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Wenwen
Huang
^{a} and
Vasily
Zaburdaev
*^{ab}
^{a}Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, D-01187 Dresden, Germany. E-mail: vzaburd@pks.mpg.de
^{b}Friedrich-Alexander Universität Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen, Germany

Received
20th November 2018
, Accepted 30th January 2019

First published on 5th February 2019

Loop geometry is a frequent encounter in synthetic and biological polymers. Here we provide an analytical theory to characterize the shapes of polymer loops subjected to an external force field. We show how to calculate the polymer density, gyration radius and its distribution. Interestingly, the distribution of the gyration radius shows a non-monotonic behavior as a function of the external force. Furthermore, we analyzed the gyration tensor of the polymer loop characterizing its overall shape. Two parameters called asphericity and the nature of asphericity derived from the gyration tensor, along with the gyration radius, can be used to quantify the shape of polymer loops in theory and experiments.

Unlike solid objects, the shape of polymers in a fluctuating environment is less visual. The typical size of a polymer can be quantified by its gyration radius.^{13,14} A number of previous studies was devoted to the description of polymer shapes.^{15–19} Dating back to 1962, Fixman calculated the gyration radius distribution of polymer chains.^{20,21} Šolc et al. used the gyration tensor to characterize the shape of a random flight chain.^{22,23} The shape of pinned polymers was investigated both theoretically^{24,25} and experimentally^{26–28} with the development of single molecule imaging techniques. The looping constraint significantly affects the polymer shape. Alim et al. studied the shape of semi-flexible polymer loops without pinning and the external force field.^{17} Crumpled or elliptical shapes were described depending on whether the polymer was flexible or stiff.^{17} To the best or our knowledge, no previous work was focused on pinned polymer loops under an external force field.

The paper is organized as follows. In the next section, we introduce the model of pulled polymer loops. We then discuss the segment density function in Section 3. Section 4 is devoted to the gyration tensor and the gyration tensor based shape descriptors, such as gyration radius, asphericity and the nature of asphericity. We discuss our results in the concluding Section 5.

An illustration of a pinned polymer loop is shown in Fig. 1. Without loss of generality, we assume the first bead, whose position is denoted by r_{0}, to be pinned at the origin point, r_{0} = 0. The looping constraint is thus r_{0} = r_{N} = 0 where N is the total number of beads. The rod length is denoted by a and the direction of the external force field is chosen along the z direction. The orientation of the jth rod is denoted by the unit vector u_{j} = (r_{j} − r_{j − 1})/a. With this we can rewrite the looping condition as

(1) |

By construction, we have . Thus the potential energy of the pinned polymer loop configuration can be written as

(2) |

The polymer is immersed in the solution which leads to thermal fluctuations. We characterize the strength of these fluctuations by temperature T. The excluded volume effect will only be discussed briefly in the concluding section. Other interactions such as Coulomb interactions and bending energy are also not considered here.^{13,14} With the help of this simple setup, we calculate several shape descriptors in the following sections.

(3a) |

(3b) |

The mean and variance of every bead position can be written as the following expressions:^{10,32}

(4a) |

(4b) |

Interestingly, the variance is not given by a simple sum, but a more elaborate construction, which is a consequence of the Brownian Bridge condition. The random walk trajectory has to return to the origin, which makes the variance to be symmetric with respect to the middle of the loop (for details see ref. 10). The mean and variance of every rod orientation u_{j} can be obtained by using the grand canonical ensemble approximation

(5) |

(6a) |

(6b) |

Using eqn (6a), the chemical potential can be readily calculated as μ = (N + 1)/2. The x and y components of rod orientations can be obtained by symmetry argument and the fact that u_{j}^{2} = 1. We thus have

〈u_{j,x}〉 = 〈u_{j,y}〉 = 0, | (7a) |

var[u_{j,x}] = var[u_{j,y}] = (1 − 〈u_{j,z}^{2}〉)/2. | (7b) |

Here 〈u_{j,z}^{2}〉 = var[u_{j,z}] + 〈u_{j,z}〉^{2}. With the results above, the distribution of the ith bead position eqn (3) can be calculated in a straightforward manner.^{10} The segment density function as the sum over individual segment distributions, is:

