The 3D organization of chromosomes is crucial for regulating gene expression and cell function. Many experimental and polymer modeling efforts are dedicated to deciphering the mechanistic principles behind chromosome folding. Chromosomes are long and densely packed—topologically constrained—polymers. The main challenges are therefore to develop adequate models and simulation methods to investigate properly the multi spatio-temporal scales of such macromolecules. Here, we proposed a generic strategy to develop efficient coarse-grained models for self-avoiding polymers on a lattice. Accounting accurately for the polymer entanglement length and the volumic density, we show that our simulation scheme not only captures the steady-state structural and dynamical properties of the system but also tracks the same dynamics at different coarse-graining. This strategy allows a strong power-law gain in numerical efficiency and offers a systematic way to define reliable coarse-grained null models for chromosomes and to go beyond the current limitations by studying long chromosomes during an extended time period with good statistics. We use our formalism to investigate in details the time evolution of the 3D organization of chromosome 3R (20 Mbp) in drosophila during one cell cycle (20 hours). We show that a combination of our coarse-graining strategy with a one-parameter block copolymer model integrating epigenomic-driven interactions quantitatively reproduce experimental data at the chromosome-scale and predict that chromatin motion is very dynamic during the cell cycle.
Though all cells of a multicellular organism contain the same genetic information, they vary widely in shapes, physiologies, and functions. These differences mainly reflect variations in gene expression between different tissues or cell types. Recent experiments have highlighted the important role of the physical organization of chromosomes inside the cell nucleus in regulating gene expression [1–3]: gene activities being modulated, not only by the local folding of the chromatin fiber but also by its higher order organization with 3D nuclear compartments favorable to gene activation or repression. During interphase, the longest phase of the cell cycle where genes are expressed and DNA is replicated, chromosomes are found to be organized hierarchically. Confocal and electron microscopy experiments have revealed that each chromosome occupies its own territory [4]. Also, the genes sharing the same transcriptional state tend to colocalize [5–7]: inactive genomic regions (the heterochromatin) being more peripheral while active regions (the euchromatin) being more central. At the sub-chromosomal level, advanced molecular biology tools, like chromosome conformation capture (Hi-C) techniques, have shown that chromosomes are partitioned into consecutive 3D interaction compartments, the so-called topologically-associated domains (TADs), [8–10]. Loci inside these domains experience enriched contact probabilities with other loci of the same domain while showing partial insulation from loci of nearest neighbor domains. These domains can be easily visualized as consecutive “squares” along the diagonal of a 2D contact frequency matrix (see section ‘Application to chromatin folding in drosophila’ for illustration). TADs formation has been associated with the local biochemical composition of chromatin, the so-called epigenome, which encodes for gene activity [7,11–14]: genes inside the same TAD tends to have the same epigenomic state, and long-range contacts may be observed between TADs of the same state. However, how genome precisely organized in space is still not fully understood and addressing this question represents one of the most exciting challenges of modern biology [15]. Lots of experimental and modeling efforts are currently dedicated to understand the mechanisms implied in chromosome folding. In particular, polymer models have been instrumental in simulating and testing different molecular and physical mechanisms and in driving new experiments [5,16–39]. An important challenge for such models is to simulate, with good precision, the behavior of long polymer chains (the typical size of a chromosome ranging from about a million base-pairs in yeast to hundreds of Mbp in human) during an extended time period (of the order of hours for a typical cell cycle). Therefore the standard strategy used in these approaches is to start from a coarse-grained “null” model for chromatin with few basic interactions [40]. Then eventually decorate it with more physical or chemical interactions driven by biological information such as the gene activity or the local epigenomic state. Chromosomes being very long polymers densely packed into the cell nucleus, topological constraints generated by polymer entanglement may play an essential role in controlling the dynamics and fluctuations of such polymers [16,41]. However, when building their null models, very few approaches account adequately of such constrained situations. As neglecting topological constraints may lead to different structural and dynamical properties of the polymer, how can one interpret the outcomes correctly of such models concerning realistic mechanisms, if the considered null model is already biased? Here, we develop a generic coarse-graining strategy for self-avoiding polymers that allows, simultaneously, to drastically reduce the computation time while maintaining the polymer in the same topological regime and thus preserving the correct structural and dynamical properties. In the first part, we explain the strategy and demonstrate its efficiency, leading to a systematic approach to develop coarse-grained null models for chromosomes. In the second part, we apply it to investigate the role of epigenomic-driven interactions in the folding and dynamics of drosophila genome. Finally, we discuss our results and their implications in the general context of chromosome modeling and chromatin biology.
