Nonequilibrium Theory of Epigenomic Microphase Separation in the Cell Nucleus

Chromatin phase separation and nuclear shape fluctuations are correlated in a polymer model of the nucleus

Abnormal cell nuclear shapes are hallmarks of diseases, including progeria, muscular dystrophy, and many cancers. Experiments have shown that disruption of heterochromatin and increases in euchromatin lead to nuclear deformations, such as blebs and ruptures. However, the physical mechanisms through which chromatin governs nuclear shape are poorly understood. To investigate how heterochromatin and euchromatin might govern nuclear morphology, we studied chromatin microphase separation in a composite coarse-grained polymer and elastic shell simulation model. By varying chromatin density, heterochromatin composition, and heterochromatin-lamina interactions, we show how the chromatin phase organization may perturb nuclear shape. Increasing chromatin density stabilizes the lamina against large fluctuations. However, increasing heterochromatin levels or heterochromatin-lamina interactions enhances nuclear shape fluctuations by a "wetting"-like interaction. In contrast, fluctuations are insensitive to heterochromatin's internal structure. Our simulations suggest that peripheral heterochromatin accumulation could perturb nuclear morphology, while nuclear shape stabilization likely occurs through mechanisms other than chromatin microphase organization.

Phase separation in soft repulsive polymer mixtures: foundation and implication for chromatin organization

Given the wide range of length scales, the analysis of polymer systems often requires coarse-graining, for which various levels of description may be possible depending on the phenomenon under consideration. Here, we provide a super-coarse grained description, where polymers are represented as a succession of mesosopic soft beads which are allowed to overlap with others. We then investigate the phase separation behaviors in a mixture of such homopolymers based on mean-field theory, and discuss universal aspects of the miscibility phase diagram in comparison with the numerical simulation. We also discuss an extension of our analysis to mixtures involving random copolymers, which might be interesting in the context of chromatin organization in a cell nucleus.

Dynamical phase transition in models that couple chromatin folding with histone modifications

Genomic regions can acquire heritable epigenetic states through unique histone modifications, which lead to stable gene expression patterns without altering the underlying DNA sequence. However, the relationship between chromatin conformational dynamics and epigenetic stability is poorly understood. In this paper, we propose kinetic models to investigate the dynamic fluctuations of histone modifications and the spatial interactions between nucleosomes. Our model explicitly incorporates the influence of chemical modifications on the structural stability of chromatin and the contribution of chromatin contacts to the cooperative nature of chemical reactions. Through stochastic simulations and analytical theory, we have discovered distinct steady-state outcomes in different kinetic regimes, resembling a dynamical phase transition. Importantly, we have validated that the emergence of this transition, which occurs on biologically relevant timescales, is robust against variations in model design and parameters. Our findings suggest that the viscoelastic properties of chromatin and the timescale at which it transitions from a gel-like to a liquidlike state significantly impact dynamic processes that occur along the one-dimensional DNA sequence.

Bottom-up data integration in polymer models of chromatin organization

Cellular functions crucially depend on the precise execution of complex biochemical reactions taking place on the chromatin fiber in the tightly packed environment of the cell nucleus. Despite the availability of large datasets probing this process from multiple angles, bottom-up frameworks that allow the incorporation of the sequence-specific nature of biochemistry in a unified model of 3D chromatin structure remain scarce. Here, we propose Sequence-Enhanced Magnetic Polymer (SEMPER), a novel stochastic polymer model that naturally incorporates observational data about sequence-driven biochemical processes, such as binding of transcription factor proteins, in a 3D model of chromatin structure. We introduce a novel approximate Bayesian algorithm to quantify a posteriori the relative importance of various factors, including the polymeric nature of DNA, in determining chromatin epigenetic state, thus providing a transparent way to generate biological hypotheses. Although accurate prediction of contact frequencies (a problem already extensively studied in the literature) is not our main aim, as a by-product of the inference procedure and without additional input from the genome 3D structure, our model can predict with reasonable accuracy some notable and nontrivial conformational features of chromatin folding within the nucleus. Our work highlights the importance of introducing physically realistic statistical models for predicting chromatin states from epigenetic data and opens the way to a new class of more systematic approaches to interpreting epigenomic data.