(8a) |

(8b) |

We show in Fig. 2 the segment density distribution of a 3D pinned polymer loop projected on the 2D x–z plane. The upper panel shows the marginal distribution of density along the force direction, while the right panel shows the marginal distribution of density perpendicular to the force direction. The red lines represent our theoretical results above and match very well with the Monte Carlo simulation data shown by blue histograms. Fig. 2(a) shows the case of the pinned polymer loop in a strong external force field. We can see that the polymer shape is fully stretched with most of fluctuations happening at the free end, which corresponds to the middle part of the loop. Fig. 2(b) shows the case of the pinned polymer loop in a moderate force field. The difference between the free end and pinned ends are clearly demonstrated here, which make the shape of the polymer loop look more like a pendulum. The case of the weak external force field is shown in Fig. 2(c), where a droplet-like shape is observed. Having an intuition of the overall shape of pinned polymer loop in an external field, we discuss the gyration tensor in the next section.

Fig. 2 The segment density distribution of pinned polymer shape indicated by the color map for (a) strong external force field with F = 1, (b) moderate external force field with F = 0.1, (c) weak external force field with F = 0.01. The unit of the force is rescaled to be k_{B}T/a. ρ indicated by the colormap is the segment density. The panels above and on the right of each density plot are the marginal distributions along the z- and x-axis respectively. The solid red lines are showing the analytical expressions eqn (8). Other parameters are set as follows k_{B}T/a = 1, N = 100. |

(9) |

(10) |

Based on the gyration tensor, the first shape indicator to be discussed is the gyration radius of the polymer, which is defined and can be calculated as follows:

(11) |

We show how the mean gyration radius 〈R_{g}^{2}〉 of the polymer loop varies with the strength of the external force field in Fig. 3. The gyration radius increases monotonically with the external force field. Note that the exact results of the two extremes can be easily obtained as 〈R_{g}^{2}〉 = Na^{2}/12 (F = 0) and 〈R_{g}^{2}〉 = N^{2}a^{2}/48 (F → ∞). In the zero external force limit, the polymer is a random coil while it is fully stretched in the strong external force limit. For the case in between, we can calculate the mean gyration radius as

(12) |

(13) |

Fig. 3 The mean of gyration radius square as a function of the strength of the external force field. N = 100, k_{B}T/a = 1. |

The mean and variance of r_{i} can be obtained from eqn (4) and the covariance can be obtained in a way similar to the variance

(14) |

Taken together we arrive at the result shown by the solid line in Fig. 3. To further investigate the fluctuations of the gyration radius, in Fig. 4, we show the normalized distributions of the gyration radius for several strengths of the external force field. Interestingly, the height of the distribution varies non-monotonically with the strength of the external force. In other words, the gyration radius fluctuates most under a moderate force field.

To explain this non-monotonic behavior, we use the concept of accessible volume,^{33,34} which can be estimated by ν = λ_{x}λ_{y}λ_{z}. We show, in the inset of Fig. 4, the variance of gyration radius, which characterizes the width of the distributions, as a function of the external force field. On top of it, we also plot the accessible volume ν. As we can see the trends of these two curves are very similar, which indicates that the accessible volume can be correlated with the non-monotonic behavior observed in Fig. 4. The discrepancy of these two curves after the maximal point is because of the symmetry break due to the pinning: as we can see in Fig. 2(b), the overall shape of the polymer is an asymmetric rather than a symmetric ellipsoid.

In the weak external force regime, the behavior of the bead-rod system is well approximated by the bead-spring model where the equilibrated spring length equals to the rod length. We have calculated characteristics of the corresponding bead-spring loop in the Appendix. An important fact for the bead-spring model is that the variance of the bead position does not depend on the external force field. Consequently, the radius of the fitted ellipsoid in the direction perpendicular to the external force direction is unchanged in different external force fields, while the radius along the external force direction grows with the external force field. Thus, a larger accessible volume is obtained, and we observe the initial increase in the inset of Fig. 4. However, the finite extensibility of rods becomes important when we further increase the strength of the external force field. The radius of the fitted ellipsoid perpendicular to the force direction is narrowed down and the accessible volume also decreases, which corresponds to the decay part in the inset of Fig. 4.

