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

E

Energy function or Hamiltonian

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(2γ−r 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 ³−c l ²+τ l)

8π(a r ⁵+b r ³−c 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)(μ−f(μ)), where \(f(\mu)=\sinh \left (\frac {1}{3}\text {arcsinh} \left (\mu \right)\right)\)

ν=f(μ)(μ−f(μ))

ν=f(μ)(μ−f(μ))

\(\nu '=\frac {\mu '^{\frac {4}{3}}}{2^{\frac {2}{3}}}\)

\(\nu '=\frac {\mu '^{\frac {4}{3}}}{2^{\frac {2}{3}}}\)

\(\nu '=\frac {\mu '^{\frac {4}{3}}}{2^{\frac {2}{3}}}\)