Glazier & Graner 1993 — full reproduction

Simulation of differential-adhesion-driven cell rearrangement (Phys. Rev. E 47, 2128). 11 studies · viva-cpm Cellular Potts engine · exact energies, initial conditions and MCS convention.

Annealing convergence Global pattern equilibration Checkerboard (negative surface tension)Cell sorting Engulfment Position reversal Partial cell sorting Dispersal - light-cell sloughing Dispersal - clusters separate Dispersal - clusters do not separate Vacancy nucleation (cavity)

Annealing convergenceannealing↑ top

J_ll=2 J_dd=2 J_ld=2 J_lM=8 J_dM=8 γ_ld=+0 γ_lM=+7 γ_dM=+7 T=0.0 λ=1.0 target: A_light=40.0 A_dark=40.0 initial condition: equilibrated_light

Finding: Bulk converges to ~5.97 within ~2 MCS with total

Fig 2 — bulk/total ⟨n⟩ and moments μ₂,μ₃,μ₄ vs MCS during T=0 annealing.

Fig 3 — cell-wall detail: unannealed, 2 MCS, 20 MCS of T=0 annealing.

Global pattern equilibrationglobal_equilibration↑ top

J_ll=2 J_dd=2 J_ld=2 J_lM=8 J_dM=8 γ_ld=+0 γ_lM=+7 γ_dM=+7 T=5.0 λ=1.0 target: A_light=40.0 A_dark=40.0 initial condition: brick_equilibrate

Finding: The tiling rounds into a 1012-cell disk with ~6.0, mu2~0.69, matching Fig 4b; statistics stable after ~400 MCS (Fig 5).

Fig 4 — rectangular tiling (0 MCS) rounding to a disk (400 MCS).

Fig 5 — total boundary length, moments, light-Medium fraction vs MCS.

Checkerboard (negative surface tension)checkerboard↑ top

J_ll=10 J_dd=8 J_ld=6 J_lM=12 J_dM=12 γ_ld=-3 γ_lM=+7 γ_dM=+8 T=10.0 λ=1.0 target: A_light=40.0 A_dark=40.0 initial condition: equilibrated_random

Finding: gamma_ld<0 yields a defect-ridden checkerboard with dominant light-dark contact and two transitions (~T=0 freeze, ~T=15 mixing), matching Figs 7-9 and Table I.

Fig 7 — checkerboard pattern at 10/100/1000/2000 MCS.

Fig 8 — total & fractional lengths, ⟨n⟩, moments.

Fig 9 — l-l, l-d, l-M interfaces vs MCS across T.

Table I — bulk moments vs temperature.

Cell sortingcell_sorting↑ top

J_ll=14 J_dd=2 J_ld=11 J_lM=16 J_dM=16 γ_ld=+3 γ_lM=+9 γ_dM=+15 T=10.0 λ=1.0 target: A_light=40.0 A_dark=40.0 initial condition: equilibrated_random

Finding: Complete cell sorting: dark cells cluster internally beneath a light monolayer; heterotypic contact falls while light-medium contact rises, reproducing Figs 12-16 and Tables II-III.

Fig 12 — cell sorting at 0→13500 MCS.

Fig 13 — total, medium-contact, cell-cell fractions, correlations.

Fig 14 — ⟨n⟩ and moments.

Fig 15 — l-d, l-M, total length vs MCS across T.

Fig 16 — total, l-d, d-M vs MCS across area constraint λ.

Table II — bulk moments vs temperature.

Table III — bulk moments vs λ.

Engulfmentengulfment↑ top

J_ll=14 J_dd=2 J_ld=11 J_lM=16 J_dM=16 γ_ld=+3 γ_lM=+9 γ_dM=+15 T=10.0 λ=1.0 target: A_light=40.0 A_dark=40.0 initial condition: half_split

Finding: Light cells engulf the dark mass and the final state matches the random-IC sort, confirming initial-condition independence (Figs 18-19).

Fig 18 — engulfment at 0/1000/5000/10000 MCS.

Fig 19 — homotypic and heterotypic fractional lengths.

Position reversalposition_reversal↑ top

J_ll=14 J_dd=2 J_ld=11 J_lM=30 J_dM=16 γ_ld=+3 γ_lM=+23 γ_dM=+15 T=10.0 λ=1.0 target: A_light=40.0 A_dark=40.0 initial condition: equilibrated_random

Finding: Dark cells sort OUTWARD, surrounding the light cells, the reversed pattern predicted for gamma_lM>gamma_dM (Figs 20-21).

Fig 20 — position reversal at 0/50/5000 MCS.

Fig 21 — fractional lengths and medium correlation.

Partial cell sortingpartial_sorting↑ top

J_ll=11 J_dd=2 J_ld=14 J_lM=16 J_dM=16 γ_ld=+7.5 γ_lM=+10.5 γ_dM=+15 T=5.0 λ=1.0 target: A_light=40.0 A_dark=40.0 initial condition: equilibrated_random

Finding: Sorting stalls partial: no light monolayer forms and clusters trap inclusions, markedly slower than normal sorting (Figs 22-24).

Fig 22 — partial cell sorting at 10/100/1000/2000 MCS.

Fig 23 — cell-cell and medium fractional lengths.

Fig 24 — partial vs normal sorting comparison.

Dispersal - light-cell sloughingdispersal_sloughing↑ top

J_ll=14 J_dd=4 J_ld=11 J_lM=2 J_dM=16 γ_ld=+2 γ_lM=-5 γ_dM=+14 T=5.0 λ=1.0 target: A_light=40.0 A_dark=40.0 initial condition: equilibrated_random

Finding: Light cells disperse as isolated cells into the medium while dark cells remain compact and round (Fig 25).

Fig 25 — light-cell sloughing/dispersal at 480 MCS.

Dispersal - clusters separatedispersal_separate↑ top

J_ll=14 J_dd=2 J_ld=35 J_lM=16 J_dM=16 γ_ld=+27 γ_lM=+9 γ_dM=+15 T=5.0 λ=1.0 target: A_light=40.0 A_dark=40.0 initial condition: equilibrated_random

Finding: Light and dark clusters separate completely; a few isolated dark cells occur (Fig 26).

Fig 26 — dispersal: clusters separate (2000 MCS).

Dispersal - clusters do not separatedispersal_no_separate↑ top

J_ll=14 J_dd=2 J_ld=29 J_lM=16 J_dM=16 γ_ld=+21 γ_lM=+9 γ_dM=+15 T=5.0 λ=1.0 target: A_light=40.0 A_dark=40.0 initial condition: equilibrated_random

Finding: Clusters remain attached, an extreme partial-sorting case just below the separation threshold (Fig 27).

Fig 27 — dispersal: clusters do not separate (2000 MCS).

Vacancy nucleation (cavity)vacancy_cavity↑ top

J_ll=14 J_dd=2 J_ld=11 J_lM=16 J_dM=16 γ_ld=+3 γ_lM=+9 γ_dM=+15 T=5.0 λ=1.0 target: A_light=20.0 A_dark=40.0 initial condition: equilibrated_random

Finding: A light-cell-lined medium cavity nucleates within the aggregate (Fig 28); reproduced approximately (the paper notes this case is delicate).

Fig 28 — vacancy nucleation / cavity (200 MCS).