Cyclic or loop polymers are distinct from their linear counterparts due to their lack of free ends. As opposed to linear chains, cyclic polymers are of relatively more compact configuration, and they can avoid topological entanglements [1, 2]. Due to these conformational and topological differences, cyclic-polymer melts exhibit a lack of a relaxation plateau in their viscoelastic spectrum [3, 4] and lower interpenetration and frictional forces if they are coated on surfaces [5–9], and they are also less prone to mix than their linear counterparts. These unique properties of cyclic polymers enable them as polymeric components in state-of-the-art polymer-base materials [10] while directing more attention towards the biological systems composed of naturally occurring cyclic-polymer structures [11].

In materials science, colloids grafted by linear polymers (i.e. polymer brushes) are common practice to avoid colloidal accumulation by harvesting the limited interpenetration between polymer brushes [12, 13]. This property emerges mainly because of the high entropic penalty for a grafted chain to diffuse through the polymer layers of an opposing colloidal particle. Suppose such colloids are confined in spherical or planar geometry. In that case, the packing geometry forces these particles to organize according to the dimension of the confinement, and particles form quasi-lattices that exhibit properties that are not conceivable in bulk systems [14]. Recent computational and experimental studies demonstrated that cyclic-polymer grafted surfaces are more effective in separating two polymer-grafted surfaces [5, 6], offering new ways of tuning such colloidal systems, mainly by changing the topology of surface-grafted polymers or the polymerization degree of grafted chains.

From a biological perspective, cyclic polymers play an essential role in understanding the mesoscale organization of genome [2, 15]. The genome is partitioned into chromosome structures, which are supramolecular protein-DNA complexes. In prokaryotes (e.g. bacteria), the tendency of circular DNA to segregate into distinct domains in the cytoplasm was explained by the topological properties of cyclic polymers [16, 17]. In eukaryotic (e.g. mammalian) cells, the cell nucleus isolates multiple chromosomes from the cellular cytoplasm. Inside the micron-size nucleus, each chromosome occupies a specific volumetric region referred to as chromosome territory (CT). CTs are robust throughout the lifetime of the cell, and they reform in sister cells after cell division [18, 19]. While the function of CTs is still under debate, experimental studies revealed that the formation of CTs is facilitated by the action of loop-forming structural maintenance of chromosomes (SMC) proteins such as condensin-family proteins [20–22]. These protein complexes can form chromosome loops of several mega-base pairs in size [23, 24], equip a single interphase chromosome with on average nearly ∼100 loop structures after assuming 100 million base pairs per chromosome. Further, these proteins' overexpression and underexpression affect the chromosome size, volume, inter-chromosome contacts, and average distance between chromosome centers in various cell types [22, 25–27]. More recent studies have revealed that the cellular concentration of SMC protein complexes generates a distinction between multiple species [28–30].

The persistent observation of chromosome territories and preservation of loop-forming proteins in humans and other species have brought up the idea that the topological properties of cyclic homo-polymers in their melt states can explain certain aspects of 3D genome organization [15, 31]. While cyclic polymers tend to mix less in melt and take a more collapsed conformation, linear polymers thoroughly mix and accept a more extended conformation (see [15] for a complete review on chromosome organization and cyclic polymers). Remarkably, a similar mechanism has been shown to contribute to the segregation of sister chromosomes in the replication process of bacterial genome [16, 17]. However, chromosomes are not homo-polymers. Instead, they are highly heterogeneous due to the cumulative effect of a dense array of structural proteins and variations in nucleotide sequences. For this reason, some chemically distinct chromosome sections containing mostly passive genes (i.e. genes not vital for the corresponding cell) are collapsed (e.g. chromocenters). In contrast, gene-active sections are relatively more swollen. In parallel, SMC proteins are more functional on the swollen sections of the chromosomes [32, 33], separating the chromosome core from loop-dominated portions of the chromosome. On a very coarse-grained level, each chromosome is a complex of many cyclic polymers attached to a compact solid-like chromosome core (i.e. chromocenter) [34].

The above depiction can also allow us to consider each chromosome as a polymer-grafted micron-size colloidal particle, such that loops of various sizes dangling from a dense chromocenter core if some loops can partially attach the core itself [21, 23, 35]. Notably, such structures are also known as rosette polymers in literature [36, 37], and previous computational models showed that they could contribute to the nuclear organization of chromosomes for a fixed polymer size [35, 38, 39]. Hence, micron-scale organization properties of chromosomes inside the cell nucleus can share a similar physical mechanism with the crystals of polymer-grafted colloids in spherical confinement. Consistent with this view, in addition to the emergence of CTs, the spatial distribution of chromocenters inside the cell nucleus is seemingly more regular than random for a wide range of species [40–43], similar to the organization of colloidal particles in spherical confinement [14, 44].