Chromosomes are long polymers confined inside a small volume, the cell nucleus [42]. As a result, the generic characteristics of these densely packed long polymers are very different from free isolated chains and exhibit distinct universal properties [41,43]. For a simple semi-flexible self-avoiding polymer, composed of N beads of size b (in nm), each bead representing ν bp, such properties are mainly determined by (i) its rigidity, characterized by its Kuhn length lk (in nm), and (ii) by its volumic density ρ (in bp/nm3). Moreover, in a non-dilute environment, topological constraints are also expected to influence the large-scale organization and long-time dynamics. Their importance depends on the ratio between the polymer contour length L ≡ Nb and Le (in nm), the so-called entanglement length. Le measures the typical subchain size above which topological confinement due to excluded volume influences configurational fluctuations, and depends on lk and ρ. It may be associated with the tube diameter in the reptation model or to the crossover time between a Rouse-like motion and a reptation-like motion [44], and may be estimated using the phenomenological relation [45] L e = l k (c ρ k l k 3) 2, (1) with c ≈ 19 a numerical factor and ρk = (ρ/ν) (b/lk) the volumic density in Kuhn segment. In the following, we define Nk = νlk/b (in bp) as the Kuhn segment size representing the number of bp in one Kuhn segment. L/Le ≪ 1 means very weak topological effects and the polymer behaves as a standard Rouse chain. If L/Le ≫ 1, the chain motion is restricted due to strong topological interactions and exploration of the available space is very slow. In this case, the equilibration time of the chain scales as N3 [46], implying that polymer dynamics remains out-of-equilibrium and the initial topological properties (presence/absence of knots) or large-scale organization features are maintained over a long time period. Using the fine scale reference model, we recover the expected structural and dynamical behaviors in the different entanglement regimes, fully consistent with previous theoretical works on knot-free and crumpled polymers [16,43,51,56,57,59] and with experiments [5,6, 47,49,53–55,60]. At this nucleosomal scale, the model has very good spatial (10 nm) and temporal (15 μsec, 30 min ≈ 108 MCS) resolutions. However, the underlying cost of this is a prohibitive computational time. For example, for long chromosomes such as in the drosophila case, simulating one 30 min long trajectory requires 83 hrs CPU time on a 3. 20 GHz CPU. To access more biologically-relevant time-scales (dozens of hours) with good statistics, we aim to develop a coarse-graining strategy of the reference model that allows to speed up the simulation of long trajectories while preserving the main physical characteristics of the original -fine scale—model. Our coarse-graining strategy is generic and has no direct connection to any particular polymer. When applied to a specific system, the question would be how to choose the optimal coarse-graining? As we observed above, regarding numerical efficiency, one wants to go for the higher CG and higher Φ (≲ 1). These two values will determine the spatial and temporal resolutions of the model. Therefore, a natural choice would be to maximize CG and Φ under the constraints of a minimal desired resolution (e. g. , determined by the experimental precision). For example, for chromosomes, we aim to be quantitative typically at a 20 kbp scale (NK ∼ 20 kbp) with a spatial precision of about 100 nm. Under this loose constraints, CG = 10 kbp and Φ = 0. 97 is a very appropriate choice (see Tab 1). Till now, throughout our study, we strategically choose the bending stiffness lk or the lattice volumic fraction Φ so that we preserve the physical properties of the system by conserving the right entanglement regime. Now the question is what happens if one uses more naive coarse-graining strategies that do not necessarily preserve the topological regime. As explained at the beginning of section ‘Relation between Kuhn length lK and lattice parameters’, a typical strategy is to “neglect” the artificial bending rigidity if the CG is higher than the Kuhn length of the reference model. Another is to consider an isolated polymer and to neglect the “confinement” of the chain. These two kinds of models may modify the L/Le ratio and therefore may change the physical properties of the system: chain motion may be slightly accelerated (g1 (t) ∼ t0. 