|
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))
|
| | |
\(\nu '=\frac {\mu '^{\frac {4}{3}}}{2^{\frac {2}{3}}}\)
|
\(\nu '=\frac {\mu '^{\frac {4}{3}}}{2^{\frac {2}{3}}}\)
|
\(\nu '=\frac {\mu '^{\frac {4}{3}}}{2^{\frac {2}{3}}}\)
|
