical studies [1]. If physical interactions and the geometry of dsDNA are modelled in detail, the range of possible dsDNA models is broadened, and an increased number of physical and geometric parameters have to be known to specify the computational model. On the other hand, for coarser models, the relation between model parameters and observable physical quantities is obscured. Choosing proper values of DNA model parameters is a fundamental problem in coarse-grained modelling of DNA. The aim of this work is to introduce and illustrate a general fitting procedure which selects parameters of coarse-grained dsDNA models to mimic global topological properties, i.e. polymer self-entanglements, of dsDNA: Parameters are chosen so that experimentally observed knotting probabilities of short dsDNA chains are in agreement with model predictions. The fitting procedure is widely applicable as it is solely based on treating dsDNA as a space curve, and does not depend on the details of a specific ⁎

polymer model. Knots in polymers [2, 3] are known to be more or less likely to occur, depending on overall physical conditions [4–9]. The probability of knot formation, seen as a fingerprint of overall system conditions, may therefore be used to gauge DNA model parameters. More specifically, the likelihood of knots in dsDNA [3, 10] sensitively depends on salt conditions: Knotting probabilities of short dsDNA strands were first measured by gel electrophoresis [11, 12], finding that, due to screening of electrostatic interactions, for high salt concentrations, the fraction of knotted chain conformations is increased. More recently, knotting probabilities of significantly longer DNA chains for high salt concentrations have been obtained by studying translocation events in solid-state nanopores [13]. The first theoretical estimate of dsDNA knotting probability was obtained in [14], simulating lattice random walks to model knot formation on chain closure, and assuming a segment length of b = 100nm. In the 1990s, the first seminal attempts to model DNA [11] based on topological information were undertaken using experimental knotting probabilities from [11] in conjunction with the known persistence length of DNA. The resulting model consists of a chain of cylinders, whose diameter and stiffness was obtained from matching knotting spectra and persistence length, respectively. This work builds upon earlier investigations [15, 16], which study the effect of excluded volume on knotting probabilities. In [17], the topological

Corresponding author at: Institute of Physics, Johannes Gutenberg University Mainz, Staudingerweg 9, D-55128 Mainz, Germany. E-mail address: [email protected] (F.C. Rieger).

https://doi.org/10.1016/j.reactfunctpolym.2018.08.002 Received 14 January 2018; Received in revised form 29 June 2018; Accepted 2 August 2018 Available online 09 August 2018 1381-5148/ © 2018 Elsevier B.V. All rights reserved.

Reactive and Functional Polymers 131 (2018) 243–250

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2.1. Mathematical models

approach has been extended to a bead-stick model. In contrast to [11], however, the knotting spectrum from [11, 12] was the only input, and self-consistently the correct persistence length of DNA (50 nm) was recovered for physiological salt conditions. In [17], simulations corresponding to dsDNA strands of up to half a million base pairs were undertaken and are seen to be in good agreement with experimental results from [13] (see Fig. 2 below). In [17], the outlined fitting procedure had been specialized to model dsDNA for physiological salt concentrations by a bead-stick chain with hard-core excluded volume interactions. In this work, ideal chain models as well as a bead-spring model with excluded volume interactions (which can in principle be used for dynamical simulations) are fitted to model dsDNA for physiological as well as for high salt concentrations of c = 1.0M NaCl: We first illustrate the method by fitting the most simple ideal chain model, the random walk or freely-jointed chain, as well as a semiflexible chain model, the Kratky-Porod model [18], which, in case that the polymer is described as a continuous space curve of fixed length, is known as the wormlike chain model [19–21]: For sufficiently long dsDNA strands, experiments show that the wormlike chain, much more so than the freely-jointed chain, accurately describes the stretching elasticity of dsDNA, with measured force-extension curves of dsDNA and predictions of the wormlike chain model in close agreement for low and intermediate forces [22]. For physiological salt concentrations, the persistence length lp of dsDNA is known to be roughly lp ≈ 50nm [23–25]. Furthermore, among the numerous experimental methods that have been employed to estimate lp, inference of lp from the measured rate of formation of DNA circles [26, 27] has been instrumental in estimating how intrinsic curvature of dsDNA contributes to lp [28], and has also been used to study the sequence dependence of DNA rigidity [29]. In (3) it is shown that the fitted Kratky-Porod model as well as the fitted bead-spring chain predict a persistence length of dsDNA in close agreement with experimental findings, for physiological as well as for high salt concentrations. Ideal chain models of dsDNA do not contain any effects of DNA selfinteraction. Arguably, the simplest approach to define a real chain model of dsDNA is to model the combined effect of excluded volume and electrostatic interactions by introducing an effective dsDNA diameter [11, 30]: In [11], dsDNA has been modelled as a sequence of impenetrable cylinders of fixed length and diameter, using a fixed value for the dsDNA Kuhn length as input to model chain bending, with the length of cylinders given by the Kuhn length. The effective diameter of dsDNA sensitively depends on salt concentration, reflecting the screening of repulsive polyelectrolyte self-interactions: In [11], experimentally measured and simulated knotting probabilities have been used to estimate the salt dependent effective dsDNA diameter, which for the first time demonstrated the use of knotting probabilities to predict model parameters. In (3.2) it is shown that the fitted beadspring model predicts an effective dsDNA diameter that is consistent with the results in [11], for physiological as well as for high salt concentrations.