5, Fig 4a) and structural properties may be strongly perturbed (Fig 4b and 4c) (see also Fig. K bottom and Fig. L in S1 Text). In particular, considering an isolated chain (Φ ≪ 1) dramatically modifies the behavior of Pc that scales within this approximation as ∼s−2, characteristics of isolated self-avoiding walks [46]. This leads to an underestimation of the contact probability by orders of magnitude compared to the reference model. In complement to coarse-graining strategies, still for the purpose of reducing the computation time, a standard approach is to simulate only small pieces of the chromosomes instead of the full length. Since the dynamical regime of the chain depends on the ratio L/Le, reducing the value of L may modify the dynamics of the chain and therefore may lead to wrong predictions. For example, instead of 20 Mbp-long polymer, if we simulate a small fragment of 2 Mbp, we found strong discrepancies. At small length scales, s < 100 kbp, it follows the reference model, but at larger length scale 100 kbp < s < 2 Mbp it deviates from reference model and behaves like an isolated Rouse polymer (see Fig 4 and Fig. M in S1 Text). All this emphasizes the need to conserve properly the ratio L/Le of the reference model if one aims to simulate the right polymeric behavior accurately. Modifying this ratio by decreasing L or by making approximations that affect Le would possibly lead to simulate a system with different physical properties than the actual system of interest. Having in hands a strategy to build an efficient coarse-grained “null” model for chromosome, we use it to study the folding of fly chromosome 3R. In drosophila, the 3D structural units, the so-called TADs, are strongly associated with the 1D epigenomic domains [11,61,62]: a locus of a given epigenomic state is likely to share its local 3D compartment with loci of the same epigenomic state. This observed correlation had recently motivated us to build a heteropolymer model accounting for the epigenome folding into interacting TADs [20,32]. Based on biochemical evidence that proteins associated with some epigenomic states have the capacity to oligomerize [63–65], hence possibly generating effective specific interactions between loci of the same state, we developed a block copolymer model of fly chromatin where each block represents a 1D epigenomic domain. By varying the strength of these specific interactions, we showed that such model well accounts for the TAD formation and inter-TAD long-range contacts. Previously, we limited our analysis to short pieces of chromatin (∼ 1 Mbp-long fragment) at equilibrium. In the previous section, we observed that simulating only small regions instead of the full system might lead to strong approximations. Here, we wonder if our conclusions on the 3D chromosome folding in drosophila remain valid and can be generalized when considering a larger genomic region and using a more realistic “null” model for chromatin. We consider the 20-Mbp region of chromosome 3R localized between 7 and 27 Mbp, that we model using an efficient coarse-graining (10 kbp and Φ = 0. 97 for L/Le = 70, ρ = 0. 009bp/nm3, see Table 1). For this region, we collect the epigenomic domains obtained by Filion et al [66] for the embryonic cell line Kc167. In this dataset, five types of epigenomic states exist: 2 euchromatic states associated with active genes that, for simplicity, we decided to merge into one “active” state; and 3 heterochromatin states: constitutive heterochromatin associated with HP1-protein and H3K9me3 histone marks, facultative heterochromatin associated with Polycomb (PcG) proteins and H3K27me3 histone marks, and the so-called “black” chromatin, the prevalent form of heterochromatin. To each 10-kbp monomer of the model, we associate the corresponding epigenomic state, and we assume that monomers of the same state may specifically interact with an energy Ei (in kB T unit) if they are spatially in contact (ie nearest-neighbor on the lattice) (see Materials and methods). For simplicity, we assume that the strength of interaction is similar for every epigenomic state.
