The intricate structural organization of the human nucleus is fundamental to cellular function and gene regulation. Recent advancements in experimental techniques, including high-throughput sequencing and microscopy, have provided valuable insights into nuclear organization. Computational modeling has played significant roles in interpreting experimental observations by reconstructing high-resolution structural ensembles and uncovering organization principles. However, the absence of standardized modeling tools poses challenges for furthering nuclear investigations. We present OpenNucleome—an open-source software designed for conducting GPU-accelerated molecular dynamics simulations of the human nucleus. OpenNucleome offers particle-based representations of chromosomes at a resolution of 100 KB, encompassing nuclear lamina, nucleoli, and speckles. This software furnishes highly accurate structural models of nuclear architecture, affording the means for dynamic simulations of condensate formation, fusion, and exploration of non-equilibrium effects. We applied OpenNucleome to uncover the mechanisms driving the emergence of ‘fixed points’ within the nucleus—signifying genomic loci robustly anchored in proximity to specific nuclear bodies for functional purposes. This anchoring remains resilient even amidst significant fluctuations in chromosome radial positions and nuclear shapes within individual cells. Our findings lend support to a nuclear zoning model that elucidates genome functionality. We anticipate OpenNucleome to serve as a valuable tool for nuclear investigations, streamlining mechanistic explorations and enhancing the interpretation of experimental observations.
3D Genomechromosome foldingnuclear compartmentsResearch organismHumanhttp://dx.doi.org/10.13039/100000057National Institute of General Medical SciencesR35GM133580ZhangBinThe funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.Author impact statementAn open-source tool for computational simulations of the human genome has been introduced, enabling the characterization of complex nuclear environments and the interpretation of experimental observations.publishing-routeprcIntroduction
The highly complex structural organization of the human nucleus plays a crucial role in the functioning and regulation of our cells (Dekker et al., 2017; Hübner et al., 2013; Bickmore, 2013; Gorkin et al., 2014; Dekker and Mirny, 2016; Furlong and Levine, 2018; Finn and Misteli, 2019; Chen and Belmont, 2019; Lin et al., 2021; Liu et al., 2024). The complexity arises from the diverse range of nuclear landmarks, such as nucleoli (Lafontaine et al., 2021), nuclear speckles (Chen and Belmont, 2019; Lamond and Spector, 2003), and the nuclear lamina (van Steensel and Belmont, 2017), each serving distinct functions. These landmarks provide specialized environments for various nuclear processes, allowing for efficient coordination and regulation of gene expression. Moreover, the spatial arrangement of chromosomes within the nucleus, intertwined with the nuclear landmarks, is critical for proper gene regulation and communication between different genome regions. Disruptions or abnormalities in the nuclear organization can have profound consequences on cellular function and can contribute to the development of diseases, including cancer and genetic disorders (Seruga et al., 2008; Schuster-Böckler and Lehner, 2012).
Recent advancements in experimental techniques have significantly enhanced our understanding of nuclear organization (Bickmore, 2013; Schmitt et al., 2016; McCord et al., 2020; Parmar et al., 2019; Jerkovic and Cavalli, 2021; Chen et al., 2016). The advent of high-throughput sequencing-based methods, such as genome-wide chromosome-conformation capture (Hi-C), has unveiled crucial structural elements of the genome (Dekker et al., 2002; Lieberman-Aiden et al., 2009), including chromatin loops (Rao et al., 2014), topologically associating domains (Dixon et al., 2016; Dekker and Heard, 2015), and compartments (Lieberman-Aiden et al., 2009). Additionally, sequencing-based techniques such as DamID (Greil et al., 2006), Chip-Seq (Park, 2009), and TSA-Seq (Chen et al., 2018) have revealed valuable information regarding interactions between chromosomes and nuclear landmarks. However, it is worth noting that these sequencing methods often offer averaged contacts, which can mask the heterogeneity present across populations, although single-cell techniques are also emerging (Wen et al., 2020; Ramani et al., 2017; Nagano et al., 2013). Moreover, translating contact data into spatial positions can be challenging, adding complexity to interpreting experimental findings.
To complement these sequencing approaches, microscopic imaging techniques directly probe the spatial positions within individual nuclei (Bickmore, 2013; van Steensel and Belmont, 2017; Chen et al., 2015; Boettiger et al., 2016; Shachar et al., 2015). Recent advancements in DNA FISH (fluorescence in situ hybridization) have enabled high-throughput imaging of thousands of loci simultaneously (Su et al., 2020; Takei et al., 2021). These imaging studies have not only confirmed the structural features observed through sequencing techniques but have also provided valuable insights into the heterogeneity present at the single-cell level.
The abundance of available experimental data in the field of nuclear organization provides a fertile ground for structural modeling (Qi et al., 2020; Qi and Zhang, 2019; Boninsegna et al., 2022; Fujishiro and Sasai, 2022; Shi and Thirumalai, 2021; Dekker et al., 2013; Jost et al., 2014; Giorgetti et al., 2014; Di Pierro et al., 2017; Buckle et al., 2018; Nuebler et al., 2018; Bianco et al., 2018; Shi et al., 2018; MacPherson et al., 2018; Shin et al., 2023; Amiad-Pavlov et al., 2021; Brahmachari et al., 2022; Jiang et al., 2022; Ganai et al., 2014; Liu et al., 2018; Laghmach et al., 2020; Chu and Wang, 2021; Lappala et al., 2021; Chu and Wang, 2022; Goychuk et al., 2023; Sun et al., 2021; Kadam et al., 2023). To make sense of this wealth of information, various computational approaches have been introduced, with polymer simulation approaches being extensively utilized. These simulation techniques aid in reconstructing structural ensembles that closely replicate experimental data, offering valuable insights into the mechanisms underlying chromosome folding. In recent studies, these approaches have also been employed to investigate the interplay between the genome and the nuclear lamina (Bajpai et al., 2021; Kamat et al., 2023; Laghmach et al., 2021; Stephens et al., 2018), as well as nucleoli (Qi and Zhang, 2021), shedding light on their dynamic relationships.
Despite the progress made in computational modeling, the absence of well-documented software with easy-to-follow tutorials pose a challenge. Many research groups develop their own independent software, which complicates cross-validation and hinders the establishment of best practices for genome modeling (Fujishiro and Sasai, 2022; Yildirim et al., 2023; Oliveira Junior et al., 2021). Moreover, comprehensive models of the entire nucleus, especially at high resolution, remain scarce. Addressing these limitations and fostering collaboration in the scientific community can be achieved through the development of open-source tools. By promoting transparency and accessibility, such tools have the potential to greatly facilitate nuclear modeling and contribute to a more unified and collaborative research environment.
We present OpenNucleome, an open-source software designed for conducting molecular dynamics (MD) simulations of the human nucleus. This software streamlines the process of setting up whole nucleus simulations through just a few lines of Python scripting. OpenNucleome can unveil intricate, high-resolution structural and dynamic chromosome arrangements at a 100KB resolution. It empowers researchers to track the kinetics of condensate formation and fusion while also exploring the influence of chemical modifications on condensate stability. Furthermore, it facilitates the examination of nuclear envelope deformation’s impact on genome organization. The software’s modular architecture enhances its adaptability and extensibility. Leveraging the power of OpenMM (Eastman et al., 2017), a GPU-accelerated MD engine, OpenNucleome ensures efficient simulations.
Our work demonstrates the fidelity of the simulated nuclear organizations by faithfully reproducing Hi-C, Lamin B DamID, TSA-Seq, and DNA-MERFISH data. The dynamic insights extracted from this model are pivotal in advancing our understanding of nuclear organization mechanisms. Our findings reveal that inherent heterogeneity in chromosome contacts naturally emerges within single cells. Interestingly, robust contacts between chromosomes and nuclear bodies can also be established due to a coupled self-assembly mechanism. Notably, the resilience of contacts involving nuclear bodies supports a nuclear zoning model for genome function. In the realm of nuclear investigations, we anticipate OpenNucleome to serve as an invaluable tool, seamlessly complementing experimental techniques.
ResultsNon-equilibrium nucleus model at 100 KB resolution
We present an open-source implementation of a computational framework that facilitates the structural and dynamical characterization of the human nucleus. This framework builds upon a previous investigation but incorporates several significant modifications. Firstly, we enhance the model resolution by a factor of 10, enabling the precise determination of the spatial positioning of each chromatin segment measuring 100KB in length. Secondly, we present a kinetic scheme for speckles that accounts for the phosphorylation of protein molecules. This inclusion captures the influence of chemical reactions on the stability and dynamics of nuclear bodies. Thirdly, we incorporate explicit nuclear envelope dynamics to explore the impact of large-scale deformations on genome organization. Finally, our implementation into OpenMM offers the advantages of Python Scripting and GPU acceleration, facilitating easy extension and customization. These features will facilitate the broad applicability and adoption of the proposed model.
The nucleus model provides particle-based representations for chromosomes, nucleoli, speckles, and the nuclear envelope. As shown in Figure 1A and B, each of the 46 chromosomes is represented as a beads-on-a-string polymer, where each bead represents a 100-KB-long genomic segment. Based on Hi-C data, we further assign each bead as compartment A, B, or C to signify euchromatin, heterochromatin, or pericentromeric regions. The lamina was modeled as a spherical enclosure with 10 µm diameter, using discrete particles arranged to represent a mesh grid with covalent bonds linking together nearest neighbors (Strom et al., 2021). We modeled nucleoli and speckles as liquid droplets that emerge through the spontaneous phase separation of coarse-grained particles, representing protein and RNA molecule aggregates (Chen and Belmont, 2019; Lafontaine et al., 2021). These particles exhibited attractive interactions within the same type to promote condensation. More details about the various components of the system can be found in the Appendix 1, section ‘Components of the whole nucleus model’.
Computer model of the human nucleus for structural and dynamical characterizations.
(A) 3D rendering of the nucleus model with particle-based representations for the 46 chromosomes shown as ribbons, the nuclear lamina (gray), nucleoli (cyan), and speckles (yellow). As shown on the right, chromosomes are modeled as beads-on-a-string polymers at a 100 KB resolution, with the beads further categorized into compartment A (red), compartment B (light blue), or centromeric regions (green). (B) Speckle particles undergo chemical modifications concurrent to their spatial dynamics, and the de-phosphorylated (dP) particles contribute to droplet formation. (C) Illustration of the ideal and compartment potential that promotes chromosome compaction and microphase separation. Specific interactions between chromosomes and nuclear landmarks are shown on the right.
The energy function of the nucleus model includes three components that account for the self-assembly of chromosomes, the assembly of nuclear bodies, and the coupling between chromosomes and nuclear landmarks. Therefore, the model approximates nuclear organization as a coupled self-assembly process. The chromosome energy function (see Equation 7 in Appendix 1, section ‘Hi-C inspired interactions for the diploid human genome’) includes terms that account for the polymer connectivity and excluded volume effect, an ideal potential, compartment-specific interactions, and specific interchromosomal interactions. As shown in Figure 1C, the ideal potential is only applied for beads from the same chromosome to approximate the effect of loop extrusion by Cohesin molecules (Sanborn et al., 2015; Fudenberg et al., 2016) for chromosome compaction and territory formation (Di Pierro et al., 2016; Zhang and Wolynes, 2017). Compartment-specific interactions, on the other hand, promote microphase separation and compartmentalization of euchromatin and heterochromatin. Finally, interchromosomal interactions account for sequence-specific effects that compartment-dependent potentials cannot capture.
Interactions among coarse-grained particles that form nuclear bodies were designed to promote and stabilize the formation of liquid droplets, as has been revealed by many experiments (Handwerger et al., 2005; Caragine et al., 2018; Caragine et al., 2019). We adopted the Lennard–Jones potential for nucleolar particles to mimic the weak, multivalent interactions that arise from protein and RNA molecules that make up the nucleoli. As a first attempt to approximate their complex dynamics, we considered two types of particles that form speckles: phosphorylated (P) and de-phosphorylated (dP). The two types can interconvert via chemical reactions (Brackley et al., 2017; Söding et al., 2020; Carrero et al., 2006) and dP particles share attractive interactions modeled with the Lennard–Jones potential.
As shown in Figure 1C, to recognize specific interactions between chromosomes and nuclear landmarks, we introduced contact potentials between them. These potentials are inspired by the experimental techniques that probe the corresponding contacts. Appendix 1, sections ‘Chromosome–nuclear landmark interactions’ and ‘Nuclear landmark–nuclear landmark interactions’ contain more details about all the nuclear landmark-related energy functions.
Optimization of model parameters with experimental data
The nucleus model was designed to be interpretable such that energy terms represent physical processes. Furthermore, the expressions of the interaction potentials were also designed such that their parameters can be determined from experimental data via the maximum entropy optimization algorithm (Lin et al., 2021; Xie and Zhang, 2019; Schuette et al., 2023). Below, we briefly outline the procedure used for parameter optimization and further details can be found in Appendix 1, section ‘Optimization of the whole nucleus model parameters.
As illustrated in Figure 2, starting from a given set of parameters, we first perform MD simulations to produce a collection of 3D structures for the diploid genome and various nuclear bodies. These structures are then transformed into a contact map or contact probabilities between chromatin beads and nuclear landmarks by averaging over homologous chromosomes. Constraints corresponding to different energy terms could be obtained from the simulated results and compared with those estimated from Hi-C, SON TSA-Seq, and Lamin B DamID profiles. Finally, the model parameters were updated based on the difference between simulated and experimental constraints using the adaptive moment estimation (Adam) optimization algorithm (Kingma and Ba, 2014). The three steps can be repeated with updated parameters to improve the simulation-experiment agreement further.
Overview of the iterative algorithm for parameterizing the nucleus model with experimental data.
Starting from an initial set of parameters, we perform molecular dynamics (MD) simulations to produce an ensemble of nuclear structures. These structures can be transformed into contacts between chromosomes or between chromosomes and nuclear landmarks for direct comparison with experimental data. Differences between simulated and experiment contacts are used to update parameters for additional rounds of optimization if needed.
The number of speckle clusters formed along a typical simulation trajectory.
