For red blood cells [23, 135, 141], blood platelets [142], their progenitors [143], but also for actin shells created in artificial vesicles [144] and cells in general [137] a cortex height in the order of hundred nanometers is reported. Therefore, the height of the plasma membrane and cortex together is small compared to a typical cell diameter in the order of several microns. As a consequence, the thin shell theory [145–147] has been established in membrane modeling, where membrane and cortex are represented by an infinitely thin shell.
In the following, we first start by introducing the important concepts of differential geometry, which is the framework of thin shell theory. Afterwards, we detail the treatment of membrane and cortex mechanics.
X s1, s2 C
ds n
e1
e2
C f
ν2 ν ν1
t2
t1
e1
e2
n
a) b)
Fig. 2.1:Membrane and cortex as thin shell. a) The thin shell is parametrized by X(s1, s2) with in-plane coordinate vectorseα, whereα= 1,2, and normal vectorn. C represents a line along the thin shell with a line element ds. b) A forcef is acting on a line element of the thin shell with in-plane normal vectorν. The force is decomposed along the in-plane coordinate vectors in terms of the surface stresstα.
2.2 Modeling a membrane in thin shell theory
2.2.1 Differential geometry
Because of their relatively small thickness, the plasma membrane and cortex are treated as a thin shell, a two dimensional manifold in three-dimensional space [148]. The thin shell representing both the plasma membrane and cortex is sometimes called membrane [pub1]. Figure 2.1 a) provides a sketch. Mathematically, the thin shell is parametrized by a vector X =X(s1, s2) depending on the two coordinates s1, s2. In-plane coordinate vectors, which point along the interface, can be derived from the parametrization by a partial derivative
eα = ∂
∂sαX =: ∂αX, (2.1)
where Greek letters refer to the thin shell coordinates and take the values α, β, γ, δ= 1,2.
The local unit normal vector on the membrane n can be deduced from the two in-plane coordinate vectors by a cross product
n= e1×e2
|e1×e2|. (2.2)
The in-plane coordinate vectors and the normal vector build a coordinate system
(e1,e2,n), (2.3)
defined at each point located on the thin shell. Therefore, they provide a local coordinate system, where the three coordinate vectors themselves are vectors in the three-dimensional space. Each general vector can be decomposed into its components along these coordinate vectors
a=aβeβ +ann, (2.4)
where the upper index refers to a contravariant component and a lower index corre-spondingly to a covariant component. Here and in the following, the Einstein sum convention is used, which implies a sum over an index occurring twice, as covariant as well as contravariant index. With Latin indices we refer to three-dimensional Cartesian coordinates, they take the valuesi, j, k = 1,2,3 and the Einstein sum convention simply applies to a double occurring index.
On the membrane, an arbitrary line C can be considered with the line element ds pointing along dX. From the first fundamental form
ds2 =g11ds1ds1+ 2g12ds1ds2+g22ds2ds2 = dX · dX, (2.5) which allows for the measurement of length and area element on the membrane, the metric tensor can be derived
gαβ =eα·eβ. (2.6)
The contravariant or inverse metric gαβ is defined by the relation gαγgγβ = δβα with δβα being the Kronecker symbol. For a symmetric tensor, i.e.,tαβ =tβα, such as the Kronecker symbol, the order of covariant and contravariant indices can be changed and therefore no distinction of the order in the mathematical symbol is necessary. Using the metric tensor the index of a tensor component can be raised or lowered
aα =gαβaβ, aα =gαβaβ, (2.7) tαβ =gβγtαγ, tαβ =gαγtγβ. (2.8)
2 Modeling of biological membranes in the context of blood flow
Partial derivatives of the metric tensor components are given by a combination of the Christoffel symbols, which in turn are defined by
Γγαβ = 1
2gγδ(∂αgβδ +∂βgδα−∂δgαβ). (2.9) The Christoffel symbols further allow for a generalization of the derivative along the in-plane coordinates to the covariant derivative. The covariant derivative in contrast to the partial one transforms in a covariant manner [149] and is given for a scalar φ, a general vector aβ, and a general tensor tαβ by
∇αφ=∂αφ, ∇αaβ =∂αaβ+ Γβαγaγ, (2.10)
∇αtβγ =∂αtβγ+ Γβαδtδγ+ Γγαδtβδ. (2.11) An important property, which strongly discriminates a deformed thin shell from the description of a three-dimensional body, is the curvature of the thin shell. It is characterized by the second fundamental form
cαβdsαdsβ = dX · dn, (2.12)
which is the variation of the normal vector projected onto a line element. Corresponding to the second fundamental form the (extrinsic) curvature tensor is defined by
cαβ =∂αX ·∂βn=−(∂αeβ)·n, (2.13) where the minus together with the second identity stems from the derivative of the identity eα·n= 0 [149]. For the curvature tensor, two scalar invariants can be defined [149], the mean curvature
H = 1
2gαβcαβ, (2.14)
and the Gaussian curvature
K = detcβα. (2.15)
The covariant derivative of an in-plane coordinate vector is given by the equation of Gauss [148, 149]
∇αeβ =−cαβn, (2.16)
the covariant derivative of the normal vector by the equation of Weingarten [148, 149]
∇αn=cβαeβ, (2.17)
and the covariant derivative of the metric vanishes
∇αgβγ = 0, ∇αgβγ = 0. (2.18)
Using equation (2.4), the covariant derivative of a general vector a can be obtained
∇αa= (∇αaβ+ancαβ)eβ + (∇αan−aβcβα)n. (2.19)
2.2 Modeling a membrane in thin shell theory
2.2.2 Mechanical properties
The thin shell theory has to cover the mechanical properties of the membrane including the cortex. Both internal membrane mechanics and interaction with the environment result in forces on the membrane. In strong analogy to the well-known Cauchy stress tensor and Cauchy’s law in three-dimensional continuum mechanics [150], the force on the membrane can be expressed in terms of stress vectors [145]
f =ναtαds, (2.20)
where f is the force on a line C, which is characterized by the in-plane normal vector ν as sketched in figure 2.1 b). The force on a local area element on the membrane is calculated by the line integral over the force in equation (2.20) along the contour of the area element. We term tα as the surface stress [pub1]. It can be decomposed in analogy to equation (2.4)
tα =tαβeβ+tαnn, (2.21)
with the components along the in-plane coordinates being the in-plane surface stress tαβ and the normal components named transverse shear stress tαn [69,151, 152].
