Model potentials

As briefly described in the main text, we employed coarse-grained molecular dynamics (MD) simulations with the Langevin thermostat. Specifically, we employed a velocity-Verlet MD integrator with a fixed time step of 0.01. In our MD simulations, we modeled chromosomes as chains consisting of spherical monomers and linearly connecting springs, and modeled condensins as point particles.

Each chromosome consists of N monomers with diameter σ = 1, mass m = 1, and friction γ = 1. The potential for chromosomes is described as

where U_excl and U_spr represent the volume exclusion among monomers and spring interactions between neighboring monomers in the chain, respectively.

The excluded volume interaction U_excl is described by a Weeks-Chandler-Andersen (WCA) potential, which corresponds to the repulsive part of the Lennard-Jones potential:

for ri,j<26σ and 0 elsewhere, where r_{i, j} denotes the distance between the centers of the i-th and j-th monomers. At r_{i, j} = σ, the interaction energy is ϵ = 1k_BT, where k_B and T are the Boltzmann constant and the temperature, respectively. To avoid numerical instability, we introduce a cut-off at a maximum energy of the potential ϵ_cut = 1000k_BT.

The spring interaction U_spr between neighboring monomers in a chain is described by the harmonic potential:

where r_{i, i+1} is the distance between the i-th and (i + 1)-th monomer centers, d_B is the natural length of the springs, and ϵ_spr is the spring coefficient. We chose the parameters d_B = σ and ϵ_spr = ϵ_cut. The spring has no excluded volume (phantom spring). Thus, spring-spring and spring-monomer can pass through each other, which is mediated by the strand-passage activity of topoisomerase II. Note that actual frequency of the strand passage was low due to the excluded volume of the monomers connected by springs (see S1 Appendix).

The potential for condensins is described as

where U_loop and U_attr represent two functions of the condensins, chromatin loop-holding and inter-condensin attractions, respectively.

With the loop-holding potential U_loop, a condensin interacts with two defined chromatin monomers to make a chromatin loop. The potential is described by the harmonic potential:

where r˜i,± is the distance between the i-th condensin and its two interacting monomers, and M is the number of condensins that interact with one chromosome by the loop-holding potential; in other words, the chromosome has M loops. Since we consider the consecutive loop structures in a chromosome by condensins, the length of the chromatin loop is L = N/M, and the i-th condensin bonds to the (i − 1)L-th and the (iL − 1)-th chromatin monomers to make a loop with length L, where the order of condensins is aligned with the order of chromatin monomers. F_loop is the strength of the interaction.

The inter-condensin attraction potential U_attr is described by the harmonic potential:

for r¯i,j<Δ and 0 elsewhere, where r¯i,j denotes the distance between the centers of the i-th and j-th condensins. Δ, M′, and F_cond are the threshold distance, total number of condensins (M′ = M for one-chromosome simulations and M′ = 2M for two-chromosome simulations), and the strength of attractions, respectively.