This plot shows that the kinetic scheme of speckle particle exchange produces a total of ~30 speckle droplets, reproducing experimental observations. See Appendix 1, section ‘Speckles as phase-separated droplets undergoing chemical modifications’ for more details of the kinetic scheme.
No quantitative experimental data exists for interactions among nuclear body particles to serve as constraints. We varied the strength of the interaction potential to produce 2–3 nucleoli and ∼30 speckle clusters during the simulations (Figure 2—figure supplement 1) while ensuring the fluidity of the resulting droplets.
Molecular dynamics simulations with GPU acceleration
We implemented the nucleus model into the MD engine OpenMM (Eastman et al., 2017). OpenMM offers an excellent interface with Python scripting, significantly improving the readability and customizability of the model. The code was designed into functional modules, with different components, such as chromosomes and nuclear landmarks, written as separate classes. This design further facilitates the introduction of additional nuclear components, if desired, with minimal changes to existing code. We provide examples of simulation set up, trajectory analysis, parameter optimization, and introducing new features in the GitHub repository.
Figure 3A illustrates the workflow for setting up and executing whole nucleus simulations. A configuration file that provides the position of individual particles in the PDB file format is needed to initialize the simulations. This file also contains topological information regarding whether a particle represents chromosomes or nuclear landmarks and the identity of specific chromosomes. The input file can be generated with provided Python scripts by randomly distributing the positions of chromosomes, speckles, and nucleoli, though optimized configurations are also included in the GitHub repository. By default, the lamina particles will be uniformly placed on a sphere of 10 μm in diameter. Upon parsing the configuration file, interactions among various components can be set up with optimized parameters. This step will produce an object that can be used for MD simulations. As shown in Figure 3B, the workflow only requires a few lines of code. The package also includes analysis scripts to compute contact maps, monitor conformational dynamics, and track nuclear bodies.
OpenNucleome facilitates GPU-accelerated simulations of the human nucleus.
(A) Illustration of workflow for setting up, performing, and analyzing molecular dynamics (MD) simulations. (B) Python scripts setting up whole nucleus simulations. (C) Performance of MD simulations on different number of CPU cores and a single GPU.
A significant benefit of OpenMM is its native support of GPU acceleration. As shown in Figure 3C, the simulation speed with one Nvidia Volta V100 GPU is 150 times faster than that of the four Intel Xeon Platinum 8260 CPU cores. Notably, this performance enhancement cannot be achieved by simply increasing the CPU core numbers. For example, the simulation speed with 32 CPU cores is less than twice that of 4 CPU cores, potentially due to the system’s heterogeneous distribution of particles.
Simulations reproduce and predict diverse experimental data
We extensively validated the parameterized nucleus model to examine its biological relevance. MD simulations initialized from 50 different initial configurations were performed to build an ensemble of structures. As mentioned in the following section, a diverse set of initial configurations is essential for reproducing interchromosomal contacts probed in Hi-C. From the simulated structures, we computed various quantities for direct comparison with experimental measurements. Given that the majority of experimental data were analyzed for the haploid genome, we adopted a similar approach by averaging over paternal and maternal chromosomes to facilitate direct comparison. More details on data analysis can be found in Appendix 1, section ‘Details of simulation data analysis’.
We compared the simulated contact probabilities among chromosomes with Hi-C data. As shown in Figure 4A and Figure 4—figure supplement 1, the simulated and experimental contact maps are highly correlated. The squares along the diagonal support the formation of chromosome territories that promote intrachromosomal contacts, and the apparent checkboard patterns follow the compartmentalization of various chromatin types. We further examined the decay of intrachromosomal contacts as a function of the sequence separation, which is known to deviate from that of an equilibrium globule (Lieberman-Aiden et al., 2009). As shown in Figure 4B, the simulated results overlap well with the Hi-C data (orange curve). In addition, the simulated average contact probabilities between various compartment types match values estimated from Hi-C data. Moreover, the simulated and experimental average contact probabilities between pairs of chromosomes agree well, and the Pearson correlation coefficient between the two datasets reaches 0.89.
Simulated structures reproduce contact frequencies between chromosomes and between chromosomes and nuclear landmarks.
(A) Comparison between simulated (top right) and experimental (bottom left) whole-genome contact probability maps with Pearson correlation coefficient r = 0.89. Zoom-ins of various regions are provided in Figure 4—figure supplement 1. (B) Comparison between simulated and experimental average contact frequencies, including average contacts between genomic loci from the same chromosomes at a given separation (top), average contacts between genomic loci classified into different compartment types (middle), and average contacts between various chromosome pairs (bottom). (C) Comparison between simulated and experimental Lamin-B DamID (top) and SON TSA-Seq signals (bottom), with Pearson correlation coefficients of haploid chromosomes shown on the right.
Zoom-in of various regions in the contact map presented in <xref ref-type="fig" rid="fig4">Figure 4</xref> further supports the agreement between simulation and experiment.
The simulated and experimental contacts are shown in the upper and lower triangles, respectively, and the Pearson correlation coefficients r between two the sets of contacts were calculated with Equation 41.
We further examined the contacts between chromosomes and nuclear landmarks. As illustrated in Figure 4C, the simulated Lamin-B DamID signals for chromosome 7 match well with the experimental results, capturing the complex contact pattern that weaves chromatin toward and away from the nuclear envelope. Similarly, SON TSA-Seq data that quantify the contact between chromosomes and speckles are well captured by simulated structures. The anti-correlation between DamID and TSA-Seq is clearly visible. The observed agreement between simulation and experimental results is not limited to any particular chromosome. Good agreements are achieved for all chromosomes.
The simulations also provide 3D representations of the nucleus that can be compared with DNA-MERFISH data (Su et al., 2020). We found that the simulated radius of gyration of individual chromosomes matches well with experimental values (Figure 4A). The simulated and experimental average normalized chromosome radial positions also correlate strongly, as shown in Figure 5B. We note that while the sequencing results presented in Figure 4 were used for model parameterization, the MERFISH data were not. Therefore, the simulation results here are de novo predictions, and their agreement with experimental data strongly supports the coupled assembly mechanism used for designing the energy function.
Structural and dynamical predictions of the nucleus model match results from microscopy imaging.
(A) Comparison between the simulated and experimental radius of gyration, Rg, for haploid chromosomes. The Pearson correlation coefficient between the two, r, is shown in the legend. (B) Comparison between the simulated and experimental normalized radial positions for haploid chromosomes, with their Pearson correlation coefficient shown in the legend. Detailed definition of the normalized radial positions is provided in Appendix 1, section ‘Computing simulated normalized chromosome radial positions’. (C) Mean-squared displacements (MSDs) as a function of time are shown for selected telomeres. (D) The probability distribution of the anomalous exponent, α, obtained from fitting the MSDs curves for all telomeres with the expression, ⟨r2(Δt)⟩=DαΔtα.
A significant advantage of MD simulation-based models is the dynamical information they naturally produce. We measured the dynamics of telomeres by tracking the mean-square displacements (MSDs), ⟨r2(Δt)⟩, as a function of time. In Figure 5C, we plot representative MSD trajectories over a 1-hr timescale. In line with previous research (Di Pierro et al., 2018; Bronstein et al., 2009; Lee et al., 2021), telomeres display anomalous subdiffusive motion. When fitted with the equation ⟨r2(Δt)⟩=DαΔtα, these trajectories yield a spectrum of α values, with a peak around 0.59. The exponent and the diffusion coefficient Dα=(27±11)×10−4μm2⋅s−α both match well with the experimental values (Bronshtein et al., 2015; Jack et al., 2022), upon setting the nucleoplasmic viscosity as 1Pa⋅s (see Appendix 1, section ‘Mapping the reduced time unit to real time’ for more details).
The good agreement in the dynamics of individual loci further inspired us to examine the diffusion of whole chromosomes. In particular, we plotted the normalized chromosome radial positions as a function of time in Figure 6A. Remarkably, we found that chromosomes appear arrested and no significant changes in their positions are observed over timescales comparable to the cell cycle (see also Figure 6—figure supplement 1). Therefore, our simulations predict that large-scale movements of chromosomes are unlikely during the G1 phase.
Heterogeneity and conserved features of nuclear organizations.
(A) Normalized chromosome radial positions as a function of simulation time. (B) Contacts between chromosomes 1 and 2 from two independent simulation trajectories show significant variations. (C) Genome-wide in silico Lamin B DamID (top) and SON TSA-Seq (bottom) profiles computed from two independent trajectories. Pearson correlation coefficients, r, are provided on each plot. (D) Pairwise Person correlation coefficients between interchromosomal contact matrices (left), genome-wide Lamin B DamID profiles (middle), and genome-wide SON TSA-Seq profiles (right) determined from independent trajectories. The averages excluding the diagonals of the three datasets are 0.06, 0.53, and 0.72.
Arrested kinetics of chromosome positions over the timescale of cell cycles.
(A) Mean-squared displacement (MSD) of chromosome center of masses as a function of time. The MSDs are much smaller than the average size of chromosomes (see Rg values in Figure 5A), supporting arrested dynamics. (B) Autocorrelation function of normalized chromosome radial positions as a function of time. The autocorrelation function was computed as C(Δt)=∑t=1N(rt−r¯)(rt+Δt−r¯)∑t=1N(rt−r¯)2, where t indexes over the trajectory frames and r¯ is the mean position.
Configurations used to initialize simulations capture the heterogeneity in interchromosomal contacts seen in DNA-MERFISH data.
(A) Comparison between the simulated and experimental distribution of Pearson correlation coefficients of interchromosomal contacts. Both distributions were computed using the pairwise correlations between genome structures from the respective configurational ensemble. The pairwise interchromosomal contacts include all unique haploid pairs and were computed by averaging the contact probabilities between all genomic segments from the four respective chromosomes. The simulated configuration ensemble includes 1000 unique structures prepared following the protocol outlined in Appendix 1, section ‘Initial configurations for simulations’. The experimental ensemble includes 5455 structures reported in Su et al., 2020 using DNA-MERFISH. (B) Pearson correlation coefficient between average experimental and simulated pairwise interchromosomal contacts as a function of the number of independent configurations used to initialize simulations. Interchromosomal contacts are similarly defined as in (A) and were computed by averaging over all configurations in the respective ensembles. To compute simulated interchromosomal contacts, we used a selection procedure as detailed in Appendix 1, section ‘Initial configurations for simulations’ to determine an ensemble of a given sample size. As the size of the ensemble increases, the resulting average interchromosomal contacts agree better with experimental results. The error bars represent standard deviation computed from 20 independent trials. (C, D) The protocol for selecting initial configurations for simulations robustly capture the heterogeneity seen in experimental configurational ensemble. To examine the experimental structural heterogeneity, we applied Uniform Manifold Approximation and Projection (UMAP) to reduce the interchromosomal contacts into two variables following the procedure detailed in Appendix 1, section ‘Interchromosomal contacts from DNA MERFISH data’. The plot in (C) shows the Pearson correlation coefficients of 50 initial configurations represented in the UMAP variables between independent trials. The average correlation coefficient is 0.98. In (D), we show three independent initial configurations (orange dots) over the distribution of UMAP variables estimated using the experimental configurational ensemble.
Heterogeneity and robustness of the simulated conformational ensemble
The lack of relaxation of chromosome radial positions suggests the importance of starting configurations used to initialize the simulations. Statistical averages of the resulting ensemble of nuclear structures depend crucially on these starting configurations. Using an optimization procedure, we selected them from 1000 configurations to maximize the agreement with experimental lamin-B DamID and interchromosomal contact probabilities. Appendix 1, section ‘Initial configurations for simulations’ provides more details on preparing the 1000 initial configurations.
We selected a total of 50 starting configurations to initiate independent simulations. Smaller sets of starting configurations are not sufficient to reproduce the interchromosomal contact probabilities, as shown in Figure 6—figure supplement 2B. Notably, different sets of 50 configurations selected from independent trials show significant overlap (Figure 6—figure supplement 2D), supporting the robustness of the selection protocol in detecting conserved features of genome organization.
While the ensemble as a whole is relatively robust, individual configurations with the ensemble exhibit significant differences. For example, the Lamin B DamID profiles produced from different trajectories are only weakly correlated (Figure 6C), with an average correlation coefficient of 0.53. These weak correlations result from significant differences in the normalized radial positions of chromosomes, as can be seen in representative configurations from two simulation trajectories (Figure 6B). The fluctuations of normalized radial positions cause changes in contacts between chromosomes as well, resulting in little correlation between interchromosomal contact matrices (Figure 6D).
We examined genome organizations reported by Su et al. and found a similar variation of interchromosomal contact probabilities across individual cells (Figure 6—figure supplement 2A and D). Notably, the simulated configurations capture the fluctuations of interchromosomal contacts observed in DNA-MERFISH data, further supporting the biological relevance of the reported in silico structures.
Despite the differences in interchromosomal contacts across trajectories, high conservation of connections between chromosomes and speckles can be observed in individual simulations. For example, the average correlation coefficient between in silico SON TSA-Seq profiles produced from different trajectories is 0.72, much higher than the corresponding value for Lamin B DamID profiles. Conservation of contacts between chromosomes and nuclear bodies (zones) across individual cells has indeed been reported in a previous study that simultaneously images chromatin and various subnuclear structures (Takei et al., 2021).
Nuclear deformation preserves chromosome–nuclear body contacts
Numerous studies have highlighted the remarkable influence of nuclear shape on the positioning of chromosomes and the regulation of gene expression (Brahmachari et al., 2022; Contessoto et al., 2023). The nucleus, once regarded as a mere compartment for DNA storage, is increasingly recognized as a dynamic and intricately structured organelle. To better understand the interplay between nuclear shape and genome organization as a fundamental mechanism that shapes the transcriptional landscape, we performed additional simulations in which the nuclear lamina was altered from a sphere into more ellipsoidal shapes by applying a force along the z-axis (Figure 7A). More details about these simulations can be found in Appendix 1, section ‘Nuclear envelope deformation simulations’.
Nuclear deformations influence genome organization while preserving chromatin-speckle contacts.