While the well-known freely-jointed chain (or random walk) model, which describes a polymer chain as a sequence of N jointed segments of fixed length b and arbitrary orientation, requires no additional model parameters to be fully defined, the Kratky-Porod model [18] also models the bending rigidity of the polymer, and therefore requires the stiffness parameter g as additional input: Starting from the Hamiltonian HWLC of the wormlike chain model [19–21], where chain conformations are taken to be continuous space curves of fixed length L in natural parametrization, discretization of the space curves as sequences of N jointed segments of fixed length b, and subsequent discretization of the integral defining HWLC, gives the Hamiltonian HKP of the Kratky-Porod model:

βκ βHWLC = βHWLC [→ r (t )] = 2 ≈

βκb 2

N −1

∑k =1

∫0

L

ds ⎜⎛ ⎝

2 ∂2→ r (s ) ⎞ ⎟ ∂s 2 ⎠

⎯⎯⎯⎯⎯→ ⎯→ ⎯ 2 ⎛ tk + 1 − tk ⎞ ⎜ ⎟ b ⎝ ⎠

(1)

with β = 1/kBT, κ = ϵb, the bending modulus of the chain [31], and ⎯→ ⎯ normalized vectors tk , k = 1, ..,N , tangent to the segments of the discretized chain. Further expanding, we have. N −1 2 βκb ⎯ 2 ⎯⎯⎯⎯⎯→ ⎯→ ⎯ ∑k =1 b−2 ( ⎯⎯⎯⎯⎯tk→+1 + ⎯→ tk − 2 tk + 1· tk ) 2 N −1 N −1 βκb−1 ∑k =1 2(1 − cos (θk )) ≕ − g ∑k =1 cos (θk ) + βE0 = 2 = β (HKP + E0)

βHWLC ≈

(2)

−1

is the dimensionless stiffness parameter, where g = βκb ⎯⎯⎯⎯⎯→ ⎯→ ⎯ cos (θk ) = tk + 1· tk , and E0 is a constant independent of chain conformation. The persistence length lp = lp(g) of the Kratky-Porod chain can be obtained analytically [32], giving.

lp (g ) = −b/ ln (coth (g ) − 1/ g )

(3)

For large g the persistence length is approximately given by lp(g) ≈ bg = βκ = βϵb. With the approximation 2(1 − cos (θ)) ≈ θ2 for angles θ = θk not too far from zero, an alternative implementation of bending rigidity is based on the quadratic bending potential UQB, giving.

βHWLC ≈ βUQB ≔ (g /2)

N −1

∑k =1

θk 2

(4)

To introduce excluded volume interactions, in a simple extension of the Kratky-Porod model, the chain beads are modelled as impenetrable spheres of diameter d = b, resulting in a bead-stick model, which has been employed in [17] to model dsDNA, choosing model parameters by application of a specialized version of the fitting procedure discussed in (2.3). As a basis for molecular dynamics (MD) simulations, the beadspring model introduced in [33] is more suitable than a bead-stick model: In this model, the angle potential of the Kratky-Porod chain is combined with a Weeks-Chandler-Anderson (WCA) potential to model excluded volume interactions, and the distance of adjacent beads is not fixed, but is kept finite by introducing a finitely extensible nonlinear elastic (FENE) potential. In this work, we also employ this bead-spring model, with all choices of constants as in [33], but chain stiffness is modelled in terms of the quadratic bending potential (4) instead. The total Hamiltonian H is therefore given by H = UWCA + UFENE + UQB, with.

2. Theory and methods To illustrate how fitting of knotting probabilities can be utilized to introduce coarse-grained models of dsDNA, in (3), the method is applied to fit model parameters of a bead-spring model, the Kratky-Porod model with quadratic bending potential, as well as the random walk model. In [17], the method has been used to introduce a bead-stick model of dsDNA. These models, as well as the implementation of computer simulations to derive knotting probabilities, are discussed in the following sections. Technical details of the fitting procedure are introduced as well.

