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

From: The biophysical nature of cells: potential cell behaviours revealed by analytical and computational studies of cell surface mechanics

Parameter Meaning General Circle Hexagon
E Energy function or Hamiltonian J p+λ p (pP)2+λ a (aA)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 \(k_{a}{L_{a}^{2}}\) \(\pi {R_{a}^{2}}\) \(\frac {3\sqrt {3}}{2}{L_{a}^{2}}\)
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(γr Π) \(6(\gamma -\frac {\sqrt {3}}{2}l\Pi)\)
γ Enterfacial tension J+2k p λ p (lL p ) J+4π λ p (rR p ) J+12λ p (lL 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 \(\overline {\Gamma }=\frac {4k_{a}-3\left (k_{a}k_{p}\overline {\Lambda }\right)^{\frac {2}{3}}}{2{k_{p}^{2}}}\) \(\overline {\Gamma }=\frac {4\pi -3\left (2\pi ^{2}\overline {\Lambda }\right)^{\frac {2}{3}}}{8\pi ^{2}}\) \(\overline {\Gamma }=\frac {2-3\sqrt [6]{3}\left (\overline {\Lambda }\right)^{\frac {2}{3}}}{8\sqrt {3}}\)