(A) Illustration of force-induced nuclear envelope deformation. The nuclear lamina is modeled as a particle mesh where neighboring lamina particles are covalently bonded together. (B) Example nucleus conformations at different strengths of applied force. (C) Pearson correlation coefficients between results from simulations of deformed nuclei and those from a spherical nucleus for interchromosomal contacts (left), DamID profiles (middle), and TSA-Seq (right). The values at zero force were computed from two independent simulations starting from the same initial configurations.
Impact of nuclear deformation on normalized chromosome radial positions.
(A) The position of chromosome 21 before and after the nuclear deformation. (B) Normalized radial positions of individual diploid chromosomes at various strengths of nuclear deformation forces. (C) Pearson correlation coefficients between normalized chromosome radial positions from simulations of deformed nuclei and those from a spherical nucleus. See Appendix 1, section ‘Nuclear envelope deformation simulations’ for simulation details.
As illustrated in Figure 7B, the presence of external forces resulted in significant alterations in nuclear shape. We conducted two independent simulations with different force strengths, leading to varying degrees of deformation in the nuclear lamina. This deformation, in turn, caused a reorganization of chromosomes, affecting their normalized radial positions and pairwise contacts (see Figure 7—figure supplement 1 and Figure 7C). We observed that more deformed nuclei exhibited lower correlation coefficients for interchromosomal contacts compared to results obtained from simulations in a spherical nucleus. Similarly, the DamID profiles exhibited significant variations upon nucleus deformation, whereas TSA-Seq signals were much less affected and remained highly correlated with the results from the spherical nucleus simulations.
Therefore, it appears that speckles, and potentially other nuclear condensates, can dynamically reorganize in response to changes in chromosome conformations to maintain contacts with genomic loci. This robustness in nuclear body contacts may be essential for ensuring the robust functioning of the genome in a population of cells with significant variability in nuclear shape.
Discussion
We introduced a computational model, OpenNucleome, to facilitate simulations for the human nucleus at high structural and temporal resolution. We conducted extensive cross-validation with experimental data to support the biological relevance of simulated 3D structures. Implementing the model into the MD package, OpenMM enables GPU acceleration for long-timescale simulations. Tutorials in the format of Python Scripts with extensive documentation are provided to facilitate the adoption of the model by the community.
Our software enhances the capabilities of existing genome simulation tools Fujishiro and Sasai, 2022; Yildirim et al., 2023; Oliveira Junior et al., 2021. Specifically, OpenNucleome aligns with the design principles of Open-MiChroM (Oliveira Junior et al., 2021), prioritizing open-source accessibility while expanding simulation capabilities to the entire nucleus. Similar to software from the Alber lab (Yildirim et al., 2023), OpenNucleome offers high-resolution genome organization that faithfully reproduces a diverse range of experimental data. Furthermore, beyond static structures, OpenNucleome facilitates dynamic simulations with explicit representations of various nuclear condensates, akin to the model developed by Fujishiro and Sasai, 2022.
A significant advantage of OpenNucleome lies in its predictive power for dynamical information. For example, the model succeeded in reproducing the subdiffusive behavior of telomeres. We further showed that the dynamics of individual chromosomes are slow and their radial positions do not relax over the time course of a cell cycle. This is consistent with previous theoretical estimations on chromosome dynamics (Rosa and Everaers, 2008) and recent observations of solid behavior of chromatin in vivo (Strickfaden et al., 2020). Live cell experiments that directly track the positions of multiple chromosomes could further validate/falsify this prediction. We anticipate the model will greatly facilitate the investigation of the dynamics of genomic loci and nuclear bodies and the interpreting of live cell imaging results.
Slow chromosome dynamics and a lack of conformational relaxation naturally result in the heterogeneity of chromosome radial positions across individual cells. This heterogeneity raises doubts about the notion that chromosome radial positions provide robust and reliable mechanisms for gene regulation (Hübner et al., 2013; Maeshima et al., 2010; Fraser and Bickmore, 2007; Takizawa et al., 2008). Instead, our results support the nuclear zoning model for gene regulation (Takei et al., 2021), where specific loci function as ‘fixed points’ anchored to certain nuclear bodies in all cells. This anchoring mechanism robustly creates the desired molecular environment surrounding these genomic segments. Unlike chromosome radial positions, contacts between genomic loci and speckles can be robustly established in individual cells, as shown in our simulations. It was achieved through a nucleation process that attracts speckle particles toward specific loci due to specific interactions. Nucleation occurs much more rapidly than chromosome rearrangement due to the smaller size of speckle particles. The coupled self-assembly mechanism for chromosomes and nuclear bodies can similarly facilitate the formation of other nuclear zones for different kinds of fixed points.
Despite the heterogeneity in chromosome positions and interchromosomal contacts, the ensemble of nuclear structures as a whole is not random and exhibits conserved features. For example, on average, certain chromosomes remain closer to the nuclear envelope than others (see Figure 5B). Similarly, the average contact frequency between certain chromosome pairs is higher than others, though this trend can be frequently violated in individual cells. How such conserved features arise as cells exit from the mitotic phase remains unclear and would be interesting for further explorations.
MethodsMolecular dynamics simulation details
We used the software package OpenMM Eastman et al., 2017 to perform MD simulations in reduced units at constant temperature (T = 1.0). Unless otherwise specified, we froze the lamina particles and only propagated the dynamics of chromatin, nucleoli, and speckles.
Two integration schemes were used with a time step of dt = 0.005to efficiently generate structural ensembles and produce realistic dynamical information, respectively. For simulations used in parameter optimization and building structural ensembles, we employed the Langevin integrator with a damping coefficient of γ−1=10.0. In the case of MSD calculations shown in Figure 6, we utilized Brownian dynamics with a damping coefficient of γ−1=0.01. The higher damping coefficient provides a better approximation to the viscous nucleus environment, while the smaller value in the Langevin integrator facilitates conformational sampling with faster diffusion rates.
We employed the semi-grand Monte Carlo technique (Sadigh et al., 2012) to simulate chemical transitions between two types of speckle particles. At every 4000 simulation steps, we attempt a total of NSp chemical reactions that converts one type of speckle particles to the other type with a probability of 0.2. NSp corresponds to the total number of speckle particles, and the switching probability was chosen to be comparable to the experimental phosphorylation rate. More details on the speckle dynamics are provided in Appendix 1, section ‘Speckles as phase-separated droplets undergoing chemical modifications’.
When deforming the nuclear envelope, we unfroze the lamina particles and evolved them dynamically as the rest of the nucleus. Bonded interactions among lamina particles held the nuclear envelope together as a particle mesh. A harmonic force along the z-axis was introduced to compress the particle mesh. More details are provided in Appendix 1, section ‘Nuclear envelope deformation simulations’.
For simulations used to optimize parameters, a total of 50 independent 3-million-step-long trajectories were performed. Configurations were recorded at every 2000 simulation steps for analysis. The first 500,000 steps of each trajectory were discarded as equilibration. For production simulations, we performed 50 independent 10-million-step long trajectories starting from different initial configurations. Nuclear structures were again recorded at every 2000 steps to determine statistical averages presented in the article. An additional eight simulations of 30million steps in length were performed to compute telomere MSDs.
We mapped the reduced units to real units with the conversion of length scale σ = 385nm and the timescale in Brownian dynamics simulations τ=0.65s. These conversions were determined as detailed in Appendix 1, section ‘Unit conversion’.
Experimental data processing and analysis
We obtained the in situ Hi-C data, SON TSA-seq data, and Lamin-B DamID data of HFF cell lines from the 4DN data portal. The intra and interchromosomal interactions were calculated at 100KB resolution with VC_SQRT normalization applied to the interaction matrices. Hi-C data extraction and normalization were performed using Juicer tools (Durand et al., 2016). We followed the same processing and normalization method described in Zhang et al., 2021 to analyze TSA-seq data. Two biological replicates of Lamin-B DamID data were merged and the normalized counts over Dam-only control were used for analysis. The SON TSA-Seq and Lamin-B DamID data were processed at the 25KB resolution and the average values at the 100KB resolution were used in Figure 4 for model validation.
Additional informationCompeting interests
No competing interests declared
Reviewing editor, eLife
Author contributions
Software, Formal analysis, Investigation, Visualization, Methodology, Writing - original draft, Writing - review and editing
Formal analysis, Methodology
Software
Conceptualization, Formal analysis, Supervision, Funding acquisition, Investigation, Visualization, Methodology, Writing - original draft, Project administration, Writing - review and editing
Additional filesData availability
Hi-C data (https://data.4dnucleome.org, accession number: 4DNFIB59T7NN). SON TSA-seq data (https://data.4dnucleome.org, accession number: pulldown data 4DNEX6U8TS3Y, control data 4DNEXI7XUWFK). LaminB DamID data (https://data.4dnucleome.org, accession number 4DNESXZ4FW4T). The software is available at https://github.com/ZhangGroup-MITChemistry/OpenNucleome (copy archived at ZhangGroup-MITChemistry, 2024).
The following previously published datasets were used:
van SteenselBNKI4DN Data Portal2017LaminB1 DamID of HFFc6 Tier 1 cells – cells were transduced with virus expressing Dam-LaminB1, gDNA was harvested after 4 days and processed for DamID-seq4DNESXZ4FW4T
ZhangLZhangYChenYGholamalamdariOWangYMaJBelmontAS4DN Data Portal2020Set of Input for SON Ab2 TSA-seq version 2 Reaction Condition 2 (PBS 50% Sucrose) Enhancement Condition E (1:300 tyramide-biotin, 30 minute reaction) on HFFc6 cells4DNEXI7XUWFK
GholamalamdariOvan SchaikTWangYKumarPZhangLZhangYHernandez GonzalezGAVouzasAEZhaoPAGilbertDAMaJvan SteenselBBelmontAS4DN Data Portal2024TSA-seq against SON protein on HFFc6 (Tier 1)4DNEX6U8TS3Y
KrietensteinNAbrahamSVenevSVAbdennurNGibcusJHsiehTSParsiKMYangLMaehrRMirnyLADekkerJRandoOJ2020Ultrastructural Details of Mammalian Chromosome Architecture4DN Data Portal4DNFIB59T7NN
van SteenselB2017DamID-seq with DAM-LMNB1 on HFFc6 (Tier 1)4DN Data Portal4DNEXFUGLVQA
Acknowledgements
This work was supported by the National Institutes of Health (grant R35GM133580).
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As outlined in the main text, the whole nucleus model consists of chromosomes, nucleoli, speckles, and the nuclear lamina. Below, we provide details on the particle-based representations of the various components, totaling 70542 coarse-grained beads. Abbreviations are frequently used for clarity in notation, with N for nucleus, La for lamina, No for nucleoli, and Sp for speckles.
Chromosomes as beads on the string polymers
We explicitly modeled the 46 human chromosomes as beads-on-a-string polymers. Each coarse-grained bead represents a 100 KB genomic segment, totaling 60642 beads for the genome. We assigned each bead as either compartment type A, B, C, or N. The compartment assignments for types A and B were extracted from the Hi-C contact matrix for HFF cells (Krietenstein et al., 2020) using the cooltools software (Venev et al., 2020), and compartment C were identified as centromeric regions based on the DNA sequence. Compartment N denotes genomic regions that cannot be assigned as A, B, or C due to a lack of Hi-C data.
The nuclear lamina as a particle-based mesh
The nuclear envelope provides an enclosure to confine DNA and a repressive environment to organize chromatin with specific interactions (Hetzer, 2010). To account for the role of the nuclear lamina while keeping our model simple, we approximate it with discrete particles uniformly placed on a sphere.
Following our previous work (Kamat et al., 2023), we used the Fibonacci grid to initialize the lamina particles, which form a uniform and almost equidistant network of lamina particles on the surface of the nucleus (Swinbank and James Purser, 2006; Li et al., 2007). The Cartesian coordinates associated with the ith lamina particles are defined asxi=2RN×(1−iNLa−1)yi=RN2−x2×cos[iΦ]zi=RN2−x2×sin[iΦ]
where NLa=8000 represents the number of lamina particles, i∈{0,1,…,NLa−2,NLa−1}, and Φ=π×(3−5) is the golden angle. We set RN=5μm as the radius of the human foreskin fibroblasts (HFF) cell nucleus.
Nucleoli as phase-separated droplets
Nucleoli have been shown to behave as liquid droplets that form through phase separation (Lafontaine et al., 2021; Pederson, 2011; Shin and Brangwynne, 2017). We modeled the droplets with coarse-grained beads. While the composition of nucleoli is rather complex, we only used one type of particle for simplicity. In our simulations, we fixed the number of nucleolus particles, NNo, based on the experimental concentration of nuclear protein NPM1, c=1μM (Qi and Zhang, 2021; Kamat et al., 2023; Zhu et al., 2019). For example,NNo=4π3⋅NA⋅(RNo)3⋅c≈300
where NA is the Avogadro constant and RNo=0.5μm is the average nucleolous size (Caragine et al., 2018; Caragine et al., 2019).
Speckles as phase-separated droplets undergoing chemical modifications
Similar to nucleoli, speckles have also been shown to behave as liquid droplets (Chen and Belmont, 2019). However, one crucial unique feature of speckles is the constant chemical modifications of protein molecules comprising them, such as splicing factors (Spector and Lamond, 2011). The phosphorylation of these molecules has been argued to be essential for the dynamics and the number of speckles. Therefore, we implemented a kinetic scheme introduced by de Vries and coworkers to account for the chemical reactions. In this scheme, we consider two types of speckle molecules: phosphorylated (Sp-P) and de-phosphorylated (Sp-dP). Only Sp-dP particles share attractive interactions.
The two protein types can inter-convert via chemical reactions with a transition probability matrix T defined asT=Sp−dPSp−PSp−dPSp−PSp−dPSp−P‖p11p12p11p12‖=Sp−dPSp−P‖0.80.20.20.8‖
For simplicity, we assume the forward transition rate from Sp-P to Sp-dP particles is identical to the reverse rate. Because of the symmetry in transition rates, the average number of dP particles 〈NSp-dP〉=0.5NSp, where NSp is the total number of speckle particles.