In this work, we extend the previous studies and offer a new view that the degree of polymerization or/and the number of cyclic polymers grafted on a colloidal particle can be used to model the activity of loop-forming SMC protein complexes to gain a polymer-physics perspective on how the genome is arranged within a nearly spherical nucleus roughly the size of several microns. In addition to its structural biological component, our study provides insights into how cyclic polymers on spherical particles cause differences in the 3D organization of colloidal particles in confined spaces compared to their linear counterparts. These differences are attributed to variations in the conformation and topology of cyclic polymers. By considering various grafting densities and polymerization degrees of grafts, we analyze the organization of at least n = 10 such particles at melt concentration using coarse-grained molecular dynamics (MD) simulations. Our analyses indicate that linear and cyclic polymer grafted particles are organized similarly in 3D within a rigid spherical volume at a polymer volume fraction relevant to eukaryotic cell nuclei (30%) [45]. However, cyclic polymers exhibit more uniform particle distribution profiles and lower contacts between the chains of neighboring particles than linear grafts for a wide range of grafting parameters. While a subgroup of our results with cyclic topology is in line with the previous predictions performed at lower polymeric volume fractions of around 10% [35, 38, 39], our simulations with linear chains suggest that the topology of grafted polymers does not change large-scale nuclear organization but rather could affect inter-chromosomal contact probability drastically at higher densities.

Using a coarse-grained model, we performed MD simulations of polymer-grafted colloidal particles in spherical confinement. To simulate flexible polymers, the bead-spring model of Kremer and Grest was utilized [46, 47]. A polymer chain was composed of N spherical beads of equal size and mass connected by massless springs. Both bonded and non-bonded monomer-monomer pairs repel each other by the truncated and shifted Lennard–Jones (LJ) potential

with the cutoff radius and . In equation (1), ε is the strength of the LJ interaction, and σ is the monomer diameter, which is taken as the units of energy and length scales, respectively. Equation (1) with the corresponding cut-off provides a purely repulsive interaction potential, and consequently, there is no attraction between any of the monomers composing our grafted particles. Correspondingly, the units of volume fraction, temperature and time are , and , where is the Boltzmann constant, and m is the monomer mass. The bonds between subsequent beads along the chain are mimicked by non-linear springs

with , and . Equations (1) and (2) together provide a Kuhn size of [46]. A polymer-grafted colloid was represented as a spherical core with f attached chains. The diameter of the colloidal particle is and its mass . To account for interactions between the polymer beads with colloids, which are incompatible in size, we incorporate the so-called expanded LJ potential [48]

with . In equation (3), the quantity δij is such that . The potential shift for monomer-colloid interaction is whereas for the colloid-colloid interaction is . The spherical confinement with radius R was modeled as a structureless, impenetrable, and repulsive rigid wall. We assume the monomer-wall interaction of the same form as in equation (1). The only difference is the replacement of the inter-particle distance r by in spherical coordinates, where the geometrical center of the system coincides with the center of the sphere.

The simulations were conducted for n = 10 confined colloids grafted by either linear or cyclic polymer chains, as shown in figure 1(a). While keeping the total number M of monomers in the systems constant, we performed simulations with different values of the degree of polymerization N of individual chains ranging from N = 15 to N = 240 and varying the number f of attached chains from f = 10 to f = 160. Initially, polymer-grafted colloids were distributed randomly inside a spherical volume with an initial radius corresponding to a density (i.e. volume fraction) (figure 1(b)). Subsequently, the initial radius was decreased to within a time window of to achieve the target density (i.e. 30% polymer content). We also tested slower compression times and various random particle positions and observed no qualitative difference in our results (cf figures 1(c) and (d)). All simulations were followed by a production run lasting .

Figure 1. (a) The structure of the chromosome models consists of f polymers grafted onto a rigid colloidal particle. (b) The model chromosomes are packed inside a sphere with an initial radius, . The volume of the sphere is decreased to to obtain the prescribed polymer volume fraction of . Each spherical volume contains n = 10 colloidal particles. The total number of chain monomers per colloidal particle is permanently fixed at M = 2400. The distribution of these monomers among the grafted chains varies based on the number N of monomers per chain in each case. (c) Various replicas lead to a similar average interparticle distance. (d) The default simulation time of volume decrease (fast) in the system preparation process is compared to a slower case.

