1. 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

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

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

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

5. 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, ...).

6. 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.

7. 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 :

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

9. 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.

10. 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.

11. 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.

12. String model

13. 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.

14. 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 :

15. 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 :

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

17. Exact study of the unbinding transition

18. 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) :

19. Plot of the q-Generalized Morse Potential :

20. 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.

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

22. Unbinding transition :
, Critical line

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

24. Contact probability :
Free energy density :

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

26. Conclusion

27. 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.

28. 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.

29. THANK YOU FOR YOUR ATTENTION