We next study the asphericity and the nature of asphericity, two other descriptors that commonly used to quantify the shape of polymers. Their definition is based on the gyration tensor.^{17} The asphericity is defined as

(15) |

(16) |

Intuitively, the asphericity Δ is a parameter related to the variance of the three eigenvalues where λ_{i} ∈ {λ_{x}, λ_{y}, λ_{z}} and = (λ_{x} + λ_{y} + λ_{y})/3. The nature of asphericity Σ is a parameter related to the product of the three eigenvalues . For visualization and comparison purposes we use the rescaled version of these two measures such that and arccosΣ/3 ∈ [0,π/3].^{17} The geometrical interpretation of these parameters is illustrated in Fig. 5(b). The limit of Δ = 0 corresponds to a fully spherical object while Δ = 1 describes the shape of the rod-like object, whereas Σ measures whether the object is prolate or oblate.

In Fig. 5, we plot the distribution of the polymer shapes on the phase diagram of asphericity and the nature of asphericity. We observe that the shapes are distributed in a smaller region of the parameter space with the increasing force. In addition, the prolate and elongated shapes are preferred largely independent of the external force. The diagrams we plotted here are similar to those obtained for semi-flexible unpinned polymer loops.^{17}

Here we focused on one freely jointed bead-rod loop. We did not take into account the excluded volume effect in our model. Our simulation results show that adding the excluded volume does not change the results presented so far qualitatively but leads to the increase in the system size by 5–10 percent (results not shown).

There are several interesting aspects we can further explore in the future. One example is the shape of multiple pinned polymer loops. Our preliminary simulation results show that the non-monotonic behavior in Fig. 4 disappears when there are two or more pinned loops in the system. Note that in fission yeast, there are three pairs of chromosomes pinned at the same point, namely six loops.^{12} Another topic to explore is the change of polymer shape in the time-dependent force field. For instance, the turning process of the chromosomes during oscillations in fission yeast is an important shape change that carries a biological function and requires further investigations.

Rouse theory is a theoretical framework to calculate the dynamics of a bead-spring polymer.^{14,30} We consider a pinned polymer loop modeled by beads and connecting springs. As in our previous discussion of the bead-rod model, the bead labeled by 0 is assumed to be pinned at the origin and there are N beads in total in the loop. Again, the periodic indexing is used. We can write the pinned condition as r_{0} = r_{L} = 0.

The dynamical equation, which is a Langevin equation in the overdamped regime, can be written as:

(17) |

(18) |

As in the main text, we do not take into account hydrodynamic interactions, bending energy and excluded volume effect in this simple model. For convenience, we use the vector notation and rewrite eqn (17) as:

(19) |

[Ω^{−1}AΩ]_{jk} = D_{jk} = λ_{k}δ_{jk}, | (20) |

(21a) |

(21b) |

Multiplying both sides of eqn (19) by Ω^{−1}, we arrive at

(22) |

(23) |

Eqn (23) can be solved easily by standard methods. The general solution can be written as:

(24) |

(25) |

Finally, the bead position can be obtained by the inverse transformation:

(26) |

Now the stationary statistics of the polymer, such as the mean and variance of each bead position can be calculated easily. Substituting _{j}(t) from eqn (25) to eqn (26) and taking the limit t → ∞, we get

(27) |

The summation over j can be calculated explicitly to get

(28) |

One can clearly see from eqn (28) that 〈r^{∞}_{i}〉 = 〈r^{∞}_{N−i}〉 as expected. In addition, the components of mean position perpendicular to the force field direction are vanished.

In order to calculate the variance of the bead position, it is nontrivial to first calculate the two-time correlation function of normal coordinate position as

(29) |

Then the second moment of the bead position can be calculated as:

(30) |

Finally, taking the limit t → ∞, we get the equilibrium variance of the bead position

(31) |

It is worth mentioning here we also have the symmetry that var[r^{∞}_{i}] = var[r^{∞}_{N−i}]. Moreover, we want to point out that the variance does not depend on the external force. It means that the statistical distance between two beads does not depend on the external force field. This is essentially because infinite extensible Hookean springs are used in the Rouse theory. Furthermore, we also want to remark that the result of eqn (31) is identical to the Brownian bridge result without external force.

Now we try to fit the Rouse theory to the pinned bead-rod loop. To do this, we take the spring in the bead-spring model as an entropic spring and then relate the spring constant to the length of the rod. If the spring in the polymer is the three-dimensional entropic spring, then the spring constant can be found by the equipartition theorem as

(32) |

(33a) |

(33b) |

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