Table 3 Parameters used in the 2D analysis and their meaning

E

Energy function or Hamiltonian

J

Adhesion energy (per contact length)

J

J

J

p

Cell perimeter

k _p l

2π r

6l

a

Cell area

k _a l ²

π r ²

\(\frac {3\sqrt {3}}{2}l^{2}\)

P

Membrane rest length

k _p L _p

2π R _p

6L _p

A

Target cell area

\(k_{a}{L_{a}^{2}}\)

\(\pi {R_{a}^{2}}\)

\(\frac {3\sqrt {3}}{2}{L_{a}^{2}}\)

λ _p

Perimeter constraint

λ _p

λ _p

λ _p

λ _a

Area constraint

λ _a

λ _a

λ _a

l

Basic length scale

l

r (radius)

l

k _p

Perimeter scaling factor

\(\frac {p}{l}\)

2π

6

k _a

Area scaling factor

\(\frac {a}{l^{2}}\)

π

\(\frac {3\sqrt {3}}{2}\)

L _p

Membrane rest length, using basic length scale

\(\frac {P}{k_{p}}\)

\(R_{p}=\frac {P}{2\pi }\)

\(\frac {P}{6}\)

L _a

Target cell area,using basic length scale

\(\sqrt {\frac {A}{k_{a}}}\)

\(R_{a}=\sqrt {\frac {A}{\pi }}\)

\(\sqrt {\frac {2A}{3\sqrt {3}}}\)

E

Energy function or Hamiltonian, using basic length scale

\({Jk}_{p}l+\lambda _{p}(k_{p}l-k_{p}L_{p})^{2} +\lambda _{a}(k_{a}l^{2}-k_{a}{L_{a}^{2}})^{2}\)

\(2\pi rJ+\lambda _{p}(2\pi r-2\pi R_{p})^{2} +\lambda _{a}(\pi r^{2}-\pi {R_{a}^{2}})^{2}\)

\(6lJ+\lambda _{p}(6l-6L_{p})^{2} +\lambda _{a}(\frac {3\sqrt {3}}{2}l^{2}-\frac {3\sqrt {3}}{2}{L_{a}^{2}})^{2}\)

\(\frac {\partial E}{\partial l}\)

Energy variation per length change

\(k_{p}(\gamma -2\frac {k_{a}}{k_{p}}l\Pi)\)

2π(γ−r Π)

\(6(\gamma -\frac {\sqrt {3}}{2}l\Pi)\)

γ

Enterfacial tension

J+2k _p λ _p (l−L _p )

J+4π λ _p (r−R _p )

J+12λ _p (l−L _p )

Π

Pressure

\(-2k_{a}\lambda _{a}(l^{2}-{L_{a}^{2}})\)

\(-2\pi \lambda _{a}(r^{2}-{R_{a}^{2}})\)

\(-3\sqrt {3}\lambda _{a}(l^{2}-{L_{a}^{2}})\)

τ

Length-independent component of interfacial tension

J−2k _p λ _p L _p

J−4π λ _p R _p

J−12λ _p L _p

ε

l ² at which \(\frac {\partial ^{2}E}{\partial l^{2}}=0\)

\(\frac {{L_{a}^{2}}}{3}-\frac {{k_{p}^{2}}\lambda _{p}}{6{k_{a}^{2}}\lambda _{a}}\)

\(\frac {{R_{a}^{2}}}{3}-\frac {2\lambda _{p}}{3\lambda _{a}}\)

\(\frac {{L_{a}^{2}}}{3}-\frac {8\lambda _{p}}{9\lambda _{a}}\)

α

\(\left.\frac {\partial E}{\partial l}\right |_{l=\sqrt {\epsilon },\tau =0}\)

\(-8{k_{a}^{2}}\lambda _{a}\left (\frac {{L_{a}^{2}}}{3}-\frac {{k_{p}^{2}}\lambda _{p}}{6{k_{a}^{2}}\lambda _{a}}\right)^{\frac {3}{2}}\)

\(-8\pi ^{2}\lambda _{a}\left (\frac {{R_{a}^{2}}}{3}-\frac {2\lambda _{p}}{3\lambda _{a}}\right)^{\frac {3}{2}}\)

