Local volume concentration, packing domains, and scaling properties of chromatin
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The authors develop a selfreturning selfavoiding polymer model of chromosome organization and show that their framework can recapitulate at the same time local density and largescale contact structural properties observed experimentally by various technologies. The presented theoretical framework and the results are valuable for the community of modelers working on 3D genomics. The work provides solid evidence that such a framework can be used, is reliable in describing chromatin organization at multiple scales, and could represent an interesting alternative to standard molecular dynamics simulations of chromatin polymer models.
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Abstract
We propose the Self Returning Excluded Volume (SREV) model for the structure of chromatin based on stochastic rules and physical interactions. The SREV rules of return generate conformationally defined domains observed by singlecell imaging techniques. From nucleosome to chromosome scales, the model captures the overall chromatin organization as a corrugated system, with dense and dilute regions alternating in a manner that resembles the mixing of two disordered bicontinuous phases. This particular organizational topology is a consequence of the multiplicity of interactions and processes occurring in the nuclei, and mimicked by the proposed return rules. Single configuration properties and ensemble averages show a robust agreement between theoretical and experimental results including chromatin volume concentration, contact probability, packing domain identification and size characterization, and packing scaling behavior. Model and experimental results suggest that there is an inherent chromatin organization regardless of the cell character and resistant to an external forcing such as RAD21 degradation.
Introduction
Chromatin is a complex macromolecular fiber that results from the assembly of DNA with histone and nonhistone proteins to form the functional organization of the genome within the eukaryotic cell nucleus. That over 2linear meters ($\sim 6\times {10}^{9}$ base pairs) is confined within human nuclei ranging between 5 and 10 µm in diameter while maintaining functionally relevant information creates a core dilemma that places a tension between efficiency of packing with information retention (Annunziato, 2008). Adding to this complexity are the rich heterogeneity of nonchromatin nuclear bodies, histone concentrations within normal cells, and chromosome copy number (and total DNA content) in malignant cells (Clapier and Cairns, 2009; Tessarz and Kouzarides, 2014; Finn et al., 2019; Mansisidor and Risca, 2022). Despite the profound degree of variability from celltocell even within microscopically normal tissues (Nagano et al., 2013), the ensemble function of organs is maintained by facilitating the preferential activation of specific gene network patterns. In these contexts, describing chromatin as a stochasticaly evolving process with constraints appears to be an alternative, complementary approach to represent the regulatory processes that couple structure with function (Sood and Misteli, 2022).
Numerous polymer models of chromatin organization have been proposed to predict possible configurations (Fujishiro and Sasai, 2022; AdameArana et al., 2023; Forte et al., 2023; Shi and Thirumalai, 2021; Tamm et al., 2015; Polovnikov et al., 2018; Mirny, 2011). Many models have been motivated by the properties observed in HiC, with some recent studies interested in recapitulating the microphase separation observed microscopically (Fujishiro and Sasai, 2022; AdameArana et al., 2023). Heteropolymer models, where the monomers are partitioned into two groups, can achieve microphase separation by introducing attractive potentials between components of each group (e.g. ‘b’ monomers are attracted to ‘b’ but repulsed by ‘a’) (Fujishiro and Sasai, 2022; AdameArana et al., 2023). Likewise, introduction of spatially defined longrange loops to approximate cohesinmediated loop extrusion can similarly regulate microporous structures (Nuebler et al., 2018). Existing homopolymer models that have been proposed either have limitations in their degree of coarsegraining (e.g. HIPPS(HiCpolymerphysicsstructures) monomers are composed of 1200 bp, ∼6 nucleosomes with a diameter of ∼60 nm) or are variations on a random walk polymer (Forte et al., 2023; Shi and Thirumalai, 2021). Homopolymer models are unable to achieve the biphasic, porous states observed on ChromEM, configurations that reliably result in contact scaling ($S$) less than −1 at supranucleosome length scales (10^{5} to 10^{6} bps) as is frequently observed in HiC (Tamm et al., 2015; Polovnikov et al., 2018; Mirny, 2011) while also producing the observed physiologic range of chromatin powerlaw massdensity organization (scaling exponent, $D$) that ranges between 2 and 3.
Regarding these last points, a fundamental issue results from solely using measures of connectivity, such as HiC, for polymer modeling of chromatin organization due to the inverse relationship in polymers between mass density and contact scaling that obey $M\propto {r}^{D}$ and $D\sim 3\u2044(S)$. Thus, the widely observed $S<1$ results in configurations with mass in excess of the volume capacity ($D>3$). Likewise, a random walk polymer model can achieve the limiting cases of $D=2$ (a polymer in a Θ solvent) and $D=3$ (a random walk in a confined volume) but these cases limit the functional role of chromatin to facilitate enzymatic processes (RNA transcription, replication, repair) with nucleosome size monomers. For example, in $D=2$, nuclear enzymes would be diffusing through very large nuclei with a fully accessible genome whereas in $D=3$, there is scant accessible space resulting in exclusive molecular activity at the surface of the genome. $D$ ranging between 2 and 3 is not achievable with existing models while accounting for volume considerations. This range has functional consequences as it produces genomic configurations that will be inaccessible (domain centers), surfaces for enzymatic activity, and lowdensity spaces for molecular mobility.
There have been important efforts to model chromatin and a comprehensive review have been recently published (Yildirim et al., 2022). Many works are based on atomistic or a nearly atomistic approach addressing different processes involving DNA, histones, and other proteins (Bishop, 2005; EslamiMossallam et al., 2016; Bowerman and Wereszczynski, 2016; Melters et al., 2019; Zhang et al., 2017; Freeman et al., 2014; Lequieu et al., 2016; Brandani, 2018; Lequieu et al., 2017; Lequieu et al., 2019; Li et al., 2023; Arya et al., 2006; Arya and Schlick, 2009; Dans et al., 2016; JimenezUseche et al., 2014; Norouzi and Zhurkin, 2015; Bajpai and Padinhateeri, 2020; Luque et al., 2014; Perišić et al., 2019; Bascom et al., 2019; Bascom et al., 2017; Wiese et al., 2019). From the other end of the chromatin length scale the aim is to use experimental results, especially from highthroughput chromatin conformation capture (HiC) (LiebermanAiden et al., 2009), to guide polymer models simulations with especial characteristics that can replicate for example, contact patterns and loop extrusion process (Banigan and Mirny, 2020; Barbieri et al., 2012; Brackley et al., 2017; Fudenberg et al., 2016; Nuebler et al., 2018; Rao et al., 2014; Sanborn et al., 2015; Chan and Rubinstein, 2023; Jost et al., 2014). Many lines of evidence support the idea of chromatin configurations as a statistical assembly that produce functional organization. First, the overwhelming majority of the genome does not code for proteins but has functional consequences at the level of regulating gene transcription. Second, HiC (LiebermanAiden et al., 2009) and similar techniques identify the presence of compartments, domains, and loops; however, these structures only become evident as distinct contact loci with millions of sequence measurements (Szabo et al., 2019; Rajderkar et al., 2023). Third, singlecell sequencing and in situ sequencing of normal tissue and malignancies has demonstrated profound heterogeneity in transcriptional patterns that were previously not appreciated under routine histological examination (Finn et al., 2019). Finally, ongoing methods investigating chromatins structure have shown that it is dynamically evolving even at the order of seconds to minutes (Nagano et al., 2017).
