In-Plane Membrane Ordering and Topological Defects

**Deviatoric Curvature Model**

One could use the so called Deviatoric Curvature model [4, 5, 6, 9] to describe the onset of nematic-type in-plane ordering due to anisotropic nature of membrane constituents. These are modelled by 2D anisotropic nanoplates of area $a_0$ [6, 24]. Their geometry is determined by the intrinsic curvature tensor

$$C^{(m)} = C_1^{(m)} \left( \vec{e}_1 \otimes \vec{e}_1 \right) + C_2^{(m)} \left( \vec{e}_2 \otimes \vec{e}_2 \right).$$

Here $\{ \vec{e}_1^{(m)}, \vec{e}_2^{(m)} \}$ and $\{ C_1^{(m)}, C_2^{(m)} \}$ label eigenvectors and eigenvalues of the tensor, respectively, and $\otimes$ marks the tensorial product. The actual membrane shape is given by the effective curvature tensor field

$$C = C_1 \left( \vec{e}_1 \otimes \vec{e}_1 \right) + C_2 \left( \vec{e}_2 \otimes \vec{e}_2 \right)$$

determined by the eigenvectors $\{ \vec{e}_1, \vec{e}_2 \}$ and eigenvalues $\{ C_1, C_2 \}$. The corresponding interaction energy density is expressed in terms of invariants formed by the mismatch curvature tensor [6, 25]

$$M = R C^{(m)} R^{-1} - C.$$

Here $R = \left( \vec{e}_1 \otimes \vec{e}_1 + \vec{e}_2 \otimes \vec{e}_2 \right) \cos \theta + \left( \vec{e}_1 \otimes \vec{e}_2 - \vec{e}_2 \otimes \vec{e}_1 \right) \sin \theta$ stands for the rotation tensor, and $\theta$ refers to the angle between $\vec{e}_1$ and $\vec{e}_2$ in the tangent plane of the membrane characterized by the surface normal $\vec{v} = \vec{e}_1 \times \vec{e}_2 = \vec{e}_1^{(m)} \times \vec{e}_2^{(m)}$. It follows [4, 5, 6]

$$w = \kappa \left( \left( H - H^{(m)} \right)^2 + D^2 - 2 D D^{(m)} \cos(2\theta) + D^{(m)^2} \right),$$

where $\kappa > 0$ is the curvature modulus. The quantities $H = \left( C_1 + C_2 \right)/2$, $H^{(m)} = \left( C_1^{(m)} + C_2^{(m)} \right)/2$, $D = \left| C_2 - C_1 \right|/2$, $D^{(m)} = \left| C_2^{(m)} - C_1^{(m)} \right|/2$ determine the mean curvature, mean intrinsic curvature, curvature deviator, and mean curvature deviator, respectively.

The ensemble average of the local orientational ordering is given by [26]

$$\eta = \left\langle \cos(2\theta) \right\rangle = \frac{\int_0^{2\pi} \cos(2\theta) e^{-w a_0 \beta} d\theta}{\int e^{-w a_0 \beta} d\theta} = \frac{I_1(x)}{I_0(x)},$$

where $I_1$ labels the Bessel function of $n$-th order and $x = 2\kappa a_0 DD^{(m)}\beta$.

Note that in the case of an isotropic intrinsic element Equation (7) yields

$$w = \kappa \left( \left( H - C_0 \right)^2 + D^4 \right),$$

where $C_1^{(m)} = C_2^{(m)} = C_0$. Taking into account terms up to the $2^{nd}$ order in the curvature tensor expansion and for $C_0=0$ one reproduces the "classic" Helfrich membrane curvature model.

**Minimal Extrinsic Curvature Model**

The nematic orientational ordering could appear also due to attached thin rod-like nano-objects, as for example BAR proteins [8]. The corresponding minimal model, where we assume that nano-objects are represented by thin flexible rod-like objects, may be given by the following expression for the free energy density $f = f_c + \phi f_n + f_{mix}$, where [8]

$$f_c = \frac{\kappa}{2} H^2,$$

$$f_n = \frac{\kappa}{2} \left( C - C_n \right)^2,$$

$$f_{mix} = \frac{k_\beta T}{a_0} \left( \phi \ln \phi + (1 - \phi) \ln(1 - \phi) \right).$$

The first contribution $f_c$ stands for the classical Helfrich spontaneous curvature term [2], where we assume isotropic intrinsic membrane constituents. The term $f_n$ is the elastic curvature density contribution of the flexible anisotropic rod-like nano-objects attached to the membrane surface. This energy term is weighted by the rigidity constant $\kappa_m > 0$. The constant $C_m$ represents the locally enforced intrinsic curvature of the attached nano-object. The third term $f_{mix}$ describes the configurational entropy contribution of the attached rod-like objects of surface area $a_0$. We refer to this model as the Minimal Extrinsic Curvature model [8].

