Des liaisons élastiques à la friction : vers des filaments adhérents polymérisants
Vuk Milisic, CNRS Researcher, LMBA, UMR 6205, Brest, UBO
Résumé
La motilité cellulaire est la capacité des cellules à générer un mouvement spontané.
Dans ce contexte, C. Schmeiser et D. Oelz ont proposé une approche pionnière
dans [6, 7].
Dans ces articles, les forces d'adhésion à mémoire font partie des éléments constitutifs
du modèle. Ces forces stabilisent le réseau d'actine et permettent l'adhésion des cellules au substrat.
Un passage à la limite formel permet de transformer ces forces en friction instantanée fournissant un ensemble d'équations aux dérivées partielles non linéaires paraboliques.
Depuis [1, 2, 3, 5, 4] beaucoup d'efforts ont été faits
afin de prouver ces calculs formels. Cette thèse de doctorat devrait aussi contribuer à cet objectif.
Pour ce faire, on propose de suivre deux directions en particuler :
les aspects treadmiling et ceux liés à la contrainte d'inextensibilité.
Dans le premier cas, ce que l'on aimerait inclure dans l'analyse de l'asymptotique, c'est le fait que les filaments d'actine polimérisent du côté qui touche l'extrémité de la cellule et dépolimérisent du côté du noyeau. Cela se traduit par un terme de transport
dans les équations. On veut pouvoir démontrer rigoureusement le passage à la limite
en tenant compte de cet effet de glissement de la configuration de référence.
Dans un deuxième volet, on veut étendre les résultats de [4] au cas d'un filament inextensible.
De manière plus précise, on se propose d'étudier la même asymptotique que dans [4], mais à la place de considérer la mappe harmonique on veut traiter le cas des équations paraboliques du quatrième ordre avec une contrainte locale (en espace) au premier ordre.
Un volet numérique pourra s'ajouter à la thèse et l'on pourra discrétiser et simuler les différents modèles et
donner des estimations d'erreur numériques.
Le stage de M2R et la thèse auraient lieu au Laboratoire de Mathématiques de Bretagne Atlantique, sous ma supervision.
Friction mediated by elastic linkages : towards treadmilling adhering filaments
Abstract
Cell motility is the ability of cells to generate a spontaneous motion.
In this context C. Schmeiser and D. Oelz proposed a pionneering model in [6, 7].
In these papers adhesion forces with memory are part of the building blocks
of the model. These forces stabilise the actin filament network and allow cell adhesion to the substrate.
A formal limit transforming them into instantaneous friction forces is made
providing a set of parabolic non-linear partial differential equations.
Since [1, 2, 3, 5, 4] a lot of efforts were made
in order to prove these formal computations. This PhD thesis should contribute to this goal.
To do this, we propose to follow two directions in particular :
the treadmiling aspects and those related to the inextensibility constraint.
In the first case, what we would like to include in the analysis of the asymptotic,
it is the fact that the actin filaments polymerize on one end
of the cell and depolimerize on the other side. This translates to a convective term
in the equations. We want to be able to rigorously prove the passage to the limit
taking into account this sliding effect of the reference configuration.
In a second part, we want to extend the results of [4] to the case of an inextensible filament.
More precisely, we propose to study the same asymptotic
than in [4], but instead of considering the harmonic map,
we want to work with parabolic equations of the fourth order
with a local constraint (in space) to the first order.
Numerical aspects can be considered as well in this work : one should discretize these models, and
compute numerically error estimates either in the long time asymtotic or in the friction regimes.
The master and PhD theses will take place at Laboratoire Mathématiques de Bretagne Atlantique under my supervision.
Références
- [1]
V. Milisic and D. Oelz.
On the asymptotic regime of a model for friction mediated by
transient elastic linkages.
J. Math. Pures Appl. (9), 96(5):484-501, 2011.
- [2]
V. Milisic and D. Oelz.
On a structured model for the load dependent reaction kinetics of
transient elastic linkages.