In the first part of this article, we introduced a new coarse-graining strategy for long and dense self-avoiding walks that conserves the entanglement regime and the volumic density. Using kinetic Monte Carlo simulations on a lattice, we demonstrated that such strategy not only leads to the accurate description of the steady-state but also the time-evolution of the expected structural and dynamical properties of the underlying fine-scale model. We showed that by introducing an effective rigidity to the coarse-grained model and by controlling the volume fraction, we could achieve very high gain in numerical (CPU-time) efficiency while maintaining a quantitative approximation and minimizing the loss in spatial and temporal resolution of the model. Using our efficient polymer model one can simulate chromosome dynamics during the whole cell cycle on a desktop computer within a day. While we illustrated our approach using chromosomes on a lattice model as toy examples, our strategy is generic and can be applied to any self-avoiding polymers and off-lattice systems. We emphasized that neglecting topological constraints may lead to an erroneous description of the fine-scale model. Therefore the effect of supplementary interactions added to the null model, to describe specific observations present in experimental data, may lead to misinterpretation. In the second part, we used our strategy to build a coarse-grained null model for chromatin and decorated it with a copolymer framework based solely on epigenomic data to investigate the folding and dynamics of a big fraction of chromosome 3R of drosophila. It is the first study trying to quantitatively describe the behavior and time evolution of such large genomic regions (20 Mbp) during one cell cycle (20h of real time) with high precision (10 kbp resolution). Our heteropolymer model has one free parameter, the strength of short-range interaction Ei between genomic loci having the same epigenomic state. Our findings are in qualitative agreement with our previous works based on shorter pieces of chromosomes [20,32], but significantly improve our description of chromosome folding in drosophila. By varying Ei, we showed that the system continuously switches from a dynamic homogeneous crumpled-like behavior to a crumpled heterogeneous micro-phased state. Interestingly, we observed that during this transition, the chromosome fluctuations characterized by the mean squared displacement conserve the same scaling behavior (g1 (t) ≈ γt0. 4) with exponents compatible with a crumpled polymer. However, the prefactor γ depends on Ei and is sharply reduced above a given strength of interaction, characteristics of a glass-like transition [67,68]. Another interesting observation was that the sequence-average contact probability Pc (s), a quantity directly comparable to experimental data, is independent of Ei (at least for weak, biologically relevant values) and is same as in the reference null model, as already observed by Gursoy et al [71]. This motivates, afterward, the validity of homopolymer models, extensively used by polymer physicists, to study the generic physical principles behind chromosome folding based on comparison with sequence-average experimental data [16,41,43,58], even if such chromatin organization is strongly heterogeneous. This also suggests that, before adding specific interactions to the system, any quantitative approach should first aim to describe such sequence-average behaviors in a null homogeneous model. Comparing our model predictions for chromosome 3R to the corresponding Hi-C data, we observed an excellent agreement at all scales, strongly suggesting that epigenomics is a primary driver of chromosome folding in drosophila. The strength of interaction compatible with the data is weak (∼ 0. 1 kT) and locate the in vivo situation in the transition zone between the homogeneous crumpled and the micro-phased states. TADs are only partially collapsed and interact dynamically with other TADs of the same main epigenomic state. This suggests a substantial stochasticity in chromosome organization, consistent with recent single-cell Hi-C or super-resolution experiments [72–77]. We also detected several discrepancies between the predicted and experimental contact maps. For example, the model predicts spurious contacts or misses some between some TADs. This could be due to a wrong annotation of the local epigenomic state or the existence of specific interactions driven by other biological processes not accounted in the model. For example, refining the model to account more precisely for the local epigenetic content (for example the relative