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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 (sS)2+λ v (vV)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(2γr 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 3c l 2+τ l) 8π(a r 5+b r 3c 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)(μf(μ)), where \(f(\mu)=\sinh \left (\frac {1}{3}\text {arcsinh} \left (\mu \right)\right)\) ν=f(μ)(μf(μ)) ν=f(μ)(μf(μ))
   \(\nu '=\frac {\mu '^{\frac {4}{3}}}{2^{\frac {2}{3}}}\) \(\nu '=\frac {\mu '^{\frac {4}{3}}}{2^{\frac {2}{3}}}\) \(\nu '=\frac {\mu '^{\frac {4}{3}}}{2^{\frac {2}{3}}}\)