SIAM J. Math. Anal., 47(3):2104-2121, 2015.
- [3]
V. Milisic and D. Oelz.
Tear-off versus global existence for a structured model of adhesion
mediated by transient elastic linkages.
Commun. Math. Sci., 14(5):1353-1372, 2016.
- [4]
Vuk Milisic.
From delayed and constrained minimizing movements to the harmonic
map heat equation.
J. Funct. Anal., 279(2):50, 2020.
Id/No 108520.
- [5]
Vuk Milisic and Dietmar Oelz.
Space dependent adhesion forces mediated by transient elastic
linkages: new convergence and global existence results.
J. Differ. Equations, 265(12):6049-6082, 2018.
- [6]
D. Oelz and C. Schmeiser.
Derivation of a model for symmetric lamellipodia with instantaneous
cross-link turnover.
Archive for Rational Mechanics and Analysis, 198(3):963-980,
2010.
- [7]
D. Oelz, C. Schmeiser, and V. Small.
Modelling of the actin-cytoskeleton in symmetric lamellipodial
fragments.
Cell Adhesion and Migration, 2:117-126, 2008.
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/TeX.pool.ltxml...
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/eTeX.pool.ltxml... 0.00 sec)
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/pdfTeX.pool.ltxml... 0.01 sec) 0.21 sec)
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/LaTeX.pool.ltxml...
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/textcomp.sty.ltxml...
(Processing definitions /usr/share/texlive/texmf-dist/tex/latex/base/ts1enc.dfu... 0.02 sec) 0.03 sec) 0.21 sec)
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/article.cls.ltxml... 0.01 sec)
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/amsmath.sty.ltxml...
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/amsbsy.sty.ltxml...
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/amsgen.sty.ltxml... 0.00 sec) 0.01 sec)
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/amstext.sty.ltxml... 0.00 sec)
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/amsopn.sty.ltxml... 0.01 sec) 0.04 sec)
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/amsthm.sty.ltxml... 0.00 sec)
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/amssymb.sty.ltxml...
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/amsfonts.sty.ltxml... 0.00 sec) 0.02 sec)
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/eucal.sty.ltxml... 0.00 sec)
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/xcolor.sty.ltxml...
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/color.sty.ltxml... 0.01 sec)
(Processing definitions /usr/share/texlive/texmf-dist/tex/latex/graphics/dvipsnam.def... 0.08 sec) 0.13 sec)
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/url.sty.ltxml... 0.00 sec)
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/hyperref.sty.ltxml...
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/ltxcmds.sty.ltxml...
(Processing definitions /usr/share/texlive/texmf-dist/tex/generic/oberdiek/ltxcmds.sty... 0.10 sec) 0.10 sec)
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/keyval.sty.ltxml...
(Processing definitions /usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty... 0.01 sec) 0.01 sec)
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/kvsetkeys.sty.ltxml...
(Processing definitions /usr/share/texlive/texmf-dist/tex/generic/oberdiek/kvsetkeys.sty...
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/infwarerr.sty.ltxml...
(Processing definitions /usr/share/texlive/texmf-dist/tex/generic/oberdiek/infwarerr.sty... 0.05 sec) 0.05 sec)
(Processing definitions /usr/share/texlive/texmf-dist/tex/generic/oberdiek/etexcmds.sty...
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/ifluatex.sty.ltxml... 0.00 sec) 0.06 sec) 0.19 sec) 0.20 sec)
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/kvdefinekeys.sty.ltxml...
(Processing definitions /usr/share/texlive/texmf-dist/tex/generic/oberdiek/kvdefinekeys.sty... 0.03 sec) 0.04 sec)
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/kvoptions.sty.ltxml...
(Processing definitions /usr/share/texlive/texmf-dist/tex/latex/oberdiek/kvoptions.sty... 0.16 sec) 0.16 sec)
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/nameref.sty.ltxml...
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/refcount.sty.ltxml...