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The simulations were conducted using the Large-scale Atomic/Molecular Massively Parallel Simulator [49]. Note that we focus on the quasi-equilibrium behavior of the grafted particles once the monomer concentration throughout the confinement is uniform. Thus, we expect Brownian or Monte Carlo simulation schemes to produce similar results. The velocity Verlet algorithm was used to integrate Newton's equations of motion, employing a time step . The temperature T was maintained by the Langevin thermostat with a friction coefficient . The simulation snapshots were rendered using the visual MD [50].

A direct one-to-one mapping of our polymer sizes to actual chromosome length is not possible due to the simplicity of the coarse-grained model. However, inspired by other pioneering modeling studies in the field [51–53], each bead can represent roughly 104 base pairs. Our choice of M = 2400 monomers per particle corresponds to base pairs per chromosome. Our shortest and longest grafts provide various loops sizes ranging between 105 and 106 base pairs, which are close to the intra-chromosome contacts controlled by condensin-family proteins [28, 54]. For n = 10 such particles, our model represents more than 107 base pairs in total. This number is smaller than human chromosomes, with ca. 100 million base pairs but close to the genome size of yeast. Note that our shorter chains are composed of N = 15 monomers, sufficient to obtain Gaussian statistics [55]. Further, these metrics provide a chromosome volume fraction of around , which is consistent with the recent electron microscopy studies [45].

In our simulations, we model chromosomes as colloidal particles, with each particle having f cyclic (or linear) flexible chains grafted onto it, where each flexible chain is composed of N monomers (cf figure 1). To create various structural configurations, we individually vary f and N while keeping the total number M = fN of chain monomers per particle fixed. In this way, we can simulate different partitioning scenarios of a single chromosome into varying numbers of loops, as observed in the cell nucleus [28], which is controlled by topological proteins such as condensin-family proteins. Our analysis considers the organization of n = 10 colloidal chromosomes inside a rigid, non-penetrable spherical shell with a radius mimicking either a cell nucleus or colloidal confinement. This confinement results in a volume fraction of [45]. The functionality f is related to the grafting density according to the equation:

Our objective is to identify the polymeric parameters, precisely the appropriate values of σg (or f) and N, that can mimic the experimentally observed effects of SMC proteins on chromosome structure and inter-chromosome interactions by using colloidal particles coated with cyclic polymers.

In figure 2, we display the organization of polymer-grafted particles for various f and N values. The snapshots depict the colloidal (core) particle (represented by red spheres) onto which polymers are attached. These cores can model either the chromocenters, which exhibit properties of a polymeric solid [34] or metal nano-particles [44]. In simulations, we observe well-separated colloidal cores repeatedly and systematically, independent of initial conditions (three replicas) and chain parameters, such that the core particles have no steric contact with each other. This suggests that the colloidal particles are kept separate from one another by the outer layers (shells), which are composed of polymers grafted onto their surfaces. However, the extent of the overlap between the polymer shells varies depending on the value of f. Through a visual examination, we observe that for systems composed of relatively longer cyclic chains (i.e. smaller f), there is a higher degree of interpenetration between the outer layers of the neighboring colloids. This trend depends on the grafting density and polymerization degree. For high grafting densities and low polymerization (e.g. f = 80 and N = 30), polymer shells overlap, but individual polymer-grafted particles are of hairy particle morphology in appearance. Consequently, mixing between polymer shells is visually absent (figure 2(a)). The shorter chains in these hairy particles can result in a glassy behavior, making them, on average, less susceptible to diffusion [56]. Further, as f increases (and N decreases), the structures of individual polymer-coated particles tend to become more spherical since shorter and densely grafted chains are more likely to adopt a stretched conformation [57]. On the contrary, for low grafting densities and high polymerization degrees (e.g. f = 10 and N = 240), the polymer shells of neighboring colloids overlap significantly (cf figure 2(a)). Notably, in this case, the pervaded volume of polymers fills the spherical volume, while the polymer of one particle can interpenetrate through the shells of neighboring particles.

Figure 2. Representative molecular dynamics snapshots of polymer-grafted colloids organized in the rigid spherical confinement. The chain monomers of each polymer-grafted particle are color-coded for clarity. The bottom frames in each case display the chains transparently to reveal the core particles, depicted as red spheres. The parameters N and f correspond to the polymerization degree of grafted chains and the number of grafted chains per particle, respectively. Each spherical volume contains n = 10 colloidal particles.