Emergence of compact disordered phase in a polymer Potts model

One of the central problems in epigenetics is how epigenetic modification patterns and chromatin structure are regulated in the cell nucleus. The polymer Potts model, a recently studied model of chromatins, is introduced with an offset in the interaction energy as a parameter, and the equilibrium properties are investigated using the mean-field analysis of the lattice model and molecular dynamics simulations of the off-lattice model. The results show that in common with both models, a phase emerges, which could be called the compact-disordered phase, in which the polymer conformation is compact and the epigenetic modification pattern is disordered, depending on the offset in the interaction energy and the fraction of the modified nucleosomes.

Dynamic ParB-DNA interactions initiate and maintain a partition condensate for bacterial chromosome segregation

In most bacteria, chromosome segregation is driven by the ParABS system where the CTPase protein ParB loads at the parS site to trigger the formation of a large partition complex. Here, we present in vitro studies of the partition complex for Bacillus subtilis ParB, using single-molecule fluorescence microscopy and AFM imaging to show that transient ParB-ParB bridges are essential for forming DNA condensates. Molecular Dynamics simulations confirm that condensation occurs abruptly at a critical concentration of ParB and show that multimerization is a prerequisite for forming the partition complex. Magnetic tweezer force spectroscopy on mutant ParB proteins demonstrates that CTP hydrolysis at the N-terminal domain is essential for DNA condensation. Finally, we show that transcribing RNA polymerases can steadily traverse the ParB-DNA partition complex. These findings uncover how ParB forms a stable yet dynamic partition complex for chromosome segregation that induces DNA condensation and segregation while enabling replication and transcription.

Critical behavior of magnetic polymers on the three-dimensional Sierpiński Gasket

We present the (numerically) exact phase diagram of a magnetic polymer on the Sierpińsky gasket embedded in three dimensions using the renormalization group method. We report distinct phases of the magnetic polymer, including paramagnetic swollen, ferromagnetic swollen, paramagnetic collapsed, and ferromagnetic collapsed states. By evaluating critical exponents associated with phase transitions, we located the phase boundaries between different phases. If the model is extended to include a four-site interaction which disfavors configurations with a single spin of a given type, we find a rich variety of critical behaviors. Notably, we uncovered a phenomenon of reentrance, where the system transitions from a collapsed (paramagnetic) state to a swollen (paramagnetic) state followed by another collapse (paramagnetic) and ultimately reaching a ferromagnetic collapsed state. These findings shed new light on the complex behavior of (lattice) magnetic polymers.

Amphiphiles Formed from Synthetic DNA-Nanomotifs Mimic the Stepwise Dispersal of Transcriptional Clusters in the Cell Nucleus

Stem cells exhibit prominent clusters controlling the transcription of genes into RNA. These clusters form by a phase-separation mechanism, and their size and shape are controlled via an amphiphilic effect of transcribed genes. Here, we construct amphiphile-nanomotifs purely from DNA, and we achieve similar size and shape control for phase-separated droplets formed from fully synthetic, self-interacting DNA-nanomotifs. Increasing amphiphile concentrations induce rounding of droplets, prevent droplet fusion, and, at high concentrations, cause full dispersal of droplets. Super-resolution microscopy data obtained from zebrafish embryo stem cells reveal a comparable transition for transcriptional clusters with increasing transcription levels. Brownian dynamics and lattice simulations further confirm that the addition of amphiphilic particles is sufficient to explain the observed changes in shape and size. Our work reproduces key aspects of transcriptional cluster formation in biological cells using relatively simple DNA sequence-programmable nanostructures, opening novel ways to control the mesoscopic organization of synthetic nanomaterials.

Polymeric nature of tandemly repeated genes enhances assembly of constitutive heterochromatin in fission yeast

Motivated by our recent experiments that demonstrate that the tandemly repeated genes become heterochromatin, here we show a theory of heterochromatin assembly by taking into account the connectivity of these genes along the chromatin in the kinetic equations of small RNA production and histone methylation, which are the key biochemical reactions involved in the heterochromatin assembly. Our theory predicts that the polymeric nature of the tandemly repeated genes ensures the steady production of small RNAs because of the stable binding of nascent RNAs produced from the genes to RDRC/Dicers at the surface of nuclear membrane. This theory also predicts that the compaction of the tandemly repeated genes suppresses the production of small RNAs, consistent with our recent experiments. This theory can be extended to the small RNA-dependent gene silencing in higher organisms.