We present herein a minimal model based purely on molecular, physical, and statistical principles which (1) preserves the efficiency of chromatin packing, (2) produces the structural heterogeneity and population diversity observed experimentally, (3) retains the capacity for functionally relevant storage of genomic information across modalities, and (4) would be sensitive to variations in density present in clinically relevant contexts (i.e. ploidy and nuclear size are frequently varied in cancer and mammalian cells have a distribution of nuclear sizes that vary by tissue function). To produce this model, we began by assuming that there is an overall statistical rule governing the spatial organization of chromatin. Inspired by known features of genome organization, (1) nucleosomes are the base structure of the chromatin polymer, (2) longrange interactions arise from a plurality of mechanisms (loop extrusion, promoter–promoter interactions, promoter–enhancer interactions, and spatial confinement), and (3) the volume fraction of chromatin depends on genomic content coupled with nuclear size which therefore varies in different tissues and states. We show by representing these processes from the interplay of stochastically occurring lowfrequency, large extrusion returns (stochasticreturns) probabilistically from multiple processes in the context that monomers occupy physical space (excluded volume) that the missing features of chromatin polymer modeling are obtained. We demonstrate first that this model recapitulates the groundtruth structure of chromatin on methods that measure structure (chromatin electron microscopy, partial wave spectroscopic [PWS] microscopy) and connectivity (HiC). The findings from the model address the deficiencies occurring in existing literature: the biphasic structures on chromatin electron microscopy is observed, scaling and spacefilling properties are preserved, and the expected population heterogeneity arises de novo from stochastically produced configurations. With a minimal model depending on just two parameters, we demonstrate the production of irregular fiber assembles with a radius of ∼60 nm while producing the average nuclear density of 2030%.
In agreement with chromatin scanning transmission electron microscopy (ChromSTEM) and many other experimental methods, we find that genomic structure has a characteristic radial dependency that can be interpreted in terms of a powerlaw with exponent $D$. Comparing Self Returning Excluded Volume (SREV) to livecell PWS microscopy, we demonstrate that the diversity in chromatin configurations observed on SREV corresponds with experimental observations. We then test the distinct roles that longrange returns and excluded volume have on structure using an auxininducible degron RAD21 cell line, allowing the depletion of a core component of the cohesin complex that can be quantitatively tested by PWS and ChromSTEM microscopy. In our model, the longrange steps arise from a confluence of processes and the inhibition of one of these processes like for example cohesinmediated loop extrusion has a limited effect on chromatin packing. Remarkably, our model demonstrates that upon RAD21 depletion, only ∼20% decrease in the number of observed domains, with the remaining domains largely unaffected at the level of their size, density, and $D$; results recapitulated directly on ChromSTEM imaging. Furthermore, depletion of RAD21 is predicted to have a minor effect on the diversity of chromatin configurations, a finding again confirmed with livecell PWS microscopy. Finally, we show that excluded volume results in a nonlinear, monotonic relationship between powerlaw organization and local density that plateaus near $D$ of 2.8, predictions observed with and without RAD21 present in ChromSTEM.
The structures predicted by our model display a porosity that result from the alternation of high and lowdensity regions. The envelope of the highdensity regions could be regarded as the separating interphase of a bicontinuous system that is a topological scenario that favors extensive mobility of proteins, mRNA, and other free crowders while providing a large accessible surface area of chromatin. The contact probability, calculated as an ensemble average, shows a good agreement with HiC results displaying a transition between intra and interdomain regimes. The intradomain contact probability scales with an exponent $S$ > −1, while the interdomain one scales with an exponent $S\sim 1$. As such, this work introduces the basis for a statistical representation of the genome structure.
Results
A minimal model for chromatin conformations
The SREV model for chromatin is derived from the Self Returning Random Walk (SRRW) model that was recently introduced by this group (Huang et al., 2020). Here, we review the SRRW model and then we introduce the modifications that lead to the SREV model.
The SRRW model is essentially a random walk with specific rules introduced to capture statistical features of chromatin organization as revealed by experiments. At each step in the SRRW generation there are two possibilities: (1) Perform a forward jump or (2) Return over the previous step to the previous position. The probability $P}_{R$ for a return step is given by
Here, $U}_{0$ is the length of the last step along the backbone over which the walk may return. The folding parameter $\alpha >1$ controls the number of returns. If the SRRW does not continue with a return step, it must continue with a forward jump. The new forward jump is chosen with an random direction and with a length U_{1} given by the following probability distribution function (pdf)
We will generally refer to Equations 1 and 2 as the return rules of the SREV model. There is a minimum size for the forward jumps that also defines the unit of length in the model. The succession of forward jumps and return steps leads to a structure than can be regarded as a linear backbone with treelike branches along its length, with the branching points representing overlaps created by the return steps. In addition to the return probability and pdf defined above, the SRRW generation algorithm (contained in Appendix 1) includes a local cutoff to avoid unrealistically long steps and a spherical global cutoff to contain the configuration. The global cutoff is applied during the generation of the conformation and is measured from the center of mass of the alreadygenerated steps.
By construction, since the SRRW includes returns over the previous steps, it contains a large number of overlaps. For α = 1.10, 1.15, and 1.20 the number of returns is 48.7%, 47.5%, and 46.2% of the total number of steps, respectively. Therefore, as a representation of a physical system, such as chromatin, the SRRW has two important drawbacks: (1) the conformations violate the principle of excluded volume and (2) it is not a linear polymer. In order to recover these two physical properties we extended the SRRW to develop the SREV model. In this new method, the overlapping points are transformed into connected clusters of beads that explicitly represent a linear chain, as shown on the scheme displayed in Figure 1. The method that we employ to remove overlaps is a lowtemperaturecontrolled molecular dynamics simulation using a soft repulsive interaction potential between initially overlapping beads, that is terminated as soon as all overlaps have been resolved, as described in the Appendix 1. An example of an SRRW configuration and its corresponding SREV are displayed in Figure 2A and E, respectively. Figure 2B, F represents a small region on the periphery of the configuration and exemplifies how structures formed by a sequence of forward and returns steps expands to a larger cluster after including excluded volume interactions. The porosity of the structure is also affected by the excluded volume introduced in SREV.
The density heterogeneity displayed by the SREV configurations can be analyzed in terms of the accessibility. One way to reveal this accessibility is by calculating the coordinations number (CN) for each nucleosome, using a coordination radius of 11.5 nm, along the SREV configuration. CN values range from 0 for an isolated nucleosome to 12 for a nucleosome immersed in a packing domain. In Figure 3, we show the SREV configuration shown in Figure 2, but colored according to CN. CN can be also considered as a measure to discriminate heterochromatin (red) and euchromatin (blue). Figure 3A shows how the density inhomogeneity is coupled to different CN, with high CN represented in red and low CN represented in blue. Figure 3B shows a 50nm thick slab obtained from the same configuration that clearly shows the nucleosomes at the center of each packing domain are almost completely inaccesible, while those outside are open and accessible. It is also clear that the surface of the packing domains is characterized by nearly white nucleosomes, i.e. coordinated toward the center of the domain and open in the opposite direction.
