SCHRÖDINGER EQUATION APPROACH TO THE UNBINDING TRANSITION OF BIOMEMBRANES AND STRINGS : RIGOROUS STUDY M. BENHAMOU, R. El KINANI, H. KAIDI ENSAM, Moulay Ismail University, Morocco © Symmetries, Differential Equations and Applications Islamabad, Pakistan, 2014

Introduction Two systems of interest : strings and bilayer membranes, called manifolds, in DG language. Strings : one-dimensional objects (DNA, ...).

Bilayer membranes : Two-dimensional sheets made of phospholipid molecules.

Phospholipid : Amphiphilic molecule possessing a hydrophilic polar-head and two hydrophobic fatty-acid chains.

Other components : Proteins, cholesterol, other lipid molecules. Cell membranes : Crucial role for life, They protects cells from their environment (barrier), and organelles inside cells Ensure exchanges of material (ions, macromolecules, drugs, ...).

Three kinds of interactions : Attractive van der Waals force Interactions : Biomembranes and strings experience mutual interactions or with solid surfaces, attractive at high-distance, and repulsive at short-distance. Three kinds of interactions : Attractive van der Waals force Repulsive shape-fluctuations force Repulsive hydration force Then : Competition between the two forces.

Unbinding transition : It occurs at some critical temperature, at which the system undergoes a phase transition from the unbind state (far each to other) to the bind state (close each to other). Unbind state : Bind state :

Similar surface transitions : Adhesion Wetting Adsorption-desorption of polymers...

Field Theoretical Renormalization-Group Variational Approach Theoretical tools : Field Theoretical Renormalization-Group Variational Approach Schrödinger Equation Method (SEM). Goal : Study of the unbinding transition thermodynamics from SEM.

Main quantities to consider : Average-separation between manifolds Roughness (fluctuations amplitude) Free energy Disjoining pressure. Used potential : More generalized Morse Potential enabling us to perform exact calculations.

Strings and bilayer membranes : Similar scaling behaviors Strings and bilayer membranes : Similar scaling behaviors. Then, it will be sufficient to consider only the problem of strings. Organization of the talk : String model Exact study of the unbinding transition Conclusion.

String model

Two interacting strings : fluctuate around a line-reference, say x-axis. Assumption : their elongations remain perpendicular to this axis. Conformation of strings : described by the local separation-field, , perpendicular to the line-reference.

Hamiltonian : Statistical Mechanics of strings is based on : : String length : Effective string tension, : Generalized Morse Potential. Analogy between the string-pair and a particle in QM :

Statistical Mechanics of the two strings from SEM : based on the resolution of a Schrödinger equation, and are eigenvalues and eigenfunctions. defines the free energy density :

Contact probability : to find the two strings at a distance, , apart, Quantities of interest : Average-separation : Average-squared separation : String roughness :

Exact study of the unbinding transition

q-Generalized Morse Potential : Introduced for the study of phase transitions from biological systems, Range-parameter : Parameter : , Potential depth : Standard MP : Generalized MP (Deng and Fan) :

Plot of the q-Generalized Morse Potential :

Remark : The q-GMP potential is bounded from below, and Katos's mathematical theorem : Schrödinger equation has only negative eigenvalues The eigenfunctions are bound states The eigenvalues spectrum is discrete.

Results and discussion : Ground state : Ground state energy : Contact probability : from which, we extracted the average-separation between strings and their roughness.

Unbinding transition : , Critical line

Average-separation : with an exact unbinding exponent : String roughness : with the same unbinding exponent .

Contact probability : Free energy density :

Disjoining pressure : The latter can be interpreted as the pressure required to maintain the two strings at the average distance .

Conclusion

Goal of this work : Analytical study of the unbinding transition undergone by strings or bilayer membranes, from a q-GMP. SEM : Exact computation of the ground state and the associated energy. Results : Identification of the unbinding temperature, Computation of contact probability, average-separation between manifolds and their roughness, Free energy and disjoining pressure.

Further considerations : Comparison with experimental data Extension of study to more than two strings Manifolds in contact with a solid surface : an extra interaction with this surface must be taken into account.

THANK YOU FOR YOUR ATTENTION