The surface stress tα covering internal mechanics is a sum of the different contributions from the different mechanical properties of the membrane [68]
tα =tαel+tαvisc+tαact, (2.22) where we here consider elastic surface stress tαel, viscous surface stress tαvisc, and actively generated surface stress tαact. Furthermore, torques can arise which are described by moments such as active moments [9] or bending moments [69, 151].
In addition to the internal mechanical stresses, external forces are acting on the membrane, which are in equilibrium with the internal membrane forces [68, 145]. When the membrane is immersed in a fluid, forces from the fluid act onto the membrane. The fluid forces acting onto a membrane are given by the projection of the three-dimensional fluid stress tensor σ on the normal vector of the interface [150]
fˆi =σijnj. (2.23)
A cell membrane is typically surrounded by an external medium (outer fluid) and encloses the cytoplasm (inner fluid). Considering the forces from both fluids ˆfout and ˆfin, respectively, leads to the traction jump across the membrane ∆f using equation (2.23)
∆f = ˆfout−fˆin=σout−σin·n. (2.24) The traction jump can again be decomposed into the components ∆fα along the in-plane coordinate vectors and the component ∆fnalong the normal vector. For negligible inertia the traction jump is in equilibrium with the membrane forces f, which are deduced from the surface stress in equation (2.22). This equilibrium results in the force balance equations
∇αtαβ+cαβtβn+ ∆fα = 0, (2.25)
∇αtαn−cαβtαβ + ∆fn = 0, (2.26)
2 Modeling of biological membranes in the context of blood flow
which are the component-wise expression of (cf. equation (2.19))
∇αtα+ ∆f = 0, (2.27)
and equivalent to the notation ∇s·T with stress tensor T and surface gradient ∇s as used by Barthès-Biesel [68].
2.2.3 Passive elasticity and viscosity of a membrane
Under the action of forces a membrane (together with the cortex) can be deformed.
The passive elasticity describes the force as response to a deformation, which drives the membrane back into an undeformed state (or towards the stress free shape). The elastic response is described in terms of a constitutive law accounting for a certain material or a certain class of materials.
For the elastic response of a red blood cell membrane towards shearing and area dilatation the Skalak law has been proposed [66]
WSK = κS
12
hI12+ 2I1−2I2
+CI22i, (2.28)
which gives the energy density WSK in terms of the deformation invariants I1 and I2, the shear modulus κS, and the area dilatation coefficient C. The deformation invariants can either be deduced from the deformation gradient [68, 153] or directly from the metric of the undeformed Gαβ and deformed membrane gαβ [145,154, 155] by
I1 =Gαβgαβ −2, (2.29)
I2 = detGαβdet (gαβ)−1, (2.30) as explicitly performed in the publications [pub1] and [pub4]. The resistance towards shearing stems from the spectrin network in red blood cells which underlines the plasma membrane and the resistance towards area dilatation stems from the plasma membrane itself. We note that for C = 1 and small deformations the Skalak law is equivalent to the Neo-Hookean law for membrane elasticity [156–158].
For the elastic response to bending deformations of lipid membranes the Helfrich law [67] is widely used [68,69, 159, 160] determining the bending energy density
WHF= 2κB(H−H0)2+κKK, (2.31) where κB is the bending modulus, κK the Gaussian curvature modulus and H0 the reference curvature. The term including the Gaussian curvature K often can be neglected because of the Gauss-Bonnet theorem [149, 161]. Therefore, a finite elastic bending energy in equation (2.31) and a corresponding restoring force arise from deviations of the mean curvature H from the reference curvature H0.
From the energy density functional of the elastic constitutive law, e.g., equations (2.28) and (2.31), the force on the membrane can be deduced in two ways: the gradient of the energy with respect to the position on the membrane can be calculated [69, 153] or in the framework of thin shell theory the contribution to the surface stress tαel is derived by derivatives with respect to the metric and curvature tensors [145]. The surface stress in