For this work, we adopted a unit length of 10 nm, similar to the diameter of a nucleosome (Maeshima et al., 2014). Therefore, each bead of the model chromatin represents a nucleosome. The spherical global cutoff was set to ${R}_{c}=$ 650 nm. From the resulting conformations, we can cut slabs spanning well over 1 µm in cross section. Excluded volume was introduced by imposing a nonoverlap radius of ${r}_{\circ}=4.9$ nm between all the beads of the SREV model. With these quantities, we defined the overall average volume fraction as $\varphi =N({r}_{\circ}/{R}_{c}{)}^{3}$, with $N$ the number of beads in the chromatin model chain. We considered four different volume fractions $\varphi =$ 0.08, 0.12, 0.16, and 0.20, which correspond to $N=$ 186,741, 280,112, 373,483, and 466,854, respectively. Each one of these four average volume fractions was studied with three different folding parameters $\alpha =$ 1.10, 1.15, and 1.20. SREV configurations, as we present them in this work, are associated to the structure of a single chromosome. Therefore, all the analysis that follows is done on the structure of a single chromosome system. For each combination of $\varphi$ and $\alpha$, we created an ensemble of 1000 different chromatin configurations. In order to introduce the genomic distance along the SREV configuration we assign 147 base pairs to each nucleosome, representing the length of DNA wrapping the histone octamers. Considering that the effective bead diameter is 9.8 nm, the average distance between adjacent base pairs in the DNA double helix, and the model bonds $U}_{i$ that are larger than 10 nm, we assign the number of base pairs in the linker DNA as the nearest integer of $({U}_{i}9.8\phantom{\rule{thinmathspace}{0ex}}\mathrm{n}\mathrm{m})/(0.34\phantom{\rule{thinmathspace}{0ex}}\mathrm{n}\mathrm{m})$. In Table 1, we summarize the 12 studied cases with the resulting mean value for the length, in base pairs, of the linker DNA between nucleosomes that slightly depends on $\varphi$ and $\alpha$. The overall average length of the linker DNA sections is 39.6 base pairs and with values of 36.3 and 44.4 for the two extreme cases. We must remark that the predicted DNA length between histone octamers agrees with the widely reported values (Beshnova et al., 2014; Wang et al., 2021; Lequieu et al., 2019; Li et al., 2023; Zhurkin and Norouzi, 2021). Finally, and in order to correlate our work with experimental examples, the longest simulated chromatin corresponds to $88\phantom{\rule{negativethinmathspace}{0ex}}\times \phantom{\rule{negativethinmathspace}{0ex}}{10}^{6}$ base pairs, which is approximately the size of human chromosome 16.
SREV reproduces the biphasic chromatin structures observed in ChromSTEM imaging
In order to start assessing whether the SREV model produces realistic configurations of chromatin it is necessary to bring the model to a representation similar to the results of imaging experiments. For example, ChromSTEM captures the chromatin density from a slab of 100 nm thickness. Then, we cut a similar slab from an SREV configuration and transform the point coordinates of the model nucleosomes to a two dimensional density that considers the nucleosomes volume. In Figure 4A, we show a representation of an SREV configuration as it result from the model and in Figure 4B the collapsed two dimensional density as a colormap highlighting the porosity of the model and the emergence of chromatin packing domains. In Figure 4C, we show a ChromSTEM image for A549 cell. Since our SREV structures represent a single chromosome, it does not cover the full field of view of 1300 nm × 1300 nm that can be appreciated in the experimental image. However, the qualitative resemblance of the theoretical and experimental chromatin densities is stunning. The quantitative characterization of the model and its agreement with experimental results is analyzed below.
SREV is a nonhomogeneous polymer model. The only physical interactions present in the model are the connectivity, the excluded volume, and the confinement that, together with the return rules induce the formation of granular structures, or packing domains, with local density variations. This granularity can be qualitatively visualized by wrapping a mesh around the chromatin conformation, as shown in Figure 2G, H. Rotating versions of Figure 2G, H are included in Figure 2—video 1 and Figure 2—video 2. It is worth noting that this representation is qualitatively similar to Figure 4, panels E, F, and G from Ou et al., 2017. At first glance, the wrapping interface between the region denser in chromatin and the region almost empty of chromatin resembles the dividing interface between two disordered bicontinuous liquid phases (Walker et al., 2014). We find this outcome from the SREV model quite interesting in view of recent claims that liquid–liquid phase separation could be related to heterochromatin and euchromatin segregation, and that chromatin domains have a liquid character (Chen et al., 2022; Itoh et al., 2021). Moreover, the bicontinuous topology offers two important functional advantages: First, the interface offers a very large surface area exposing the a significant fraction of the genome and second, the continuity of the dilute phase allows for the migration of free crowders (including proteins, transcription agents, mRNA, etc.) to any region in the nucleus.
SREV demonstrates that genome connectivity decouples from domain structure
The granularity of chromatin manifest itself in the polymeric properties of the model. Chromatin is a special type of polymer, and requires a careful analysis. The scaling relationship between the endtoend distance and the polymer contour length, in this case the genomic distance, cannot be described in general with a single powerlaw relationship, i.e. a single Flory exponent, as it is the case for synthetic polymers. In Figure 5A, we display the ensemble averaged endtoend distance, $\u27e8{R}^{2}(n){\u27e9}^{0.5}$ as a function of the genomic distance $n$. All the studied cases are included in the plot, but they coalesce in three distinct groups according to the folding parameter $\alpha$ and with almost no effect of the overall volume fraction. The figure also shows a transition occurring for $n\sim 4\phantom{\rule{negativethinmathspace}{0ex}}\times \phantom{\rule{negativethinmathspace}{0ex}}{10}^{4}$ base pairs, from a local or intradomain regime that corresponds with distances up to 100 nm, to a longrange or interdomain one. The Flory exponent in the intradomain regime (0.342, 0.347, and 0.354 for $\alpha =1.10$, 1.15, and 1.20, respectively) is consistent with a nearly spacefilling cluster and slightly smaller than in the interdomain regime (0.353, 0.394, and 0.396). For $n$ values larger than 10^{6} the curves level off due to the effect of the spherical confinement. The analysis can also be applied to the ensemble average contact probability, $\u27e8{C}_{p}(n)\u27e9$, which is defined as the probability for two base pairs, separated along the polymer by a genomic distance $n$, of being in contact with each other (or being at a distance smaller than a cutoff). In Figure 5B, we display $\u27e8{C}_{p}(n)\u27e9$ for all studied cases, using a cutoff distance of 35 nm. We see in this figure that the contact curves depend only marginally on volume fraction as the four distinct cases for each $\alpha$ are nearly indistinguishable. Thus, this indicates that measures of connectivity observed in HiC would not depend on nuclear volume concentrations. This finding is in strong agreement with the results reported in Liu and Dekker, 2022 where expansion and contraction of isolated nuclei has minimal effects on contact scaling, $S$. As in the endtoend distance, in $\u27e8{C}_{p}(n)\u27e9$ we can also distinguish a transition between intra and interdomain regimes. In general, the slope $S$ of $\u27e8{C}_{p}(n)\u27e9$ in log–log representation is larger than −1 in the interdomain regime, and fluctuate around −1 for interdomain genomic distances. Figure 5C shows the contact probability determined from HiC experiments. The blue dots correspond to chromosome 1 of HCT116 cells and the behavior between 10^{5} and 10^{6} base pairs is well described by a slope $S$ very close to −1. The experimental data also show a change at intermediate separations. It is important to note to the agreement is relatively good even in quantitative terms, with the transition occurring at similar genomic distance and value of ${C}_{p}(n)$. Since the model does not have a genomic identity or any specific architectural modifiers (e.g. CTCF(CCCTCbinding factor) and/or cohesin), the contact probability curves do not represent a particular cell or chromosome. We must mention that the other chromosomes from the HCT116 cells have a qualitatively similar contact probability, with a powerlaw fitting having slopes $S$ varying from −0.85 to −1.10, depending the case.