We chose the transition probability as 0.2 to be consistent with the phosphorylation rate. In particular, we estimate the rate ask12=p12×τ−1=0.2×14000×0.005×0.65s=0.0154s−1
where τ is the time interval between consecutive attempts of chemical reactions. As detailed in section ‘Molecular dynamics simulation details’, the reactions were attempted every 4000 simulation steps, with a time step of 0.005. The time unit in our simulations is 0.65 s (see section ‘Mapping the reduced time unit to real time’). The estimated value for k12 is in the same order as the experimental phosphorylation rate (Velazquez-Dones et al., 2005).
We estimated the total number of speckle particles as follows. Assuming that there is a total of 30 speckles (Galganski et al., 2017), we have NSp-dP=30×Ns, where Ns is the number of Sp-dP particles in each cluster. This estimation assumes that only Sp-dP particles share attractive interactions and contribute to cluster formation. From the experimentally estimated relative mass densities of the protein concentrations in the speckle and nucleolus droplet as 170203 (Handwerger et al., 2005), we haveNs×m/0.33100×m/0.53=170203.
We assumed that speckle and nucleolus particles have identical mass and each nucleolus has 100 particles. The radius for speckle and nucleolus was approximated as 0.3 and 0.5μm, yielding Ns≈20 and NSp-dP≈600. Because of the kinetic scheme defined in Equation 3, only parts of Sp-dP particles will participate in droplet formation during the simulations. Therefore, we increase the particle number and set 〈NSp-dP〉=800, which yields NSp=1600.
Energy function of the whole nucleus model
As detailed below, the energy function of the whole nucleus, UNucleus, consists of interactions among chromosomes, among nuclear landmarks, and cross interactions between the two. Therefore,UNucleus=UGenome+UNL+UGN
Hi-C inspired interactions for the diploid human genome
The energy function of the genome model is defined asUGenome=Uhomo(r)+Uideal(r)+Ucompt(r)+Uinter(r).
Uhomo(r) determines a generic polymeric topology of chromosomes with excluded volume effect:Uhomo(r)=∑i[ubond(ri,i+1)+uangle(r→i,i+1,r→i+1,i+2)]+Usc(r)
where the subscripts i,i+1, and i+2 represent the index of ith,(i+1)th, and (i+2)th beads, respectively, and ubond(ri,i+1) and uangle(ri,i+1,ri+1,i+2) denote the bonding and angular potential applied for neighboring beads to ensure the connectivity of the chromatin chain and follow:ubond(ri,i+1)=K2(r−r0)2+K3(r−r0)3+K4(r−r0)4,K2=K3=K4=20ϵuangle(r→i,i+1,r→i+1,i+2)=Ka[1−cos(θ−π)],Ka=2ϵ,cosθ=r→i,i+1⋅r→i+1,i+2|r→i,i+1|⋅|r→i+1,i+2|
where, as discussed in Equation 32, r0=0.5σ represents the size of the chromatin bead. The soft-core potential provides excluded volume effects for pairs of beads from the same or different chromosomes and follows:Usc(r)=∑j>iusc(rij)
where usc(rij) denotes a soft-core potential added to each pair formed by beads index i and j to account for the excluded volume effect while allowing the finite probability of cross-over of polymer chains.usc(rij)={0.5Ecut(1+tanh[2ULJ(rij)Ecut−1]),rij≤rcutULJ(rij),rcut<rij≤0.5×21/6σ0,rij>0.5×21/6σ
which corresponds to the Lennard–Jones potential capped off at a finite volume within a repulsive core to allow for chain crossing at a finite energy cost. Ecut=4ϵ and rcut is chosen as the distance at which ULJ(r)=0.5Ecut.
Uideal(r) is the intra-chromosomal potential applied to genomic loci within the same chromosome, while Ucompt(r) is the compartment-specific interaction potential. The ideal potential, which can be rigorously derived following the maximum entropy principle (Roux and Weare, 2013; Zhang and Wolynes, 2015), adopts the following form:Uideal(r)=∑I∑i,j∈Iαideal(|i−j|)f(rij)
where I indexes over each chromosome and i and j index over pair of beads on that chromosome. αideal(|i−j|) depends only on the sequence separation between two beads i and j. f(rij) measures the probability of contact formation for two loci separated by a distance of rij, and its ensemble average corresponds to the contact probability measured in Hi-C experiments. f(rij) adopts the formf(rij)=12(1+tanh[(rc−ri,j)−5+5(rc−ri,j)])+12(1+tanh[(rc−ri,j)−5+5(rc−ri,j)])×(rcri,j4).
The numerical value of rc was determined from the Hi-C contact map, as detailed in the next section. This contact probability function depicts that when r<rc,f≈1 but when r>rc,f≈(rc/r)4. The power-law decay with an exponent of 4 is consistent with the relationship between contact probability and spatial distances revealed in imaging studies (Qi and Zhang, 2019; Wang et al., 2017). The tanh function ensures the continuity of the function and its derivative around rc (Appendix 1—figure 1). Additionally, we truncated the ideal potential to be applicable for a sequence separation less than or equal to 100 MB and set the parameters for larger sequence separations to be zero. As shown in Figure 4 of the main text, our parameterized ideal potential produced chromosomes with sizes comparable to imaging results. Incorporating longer-range interactions to improve the model further is straightforward but would also significantly increase the number of parameters.
Similar to the ideal potential discussed above, we haveUcompt(r)=∑i,jαcompt(Ti,Tj)f(rij),
where Ti and Tj denote the compartment types of beads i and j which can be A, B, or C. Therefore, CG beads of the same compartment types will share the same interaction parameter αcompt(Ti,Tj), which will be derived from average Hi-C contact frequencies as detailed in the following sections.
To account for specific interactions between chromosomes, we introduced the interchromosomal potential asUinter(r)=∑I,J>I∑i∈I,j∈Jαinter(I,J)f(rij).
I,J∈{1,2,…,23} index the haploid chromosomes, and parental and maternal chromosomes share identical parameters. This potential allows the model to capture interactions beyond those arising purely from compartmentalization as defined in Equation 14.
All parameters in the energy function are summarized in Appendix 1—table 1. The procedure used for parameter optimization is detailed in the following sections.
Summary of the various terms of the chromosome energy function and the algorithms used for parameter optimization.
See also Appendix 1, section ‘Hi-C inspired interactions for the diploid human genome’ for detailed expression of the energy function and ‘Adam optimizer for chromosome interaction parameters’ for details on the optimization algorithm.
Potentials
Functional forms
Parameter values
Bonding potential
ubond(ri,i+1) in Equation 9
Standard values in coarse-grained polymer models
Angular Potential
uangle(r→i,i+1,r→i+1,i+2) in Equation 9
Standard values in coarse-grained polymer models
Soft-core potential
usc(rij) in Equation 11
Standard values in coarse-grained polymer models
Ideal potential
Uideal(r) in Equation 12
Values for αideal were obtained from optimizations against Hi-C data (see Appendix 1—figure 3).
Compartment potential
Ucompt(r) in Equation 14
Values for αcompt were obtained from optimizations against Hi-C data (see Appendix 1—table 2)
Inter potential
Uinter(r) in Equation 15
Values for αinter were obtained from optimizations against Hi-C data (see Appendix 1—figure 4)
Summary of interaction parameters between various compartment types, that is, <inline-formula><mml:math id="inf62"><mml:mrow><mml:msub><mml:mi>α</mml:mi><mml:mrow><mml:mtext>compt</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> defined in <xref ref-type="disp-formula" rid="equ14">Equation 14</xref>.
αAA
–0.074185
αAB
0.112285
αAC
0.009947
αBB
0.059981
αBC
0.072481
αCC
0.088825
Nuclear landmark–nuclear landmark interactions
The general energy function for interactions among nuclear landmark particles is defined asUNL=ULa(r)+UNo(r)+USp−dP(r)+UEV(r).
The nuclear lamina was modeled as a particle mesh, and bonded potentials were introduced for nearest neighbor particles defined asULa(r)=∑i∑j∈n.n.,j>iK2(r−ro)2+K3(r−ro)3+K4(r−ro)4,K2=K3=K4=20ϵ
with ro=0.5σ. i indices all the lamina particles, and j represents the nearest four neighbors around i determined from the initial configuration for which the particles were placed on a Fibonacci grid. To avoid pairs (i, j) being counted twice or more, we set j always larger than i.
Short-ranged, non-bonded interactions were introduced among nuclear landmark particles to account for attractions that promote phase separation and the excluded volume effect. These interactions were modeled with a cut and shifted Lennard–Jones (LJ) potential defined asULJ(rij)={4ϵLJ((σLJrij)12−(σLJrij)6−Ecut),forr<=rcut0,forr>rcut
with Ecut=4ϵ((σrcut)12−(σrcut)6). We note that when rcut was set as σLJ×21/6, the potential has no attractive regime and only serves to prevent the overlap among particles, that is, the excluded volume effect.
For attractive interactions between nucleolus particles, and between type dP speckle particles, we set the parameters as ϵLJ=3.0,σLJ=0.5, and rcut=1.5. Therefore,UNo(r)=∑j>i∈NoULJ(rij,ϵLJ=3.0,σLJ=0.5,rcut=1.5)USp-dP(r)=∑j>i∈Sp-dPULJ(rij,ϵLJ=3.0,σLJ=0.5,rcut=1.5),
where the sums iterate over pairs of nucleolus particles and speckle dP particles.
For the excluded volume effect between nucleolus and speckle particles, between dP and P particles, and between P particles, we set the parameters as ϵLJ=1.0,σLJ=0.5, and rcut=0.5×21/6. These potentials are consistent with the estimated size of 0.5 σ for speckle and nucleolus particles.
The excluded volume effect was also introduced between lamina and nucleolus particles and between the lamina and speckle particles to confine the nuclear bodies inside the nuclear envelope. We set the parameters as ϵLJ=1.0,σLJ=0.75, and rcut=0.75×21/6. The value for σLJ was chosen based on a linear combination of the lamina particle size (1.0 σ) and the speckle/nucleolus particle size (0.5 σ).
Therefore, the excluded volume potential can be written asUEV(r)=∑i∈La∑j∈NoULJ(rij,ϵLJ=1.0,σLJ=0.75,rcut=0.75×21/6)+∑i∈La∑j∈SpULJ(rij,ϵLJ=1.0,σLJ=0.75,rcut=0.75×21/6)+∑i∈No∑j∈SpULJ(rij,ϵLJ=1.0,σLJ=0.5,rcut=0.5×21/6)+∑i∈Sp-P∑j∈Sp-dPULJ(rij,ϵLJ=1.0,σLJ=0.5,rcut=0.5×21/6)+∑j>i∈Sp-dPULJ(rij,ϵLJ=1.0,σLJ=0.5,rcut=0.5×21/6)+∑j>i∈LaULJ(rij,ϵLJ=1.0,σLJ=0.5,rcut=0.5×21/6).
We used abbreviations to denote various nuclear landmarks, with La for the nuclear lamina, Sp-P for P-type speckle particles, Sp-dP for dP-type speckle particles, and No for nucleolus particles. All the interaction parameters for the nuclear landmarks are listed in Appendix 1—table 3 for convenient reference.
Summary of the interaction potentials among particles that make up the nuclear landmarks and their corresponding parameter values.
See also Appendix 1, section ‘Nuclear landmark–nuclear landmark interactions’ for further discussion and ‘Unit conversion’ for choosing the size of various particles, from which the σLJ were derived with a linear combination rule.
Potentials
Function forms
Parameter values
Nucleolus/nucleolus
ULJ(rij) in Equation 18
ϵLJ=3.0 and rcut=1.5 were chosen to mimic short-range attractions that produce an average of two nucleoli per cell. σLJ=σNo=0.5.
Sp-dP/Sp-dP (speckles)
ULJ(rij) in Equation 18
ϵLJ=3.0 and rcut=1.5 were chosen to mimic short-range attractions that produce around 30 speckle droplets. σLJ=σSp-dP=0.5.
Sp-dP/Sp-P (speckles)
ULJ(rij) in Equation 18
ϵLJ=1.0 and rcut=0.5×21/6 were chosen as standard values to provide the excluded volume effect. σLJ=σSp-dP+σSp-P2=0.5.
Sp-P/Sp-P (speckles)
ULJ(rij) in Equation 18
ϵLJ=1.0 and rcut=0.5×21/6 were chosen as standard values to provide the excluded volume effect. σLJ=σSp-P=0.5.
Nucleolus/speckle
ULJ(rij) in Equation 18
ϵLJ=1.0 and rcut=0.5×21/6 were chosen as standard values to provide the excluded volume effect. σLJ=σNo+σSp2=0.5.
Nucleolus/lamina
ULJ(rij) in Equation 18
ϵLJ=1.0 and rcut=0.75×21/6 were chosen as standard values to provide the excluded volume effect. σLJ=σNo+σLa∗2=0.75.
Speckle/lamina
ULJ(rij) in Equation 18
ϵLJ=1.0 and rcut=0.75×21/6 were chosen as standard values to provide the excluded volume effect. σLJ=σSp+σLa∗2=0.75.
Lamina/lamina
ULJ(rij) in Equation 18
ϵLJ=1.0 and rcut=0.5×21/6 were chosen as standard values to provide the excluded volume effect. σLJ=σLa=0.5.
* As mentioned in Appendix 1, section ‘Mapping lamina bead size to real unit’, a larger value for σLa was used here to provide a stronger excluded volume effect that prevents these particles from crossing the nucleus boundary or getting stuck in the space of the lamina particle mesh grid.
Chromosome–nuclear landmark interactions
The energy function for interactions between chromosome and nuclear landmark particles is defined asUGN=UC-La(r)+UC-Sp(r)+UC-No(r).
The functional form of the potential used to describe interactions between chromosomes and nuclear landmarks is inspired by experimental techniques that probe their contacts, such as Lamin B DamID and SON TSA-Seq. For example, the average contact probability between a chromatin bead i and the nuclear lamina can be estimated aspiL=⟨∑j∈Lac(rij)⟩,
where j indexes over the lamina particles. c(rij) is defined asc(rij)=12(1+tanh[η(rc−rij)]).
