Morphology change in nematic membranes induced by defects

The cell membrane is one of the most important structures of living organisms. This is due to the many functions attributed to it such as permeable selectivity, protection, anchoring to the cytoskeleton and so many others. Any change in the shape of the cell membrane may affect directly the properties and abilities. In this article, we study how defects in the liquid crystalline organization of a membrane can change its shape. For performing this, we consider a membrane with orientational order, i.e., a nematic membrane, which can happen in biological membranes, nematic films and other systems and study how a defect in this order can change the shape of the membrane when the bending rigidity is considered. We find that depending on the ratio of rigidity and elastic constant, buckling of this membrane may happens and turn it into pseudo-spheres.


Introduction
The biomembrane is one of the most intriguing structures in nature.The plasma membrane is made of lipids, having amphiphilic nature, just as liotropic liquid crystals (JAKLI; SAUPE, 2006).There are several functions attributed to it, such as permeable selectivity, protection, anchoring to the cytoskeleton and so many others (ALBERTS et al., 2002).All these functions can be greatly affected by external factors, which include changes in the membrane morphology, often described by geometry and topology (FRANK;KARDAR, 2008).In fact, shape change in membranes has been subject of great effort in condensed matter.Very often, these shapes are investigated by analyzing the coupling between geometrical shape and orientational order.These studies include orientional order and defects in deformable vesicles (LUBENSKY; PROST, 1992;HIRST et al., 2013;JIANG et al., 2007;NYUGEN et al., 2013;PARK et al., 1992;RAMAKRISHNAN et al., 2010) and cytoskeletal filaments (NÉDÉLEC et al., 1997;SURREY et al., 2001).Furthermore, vector fields on a surface or nematic membranes, are considered simplified models to describe membranes that are more complex.These include any flexible sheet with ordered road-like constituents (SHALAGINOV, 1996;SPECTOR et al., 1993;YOUNG et al., 1978).
In this work, we investigate the problem of a flat membrane with coupled nematic order when defects of the +1 kind appear.In regular nematic cells, it is expected that the bulk will buckle into the third dimension when defects are present (CLADIS;   , 1972;JAKLI;SAUPE, 2006;MEYER, 1973).By using a simple model composed of nematic order and bending rigidity in covariant form, we are able to determine that a secondorder like transition may guide the buckling of the membrane depending on the ratio of the parameters involved.Furthermore, we obtain an analytical result for the shape of the membrane, which buckles into pseudo-spheres.

Material and methods
In order to model the membrane, the Monge parameterization (NELSON, 2004) was used, which defines a surface, mapped on a plane defined by the variables 1  and 2  with height . The position vector then can be written , which defines the tangent vectors as ), , ( We consider the free energy of the system to come from two contributions: the field, which wants to make all the vectors on the surface of the membrane parallel one to each order, supposing initial flat membrane, and the bending rigidity term.In a configuration with a defect, the flat configuration cannot minimize the free energy of the system, and the membrane has to bend, changing its shape.However, it has to pay a price for bending.The total free energy of the system, in the absence of surface tension (RAMAKRISHNAN et al., 2011), can be written in covariant form as: where:  is the bending rigidity, H is the mean curvature, A K is the nematic elastic constant (elastic isotropy approximation), y n represent the components of the director field and g is the metric tensor.The director field, in the presence of defects of +1 kind can be written as where:  is a parameter used to distinguished defects asters ( ) and vortices ( ), which is shown in Figure 1.

