(A) To investigate the possible nature of buttons, we developed a more general model in which buttons are made of nsite binding sites for specific architectural proteins. Every site is occupied by one type of architectural proteins randomly chosen among narchi types. Two buttons may interact if they have common binding sites for some architectural proteins. The energy of a configuration is given by.
with parameters defined as in Equation S1, being the number of common binding sites between buttons (chr,i) and (chr’,j), and being the strength of interaction between binding sites bound to the same architectural proteins. Homologs share the same pattern of binding sites, that is if chr and chr’ are homologous and if i is a button. For example, the case nsite = 1, narchi = 1 ( = 1 ,) corresponds to the non-specific button model described in Figure 2—figure supplement 2E. To simplify and to avoid potential issues arising from periodic boundary conditions, we focused on favorable situations for pairing where homologs are initially aligned and close to each other (~640 nm between their respective centers of mass, see inset in A) and where the monomers evolve in a closed box (rigid wall conditions). Using this model, we first investigated how specific a button should be in order to lead to pairing. We fixed nsite = 1 and varied narchi and for a button density of 60%. In these cases, narchi represents the number of button types. In (B), we plotted, for each narchi, the time evolution of the average pairing probability between homologous sites (paired if distance ) for the value that leads to maximal pairing. As a point of comparison, we also plotted the corresponding curves in the absence of buttons (black line) and for the homologous button model investigated in the main text (red line) for the value (–1.5kT) consistent with experiments at the corresponding button density. We observed that as narchi increases (as the buttons become more specific) the pairing efficiency increases. The full specific model (red) becomes well approximated by our combinatorial button model for narchi > 200, i.e., when the number of buttons for one type is less than 10 per chromosome at 10kbp resolution. This suggests that pairing needs a significant degree of specificity via a large number of button types but each button type may be present in a small amount. However, we consider it unlikely that there are enough different architectural proteins in Drosophila to reach such single-site specificity. A possibility to increase specificity from a small number of proteins is to allow more than one binding site per button (nsite > 1). The number of different buttons is then (nsite)!/[(narchi-nsite)!(nsite)!]. In (C), we fixed narchi = 50, an upper maximal number of architectural proteins in flies, and varied nsite. For each nsite, we plotted the pairing probability for the value that leads to maximal pairing. We observed that there exists an optimal number of sites (here ~5) for which the pairing is close to the one obtained with the homologous button model. This corresponds to a value that leads to a large diversity of buttons while maintaining a low number of spurious interactions between non-homologous buttons, which is of the order of nsite/narchi~0.1. (D–E) In the main text, we assumed that the size of a specific button corresponded to one monomer in our simulations (10 kbp). Recently, Viets et al., 2019 suggested that homologous TADs (average size ~100 kbp [Haddad et al., 2017]) may be the basic units of pairing. (D) To test this hypothesis, we developed a variant model that assumed that one pairing unit is composed by consecutive buttons along the genome. These buttons can interact specifically with each other and with their homologs at an interaction energy . For example, the situation depicted in the main text corresponds to . Formally this model is equivalent to the combinatorial model with = 1 and with narchi = (total number of buttons)/ where buttons of the same type were placed consecutively along the chain (instead of randomly in the combinatorial model). As in (A–C), we simulated favorable situations where homologs are initially aligned and close to each other and where the monomers evolve in a closed box, using the same Hamiltonian as the combinatorial model. (E) Time evolution of the average pairing probability between homologous sites (paired if distance ) for different values and the value that leads to maximal pairing for a fixed global button density of 60%. We observed that the level of pairing obtained for the full specific button model at an interaction strength compatible with experiments (red) can only be achieved if is smaller than ~40–50 buttons. This would correspond to pairing units of maximal size of about 750 kbp. For larger units, cis interactions (between buttons of the same unit on the same chromosome) dominate over trans interactions (between homologous buttons of the same unit) leading to less efficient pairing. TADs, that would correspond to ~ 5–15, are therefore possible pairing units compatible with pairing.