Size distribution of ring polymers

ABSTRACT We present an exact solution for the distribution of sample averaged monomer to monomer distance of ring polymers. For non-interacting and local-interaction models these

distributions correspond to the distribution of the area under the reflected Bessel bridge and the Bessel excursion respectively and are shown to be identical in dimension _d_ ≥ 2, albeit

with pronounced finite size effects at the critical dimension, _d_ = 2. A symmetry of the problem reveals that dimension _d_ and 4 − _d_ are equivalent, thus the celebrated Airy distribution

describing the areal distribution of the _d_ = 1 Brownian excursion describes also a polymer in three dimensions. For a self-avoiding polymer in dimension _d_ we find numerically that the

fluctuations of the scaled averaged distance are nearly identical in dimension _d_ = 2, 3 and are well described to a first approximation by the non-interacting excursion model in dimension

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WALKS Article Open access 09 September 2022 INTRODUCTION In nature, chemically identical copies of a random polymer come in different shapes and sizes due to their vast number of

conformational states. Traditionally, ensemble averaged observables, like the averaged radius of gyration, are used to quantify the size of a polymer, concealing the ever-present

fluctuations1,2. In recent years a new set of single polymer experiments have succeeded in determining the structure fluctuations3,4,5,6 of a polymer, one snapshot at a time. This has led to

the investigation of fluctuations of single polymers. For example based on a random walk picture, a single realization of a polymer is prolate, while from symmetry, the ensemble average

shape is spherical7. The conformational fluctuations of open polymers can be analyzed using the theory of random walks (RW)1,8,9. While much is known about fluctuations in linear polymer

chains, much less is known about ring polymers (see, however, Refs. 6,10 which studies the size distribution of small unknotted vs. knotted rings of different complexities). It is therefore

interesting to study the fluctuation properties of these objects, to test how the additional constraint of closure modifies the size statistics. Ring polymers2 are commonly found in many

biological systems, e.g., bacterial and mitochondrial genomes, as well as DNA plasmids10,11,12 used in many molecular biology experiments13,14,15. Recently, ring polymers were also studied

in the context of a model for chromosome territories in the nucleus of eukaryotic cells16 and of a model for cell-division in bacteria17. In particular, the fluctuations of the polymer size

are of physical and biological importance, since size controls packing properties and reaction rates, among other things. In the current paper we study the distribution of sample-averaged

monomer-to-monomer distance of ring polymers, both for ideal, noninteracting polymers as well as for polymers with excluded volume. In particular, we exploit recent mathematical developments

on _d_ dimensional constrained Brownian motion (defined below)18 to find an exact expression for the distribution of the sample-averaged monomer to monomer distance, both for ideal ring

polymers and those with an additional excluded volume interaction applied at a single point. This observable yields insight into the size fluctuations of polymer rings. The resulting

distributions are then compared numerically to those measured in simulations for both the above two cases and, in addition, to simulations with full excluded volume constraints. We find that

the excluded volume effects change, as expected, the overall size of the ring polymer, but also the distribution shape. Surprisingly, we find that interactions act as an effective shift of

dimensionality, reducing the fluctuations considerably. As we have noted, constrained random walks lie at the heart of our analysis. For rings, the primary constraint is that the path

returns to the origin after _N_ steps. Statistics of such constrained Brownian paths in one dimension have been the subject of much mathematical research19,20,21,22. These constrained paths

have been given various names, depending on the additional constraints imposed. The basic case is that of a Brownian bridge, where the return to the origin is the sole constraint. Paths

which are also forbidden from reaching the origin in between the endpoints are called Brownian excursions. Majumdar and Comtet used Brownian excursions to determine statistical properties of

the fluctuating Edwards-Wilkinson interface in an interval23,24. The focus of most previous work has been on constrained one dimensional Brownian paths (note that the problems of

non-intersecting Brownian excursions25 or vicious random walkers26 in higher dimensions are exceptions). The case of the fluctuations of ring polymers requires the extension of the theory of