(Processing definitions /usr/share/texlive/texmf-dist/tex/latex/oberdiek/refcount.sty... 0.07 sec) 0.07 sec)
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/gettitlestring.sty.ltxml...
(Processing definitions /usr/share/texlive/texmf-dist/tex/generic/oberdiek/gettitlestring.sty... 0.29 sec) 0.29 sec) 0.39 sec)
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/bitset.sty.ltxml...
(Processing definitions /usr/share/texlive/texmf-dist/tex/generic/oberdiek/bitset.sty...
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/intcalc.sty.ltxml...
(Processing definitions /usr/share/texlive/texmf-dist/tex/generic/oberdiek/intcalc.sty... 0.06 sec) 0.06 sec)
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/bigintcalc.sty.ltxml...
(Processing definitions /usr/share/texlive/texmf-dist/tex/generic/oberdiek/bigintcalc.sty...
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/pdftexcmds.sty.ltxml...
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/iftex.sty.ltxml... 0.00 sec) 0.01 sec) 0.19 sec) 0.19 sec) 0.52 sec) 0.52 sec) 1.50 sec)
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/latexml.sty.ltxml... 0.01 sec)
ltxmojo (LaTeXML version 0.8.6; revision 87f8706)
invoked as [/home/deginev/LaTeXML-Plugin-ltxmojo/script/ltxmojo ]
processing started Mon Jan 24 18:20:49 2022
(Digesting TeX Anonymous String...
(Processing content Literal String...
(Processing content Literal String...
(Processing content Literal String... 0.01 sec)
Error:unexpected:\usepackage The current command '\usepackage' can only appear in the preamble
at String; line 1 col 11 - line 1 col 11
In Core::Definition::Constructor[\usepac... /home/deginev/perl5/lib/perl5/LaTeXML/Package/LaTeX.pool.ltxml; line 808
<= Core::Stomach[@0xafb5fe8] <= ...
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/babel.sty.ltxml...
(Processing definitions /usr/share/texlive/texmf-dist/tex/generic/babel/babel.sty...
(Processing definitions /usr/share/texlive/texmf-dist/tex/generic/babel/switch.def... 0.05 sec)
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/frenchb.ldf.ltxml...
(Processing definitions /usr/share/texlive/texmf-dist/tex/generic/babel-french/frenchb.ldf...
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/babel.def.ltxml...
(Processing definitions /usr/share/texlive/texmf-dist/tex/generic/babel/babel.def...
(Processing definitions /usr/share/texlive/texmf-dist/tex/generic/config/language.def... 0.07 sec)
Error:unexpected:\@onlypreamble The current command '\@onlypreamble' can only appear in the preamble
at babel.def; line 146 col 0 - line 146 col 27
In Core::Definition::Primitive[\@onlypre... /home/deginev/perl5/lib/perl5/LaTeXML/Package/LaTeX.pool.ltxml; line 5398
<= Core::Stomach[@0xafb5fe8] <= Core::Gullet[@0xba09b80] <= Core::Definition::Expandable[\input T... <= ...
Error:unexpected:\@onlypreamble The current command '\@onlypreamble' can only appear in the preamble
at babel.def; line 151 col 0 - line 151 col 32
In Core::Definition::Primitive[\@onlypre... /home/deginev/perl5/lib/perl5/LaTeXML/Package/LaTeX.pool.ltxml; line 5398
<= Core::Stomach[@0xafb5fe8] <= Core::Gullet[@0xba09b80] <= Core::Definition::Expandable[\input T... <= ...
Error:unexpected:\@onlypreamble The current command '\@onlypreamble' can only appear in the preamble
at babel.def; line 163 col 0 - line 163 col 33
In Core::Definition::Primitive[\@onlypre... /home/deginev/perl5/lib/perl5/LaTeXML/Package/LaTeX.pool.ltxml; line 5398
<= Core::Stomach[@0xafb5fe8] <= Core::Gullet[@0xba09b80] <= Core::Definition::Expandable[\input T... <= ...