Chromatin volume concentrations couple with longrange returns to determine 3D structure
The heterogeneous character of chromatin revealed by experiments is captured, as we have qualitatively shown above, by the SREV model. A straightforward characterization of this heterogeneity is the distribution of local volume fraction calculated with a probing volume of adequate size. In the language common in chromatin experiments, this volume fraction is referred to as the chromatin volume concentration (CVC) and the probing volume is, for example, a cube with an edge of 120 nm. Using electron microscopy and tomography techniques (ChromEMT), the group of Dr Clodagh O’Shea (Ou et al., 2017) reconstructed the conformation of chromatin on a 120nm thick slab with an area of 963 nm × 963 nm, which allowed them to measure the CVC distribution using a 8 × 8 × 1 grid with cubic cells of 120 nm edge size. To calculate the CVC from the SREV configuration ensembles we followed the same methodology employed in the experiments. Since we have the full 3D structure of the model chromatin we are not restricted to a slab, then we used a 6 × 6 × 6 cubic grid of (120 nm)^{3} probing volumes. Moreover, our results represent ensemble averages over the populations of 1000 replicates for each of the Φ and α combinations. The results for each case are summarized in Figure 6 revealing that both SREV parameters, Φ and α, are important in determining the CVC distributions. We see that overall the volume fraction take values up to 0.6, which is consistent with our model representing the nucleosomes as spheres that can achieve a maximum volume fraction of 0.74 as a crystal and 0.64 in the jamming limit (Jin and Yoshino, 2021). The peak of the CVC distribution increases as the overall volume fraction α increases. The recent ChromEMT results reveal a CVC distribution covering a nearly identical range to our SREV results. Comparing with the experimental results, for the lowest overall volume fraction the distribution has an excessive proportion of lowdensity regions. Consequently, although we show that all regimes will result in domain formation with $\alpha =1.15$ and $\varphi =0.16$ being the closest to what is observed in A549 cells (Figure 7C), this would indicate that the variation in chromatin density that arises in mammalian cells would be predicted to have distinct functional consequences that would not be captured by connectivity. As we show below, chromatin packing domain organization will be weakly and inversely related to the probability of return events but strongly associated with the local volume concentrations.
Since the CVC is a measure using a relative large probing volume its distribution with values ranging from 0 to 0.6 may be achieved by a (dynamic) smooth continuous modulation of the chromatin density or by a (also dynamic) mixing of distinct high and lowdensity regions. The latter scheme gives rise to the concept of packing domains, as it has been recently proposed from the analysis of imaging experiments (Li et al., 2022; Li et al., 2021; Miron et al., 2020). The formation of domains is also consistent with the possibility of a microphase separation process dynamically occurring in chromatin (Strom et al., 2017; Larson et al., 2017; Falk et al., 2019; Hilbert et al., 2021). Moreover, a dynamic disordered bicontinuous phase separation is also in line with all the mentioned scenarios, especially considering that all imaging experiments are restricted to a quasi 2D slab of the system that could be insufficient to reveal a full 3D topology.
For the analysis of the SREV configurations, we take advantage of the methodology developed by our experimental collaborators and transform our coordinates to a stack of images (Li et al., 2022; Li et al., 2021). For this transformation, each bead is represented by a normal distribution and its contribution to a given voxel of the tomogram is the integral of the normal distribution over the voxel volume. We include Figure 4—video 1 that is an example of the resulting volumetric image stack. As we display in Figure 4B, the image representation of the SREV conformations immediately reveals, in 2D, the inhomogeneity of the chromatin density that includes multiple regions of high density that we identify as packing domains. We analyzed the distribution of packing domain radii using the procedure outlined in Figure 7—figure supplements 1 and 2, which is essentially the same as the experimental one. In Figure 7A, we display the distribution of domain radii for all simulated conditions and the mean value for the 12 cases is displayed in Figure 7B. For comparison, we include in Figure 7C the results from our experiments on an A549 cell line (Li et al., 2022) obtained with ChromSTEM that agree very well with the theoretical values in general, and in particular the agreement is excellent with the case corresponding to $\alpha =1.15$ and $\varphi =0.16$.
In order to further characterize the structure of the model chromatin we calculated the pair correlation function between the model nucleosomes, i.e. $g(r)$. From the model definition and previous analysis, we know that $g(r)$ must reveal different features at different length scales. At short distances, $r\lesssim$ 40 nm, $g(r)$ shows the structure of the dense packing domains through the typical maxima and minima, at the intermediate distances corresponding to the average size of the packing domains and the transition between intra and interdomains $g(r)$ is a decreasing function of $r$ approaching the expected plateau for large distances. Motivated by the mass scaling analysis introduced in ChromSTEM experiments (Li et al., 2022; Li et al., 2021) we will use the integral form of the pair correlation function: $G(r)={\int}_{0}^{r}4\pi {r}^{\mathrm{\prime}2}g({r}^{\mathrm{\prime}})d{r}^{\mathrm{\prime}}$. G(r) smoothes out the short distance oscillations of $g(r)$ and reflects the intermediate regime as a power law with exponent $D<$ 3.
In Figure 8A, we show in a log–log representation, as an example, the ensemble average $\u27e8G(r)\u27e9$ corresponding to the global volume fraction $\varphi =0.16$ and the three values of $\alpha$. Between 40 and 120 nm we found that the $\u27e8G(r)\u27e9$ is essentially a perfect straight line, i.e. $\u27e8G(r)\u27e9\propto {r}^{D}$. We define $D$ as the packing parameter that we calculate for 40 $<r/\text{nm}<$ 120. The slopes for the three displayed cases are slightly different, with $D$ values ranging between 2.75 and 2.80 as $\alpha$ decreases from 1.20 to 1.10. In Figure 8B, we summarize the results for $D$ for all the simulated conditions, which shows that $D$ has a positive correlation with $\varphi$, the overall volume fraction of the whole configuration, and a weaker inverse dependence on the folding parameter $\alpha$.
A similar powerlaw regression can be applied on the ${G}_{i}(r)$ obtained for each configuration. We use the subscript $i$ to distinguish that the quantity corresponds to a single configuration $i$. Since the configurations are obtained using a stochastic procedure, there is a large variability in the powerlaw fits obtained from them and some examples are included in Figure 8—figure supplement 1. In Figure 8C, we show the distributions of $D}_{i$ values for all 12 simulated conditions. Notice that individual $D}_{i$ can be larger than 3. To understand this in the context of population heterogeneity of chromatin structure, we performed livecell PWS microscopy on U2OS cells and measured the distribution of chromatin packing states observed. As demonstrated from SREV, variations packing arise from the same $\alpha$ and $\varphi$ conditions due to the probabilistic nature of the model. Consequently, this demonstrates that population heterogeneity arises intrinsically from our model, a finding consistent with experimental results (Figure 8C). This heterogeneity arises in simulated conditions and is in best agreement for U2OS cells in the condition of $\alpha$ of 1.15 and $\varphi$ of 0.16.