Brownian excursions and bridges to other dimensions. We address the influence of different kinds of interactions on the polymer structure, both analytically (for a single point interaction)

and numerically (for a polymer with excluded volume interactions). These models yield rich physical behaviors and open new questions. POLYMER MODELS We consider three lattice models of ring

polymers with _N_ bonds, each of length _b_, in _d_ dimensions. The simplest polymer model is an “ideal ring” - a closed chain without excluded volume8, where different monomers can occupy

the same lattice site. An ideal ring (IR) corresponds exactly to an unbiased random walk in _d_ dimensions which starts and ends at the origin, i.e. a _d_-dimensional bridge. In the “local

interaction ring (LIR) polymer” model, the first and last monomers are tied to the origin and no other monomer is allowed to occupy this lattice site. The differences between these two

models is exemplified in Fig. 1. The LIR case is equivalent to that of _d_-dimensional excursions. The third model is a ring polymer with excluded volume interactions, also called a

self-avoiding walk (SAW). Further details on the models and simulation methods are provided in Supplementary A. We first consider the ideal and local interaction ring models, for which we

can provide analytic solutions. BESSEL PROCESS In the analogy between statistics of an ideal polymer and a RW, the position of the _i_th monomer, R_i_, corresponds to the position R_i_ of

the random walker after _i_ time steps. In the Brownian (large number of steps, _N_) limit, _N_ is proportional to the total observation time. The Bessel process27,28 describes the dynamics

of the distance _r_ = |R| from the origin of a Brownian particle in _d_ dimensions. This process is described by the following Langevin equation: where η(_t_) is Gaussian white noise

satisfying 〈η〉 = 0 and 〈η(_t_)η(_t_′)〉 = δ(_t_ − _t_′). One may map the polymer models to the Bessel process using 〈R(_t_)2〉 = _dt_ = _Nb_2 = 〈R2〉, where R2 is the mean squared end-to-end

distance of an ideal linear polymer chain without constraints. In what follows we choose _b_2 = _d_ and _t_ = _N_. BESSEL EXCURSIONS AND REFLECTED BRIDGES The process _r_(_t_) with the

additional constraint of starting and ending at the origin, is called a reflected (since _r_ ≥ 0) Bessel bridge. This process corresponds to an ideal (non-interacting) ring chain. Bessel

excursions are paths still described by Eq. (1), with the additional constraint that any path that reaches the origin (not counting the endpoints _t_ = 0 and _t_ = _N_) is excluded. The

Bessel excursion corresponds to what we have called the “local interaction” ring, where a multiple occupation of the origin is not allowed. The mapping of the polymer models to the Bessel

process, allows us to extract statistical properties of the former with new tools developed in the stochastic community18,24,29. THE OBSERVABLE A For linear polymers, two observables are

traditionally studied. The first is the radius of gyration, _R__g_. This is the observable most easily accessed in experimental studies of polymer ensembles. However, for studies of the

fluctuations, it is preferable to consider the end-to-end distance, which is trivially Gausian in the ideal case. Similarly, for a ring polymer, it is difficult to obtain analytically the

distribution for _R__g_. While the end-to-end distance obviously has no meaning for a ring polymer, there is a new observable whose distribution, as we will see, is analytically tractable.

Let R_i_ be the position of the _i_th, (_i_ = 0, …, _N_) monomer in space and we place the origin at the position of the zeroth monomer, _r_0. In the LIR model, this monomer is also the

excluding one. This new observable, _A_, is defined by In the RW language, _A_ is the area under a random (non-negative) process and hence is a random variable itself. Clearly is the sample

averaged distance of the monomers from the origin. Specifically, let the area under the random Bessel curve _r_(_t_) be denoted by (the subscript _B_ is for Bessel). More generally, the

mapping of the processes implies that, in the limit of large _N_, the distribution of _A__B_/〈_A__B_〉 is identical to the distribution of _A_/〈_A_〉 (or ), with the corresponding constraints.

We note that unlike some other observables, such as the distance to the (_N_/2)_th_ mer15, which have been suggested as a measure of polymer size, _A_ can only be small if the entire

polymer ring is collapsed, similar to what happens in the case of the radius of gyration. RESULTS NUMERICAL RESULTS To build some intuition for the problem we first present numerical

results, which will be later compared to theory. In Fig. 2 we plot the probability density function (PDF) of the scaled random variable _A_/〈_A_〉. For reference, one can compare it to the

PDF of the radius of gyration of open chains30. For dimensions _d_ = 2, …, 5, there is a clear trend of narrowing of the PDF for increasing _d_. This trend is explained by examining Eq. (1):

As _d_ increases, the noise term becomes negligible compared with the force term, resulting in smaller fluctuations and narrower tails. Against this expected trend are the results in _d_ = 

1 for the local interaction model. Figure 2 exhibits a second surprising feature. As our analytical calculations will confirm, the local interaction model in dimensions one and three are

seen to have the same distribution even though _d_ = 1 has a vanishing deterministic force term in Eq. (1) while for _d_ = 3 the force is clearly not zero. In addition, we observe that for