UWCA (rij ) = 4ε ((σ / rij )12 − (σ / rij )6) + ε

(5)

for two beads at a distance rij ≤ 21/6σ, and UWCA(rij) = 0 in case that rij > 21/6σ. For adjacent beads at distance rij, the potential UFENE adds a non-vanishing contribution of. 244

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UFENE = −0.5kR 02 ln (1 − (rij / R 0 )2)

12], in (3), knot fitting is illustrated for the case of physiological salt concentrations c = 0.15M NaCl, as well as for a salt concentration of c = 1.0M NaCl. To gauge a coarse-grained dsDNA model with model parameters N, σ1, …, σn, with N being the number of (coarse-grained) chain segments, parameter values are to be selected such that the model reproduces a known set of dsDNA knotting probabilities. If the probability pK(N, σ1, …, σn) of finding sampled chain conformations to be of a particular knot type K was known analytically, fitting of the analytical curve would yield a set of model parameters to gauge the polymer model. In [41, 42], knotting of closed random polygons and closed self-avoiding polygons was studied, to establish an empirical formula for pK. For more complex polymer models, e.g. computational models including stiffness and other interaction potentials, to be employed in MD simulations, no such formulas are known. The procedure of knot fitting, as described below, is independent of any assumptions on the analytical form of pK(N, σ1, …, σn), and can be applied to fit any polymer model, given a prescribed set of known knotting probabilities. This set of prescribed knotting probabilities could either result from experimental measurements or it could have been derived from computer simulations of other, possibly more detailed, dsDNA models. To fit a dsDNA model with parameters N, σ1, …, σn, we associate a physical length scale ls of unknown value with a single segment of the coarse-grained chain. If the value of ls has been selected, a dsDNA chain of B base pairs is to be modelled as a coarse-grained chain of.

(6)

in case that rij < R0, while bead-bead distances rij ≥ R0 are not allowed. The values of the constants are chosen to be [33] ε = kBT, R0 = 3/2σ, and k = 30ε/σ2, leaving only σ as additional model parameter. 2.2. Simulation methods and knot detection To compute knotting probabilities of model chains introduced in (2.1), for the respective chain model and selected set of model parameters, the Markov chain Monte Carlo (MCMC) method is employed to estimate ensemble averages. MCMC standard errors are estimated with the method of non-overlapping batch means [34]. The Monte Carlo move set consists of pivot [35], crank-shaft [35] and generalized MOS moves (inversion, reflection and interchange) [36], which are particularly well-suited for MCMC simulations of chains with stiffness. For simulations of the bead-spring model discussed in (2.1), bead translations are added to the set of Monte Carlo moves. From a mathematical point of view, a knot is an embedding of the circle in three-dimensional space, i.e. knots of a particular knot type are closed space curves which can be transformed into each other by a properly defined smooth deformation of the ambient continuum [37]. Besides the unknot, there is an infinite but discrete set of prime knots, and every knot is either prime or a composite knot (connected sum of prime knots) [38]. Prime knots are categorized by their minimal number of self-crossings in planar projection, the simplest non-trivial prime knot being the trefoil knot, with a minimal number of three crossings. To discriminate the knot type (unknot, prime knot, or a particular composite knot) of a given closed space curve, the Alexander polynomial of the curve is evaluated, following the procedure described in [39]. Given a particular sampled chain conformation, the space curve to be tested is derived from the chain by joining its ends in a well-defined manner: First, the chain's center of mass is calculated. The chain is then extended by adding two more line segments to its ends: The first segment starts at the first endpoint of the chain, the second segment starts at the second endpoint of the chain, and both lines are extended away from the center of mass by a distance of 20Rg, where Rg is the chain's radius of gyration. The two new endpoints of the chain are joined to a third point, adding two more line segments to the chain and closing it up. To minimize interference with the original chain volume, the third point is chosen so that these three points define the vertices of an isosceles triangle, the three points lie on a circle with the center of mass as origin. To perform the knot analysis, other methods of chain closure can be employed as well [39, 40], and furthermore, closing up an open chain may create a knot in the first place. In deriving ensemble averages, the particular choice of chain closure is likely negligible, as was demonstrated for random walks [39] and self-avoiding chains [5].

N = B · a / ls

(7)

segments, where a = 0.34nm is the known distance of adjacent dsDNA base pairs. Selecting a value for ls is part of the fitting procedure. For the model parameters σ1, …, σn, one may either assign a pre-defined constant value, assign a value by an expression of the form σk = fk(ls), with some function fk, or assign an optimal value to the parameter, i.e. select its value by fitting. For example, the random walk model introduced in (2.1) requires parameters N, b, so we have n = 1 with σ1 = b, and since b is the length of a coarse-grained chain segment, we identify b = f (ls) ≔ ls. So if ls, σ1, …, σm are the quantities to be fitted, and σm+1 = σm+1(ls) = fm+1(ls), …, σn = σn(ls) = fn(ls), with some of the fk possibly being constant, the goal is to find an optimal tuple ls, σ1, …, σm, so that in some sense, computed knotting probabilities of the corresponding model chain are in best agreement with experimentally measured knotting probabilities. For each experimentally measured knotting probability Pj, the base pair count Bj of the dsDNA chain is known. For a given tuple ls, σ1, …, σm, the value Pj has to be compared to the knotting probability p(Nj, σ1, …, σm, σm+1(ls), …, σn(ls)) of a model chain with Nj = N(Bj) = Bj · a/ls segments, and with model parameters σ1, …, σm, σm+1(ls), …, σn(ls). A tuple ls, σ1, …, σm is taken to be optimal if the vector of differences.