It is a switching function that approaches one for rij<rc, a threshold distance at which we set chromatin and the lamina as in contact. We chose η=4.0 to obtain a reasonable decay of contact probability between chromosomes and nuclear landmarks. rc=0.75 was selected as the average size of the lamina (1.0 σ) and chromatin (0.5 σ) particles.
For the computational model to reproduce the experimental contact probability, following the maximum entropy argument (Roux and Weare, 2013; Zhang and Wolynes, 2015), the interaction potential between chromosomes and the nuclear lamina adopts the following form:UC-La(r)=∑i∈Chr∑j∈La{12αiC-La(1+tanh[η(rc−rij)])+ULJ(rij,ϵLJ=1.0,σLJ=0.75,rcut=0.75×21/6)}.
A similar argument to the one outlined above was used to derive the interactions among chromosomes from Hi-C data, that is, Equations 12, 14, and 15 (Hetzer, 2010). The individual parameters αiC-La were optimized to ensure a match between simulated and experimental Lamin B DamID data. The second term was included to account for the excluded volume effect and prevent chromatin from moving outside the envelope.
The interaction potential between chromosomes and the speckles adopts a similar form defined asUC-Sp(r)=∑i∈Chr∑j∈SP-dP12αiC-Sp(1+tanh[η(rc−rij)]).
The second sum for j only includes dP-type speckle particles. The individual parameters αiC-Sp were optimized to ensure a match between simulated and experimental SON TSA-seq data.
Finally, the interaction potential between chromosomes and nucleoli is defined asUC-No(r)=∑i∈Chr∑j∈No12αiC-No(1+tanh[η(rc−rij)]).
Because of the low data quality for the ChIP-Seq experiments for detecting chromatin-nucleoli contacts, we did not perform systematic optimizations for αiC-No. Instead, we simply set them as αiC-No=PiNϵ, with ϵ=1.0. PiN is the probability for the chromatin bead i to contact nucleoli as quantified by the software SPIN (Wang et al., 2021).
We list all the interaction parameters between chromosomes and the nuclear landmarks in Appendix 1—table 4.
Summary of the interaction potentials between chromatin particles and nuclear landmarks and their corresponding parameter values.
See also Appendix 1, section ‘Chromosome–nuclear landmark interactions’ for further discussion and ‘Adam optimizer for chromosome–nuclear body interaction parameters’ for details on the optimization algorithm.
Potentials
Functional forms
Parameter values
Chromatin-nucleolus
UC-No(rij) in Equation 26
η=4.0 provides a smooth transition in the tanh function for contacts. rc=0.75 reflects the minimal distances between chromatin and nucleolus beads as reflected in the excluded volume potential defined in Appendix 1—table 3. The interaction strength of the ith chromatin bead αiC-No=PiN, where PiN is the probability for the chromatin bead i to contact nucleoli as quantified by the software SPIN (Wang et al., 2021).
Chromatin-speckle
UC-Sp(rij) in Equation 25
η=4.0 and rc=0.75 were similarly determined as in UC-No(rij). Value for the interaction strength of the ith chromatin bead αiC-No was obtained from optimizations against SON TSA-Seq data.
Chromatin-lamina
UC-La(rij) in Equation 24
η=4.0 and rc=0.75 were similarly determined as in UC-No(rij). Value for the interaction strength of the ith chromatin bead αiC-No was obtained from optimizations against Lamin B DamID data. The extra Lennard Jones potential was included to provide the excluded volume effect, with ϵLJ=1.0 and rcut=0.75×21/6 as standard values. σLJ=σC+σLa∗2=0.75.
* As mentioned in Appendix 1, section ‘Mapping lamina bead size to real unit’, a larger value for σLa was used here to provide a stronger excluded volume effect that prevents these particles from crossing the nucleus boundary or getting stuck in the space of the lamina particle mesh grid.
Optimization of the whole nucleus model parameters
Below, we describe the procedures used to derive model parameters.
Connecting imaging and Hi-C data with the contact function
The function f(r) defined in Equation 13 was used to determine the chromatin contact probabilities. The availability of spatial positions and Hi-C data makes possible the definition of a contact function, f(r), that converts distances into contact probabilities. In particular, we determined rc as the value at which the simulated average interchromosomal contact probability 〈f(rc)〉intersim matches the experimental value, that is,⟨f(r)⟩intersim=finterexp.
The angular brackets represent ensemble averaging, performed using the structures at 100 KB resolution reported in our previous work (Kamat et al., 2023). Matching simulation and experimental values produced rc=0.54σ≈208 nm. We note that this estimation for rc is comparable to the average bond length (0.5 σ), thus ensuring that nearest neighbor genomic regions with contact probability close to 1, that is, ⟨f(ri,i+1)⟩≈1.
Adam optimizer for chromosome interaction parameters
Mathematical expressions for the various energy terms in UGenome were designed such that their ensemble averages can be mapped onto combinations of contact frequencies measured in Hi-C. The correspondence between the energy functions and Hi-C measurements allows model parameterization with an efficient adaptive moment (Adam) algorithm (Kingma and Ba, 2014). Specifically, αideal(|i−j|),αcompt(Ti,Tj), and αinter(I,J) were tuned to satisfy the following constraints:⟨∑I∑i,j∈If(rij)δ|i−j|,s⟩=∑I∑i,j∈Ifijexpδ|i−j|,s,fors=1,⋯,n−1⟨∑i,jf(rij)δTi,T1δTj,T2⟩=∑i,jfijexpδTi,T1δTj,T2,forT1,T2∈{A,B,C}⟨∑i∈I,j∈Jf(rij)δI,K1δJ,K2⟩∑i∈I,j∈JfijexpδI,K1δJ,K2,forK1,K2∈{1,…,23}
where δTi,T1 is the Kronecker delta function with the following definition:δTi,T1={1,ifTi=T10,otherwise
The angular bracket represents the ensemble average, and fijexp is the corresponding experimental contact frequency.
During the optimization process, our aim was to minimize the disparity between experimental findings and simulated data. To achieve this, we defined the cost function as follows:L=∑i(⟨fi⟩−fiexp)2,
where the index i iterates over all the constraints defined in Equation 28.
The details of the algorithm for parameter optimization are as follows:
Starting with a set of values for αideal(|i−j|),αcompt(Ti,Tj), and αinter(I,J), we performed 50 independent 3-million-step long MD simulations to obtain an ensemble of nuclear configurations. The 500K steps of each trajectory are discarded as equilibration. We collected the configurations at every 2000 simulation steps from the rest of the simulation trajectories to compute the ensemble averages defined on the left-hand side of Equationi 13.
Check the convergence of the optimization by calculating the percentage of error defined as ∑i(⟨fi⟩−fiexp)/∑ifiexp. The summation over i includes all the average contact probabilities defined in Equation 28.
If the error is less than a tolerance value etol, the optimization has converged, and we stop the simulations. Otherwise, we update the parameters, α, using the Adam optimizer (Kingma and Ba, 2014). With the new parameter values, we return to step one and restart the iteration.
Adam optimizer for chromosome–nuclear body interaction parameters
Similar to those among chromatin particles, the interaction parameters between chromatin and nuclear landmarks were optimized with Adam’s algorithm to reproduce experimental constraints.
The constraints that we aimed to reproduce were defined as follows:⟨CiLa⟩=LAFiexp,fori=1,⋯,N⟨CiSp⟩=SAFiexp,fori=1,⋯,N,
where CiLa and CiSp measure the contacts between chromatin bead i and nuclear lamina and speckles, respectively, as defined in Equations 42 and 45. LAFi and LAFi denote the lamina and speckle association frequency for chromatin bead i as measured in Lamin B DamID and SON TSA-Seq experiments. N denotes the number of chromatin beads. We combined the constraints defined in Equation 31 with those in Equation 28 to simultaneously optimize the parameters using the iterative algorithm outlined in the previous section. We note that the interaction potential between chromatin and speckles defined in Equation 25 did not use precisely the same function as in CiSp. We chose to sum over all speckle dP particles, rather than identifying the droplets, which is difficult to do during the simulations.
Parameter optimization for nuclear body–nuclear body interactions
As much remains to be known about the organization of nuclear bodies, we designed the interaction potentials and parameters based on qualitative observations without extensive fine-tuning. For example, we used the standard Lennard–Jones potential (Equation 18) to mimic short-range interactions. The lengthscales, σLJ, in these potentials, were chosen based on a linear combination of the size of interacting particles, as discussed in section ‘Unit conversion’.
The interaction strength, ϵLJ, was set as 1.0 to be on the same order as thermal energy (kBT), when the potential was used to account for the excluded volume effect.
For attractive interactions that promote phase separation and nuclear body formation, we set ϵLJ=3.0. Smaller values failed to produce clustered nucleoli, while much larger values significantly decreased the fluidity of the resulting droplets. The same value was used for speckle dP particles and produced droplet numbers comparable to experimental observations (Figure 2—figure supplement 1).
Unit conversion
The reduced unit for length scale is noted as σ. We set the nucleus radii as 13σ. Assuming a nucleus with an average size of 5 μm, we have σ = 385 nm.
Mapping chromatin bead size to real unit
We estimated the size of the chromosome bead as 192.5 nm based on super-resolution imaging data as follows. The median radius of gyration has been shown to follow a power-law scaling as a function of domain length with an exponent of 0.3 (Boettiger et al., 2016). Assuming that the radius of a domain is proportional to the radius of gyration, we haveR∝Rg∝L0.3⇒R1MBR100KB=(1MB100KB)0.3.
We previously estimated the size of 1 MB bead as R1MB=σ=385 nm, and Equation 32 yields the size of 100 KB as R100KB=0.5σ.
Mapping lamina bead size to real unit
We chose the number and the diameter of lamina beads NLa,σLa by estimating the distance between nearest neighbor lamina beads. We found that at NLa=8000, when the lamina particles were placed on the Fibonacci grid over the spherical surface, the average nearest neighbor distance was 0.52. Therefore, we set σLa=0.5σ when considering the excluded volume effect between lamina particles. However, when modeling the excluded volume effect between lamina and chromatin, nucleolus, or speckle particles, we used σLa=1.0 (see Equation 20). A larger value provides a stronger excluded volume effect that prevents these particles from crossing the nucleus boundary or getting stuck in the space of the lamina particle mesh grid.
Mapping nucleoli bead size to real unit
The size of nucleolus particles (σNo) was estimated as follows. Since the average number of nucleoli inside a cell nucleus ranges from 2 to 5, we approximate the number of particles comprising individual droplets as NNo/3, assuming a total of three nucleoli. NNo corresponds to the total number of nucleolus particles. With a space-filling model, the ratio of the volume between one nucleolus and the cell nucleus can be estimated as(4π/3)(21/6σn/2)3(NNo/3)(4π/3)RN3=(RNoRN)3
where 21/6σn/2 denotes the effective radius of a nucleolus particle, and RN is the nucleus size. Using experimental values for the nucleolus and nucleus size (Caragine et al., 2018; Caragine et al., 2019) as RNo=0.5μm and RN=5μm, we have σNo=0.5.
Mapping speckle bead size to real unit
A similar procedure as in the previous section was used to estimate the size of speckle particles σSp. Since approximately 600 dP-type speckle particles form speckle clusters, each speckle cluster consists of around 20 particles. This estimation assumes a total of 30 speckle droplets in the system, consistent with the experimentally reported range of 20–50 speckles.
With a space-filling model, the ratio of the volume between one speckle and the cell nucleus can be estimated as(4π/3)(21/6σSp/2)3(NSp)(4π/3)RN3=(RSpRN)3
where NSp=20. Using experimental values for the speckle and nucleus size
(Handwerger et al., 2005) as RSp=0.3μm and RN=5μm, we have σSp=0.5.
Mapping the reduced time unit to real time
We determined the timescale mapping by matching the simulated diffusion coefficient of chromatin particles with experimental values. The diffusion coefficient in our simulations can be estimated from the fluctuation-dissipation theorem (Kubo, 1966) as D=kBTζ, where the friction coefficient ζ=mγ. Using the conversion from kBTm=σ2τB2, we haveD=kBTζ=kBTmγ=σ2τB2γ=10−2σ2τB.
We used the simulation setup γ−1=10−2τB when deriving the last equation.
In the meantime, from the Stokes–Einstein (SE) equation, we have D=kBT6πηr, where η is the viscosity and r=0.25σ is the radius of chromatin beads. Therefore,kBT6πηr=10−2σ2τB,
andτB=10−2σ2⋅6πηrkBT=1.5×10−2πησ3kBT.
Setting the nucleoplasmic viscosity as 1Pa⋅s produces τB≈0.65s. This mapping produced diffusion coefficients and MSD curves that match well with experimental measurements presented in Bronshtein et al., 2015, as discussed in the main text. We note that the chosen value for the nucleoplasmic viscosity indeed falls into the range of reported experimental values from 10−1Pa⋅s to 102Pa⋅s (Platani et al., 2002; Tseng et al., 2004).
Molecular dynamics simulation detailsInitial configurations for simulations
Due to the slow relaxation dynamics of whole chromosomes relative to the simulation timescale, the reported results are sensitive to the configurations used to initialize the simulations. Therefore, we designed the following protocol to prepare the initial configurations and ensure the biological relevance of simulation results.
We first created a total of 1000 configurations for the genome by sequentially generating the conformation of each one of the 46 chromosomes as follows. For a given chromosome, we start by placing the first bead at the center (origin) of the nucleus. The positions of the following beads, i, were determined from the (i−1)-th bead as ri=ri−1+0.5v. v is a normalized random vector, and 0.5 was selected as the bond length between neighboring beads. To produce globular chromosome conformations, we rejected vectors, v, that led to bead positions with distance from the center larger than 4σ. Upon creating the conformation of a chromosome i, we shift its center of mass to a value rcomi determined as follows. We first compute a mean radial distance, roi with the following equation:6σ−roiroi−2σ=Dhi−DD−Dlo,
where Di is the average value of Lamin B DamID profile for chromosome i. Dhi and Dlo represent the highest and lowest average DamID values of all chromosomes, and 6σ and 2σ represent the upper and lower bound in radial positions for chromosomes. As shown in Appendix 1—figure 2, the average Lamin B DamID profiles are highly correlated with normalized chromosome radial positions as reported by DNA MERFISH (Su et al., 2020), supporting their use as a proxy for estimating normalized chromosome radial positions. We then select rcomi as a uniformly distributed random variable within the range [roi−2σ,roi+2σ]. Without loss of generality, we randomly chose the directions for shifting all 46 chromosomes.
