Stochastic epigenetic dynamics of gene switching

Irreversibility in bacterial regulatory networks

Irreversibility, in which a transient perturbation leaves a system in a new state, is an emergent property in systems of interacting entities. This property has well-established implications in statistical physics but remains underexplored in biological networks, especially for bacteria and other prokaryotes whose regulation of gene expression occurs predominantly at the transcriptional level. Focusing on the reconstructed regulatory network of Escherichia coli, we examine network responses to transient single-gene perturbations. We predict irreversibility in numerous cases and find that the incidence of irreversibility increases with the proximity of the perturbed gene to positive circuits in the network. Comparison with experimental data suggests a connection between the predicted irreversibility to transient perturbations and the evolutionary response to permanent perturbations.

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.

Sampling rare trajectories using stochastic bridges

The numerical quantification of the statistics of rare events in stochastic processes is a challenging computational problem. We present a sampling method that constructs an ensemble of stochastic trajectories that are constrained to have fixed start and end points (so-called stochastic bridges). We then show that by carefully choosing a set of such bridges and assigning an appropriate statistical weight to each bridge, one can focus more processing power on the rare events of a target stochastic process while faithfully preserving the statistics of these rare trajectories. Further, we also compare the stochastic bridges we produce to the Wentzel-Kramers-Brillouin (WKB) optimal paths of the target process, derived in the limit of low noise. We see that the generated paths, encoding the full statistics of the process, collapse onto the WKB optimal path as the level of noise is reduced. We propose that the method can also be used to judge the accuracy of the WKB approximation at finite levels of noise.

Promoter methylation in a mixed feedback loop circadian clock model

We investigate how epigenetic modifications to clock gene promoters affect transcriptomic activity in the circadian clock. Motivated by experimental observations that link DNA methylation with the behavior of the clock, we introduce and analyze an extension of the mixed feedback loop (MFL) model of François and Hakim. We extend the original model to include an additional methylated promoter state and allow for reversible protein sequestration, an important feature for circadian applications. First, working with the general form of the MFL model, we find that the qualitative behavior of the model is dictated by the promoter state with the highest transcription rate. We then build on the work of Kim and Forger, who analyzed the stability of the mammalian circadian clock by using a reduced form of the MFL model. We present a rigorous procedure for translating between the MFL model and the reduction of Kim and Forger. We then propose a model reduction more appropriate for the study of oscillatory promoter states, making use of a fully coupled quasi-steady-state approximation rather than the standard partially uncoupled quasi-steady-state approach. Working with the novel reduced form of the model, we find substantial differences in the transcription function and show that, although methylation contributes to period control, excessive methylation can abolish rhythmicity. Altogether our results show that even in a minimal clock model, DNA methylation has a nontrivial influence on the system's ability to oscillate.

Late stage of the formation of a protein corona around nanoparticles in biofluids

In biofluids containing various proteins, nanoparticles rapidly come to be surrounded by a nanometer-thick protein layer referred to as a protein corona. The late stage of this process occurs via replacement of proteins already bound to a nanoparticle by new ones. In the available kinetic models, this process is considered to include independent acts of protein detachment and attachment. It can, however, occur also at the level of protein pairs via exchange, i.e., concerted replacement of an attached protein by a newly arrived one. I argue that the exchange channel can be more important than the conventional one. To illustrate the likely specifics of the exchange channel, I present a kinetic model focused exclusively on this channel and based on the Evans-Polanyi-type relation between the activation energies of the protein-exchange steps and the protein binding energies. The corresponding kinetics were calculated for three qualitatively different distributions of proteins in solution over binding energy (with a maximum or monotonously decreasing or increasing, respectively) and are found to be similar, with relatively rapid replacement of weakly bound proteins and slow redistribution of strongly bound proteins. The ratio of the timescales characterizing the evolution of weakly and strongly bound proteins is found to depend on the type of the binding-energy distribution.

Scalable control of developmental timetables by epigenetic switching networks

During development, progenitor cells follow timetables for differentiation that span many cell generations. These developmental timetables are robustly encoded by the embryo, yet scalably adjustable by evolution, facilitating variation in organism size and form. Epigenetic switches, involving rate-limiting activation steps at regulatory gene loci, control gene activation timing in diverse contexts, and could profoundly impact the dynamics of gene regulatory networks controlling developmental lineage specification. Here, we develop a mathematical framework to model regulatory networks with genes controlled by epigenetic switches. Using this framework, we show that such epigenetic switching networks uphold developmental timetables that robustly span many cell generations, and enable the generation of differentiated cells in precisely defined numbers and fractions. Changes to epigenetic switching networks can readily alter the timing of developmental events within a timetable, or alter the overall speed at which timetables unfold, enabling scalable control over differentiated population sizes. With their robust, yet flexibly adjustable nature, epigenetic switching networks could represent central targets on which evolution acts to manufacture diversity in organism size and form.

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.