Magnetic polymer models for epigenetics-driven chromosome folding

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.

Epigenetic regulatory layers in the 3D nucleus

Nearly 7 decades have elapsed since Francis Crick introduced the central dogma of molecular biology, as part of his ideas on protein synthesis, setting the fundamental rules of sequence information transfer from DNA to RNAs and proteins. We have since learned that gene expression is finely tuned in time and space, due to the activities of RNAs and proteins on regulatory DNA elements, and through cell-type-specific three-dimensional conformations of the genome. Here, we review major advances in genome biology and discuss a set of ideas on gene regulation and highlight how various biomolecular assemblies lead to the formation of structural and regulatory features within the nucleus, with roles in transcriptional control. We conclude by suggesting further developments that will help capture the complex, dynamic, and often spatially restricted events that govern gene expression in mammalian cells.

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.

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.

Epigenetic switching with asymmetric bridging interactions

Gene expression states are often stably sustained in cis despite massively disruptive events like DNA replication. This is achieved by on-going enzymatic activity that maintains parts of the DNA in either heterochromatic (packed) or euchromatic (free) states, each of which is stabilized by both positive feedback and bridging interactions between individual nucleosomes. In contrast to condensed matter, however, the dynamics is not only governed by equilibrium binding interactions but is also mediated by enzymes that recognize and act on specific amino acid tails of the nucleosomes. The mechanical result is that some nucleosomes can bind to one another and form tightly packed polymer configurations, whereas others remain unbound and form free, noncompact polymer configurations. Here, we study the consequences of such an asymmetric interaction pattern on the dynamics of epigenetic switching. We develop a 3D polymer model and show that traits associated with epigenetic switching, such as bistability and epigenetic memory, are permitted by such a model. We find, however, that the experimentally observed burst-like nature of some epigenetic switches is difficult to reproduce by this biologically motivated interaction. Instead, the behavior seen in experiments can be explained by introducing partial confinement, which particularly affects the euchromatic regions of the chromosome.

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.

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.

Competition between local erasure and long-range spreading of a single biochemical mark leads to epigenetic bistability

The mechanism through which cells determine their fate is intimately related to the spreading of certain biochemical (so-called epigenetic) marks along their genome. The mechanisms behind mark spreading and maintenance are not yet fully understood, and current models often assume a long-range infectionlike process for the dynamics of marks, due to the polymeric nature of the chromatin fiber which allows looping between distant sites. While these existing models typically consider antagonizing marks, here we propose a qualitatively different scenario which analyses the spreading of a single mark. We define a one-dimensional stochastic model in which mark spreading/infection occurs as a long-range process whereas mark erasure/recovery is a local process, with an enhanced rate at boundaries of infected domains. In the limiting case where our model exhibits absorbing states, we find a first-order-like transition separating the marked/infected phase from the unmarked/recovered phase. This suggests that our model, in this limit, belongs to the long-range compact directed percolation universality class. The abrupt nature of the transition is retained in a more biophysically realistic situation when a basal infection/recovery rate is introduced (thereby removing absorbing states). Close to the transition there is a range of bistability where both the marked/infected and unmarked/recovered states are metastable and long lived, which provides a possible avenue for controlling fate decisions in cells. Increasing the basal infection/recovery rate, we find a second transition between a coherent (marked or unmarked) phase, and a mixed, or random, one.

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.

Nonequilibrium Theory of Epigenomic Microphase Separation in the Cell Nucleus

Understanding the spatial organization of the genome in the cell nucleus is one of the current grand challenges in biophysics. Certain biochemical-or epigenetic-marks that are deposited along the genome are thought to play an important, yet poorly understood, role in determining genome organization and cell identity. The physical principles underlying the interplay between epigenetic dynamics and genome folding remain elusive. Here we propose and study a theory that assumes a coupling between epigenetic mark and genome densities, and which can be applied at the scale of the whole nucleus. We show that equilibrium models are not compatible with experiments and a qualitative agreement is recovered by accounting for nonequilibrium processes that can stabilize microphase separated epigenomic domains. We finally discuss the potential biophysical origin of these terms.