levels of histone modifications or chromatin-binding proteins) or differences in interaction strengths between different states would certainly lead to a better correspondence. We also observed that TADs are more sharply defined in the experiments, particularly in the corners of large TADs. This might be the results of the presence of cis-interacting mechanisms, like supercoiling [23,78] or the recently proposed loop extrusion model in mammals [26,27], that enhance the contact frequency along the genome. To exploit the capacity of the model to simulate long trajectories (20 hrs), we analyzed the time evolution of chromosome organization. TADs formed very quickly (within minutes), entirely consistent with Hi-C data made on synchronized mammalian cells showing that, in early G1, TADs are already observable in the data [76,79]. Formation of long-range contacts require more time and eventually appear hours after the mitotic exit, also consistent with the time evolution of Hi-C data during the cell cycle in human [76,79] and yeast [80]. To go deeper into this characterization and get insights into the dynamics of contact formation, we tracked pairs of loci. At the investigated resolution (10 kbp), contacts are transient and their typical lifetime is ∼ 10 seconds, indicating a very dynamic situation, consistent with many experiments performed on living cells [81]. Probability distribution functions of the first encounter time, the contact time, or the search time are polynomial, suggesting a possible connection with fractional brownian motion physics, as for bacterial chromosomes [82]. Interestingly, we predicted that a significant proportion (5 − 15%) of long-range contacts (> 1 Mbp) are not established within one cell cycle. This suggests that the genomic distance between regulatory elements should not exceed 1 Mbp to ensure that physical contact between the elements, prerequisite to an activation or repression event, would happen at least once in the cell cycle in order to maintain a stable regulation or gene expression. With the recent progress in genome editing [83], it would be interesting to experimentally test such predictions by simultaneously tracking the distance between a promoter and its enhancer and the current gene activity [84], for various genomic distances between the two elements. All this suggests that the 3D chromosome organization in higher eucaryotes is out-of-equilibrium and the chromatin is very dynamic. This emphasizes the necessity to adequately account for the time evolution of such organization in quantitative models of chromosomes, especially for higher eucaryotes where chromosomes are strongly topologically constrained. As a proof of concept, we demonstrated the utility of our coarse-graining approach to study chromatin organization in drosophila. However, the numerical efficiency of the method opens new perspectives to investigate the physical and mechanistic principles behind chromosome folding more deeply with many aspects remain to be understood. For example, extrapolating to the whole human genome, it would require ∼120h of CPU time with our strategy at 10kbp resolution to simulate one cell cycle while it remains elusive to do it with the fine-scale model (> 100 years of CPU time). The possibility to easily simulate the dynamics of chromosomes or full genomes during long biologically relevant time period would allow to quantitatively investigate in the future the role of other types of interactions, like those associated with the nuclear membrane, another important player in organizing chromosomes inside the nucleus [25,85] and the crosstalk with epigenomic-driven interaction as presented here. This situation seems particularly attractive to describe the reorganization of chromatin during senescence where constitutive heterochromatin detaches from the membrane to form large foci at the interior of the nucleus [86].
The chromatin volumic density ρ is defined as the ratio between the genome size G and the average volume V of the nucleus. For haploid yeast, G = 12. 2Mbp and V ≈ 2. 6μm3 [69,87], leading to ρ ≈ 0. 005 bp/nm3. For drosophila (diploid) late embryos, nuclei have a typical diameter of 4μm [88] and contains about 300Mbp of genomic DNA, thus ρ ≈ 0. 009 bp/nm3. In mammals, the size of the nucleus depends strongly on the cell type typically ranging from 5 to 15μm in diameter [69]. For example, for a human nucleus (G ≈ 6Gbp) of diameter 9μm, ρ ≈ 0. 015 bp/nm3. For larger nuclei (12μm in diameter), as measured by Muller et al [89], ρ may be weaker (≈ 0. 