We further relaxed the 1000 configurations to build more realistic genome structures. Following an energy minimization process, 1-million-step MD simulations were performed starting from each configuration. Simulations were performed with the following energy function:URelax=UGenome+UC-LaEV,
where UGenome is defined as in Equation 7. UG-La is the excluded volume potential between chromosomes and lamina, that is, only the second term in Equation 24. Parameters in UGenome were from a preliminary optimization. The end configurations of the MD simulations were collected to build the final configuration ensemble (FCE).
We further computed the Pearson correlation coefficient of pairwise interchromosomal contacts between different structures in FCE (see section ‘Computing pairwise interchromosomal contact probabilities’). As shown in Figure 6—figure supplement 2A, the probability distribution of these correlation coefficients is comparable with that determined from DNA-MERFISH structures, supporting the biological relevance of the structural diversity in the constructed ensemble.
From 1000 relaxed configurations, we selected a subset of structures to initialize simulations presented in the main text. An optimization procedure was introduced for structure selection. We start this procedure by randomly select N structures to build the initial configuration ensemble (ICE). We then iteratively go through every configuration in ICE and replace with a structure from FCE that’s not already included in ICE. We then compute the Pearson correlation coefficient between new average ICE interchromosomal contact probabilities and experimental values. If the Pearson correlation coefficient is higher than the value determined from the original ICE, the new structure is accepted and the ICE is updated. Otherwise, the new structure is rejected. We stop the selection process for when the Pearson correlation coefficient stops improving.
We found that as N increases, the agreement between ICE interchromosomal contact probabilities and experimental values continue to increase (Figure 6—figure supplement 2B). We set N = 50, which produces a Pearson correlation coefficient between ICE and experimental interchromosomal contact probabilities of 0.9. Further increasing N does not significantly improve the agreement but incurs more computational cost.
It is worth noting that the outcomes of the selection procedure depend on the initial set of configurations included in ICE at the beginning. However, we found that the ICEs produced from 20 independent trials are highly correlated (Figure 6—figure supplement 2C) and all reproduce the heterogeneity in interchromosomal contacts seen in DNA MERFISH data (Figure 6—figure supplement 2D). Therefore, the selection procedure is robust and can produce biologically meaningful configurations to initialize simulations.
With the chromosome positions prepared, we randomly placed 300 nucleoli and 1600 speckle particles inside the nucleus to complete the set up of initial configurations.
Langevin dynamics simulations
We used the Langevin integrator with the damping coefficient γ−1=10 to control the temperature at T = 1.0 for simulations used for parameter optimization and for producing an ensemble of nucleus structures. Langevin dynamics simulations allow faster chromosome movements, compared to Brownian dynamics simulations, facilitating the conformational sampling. In these simulations, the lamina particles were frozen and no explicit dynamics were considered for the nuclear envelope.
Brownian dynamics simulations
We also performed Brownian dynamics simulations with damping coefficient γ−1=10−2 to control the temperature at T = 1.0. These simulations provide better approximations of the overdamped dynamics of chromatin for direct comparison with live cell imaging studies. As detailed in section ‘Unit conversion’, upon mapping the coarse-grained timescale to the physical unit, Brownian dynamics simulations produce diffusion coefficients for telomeres comparable to experimental values (see Figure 5).
Nuclear envelope deformation simulations
We performed Langevin dynamics simulations to investigate the impact of nuclear envelope deformation on genome organization. To induce a compressing force along the z-axis, we introduced a harmonic potential in the form ofUcompress=∑i=1NLak×zi2RN.
where zi is the z coordinate of the ith lamina bead, and NLa represents the total number of lamina beads. The particles in the system evolve under the combined effect of Ucompress and UNucleus defined in Equation 6.
Details of simulation data analysis
The computer simulations yield 3D coordinates of the diploid genome. However, when comparing directly with experimental data processed for the haploid genome, unless stated otherwise, we computed averages across paternal and maternal chromosomes to ascertain various genome-wide properties as listed below.
Computing simulated contact probabilities
Simulated contact probability maps were computed by averaging over chromosome configurations collected from all trajectories. For a given configuration, the contact probability between two chromatin segments (i and j) was evaluated using the contact function defined in Equation 13.
Computing the Pearson correlation coefficients between experimental and simulated contact maps
We computed the Pearson correlation coefficients (PCCs) between experimental and simulated contact maps in Figure 4A and Figure 4—figure supplement 1 asr=∑i=1n(xi−x¯)(yi−y¯)∑i=1n(xi−x¯)2∑i=1n(yi−y¯)2.
where xi and yi represent the experimental and simulated contact probabilities, and n is the total number of data points. Only non-redundant data points, that is, half of the pairwise contacts, are used in the PCC calculation.
For a given genome structure, we computed the pairwise interchromosomal contacts as follows. For every pair of chromosomes, we determined their contact probability by averaging all genomic pairs from two chromosomes using Equation 13. We then averaged over all four pairs of diploid chromosomes to compute the haploid average contacts. In total, there are C222=231 contact pairs between haploid chromosomes excluding the sex chromosomes.
Distances from nuclear bodies and association frequencies
The contacts of a chromatin bead i with the nuclear lamina were evaluated asCiLa=1Nt∑t∑j∈La12(1+tanh[η(rc−ri,j)])
with rc=0.75σ. We average over the ensemble of nuclear configurations and homologs to compute the in silico Lamin B DamID signal asDamIDi=log2(〈CiLa〉C¯La),
where the angular brackets indicate ensemble averaging. C¯La is defined as the genome wide average of 〈CiLa〉.
For chromatin-speckle contacts, we first identified the speckles formed at any given structure using the density-based spatial clustering algorithm DBSCAN (Ester et al., 1996) as implemented in the scikit library for Python (Pedregosa et al., 2011). For the identified droplets, we computed their center of mass coordinates, r→com and the radius of gyration, R. With the identified clusters, we then determined the distance from the ith chromatin bead to the sth speckle asdi,s=||r→i−r→scom||−Rs,
where ||⋅|| represents the L2 norm. We subtract the radius of the speckle cluster in the above equation to determine the distance to the droplet surface. From the list of distances to different speckles, the contact between chromatin bead i and speckles is computed asCiSp=1Ns∑s12(1+tanh[η(dc−di,s)]),
where we sum over all the Ns speckle clusters. A similar expression was used for determining the contacts between chromatin and nucleoli.
Finally, we average over the ensemble of nuclear configurations and homologs to compute the in silico SON TSA-Seq signal asTSAi=log2(⟨CiSp⟩C¯),
where the angular brackets indicate ensemble averaging. C¯Sp is defined as the genome wide average of 〈CiSp〉.
For a given chromosome i, we first determined its center of mass position denoted as Ci. Starting from the center of the nucleus, O, we extend the vector vOC to identify the intersection point with the nuclear lamina as Pi. The normalized radial position of chromosome i is then defined as ||vOCi||||vOPi||, where ||.|| represents the L2 norm.
Computing simulated chromosome radii of gyration
The radius of gyration for a chromosome is computed asRg=∑in||ri−rcom||n,
where rcom and n are the center of mass and the number of beads of the chromosome. i indices over all the chromosome beads and ri correspond to the Cartesian coordinates of bead i. ||.|| represents the L2 norm.
Computing simulated mean-square displacement
MSD for telomeres were computed as⟨r2(Δt)⟩=1Ntraj∑t=1Ntraj1Nstep∑i=1Nstep[rt((i−1)δt+Δt)−rt((i−1)δt)]2,
where Δt,δt, and Nstep represent the time interval, the time step, and the total number of steps, respectively. The summation over t corresponds to averaging over eight independent trajectories. MSDs telomeres from paternal and maternal chromosomes are separately computed and analyzed.
Details of experimental data analysisInterchromosomal contacts from DNA MERFISH data
We collected the DNA MERFISH data reported in Su et al., 2020 to construct the experimental ensemble of 5455 genome structures. For each structure, we computed the pairwise interchromosomal contacts following the procedure outlined in section ‘Computing pairwise interchromosomal contact probabilities’.
To better visualize and analyze interchromosomal contacts, we applied the Uniform Manifold Approximation and Projection (UMAP) technique as implemented in software package umap-learn (McInnes et al., 2018; Moshtagh, 2005), with default parameters to reduce the 231 haploid contacts into two dimensions. All 5455 DNA MERFISH structures were included in this analysis.
The same transformations produced from the UMAP analysis of experimental structures were applied to in silico configurations to produce results shown in Figure 6—figure supplement 2C and D .
We followed the same procedure outlined in section ‘Computing simulated normalized chromosome radial positions’ to compute the experimental values. To determine the center of the nucleus using DNA MERFISH data, we used the algorithm, minimum volume enclosing ellipsoid (MVEE) (Moshtagh, 2005), to fit an ellipsoid for each genome structure. The optimal ellipsoid defined as (x−c)TA(x−c)≡1 is obtained by optimizing min(log(det[A])) subjecting to the constraint that (xi−c)TA(xi−c)≤1. xi correspond to the list of chromatin positions determined experimentally.
Computing experimental radii of gyration
We computed the experimental radii of gyration with using the same expression as that for analyzing simulated structures (Equation 47).
The function defined in <xref ref-type="disp-formula" rid="equ13">Equation 13</xref> smoothly switches from high to low contact probabilities.
The left and right panels plot the function and its derivative as a function of the distance, r.
Correlation between average DamID profiles of individual chromosomes with their normalized radial positions.
The normalized radial positions were determined using the average value of all cells reported from DNA MERFISH data (Su et al., 2020). The correlation coefficient between the two datasets is 0.8.
Parameters of the ideal potential, <inline-formula><mml:math id="inf226"><mml:mrow><mml:msub><mml:mi>α</mml:mi><mml:mrow><mml:mtext>ideal</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>, as defined in <xref ref-type="disp-formula" rid="equ12">Equation 12</xref>.
Numerical values for αideal are included in the software’s GitHub repository.
Parameters of the inter potential, <inline-formula><mml:math id="inf228"><mml:mrow><mml:msub><mml:mi>α</mml:mi><mml:mrow><mml:mtext>inter</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>, as defined in <xref ref-type="disp-formula" rid="equ15">Equation 15</xref>.
Numerical values for αinter are included in the software’s GitHub repository.
Parameters of the chromosome-lamina potential, <inline-formula><mml:math id="inf230"><mml:mrow><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mi>C</mml:mi><mml:mo>−</mml:mo><mml:mi>L</mml:mi><mml:mi>a</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>, as defined in <xref ref-type="disp-formula" rid="equ24">Equation 24</xref>.
Numerical values for αC−La are included in the software’s GitHub repository.
Parameters of the chromosome-lamina potential, <inline-formula><mml:math id="inf232"><mml:mrow><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mi>C</mml:mi><mml:mo>−</mml:mo><mml:mi>S</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>, as defined in <xref ref-type="disp-formula" rid="equ25">Equation 25</xref>.
Numerical values for αC−Sp are included in the software’s GitHub repository.
Pearson correlation coefficients between experimental and simulated contact probabilities at various sequence separations within specific chromosomes.
For each chromosome, we first gathered a set of experimental contacts alongside a matching set of simulated ones for genomic pairs within a particular separation range. The Pearson correlation coefficient at the corresponding sequence separation was then determined using Equation 41. We limited the calculations to half of the chromosome length to ensure the availability of sufficient data.
10.7554/eLife.93223.3.sa0eLife assessmentCollepardoRosanaReviewing EditorUniversity of CambridgeUnited KingdomCompellingImportant
This important work significantly advances the field of computational modeling of genome organization through the development of OpenNucleome. The evidence supporting the tool's effectiveness is compelling as the authors compare their predictions with experimental data. It is anticipated that OpenNucleome will attract significant interest from the biophysics and genomics communities.
In this paper the authors develop a comprehensive program to investigate the organization of chromosome structures at 100 kb resolution. It is extremely well executed. The authors have thought through all aspects of the problem. The resulting software will be most useful to the community. Interestingly they capture many experimental observations accurately. I have very little complaints.
Strengths:
A lot of details are provided. The success of the method is well illustrated. Software is easily available,
Weaknesses:
The number of parameters in the energy function is very large. Any justification? Could they simply be the functions?
What would the modification be if the resolution is increased?
They should state that the extracted physical values are scale dependent. Example, viscosity.
In this work, Lao et al. develop an open-source software (OpenNucleome) for GPU-accelerated molecular dynamics simulation of the human nucleus accounting for chromatin, nucleoli, nuclear speckles, etc. Using this, the authors investigate the steady-state organization and dynamics of many of the nuclear components.
Strengths:
This is a comprehensive open-source tool to study several aspects of the nucleus, including chromatin organization, interactions with lamins and organization, and interactions with nuclear speckles and nucleoli. The model is built carefully, accounting for several important factors and optimizing the parameters iteratively to achieve experimentally known results. Authors have simulated the entire genome at 100kb resolution (which is a very good resolution to simulate and study the entire diploid genome) and predict several static quantities such as the radius of gyration and radial positions of all chromosomes, and time-dependent quantities like the mean-square displacement of important genomic regions.
Weaknesses:
One weakness of the model is that it has several parameters. Some of them are constrained by the experiments. However, the role of every parameter is not clear in the manuscript.
The authors present OpenNucleome, a computational tool for simulating the structure and dynamics of the human nucleus. The software models nuclear components, including chromosomes and nuclear bodies, and incorporates GPU acceleration for potential performance gains. The authors aim to advance the understanding of nuclear organization by providing a tool that aligns with experimental data and is accessible to the genome architecture research community.
Strengths:
OpenNucleome provides a model of the nucleus, contributing to the advancement of computational biology.
Utilizing GPU acceleration with OpenMM may offer potential performance improvements.
Weaknesses:
It could still take advantage of clearer explanations regarding the generation and usage of input and output files and compatibility with other tools.