Dj ≡ {Pj − p (Nj , σ1,…, σm , σm + 1 (ls ),…, σn (ls ) ) }j

(8)

has minimal Euclidean norm. If the tuple ls, σ1, …, σm is to be found within a certain range S = L × Σ1 × … × Σm of values, the value of p (Nj, σ1, …, σm, σm+1(ls), …, σn(ls)) has to be known for every tuple Nj, σ1, …, σm, σm+1(ls), …, σn(ls) that can be formed by selecting one of the dsDNA base pair counts Bj, and by varying ls, σ1, …, σm in S. An optimal tuple ls, σ1, …, σm is identified by application of the LevenbergMarquardt algorithm (LMA) [43], which finds ls, σ1, …, σm (in S) so that Dj, taken as a function of the ls, σ1, …, σm, is minimized in the least squares sense: This is the second and last step in knot fitting as described here. The first step in knot fitting is to compute knotting probabilities of coarse-grained model chains for a discrete set of parameter tuples N, σ1, ∼ ∼ …, σn, covering a suitable range R = [Na , Nb] × Σ1 × …Σn of values, where the range R has to be chosen so that for every tuple ls, σ1, …, σm in S, and for every dsDNA base pair count Bj, Nj(ls), σ1, …, σm, σm+1(ls), …, σn(ls) lies in R. So given that, by means of the MCMC method, p

2.3. Knot fitting procedure The objective of this section is to discuss how computed knotting probabilities can be compared to experimental observations, and how model parameters are then selected to optimize the agreement of theory and experiment. The resulting parameter fitting procedure is referred to as knot fitting. Knotting probabilities of dsDNA have been studied experimentally as a function of salt concentration for strands of 5.6kbp and 8.6kbp [12] (total knotting probabilities), as well as for strands of 10kbp [11] (trefoil knotting probabilities). For high salt concentrations, dsDNA knots are more likely to occur, as the effective diameter of dsDNA [11] is decreased by screening of electrostatic interactions, with [11, 12] both having obtained dsDNA knotting probabilities for a salt concentration of c = 1.0M NaCl. So using the experimental data from [11, 245

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(N, σ1, …, σn) has been computed for a discrete grid of parameter values covering R, this discrete dataset can then be interpolated by means of a local polynomial regression [44], estimating p at arbitrary points in R. This translates to a difference vector Dj = Dj(ls, σ1, …, σm), with arguments that vary continuously in S (see (8)). Local polynomial regression is performed by application of loess fitting [44]: For inputs xi and noisy responses Yi, loess estimates the function F in the functional relationship y = F(x), assuming that the noisy data is related to the inputs via Yi = F(xi) + εi, where the εi are assumed to be independent and identically distributed random variables, with zero mean, E(εi) = 0, and finite variance, E(εi2) = σ2. Loess is a nonparametric regression method, as no specific functional form is assumed for F. Loess essentially smooths the noisy input data, allowing  of F for continuously varying input for the evaluation of the estimate F values x. Loess can be generalized to an arbitrary number of dimensions, to smooth a noisy dataset {(xi, 1, …, xi, n, Yi)}i, and subsequently  (x 0,1,…, x 0, n ) at arbitrary points (x0, 1, …, x0, n). evaluate y0 = F With the knotting probabilities p(N, σ1, …, σn) of model chains being computed by means of the MCMC method, simulated knotting probabilities are noisy responses to the input values N, σ1, …, σn: By application of the loess method, the true functional form of the knotting p , and  p is evaluated for continuously probability p is estimated as  varying input arguments N, σ1, …, σn. As discussed above, the estimate  p is then used to define a least squares minimization procedure, allowing for the computation of an optimal length scale ls and an optimal set of model parameters σ1, …, σm. The knot fitting procedure is implemented in R [45], where the R library method loess is used to perform local polynomial regression. The difference vector (8) is defined in terms of the regression, and subsequently minimized by application of the Levenberg-Marquardt algorithm, i.e. the R library method nls.lm (package minpack.lm) is used to find the length scale ls, and model parameters σ1, …, σm.

