. 2015 May 12;8:8. doi: 10.1186/s13628-015-0022-x

Table 3.

Parameters used in the 2D analysis and their meaning

Parameter	Meaning	General	Circle	Hexagon
E	Energy function or Hamiltonian	J p+λ _p(p−P)²+λ _a(a−A)²
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	$k_{a} L_{a}^{2}$	$π R_{a}^{2}$	$\frac{3 \sqrt{3}}{2} L_{a}^{2}$
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	$\frac{P}{k_{p}}$	$R_{p} = \frac{P}{2 π}$	$\frac{P}{6}$
L _a	Target cell area,using basic length scale	$\sqrt{\frac{A}{k_{a}}}$	$R_{a} = \sqrt{\frac{A}{π}}$	$\sqrt{\frac{2 A}{3 \sqrt{3}}}$
E	Energy function or Hamiltonian, using basic length scale	${Jk}_{p} l + λ_{p} {(k_{p} l - k_{p} L_{p})}^{2} + λ_{a} {(k_{a} l^{2} - k_{a} L_{a}^{2})}^{2}$	$2 πrJ + λ_{p} {(2 πr - 2 π R_{p})}^{2} + λ_{a} {(π r^{2} - π R_{a}^{2})}^{2}$	$6 lJ + λ_{p} {(6 l - 6 L_{p})}^{2} + λ_{a} {(\frac{3 \sqrt{3}}{2} l^{2} - \frac{3 \sqrt{3}}{2} L_{a}^{2})}^{2}$
$\frac{∂E}{∂l}$	Energy variation per length change	$k_{p} (γ - 2 \frac{k_{a}}{k_{p}} lΠ)$	2π(γ−r Π)	$6 (γ - \frac{\sqrt{3}}{2} lΠ)$
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	$- 2 k_{a} λ_{a} (l^{2} - L_{a}^{2})$	$- 2 π λ_{a} (r^{2} - R_{a}^{2})$	$- 3 \sqrt{3} λ_{a} (l^{2} - L_{a}^{2})$
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 $\frac{\partial^{2} E}{\partial l^{2}} = 0$	$\frac{L_{a}^{2}}{3} - \frac{k_{p}^{2} λ_{p}}{6 k_{a}^{2} λ_{a}}$	$\frac{R_{a}^{2}}{3} - \frac{2 λ_{p}}{3 λ_{a}}$	$\frac{L_{a}^{2}}{3} - \frac{8 λ_{p}}{9 λ_{a}}$
α	${\frac{∂E}{∂l}\|}_{l = \sqrt{ε}, τ = 0}$	$- 8 k_{a}^{2} λ_{a} {(\frac{L_{a}^{2}}{3} - \frac{k_{p}^{2} λ_{p}}{6 k_{a}^{2} λ_{a}})}^{\frac{3}{2}}$	$- 8 π^{2} λ_{a} {(\frac{R_{a}^{2}}{3} - \frac{2 λ_{p}}{3 λ_{a}})}^{\frac{3}{2}}$	$- 54 λ_{a} {(\frac{L_{a}^{2}}{3} - \frac{8 λ_{p}}{9 λ_{a}})}^{\frac{3}{2}}$
β	Aggregate parameter	$4 \frac{k_{a}^{2} λ_{a}}{k_{p}}$	2π λ _a	$\frac{9 λ_{a}}{2}$
ζ	Aggregate parameter	$\frac{64 k_{a}^{4} λ_{a}^{2}}{k_{p}^{2}} {(\frac{k_{p}^{2} λ_{p}}{6 k_{a}^{2} λ_{a}} - \frac{L_{a}^{2}}{3})}^{3}$	$16 π^{2} λ_{a}^{2} {(\frac{2 λ_{p}}{3 λ_{a}} - \frac{R_{a}^{2}}{3})}^{3}$	$81 λ_{a}^{2} {(\frac{8 λ_{p}}{9 λ_{a}} - \frac{L_{a}^{2}}{3})}^{3}$
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	$τ = \frac{8 k_{a}^{2} λ_{a}}{k_{p}} ε^{\frac{3}{2}}$	$τ = 4 π λ_{a} ε^{\frac{3}{2}}$	$τ = 9 λ_{a} ε^{\frac{3}{2}}$
$\bar{Λ}$	Normalised tension,as used in [25]	$\frac{J}{k_{a}^{\frac{3}{2}} λ_{a} L_{a}^{3}}$	$\frac{J}{π^{\frac{3}{2}} λ_{a} R_{a}^{3}}$	$\frac{2 \sqrt{2} J}{9 \sqrt[4]{3} λ_{a} L_{a}^{3}}$
$\bar{Γ}$	Normalised contractility,as used in [25]	$\frac{λ_{p}}{k_{a} λ_{a} L_{a}^{2}}$	$\frac{λ_{p}}{π λ_{a} R_{a}^{2}}$	$\frac{2 λ_{p}}{3 \sqrt{3} λ_{a} L_{a}^{2}}$
Bifurcation 1 (γ(l ^∗)=0)	Transition from negative to positive interfacial tension at equilibrium	$\bar{Γ} = - \frac{\sqrt{k_{a}}}{2 k_{p}} \bar{Λ}$	$\bar{Γ} = - \frac{1}{4 \sqrt{π}} \bar{Λ}$	$\bar{Γ} = - \frac{1}{4 \sqrt{2} \sqrt[4]{3}} \bar{Λ}$
Bifurcation 2 (pseudo-transcritical)	Transition of l ^∗=0 from unstable to stable	$\bar{Λ} = 0$	$\bar{Λ} = 0$	$\bar{Λ} = 0$
Bifurcation 3 (fold)	Transition from 2 to 0non-trivial equilibria	$\bar{Γ} = \frac{4 k_{a} - 3 {(k_{a} k_{p} \bar{Λ})}^{\frac{2}{3}}}{2 k_{p}^{2}}$	$\bar{Γ} = \frac{4 π - 3 {(2 π^{2} \bar{Λ})}^{\frac{2}{3}}}{8 π^{2}}$	$\bar{Γ} = \frac{2 - 3 \sqrt[6]{3} {(\bar{Λ})}^{\frac{2}{3}}}{8 \sqrt{3}}$