_d_ ≥ 3 the shape of the distributions of locally interacting and ideal ring chains coincide, indicating that weak interactions are negligible (when _N_ → ∞). As we shall see, this is also

observed in the theory. Indeed the theory discussed below suggests that these two distributions are already identical for _d_ = 2. However, this is a critical dimension, with extremely slow

convergence, due to logarithmic corrections31 to scaling for the LIR case, so we don’t see this behavior in the simulations32. This is treated in more detail below. As for the SAW polymer,

we see that the fluctuations are considerably reduced compared to the other models. This is due to the fact that small conformations are rejected from the sample, since they overlap and thus

fluctuations are reduced. A striking observation is that the two and three dimensional SAW results are identical, both being equal to the simulations of the _d_ = 5 IR and LIR models. We

now present a theoretical analysis in order to address these observations. FUNCTIONALS OF CONSTRAINED BESSEL PROCESSES Our goal is to find the PDF _P_(_A__B_, _t_) of the functional of the

Bessel process, constrained to start and end at the origin. The advantage of the observable _A_ is that it is a linear functional of the path, unlike other observables such as the radius of

gyration. We show that the difference between the local interaction model (the Bessel excursion) and the ideal polymer (the reflected Bessel bridge) enters through the boundary condition in

the Feynman-Kac type of equation describing these functionals. The choice of boundary condition turns out to be non-trivial and controls the solution. Other aspects of the solution follow

the steps in Ref. 18 and so our description here will be brief. It is useful to find first the Laplace transform of _P_(_A__B_, _t_), i.e., to solve the Feynman-Kac equations and invert back

to _P_(_A__B_, _t_). Let _G__t_(_r, A__B_|_r_0) be the joint PDF of the random pair (_r, A__B_) with initial condition _G_0(_r, A__B_|_r_0) = δ(_A__B_)δ(_r_ − _r_0) and its Laplace pair.

The Feynman-Kac equation deals with fluctuations of functionals of Brownian motion (_d_ = 1 in Eq. (1)). Here, we have to deal with the effective restoring force introduced by higher

dimensionality and this requires a generalized Feynman-Kac equation. The modified Feynman-Kac equation reads33: with and _r_0 is the initial radius which serves to regularize the calculation

and is eventually taken to zero, since excursions are defined to start at the origin. For _d_ = 1, the force in Eq. 1 is absent and so the second term on the left hand side vanishes and we

get the celebrated Feynman-Kac equation corresponding to Brownian functionals34. The third, linear, term stems from the choice of our observable, namely our functional _A__B_ is linear in

_r_33. Since we are describing a ring polymer, the Bessel process must start and end at the origin and so, following Ref. 23, we need to calculate The denominator gives the proper

normalization condition. The first step in the calculation is to perform a similarity transformation: Using Eq. (3), is the imaginary time propagator of a Schrödinger operator: with the

effective Hamiltonian: The effective Hamiltonian reveals a subtle symmetry, namely two systems in dimensions _d_1 and _d_2 satisfying _d_1 + _d_2 = 4 behave identically. This explains the

above-mentioned identity of the _d_ = 1 and _d_ = 3 PDFs observed in the simulations, a symmetry noted previously in the mathematical literature35. Note that this symmetry is not affected by

the choice of functional (or observable) since the latter only modifies the last term in . BOUNDARY CONDITIONS FOR IDEAL AND LOCAL INTERACTION MODELS The solution of Eq. (6) is

constructed18 from the eigenfunctions ϕ_k_ of where λ_k_ is the _k_th eigenvalue and the normalization condition is . The subtle point in the analysis is the assignment of the appropriate

boundary condition corresponding to the underlying polymer models we consider. The eigenfunctions at small _r_ exhibit for  _d_ ≠ 2 one of two behaviors: From the normalization condition,

the ϕ− solution cannot be valid for _d_ ≥ 4 and _d_ ≤ 0. For the critical dimension _d_ = 2 the two solutions are: or ln _r_. We now solve the problem for the two possible leading boundary

behaviors and then show how to choose the relevant one for the physical models under investigation. THE DISTRIBUTION OF _A_/〈_A_〉 Following the Feynman-Kac formalism described above and

performing the inverse Laplace transform along the lines of ref. 18, (and for convenience dropping from now on the subscript _B_ from _A_, since _A__B_ = _A_ in the large _N_ limit) we find

two solutions for the PDF of the scaled variable χ ≡ _A_/〈_A_〉 where and we have substituted _t_ = _N_ as explained above. This solution is independent of _N_ and valid in the limit of _N_ →