length of dsDNA is known to be lp ≈ 50nm, and, as discussed below, this value is also found from knot fitting of different chain models with stiffness. For high monovalent salt concentrations c, no conclusion has been reached how dsDNA persistence length depends on c: Experimental studies, employing a range of different techniques to measure lp, either predict no significant change of lp for high salt concentrations, or find a further drop of lp of roughly 25% to 30% for c = 1.0M NaCl [49, 50]. This drop of lp has also been reproduced in MD simulations of dsDNA [49, 51], indicating that electrostatic interactions have a significant effect on dsDNA stiffness. For high salt concentrations of c = 1.0M NaCl, knot fitting of chain models with stiffness predicts a drop of lp of almost exactly 30% for the (ideal) Kratky-Porod chain, and a drop of roughly 26.5% for the real chain model introduced in (2.1). Therefore, the analysis in (3.1) and (3.2) supports the experimental and theoretical findings which predict a noticeable continued decrease of lp for high monovalent salt concentrations. 3.1. Knot fitting of simple ideal chain models To apply the fitting procedure discussed in (2.3) to fit the freelyjointed chain (see (2.1)) to model dsDNA, the model parameter b has to be assigned the value b = f(ls) ≔ ls. So in the simple case of the random walk, the only additional model parameter b besides N is identical to the length scale ls, leaving no model parameters to be fitted. As the knotting probabilities p(N, b) of model chains are independent of b, knotting probabilities have to be computed for a proper range of segment counts N only, leaving a one-dimensional optimization problem in ls. A classical coarse-grained dsDNA model is the (ideal) wormlike chain model, which, to keep things simple, is modelled as a KratkyPorod chain with quadratic bending potential (4), choosing a value of g = 0.5 for the dimensionless stiffness parameter. This amounts to modelling each Kuhn length by about one coarse-grained model segment: Using that, approximately, the Kuhn length aK of a wormlike chain is equal to twice its persistence length [52], aK ≈ 2lp, and lp ≈ gb (see (2.1)), we have aK ≈ b. More precisely, using the exact result (3) for the persistence length of a Kratky-Porod chain, we have lp(g = 0.5) ≈ 0.553b. As in the case of the freely-jointed chain, we have b = ls. Since the value of g is fixed, and since p(N, b, g) = p(N, g) is independent of b, after computing p(N, g = 0.5) for a proper range of chain segment counts N, again, we are left with a one-dimensional optimization problem in ls. Simulations of random walks and Kratky-Porod chains with quadratic bending potential, g = 0.5, are performed for chain lengths in the range N = [10, 55], with grid spacing ΔN = 1. Computed knotting probabilities are compared to experimental values as described in (2.3), for physiological salt concentrations of c = 0.15M NaCl, as well as for high salt concentrations of c = 1.0M NaCl. Results of the fitting procedure are displayed in Fig. 1. Knot fitting gives ls, rw(c = 0.15M NaCl) ≈ 144.8nm and ls, rw(c = 1.0M NaCl) ≈ 102.5nm for the random walk model, as well as ls, KP(c = 0.15M NaCl) ≈ 90.71nm and ls, KP(c = 1.0M NaCl) ≈ 63.47nm for the Kratky-Porod chain. With b = ls in formula (3), knot fitting of the Kratky-Porod chain predicts a dsDNA persistence length of lp(c = 0.15M NaCl) ≈ 50.17nm for physiological salt concentrations, and a persistence length of lp(c = 1.0M NaCl) ≈ 35.10nm for high salt concentrations. These predictions are in good agreement with experimental data on the salt dependence of the dsDNA persistence length, as discussed above. This indicates that knot fitting captures essential physical features of dsDNA over a range of different salt concentrations. It also indicates that the Kratky-Porod wormlike chain gives a good coarse-grained description of dsDNA, which lines up with the observation that force-extension curves of wormlike chains are in good agreement with the measured force-extension curves of dsDNA [22]. In [17], knot fitting has been used to fit the bead-stick model (see

3. Results and discussion Following the discussion of coarse-grained model chains in (2.1), to model dsDNA for physiological as well as for high salt concentrations of c = 1.0M NaCl, in the following sections, the freely-jointed chain, the (ideal) Kratky-Porod chain with quadratic bending potential, as well as the bead-spring model chain introduced in (2.1), are fitted, applying the knot fitting procedure discussed in (2.3). The fitted models are expected to thoroughly reproduce polymer entanglements, as, in terms of the knot spectrum, model chain topology matches the experimentally observed topology. As discussed below, for the fitted chain models with stiffness, knot fitting reproduces the persistence length of dsDNA, and for the fitted real chain models, knot fitting reproduces the effective diameter of dsDNA, for physiological as well as for high salt concentrations of c = 1.0M NaCl. As knot statistics are the only input to gauge the models, these results indicate that in principle, physical features of polymer chains can be captured if knotting is described self-consistently, given that a proper set of adjustable model parameters has been selected in the first place: The excluded volume parameters of the chain models discussed in (2.1) are closely related to the effective diameter of dsDNA, and it has been known for a long time that knotting probabilities sensitively depend on chain diameter [6, 11, 15, 16, 41]. Furthermore, as was first seen for lattice polygons [46], and later shown for real chains [47], knotting probabilities of semiflexible chains with excluded volume interactions depend on the stiffness parameter g in a non-trivial manner, which is also apparent from Fig. 3a below. In a recent study [48], the phenomenon has been studied in much greater detail. As a consequence, due to sensitive responses of the knot spectrum to changes of either the stiffness parameter or the excluded volume parameter, as discussed in (3.2), knot fitting allows for determining the persistence length and effective diameter of dsDNA at the same time. For physiologically relevant salt concentrations, the persistence 246