\(-54\lambda _{a}\left (\frac {{L_{a}^{2}}}{3}-\frac {8\lambda _{p}}{9\lambda _{a}}\right)^{\frac {3}{2}}\)

β

Aggregate parameter

\(4\frac {{k_{a}^{2}}\lambda _{a}}{k_{p}}\)

2π λ _a

\(\frac {9\lambda _{a}}{2}\)

ζ

Aggregate parameter

\(\frac {64{k_{a}^{4}}{\lambda _{a}^{2}}}{{k_{p}^{2}}}\left (\frac {{k_{p}^{2}}\lambda _{p}}{6{k_{a}^{2}}\lambda _{a}}-\frac {{L_{a}^{2}}}{3}\right)^{3{\vphantom {\frac {1}{2}}}}\)

\(16\pi ^{2}{\lambda _{a}^{2}}\left (\frac {2\lambda _{p}}{3\lambda _{a}}-\frac {{R_{a}^{2}}}{3}\right)^{3}\)

\(81{\lambda _{a}^{2}}\left (\frac {8\lambda _{p}}{9\lambda _{a}}-\frac {{L_{a}^{2}}}{3}\right)^{3}\)

Bifurcation 1 (γ(l ^∗)=0)

Transition from negative to positive interfacial tension at equilibrium

τ=−2k _p λ _p L _a

τ=−4π λ _p R _a

τ=−12λ _p L _a

Bifurcation 2 (pseudo-transcritical)

Transition of l ^∗=0 from unstable to stable

τ=0

τ=0

τ=0

Bifurcation 3 (fold)

Transition from 2 to 0 non-trivial equilibria

\(\tau =\frac {8{k_{a}^{2}}\lambda _{a}}{k_{p}}\epsilon ^{\frac {3}{2}}\)

\(\tau =4\pi \lambda _{a}\epsilon ^{\frac {3}{2}}\)

\(\tau =9\lambda _{a}\epsilon ^{\frac {3}{2}}\)

\(\overline {\Lambda }\)

Normalised tension,as used in [25]

\(\frac {J}{k_{a}^{\frac {3}{2}}\lambda _{a}{L_{a}^{3}}}\)

\(\frac {J}{\pi ^{\frac {3}{2}}\lambda _{a}{R_{a}^{3}}}\)

\(\frac {2\sqrt {2}J}{9\sqrt [4]{3}\lambda _{a}{L_{a}^{3}}}\)

\(\overline {\Gamma }\)

Normalised contractility,as used in [25]

\(\frac {\lambda _{p}}{k_{a}\lambda _{a}{L_{a}^{2}}}\)

\(\frac {\lambda _{p}}{\pi \lambda _{a}{R_{a}^{2}}}\)

\(\frac {2\lambda _{p}}{3\sqrt {3}\lambda _{a}{L_{a}^{2}}}\)

Bifurcation 1 (γ(l ^∗)=0)

Transition from negative to positive interfacial tension at equilibrium

\(\overline {\Gamma }=-\frac {\sqrt {k_{a}}}{2k_{p}}\overline {\Lambda }\)

\(\overline {\Gamma }=-\frac {1}{4\sqrt {\pi }}\overline {\Lambda }\)

\(\overline {\Gamma }=-\frac {1}{4\sqrt {2}\sqrt [4]{3}}\overline {\Lambda }\)

Bifurcation 2 (pseudo-transcritical)

Transition of l ^∗=0 from unstable to stable

\(\overline {\Lambda }=0\)

\(\overline {\Lambda }=0\)

\(\overline {\Lambda }=0\)

Bifurcation 3 (fold)

Transition from 2 to 0non-trivial equilibria

\(\overline {\Gamma }=\frac {4k_{a}-3\left (k_{a}k_{p}\overline {\Lambda }\right)^{\frac {2}{3}}}{2{k_{p}^{2}}}\)

\(\overline {\Gamma }=\frac {4\pi -3\left (2\pi ^{2}\overline {\Lambda }\right)^{\frac {2}{3}}}{8\pi ^{2}}\)

\(\overline {\Gamma }=\frac {2-3\sqrt [6]{3}\left (\overline {\Lambda }\right)^{\frac {2}{3}}}{8\sqrt {3}}\)