# Différences

Ci-dessous, les différences entre deux révisions de la page.

Les deux révisions précédentes Révision précédente Prochaine révision | Révision précédente Prochaine révision Les deux révisions suivantes | ||

users:giacomin:index [2018/04/19 15:48] giacomin |
users:giacomin:index [2019/11/01 16:31] giacomin |
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//E-mail// <giambattista.giacomin*@univ-paris-diderot.fr> | //E-mail// <giambattista.giacomin*@univ-paris-diderot.fr> | ||

\\ | \\ | ||

- | [[users:giacomin:more|If you really insist on a more realistic image...]] | + | //E-mail// <giacomin*@lpsm.paris> |

+ | \\ | ||

+ | [[users:giacomin:more|And if you really insist on a more realistic image...]] | ||

\\ | \\ | ||

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<fs large> [[https://www.lpsm.paris//|LPSM]] </fs> | <fs large> [[https://www.lpsm.paris//|LPSM]] </fs> | ||

<fs large> [[https://www.math.univ-paris-diderot.fr/espaces/personnels/index|local links]] </fs> | <fs large> [[https://www.math.univ-paris-diderot.fr/espaces/personnels/index|local links]] </fs> | ||

+ | |||

---- | ---- | ||

- | <note><fs large> June 18<sup>th</sup> to 20<sup>th</sup> 2018: [[http://www.lpsm.paris/conf_lpsm/|LPSM | + | <note><fs large>[[https://www.lpsm.paris/semoa/sem-lpsm|Le Séminaire du LPSM]]</fs></note> |

- | Kick-off Conference]]</fs></note> | + | |

+ | <fs large> June 18<sup>th</sup> to 20<sup>th</sup> 2018: [[http://www.lpsm.paris/conf_lpsm/|LPSM | ||

+ | Kick-off Conference]]</fs> | ||

| | ||

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<fs large> [[:users:giacomin:publications|Publications (books, articles, preprints,...)]] </fs> | <fs large> [[:users:giacomin:publications|Publications (books, articles, preprints,...)]] </fs> | ||

+ | <fs large> and [[users:giacomin:teaching|Teaching material]]</fs> | ||

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| [[http://www.cpt.univ-mrs.fr/~bastien/Home.html|Bastien Férnandez]] | [[http://webusers.imj-prg.fr/~david.gerard-varet/|David Gérard-Varet]]| Maha Khatib | | | [[http://www.cpt.univ-mrs.fr/~bastien/Home.html|Bastien Férnandez]] | [[http://webusers.imj-prg.fr/~david.gerard-varet/|David Gérard-Varet]]| Maha Khatib | | ||

| Assaf Shapira | [[https://sites.google.com/site/giuseppegenovesemathphys/|Giuseppe Genovese]] |[[http://ricerca.mat.uniroma3.it/users/greenbla/|Rafael L. Greenblatt]] | | | Assaf Shapira | [[https://sites.google.com/site/giuseppegenovesemathphys/|Giuseppe Genovese]] |[[http://ricerca.mat.uniroma3.it/users/greenbla/|Rafael L. Greenblatt]] | | ||

- | | [[http://www.proba.jussieu.fr/pageperso/delattre/|Sylvan Delattre]] |[[http://www.lpma-paris.fr/dw/doku.php?id=users:berger:index|Quentin Berger]] | | | + | | [[http://www.proba.jussieu.fr/pageperso/delattre/|Sylvan Delattre]] |[[http://www.lpma-paris.fr/dw/doku.php?id=users:berger:index|Quentin Berger]] | [[http://pensierolibero.me/mathematics/|Fabio Coppini]] | |