Reactive and Functional Polymers 131 (2018) 243–250

0.10

F.C. Rieger, P. Virnau

0.08 0.00

0.02

0.04

0.06

knotting probability

Fig. 1. Knot fitting of ideal chains to model dsDNA: Total knotting probability and trefoil knotting probability are shown for the random walk and the Kratky-Porod chain (with quadratic bending potential) as a function of the number of coarsegrained chain segments N. Estimated knotting probabilities are included as solid/dashed lines (total knotting probability/ trefoil knotting probability). It is shown how experimentally obtained knotting probabilities [11, 12] compare to model predictions if knot fitting is applied to fit the models, for physiological salt concentrations of c = 0.15M NaCl as well as for high salt concentrations of c = 1.0M NaCl. As discussed in (2.3), experimental knotting probabilities Pj are mapped to model chains by application of formula (7).

all knots, random walk trefoils, random walk all knots, Kraty−Porod chain trefoils, Kratky−Porod chain experiment [11,12], physiological salt, random walk fit experiment [11,12], high salt, random walk fit experiment [11,12], physiological salt, Kratky−Porod chain fit experiment [11,12], high salt, Kratky−Porod chain fit

10

20

30

40

50

N

obtained experimentally for high salt concentrations. The comparison of computed and measured knotting probabilities indicates that predictions of the bead-stick model, which includes excluded volume interactions, are more accurate, but overestimate the probability of dsDNA knots for small strands, as predicted knotting probabilities for physiological salt concentrations are close to experimental results that have been obtained for high salt concentrations of c = 1.0M KCl.

1.0

(2.1)), to model dsDNA for physiological salt concentrations of c = 0.15M NaCl. Again, using formula (3), the fitted model predicts a dsDNA persistence length of lp(c = 0.15M NaCl) ≈ 49.85nm [17]. As depicted in Fig. 2, for physiological salt concentrations of c = 0.15M NaCl, predictions of knotting probabilities for long dsDNA chains that are based on the fitted bead-stick model differ from predictions that are based on the fitted ideal chain models. Predictions of the random walk and Kratky-Porod chain are in close agreement. As can be seen, ideal chain models predict a larger fraction of knots that are not trefoil knots. Depicted knotting probabilities of bead-stick chains are taken from [17]. In Fig. 2, computed knotting probabilities are compared to experimental results from [13] as well: By analysing translocation events in solid-state nanopores, in [13], knotting probabilities have been

Parameters of the bead-spring model introduced in (2.1) are, besides the number of coarse-grained chain segments N, the dimensionless stiffness parameter g and the parameter σ, which determines both the WCA interaction potential as well as the FENE spring potential. The Fig. 2. Comparision of predicted knotting probabilities for long dsDNA chains: Predicted total knotting probabilities and trefoil knotting probabilities for long dsDNA chains are shown for the random walk and the Kratky-Porod chain (with quadratic bending potential), as well as for the bead-stick chain. All models have been fitted to model dsDNA by application of the knot fitting procedure discussed in (2.3), for physiological salt concentrations of c = 0.15M NaCl. Predicted knotting probabilities of bead-stick chains are taken from [17]. Knotting probabilities of dsDNA in high salt have been determined experimentally by studying translocation events in solid-state nanopores [13], and are compared to the theoretical predictions: Depicted experimental values are the means of knotting probabilities that have been obtained for different applied voltages [13]. As can be seen, chain models without excluded volume interactions largely overestimate dsDNA knotting probabilities, especially for knots more complex than the trefoil knot. Predictions of the fitted ideal chain models are in close agreement.

0.2

0.4

0.6

0.8

all knots, bead−stick chain [17] trefoils, bead−stick chain [17] all knots, random walk trefoils, random walk all knots, Kratky−Porod chain trefoils, Kratky−Porod chain experiment [13], 1M KCl experiment [13], 2M LiCl experiment [13], 4M LiCl

0.0

knotting probability

3.2. Knot fitting of bead-spring model with excluded volume interactions

0

100,000

200,000

300,000

400,000

DNA size in base pairs

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Fig. 3. Knot fitting of bead-spring chain to model dsDNA, interpolation and errors: (a) Estimated total knotting probabilities of bead-spring chains, as a function of stiffness parameter g and number of coarse-grained chain segments N. It is shown how experimentally obtained knotting probabilities [11, 12] compare to model predictions if knot fitting is applied to fit the bead-spring chain to model dsDNA, for physiological salt concentrations of c = 0.15M NaCl (circles) as well as for high salt concentrations of c = 1.0M NaCl (triangles). Indicated surface cuts are displayed in Fig. 4. (b) Error estimate for optimization of parameters ls, g in knot fitting procedure, given as the norm of the difference vector ‖Dj‖, for physiological salt concentrations of c = 0.15M NaCl. The computed tuple of optimal parameter values is indicated by a circle. A line highlights all data points which predict a dsDNA persistence length of lp = 50nm according to formula (3), with b substituted by E (b) = ls (see main text).