007 bp/nm3). The polymer is modeled as a semi-flexible self-avoiding walk, consisting of N beads, on a face centered cubic (fcc) lattice of size S × S × S (each unit cell contains 4 lattice sites) following the model developed by Hugouvieux et al [90] (Fig 8) (more details can be found in [32,90,91]). As in the elastic lattice model introduced recently by Schram and Barkema [92], we authorize at maximum two monomers to occupy the same lattice site if and only if they are consecutive along the chain [90,92] (Fig 8). Otherwise, due to excluded volume, two non-consecutive monomers cannot be located at the same site. When two successive monomers occupy a lattice site, an extra bond length is accumulated in that node as a ‘stored length’ [92–94] (Fig. P (b) in S1 Text). The concept of stored length was first introduced by Rubinstein in his pioneering article on the implementation of repton model on a lattice [93]. Such double occupancy of consecutive monomers accounts for the effect of contour length fluctuations [94]. Rigidity is accounted using a standard formulation [95]: H = κ 2 ∑ i = 1 N - 1 (1 - cos θ i), (4) where κ is the bending rigidity and is directly related to lk/b (see below), and θi is the angle between the bond vectors i and i + 1. Confinement and effect of other chains are approximated using periodic boundary conditions, the corresponding lattice volume fraction being Φ = N/ (4S3). Note that such periodic conditions do not confine the chain to the finite volume of the simulation box. Using correct unfolded coordinates, chains can extend over any large distances. The dynamics of the chain follows a kinetic Monte-Carlo (KMC) scheme with simple local moves [90]: one Monte Carlo step (MCS) consists of N trial moves where a monomer is randomly chosen and randomly displaced to a nearest-neighbor site on the lattice (Fig 8). Trial moves are accepted according to a Metropolis criterion applied to H and if the chain connectivity is maintained. Compared to standard Monte Carlo methods used to study systems at thermal equilibrium [96], KMC has the advantage to track the equilibrium or out-of-equilibrium dynamical evolution of a system. The transition rates in the KMC are assumed to be Poissonian which is likely to be an approximation of the exact dynamics at the relevant time scales. However, the connection between the real-time and KMC steps can be established precisely within the framework of Poissonian processes [97]. In our model, due to the approximated transition rates, accuracy is not guaranteed for time-scales below few MCS (temporal resolution) and length-scales below few b (spatial resolution). This KMC scheme coupled to the notion of stored length (see above) allows efficient simulations of reptation motion in dense—topologically constrained—systems, while still accounting for the main characteristics of polymer dynamics like polymer connectivity, excluded volume, and non-crossability of polymer strands [92,93]. Due to the simplicity and efficiency of such frameworks, they have been widely used in the literature to investigate various properties of polymeric systems [32,90–92,94,98–100]. In the entangled regime (L/Le ≫ 1), dynamics could be very slow and the system may keep the ‘memory’ of its initial configuration and topology for a long time. Chromosomes are thought to be mostly knot-free structures [16,43,101]. Therefore, we initiate our simulations by knot-free configurations generated using the ‘hedgehog’ algorithm [59,102,103]: starting from a central unknotted scaffold, configurations are iteratively grown by randomly inserting monomers at nearest-neighbor sites common to two already placed consecutive monomers (Fig. A (a) in S1 Text). We verified that, starting from other types of unknotted configurations such as Rosette (Fig. A (e) in S1 Text) and Cylindrical (Fig. A (f) in S1 Text), we recovered the same scaling laws for the null model (Fig. C in S1 Text). Starting from a given initial configuration, we then normally simulate 107 − 108 MCS and store the configurations after each 103 MCS. In some special cases where we are specifically interested in small time scales, we collect configurations more frequently. From these snapshots taken from 102 simulated trajectories, we then estimate several structural and dynamical quantities of interest. We focus on the time-evolution of the mean squared displacements (MSD) of individual monomers (g1 (t) ) and of the center of mass of the chain (g3 (t) ), as well as the average squared distance 〈R2 (s) 〉 and contact probability Pc (s) between monomers separated by a linear distance s along the chain. For the latter, a contact is defined if the physical distance between a pair of genomic loci is less than a particular distance