Results and discussion
The first approach for solving this problem is to try some specific shape for the membrane that would minimize the energy of the membrane (SEUNG; NELSON, 1998).One could argue that the actually shape for minimizing a +1 defect configuration would be a cone shape, and we can find how deep the cone goes by minimizing the free energy with respect to the angle  between the cone side and the z axis (in cylindrical coordinates), going from 2 /    for a flat membrane until 0   . For this configuration, the tangent vectors may be defined everywhere in terms of  as: The element of area for a conic surface is given by ] csc[  rdrd dS  . Furthermore, the mean curvature of a cone is given by .
Therefore, the free energy's first term is described by the following equation In equation ( 5), min r and max r are the distances from the core of the defect to the size of the membrane.The second term of the free energy becomes Therefore, the total free energy is Let us first examine the case for a defect of the "aster" configuration, which means, 0   .In this case, the equilibrium situation is found by setting is solution, which means a flat membrane.Nonetheless, the solution for  is which is a minimum only when Therefore, there is a ratio between elastic constant and bend rigidity that determines the buckling of the membrane.If the ratio is smaller than 2 / 1 , than the stable configuration is the one where the membrane remains flat.However, as the ratio grows larger than , 2 / 1 the system smoothly changes from flat to buckled, in a second order-like change transition.In Figure 2 In such case, the total free energy, equation (8), becomes:  Now, we shall look for profiles that are different from the conical shape.However, it is straightforward to foresee that the actual morphology of a buckled membrane in the presence of +1 defects should not differ much from the conical shape.This is indeed expected from the famous scape to the third dimension in bulk nematic (CLADIS; KLEMAN, 1972;JAKLI;SAUPE, 2006;MEYER, 1973).In fact, our first approach is to solve the problem imagining the following situation: where  is a coefficient to be determined.By numerically integrating and finding the minimum of the free energy, equation ( 11), we encounter that the stable configuration of the buckled membrane happens for   In order to seek for a more general solution of the problem, we assume that height of the surface depends only on the radial distance from the core, or,

). (r h h 
In this situation, the tangent vectors, as well as the normal vector k ˆ are given by The total free energy, equation ( 1), is then written as Then, equation (11) becomes: The profile that minimizes the free energy is the one that satisfies the Euler-Lagrange equation where: f is the integrand of the total free energy in equation ( 12).Hence which corresponds to the conical solution.The next step requires solving equation ( 14).Unfortunately, equation ( 14) is very complicated and has no analytical solution.Nonetheless, we can use our previous results to infer about solving it.The numerical minimization showed that the actual minimum of energy is slightly different from the conical configuration.Therefore, we assume the solution for equation ( 16) can be written as where: ) (r m is a function to be determined, considered small.By replacing equation ( 17) in ( 14), we get an equation depending only on ).

(r m
Since it is small, we can expand this equation and take only linear terms on ).

(r m
The usual minimization procedure allows the following equation for where: ./ dr dm m r  Equation ( 18) is solved by setting where  is given by equation ( 10), so near the core the membrane remains flat; and , 0 ) ( max  r m meaning that far away from the core the configuration is basically the same as a cone.Therefore, we find that and Figure 4 shows a graphic of the height profile, equation ( 20

Conclusion
In conclusion, the problem of a membrane with bend rigidity and a nematic vector field (nematic membrane) has been studied when defects of +1 kind exist in the director.We were able to show that the conical shape minimizes, in a first approximation, the free energy of the system when the ratio between the bend rigidity and elastic constant are in the appropriate range and that the transition between the flat and buckled state is second-order like.Further analyzes have shown that without any approximation the lowest energy is slightly different than the cone, which was first numerically calculated.Then, by using a perturbation method, we were able to analytically calculate the shape of the buckled membrane and show that it is a pseudosphere.


The behavior of F against  is shown in Figure3(b).

Figure
Figure 3. (a) ]) / ln[ /( 2 ' min max r r F F  that the  coefficient slightly smaller than one leads to lower energy. it is a function of r ).Furthermore, based on previous results, we look for solutions in the case where .0  


Figure 4 shows a graphic of the height profile, equation (20) as a function of r for 1 /   A K

Figure 4 .
Figure 4. height profile calculated when the membrane buckles due to a +1 defect in the director field.The pseudosphere morphology is the one that minimizes the free energy.