this first nontrivial assumption. Moreover, in actual biomembranes, none of this
need be true because active and passive processes can maintain an asymmetric lipid
composition across the two leaflets [264, 265]. Finally, the term involving the
Gaussian curvature can be dropped here because we will encounter neither edges
nor topology changes, and so the Gauss–Bonnet theorem will work in our favor.
What remains is the simpler Hamiltonian Eq. (2), but this looks quite formidable
in Monge parametrization. To make any progress with something as forbidding as
this appears quite unlikely. And yet, not all hope is lost. For a spherical particle
attached to an asymptotically flat membrane, the nonlinear shape equation has an
exact solution, namely, a catenoid. This is an axisymmetric minimal surface with
K 0 and hence obviously minimizes the left-hand side of Eq. (2). If one adds
additional lateral membrane tension, the exact shape of the membrane around a
single adhering spherical particle can no longer be calculated analytically, but
numerical solutions are relatively easy to come by using an angle–arc length
parametrization [266]. Unfortunately, we need to know the solution for two particles, and in the absence of axisymmetry this is difficult, even numerically. It has
been done [267], but before we discuss this approach, let us first see what results we
can analytically wrest from these equations.
Even for the full nonlinear problem, the tight link between geometry and surface
stress permits one to express mediated interactions as line integrals over the
equilibrium membrane geometry. For instance, picture two spherical particles
bound to a membrane, held at some mutual distance. If the particles are identical,
then this will give rise to a mirror-symmetric membrane shape, and it can be shown
that the force between these particles can be written as [158, 159]:
F ¼
1
2
κ
ð
ds K
2
⊥ À K
2
jj
n
o
,
ð14Þ
where for simplicity we restrict to the tensionless case. The integral runs across the
symmetry curve (the intersection of the membrane with the mirror plane), K || is the
local curvature of that curve and K ⊥ the local curvature perpendicular to that curve.
The sign convention is such that a negative sign implies attraction. To obtain an
interaction strength out of Eq. (14) we need these curvatures, for which we need to
solve the shape equations after all. Unfortunately, not even the sign of the interaction is evident from Eq. (14), since the difference of two squares enters the
integrals. Had we been curious instead about the interaction (per unit length)
between two parallel rods on the membrane, we would have been in a better
position: Now K || would be zero and the interaction would be clearly repulsive
(even though we still do not quite know how strong it is). It seems that in order to
make headway, we must solve the shape equation. The only hope of doing this in
reasonable generality using analytical tools is to linearize it.
Computational Studies of Biomembrane Systems: Theoretical Considerations. . .
259
need be true because active and passive processes can maintain an asymmetric lipid
composition across the two leaflets [264, 265]. Finally, the term involving the
Gaussian curvature can be dropped here because we will encounter neither edges
nor topology changes, and so the Gauss–Bonnet theorem will work in our favor.
What remains is the simpler Hamiltonian Eq. (2), but this looks quite formidable
in Monge parametrization. To make any progress with something as forbidding as
this appears quite unlikely. And yet, not all hope is lost. For a spherical particle
attached to an asymptotically flat membrane, the nonlinear shape equation has an
exact solution, namely, a catenoid. This is an axisymmetric minimal surface with
K 0 and hence obviously minimizes the left-hand side of Eq. (2). If one adds
additional lateral membrane tension, the exact shape of the membrane around a
single adhering spherical particle can no longer be calculated analytically, but
numerical solutions are relatively easy to come by using an angle–arc length
parametrization [266]. Unfortunately, we need to know the solution for two particles, and in the absence of axisymmetry this is difficult, even numerically. It has
been done [267], but before we discuss this approach, let us first see what results we
can analytically wrest from these equations.
Even for the full nonlinear problem, the tight link between geometry and surface
stress permits one to express mediated interactions as line integrals over the
equilibrium membrane geometry. For instance, picture two spherical particles
bound to a membrane, held at some mutual distance. If the particles are identical,
then this will give rise to a mirror-symmetric membrane shape, and it can be shown
that the force between these particles can be written as [158, 159]:
F ¼
1
2
κ
ð
ds K
2
⊥ À K
2
jj
n
o
,
ð14Þ
where for simplicity we restrict to the tensionless case. The integral runs across the
symmetry curve (the intersection of the membrane with the mirror plane), K || is the
local curvature of that curve and K ⊥ the local curvature perpendicular to that curve.
The sign convention is such that a negative sign implies attraction. To obtain an
interaction strength out of Eq. (14) we need these curvatures, for which we need to
solve the shape equations after all. Unfortunately, not even the sign of the interaction is evident from Eq. (14), since the difference of two squares enters the
integrals. Had we been curious instead about the interaction (per unit length)
between two parallel rods on the membrane, we would have been in a better
position: Now K || would be zero and the interaction would be clearly repulsive
(even though we still do not quite know how strong it is). It seems that in order to
make headway, we must solve the shape equation. The only hope of doing this in
reasonable generality using analytical tools is to linearize it.
Computational Studies of Biomembrane Systems: Theoretical Considerations. . .
259