4D epigenomics: deciphering the coupling between genome folding and epigenomic regulation with biophysical modeling

Recent experimental observations suggest a strong coupling between the 3D nuclear chromosome organization and epigenomics. However, the mechanistic and functional bases of such interplay remain elusive. In this review, we describe how biophysical modeling has been instrumental in characterizing how genome folding may impact the formation of epigenomic domains and, conversely, how epigenomic marks may affect chromosome conformation. Finally, we discuss how this mutual feedback loop between chromatin organization and epigenome regulation, via the formation of physicochemical nanoreactors, may represent a key functional role of 3D compartmentalization in the assembly and maintenance of stable - but yet plastic - epigenomic landscapes.

Semianalytical solutions of Ising-like and Potts-like magnetic polymers on the Bethe lattice

We study magnetic polymers, defined as self-avoiding walks where each monomer i carries a "spin" s_{i} and interacts with its first neighbor monomers, let us say j, via a coupling constant J(s_{i},s_{j}). Ising-like [s_{i}=±1, with J(s_{i},s_{j})=ɛs_{i}s_{j}] and Potts-like [s_{i}=1,...,q, with J(s_{i},s_{j})=ɛ(s_{i})δ(s_{i},s_{j})] models are investigated. Some particular cases of these systems have recently been studied in the continuum and on regular lattices and are related to interesting applications. Here, we solve these models on Bethe lattices of branching number σ, focusing on the ferromagnetic case in zero external magnetic field. In most cases, the phase diagrams present a nonpolymerized (NP) and two polymerized phases: a paramagnetic (PP) and a ferromagnetic (FP) one. However, quite different thermodynamic properties are found depending on q in the Potts-like polymers and on whether one uses the Ising or Potts coupling in the two-state systems. For instance, for the standard Potts model [where ɛ(1)=⋯=ɛ(q)] with q=2, beyond the θ-point (where the critical and discontinuous NP-PP transition lines meet), a second tricritical point connecting a critical and a discontinuous transition line between the PP-FP phases is found in the system. A triple point where the NP-PP, NP-FP, and PP-FP first-order transition lines meet is also present in the phase diagram. For q≥3 the PP-FP transition is always discontinuous, and the scenario with the triple and θ points appears for q≤6. Interestingly, for q≥7, as well as for the Ising-like model the θ point becomes metastable and the critical NP-PP transition line ends at a critical end point (CEP), where it meets the NP-FP and PP-FP coexistence lines. Importantly, these results indicate that when q≤6 the spin ordering transition is preceded by the polymer collapse transition, whereas for q≥7 and in the Ising case these transitions happen together at the CEPs. Some interesting nonstandard Potts models are also studied, such as the lattice version of the model for epigenetic marks in the chromatin introduced by Michieletto et al. [Phys. Rev. X 6, 041047 (2016)2160-330810.1103/PhysRevX.6.041047]. In addition, the solution of the dilute Ising and dilute Potts models on the Bethe lattice are also presented here, since they are important to understand the PP-FP transitions.

Rheology and Viscoelasticity of Proteins and Nucleic Acids Condensates

Phase separation is as familiar as watching vinegar separating from oil in vinaigrette. The observation that phase separation of proteins and nucleic acids is widespread in living cells has opened an entire field of research into the biological significance and the biophysical mechanisms of phase separation and protein condensation in biology. Recent evidence indicates that certain proteins and nucleic acids condensates are not simple liquids and instead display both viscous and elastic behaviors, which in turn may have biological significance. The aim of this Perspective is to review the state-of-the-art of this quickly emerging field focusing on the material and rheological properties of protein condensates. Finally, we discuss the different techniques that can be employed to quantify the viscoelasticity of condensates and highlight potential future directions and opportunities for interdisciplinary cross-talk between chemists, physicists, and biologists.

