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Table 6 Parameters used in the 3D 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 Sphere Rhombic dodecahedron
E Energy function or Hamiltonian J s+λ s (s−S)2+λ v (v−V)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(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))
   $\nu '=\frac {\mu '^{\frac {4}{3}}}{2^{\frac {2}{3}}}$ $\nu '=\frac {\mu '^{\frac {4}{3}}}{2^{\frac {2}{3}}}$ $\nu '=\frac {\mu '^{\frac {4}{3}}}{2^{\frac {2}{3}}}$
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