 ∞. Here, |α| = |_d_ − 2|/2, |ν| = 2|α|/3 and 2_F_2(⋅) refers to a generalized hypergeometric function. Supplementary B provides a list of λ_k_ and _d__k_ values for _d_ = 1,…,4. For _d_ = 1

the solution agrees with the known results23,24,36,37, where the + solution is the celebrated Airy distribution23,24. The average of _A_ is The scaling 〈_A_〉 ∝ _N_3/2 is expected since _r_

scales with the square root of _N_ as for Brownian motion, so the integral over the random processes _r_ scales like _N_3/2. The question now is how to choose the solution corresponding to

our various polymer models. Clearly, for _d_ = 2, 〈_A_〉+ = 〈_A_〉−, indicating that this is a critical dimension. Further 〈_A_〉+ in 1 and 3 dimensions are identical and so is 〈_A_〉− as the

result of the symmetry around _d_ = 2 in Eq. (7). In Fig. 3, we plot the scaled average area 〈_A_〉/_N_3/2, found from the simulation of IR and LIR polymers in different dimensions d = 1, 2,

3, 4 versus the theory provided in Eqs (11, 12). Notice the simulation results for the IR case fit nicely the + branch of Eq. (12), whereas the LIR case agrees with + solution for _d_ > 2

and the − solution for _d_ < 2. The reason for this agreement is explained immediately below. In _d_ = 2, the simulation results for the LIR polymer did not converge even for _N_ = 106,

which is a direct result of the extremely slow convergence at the critical dimension. We discuss that as well in more detail below. We investigate the physical interpretation of the two

possible boundary conditions. A mathematical classification of boundary conditions was provided in Refs. 28,38 and here we find the physical situations where these conditions apply. We

examine the behavior of the probability current associated with the _k__th_ mode through the continuity equation for the _kth_ mode of the total probability, to reach _r_ at time t, starting

from _r_0. We obtain, using Eq. (3), Near the boundary , the function is given by Eqs (5) and (8), so that, for small _r, r_0, (see also ref. 28) The behavior of this has to analyzed

separately for the ± modes, for each value of _d_. This analysis is summarized in Table 1. We see that in dimension two and higher, the current at the origin is either zero or positive. A

positive current at the boundary means that probability is flowing into the system, which is an unphysical situation in our system. Hence we conclude that in dimension two and higher, the −

solution is not relevant. This implies that the statistics of excursions and reflected bridges (and equivalently, ideal and locally interacting ring polymers) are identical for _d_ ≥ 2 and

correspond to the + solution. In 1 dimension, the negative current means an absorbing boundary, hence the the + solution describes the excursion (and the LIR polymer). Zero current means a

reflecting boundary, so the − solution describes the reflected bridge (IR). In Fig. 2 we compare the results of the simulations of the ideal and locally interacting polymer models with our

theoretical results for _P_+(_A_/〈_A_〉), as given in Eq. (10). As noted above, for _d_ ≥ 3, we see that even for finite size chains the local interaction is not important and that the theory

and simulations perfectly match, while for _d_ = 2 there are strong finite size effects in the locally interacting case. This critical slowing down in _d_ = 2 is analyzed in Supplementary A

and we find logarithmic convergence to our asymptotic result, as can be seen in Fig. 4. The fact that _d_ = 2 is critical is surely related to Polya’s problem, since in _d_ = 3 the random

walk is not recurrent, indicating that local interaction is asymptotically of no importance. SELF-AVOIDING POLYMERS Extensive simulations of ring SAWs were performed on cubic lattices. The

global expansion of a polymer of length _N_ scales like , where ν_p_ is a dimension dependent critical exponent. Since _A_ constitutes a measure of the overall size of a polymer, its

behavior for large _N_ should follow . For the ideal and locally interacting chains ν_p_ = 1/2 and _A_ ~ _N_3/2. For SAWs the exact value of the exponent depends on _d_, i.e., ν_p_ = 