Up to this point, we have performed our analysis based on the SREV parameters $\alpha$ and to distinguish the different ensembles of configurations. However, the local volume fraction, as it has been shown in Figures 2 and 6, fluctuates at the scale of the packing domain size. This inhomogeneity makes the representation of a configuration by its overall SREV parameter $\varphi$ not completely meaningful when we study a local or mesoscopic property, such as the packing parameter $D}_{i$. Therefore, it is convenient to introduce the local average chromatin volume fraction $\u27e8{\varphi}_{i}\u27e9$ calculated in exactly the same 240 nm sphere that we use to calculate $D}_{i$. The correlation between these two mesoscopic quantities is plotted in Figure 9 and includes every one of the SREV 12,000 configurations. There is a very clear and interesting correlation between $D}_{i$ and $\u27e8{\varphi}_{i}\u27e9$. For high $\u27e8{\varphi}_{i}\u27e9$, the local $D}_{i$ approaches to 3, which is the theoretical upper limit for $\u27e8D\u27e9$. For intermediate and small $\u27e8{\varphi}_{i}\u27e9$, there is a quite wide distribution of $D}_{i$ values, consistent with the violin plots of Figure 8C. Nevertheless, the local chromatin volume fraction is the main factor determining the corresponding packing parameter. In the next portion, we will demonstrate the predictions of SREV by affecting the probability of returns and the contribution of excluded volume by depleting RAD21 in HCT116 cells.
SREV predicts that loss of cohesinmediated loops have a limited impact on packing domain formation
So far, we have presented the integration of $\alpha$ with excluded volume effects on representative, and distinct, methods to measure chromatin structure (HiC, livecell PWS microscopy, and ChromSTEM). It is evident from SREV that these methods probe different features of genomic organization, which using SREV, could potentially be converged. To test the role of $\alpha$ in the regulation of genome structure, we target processes that govern stochastic returns. Recent work has demonstrated that cohesinmediated loops are short lived, with an individual loop existing in that configuration ∼6 % of the time (Gabriele et al., 2022). Likewise, the process of forming longrange returns is not exclusive to cohesin loop extrusion and arises from the confinement of a polymer in a crowded space as well as from transcriptioninduced promoter–promoter or promoter–enhancer interactions. Thus, even in the absence of cohesin or transcription, entropic loops would exist in chromatin. Consequently, all of these processes converge as components that produce a probability of a longrange step with a reciprocal probability of return at each individual loci across any individual cell. The returns and forward steps exist on a monomer scaffold and therefore individual nucleosome monomers cannot overlap in the same space.
Quantitively, SREV models the loss of cohesinmediated loops as a decrease in $\alpha$ (e.g. from 1.20 to 1.15 or 1.10) without a change in nuclear volume at shorttime scales ($\varphi$ remains constant). SREV predicts (1) that domains will still exist on ChromSTEM (Figure 7A), (2) there will be a ∼20% decrease in the number of domains, (3) the remaining domains will have similar sizes, densities, and mass scaling (Figure 8C), (4) population heterogeneity would be largely unaffected, and (5) $D$ in each domain will be predominantly determined by the local volume fractions (Figures 8B and 9). This view is in contrast to an alternative polymer model that produces biphasic structures from attractions coupled with loop extrusion (Nuebler et al., 2018), as the loss of cohesin in this model would results in microphases occurring only at large (∼500 kbp) length scales. To test this experimentally, we degraded RAD21 (Nishimura et al., 2009; Yesbolatova et al., 2020) using HCT116Rad21mAID2 cell line and performed ChromSTEM and livecell PWS imaging of these cells in comparison to a vehicle treated control at 4 hr. As predicted and in contrast to expectation from existing chromatin polymer models, we found that (see Li, 2024) the majority of chromatin packing domains are retained (∼80% 62/78) with small changes in density (CVC 0.4 → 0.43), radius (84 → 89 nm), and (2.61 → 2.60). At the level of the heterogeneity of chromatin states observed in live cells, we performed livecell PWS microscopy on cells with and without RAD21 depletion and find that it has a minimal impact on chromatin population diversity. Finally, we tested the prediction that local excluded volumes will predominantly determine the powerlaw geometry of chromatin within the nucleus. As observed on ChromSTEM, we find that the local volume concentrations will nonlinearly relate to the scaling behavior of the chromatin polymer while being minimally influenced by the change in $\alpha$ (Figure 10).
Discussion
We have presented the SREV model based on stochastic rules of return and excluded volume interactions. Remarkably, the proposed rules of return are sufficient to generate polymer configurations having 3D packing domains that are observed in singlecell imaging experiments. We demonstrate that the SREV model produces chromosomesize configurations with nucleosome size monomers (200 bp) that agree with multiple distinct experimental methodologies spanning both live (PWS microscopy) and fixed cells (HiC, ChromSTEM) without the need for constraints from other omic methodologies (e.g. ATACseq, ChIPSeq). Without the need for these biological inputs, SREV configurations have contact $S<1$, biphasic heterogeneous packing domains with a continuous distribution of sizes and densities, and the population heterogeneity innate to cellular systems. Initially, this seemed like a disquieting feature of the SREV model: stochastic returns based on a mathematical framework disregarding many longheld assumptions about hierarchical chromatin assembly produced strong experimental agreement, suggesting that genome organization is disordered at supranucleosome scales. This could create a paradox of how nonrandom features arise within organs (e.g. muscle is distinctly not the same as an eye) if one were to incorrectly equate stochasticity with randomness. Instead, we posit that stochastic returns are not synonymous solely with cohesinmediated loop extrusion but are an agnostic event arising from multiple possible biochemical processes: be that shortrange monomeric attractions, promoter–enhancer interactions, loop extrusion, etc. As such, stochastic returns will still occur in the event of the loss of any one of these mechanisms, thereby acting as a failsafe to maintain a degree of organizational integrity. That nonrandom tissues arise from a stochastic polymer would be viewed as an emergent, but not well understood, phenomenon from the interactions between stimuli/transcription factor signaling and disordered genome structure that requires further investigation (Oberbeckmann et al., 2024). This permits tissues to both have ensemble functions and the diversity of transcriptional states that are observed experimentally; microscopically identical cells (immune cells, muscle cells, bone cells, etc.) having a welldescribed distribution of transcriptional states in multiple organs.
An unexpected and testable prediction from the SREV model that is not present in existing polymermodeling frameworks is that methods that measure connectivity (e.g. HiC) would be relatively insensitive to the effects of volume concentrations (a finding in line with on recent experimental results by Yiu and Dekker) whereas methods that measure chromatin density (ChromEM, superresolution imaging) would show profound changes in chromatin response to density. This latter point is of particular significance in multiple clinical contexts. For instance, in cancer, variations in nuclear size and density are the oldest, most widely preserved hallmark of malignancy whose functional consequence remain poorly understood (Hansemann, 1890). In the context of SREV, these features result in significant structural heterogeneity within a cell population that could not be predicted by existing models due to their muted effects in methods that measure connectivity. Likewise, cells with low densities (neurons, oocytes, senescent cells) would contrast to cells with high densities (sperm, lymphocytes) in their domain structure but would have relatively similar contact scaling behaviors. SREV would therefore predict domain organization arises from chromatin concentration and nuclear size directly, indicating that a global feature (nuclear size, CVC) would have mechanistic consequences that would otherwise be missed in the existing polymer framework of genome organization.
