Field-Theoretic Treatment of Interactions

Our treatment of interactions uses a field-theoretic treatment of the densities to determine the interactions between polymer segments. Following work by Pike, et al. (Refs. [Pike2009a], [Pike2009b]), we define

The simulation has a fixed volume with sides lengths \(L_{x}\), \(L_{y}\), and \(L_{z}\). These lengths are discretize into \(M_{x}\), \(M_{y}\), and \(M_{z}\) bins of length \(\Delta_{x} = L_{x}/M_{x}\), \(\Delta_{y} = L_{y}/M_{y}\), and \(\Delta_{z} = L_{z}/M_{z}\). The bins are defined by the three indices \(i_{x}\), \(i_{y}\), and \(i_{z}\) that run from zero to \(M_{x}-1\), \(M_{y}-1\), and \(M_{z}-1\), respectively.

We consider the \(n\) th bead located at position \(\vec{r}^{(n)}\). We define a weight function \(w_{I}(\vec{r}^{(n)})\) within the \(I`th bin. The :math:`I`th index is defined to be a superindex that combines :math:`i_{x}\), \(i_{y}\), and \(i_{z}\) into a single unique index \(I= i_{x} + M_{x} i_{y} + M_{x}M_{z} i_{z}\) that runs from zero to \(M_{x}M_{y}M_{z}-1\) (total of \(M_{x}M_{y}M_{z}\) unique indices) The total weight on the \(I`th bin is given by the contributions from the three cartesian directions, `i.e.\) \(w_{I}(\vec{r}^{(n)}) = w_{i_{x}}^{(x)}(x^{(n)}) w_{i_{y}}^{(y)}(y^{(n)}) w_{i_{z}}^{(z)}(z^{(n)})\). The figure below shows a schematic of the \(x\)-direction weight function (same method for \(y\) and \(z\)). This shows a linear interpolation weighting method, consistent with Refs. [Pike2009a], [Pike2009b].

Schematic of the weight function \(w_{i_{x}}^{(x)}\) that gives the weighting of the particle in the \(i_{x}\) site in the \(x\)-direction based on a linear interpolation method

The number of epigenetic proteins (e.g. HP1) to the \(n`th site is given by :math:`N_{I}^{(\alpha)}\), where \(\alpha\) determines the type of epigenetic mark. The \(\alpha\)-protein density within the :math:`I`th bin is given by

\[\rho_{I}^{(\alpha)} = \frac{1}{v_{\mathrm{bin}}} \sum_{n=0}^{n_{b} - 1} w_{I}(\vec{r}^{(n)}) N_{I}^{(\alpha)}\]

where \(v_{\mathrm{bin}} = \Delta_{x} \Delta_{y} \Delta_{z}\) is the volume of a bin. The maximum number of epigenetic proteins bound \(N_{\mathrm{max}}^{(\alpha)}\) gives an upper bound on the number of proteins that can bind to a bead, accounting for coarse graining of a bead to represent multiple nucleosomes. For discretization of one nucleosome per bead, the maximum \(N_{\mathrm{max}}^{(\alpha)} = 2\) implies binding of a protein to the two histone tail proteins for the \(\alpha\) epigenetic mark. We define the number of \(\alpha\) marks on the \(I`th bead as :math:`M_{I}^{(\alpha)}\), which can take values from zero to \(N_{\mathrm{max}}^{(\alpha)}\).

Protein binding to a marked tail results in energy \(-\beta \epsilon_{m}\) [non-dimensionalized by \(\beta = 1/(k_{B}T)\)], and protein binding to an unmarked tail is associated with energy \(-\beta \epsilon_{u}\). The chemical potential of the \(\alpha\) protein is defined as \(\beta \mu^{(\alpha)}\). The binding of \(N_{I}^{(\alpha)}\) proteins to a bead with \(M_{I}^{(\alpha)}\) marks results in a free energy that accounts for all of the combinatoric ways of binding.