A Liquid State Perspective on Dynamics of Chromatin Compartments

The interior of the eukaryotic cell nucleus has a crowded and heterogeneous environment packed with chromatin polymers, regulatory proteins, and RNA molecules. Chromatin polymer, assisted by epigenetic modifications, protein and RNA binders, forms multi-scale compartments which help regulate genes in response to cellular signals. Furthermore, chromatin compartments are dynamic and tend to evolve in size and composition in ways that are not fully understood. The latest super-resolution imaging experiments have revealed a much more dynamic and stochastic nature of chromatin compartments than was appreciated before. An emerging mechanism explaining chromatin compartmentalization dynamics is the phase separation of protein and nucleic acids into membraneless liquid condensates. Consequently, concepts and ideas from soft matter and polymer systems have been rapidly entering the lexicon of cell biology. In this respect, the role of computational models is crucial for establishing a rigorous and quantitative foundation for the new concepts and disentangling the complex interplay of forces that contribute to the emergent patterns of chromatin dynamics and organization. Several multi-scale models have emerged to address various aspects of chromatin dynamics, ranging from equilibrium polymer simulations, hybrid non-equilibrium simulations coupling protein binding and chromatin folding, and mesoscopic field-theoretic models. Here, we review these emerging theoretical paradigms and computational models with a particular focus on chromatin’s phase separation and liquid-like properties as a basis for nuclear organization and dynamics.

Critical behavior of magnetic polymers in two and three dimensions

We explore the critical behavior of two- and three-dimensional lattice models of polymers in dilute solution where the monomers carry a magnetic moment which interacts ferromagnetically with near-neighbor monomers. Specifically, the model explored consists of a self-avoiding walk on a square or cubic lattice with Ising spins on the visited sites. In three dimensions we confirm and extend previous numerical work, showing clearly the first-order character of both the magnetic transition and the polymer collapse, which happen together. We present results in two dimensions, where the transition is seen to be continuous. Finite-size scaling is used to extract estimates for the critical exponents and the transition temperature in the absence of an external magnetic field.

Solution of disordered microphases in the Bethe approximation

The periodic microphases that self-assemble in systems with competing short-range attractive and long-range repulsive (SALR) interactions are structurally both rich and elegant. Significant theoretical and computational efforts have thus been dedicated to untangling their properties. By contrast, disordered microphases, which are structurally just as rich but nowhere near as elegant, have not been as carefully considered. Part of the difficulty is that simple mean-field descriptions make a homogeneity assumption that washes away all of their structural features. Here, we study disordered microphases by exactly solving a SALR model on the Bethe lattice. By sidestepping the homogenization assumption, this treatment recapitulates many of the key structural regimes of disordered microphases, including particle and void cluster fluids as well as gelation. This analysis also provides physical insight into the relationship between various structural and thermal observables, between criticality and physical percolation, and between glassiness and microphase ordering.

The hierarchical packing of euchromatin domains can be described as multiplicative cascades

The genome is packed into the cell nucleus in the form of chromatin. Biochemical approaches have revealed that chromatin is packed within domains, which group into larger domains, and so forth. Such hierarchical packing is equally visible in super-resolution microscopy images of large-scale chromatin organization. While previous work has suggested that chromatin is partitioned into distinct domains via microphase separation, it is unclear how these domains organize into this hierarchical packing. A particular challenge is to find an image analysis approach that fully incorporates such hierarchical packing, so that hypothetical governing mechanisms of euchromatin packing can be compared against the results of such an analysis. Here, we obtain 3D STED super-resolution images from pluripotent zebrafish embryos labeled with improved DNA fluorescence stains, and demonstrate how the hierarchical packing of euchromatin in these images can be described as multiplicative cascades. Multiplicative cascades are an established theoretical concept to describe the placement of ever-smaller structures within bigger structures. Importantly, these cascades can generate artificial image data by applying a single rule again and again, and can be fully specified using only four parameters. Here, we show how the typical patterns of euchromatin organization are reflected in the values of these four parameters. Specifically, we can pinpoint the values required to mimic a microphase-separated state of euchromatin. We suggest that the concept of multiplicative cascades can also be applied to images of other types of chromatin. Here, cascade parameters could serve as test quantities to assess whether microphase separation or other theoretical models accurately reproduce the hierarchical packing of chromatin.

Transcription organizes euchromatin via microphase separation

In eukaryotes, DNA is packed inside the cell nucleus in the form of chromatin, which consists of DNA, proteins such as histones, and RNA. Euchromatin, which is permissive for transcription, is spatially organized into transcriptionally inactive domains interspersed with pockets of transcriptional activity. While transcription and RNA have been implicated in euchromatin organization, it remains unclear how their interplay forms and maintains transcription pockets. Here we combine theory and experiment to analyze the dynamics of euchromatin organization as pluripotent zebrafish cells exit mitosis and begin transcription. We show that accumulation of RNA induces formation of transcription pockets which displace transcriptionally inactive chromatin. We propose that the accumulating RNA recruits RNA-binding proteins that together tend to separate from transcriptionally inactive euchromatin. Full phase separation is prevented because RNA remains tethered to transcribed euchromatin through RNA polymerases. Instead, smaller scale microphases emerge that do not grow further and form the typical pattern of euchromatin organization.

