Multistability with a metastable mixed state

Role of diffusion and reaction of the constituents in spreading of histone modification marks

Cells switch genes ON or OFF by altering the state of chromatin via histone modifications at specific regulatory locations along the chromatin polymer. These gene regulation processes are carried out by a network of reactions in which the histone marks spread to neighboring regions with the help of enzymes. In the literature, this spreading has been studied as a purely kinetic, non-diffusive process considering the interactions between neighboring nucleosomes. In this work, we go beyond this framework and study the spreading of modifications using a reaction-diffusion (RD) model accounting for the diffusion of the constituents. We quantitatively segregate the modification profiles generated from kinetic and RD models. The diffusion and degradation of enzymes set a natural length scale for limiting the domain size of modification spreading, and the resulting enzyme limitation is inherent in our model. We also demonstrate the emergence of confined modification domains without the explicit requirement of a nucleation site. We explore polymer compaction effects on spreading and show that single-cell domains may differ from averaged profiles. We find that the modification profiles from our model are comparable with existing H3K9me3 data of S. pombe.

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.

Quantifying the Stability of Coupled Genetic and Epigenetic Switches With Variational Methods

The Waddington landscape provides an intuitive metaphor to view development as a ball rolling down the hill, with distinct phenotypes as basins and differentiation pathways as valleys. Since, at a molecular level, cell differentiation arises from interactions among the genes, a mathematical definition for the Waddington landscape can, in principle, be obtained by studying the gene regulatory networks. For eukaryotes, gene regulation is inextricably and intimately linked to histone modifications. However, the impact of such modifications on both landscape topography and stability of attractor states is not fully understood. In this work, we introduced a minimal kinetic model for gene regulation that combines the impact of both histone modifications and transcription factors. We further developed an approximation scheme based on variational principles to solve the corresponding master equation in a second quantized framework. By analyzing the steady-state solutions at various parameter regimes, we found that histone modification kinetics can significantly alter the behavior of a genetic network, resulting in qualitative changes in gene expression profiles. The emerging epigenetic landscape captures the delicate interplay between transcription factors and histone modifications in driving cell-fate decisions.

Quantifying epigenetic stability with minimum action paths

Chromatin can adopt multiple stable, heritable states with distinct histone modifications and varying levels of gene expression. Insight on the stability and maintenance of such epigenetic states can be gained by mathematical modeling of stochastic reaction networks for histone modifications. Analytical results for the kinetic networks are particularly valuable. Compared to computationally demanding numerical simulations, they often are more convenient at evaluating the robustness of conclusions with respect to model parameters. In this communication, we developed a second-quantization-based approach that can be used to analyze discrete stochastic models with a fixed, finite number of particles using a representation of the SU(2) algebra. We applied the approach to a kinetic model of chromatin states that captures the feedback between nucleosomes and the enzymes conferring histone modifications. Using a path-integral expression for the transition probability, we computed the epigenetic landscape that helps to identify the emergence of bistability and the most probable path connecting the two steady states. We anticipate the generalizability of the approach will make it useful for studying more complicated models that couple epigenetic modifications with transcription factors and chromatin structure.

Reliable target ligand detection by noise-induced receptor cluster formation

Intracellular reactions are intrinsically stochastic. Nonetheless, cells can reliably respond to the changing environment by sensing their target molecules sensitively and specifically, even with the existence of abundant structurally-similar non-target molecules. The mechanism of how the cells can balance and achieve such different characteristics is not yet fully understood. In this work, we demonstrate that these characteristics can be attained by a ligand-induced stochastic cluster formation of receptors via the noise-induced symmetry breaking, in which the intrinsic stochasticity works to enhance sensitivity and specificity. We also show that the noise-induced cluster formation enables cells to detect the target ligand reliably by compensating the abundant non-target ligands in the environment. The proposed mechanism may lead to a deeper understanding of a biological function of the receptor clustering and provide an alternative candidate for the reliable ligand detection to the kinetic proofreading.

Epigenomics in 3D: importance of long-range spreading and specific interactions in epigenomic maintenance

Recent progresses of genome-wide chromatin conformation capture techniques have shown that the genome is segmented into hierarchically organized spatial compartments. However, whether this non-random 3D organization only reflects or indeed contributes—and how—to the regulation of genome function remain to be elucidated. The observation in many species that 3D domains correlate strongly with the 1D epigenomic information along the genome suggests a dynamic coupling between chromatin organization and epigenetic regulation. Here, we posit that chromosome folding may contribute to the maintenance of a robust epigenomic identity via the formation of spatial compartments like topologically-associating domains. Using a novel theoretical framework, the living chromatin model, we show that 3D compartmentalization leads to the spatial colocalization of epigenome regulators, thus increasing their local concentration and enhancing their ability to spread an epigenomic signal at long-range. Interestingly, we find that the presence of 1D insulator elements, like CTCF, may contribute greatly to the stable maintenance of adjacent antagonistic epigenomic domains. We discuss the generic implications of our findings in the light of various biological contexts from yeast to human. Our approach provides a modular framework to improve our understanding and to investigate in details the coupling between the structure and function of chromatin.