We present a novel model of chromatin based on stochastic returns and physical interactions that captures the groundtruth structures observed across both imaging and sequencing based measures of chromatin organization. By maintaining in SREV the possibility of selfreturning extensions that are presented in SRRW, several features arise. (1) High frequency, short return events lead to the formation of individual packing domains. (2) Low frequency, large steps give rise to a corrugated chromatin structure at intermediate length scales (∼100 nm) that allows genomic accessibility to arise (Figure 2). Expanding on the theory originally presented by Huang et al., 2020, we now can account for excluded volume interactions between single nucleosomes to quantitively and qualitatively represent chromatin configurations. This extension is crucial as it allows for the accurate reconstruction of the occupied volumes within chromatin and to calculate the physical properties of genomic organization. Pairing the excluded volume representation of the individual monomer units (nucleosomes) with stochastic returns produces a continuous heterogeneous polymer chain with a random distribution of spacefilling domains. In comparing the effects of the folding parameter, $\alpha$, with the overall chromatin volume fraction, $\varphi$, we show that just two parameters can recapture the heterogeneous nature of chromatin observed in electron microscopy, the variations in CVCs, the formation of packing domains with appropriate sizes, that powerlaw distributions are present at intermediate length scales (quantified by $D$), and the heterogeneity observed experimentally in livecell measurements of chromatin structure.
Crucially, the SREV model is grounded in the stochastic description of genome organization which allows capturing both the description of ensemble properties (e.g. populations of cells/chromosomes) and individual chromosomes. This feature is what allows both the accurate representation of individual experiments (such as the visualized 3D structure in ChromSTEM) as well as features that only become apparent over numerous realizations (such as contact scaling observed in HiC, population heterogeneity observed in PWS microscopy). The model unit length coincides with the size of a nucleosome and owing to physical principles, the linker unit produced is concordant with reported experimental values of 35–45 bp (Table 1). The present length of the model polymer is comparable with the size of human chromosome 16 or smaller; but could be expanded with additional computational resources. Therefore, the SREV configurations span over a large range of spatial dimensions (∼10 nm to ∼1 µm). The agreement with the experimentally found CVC distributions gives us a first confirmation on the validity of the model, and an indication of the relevant values for $\alpha$ and $\varphi$ present physiologically. The quantitative agreement of the packing domain radii distribution with the outcome of ChromSTEM reinforce the confidence in the theory. The packing parameter $D$ is defined in terms of the incremental pair correlation function between model nucleosomes; a definition that is similar (but not exactly the same) as the one proposed in ChromSTEM studies. The value of $D$ is consistently found between 2 and 3 for all simulated conditions. $D$ is calculated on a mesoscopic region of 240 nm in radius, which is completely independent of the location of the packing domains. However, since we show that there is a strong positive correlation between $D}_{i$ and the corresponding local volume fraction $\u27e8{\varphi}_{i}\u27e9$ we can infer that regions containing large packing domains will be associated with a large $D$. The distribution of $D}_{i$ values span over the same range of values observed in PWS experiments. In particular, we show a case in excellent quantitative agreement with PWS results for U2OS cell line (noting that similar distributions are observed independently of this cancer cell line). The incorporation of genomic character to the SREV model will allow us to study all individual single chromosomes properties, and also topological associated domains and A/B compartmentalization from ensemble of configurations as in HiC experiments.
Finally, we view the simplicity of our model as a core strength as it already captures key details about genome organization without introducing many of the constraints present within existing frameworks. Currently, we could generate 12,000 independent configurations of a 500,000 nucleosome (75 Mbp, approximately the size of chromosome 16) within a short period of time. Likewise, we envision that future work can incorporate some of the myriad molecular features known to exist within chromatin organization to be able to interrogate how key components (e.g. sparse, focal constraint such as CTCFbinding sites or heterochromatin modifying enzymes) would alter the observed physical structures. As with any modeling work, there will always be the tension between the addition of details for fidelity and the ability to capture the properties of genome organization. As the SREV already captures many key properties seen within chromatin, we anticipate that it can serve as the basis model of stochastically configured genome organization within the wider field.
Materials and methods
Cell culture
Request a detailed protocolHuman cell line U2OS cells (ATCC, #HTB96) used for experimental validation of the model were cultured in McCoy’s 5A Modified Medium (Thermo Fisher Scientific, #16600082) supplemented with 10% fetal bovine serum (FBS) (Thermo Fisher Scientific, #16000044) and 100 µg/ml penicillin–streptomycin antibiotics (Thermo Fisher Scientific, #15140122). Human cell line A549 cells (ATCC, #CCL185) used for experimental validation of the model were cultured in Dulbecco’s modified Eagle’s medium (Thermo Fisher Scientific, #11965092) supplemented with 10% FBS (Thermo Fisher Scientific, #16000044) and 100 µg/ml penicillin–streptomycin antibiotics (Thermo Fisher Scientific, #15140122). Experiments were performed on cells from passages 5 to 10. All cells were maintained under recommended conditions at 37°C and 5% CO_{2}. Cells were verified to have no detectable mycoplasma contamination (ATCC, #301012K) prior to starting experiments.
PWS sample preparation
Request a detailed protocolPrior to imaging, cells were cultured in 35 mm glassbottom Petri dishes. All cells were allowed a minimum of 24 hr to readhere and recover from trypsininduced detachment. PWS imaging was performed when the surface confluence of the dish was approximately 70%.
PWS imaging
Request a detailed protocolThe PWS optical instrument consists of a commercial inverted microscope (Leica, DMIRB) equipped with a broadspectrum white light LED source (Xcite120 lightemitting diode lamp, Excelitas), ×63 oil immersion objective (Leica HCX PL APO, NA1.4 or 0.6), long pass filter (Semrock, BLP01405R25), and Hamamatsu ImageEM CCD camera C910013 coupled to an LCTF (CRi VariSpec). Live cells were imaged and maintained under physiological conditions (37°C and 5% CO_{2}) using a stage top incubator (In Vivo Scientific, Stage Top Systems). Briefly, PWS directly measures the variations in spectral light interference that results from internal light scattering within the cell, due to heterogeneities in chromatin density, with sensitivity to length scales between 20 and 300 nm (Li et al., 2021). Variations in the refractive index distribution are characterized by the mass scaling (chromatin packing scaling) parameter, $D$. A detailed description of these methods is reported in several publications (Subramanian et al., 2009; Almassalha et al., 2016; Gladstein et al., 2018; Eid et al., 2020).