Error:unexpected:\@onlypreamble The current command '\@onlypreamble' can only appear in the preamble
at babel.def; line 283 col 0 - line 283 col 22
In Core::Definition::Primitive[\@onlypre... /home/deginev/perl5/lib/perl5/LaTeXML/Package/LaTeX.pool.ltxml; line 5398
<= Core::Stomach[@0xafb5fe8] <= Core::Gullet[@0xba09b80] <= Core::Definition::Expandable[\input T... <= ...
Error:unexpected:\@onlypreamble The current command '\@onlypreamble' can only appear in the preamble
at babel.def; line 284 col 0 - line 284 col 23
In Core::Definition::Primitive[\@onlypre... /home/deginev/perl5/lib/perl5/LaTeXML/Package/LaTeX.pool.ltxml; line 5398
<= Core::Stomach[@0xafb5fe8] <= Core::Gullet[@0xba09b80] <= Core::Definition::Expandable[\input T... <= ...
Error:unexpected:\@onlypreamble The current command '\@onlypreamble' can only appear in the preamble
at babel.def; line 285 col 0 - line 285 col 25
In Core::Definition::Primitive[\@onlypre... /home/deginev/perl5/lib/perl5/LaTeXML/Package/LaTeX.pool.ltxml; line 5398
<= Core::Stomach[@0xafb5fe8] <= Core::Gullet[@0xba09b80] <= Core::Definition::Expandable[\input T... <= ...
Error:unexpected:\@onlypreamble The current command '\@onlypreamble' can only appear in the preamble
at babel.def; line 631 col 0 - line 631 col 32
In Core::Definition::Primitive[\@onlypre... /home/deginev/perl5/lib/perl5/LaTeXML/Package/LaTeX.pool.ltxml; line 5398
<= Core::Stomach[@0xafb5fe8] <= Core::Gullet[@0xba09b80] <= Core::Definition::Expandable[\input T... <= ...
Error:unexpected:\@onlypreamble The current command '\@onlypreamble' can only appear in the preamble
at babel.def; line 726 col 0 - line 726 col 31
In Core::Definition::Primitive[\@onlypre... /home/deginev/perl5/lib/perl5/LaTeXML/Package/LaTeX.pool.ltxml; line 5398
<= Core::Stomach[@0xafb5fe8] <= Core::Gullet[@0xba09b80] <= Core::Definition::Expandable[\input T... <= ...
Error:unexpected:\@onlypreamble The current command '\@onlypreamble' can only appear in the preamble
at babel.def; line 814 col 0 - line 814 col 33
In Core::Definition::Primitive[\@onlypre... /home/deginev/perl5/lib/perl5/LaTeXML/Package/LaTeX.pool.ltxml; line 5398
<= Core::Stomach[@0xafb5fe8] <= Core::Gullet[@0xba09b80] <= Core::Definition::Expandable[\input T... <= ...
Error:unexpected:\@onlypreamble The current command '\@onlypreamble' can only appear in the preamble
at babel.def; line 948 col 0 - line 948 col 31
In Core::Definition::Primitive[\@onlypre... /home/deginev/perl5/lib/perl5/LaTeXML/Package/LaTeX.pool.ltxml; line 5398
<= Core::Stomach[@0xafb5fe8] <= Core::Gullet[@0xba09b80] <= Core::Definition::Expandable[\input T... <= ...
Error:unexpected:\@onlypreamble The current command '\@onlypreamble' can only appear in the preamble
at babel.def; line 1393 col 0 - line 1393 col 35
In Core::Definition::Primitive[\@onlypre... /home/deginev/perl5/lib/perl5/LaTeXML/Package/LaTeX.pool.ltxml; line 5398
<= Core::Stomach[@0xafb5fe8] <= Core::Gullet[@0xba09b80] <= Core::Definition::Expandable[\input T... <= ...
