A couple of DNA loci colocalizes by increasing the MF-BS interaction energy E. (a) The normalized mean square distance is shown as function of for two values of the interaction energy between the Molecular Factors (MFs) and the Binding Sites (BSs). DNA loci start at a distance of the order of the containing volume size (). Then, for (orange square markers), saturates at the Brownian value (), while, at (blue circle markers), an effective attraction appears, so strong to produce DNA colocalization at equilibrium (). (b) Together with the fit function for (dashed lines), the mean fraction of MFs attached to a single DNA locus, , is plotted. does not depend on , since MFs were initially distributed at random in the lattice. Subsequently, reaches a first plateau, which is due to the MF binding to the single DNA locus and is, obviously, higher with larger . Yet, while at lower energy (, orange square markers) this first plateau is also the thermodynamic equilibrium value for ; if (blue circle markers), rises to a second plateau, just when exponentially decreases to . Actually, at this value of , at equilibrium, some additional MFs bind both DNA loci to form thermodynamic stable “bridges”(see Figure 1), which keep them together. This is illustrated in (c), where the MF fraction bind to both DNA loci (gray diamonds) and to a single DNA locus (green triangles) is plotted as function of for (the sum of these two curves gives for , see (b)). Here an MF concentration is present, with BSs within each DNA locus.
