Exploring coupled epigenetic and genetic switches with variational methods

For eukaryotes, gene regulation is intimately linked to epigenetic modifications such as histone marks and corresponding change in nucleosome structure. In this work, we introduce a minimal kinetic model for gene regulation that accounts for the impact of histone modifications on protein production by a self-activating gene. We develop an analytical approach using the variational method, a useful approximation scheme, to study the exact master equation in a second quantized framework. Following our previous work wherein we exploited the symmetry of SU(2) operators to construct the action for an isolated epigenetic switch, we once again utilize the connection between SU(n) operators and probability distributions to arrive at a generalized variational ansatz that allows us to treat the coupled genetic and epigenetic switches. The emerging epigenetic landscape captures the delicate interplay between transcription factors and epigenetic regulation vis-à-vis histone modifications. We notice that by changing a single parameter, and quenching the noise in the epigenetic system, we are able to induce switching behaviour in the coupled system.