0.71 sec)
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/babel_support.sty.ltxml... 0.00 sec) 0.72 sec)
Info:latex:(babel) Package babel Info: Making : an active character
at frenchb.ldf; line 342 col 26 - line 342 col 26
Info:latex:(babel) Package babel Info: Making ; an active character
at frenchb.ldf; line 343 col 26 - line 343 col 26
Info:latex:(babel) Package babel Info: Making ! an active character
at frenchb.ldf; line 344 col 26 - line 344 col 26
Info:latex:(babel) Package babel Info: Making ? an active character
at frenchb.ldf; line 345 col 26 - line 345 col 26
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/t1.fontmap.ltxml... 0.00 sec)
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/ly1.fontmap.ltxml... 0.00 sec)
Error:unexpected:\@onlypreamble The current command '\@onlypreamble' can only appear in the preamble
at frenchb.ldf; line 966 col 0 - line 966 col 27
In Core::Definition::Primitive[\@onlypre... /home/deginev/perl5/lib/perl5/LaTeXML/Package/LaTeX.pool.ltxml; line 5398
<= Core::Stomach[@0xafb5fe8] <= Core::Gullet[@0xba09b80] <= Core::Definition::Expandable[\InputIf... <= ...
2.18 sec) 2.20 sec)
Info:unexpected:main=english Unexpected option 'main=english' passed to babel.sty
at babel.sty; line 310 col 0 - line 310 col 16
2.97 sec) 2.97 sec)
Error:unexpected:\RequirePackage The current command '\RequirePackage' can only appear in the preamble
at String; line 4 col 6 - line 4 col 6
In Core::Definition::Constructor[\Requir... /home/deginev/perl5/lib/perl5/LaTeXML/Package/LaTeX.pool.ltxml; line 821
<= Core::Stomach[@0xafb5fe8] <= Core::Gullet[@0xba09b80] <= Core::Stomach[@0xafb5fe8] <= ...
(Loading /home/deginev/perl5/lib/perl5/LaTeXML/Package/scalefnt.sty.ltxml... 0.00 sec)
Error:unexpected:\RequirePackage The current command '\RequirePackage' can only appear in the preamble
at String; line 4 col 6 - line 4 col 6
In Core::Definition::Constructor[\Requir... /home/deginev/perl5/lib/perl5/LaTeXML/Package/LaTeX.pool.ltxml; line 821
<= Core::Stomach[@0xafb5fe8] <= Core::Gullet[@0xba09b80] <= Core::Stomach[@0xafb5fe8] <= ...
Warning:latex:(frenchb.ldf) Package frenchb.ldf Warning: OT1 encoding should not be used for French.\MessageBreakAdd \usepackage[T1]{fontenc} to the
at String; line 8 col 0 - line 8 col 16
preamble\MessageBreakof your document,
In Core::Definition::Primitive[\GenericW... /home/deginev/perl5/lib/perl5/LaTeXML/Package/LaTeX.pool.ltxml; line 5400
3.61 sec) 3.61 sec)
Error:expected:\end{document} Input ended while environment document was open
at String; line 1 col 0 - line 1 col 0
3.61 sec)
(Building...
(Loading compiled schema /home/deginev/perl5/lib/perl5/LaTeXML/resources/RelaxNG/LaTeXML.model... 0.01 sec) 0.32 sec)
(Rewriting... 0.00 sec)
(Finalizing... 0.01 sec)
Conversion complete: 1 warning; 16 errors
(post-processing...
(Scan processing...
Scan: DBStatus: 41/0 objects
0.00 sec)
(MakeBibliography processing... 0.00 sec)
(CrossRef processing...
Warning:expected:ids Missing File location for ID: bib.bib3,bib.bib6,bib.bib5,bib.bib1,bib.bib4,bib.bib2,bib.bib7
In Post::CrossRef[@0xfb7f238] ->process
0.00 sec)
(XSLT[using LaTeXML-html5.xsl] processing... 0.01 sec) 0.02 sec)
Post-processing complete: 1 warning
processing finished Mon Jan 24 18:20:53 2022
Status:conversion:2