Table 6 Parameters used in the 3D analysis and their meaning

Parameter Meaning General Sphere Rhombic dodecahedron
E Energy function or Hamiltonian J s+λ s (sS)2+λ v (vV)2
J Edhesion energy (per contact length) J J J
s Cell surface k s l 2 4π r 2 $$8\sqrt {2}l^{2}$$
v Cell volume k v l 3 $$\frac {4}{3}\pi r^{3}$$ $$\frac {16}{3\sqrt {3}}l^{3}$$
S Rest surface area $$k_{s}{L_{s}^{2}}$$ $$4\pi {R_{s}^{2}}$$ $$8\sqrt {2}{L_{s}^{2}}$$
V Ttarget cell volume $$k_{v}{L_{v}^{3}}$$ $$\frac {4}{3}\pi {R_{v}^{3}}$$ $$\frac {16}{3\sqrt {3}}{L_{v}^{3}}$$
λ s Surface constraint λ s λ s λ s
λ v Volume constraint λ v λ v λ v
l Basic length scale l r (radius) l
k s Surface scaling factor $$\frac {s}{l^{2}}$$ 4π $$8\sqrt {2}$$
k v Volume scaling factor $$\frac {v}{l^{3}}$$ $$\frac {4}{3}\pi$$ $$\frac {16}{3\sqrt {3}}$$
L s Rest surface area, using basic length scale $$\sqrt {\frac {S}{k_{s}}}$$ $$R_{s}=\frac {1}{2}\sqrt {\frac {S}{\pi }}$$ $$\sqrt {\frac {S}{8\sqrt {2}}}$$
L v Target cell volume,using basic length scale $$\sqrt [3]{\frac {V}{k_{v}}}$$ $$R_{v}=\sqrt [3]{\frac {3V}{4\pi }}$$ $$\sqrt [3]{\frac {3\sqrt {3}V}{16}}$$
E Energy function or Hamiltonian, using basic length scale $${Jk}_{s}l^{2}+\lambda _{s}(k_{s}l^{2}-k_{s}{L_{s}^{2}})^{2} +\lambda _{v}(k_{v}l^{3}-k_{v}{L_{v}^{3}})^{2}$$ $$4\pi Jr^{2}+\lambda _{s}(4\pi r^{2}-4\pi {R_{s}^{2}})^{2} +\lambda _{v}(\frac {4}{3}\pi r^{3}-\frac {4}{3}\pi {R_{v}^{3}})^{2}$$ $$8\sqrt {2}Jl^{2}$$ $$+\lambda _{s}(8\sqrt {2}l^{2}-8\sqrt {2}{L_{s}^{2}})^{2} +\lambda _{v}(\frac {16}{3\sqrt {3}}l^{3}-\frac {16}{3\sqrt {3}}{L_{v}^{3}})^{2}$$
$$\frac {\partial E}{\partial l}$$ Energy variation per length change $$2k_{s}l\left (\gamma -\frac {3k_{v}}{2k_{s}}l\Pi \right)$$ 4π r(2γr Π) $$16\sqrt {2}l\left (\gamma -\frac {l}{\sqrt {6}}\Pi \right)$$
γ Interfacial tension $$J+2k_{s}\lambda _{s}(l^{2}-{L_{s}^{2}})$$ $$J+8\pi \lambda _{s}(r^{2}-{R_{s}^{2}})$$ $$J+16\sqrt {2}\lambda _{s}(l^{2}-{L_{s}^{2}})$$
Π Pressure $$-2k_{v}\lambda _{v}(l^{3}-{L_{v}^{3}})$$ $$-\frac {8}{3}\pi \lambda _{v}(r^{3}-{R_{v}^{3}})$$ $$-\frac {32}{3\sqrt {3}}\lambda _{v}(l^{3}-{L_{v}^{3}})$$
$$\frac {\partial E}{\partial l}$$ Energy variation per length change, full expansion 2k s (a l 5+b l 3c l 2+τ l) 8π(a r 5+b r 3c r 2+τ r) $$16\sqrt {2}\left (al^{5}+bl^{3}-cl^{2}+\tau l\right)$$
a Aggregate parameterin $$\frac {\partial E}{\partial l}$$ equation $$\frac {3{k_{v}^{2}}\lambda _{v}}{k_{s}}$$ $$\frac {4}{3}\pi \lambda _{v}$$ $$\frac {16}{9}\sqrt {2}\lambda _{v}$$
