PDEs on Manifolds

Authors: Lloyd Connellan, Adam Nixon, and Wojciech Ozanski
Supervisors: Charles Elliott, Hans Fritz

Our research group is concerned with the modeling of protein inclusions into a cellular membrane.

Both biologically and physically, proteins are important factors when considering the bilayer of a cell membrane. They are larger molecules that are used for a variety of functions involving the cell. The problem of modelling the cell's morphology is important to understanding how the cell interacts with its environment. Transmembrane proteins are embedded in the membrane (see [4] and [5]) and play an important role in transport, adhesion and signalling.

As shown below, a protein inclusion causes a deformation of the cell membrane, which we will denote by $u$ . In this project we sought to model this deformation, along with the variable $\phi$ which represents the lipid composition, a physical property of the membrane.

Deformation of the cell membrane due to a protein inclusion

Here we use a simplified model for the cell membrane, as a construction of pairs of lipids. Due to the relative size of the proteins and the width of the lipid bilayer to the overall cell, we will consider the bilayer as a surface.

The variables $u$ and $\phi$ are shown to satisfy the minimisation of the following energy functional.

$\mathcal{F}(u,\phi)=\frac{1}{2}\kappa \int_{\Omega} |{\Delta}u|^2 + \frac{1}{2}\sigma\int_{\Omega}|{\nabla}u|^2 + \frac{1}{2}a\int_{\Omega} \phi^2 + \frac{1}{2} b \int_{\Omega} |{\nabla}\phi|^2 + c\int_{\Omega}\phi {\Delta}u$

Here the $\kappa$ term comes from the bending energy, the $\sigma$ term comes from the surface tension, the $a$ and $b$ terms relate to the lipid composition and the $c$ term is the coupling energy. We also consider two types of boundary conditions.

$1. \text{Dirichlet:} u|_{\partial\Omega} = g, \frac{{\partial}u}{{\partial}n}|_{\partial\Omega} = f \text{and} \phi|_{\partial\Omega} = p,$

$2. \text{Neumann:} u|_{\partial\Omega} = g, {\Delta}u|_{\partial\Omega} = f \text{and} \phi|_{\partial\Omega} = p,$

By computing the minimiser to this functional using Dirichlet boundary conditions, we get the following Euler-Lagrange equations

$\kappa {\Delta^2}u - \sigma {\Delta}u + c{\Nabla}\phi = 0,$

$a\phi - b{\Delta}\phi + c{\Delta}u = 0.$

The next step is to solve this equation numerically, which involves discretising the problem. More of this is discussed in the section on the analysis, as well as our choice of finite elements. Our weak Euler-Lagrange equations take the form

$\kappa\int_{\Omega} {\Delta}u{\Delta}v + \sigma\int_{\Omega}{\nabla}u\cdot{\nabla}v + c\int_{\Omega}\psi{\Delta}u + a\int_{\Omega}\phi\psi + b\int_{\Omega}\nabla\phi\cdot\nabla\psi + c\int_{\Omega}\phi{\Delta}v = 0.$

We discretise this using our nodal basis functions for $u$ and for $\phi$ , which we will denote by $\theta_i$ and $\eta_i$ . The resulting equation is as follows

$\sum_j U_j \int_{\Omega} \kappa\Delta\theta_j\Delta\theta_i + \sigma\nabla\theta_j\cdot\nabla\theta_k + c\eta_k\Delta\theta_j + \sum_l \Phi_l \int_{\Omega}a\eta_l\eta_k + b\nabla\eta_l\cdot\nabla\eta_k + c\eta_l\Delta\theta_i = 0.$

We will use this equation in matrix form in the section on numerical results.

We thus have a model for the compositions of lipids around a membrane that are affected by the inclusions we insert in the domain. We have analysed this model for various boundary conditions and found a restriction on the existence of a unique solution relating to the value $c$ . A similar inequality is proved in the discrete case but we suspect this is too strong of a bound.

We have implemented the code for the Dirichlet case and investigated the effect of distance between two inclusions with various boundary values. This was done in Dune using a combination of Morley element (for $u$ ) and linear elements (for $\phi$ ). From this we have seen that for inclusions of the same type (same boundary conditions) we observe an ideal attraction distance between them. For inclusions of the opposite type we instead see only a repulsion between the two proteins. This suggests that biologically proteins of a similar type will naturally try to group with other proteins of the same type.

References

[1] G. Hobbs. Monge gauge with point particles. 2013. MSc thesis.
[2] S. A. Rautu, G. Rowlands and M. S. Turner. Composition Variation and Underdamped Mechanics near Membrane Proteins and Coats, 2014. http://arxiv.org/abs/1407.1672.
[3] M. Wang and J. Xu. The Morley element for fourth order elliptic equations in any dimensions. Numerische Mathematik, 2004.
[4] S. J. Singer and G. L. Nicolson. Science, 1972.
[5] D. M. Engelman. Nature, 2005.

Acknowledgements

We acknowledge and thank the help of our supervisors Prof Charles Elliott and Dr Hans Fritz.
We also acknowledge the funding body EPSRC and the support from MASDOC CDT.

Contact: L.Connellan at warwick.ac.uk, Adam.Nixon at warwick.ac.uk, W.S.Ozanski at warwick.ac.uk.

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Authors: Lloyd Connellan, Adam Nixon, and Wojciech Ozanski Supervisors: Charles Elliott, Hans Fritz

Acknowledgements

Authors: Lloyd Connellan, Adam Nixon, and Wojciech Ozanski
Supervisors: Charles Elliott, Hans Fritz