+ | |[[https://hdietert.github.io|Helge Dietert]] | | | | ||

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The Poland-Scheraga (PS) is the standard basic model for DNA denaturation, that is the transition that happens at high temperature from two complementary DNA strands that are bind together (//localized state//) to two free strands (//delocalized state//). {{ :users:giacomin:dna_col.jpg?nolink&450|}} The original PS model is limited to exact complementarity of the two strands - equal length strands and no mismatches are allowed - and it boils down to a sequence of bind pairs and symmetric loops (i.e., the number of bases contributed by each strand is the same). This model enjoyed and still enjoys a large popularity also because, in its homogeneous version, it is exactly solvable (in the sense of statistical mechanics: the PS model is a Gibbs measure) and the denaturation transition can be understood in detail. This solvable character is ultimately related to the fact that the homogeneous PS model is very closely related to a class of discrete renewal processes (it can even be mapped to a renewal).{{:users:giacomin:dna-ren_col.jpg?nolink&250 |}} Very remarkably, a natural generalization of the PS model - the generalized PS (gPS) model - allowing unequal length strands and asymmetric loops turns out to retain the solvable character of the PS model. A direct representation of a trajectory of the model with two strands of respective lengths twelve and nineteen bases is shown in the first figure. In [[http://arxiv.org/abs/1510.07996|Generalized Poland-Scheraga denaturation model and two-dimensional renewal processes]] (joint work with M. Khatib) we considered the homogeneous gPS model and we have exploited the fact that this model can be mapped to a two dimensional (or bivariate) renewal. The mapping at the level of trajectory transforms base pairs into points in the plane: the base pairs (1,1), (2,2), (3,6),... become an //increasing// sequence in the plane (see the second figure: we made the conventional choice to shift all coordinates down by one). This allows an analysis which is parallel to the original PS model, with the novelties introduced by the higher dimensional character. These novelties are not only technical, because the gPS has a phenomenology that is substantially richer than the PS model. The most evident novelty is the appearance of transitions inside the localized regime. From a mathematical viewpoint these transitions between //different// localized states can be interpreted as the switching of the underlying renewal (that is under the effect of pinning potentials) from Large Deviations regimes that are of Cramér type, i.e. that correspond to tilting the measure, to regimes that are not. In [[https://arxiv.org/abs/1703.10343|DNA melting structures in the generalized Poland-Scheraga model]] (joint work with Q. Berger and M. Khatib) we obtain sharp estimates on the partition function in the non-Cramér regime and this allows to pin down the precise behavior of the trajectories of the system and unravel the geometric richness of the model. | The Poland-Scheraga (PS) is the standard basic model for DNA denaturation, that is the transition that happens at high temperature from two complementary DNA strands that are bind together (//localized state//) to two free strands (//delocalized state//). {{ :users:giacomin:dna_col.jpg?nolink&450|}} The original PS model is limited to exact complementarity of the two strands - equal length strands and no mismatches are allowed - and it boils down to a sequence of bind pairs and symmetric loops (i.e., the number of bases contributed by each strand is the same). This model enjoyed and still enjoys a large popularity also because, in its homogeneous version, it is exactly solvable (in the sense of statistical mechanics: the PS model is a Gibbs measure) and the denaturation transition can be understood in detail. This solvable character is ultimately related to the fact that the homogeneous PS model is very closely related to a class of discrete renewal processes (it can even be mapped to a renewal).{{:users:giacomin:dna-ren_col.jpg?nolink&250 |}} Very remarkably, a natural generalization of the PS model - the generalized PS (gPS) model - allowing unequal length strands and asymmetric loops turns out to retain the solvable character of the PS model. A direct representation of a trajectory of the model with two strands of respective lengths twelve and nineteen bases is shown in the first figure. In [[http://arxiv.org/abs/1510.07996|Generalized Poland-Scheraga denaturation model and two-dimensional renewal processes]] (joint work with M. Khatib) we considered the homogeneous gPS model and we have exploited the fact that this model can be mapped to a two dimensional (or bivariate) renewal. The mapping at the level of trajectory transforms base pairs into points in the plane: the base pairs (1,1), (2,2), (3,6),... become an //increasing// sequence in the plane (see the second figure: we made the conventional choice to shift all coordinates down by one). This allows an analysis which is parallel to the original PS model, with the novelties introduced by the higher dimensional character. These novelties are not only technical, because the gPS has a phenomenology that is substantially richer than the PS model. The most evident novelty is the appearance of transitions inside the localized regime. From a mathematical viewpoint these transitions between //different// localized states can be interpreted as the switching of the underlying renewal (that is under the effect of pinning potentials) from Large Deviations regimes that are of Cramér type, i.e. that correspond to tilting the measure, to regimes that are not. In [[https://arxiv.org/abs/1703.10343|DNA melting structures in the generalized Poland-Scheraga model]] (joint work with Q. Berger and M. Khatib) we obtain sharp estimates on the partition function in the non-Cramér regime and this allows to pin down the precise behavior of the trajectories of the system and unravel the geometric richness of the model. | ||

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