ν_p_(_d_). ν_p_ = 1, 0.75 and 0.5 are known exactly for _d_ =1, 2 and 439, respectively. For _d_ = 3, based on renormalization group considerations and Monte Carlo simulations, 40. These

predictions were extensively tested numerically for the observable of interest _A_ with a critical dimension of _d_ = 4, characterized by a very slow convergence of the locally interacting

model (see Supplementary B). While field theoretic calculations includes knotted as well as unknotted rings41, the scaling exponent is the same for the subensembles of knotted and unknotted

rings (though the corrections to scaling differ42). While the scaling behavior of the SAW model is different from that of the other two models (as reflected in ν_p_), as we have noted, the

scaled PDFs, _P_(_A_/〈_A_〉) are nevertheless similar. A striking observation is that the SAWs in _d_ = 2 and 3 coincide to the precision of our measurements with the comparably narrow PDF of

the _d_ = 5 non-interacting model, (see Fig. 2). That these distributions are narrower than the non-self-avoiding case can be qualitatively explained by the fact that small rings are

suppressed in the SAW case due to the need to avoid overlaps. GENERALITY OF THE METHOD The mapping of ring polymer models to the reflected Bessel bridge and excursion is very promising since

it implies that not only the distribution of the size, i.e., _A_, can be analytically computed, but also the statistics of other properties of ring polymers. An example would be the maximal

distance from one of the monomers to any other monomer, since that would relate to extreme value statistics of a correlated process. As mentioned in the introduction, the shape of a polymer

is prolate7. To investigate fluctuations in shape one may consider, for a 2 − _d_ polymer, the distribution of the order parameter, , where cos(θ_i_) = _x__i_/_r__i_, measured from some

point on the polymer, which for a circular object is zero. In the language of Brownian motion this quantity is a functional of the random path. Hence we may write a modified Feynman-Kac

equation for such an observable and map the problem to a Schrödinger-like Eq. (7). We leave the detailed analysis to future work. Clearly the approach presented here is valuable for

calculation of different types of single polymer fluctuations, which as mentioned in the introduction are nowadays accessible in the laboratory. SUMMARY We have quantified analytically and

numerically the distribution of the fluctuations of the size of ring polymers, with the hope that these can now be experimentally tested. The famed Airy distribution describes both the one

dimensional ideal ring polymer, as well as the three dimensional one, due to the symmetry we have found in the underlying Hamiltonian. The case of _d_ = 2 is critical in the sense that

interaction at the origin becomes negligible, though for finite size chains it is still important. Boundary conditions of the Feynman-Kac equation were related to physical models, which

allowed us to select the solutions relevant for physical models. The SAW polymer exhibits interesting behavior; the distribution of _A_/〈_A_〉 is identical (up to numerical precision) in

dimension 2 and 3 and corresponds to the non-interacting models in dimension 5. In general, interactions are seen to reduce the fluctuations. ADDITIONAL INFORMATION HOW TO CITE THIS ARTICLE:

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Science Foundation (ISF). Append (Grant number 376/12). AUTHOR INFORMATION Author notes * Medalion Shlomi, Aghion Erez, Meirovitch Hagai, Barkai Eli and Kessler David A. contributed equally

to this work. AUTHORS AND AFFILIATIONS * Department of Physics, Bar-Ilan University, Ramat-Gan, 52900, Israel Shlomi Medalion, Erez Aghion, Hagai Meirovitch, Eli Barkai & David A.

Kessler * Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan, 52900, Israel Shlomi Medalion, Erez Aghion & Eli Barkai Authors * Shlomi Medalion View

author publications You can also search for this author inPubMed Google Scholar * Erez Aghion View author publications You can also search for this author inPubMed Google Scholar * Hagai

Meirovitch View author publications You can also search for this author inPubMed Google Scholar * Eli Barkai View author publications You can also search for this author inPubMed Google

Scholar * David A. Kessler View author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS The research was designed by E.B. and D.A.K. The analytic

calculations were performed by E.A., E.B. and D.A.K. The numerical simulations were performed by S.M., E.A. and H.M. All authors contributed to the analysis of the results and the writing of

the manuscript. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no competing financial interests. ELECTRONIC SUPPLEMENTARY MATERIAL SUPPLEMENTARY INFORMATION RIGHTS AND

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Medalion, S., Aghion, E., Meirovitch, H. _et al._ Size distribution of ring polymers. _Sci Rep_ 6, 27661 (2016). https://doi.org/10.1038/srep27661 Download citation * Received: 01 February

2016 * Accepted: 16 May 2016 * Published: 15 June 2016 * DOI: https://doi.org/10.1038/srep27661 SHARE THIS ARTICLE Anyone you share the following link with will be able to read this content:

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