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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
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Introduction Two systems of interest : strings and bilayer membranes, called manifolds, in DG language. Strings : one-dimensional objects (DNA, ...).
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Bilayer membranes : Two-dimensional sheets made of phospholipid molecules.
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Phospholipid : Amphiphilic molecule possessing a hydrophilic polar-head and two hydrophobic fatty-acid chains.
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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, ...).
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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.
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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 :
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Similar surface transitions :
Adhesion Wetting Adsorption-desorption of polymers...
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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.
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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.
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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.
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String model
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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.
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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 :
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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 :
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Contact probability : to find the two strings at a distance, , apart,
Quantities of interest : Average-separation : Average-squared separation : String roughness :
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Exact study of the unbinding transition
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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) :
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Plot of the q-Generalized Morse Potential :
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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.
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Results and discussion :
Ground state : Ground state energy : Contact probability : from which, we extracted the average-separation between strings and their roughness.
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Unbinding transition :
, Critical line
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Average-separation : with an exact unbinding exponent : String roughness : with the same unbinding exponent
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Contact probability : Free energy density :
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Disjoining pressure : The latter can be interpreted as the pressure required to maintain the two strings at the average distance .
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Conclusion
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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.
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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.
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THANK YOU FOR YOUR ATTENTION
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