The local curvature $C$, experienced by the attached flexible nano-rod, can be expressed as

$$C = C_1 \cos^2 \theta + C_2 \sin^2 \theta = H + D \cos(2\theta).$$

Here $\theta$ describes the angle between the normal plane of the 1st principal curvature and the normal plane in which the nano-rod is lying.

The above described models are usually solved for axial membrane shape symmetry and are mainly used to determine qualitatively different membrane shapes [1, 2, 6, 8]. A membrane shape is in the Cartesian coordinate frame $(\vec{e}_x, \vec{e}_y, \vec{e}_z)$ commonly parametrized as

$$r = \rho(s) \cos(\varphi) e_x + \rho(s) \sin(\varphi) \vec{e}_y + \rho(s) \vec{e}_z.$$

where $\varphi \in [0,2\pi]$, and $s$ describes the arc-length of the membrane profile in the $(x,z)$ plane. Numerical details are described in [21]. The above model for orientational ordering of thin rod-like molecules can be further generalized by considering the orientational ordering of small plate-like flexible nanodomains embedded in the membrane [24] or flexible plate-like nano-objects attached to the membrane [27].

**Landau-Type Mesoscopic Model**

In order to emphasize role of orientational degrees of freedom the Landau-type modelling is commonly used [21, 28, 29]. To take into account the head-to-tail invariance it is necessary to introduce the tensor nematic order parameter [29]

$$Q = \lambda (\vec{n} \otimes \vec{n} - \vec{n}_\perp \otimes \vec{n}_\perp).$$

The quantity $\lambda$ determines the amplitude field, the unit director field $\vec{n}$ determines mesoscopic local membrane in-plane orientation, $\vec{n} \cdot \vec{n}\perp = 0$, and the membrane surface normal is defined as $\vec{v} = \vec{n} \times \vec{n}\perp$.

In terms of invariants in terms of $C$ and $Q$ we express the free energy density $f = f_c + f_e^{int} + f_e^{(ext)}$, where we take into account only the most essential terms of our interest. Quantities $f_c$, $f_e^{int}$, $f_e^{(ext)}$ stand for condensation, intrinsic elastic and extrinsic elastic term, respectively. We express them as [21, 29, 30]

$$f_c = -A \text{Tr} Q^2 + B \left( \text{Tr} Q^2 \right)^2,$$

$$f_e^{int} = k_i \text{Tr} \left( \nabla_s Q \right)^2,$$

$$f_e^{(ext)} = k_e \text{Tr} \left( Q C^2 \right).$$

The coefficients $A$ and $B$ are positive material constants, $k_i$ and $k_e$ stand for the representative intrinsic and extrinsic curvature elastic constants, $\nabla_s = (L - v \otimes v) \nabla$ is the surface gradient operator [29] and $\nabla$ represents the conventional gradient operator.

The model introduces several intrinsic (material dependent) length scales. The most important is the order parameter correlation length which we define as

$$\xi = \sqrt{\frac{k_i}{A}}.$$

**Membrane Nematic Order and Topological Defects**

In the following we present some demonstrative results of presented mesoscopic continuum models focusing on in-plane membrane orientational ordering.

The Deviatoric Elastic model enforces orientational ordering if deviatoric terms $D$ and $D_m$ are different from zero. For simplicity we restrict to axially symmetric configurations. The ordering term in Equation (7) displays external-field like structure. The corresponding field strength monotonously increases with increasing value of $D$ and $D_m$. Anisotropic membrane constituents orient along the direction for which eigenframes of tensors $C$ and $C^{(m)}$ coincide. The degree of nematic ordering $\eta = \left\langle \cos(2\theta) \right\rangle$ is plotted in Figure 2 as a function of dimensionless quantity $x = 2\kappa a_0 DD^{(m)} \beta$. The nematic ordering is absent if $\eta = 0$. Within this model nematic ordering is absent at umbilical points where $C_1 = C_2$. Therefore, the essential message that this model reveals is that anisotropic membrane constituents introduce an external field-type contribution in the interaction potential which is present in membrane areas.

$$ \mathrm {w h e r e} C _ {1} \neq C _ {2} $$

. Note that the model yields only

information on local degree of ordering and does not take

into account global topologically imposed constrains.

Note that in general the distribution of the orientational

ordering of the membrane components over the whole

membrane and the global shape of the membrane are

interdependent and determined by the minimum of the

membrane free energy [5].

