# Physics of Liquid-crystal Membranes

Mentor: | Sergei Akimov |
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Revision: | 7 Dec 2013 |

## Course Summary

The goal of this course is to expose students to various aspects of modern physics of liquid crystals. Membranes are bilayer liquid-crystal films, which are macroscopic in the lateral direction. The course contains foundations of theory of elasticity of liquid crystals initially developed for 3D systems, and then adapted for description of bilayer membranes. Thermodynamics and kinetics of phase transitions in multicomponent systems are recounted concerning biological membranes, which contain proteins and several lipid components. Gibbs’ phase diagrams are introduced, and various 2D lattice models are considered. Local phase transitions, occurring in the vicinity of protein complexes due to so-called wetting phenomenon under conditions of absence of global phase transition in the surrounding membrane, are described in the framework of wetting theory adapted for biological membranes. Various mechanisms of protein-lipid interactions as well as conditions for macroscopic wetting films formation are considered. Dependence of cellular processes rates on energetic of formed membrane structures is illustrated for exo- and endocytosis.

## Course Format

Instruction will consist of 34 hours of lecture and 17 hours of discussion. Assigned problem set will be provided.

Hours of lecture | Hours of discussion | Hours of class preparation | Hours total |
---|---|---|---|

34 | 17 | 45 | 96 |

**Please note** that students are expected to study outside of class for three hours for every hour in class.

## Course Content

The plan is to work through the following topics:

- Properties of 3D liquid crystals
- Elements of the classical theory of elasticity
- Classification: nematics, cholesterics, and smectics
- Smectics
- Phase diagrams “lipid-water”

- Membrane mechanics
- Structure of bilayer lipid membranes
- Features of membrane elasticity theory
- Fundamental deformations in membranes
- Equation of local volumetric incompressibility
- Painted membranes as the systems with variable number of particles
- Energy of contact of two planar bilayers of different thickness
- Phenomenological Flory model
- Interaction of membrane inclusions, mediated by membrane deformations
- A pore in lipid bilayer
- Life cycle of an enveloped virus
- Fission of model membranes
- The basic models of the dynamin function

- Phase transitions in multicomponent membranes
- Hierarchy of lipid-protein structures in biological membranes
- Some kinds of lipidic fluorophores
- The analysis of diffusion for determination of raft size in cells
- Method of membrane nanotube pulling
- Phase diagrams of two- three-, and multicomponent membranes
- A problem of stability of small domains
- Kinetic stabilization of small domains ensemble
- Methods of measurement of line tension of rafts boundary
- The mechanical model based on hydrophobic mismatch.
- Hybrid mechanical-chemical model
- Stability of small rafts in cellular membranes
- Bilayer structure of rafts

- Electrical properties of membranes
- Electrolyte solution and Debye screening
- A double electrical layer
- Membrane electrostatics
- Methods of measurement of membrane potential components.
- Electroporation of membranes
- Practical applications of electroporation

## Textbooks

Primary textbooks:

- B.V. Derjaguin, N.V. Churaev, V.M. Muller, Surface forces (Springer, 1987).
- L.D. Landau, E.M. Lifshitz, Theory of elasticity (Butterworth-Heinemann, 1986).
- D.L. Landau, E.M. Lifshitz, Statistical physics (Addison-Wesley, 1969).

Additional textbooks:

- H.T. Davis, Statistical mechanics of phases, interfaces and thin films (Wiley-VCH, 1996).
- J.N. Israelachvili, Intermolecular and surface forces (Academic Press, 2006).
- P.M. Chaikin, T.C. Lubensky, Principles of condensed matter physics (Cambridge University Press, 2000)
- E.M. Lifshitz, L.P. Pitaevskii, Physical Kinetics (Butterworth-Heinemann, 1981)

## Homework Assignments

Weekly, 1 problem set in total, due at the beginning of the lecture. You may also submit via e-mail before the due date/time. It is of outmost importance that you invest your own effort into solving problems. Should you consult any sources, please provide references. Homework assignments should be typed. Legible handwritten assignments are acceptable.

## Grading

Class participation | 10% |

Homework assignments | 20% |

Midterm exam | 20% |

Final exam | 50% |