ChromSTEM sample preparation and imaging
Request a detailed protocolCell samples were prepared as reported in Li et al., 2022. Cells were first washed with Hank’s Balanced Salt Solution without calcium and magnesium (Thermo Fisher Scientific, #14170112) three times, 2 min each. Fixation, blocking, DNA staining and 3,3′diaminobenzidine tetrahydrochloride (DAB) solutions were prepared with 0.1 M sodium cacodylate buffer (pH = 7.4). Cells were fixed with 2% paraformaldehyde, 2.5% glutaraldehyde, 2 mM calcium chloride for 5 min in room temperature and 1 hr on ice and all the following steps were performed on ice or in cold temperature unless otherwise specified. After fixation, cells were washed with 0.1 M sodium cacodylate buffer five times, 2 min each. Cells were then blocked with 10 mM glycine, 10 mM potassium cyanide for 15 min. Cells were stained with 10 µM DRAQ5, 0.1% Saponin for 10 min and washed with the blocking solution three times 5 min each. Cells were bathed in 2.5 mM DAB and exposed to 150 W Xenon Lamp with ×100 objective lens and a Cy5 filter for 7 min. Cells were washed with 0.1 M sodium cacodylate buffer five times, 2 min each, followed by staining with 2% osmium tetroxide, 1.5% potassium ferrocyanide, 2 mM calcium chloride, 0.15 M sodium cacodylate buffer for 30 min. After osmium staining, cells were washed with double distilled water five times, 2 min each and sequentially dehydrated with 30%, 50%, 70%, 85%, 95%, 100% twice, ethanol, 2 min each. Cells were then washed with 100% ethanol for 2 min and infiltrated with Durcupan ACM ethanol solutions (1:1 for 20 min, and 2:1 for 2 hr) at room temperature. Cells were then infiltrated with resin mixture for 1 hr, resin mixture with accelerator for 1 hr in 50°C dry oven and embedded in BEEM capsule with the resin mixture at 60°C dry oven for 48 hr.
Resin sections with thickness around 100 nm were prepared with a Leica UC7 ultramicrotome and a 35°C DiATOME diamond knife. The sections were collected on copper slot grids with carbon/Formvar film and 10 nm colloidal gold nanoparticles were deposited on both sides of the section as fiducial markers. HAADF(Highangle annular darkfield imaging) images collected by a 200kV cFEG Hitachi HD2300 scanning transmission electron microscope. For each sample, projections were collected from −60 to +60°C with 2°C increments, along two roughly perpendicular axes.
Each projection series along one rotation axis was aligned with IMOD using gold nanoparticle fiducial markers. After image alignment, penalized maximum likelihood algorithm in Tomopy was used to reconstruct the images with 40 iterations. IMOD was used to combine tomograms from different rotation axis of the same sample.
Chromatin domain radius measured from experiment
Request a detailed protocolThe chromatin domains were identified using FIJI. 2D chromatin density distributions were obtained by reprojection of the tomogram along zaxis, followed by Gaussian filtering with 5 pixels radius and CLAHE contrast enhancements with block size of 120 pixels. Chromatin domain centers were selected as the local maxima of chromatin density.
To evaluate the size of a domain, two properties were analyzed for each domain, which are the mass scaling properties and radial volume chromatin concentration (CVC). For mass scaling, multiple mass scaling curves were sampled by using pixels (a 11pixel × 11pixel window) around the center of an identified domain and they were averaged by the weight of the pixel values of the selected center pixel. A size of domain is defined by the length scale that the domain meets any of the following three criteria: (1) it deviates from the powerlaw mass scaling relationship $M(r)\propto {r}^{D}$ by 5%; (2) the local fitting of $D$ reaches 3; (3) the radial CVC reaches a local minimum and begins to increase for longer length scale.
Experimental validation plots
Request a detailed protocolGraphPad Prism 10.0.0 was used to make the violin plots in Figures 7C and 8C. The violin plots are represented as individual data points, with lines at the median and quartiles.
Appendix 1
Supplementary algorithms
Algorithm for the generation of an SRRW in free space
Here, we describe a recursive Monte Carlo algorithm to generate an SRRW parameterized by folding parameter $\alpha >1$ and local cutoff $U}_{\mathrm{m}\mathrm{a}\mathrm{x}$, which represents the maximum bond length. Within the algorithms to be described, the length unit is the minimum bond length ${U}_{\mathrm{m}\mathrm{i}\mathrm{n}}=10$ nm, so that all length are dimensionless, and taken relative to the minimum bond length $U}_{\mathrm{m}\mathrm{i}\mathrm{n}$. The conformation of the SRRW, emanating from the origin, is defined by $N$ bond vectors $\mathbf{U}}_{1$, $\mathbf{U}}_{2$, …, $\mathbf{U}}_{N$. In the following, the symbol $\xi$ stands for an independent random number drawn with equal probability from the interval $[0,1]$, and has to be recreated whenever it occurs below.
(A1) Define $\beta \equiv 1{U}_{\mathrm{m}\mathrm{a}\mathrm{x}}^{(1+\alpha )}$.
(A2) Generate a set of $2N$ bond vectors $\mathbf{B}}_{n$ with $n=1,2,..,2N$ for eventual later use. Each $\mathbf{B}}_{n$ is given by $\mathbf{B}}_{n}=\ell \mathbf{u$, where $\mathbf{u}$ is a random unit vector and $\ell =(1\beta \xi {)}^{1/(1+\alpha )}$ its bond length. A random unit vector, we create via $\mathbf{u}=(\sqrt{1{z}^{2}}\mathrm{cos}\varphi ,\sqrt{1{z}^{2}}\mathrm{sin}\varphi ,z)$, where $\varphi =2\pi \xi$ and $z=2\xi 1$. The generation of the set $\{\mathbf{B}\}$ hence requires $6N$ random numbers $\xi$ and if not otherwise mentioned, the $\{\mathbf{B}\}$ will remain unchanged during the course of the algorithm.
(A3) Initialize $n=N$, set $\mathbf{U}}_{1}={\mathbf{B}}_{n$.
(A4) Increase $n$ by one, set $\mathbf{U}}_{2}={\mathbf{B}}_{n$, and initialize step $s=2$.
(A5) Call a recursive routine that takes the existing sets $\{\mathbf{B}\}$, $\{\mathbf{U}\}$, $n$, and $s$ as arguments, and returns new sets $\{\mathbf{B}\}$, $\{\mathbf{U}\}$, and $n$. This routine does the following:
If $s=N$, just return from the routine.
Calculate return probability ${P}_{R}={\mathbf{U}}_{s}{}^{\alpha}/\alpha$.
If $\xi <{P}_{R}$, then $\mathbf{U}}_{s+1}={\mathbf{B}}_{n$ and $n$ is decreased by one. Otherwise, $n$ is increased by one, the single $\mathbf{B}}_{n$ is recreated using the above procedure (A2), and $\mathbf{U}}_{s+1}={\mathbf{B}}_{n$.
Routine calls itself with identical arguments as before, with the exception of $s+1$ instead of $s$.
The described algorithm terminates automatically as soon as $N$ bond vectors $\mathbf{U}}_{1$, $\mathbf{U}}_{2$, …, $\mathbf{U}}_{N$ have been created. The coordinates $\{\mathbf{x}\}$ of nodes are simply given by the cumulative sum over the set of bond vectors $\{\mathbf{U}\}$, i.e. $\mathbf{x}}_{j+1}={\mathbf{x}}_{j}+{\mathbf{U}}_{j$. Note that using this algorithm the return probabilities satisfy Equation 1 and that all bond lengths $\ell$ are automatically confined to the interval $[{U}_{\mathrm{m}\mathrm{i}\mathrm{n}},{U}_{\mathrm{m}\mathrm{a}\mathrm{x}}]$ and distributed according to Equation 2. The proof is provided in the next section.