10.7554/eLife.93223.3.sa4Author responseLaoZhuohanAuthorMassachusetts Institute of TechnologyCambridgeUnited StatesKamatKartik DAuthorMassachusetts Institute of TechnologyCambridgeUnited StatesJiangZhonglingAuthorMassachusetts Institute of TechnologyCambridgeUnited StatesZhangBinAuthorMassachusetts Institute of TechnologyCambridgeUnited States
The following is the authors’ response to the original reviews.
Reviewer 1:
Comment 0: In this paper, the authors develop a comprehensive program to investigate the organization of chromosome structures at 100 kb resolution. It is extremely well executed. The authors have thought through all aspects of the problem. The resulting software will be most useful to the community. Interestingly they capture many experimental observations accurately.
I have very few complaints.
We appreciate the reviewer’s strong assessment of the paper’s significance, novelty, and broad interest, and we thank them for the detailed suggestions and comments.
Comment 1: The number of parameters in the energy function is very large. Is there any justification for this? Could they simplify the functions?
We extend our gratitude to the reviewer for their insightful remarks. The parameters within our model can be categorized into two groups: those governing chromosome-chromosome interactions and those governing chromosome-nuclear landmark interactions.
In terms of chromosome-chromosome interactions, the parameter count is relatively modest compared to the vast amount of Hi-C data available. For instance, while the whole-genome Hi-C matrix at the 100KB resolution encompasses approximately 303212 contacts, our model comprises merely six parameters for interactions among different compartments, along with 1000 parameters for the ideal potential. As outlined in the supporting information, the ideal potential is contingent upon sequence separation, with 1000 chosen to encompass bead separations of up to 100MB. While it is theoretically plausible to reduce the number of parameters by assuming interactions cease beyond a certain sequence separation, determining this scale a priori presents a challenge.
During the parameterization process, we observed that interchromosomal contacts predicted solely based on compartmental interactions inadequately mirrored Hi-C data. Consequently, we introduced 231 additional parameters to more accurately capture interactions between distinct pairs of autosomes. These interactions may stem from factors such as non-coding RNA or proteins not explicable by simple, non-specific compartmental interactions.
Regarding parameters concerning chromosome-nuclear landmark interactions, we have 30321 parameters for speckles and 30321 for the nuclear lamina. To streamline the model, we opted to assign a unique parameter to each chromatin bead. However, it is conceivable that many chromatin beads share a similar mechanism for interacting with nuclear lamina or speckles, potentially allowing for a common parameter assignment. Nonetheless, implementing such simplification necessitates a deeper mechanistic understanding of chromosome-nuclear landmark interactions, an aspect currently lacking.
As our comprehension of nuclear organization progresses, the interpretability of parameter counts may improve, facilitating their reduction.
Comment 2: What would the modification be if the resolution is increased?
To increase the resolution of chromatin, we can in principle keep the same energy function as defined in Eq. 6. In this case, we only need to carry out further parameter optimization.
However, transitioning to higher resolutions may unveil additional features not readily apparent at 100kb. Notably, chromatin loops with an average size of 200kb or smaller have been identified in high-resolution Hi-C data [1]. To effectively capture these loops, new terms in the energy function must be incorporated. For instance, Qi and Zhang [2] employed additional contact potentials between CTCF sites to account for loop formation. Alternatively, an explicit loop-extrusion process could be introduced to model loop formation more accurately.
Comment 3: They should state that the extracted physical values are scale-dependent. For example, viscosity.
We thank the reviewer for the comment and would like to clarify that our model does not predict the viscosity. The nucleoplasmic viscosity was set as 1Pa · s to produce a diffusion coefficient that reproduces experimental value. The exact value for the nucleoplasmic viscosity is still rather controversial, and our selected value falls in the range of reported experimental values from 10−1Pa·s to 102Pa · s.
We have modified the main text to clarify the calculation of the diffusion coefficient.
“The exponent and the diffusion coefficient Dα = (27±11)×10−4μm2 · s−α both match well with the experimental values [cite], upon setting the nucleoplasmic viscosity as 1Pa · s (see Supporting Information Section: Mapping the reduced time unit to real time for more details).”
Reviewer 2:
Comment 0: In this work, Lao et al. develop an open-source software (OpenNucleome) for GPU-accelerated molecular dynamics simulation of the human nucleus accounting for chromatin, nucleoli, nuclear speckles, etc. Using this, the authors investigate the steady-state organization and dynamics of many of the nuclear components.
We thank the reviewer for summary of our work.
Comment 1: The authors could introduce a table having every parameter and the optimal parameter value used. This would greatly help the reader.
We would like to point out that model parameters are indeed provided in Table 1, 2, 3, 4, and Appendix 1 - figure 3. In these tables, we further provided details on how the parameters were determined.
Given the large number of parameters for the ideal potential (1000), we opted to plot it rather than listing out all the numbers. We added three new figures to plot the interaction parameters between chromosomes, between chromosomes and speckles, and between chromosomes and the nuclear lamina. Numerical values can be found online in the GitHub repository (parameters).
Comment 2: How many total beads are simulated? Do all beads have the same size?
The total number of the coarse-grained beads is 70542, including 60642 chromatin beads, 300 nucleolus beads, 1600 speckle beads, and 8000 nuclear lamina beads. The radius of the chromatin, nucleolus, and speckle beads is 0.25, while that of the lamina bead is 0.5. More information of the size and number of the beads are discussed in the Section: Components of the whole nucleus model.
Comment 3: In 17, what is the 3rd and 4th powers mean? What necessitates it?
The potential defined in Equation 17 follows the definition of class2 bond in the LAMMPS package (LAMMPS docs). Compared to a typical harmonic potential, the presence of higher order terms produces sharper increase in the energy at large distances (Author response image 1). This essentially reduces the flucatuation of bond length in simulations.
Comparison between the Class2 potential (defined in Eq. 17) and the Harmonic potential (K(r − r0)2, with K = 20 and r0 = 0.5).
Comment 4: What do the X-axis and Y-axis numbers in Figure 5A and 5B mean? What are their units?
We apologize for the lack of clarify in our original figure. In Fig. 5A, the X and Y axis depicts the simulated and experimental radius of gyration (Rg) for individual chromosomes, as indicated in the title of the figure. Similarly, in Fig. 5B, the X and Y axis depicts the simulated and experimental radial position of individual chromosomes.
We have converted the chromosome Rg values into reduced units and labeled the corresponding axes in the updated figure (Fig. 5). The normalized radial position is unitless and its detailed definition is included in the supporting information Section: Computing simulated normalized chromosome radial positions. We updated the figure caption to provide an explicit reference to the SI text.
Reviewer 3:
Comment 0: In this work, the authors present the development of OpenNucleome, a software for simulating the structure and dynamics of the human nucleus. It provides a detailed model of nuclear components such as chromosomes and nuclear bodies, and uses GPU acceleration for better performance based on the OpenMM package. The work also shows the model’s accuracy in comparisons with experimental data and highlights the utility in the understanding of nuclear organization. While I consider this work a good tool for the genome architecture scientific community, I have some comments and questions that could further clarify the usage of this tool and help potential users. I also have a few questions that would help to clarify the technique and results and some suggestions for references.
We appreciate the reviewer’s strong assessment of the paper’s significance, novelty, and broad interest, and we thank them for the detailed suggestions and comments.
Comment 1: Could the authors elaborate on what they consider to be ’well-established and easily adoptable modeling tools’?
By well established, we meant that models that have been extensively validated and verified, and are highly regarded by the community.
By easily adoptable, we meant that tools that are well documented and can be relatively easily learned by new groups without help from the developers.
We have revised the text to clarify our meaning.
“Despite the progress made in computational modeling, the absence of well-documented software with easy-to-follow tutorials pose a challenge.”
Comment 2: Recognizing the value of a diverse range of tools in the community, the Open-MiChroM tool is also an open-source platform built on top of OpenMM. The documentation shows various modeling approaches and many tutorials that contain different approaches besides the MiChroM energy function. How does OpenNucleome compare in terms of facilitating crossvalidation and user accessibility? The two tools seem to be complementary, which is a gain to the field. I recommend adding one or two sentences in the matter. Also, while navigating the OpenNucleome GitHub, I have not found the tutorials mentioned in the text. I also consider a barrier in the process of generating necessary input files. I would suggest expanding the tutorials and documentation to help potential users.
We thank the reviewer for the excellent comments. We agree that while many of the tutorials were included in the original package, they were not as clearly documented. We have revised them extensively to to now present:
• A tutorial for optimizing chromosome chromosome interactions.
• A tutorial for optimizing chromosome nuclear landmark interactions.
• A tutorial for building initial configurations.
• A tutorial for relaxing the initial configurations.
• A tutorial for selecting the initial configurations.
• A tutorial for setting up performing Langevin dynamics simulations.
• A tutorial for setting up performing Brownian dynamics simulations.
• A tutorial for setting up performing simulations with deformed nucleus.
• A tutorial for analyzing simulation trajectories.
• A tutorial for introducing new features to the model.
These tutorials and our well-documented and open source code (https://zhanggroup-mitchemistry.github.io/OpenNucleome) should significantly promote user accessibility. Our inclusion of python scripts for analyzing simulation trajectorials shall allow users to compute various quantities for evaluating and comparing model quality.
We added a new paragraph in the Section: Conclusions and Dicussion of the main text to compare OpenNucleosome with existing software for genome modeling.
“Our software enhances the capabilities of existing genome simulation tools [cite]. Specifically, OpenNucleome aligns with the design principles of Open-MiChroM [cite], prioritizing open-source accessibility while expanding simulation capabilities to the entire nucleus. Similar to software from the Alber lab [cite], OpenNucleome offers highresolution genome organization that faithfully reproduces a diverse range of experimental data. Furthermore, beyond static structures, OpenNucleome facilitates dynamic simulations with explicit representations of various nuclear condensates, akin to the model developed by [citet].”
Comment 3: Lastly, I would appreciate it if the authors could expand their definition of ’standardized practices’.
We apologize for any confusion caused. By ”standardized practices,” we refer to the fact that different groups often employ unique procedures for structural modeling. These procedures differ in the representation of chromosomes, the nucleus environment, and the algorithms for parameter optimization. This absence of a consensus on the optimal practices for genome modeling can be daunting for newcomers to the field.
We have revised the text to the following to avoid confusion:
“Many research groups develop their own independent software, which complicates crossvalidation and hinders the establishment of best practices for genome modeling [3–5].”
Comment 4: On page 7, the authors refer to the SI Section: Components of the whole nucleus model for further details. Could the authors provide more information on the simulated density of nuclear bodies? Is there experimental data available that details the ratio of chromatin to other nuclear components, which was used as a reference in the simulation?
We thank the reviewer for the comment. Imaging studies have provided quantitative measures about the size and number of various nuclear bodies. For example, there are 2 ∼ 5 nucleoli per nucleus, with the typical size RNo ≈ 0.5μm [6–10]. In the review by Spector and Lamond [11], the authors showed that there are 20 ∼ 50 speckles, with the typical size RSp ≈ 0.3μm. We used these numbers to guide our simulation of nuclear bodies. These information was mentioned in the Section: Chromosomes as beads on the string polymers of the supporting information.
The chromatin density is fixed by the average size of chromatin bead and the nucleus size. We chose the size of chromatin based on imaging studies as detailed in the Subsection: Mapping chromatin bead size to real unit of the supporting information. Upon fixing the bead size, the chromatin volume is determined.
Comment 5: In the statement, ’the ideal potential is only applied for beads from the same chromosome to approximate the effect of loop extrusion by Cohesin molecules for chromosome compaction and territory formation,’ it would be helpful if the authors could clarify the scope of this potential. Specifically, the code indicates that the variable ’dend ideal’ is set at 1000, suggesting an interaction along a 100Mb polymer chain at a resolution of 100Kb per bead. Could the authors elaborate on their motivation for the Cohesin complex’s activity having a significant effect over such long distances within the polymer chain?
We thank the reviewer for the insight comment. They are correct that the ideal potential was introduced to capture chromosome folding beyond the interactions between compartments, including loop extrusion. Practically, we parameterized the ideal potential such that the simulated average contact probabilities as a function of sequence separation match the experimental values. The reviewer is correct that beyond a specific value of sequence separation, one would expect the impact of loop extrusion on chromosome folding should be negligible, due to Cohesin dissociation. Correspondingly, the interaction potential should be zero at large sequence separations.
However, it is important to note that the precise separation scale cannot be known a priori. We chose 100Mb as a conservative estimation. However, as we can see from Fig. S7, our parameterization scheme indeed produced interaction parameters are mainly zero at large sequence separations. Interesting, the scale at which the potential approaches 0 (∼ 500KB), indeed agree with the estimated length traveled by Cohesin molecules before dissociation [12].
Comment 6: On pages 8 and 9, the authors discuss the optimization process. However, in reviewing the code and documentation available on the GitHub page, I could not find specific sections related to the optimization procedure described in the paper. In this context, I have a few questions: Could the authors provide more details or direct me to the parts of the documentation and the text/SI that address the optimization procedure used in their study? Additional clarification on the cost/objective function employed during the optimization process would be highly beneficial, as this was not readily apparent in the text.
We thank the reviewer for the comment. We revised the SI to include the definition of the cost function for the Adam optimizer.
“During the optimization process, our aim was to minimize the disparity between experimental findings and simulated data. To achieve this, we defined the cost function as follows:L=∑i(⟨fi⟩−fiexp)2
where the index i iterates over all the constraints defined in Eq. 28.”
The detailed optimization procedure was included in the SI as quoted below
“The details of the algorithm for parameter optimization are as follows
(1) Starting with a set of values for αideal (|i−j|),αcompt (Ti,Tj) and αinter (I,J) we performed 50 independent 3-million-step long MD simulations to obtain an ensemble of nuclear configurations. The 500K steps of each trajectory are discarded
as equilibration. We collected the configurations at every 2000 simulation steps from the rest of the simulation trajectories to compute the ensemble averages defined on the left-hand side of Eq. 13.
(2) Check the convergence of the optimization by calculating the percentage of error
defined as ∑i(⟨fi⟩−fiexp)/∑ifiexp. The summation over i includes all the average contact probabilities defined in Eq. 28.