dc. Note that all such properties are estimated using the ‘unfolded’ polymer conformations. In this section we derived the relation between Kuhn length lK and different lattice parameters, expressed in Eq 3. We consider an arbitrary fine-scale model (FSM) of a semi-flexible self-avoiding walk defined by N0, l k 0, b0. We note N, lk and b, the corresponding values of a coarse-grained model (CGM) of the FSM. Each CGM monomer encompasses n = N0/N > 1 FSM monomers. Using Eq 1, conservation of L/Le and conservation of the volumic density in FSM monomer ρFS leads to L L e = N b l K (ρ F S b l k 2 c n) 2 (5) ≡ L 0 L e 0 = N 0 b 0 L e 0, (6) (l K b) 3 = (n 3 c 2 ρ F S 2 b 6) (b 0 L e 0), (7) ρFS = (N0/V) (= ρ/ν0 in the context of chromosome) with V the volume of the box. Conservation of the volume implies that ρ F S = N n 2 N s b 3 = 2 Φ n b 3, (8) with Ns the number of lattice sites and Φ = N/Ns the lattice volumic fraction. Incorporating Eq 8 into Eq 7 leads to l k b = [ (b 0 L e 0) (n c 2 2 Φ 2) ] 1 / 3. (9) Practically, knowing ρFS, b0 and L e 0 (from Eq 1 of the main text) for the reference model, for a given coarse-graining (defined by n), Eq 9 gives a relation between lk/b (which is related to the bending energy of the model, see below) and the lattice volumic fraction Φ. Eq 8 is used to compute the corresponding value for b. For a lattice phantom chain with N beads, the mean squared end-to-end distance 〈 R e 2 〉 is given by [104] ⟨ R e 2 ⟩ = 12 13 b 2 [ (N - 1) (1 + x 1 - x) - 2 x (1 - x N - 1) (1 - x) 2 ], (10) where x = 12 13 ⟨ cos θ ⟩ (11) = 12 13 (1 + 2 e x p [ - κ / 2 ] - 2 e x p [ - 3 κ / 2 ] - e x p [ - 2 κ ] 1 + 4 exp [ - κ / 2 ] + 2 e x p [ - k ] + 4 exp [ - 3 κ / 2 ] + e x p [ - 2 κ ]), (12) where θ is the angle between two monomers and κ is the bending rigidity correspond to the bending energy E (θ) = κ (1 − cosθ). Now we have lK/b = (1 + x) / (1 − x), which relates NK to κ. In the block copolymer model, the energy of a given configuration is given by H = κ 2 ∑ m = 1 N - 1 (1 - cos θ m) + ∑ l, m U e (l), e (m) δ l, m. (13) The first contribution accounts for the null model. The second contribution accounts for epigenomic-driven interactions with δl, m = 1 if monomers l and m occupy nearest-neighbor (NN) sites on the lattice (δl, m = 0 otherwise), e (l) the epigenomic state of monomer l and Ue, e′ the strength of interaction between a pair of spatially neighbor beads of epigenomic states e and e′. For simplicity, we assume that interactions occur only between monomers of the same chromatin state (Ue, e′ = 0 if e ≠ e′) and that the strength of interaction (that we note Ei) is the same whatever the chromatin state (Ue, e ≡ Ei for all e). Dynamics of the chain follows the same KMC scheme as the null model using a Metropolis criterion applied to H. For various values of Ei, we simulate 400 trajectories during 107 MCS starting from a random unknotted “hedgehog” configuration (as in Fig. A (a, c) in S1 Text) at a standard in vivo bp-density (ρ = 0. 009bp/nm3) (see the Movie for example). Note that such initial configurations might not reflect exactly the post-mitotic organization of fly chromosome and may impact the very large-scale—out-of-equilibrium—behavior predicted by the model.
The chromosome architecture inside cell nuclei plays important roles in regulating cell functions. Many experimental and modeling efforts are dedicated to deciphering the mechanisms controlling such organization. There are proliferations of experimental studies which report the hierarchical structure of chromosomes but how exactly they physically organize in 3D is not fully understood. In modeling, the main challenges are to develop adequate models and simulation methods to investigate correctly these highly dense long polymer chains. Taken into consideration the fundamental physical characteristics of chromosomes, we developed robust and numerically efficient polymer models that enabled us to explore the dynamics of long chromosomes over long time periods with good statistics. We applied this framework to investigate the dynamical folding of chromosome in drosophila. Accounting for the local biochemical information, we were able to reproduce the experimentally-measured contact frequencies between any pairs of genomic loci quantitatively and to track the hierarchical chromosome structure throughout the cell cycle. Our results further support the picture of a very dynamic chromosome organization driven by weak short-range interactions.
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How epigenome drives chromatin folding and dynamics, insights from efficient coarse-grained models of chromosomes