Algorithm for the generation of an SRRW subject to global cutoff
The idea of an SRRW with global cutoff $R}_{c$ is to make sure the SRRW will tend to grow within a certain spherical volume of radius $\approx {R}_{c}$. To this end the above algorithm is slightly modified as follows. Instead of the earlier (ii) calculate the geometric center $\mathbf{C}$ of the existing nodes from $\{\mathbf{x}\}$. If ${\mathbf{x}}_{s}\mathbf{C}>{R}_{c}$, then set, otherwise calculate ${P}_{R}={\mathbf{U}}_{s}{}^{\alpha}/\alpha$ as before.
Molecular dynamics protocol for the generation of an SREV
An SRRW conformation subject to global cutoff is produced via Monte Carlo as just described; such a conformation usually exhibits a large number of nodes (points) with identical coordinates. All these points need to be turned into beads, i.e., receive a finite spherical volume within the final SREV configuration, that should preserve all large scale features and domain characteristics of the SRRW. We alter the local structure to avoid bead–bead overlap, while operating at (ideally) minimal displacement effort. To this end we use the original node coordinates $\{\mathbf{x}\}$ as initial center positions of spherical beads of radius ${r}_{\circ}=0.49$ and unit mass $m$. In a first step, to allow for a random element, and to avoid center–center distances that are exactly zero up to numerical precision, we displace all overlapping beads randomly by 1% of the bead diameter. Afterwards we employ LAMMPS (Thompson et al., 2022) to run a molecular dynamics simulation on the modified SRRW systems composed of spherical beads. We let all beads interact via a soft repulsive radially symmetric pair potential $V(r)=20\u03f5[1+\mathrm{cos}(\pi r/{r}_{c})]$ for $r\le {r}_{c}$, and $V(r)=0$ otherwise, where $r$ denotes the center–center distance between pairs of beads, $\u03f5$ the irrelvant energy unit, and the cutoff distance ${r}_{c}=1.03$ is chosen slightly larger than the bead diameter. The system is thermostatted via the Nosé–Hoover scheme at $T=0.001\phantom{\rule{thinmathspace}{0ex}}\u03f5/{k}_{\mathrm{B}}$, and run using a time step $\mathrm{\Delta}t=0.005\phantom{\rule{thinmathspace}{0ex}}{U}_{\mathrm{m}\mathrm{i}\mathrm{n}}\sqrt{m/\u03f5}$. During runtime, the bead–bead pair correlation function $g(r)$ is evaluated at each time step and averaged for a duration of 200 time steps. Each time unit (200 time steps) we inspect the averaged $g(r)$, integrated up to r_{c}, as this quantity informs about the amount of remaining overlap. In rare cases, the integral did not decrease with time, in that case we start over using another seed value for the random number generator. While the integral keeps decreasing, we monitor the potential energy of the system. As soon as the potential energy has reached a minimum, which happens if the energy is close to zero, we terminate the molecular dynamics run and save the resulting SREV coordinates. The minimum center–center distance between pairs of beads in the SREV configuration exceeds $2{r}_{\circ}$, as we verified. Note that the distribution of bond lengths is significantly different for SREV and SRRW conformations.
Proof of the validity of the SRRW algorithm
The forward jump probability $P}_{J}(U)=(\alpha +1){U}^{(\alpha +2)$ was stated in the manuscript. It was furthermore mentioned that new bonds of length U_{1} should not exceed a local dimensionless cutoff length $U}_{\mathrm{m}\mathrm{a}\mathrm{x}$, while ${U}_{\mathrm{m}\mathrm{i}\mathrm{n}}=1$ within these units. Because $P}_{J$ is a probability distribution, it must fulfill ${\int}_{1}^{{U}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{P}_{J}(U)dU=1$ and the properly normalized version thus reads
To efficiently create bond lengths $U$ distributed according to Equation 3 using equally distributed random numbers $\xi \in [0,1]$, one has to solve the differential equation ${\xi}^{\mathrm{\prime}}(U)={P}_{J}(U)$ with initial condition $\xi (1)=0$, and then invert the solution. The solution of the differential equation is $\xi (U)=({U}_{\mathrm{m}\mathrm{a}\mathrm{x}}/U{)}^{1+\alpha}({U}^{1+\alpha}1)/({U}_{\mathrm{m}\mathrm{a}\mathrm{x}}^{1+\alpha}1)$. Solving this expression for $U$ gives $U=(1\beta \xi {)}^{1/(1+\alpha )}$ with the constant $\beta \equiv 1{U}_{\mathrm{m}\mathrm{a}\mathrm{x}}^{(1+\alpha )}$, so that $U=1$ and $U={U}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ for $\xi =0$ and $\xi =1$, respectively. This completes the proof of item (A2) with (A1).
It might be just interesting to mention that one has access to some statistical properties of the chain conformation from $\beta \equiv 1{U}_{\mathrm{m}\mathrm{a}\mathrm{x}}^{(1+\alpha )}$, while $P}_{R$ has to be taken into account for the exact calculation. For sufficiently large $U}_{\mathrm{m}\mathrm{a}\mathrm{x}$ the mean bond length is
For $\alpha \in \{1.1,1.15,1.2\}$ the mean bond length is hence $\u27e8U\u27e9\in \{1.91,1.87,1.83\}$. Similarly, the mean return probability is approximately
i.e. $\u27e8{P}_{R}\u27e9=\{0.597,0.567,0.539\}$ for $\alpha =\{1.1,1.15,1.2\}$. While for $\alpha \le 1.03$ the SRRW basically collapses to a small region in space, beyond this value the effective number of forward steps is approximately $[0.49(\alpha 1)0.02]N\approx (\alpha 1)N/2$.
Data availability
The current manuscript is a computational study based on our own software.
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Article and author information
Author details
Funding
National Cancer Institute (U54CA268084)
 Vadim Backman
 Igal Szleifer
National Cancer Institute (U54CA261694)
 Vadim Backman
National Cancer Institute (R01CA228272)
 Vadim Backman
National Cancer Institute (R01CA224911)
 Vadim Backman
National Cancer Institute (R01CA225002)
 Vadim Backman
National Institute of General Medical Sciences (T32GM142604)
 Vadim Backman
National Science Foundation (EFMA1830961)
 Vadim Backman
 Igal Szleifer
National Institute of Allergy and Infectious Diseases (T32AI083216)
 Luay M Almassalha
The funders had no role in study design, data collection, and interpretation, or the decision to submit the work for publication.
Acknowledgements
We acknowledge funding from the National Institutes of Health (NIH) grants U54CA268084, U54CA261694, R01CA228272, R01CA224911, R01CA225002, T32GM142604, NSF grant EFMA1830961, and philanthropic support from K Hudson and R Goldman, S Brice and J Esteve, ME Holliday and I Schneider, the Christina Carinato Charitable Foundation, and D Sachs. Luay Almassalha acknowledges the support of NIH training grant T32AI083216. This research was supported in part through the computational resources and staff contributions provided for the Quest high performance computing facility at Northwestern University which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology.
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© 2024, Carignano, Kroeger, Almassalha et al.
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