b Aggregate parameterin $$\frac {\partial E}{\partial l}$$ equation 2k s λ s 8π λ s $$16\sqrt {2}\lambda _{s}$$
c Aggregate parameterin $$\frac {\partial E}{\partial l}$$ equation $$\frac {3{k_{v}^{2}}\lambda _{v}{L_{v}^{3}}}{k_{s}}$$ $$\frac {4}{3}\pi \lambda _{v}{R_{v}^{3}}$$ $$\frac {16}{9}\sqrt {2}\lambda _{v}{L_{v}^{3}}$$
τ Length-independent component of interfacial tension $$J-2k_{s}\lambda _{s}{L_{s}^{2}}$$ $$J-8\pi \lambda _{s}{R_{s}^{2}}$$ $$J-16\sqrt {2}\lambda _{s}{L_{s}^{2}}$$
ϕ $$\frac {b}{6a}$$ $$\frac {{k_{s}^{2}}\lambda _{s}}{9{k_{v}^{2}}\lambda _{v}}$$ $$\frac {\lambda _{s}}{\lambda _{v}}$$ $$\frac {3\lambda _{s}}{2\lambda _{v}}$$
ψ $$\frac {c}{8a}$$ $$\frac {{L_{v}^{3}}}{8}$$ $$\frac {{R_{v}^{3}}}{8}$$ $$\frac {{L_{v}^{3}}}{8}$$
ν (when ϕ>0) $$\frac {\tau }{12a\phi ^{2}}$$ $$\frac {\left (J-2k_{s}\lambda _{s}{L_{s}^{2}}\right)9{k_{v}^{2}}\lambda _{v}}{4{k_{s}^{3}}{\lambda _{s}^{2}}}$$ $$\frac {\left (J-8\pi \lambda _{s}{R_{s}^{2}}\right)\lambda _{v}}{16{\pi \lambda _{s}^{2}}}$$ $$\frac {\left (J-16\sqrt {2}\lambda _{s}{L_{s}^{2}}\right)\lambda _{v}}{48\sqrt {2}{\lambda _{s}^{2}}}$$
μ(when ϕ>0) $$\frac {\psi }{\phi ^{\frac {3}{2}}}$$ $$\frac {27{k_{v}^{3}}\lambda _{v}^{\frac {3}{2}}{L_{v}^{3}}}{8{k_{s}^{3}}\lambda _{s}^{\frac {3}{2}}}$$ $$\frac {{R_{v}^{3}}\lambda _{v}^{\frac {3}{2}}}{8\lambda _{s}^{\frac {3}{2}}}$$ $$\frac {{L_{v}^{3}}\lambda _{v}^{\frac {3}{2}}}{6\sqrt {6}\lambda _{s}^{\frac {3}{2}}}$$
ν (when ϕ=0) $$\frac {\tau }{12a}$$ $$\frac {{Jk}_{s}}{36{k_{v}^{2}}\lambda _{v}{\vphantom {\frac {1}{2}}}}$$ $$\frac {J}{16\pi \lambda _{v}}$$ $$\frac {3J}{64\sqrt {2}\lambda _{v}}$$
μ (when ϕ=0) ψ $$\frac {{L_{v}^{3}}}{8}$$ $$\frac {{R_{v}^{3}}}{8}$$ $$\frac {{L_{v}^{3}}}{8}$$
Bifurcation 1 (γ(l )=0) Transition from negative to positive interfacial tension at equilibrium $$\nu =-\frac {9{k_{v}^{2}}\lambda _{v}{L_{v}^{2}}}{2{k_{s}^{2}}\lambda _{s}}$$ $$\nu =-\frac {\lambda _{v}{R_{v}^{2}}}{2\lambda _{s}}$$ $$\nu =-\frac {\lambda _{v}{L_{v}^{2}}}{3\lambda _{s}}$$
ν =0 ν =0 ν =0
Bifurcation 2 (pseudo-transcritical) Transition of l =0 from unstable to stable ν=0 ν=0 ν=0
ν =0 ν =0 ν =0
Bifurcation 3 (fold) Transition from 2 to 0 non-trivial equilibria ν=f(μ)(μf(μ)), where $$f(\mu)=\sinh \left (\frac {1}{3}\text {arcsinh} \left (\mu \right)\right)$$ ν=f(μ)(μf(μ)) ν=f(μ)(μf(μ))
$$\nu '=\frac {\mu '^{\frac {4}{3}}}{2^{\frac {2}{3}}}$$ $$\nu '=\frac {\mu '^{\frac {4}{3}}}{2^{\frac {2}{3}}}$$ $$\nu '=\frac {\mu '^{\frac {4}{3}}}{2^{\frac {2}{3}}}$$