# Polymer Models of Chromatin Imaging Data in Single Cells
## 2 Methods  
In the studied models, chromatin is represented as a polymer chain of non-overlapping beads, subject to standard physical potentials derived from classical studies of polymer simulations [41; 42]. Specifically, adjacent beads along the polymer are tethered by an elastic FENE potential and their overlap is prevented by a repulsive Weeks-Chandler-Anderson (WCA) potential. In the loop-extrusion (LE) model, the extruding motors are modelled as harmonic springs that can extrude loops by a translocation along the polymer chain, i.e., at each simulation step the spring is updated from the bead pair (i,j) to \((\mathrm{i}-1,j+1)\). As broadly reported in the literature [18; 19; 20; 21; 22; 23], those springs cannot pass through each other, their number is fixed and they halt translocation when they collide with another extruding motor or anchor sites with opposite orientation, or they stochastically unbind from the polymer. A typical parameter choice in the simulations is to set the LE spring energy constant equal to 10 \(\mathrm{k_{B}T}\) (\(\mathrm{k_{B}}\) is the Boltzmann constant and T the temperature) and its rest length to 1.1\(\sigma\) (\(\sigma\) is the bead diameter) [18; 19; 37]. Additionally, to unveil the roles of the LE ingredients beyond its minimal implementation, we examined a more refined version where LE boundaries are chosen to best reproduce population-averaged contact data and the model anchor sites are present with a specific, finite probability in a single-polymer molecule to model cell-to-cell variation [28; 37]. The average domain structure of the model is thus reproduced from bulk data, but the ensemble of its single-molecule structures is validated against independent single-cell microscopy conformations (see below). Another key parameter controlling LE dynamics is the extruder processivity, i.e., the ratio between the extrusion velocity and the unbinding rate from the chain, whose values can range, e.g., from 80 kb up to 750 kb [19; 23; 37]. In the class of phase-separation based models, we focused on the strings and binders (SBS) model [30; 39], in which a chromosome region is modeled as a self-avoiding polymer chain where different specific types of binding sites are located for diffusing cognate molecular particles (called binders). Binders and polymer sites are subject to a Brownian motion regulated by the Langevin equation with standard parameters (i.e., friction coefficient \(\zeta=0.5\)[41]). The binders, via specific attractive interactions represented by a truncated Lennard-Jones potential [34], can bridge their cognate sites on the chain, hence guiding a micro-phase-separation of the polymer into different globules. As the number of the binders (or their energy affinity) increases above a given threshold, the system undergoes a thermodynamic phase-transition from a coil (i.e., randomly folded) to a phase-separated globule state where the polymer self-assembles into distinct, spatially segregated globules via a phase-separation mechanism [39]. Critical binder concentrations, for weak biochemical affinities, fall in the fractions of \(\mathrm{\SIUnitSymbolMicro mol/L}\) range [39], which is consistent with typical transcription factor concentrations. The genomic locations of the binding sites of the SBS model are inferred by a machine learning procedure based on the PRISMA algorithm [12]. In brief, PRISMA is a recursive Monte Carlo procedure that identifies the minimal set of binding sites to best match input bulk (e.g., ensemble-averaged Hi-C or microscopy) data, as fully detailed in [12; 29; 39]. Finally, we also discuss a model where the LE and SBS modelsare combined and act simultaneously in a single-polymer molecule (hereafter indicated as the LE+SBS model).