NaCl) ≈ 13.529, and ls(c = 1.0M NaCl) ≈ 2.283nm, g(c = 1.0M NaCl) ≈ 16.919. With E(b) = ls substituted for b in formula (3), the fitted bead-spring chain predicts a dsDNA persistence length of lp(c = 0.15M NaCl) ≈ 50.99nm for physiological salt concentrations, and lp(c = 1.0M NaCl) ≈ 37.47nm for high salt concentrations, in good agreement with experimental findings as discussed above, and in agreement with predictions of the fitted ideal Kratky-Porod chain discussed in (3.1). As the bead-spring model includes excluded volume interactions, the fitted bead-spring chain predicts the effective (salt dependent) diameter of dsDNA: Identifying the cutoff d ≔ 21/6σ = 21/6ls/0.97 of the WCA potential with the effective dsDNA diameter, knot fitting predicts d(c = 0.15M NaCl) ≈ 4.531nm and d(c = 1.0M NaCl) ≈ 2.641nm. This is consistent with the analysis in [11], predicting d(c = 0.15M NaCl) ≈ 5nm and d(c = 1.0M NaCl) ≈ 2.7nm, where dsDNA has been modelled as a sequence of joint impenetrable cylinders of diameter d and length given by the Kuhn length aK of dsDNA (using aK = 100nm as input parameter). The bead-stick model introduced in (2.1) includes hard-core excluded volume interactions, as beads are modelled as impenetrable spheres of diameter d. Knot fitting of the bead-stick model predicts d = 4.465nm for physiological salt concentrations of c = 0.15M NaCl [17], in good agreement with predictions of the fitted bead-spring model. The fitted bead-spring chain is a simple coarse-grained dsDNA model that can be employed to study dsDNA in MD simulations: In [53] the bead-spring model introduced in (2.1) has been used to study how a trefoil and a figure-eight knot, as parts of a composite dsDNA knot, can pass through each other. In [53] model parameters g, σ have been selected by first identifying σ with the effective dsDNA diameter, with value taken from [11], and by identifying the product lp ≈ gσ with the dsDNA persistence length, assuming lp = 50nm. Resulting values for g, σ are similar to the ones obtained by knot fitting. In this and the preceding section, knot fitting is applied to fit models of open chains. Experimentally obtained knotting probabilities [11, 12] are found for closed chains, which raises the question if closed chain

details of the simulation do not depend on the value of σ, as σ defines the scales only. The length b of chain segments fluctuates, but its expectation E(b) is found to be E(b) ≈ 0.97σ in simulations. Following the procedure outlined in (2.3), to fit the bead-spring chain to model dsDNA, we identify ls, the length scale to be associated with individual coarse-grained chain segments, with E(b), i.e. we have σ = σ(ls) = ls/ 0.97. The value of g is fitted as well, i.e. knot fitting of the bead-spring model as described here leads to a two-dimensional optimization problem in ls, g. Hence, as the values ls, g are selected by fitting, besides the distance a = 0.34nm of adjacent base pairs (compare with (2.3)), experimentally obtained knotting probabilities are the only input to fit the bead-spring model. To identify an optimal set of parameters ls, g, knotting probabilities p(N, g, σ) = p(N, g) of model chains are obtained for an equidistant grid of parameter values, with N = [100, 2000], g = [5, 30], and grid spacing ΔN = 50, Δg = 0.5. Computed knotting probabilities are compared to experimental values as described in (2.3), for physiological salt concentrations of c = 0.15M NaCl, as well as for high salt concentrations of c = 1.0M NaCl. In Fig. 3a, the regression surface of estimated p (N , g ) is depicted. It is shown how extotal knotting probabilities  perimentally obtained knotting probabilities compare to estimated (smoothed, interpolated) knotting probabilities of bead-spring chains for the optimal parameter tuple ls, g, for physiological as well as for high salt concentrations. This comparison is further detailed in Fig. 4, where in Fig. 3a, surface cuts displayed in Fig. 4 are highlighted as well. In Fig. 3b, the error estimate ‖Dj‖ = ‖Dj‖(ls, g) (see (2.3)) is displayed for the case of physiological salt concentrations, and the location of the optimal parameter tuple ls, g is indicated. p (N , g ) on g is not As can be seen in Fig. 3a, the dependency of  monotone [47], i.e. for semiflexible polymers, knotting probabilities increase if the chain stiffness is increased. As a consequence, using MCMC simulations to estimate knotting probabilities of chains with stiffness potential and excluded volume interactions, the number of sampled chain configurations has to be large enough to accurately resolve the non-trivial surface structure p(N, g). Knot fitting gives ls(c = 0.15M NaCl) ≈ 3.916nm, g(c = 0.15M 248