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Figure 2: The degree of local nematic ordering η as a function of ( ) 0 2 m x a DD κ β = . We next consider the model described by the free energy density Eq.(10) within the Minimal Extrinsic Curvature model. In this treatment the nematic ordering arises due to the ordering of attached flexible rod-like nano-objects, to which we henceforth refer to as the rods. In this modelling one restricts to axially symmetric configurations of membranes. The variational parameters are the membrane shape, in-plane orientational ordering and spatially dependent concentration φ of the rods. We calculate the parameters by minimizing the free energy functional (see Eq.(10)) for a given average concentration ave φ of nano-rods assuming constant membrane volume and its surface area. Some representative results are shown in Figures 3 and 4. In Figure 3 we plot membrane shapes with superimposed orientational ordering for a fixed relative volume v and Cn on increasing the average concentration of rods ave φ . As it can be seen the bottom part of membranes becomes increasingly spherical on expense of pronounced tubular extrusion. The rods align along the meridians. The Gauss-Bonnet and Poincare-Hopf theorems are obeyed: the total topological charge of structures equals two. The vortex-like topological defect of strength m=1 is localized at the poles of the structures.

Furthermore, on increasing Cn the rods become increasingly attracted to the tubular membrane extrusion. This is even more evident in Figures 4 where we increase the rods’ imposed curvature Cn. for a fixed average concentrationφave . Namely, in the tubular part of the membrane the mismatch between preferred curvature of rods and local curvature of membrane could be efficiently minimized. Consequently, the rods are expelled from the spherical part where curvature is significantly lower with respect to Cn.. We remark that also in extreme case shown in Figure 4c the topological theorems are obeyed. A defect of strength m=1 resides at the top pole of the structure. The other m=1 TD residing in the bottom part of the structure is not visible because the nematic ordering there is absent.

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Figure 3: Membrane shapes with superimposed orientational ordering for a fixed relative volume v and Cn on increasing the average concentration of rods ave φ .(a) 𝜙𝑎𝑣𝑒= 0.0 , (b) 𝜙𝑎𝑣𝑒= 0.25 , (c) 𝜙𝑎𝑣𝑒= 0.5 . 𝐶0 = 0 , 𝑣= 0.66 , 𝐶𝑛= 8 , 𝜅𝑛= 𝜅, 𝜅= 30𝑘𝐵𝑇, 𝑅0 = 250 nm , 𝑎0 = 100 nm2. 𝑅0 stands for the radius of a sphere which has the same surface as the membrane. 𝐶𝑛 is expressed in units of 1/R0.

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on increasing 𝐶𝑛. (a) 𝐶𝑛= 10 , (b) 𝐶𝑛= 12 , (c)

𝐶𝑛= 14. 𝐶0 = 0, 𝑣= 0.85, 𝜙𝑎𝑣𝑒= 0.1, 𝜅𝑛= 𝜅, 𝜅= 30𝑘𝐵𝑇,

𝑅0 = 250 nm, 𝑎0 = 100 nm2.

Finally, we discuss the model given by Equation (14)

which focuses mainly on nematic ordering for a given

membrane shape. Note that the term weighted by the

constant ke mimics deviatoric-type term that was first

introduced via the deviatoric elastic model [6]. In Figure 5

we show typical nematic ordering for a tubular-type

nematic structure. In Figure 5a we depict structure in the

{s, φ } parameter plane, where s determines the

membrane arc-length and v is the azimuthal angle in the

(x,y) plane of the Cartesian coordinate system (x,y,z) of

the axially symmetric membrane shape. In this modelling

we allow solutions which break the axial symmetry. In the

case shown three m=1/2 TDs are formed in the spherical

part of the membrane exhibiting 𝐾 > 0, one m=-1/2 TD in

the neck area where 𝐾 < 0, and m=1 TD at the bottom

pole of the membrane, where 𝐾 > 0. The total charge

equals two in accordance with Eq.(1). Furthermore, the

distribution of TDs is in line with the ETCC mechanism,

although the structure is not totally “topologically

neutralized”. For this purpose one would need stronger

local distortions which would enable formation of an

$$ \text{additional pair} \quad \{\text{defect, antidefect}\} = \{1 / 2, - 1 / 2\}. \mathrm {The} $$

resulting structure would possess four m=1/2, two m=-

1/2 and one m=1 topological defects in the top spherical-

like part, neck region, and bottom tubular extrusion,

respectively.

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Figure 5: (a) The nematic ordering scaled with respct to its bulk value 0 λ in the {s, φ } parameter plane. Ls determines the total membrane’s shape length. b The membrane shape where we indicate positions of TDs. 𝐶0 = 0 , 𝑣= 0.85 , 𝐶𝑛= 12 , 𝜙𝑎𝑣𝑒= 0.1𝜅𝑛= 𝜅, 𝜅= 30𝑘𝐵𝑇, 𝑅0 = 250 nm, 𝑎0 = 100 nm2, ki= 𝜅, ke=0, 𝑅0/𝜉= 60.