Nonlinear dynamics with Hopf bifurcations by targeted mutation in the system of rock-paper-scissors metaphor

The role of mutation, which is an error process in gene evolution, in systems of cyclically competing species has been studied from various perspectives, and it is regarded as one of the key factors for promoting coexistence of all species. In addition to naturally occurring mutations, many experiments in genetic engineering have involved targeted mutation techniques such as recombination between DNA and somatic cell sequences and have studied genetic modifications through loss or augmentation of cell functions. In this paper, we investigate nonlinear dynamics with targeted mutation in cyclically competing species. In different ways to classic approaches of mutation in cyclic games, we assume that mutation may occur in targeted individuals who have been removed from intraspecific competition. By investigating each scenario depending on the number of objects for targeted mutation analytically and numerically, we found that targeted mutation can lead to persistent coexistence of all species. In addition, under the specific condition of targeted mutation, we found that targeted mutation can lead to emergences of bistable states for species survival. Through the linear stability analysis of rate equations, we found that those phenomena are accompanied by Hopf bifurcation which is supercritical. Our findings may provide more global perspectives on understanding underlying mechanisms to control biodiversity in ecological/biological sciences, and evidences with mathematical foundations to resolve social dilemmas such as a turnover of group members by resigning with intragroup conflicts in social sciences.

Changes in political party systems arising from conflict and transfer among political parties

Conflict that arises between two groups of different paradigms is an inevitable phenomenon, and a representative example of the conflict among different groups is a conflict phenomenon caused by competition among political parties. In this paper, we study the dynamical behavior of a political party system. Considering three major political parties, we investigate how political party systems can be changed by employing a mathematical model. By considering the transfer mechanism of recruitment as well as conflict of competition between political parties, we found that all parties are likely to coexist when both the competition and transfer between the parties are weak, or if either mechanism can occur at a relatively low level. Otherwise, a political party system is changed to a single-party system. In addition, we found that when a party system was changed into a single-party system, it appeared to be either bistable or multistable, and has been elucidate by linear stability analysis. Our results may provide insights to understand mechanisms how political party systems can be changed by conflict and transfer.

Basin stability measure of different steady states in coupled oscillators

In this report, we investigate the stabilization of saddle fixed points in coupled oscillators where individual oscillators exhibit the saddle fixed points. The coupled oscillators may have two structurally different types of suppressed states, namely amplitude death and oscillation death. The stabilization of saddle equilibrium point refers to the amplitude death state where oscillations are ceased and all the oscillators converge to the single stable steady state via inverse pitchfork bifurcation. Due to multistability features of oscillation death states, linear stability theory fails to analyze the stability of such states analytically, so we quantify all the states by basin stability measurement which is an universal nonlocal nonlinear concept and it interplays with the volume of basins of attractions. We also observe multi-clustered oscillation death states in a random network and measure them using basin stability framework. To explore such phenomena we choose a network of coupled Duffing-Holmes and Lorenz oscillators which are interacting through mean-field coupling. We investigate how basin stability for different steady states depends on mean-field density and coupling strength. We also analytically derive stability conditions for different steady states and confirm by rigorous bifurcation analysis.

Models of life: epigenetics, diversity and cycles

This review emphasizes aspects of biology that can be understood through repeated applications of simple causal rules. The selected topics include perspectives on gene regulation, phage lambda development, epigenetics, microbial ecology, as well as model approaches to diversity and to punctuated equilibrium in evolution. Two outstanding features are repeatedly described. One is the minimal number of rules to sustain specific states of complex systems for a long time. The other is the collapse of such states and the subsequent dynamical cycle of situations that restitute the system to a potentially new metastable state.

Feedbacks, Bifurcations, and Cell Fate Decision-Making in the p53 System

The p53 transcription factor is a regulator of key cellular processes including DNA repair, cell cycle arrest, and apoptosis. In this theoretical study, we investigate how the complex circuitry of the p53 network allows for stochastic yet unambiguous cell fate decision-making. The proposed Markov chain model consists of the regulatory core and two subordinated bistable modules responsible for cell cycle arrest and apoptosis. The regulatory core is controlled by two negative feedback loops (regulated by Mdm2 and Wip1) responsible for oscillations, and two antagonistic positive feedback loops (regulated by phosphatases Wip1 and PTEN) responsible for bistability. By means of bifurcation analysis of the deterministic approximation we capture the recurrent solutions (i.e., steady states and limit cycles) that delineate temporal responses of the stochastic system. Direct switching from the limit-cycle oscillations to the “apoptotic” steady state is enabled by the existence of a subcritical Neimark—Sacker bifurcation in which the limit cycle loses its stability by merging with an unstable invariant torus. Our analysis provides an explanation why cancer cell lines known to have vastly diverse expression levels of Wip1 and PTEN exhibit a broad spectrum of responses to DNA damage: from a fast transition to a high level of p53 killer (a p53 phosphoform which promotes commitment to apoptosis) in cells characterized by high PTEN and low Wip1 levels to long-lasting p53 level oscillations in cells having PTEN promoter methylated (as in, e.g., MCF-7 cell line).

Cancers are diseases of signaling networks. Transcription factor p53 is a pivotal node of a network that integrates a variety of stress signals and governs critical processes of DNA repair, cell cycle arrest, and apoptosis. Somewhat paradoxically, despite the fact that carcinogenesis is prevalently caused by p53 network malfunction, most of our knowledge about p53 signaling is based on cancer or immortalized cell lines. In this paper, we construct a mathematical model of intact p53 network to understand dynamics of non-cancerous cells and then dynamics of cancerous cells by introducing perturbations to the regulatory system. Cell fate decisions are enabled by the presence of interlinked feedback loops which give rise to a rich repertoire of behaviors. We explain and analyze by means of numerical simulations how the dynamical structure of the regulatory system allows for generating unambiguous single-cell fate decisions, also in the case when the cell population splits into an apoptotic and a surviving subpopulation. Perturbation analysis provides an explanation why cancer cell lines known to have vastly diverse expression levels of p53 regulators can exhibit a broad spectrum of responses to DNA damage.