Genome-in-a-Box: Building a Chromosome from the Bottom Up

Chromosome structure and dynamics are essential for life, as the way that our genomes are spatially organized within cells is crucial for gene expression, differentiation, and genome transfer to daughter cells. There is a wide variety of methods available to study chromosomes, ranging from live-cell studies to single-molecule biophysics, which we briefly review. While these technologies have yielded a wealth of data, such studies still leave a significant gap between top-down experiments on live cells and bottom-up in vitro single-molecule studies of DNA-protein interactions. Here, we introduce "genome-in-a-box" (GenBox) as an alternative in vitro approach to build and study chromosomes, which bridges this gap. The concept is to assemble a chromosome from the bottom up by taking deproteinated genome-sized DNA isolated from live cells and subsequently add purified DNA-organizing elements, followed by encapsulation in cell-sized containers using microfluidics. Grounded in the rationale of synthetic cell research, the approach would enable to experimentally study emergent effects at the global genome level that arise from the collective action of local DNA-structuring elements. We review the various DNA-structuring elements present in nature, from nucleoid-associated proteins and SMC complexes to phase separation and macromolecular crowders. Finally, we discuss how GenBox can contribute to several open questions on chromosome structure and dynamics.

Bottom-Up Meets Top-Down: The Crossroads of Multiscale Chromatin Modeling

Chromatin can be viewed as a hierarchically structured fiber that regulates gene expression. It consists of a complex network of DNA and proteins whose characteristic dynamical modes facilitate compaction and rearrangement in the cell nucleus. These modes stem from chromatin's fundamental unit, the nucleosome, and their effects are propagated across length scales. Understanding the effects of nucleosome dynamics on the chromatin fiber, primarily through post-translational modifications that occur on the histones, is of central importance to epigenetics. Within the last decade, imaging and chromosome conformation capture techniques have revealed a number of structural and statistical features of the packaged chromatin fiber at a hitherto unavailable level of resolution. Such experiments have led to increased efforts to develop polymer models that aim to reproduce, explain, and predict the contact probability scaling and density heterogeneity. At nanometer scales, available models have focused on the role of the nucleosome and epigenetic marks on local chromatin structure. At micrometer scales, existing models have sought to explain scaling laws and density heterogeneity. Less work, however, has been done to reconcile these two approaches: bottom-up and top-down models of chromatin. In this perspective, we highlight the multiscale simulation models that are driving toward an understanding of chromatin structure and function, from the nanometer to the micron scale, and we highlight areas of opportunity and some of the prospects for new frameworks that bridge these two scales. Taken together, experimental and modeling advances over the last few years have established a robust platform for the study of chromatin fiber structure and dynamics, which will be of considerable use to the chromatin community in developing an understanding of the interplay between epigenomic regulation and molecular structure.

Polymer models for the mechanisms of chromatin 3D folding: review and perspective

In this perspective paper, classical physical models for mammalian interphase chromatin folding are reviewed.

Magnetic polymer models for epigenetics-driven chromosome folding

Epigenetics is a driving force of important and ubiquitous phenomena in nature such as cell differentiation or even metamorphosis. Opposite to its widespread role, understanding the biophysical principles that allow epigenetics to control and rewire gene regulatory networks remains an open challenge. In this work we study the effects of epigenetic modifications on the spatial folding of chromosomes-and hence on the expression of the underlying genes-by mapping the problem to a class of models known as magnetic polymers. In this work we show that a first order phase transition underlies the simultaneous spreading of certain epigenetic marks and the conformational collapse of a chromosome. Further, we describe Brownian dynamics simulations of the model in which the topology of the polymer and thermal fluctuations are fully taken into account and that confirm our mean field predictions. Extending our models to allow for nonequilibrium terms yields new stable phases which qualitatively agrees with observations in vivo. Our results show that statistical mechanics techniques applied to models of magnetic polymers can be successfully exploited to rationalize the outcomes of experiments designed to probe the interplay between a dynamic epigenetic landscape and chromatin organization.