(3) If the error is less than a tolerance value etol, the optimization has converged, and we stop the simulations. Otherwise, we update the parameters, α, using the Adam optimizer [13]. With the new parameter values, we return to step one and restart the iteration.”
Previously, the optimization code was included as part of the analysis folder. To avoid confusion and improve readability, a separate folder named optimization has been created. This folder provides the Adam optimization of chromosome-chromosome interactions (chr-chr optimization) and chromosome-nuclear landmarks interactions (chr-NL optimization).
Comment 7: What was the motivation for choosing the Adam algorithm for optimization? Adam is designed for training on stochastic objective functions. Could the authors elucidate on the ’stochastic’ aspect of their function to be optimized? Why the Adam algorithm was considered the most appropriate choice for this application?
We thank the reviewer for the comment. As defined in Eq. R1, the cost function measures the difference between the simulated constraints with corresponding experimental values. The estimation of simulation values, by averaging over an ensemble of chromosome configurations, is inherently noisy and stochastic. Exact ensemble averages can only be achieved with unlimited samples obtained from infinite long simulations.
In the past, we have used the Newton’s method for parameterization, and the detailed algorithm can be found in the SI of Ref 14. However, we found that Adam is more efficient as it is a first-order approximation method. The Newton’s method, on the other hand, is second-order approximation method and requires estimation of the Hessian matrix. When the number of constraints is large, as is in our case, the computational cost for estimating the Hessian matrix can be significant. Another advantage of the Adam algorithm lies in its adjustment of the learning rate along the optimization to further speedup convergence.
Comment 8: The authors mention that examples of setting up simulations, parameter optimization, and introducing new features are provided in the GitHub repository. However, I was unable to locate these examples. Could the authors guide me to these specific resources or consider adding them if they are not currently available?
We thank the reviewer for the comment. We have improved the GitHub repository and all the tutorials can be found using the links provided in Response to Comment 2.
Comment 9: Furthermore, the paper states that ’a configuration file that provides the position of individual particles in the PDB file format is needed to initialize the simulations.’ It would be beneficial for new users if the authors could elaborate on how this file is generated. And all other input files in general. Detailing the procedures for a new user to run their system using OpenNucleome would be helpful.
We thank the reviewer for the comment. The procedure for generating initial configurations was explained in the SI Section: Initial configurations for simulations and quoted below.
“We first created a total of 1000 configurations for the genome by sequentially generating the conformation of each one of the 46 chromosomes as follows. For a given chromosome, we start by placing the first bead at the center (origin) of the nucleus. The positions of the following beads, i, were determined from the (i − 1)-th bead as ri=ri−1+0.5v. v is a normalized random vector, and 0.5 was selected as the bond length between neighboring beads. To produce globular chromosome conformations, we rejected vectors, v, that led to bead positions with distance from the center larger than 4σ. Upon creating the conformation of a chromosome i, we shift its center of mass to a value ri com determined as follows. We first compute a mean radial distance, TO2 with the following equation6σ−roiroi−2σ=Dhi−DD−Dlo
where Di is the average value of Lamin B DamID profile for chromosome i. Dhi and Dlo represent the highest and lowest average DamID values of all chromosomes, and 6σ and 2σ represent the upper and lower bound in radial positions for chromosomes. As shown in Fig. S6, the average Lamin B DamID profiles are highly correlated with normalized chromosome radial positions as reported by DNA MERFISH [cite], supporting their use as a proxy for estimating normalized chromosome radial positions. We then select rcomi as a uniformly distributed random variable within the range [roi−2σ,roi+2σ]. Without loss of generality, we randomly chose the directions for shifting all 46 chromosomes.
We further relaxed the 1000 configurations to build more realistic genome structures. Following an energy minimization process, one-million-step molecular dynamics (MD) simulations were performed starting from each configuration. Simulations were performed with the following energy functionURelax =UGenome +UC−LaEV
where UGenome is defined as in Eq. 7. UG-La is the excluded volume potential between chromosomes and lamina, i.e, only the second term in Eq. 24. Parameters in UGenome were from a preliminary optimization. The end configurations of the MD simulations were collected to build the final configuration ensemble (FCE).”
The tutorial for preparing initial configurations can be found at this link.
Comment 10: In the section discussing the correlation between simulated and experimental contact maps, as referenced in Figure 4A and Figure S2, the authors mention a high degree of correlation. Could the authors specify the exact value of this correlation and explain the method used for its computation? Considering that comparing two Hi-C matrices involves a large number of data points, it would be helpful to know if all data points were included in this analysis.
We have updated Fig 4A and S2 to include Pearson correlation coefficients next to the contact maps. The reviewer is correct in that all the non-redundant data points of the contact maps are included in computing the correlation coefficients.
For improved clarity, we added a new section in the supporting information to detail the calculations. The section is titled Computing Pearson correlation coefficients between experimental and simulated contact maps, and the relevant text is quoted below.
“We computed the Pearson correlation coefficients (PCC) between experimental and simulated contact maps in Fig. 4A and Fig. S2 asr=∑i=1n(xi−x¯)(yi−y¯)∑i=1n(xi−x¯)2∑i=1n(yi−y¯)2
xi and yi represent the experimental and simulated contact probabilities, and n is the total number of data points. Only non-redundant data points, i.e., half of the pairwise contacts, are used in the PCC calculation.”
Comment 11: In addition, the author said: ”Moreover, the simulated and experimental average contact probabilities between pairs of chromosomes agree well, and the Pearson correlation coefficient between the two datasets reaches 0.89.” How does this correlation behave when not accounting for polymer compaction or scaling? An analysis presenting the correlation as a function of genomic distance would be interesting.
Pearson correlation coefficient between experimental and simulated contact probabilities as a function of the sequence separation within specific chromosomes.
For each chromosome, we first gathered a set of experimental contacts alongside a matching set of simulated ones for genomic pairs within a particular separation range. The Pearson correlation coefficient at the corresponding sequence separation was then determined using Equation R4. We limited the calculations to half of the chromosome length to ensure the availability of sufficient data.
We thank the reviewer for the comment. The analysis presenting the correlation as a function of genomic distance (sequence separation) for each chromosome is shown in Figure S12 and also included in the SI. While the correlation coefficients decreases at larger separation, the values around 0.5 is quite reasonable and comparable to results obtained using Open-Michrom.
We also computed the correlation of whole genome contact maps after excluding intra-chromosomal contacts. The PCC decreased from 0.89 to 0.4. Again, the correlation coefficient is quite reasonable considering that these contacts are purely predicted by the compartmental interactions and were not directly optimized.
Comment 12: I recommend using the web-server that is familiar to the authors to benchmark the OpenNucleome tool/model: ”3DGenBench: A Web-Server to Benchmark Computational Models for 3D Genomics.” Nucleic Acids Research, vol. 50, no. W1, July 2022, pp. W4-12.
We appreciate the reviewer’s suggestion. Unfortunately, the website is no longer active during the time of the revision. However, as detailed in Response to comment 11, we used the one of the popular metrics to exclude polymer compact effect and evaluate the agreement between simulation and experiments.
Comment 13: Regarding the comparison of simulation results with microscopy data from reference 34. Given their different resolutions and data point/space groupings, how do the authors align these datasets? Could the authors describe how they performed this comparison? How were the radial positions calculated in both the simulations and experiments? Since the data from reference 34 indicates a non-globular shape of the nucleus; how did this factor into the calculation of radial distributions?
We thank the reviewer for the comment and apologize for the confusion. First, the average properties we examined, including radial positions and interchromosomal contacts, were averaged over all genomic loci. Therefore, they are independent of data resolution.
Secondly, instead of calculating the absolute radial positions, which are subject to variations in nucleus shape and size, we defined the normalized radial positions. They measure the ratio between the distance from the nucleus center to the chromosome center and the distance from the nucleus center to the lamina. This definition was frequently used in prior imaging studies to measure chromosome radial positions.
The calculation of the simulated normalized radial positions and the experimental normalized radial positions are discussed in the Section: Computing simulated normalized chromosome radial positions
“For a given chromosome i, we first determined its center of mass position denoted as Ci. Starting from the center of the nucleus, O, we extend the the vector vOC to identify the intersection point with the nuclear lamina as Pi. The normalized chromosome radial position i is then defined as ‖vOCi‖‖vOPi‖, where ||·|| represents the L2 norm.
and Section: Computing experimental normalized chromosome radial positions.
“We followed the same procedure outlined in Section: Computing simulated normalized chromosome radial positions to compute the experimental values. To determine the center of the nucleus using DNA MERFISH data, we used the algorithm, minimum volume enclosing ellipsoid (MVEE)[15], to fit an ellipsoid for each genome structure. The optimal ellipsoid defined as (x−c)TA(x−c)≡1 is obtained by optimizing min(log(det[A])) subjecting to the constraint that (xi−c)TA(xi−c)≤. xi correspond to the list of chromatin positions determined experimentally.”
Comment 14: In the sentence: ”It is evident that telomeres exhibit anomalous subdiffusive motion.” I recommend mentioning the work ”Di Pierro, Michele, et al., ”Anomalous Diffusion, Spatial Coherence, and Viscoelasticity from the Energy Landscape of Human Chromosomes.” Proceedings of the National Academy of Sciences, vol. 115, no. 30, July 2018, pp. 7753-58.”.
We have revised the sentence to include the citation as follows.
“In line with previous research [cite], telomeres display anomalous subdiffusive motion. When fitted with the equation ⟨r2(Δt)⟩=DαΔtα, these trajectories yield a spectrum of α values, with a peak around 0.59.”
Comment 15: Regarding the observation that ’chromosomes appear arrested and no significant changes in their radial positions are observed over timescales comparable to the cell cycle,’ could the authors provide more details on the calculations or analyses that led to this conclusion? Specifically, information on the equilibration/relaxation time of chromosome territories relative to rearrangements within a cell cycle would be interesting.
Our conclusion here was mostly based on the time trace of normalized radial positions shown in Figure 6A of the main text. Over the timescale of an entire cell cycle (24 hours), the relatively little to no changes in the radial positions supports glassy dynamics of chromosomes. We further determined the mean squared displacement (MSD) for chromosome center of masses. As shown in the left panel of Fig. S12, the MSDs are much smaller than the average size of chromosomes (see Rg values in Fig. 5A), supporting arrested dynamics.
We further computed the auto-correlation function of the normalized chromosome radial position asC(Δt)=∑t=1N(rt−r¯)(rt+Δt−r¯)∑t=1N(rt−r¯)2
where t indexes over the trajectory frames and ¯r is the mean position. As shown in Fig. S12, the positions are not completely decorrelated over 10 hours, again supporting slow dynamics. It would be interesting to examine the relaxation timescale more closely in future studies.
Comment 16: The authors also comment on the SI ”Section: Initial configurations for simulations provides more details on preparing the 1000 initial configurations.” and related to reference 34 mentioning that ”the average Lamin B DamID profiles are highly correlated with chromosome radial positions as reported by DNA MERFISH”. How do the authors account for situations where homologous chromosomes are neighbors or have an interacting interface? Ref 34 indicates that distinguishing between these scenarios can be challenging, potentially leading to ’invalid distributions’ that are filtered out. Clarification on how such cases were handled in the simulations would be helpful.
We would like to first clarify that when comparing with experimental data, we averaged over the homologous chromosomes to obtain haploid data. We added the following text in the manuscript to emphasize this point
“Given that the majority of experimental data were analyzed for the haploid genome, we adopted a similar approach by averaging over paternal and maternal chromosomes to facilitate direct comparison. More details on data analysis can be found in the Supporting Information Section: Details of simulation data analysis.”
Furthermore, we used the processed DNA MERFISH data from the Zhuang lab, which unambiguously assigns a chromosome ID to each data point. Therefore, the issue mentioned by the reviewer is not present in the procssed data. In our simulations, since we keep track of the explicit connection between genomic segments, the trace of individual chromosomes can be determined for any configuration. Therefore, there is no ambiguity in terms of simulation data.
Comment 17: When discussing the interaction with nuclear lamina and nuclear envelop deformation, I suggest mentioning the following studies: The already cited ref 52 and ”Contessoto, Vin´ıcius G., et al. ”Interphase Chromosomes of the Aedes Aegypti Mosquito Are Liquid Crystalline and Can Sense Mechanical Cues.” Nature Communications, vol. 14, no. 1, Jan. 2023, p. 326.”
We updated the text to include the suggested reference.
“Numerous studies have highlighted the remarkable influence of nuclear shape on the positioning of chromosomes and the regulation of gene expression [16, 17].”
Comment 18: The authors state that ’Tutorials in the format of Python Scripts with extensive documentation are provided to facilitate the adoption of the model by the community.’ However, as I mentioned, the documentation appears to be limited, and the available tutorials could benefit from further expansion. I suggest that the authors consider enhancing these resources to better assist users in adopting and understanding the model.
As detailed in the Response to Comment 2, we have updated the GitHub repository to better document the included Jupyter notebooks and tutorials.
Comment 19: In the Methods section, the authors discuss using Langevin dynamics for certain simulations and Brownian dynamics for others. Could the authors provide more detailed reasoning behind the choice of these different dynamics for different aspects of the simulation? Furthermore, it would be insightful to know how the results might vary if only one of these dynamics was utilized throughout the study. Such clarification would help in understanding the implications of these methodological choices on the outcomes of the simulations.
We thank the reviewer for the comment. As detailed in the supporting information Section: Mapping the Reduced Time Unit to Real Time, the Brownian dynamics simulations provide a rigorous mapping to the biological timescale. By choosing a specific value for the nucleoplasmic viscosity, we determined the time unit in simulations as τ = 0.65s. With this time conversion, the simulated diffusion coefficients of telomeres match well with experimental values. Therefore, Brownian dynamics simulations are recommended for computing time dependent quantities and the large damping coefficients mimics the complex nuclear environment well.
On the other hand, the large damping coefficient slows down the configuration relaxation of the system significantly. For computing equilibrium statistical properties, it is useful to use a small coefficient and the Langevin integrator with large time steps to facilitate conformational relaxation.
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