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Previous experiments showed that overexpression of condensin II in Drosophila cells leads to a lower chromosomal volume [22, 25]. The knockdown of various SMC proteins in embryonic stem cells also led to a nuclear chromosome decompaction and nuclear volume increase [26]. In our model, the low and high loop-forming activity could be described by the polymerization degree of grafted chains and provide insights into chromosome volume changes. To observe the effect of chain parameters on the total size of the particles (polymer+core), we calculate the average volume of individual particles by calculating the radius of gyration of all grafted polymers as a function of N. The average volume decreases as N decreases and f increases, which may correspond to a high condensin activity [25] (figure 3). This effect is more dramatic if linear chains replace cyclic polymers, as we will discuss further in the following paragraphs. Notably, the stiff chromosome structures also change the collective morphology of all confined chromosomes, specifically from a spherical shape to having protrusions, and polymer-free voids emerge between the chromosomes and the boundary of the spherical confinement.

Figure 3. The average volume per particle for linear and cyclic grafts as a function of polymerization degree. The volume is calculated via the radius of gyration of grafted polymers and normalized by the volume of the core.

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To further investigate cyclic topology's effect on particle organization, we analyze colloidal particles grafted by linear polymers. To construct the linear systems while keeping the total number of beads per core identical, we follow two strategies: (i) we cut the th bond of a cyclic polymer and effectively double the functionality while halving the chain length per core (figure 1(a)), (ii) we replace each cyclic chain by a linear chain without changing its functionality (figure 1(a)). Throughout this article, we refer to these two models as Linear I and Linear II. In principle, the two linear models explore a similar range of f and N values; however, the chains in the linear I case are relatively more densely grafted than those in the linear II cases (cf figure 1(a)). For completeness and direct comparison with corresponding cyclic-polymer cases, we will discuss these two cases in parallel throughout this work. In these simulations with linear chains, the arrangement of colloids exhibits comparable patterns similar to those observed in the cyclic-polymer grafted particles in relation to f (or N) values (figures 2(b) and (c)). However, a visual inspection reveals that polymer layers in the two linear cases can mix more drastically. The most drastic change occurs for the Linear II cases, in which even the highly functionalized colloids (e.g. f = 80-case shown in (figure 2) do not display a quasi-lattice order that is observed for Cyclic and Linear I cases. Notably, the Linear I case with f = 160 (i.e. N = 15) is quite distinct from other linear-polymer grafted cases and exhibits a hairy particle morphology due to its relatively low polymerization degree.

Earlier studies also reported a correlation between inter-chromosome interactions and gene translocation in Drosophila and human cells [25, 58]. To quantify the amount of overlap between our polymer-grafted colloids, we calculate the number of steric contacts between polymer chains of neighboring colloids by assigning a contact distance of between the beads of any two colloidal particles (figure 4). This distance corresponds to the occurrence of a steric contact between any two beads (cf equation (1)). This calculation confirms the effect of increasing contacts with decreasing f (and increasing N) (figure 4). However, the number of contacts between the beads of cyclic chains is systematically lower than those of linear cases (figure 4). Since both linear and cyclic cases have an equal number of beads per colloid, and f and N are related via for a fixed M, the difference between the linear and cyclic cases in figure 4 indicates that the main effect controlling the number of contacts is the chain topology. This significant difference is due to the lack of free ends in the cyclic polymers that are known to suppress the inter-digitation between two polymer brushes composed of cyclic polymers [5]. Notably, in figure 4, we observe a saturation behavior for large N regardless of the topology. The trend remains unchanged even when plotted on a logarithmic scale (shown in the inset of figure 4). We attribute this behavior to the finite size of the confinement; as the size of the polymer chains increases, the walls of the confinement can restrict the amount of inter-polymer contacts. Overall, our simulations show that shorter loops (more condensin activity) can reduce the average size of chromosomes while decreasing inter-chromosome intermixing in accord with the experiments [22, 25].

Figure 4. The average number of pairwise contacts between the shell beads of adjacent colloids for linear and cyclic grafts defined in figure 1. The inset shows the same data on a logarithmic scale.

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The number of contacts between the chains of two colloids could be related to the polymerization degree of grafted chains by utilizing the scaling arguments originally suggested for two weakly interacting planar brushes [59]. The thickness of the overlap region between the polymer shells of two neighboring colloids separated by a distance d can be expressed as , where is the characteristic size of the polymer chain, and is the corresponding fractal dimension of the polymer chains [5, 57, 59]. Assuming that the inter-particle distance is constant, we can write

where for linear polymers and concatenated cyclic polymers [4, 31], respectively. If the number of contacts is assumed to increase with the core-core distance d, both linear and cyclic polymers exhibit an increase as vs. , respectively. We should note that for cyclic polymers,