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Fig. 4. Knot fitting of bead-spring chain to model dsDNA, detailed comparison to experiment: Both plots display estimated total knotting probabilities (solid line) and estimated trefoil knotting probabilities (dashed line) of bead-spring chains as a function of N, the number of coarse-grained chain segments. It is shown how experimentally obtained knotting probabilities [11, 12] compare to model predictions if knot fitting is applied to fit the bead-spring chain to model dsDNA, (a) for physiological salt concentrations of c = 0.15M NaCl, (b) for high salt concentrations of c = 1.0M NaCl. I.e. in (a), experimental values are compared to  p (N , g (c = 0.15M NaCl) ) , while in (b), experimental values are compared to  p (N , g (c = 1.0M NaCl) ) , where g(c = 0.15M NaCl) and g(c = 1.0M NaCl) are fitted values of the dimensionless stiffness parameter g. As discussed in (2.3), experimental knotting probabilities Pj are mapped to model chains by application of formula (7).

models should be fitted instead: In [48], ring polymers are modelled in terms of the bead-stick model which has also been employed in [17] to model dsDNA (see (2.1)). In [48], knotting probabilities of closed chains are computed for a range of different values of the stiffness parameter, so experimental knotting probabilities from [11, 12] and theoretical results from [48] can be compared. For example, for a salt concentration of c = 0.1M NaCl, in [11], the dsDNA diameter is found to be roughly d ≈ 5nm, and this result is based on fitting simulated knotting probabilities of ring polymers to the experimental data. If the persistence length of dsDNA is taken to be lp = 50nm, ten beads of a closed model chain from [48] represent one persistence length of dsDNA, identifying the bead diameter, which also equals the distance b of adjacent beads, with the effective dsDNA diameter. To adjust the bending potential, the stiffness parameter is taken to be g = 10 as lp ≈ gb. Results from [48] are most easily compared to results from [12] as both studies depict total knotting probabilities, whereas [11] depicts measured trefoil fractions: In [12], for a dsDNA strand of 8.6kbp and c = 0.1M NaCl, the total dsDNA knotting probability is found to be slightly higher than 1%. A dsDNA strand of 8.6kbp should now be modelled as a chain of roughly 585 beads, assuming a dsDNA length of L = 8600 · 0.34nm = 2924nm. From the results in [48] it can be seen that for g = 10, a closed chain of N = 500 beads has a total knotting probability of almost 2%, and the total knotting probability will be significantly higher for a chain of length N = 585. As part of the knot fitting procedure discussed in this section, the total knotting probability of an open chain was calculated for g = 10 and N = 600, corresponding to a point on the surface depicted in Fig. 3a. For these parameter values, roughly 1.33% of all chains are knotted, which is close to the experimental value found in [12]. Similar comparisons can be made for other chain lengths, which strongly suggests that experimental knotting probabilities from [11, 12] are best understood in terms of open chains: In [11, 12], linear DNA was prepared and then cyclized. Once the DNA is cyclized, the topological state, which was formed in the linear strand can no longer be changed. So knotting probabilities given in [11, 12]

likely reflect probabilities in linear (uncyclized) DNA. Of course, the cyclization event may have a minor influence on that probability as well, but this effect is difficult if not impossible to gauge. 4. Conclusions To select parameters of coarse-grained polymer models to describe dsDNA in the best possible way, in this work, a general knot fitting procedure was developed: Parameters of coarse-grained chain models are selected such that experimentally obtained knotting probabilities of dsDNA are in agreement with theoretical predictions. Application of knot fitting to select model parameters for the random walk, the (ideal) Kratky-Porod chain with quadratic bending potential, as well as for a bead-spring chain with excluded volume interactions, leads to dsDNA models which correctly predict the persistence length and the effective diameter of dsDNA, for physiological as well as for high salt concentrations. In particular, knot fitting predicts a dsDNA persistence length of roughly lp ≈ 50nm for physiological salt concentrations. For high salt concentration of c = 1.0M NaCl, knot fitting predicts that the persistence length of dsDNA will drop by roughly 30%, comparing the value at c = 1.0M NaCl to the value seen at c = 0.15M NaCl. This finding supports experimental and theoretical studies which have reported the same continued decrease of dsDNA persistence length for high salt concentrations. The documented discrepancies in measurements of lp for high salt concentrations [49, 50] show that in general, it is a difficult problem to establish a clear connection between parameters of coarse-grained DNA models and experimentally measurable quantities. As knot fitting of dsDNA models is solely based on treating DNA as a space curve and does not require additional theory to connect theoretical and experimental data on dsDNA knotting probabilities, it is a quite universal method of DNA model fitting: The method is designed to select model parameters by fitting chain topologies instead of fitting metric quantities and the fitting procedure does not depend on the details of a particular polymer model. Therefore, knot fitting, as 249

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described here, is widely applicable should corresponding knot spectra become available. Knot fitting of real chains reproduces at the same time the dsDNA persistence length as well as the effective dsDNA diameter, for physiological as well as for high salt concentrations. Therefore, since knotting probabilities are the only input to gauge the model, we conclude that self-consistent modelling of knot statistics can successfully capture physical properties of linear polymers. Although we have focused on the problem of selecting model parameters to properly describe dsDNA, knot fitting can also be used as a more general tool in polymer physics: Instead of comparing experimental results and theoretical predictions, the method can be applied to compare the knot spectra of different chain models. In this way, the method could be used to coarse-grain model chains in a systematic manner.

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