Multi-scale tissular-cellular model for wound healing

Patrizia Bagnerini (Dime - Università di Genova, Italy)Luis Almeida (CNRS, Laboratoire J.L. Lions – Univ. Pierre et Marie Curie, France)

Collaborators: S. Noselli (Institut Valrose Biology Nice) A. Jacinto (Instituto de Medicina Molecular Lisbon)S. Vincent (Ecole Normale Superieure Lyon)

Dorsal closure Drosophila embryo

Wound healing in Pupa

Epiboly in Zebrafish

Models for wound healing and regenerative repair

Embryos (also human ones up to the 6th month) and adults of some species like Zebrafish can make perfect regenerative repair

Adult wound healing produces fibrotic tissue known as a scar: heart attacks, burns, cirrhosis

Why study embryo wound healing?

1. No inflammatory response2. Perfect scarless repair3. Many similarities concerning the genetic pathways of adults and embryos4. Conserved pathways across species

10 days

Drosophila melanogaster

ANR project REGENR (2011) on Zebrafish

Gurtner, & al, Wound repair and regeneration, Nature. 2008

Extension of an epithelial membrane to close a hole appears both in morphogenesis and in tissue repair Purse-string contraction

Therapies that interfere with the purse-string: cancer metastasis, acute lung injury, inflammatory bowel diseases Garcia-Fernandez & al, Int. J. Dev. Biol, 2009

Contractive structure: filamentous of actin and myosin II (a motor protein)

Mean Curvature flow + other terms

Wound healing in Drosophila embryo Dorsal closure in Drosophila embryo

Wound healing in pupa, epiboly in zebrafish, etc.

Purse-string contraction

Wound healing in Drosophila embryo Wound healing in Drosophila Pupa

Actin cable tension + Epidermal tension Actin cable tension + Lamellipodial crawling

Curvature flow + Vertical force Curvature flow + normal flow

•L. Almeida, P. Bagnerini, A. Habbal, S. Noselli, F. Serman, Singularities in nonlinear evolution phenomena and applications, 2009•L. Almeida, P. Bagnerini, A. Habbal, S. Noselli, F. Serman, Journal of Theoretical Biology 2011•L. Almeida, P. Bagnerini, A. Habbal, Computer and Mathematics with Applications 2012•L. Almeida, A. chambolle, A. Novaga, preprint 2012

Continuum PDE model for tissue: purse-string + terms depending on application

i

Wi

Mt

Mb

Di

epidermal stretching

actin cable

connective tissue contraction

wi

M

M

t

b

• Simplicity• Small number of parameters to fit• Linearity (separation of contributions during optimization)

Quasi-static approach : succession of equilibria

Simple linear elastic model -> more realistic visco-elastic model

Purse-string model: epidermal wound in Drosophila embryo

ui displacement vector at time step i

Purse-string model: epidermal wound in Drosophila embryo

1. We compute the displacement field {ui} (solution of the PDE) outside the hole (and extend it inside)

Succession of equilibria at each time step i

1. We compute the displacement field {ui} (solution of the PDE) outside the hole (and extend it inside)

2. We evolve wi based on {ui} by Level Set methods (Osher-Sethian)

Purse-string model: epidermal wound in Drosophila embryo

Succession of equilibria at each time step i

1. We compute the displacement field {ui} (solution of the PDE) outside the hole (and extend it inside)

2. We evolve wi based on {ui} by Level Set methods (Osher-Sethian)

3. We obtain the new contour at time i+1

Purse-string model: epidermal wound in Drosophila embryo

Succession of equilibria at each time step i

We assume parameters C1,C2,C3 constant in time. We optimize parameters minimizing the area of the symmetric difference between computed and experimental contours

Some results for wound healing

For wound healing : the model is predictive(optimization made on first 9 contours and validated by the following 9 images)

Caco2BBE gut cells (Jacinto et al., 2000)

Human keratinocytes

Extracted

Simulated

Global Process of Dorsal Closure

PA

amnioserosa

ectoderm

hPrevious work: ODE modelHutson & al, Science, 2003

• ectoderm resistance C1 • actin cable tension C2• amnioserosa contraction C3• zipping C4• head and tail elongation

Wild type dorsal closure

We add a zipping parameter C4 in the model.

Beyond wild-type

Spastin: Micro-Tubule severing protein – reduces Filopodia density and activity (Jankovics, Brunner, Dev Cell 2006)

10 wild-type and 10 spastin mutants

Aim: understand genetic pathways and their effects

We couple the continuum PDE model with cellular one

1. Experiments involving group of cells : we modify genes by UAS-GAL4 (1993) • thousand of varietes which express GAL4 in subset of tissue fluorescent in• lines with UAS region next to fluorescent GFP or RFP gene a subset• mutants

2. Multiscale: high number of cells• Continuum model in a part of tissue and continuum + cellular or only cellular in another

3. Cell-sorting: spatial control of tissue and compartment boundary maintenance• Two hypothesis: differential cell adhesion or differential interfacial tension. Which protein?

S. Schilling & al, Plos Computational Biology 2012

3. Cell called Mixer cell which transdifferentiates (patched becomes engrailed) and moves breaking the compartment boundary:

M. Gettings, F. Serman, R. Rousset, P. Bagnerini, L. Almeida, S. Noselli, Plos Biology 2010

We generate initial configuration

1. Quadrilater mesh2. Baricenters3. Jiggle4. Voronoi diagram5. Cut cells

Matlab code

1. We solve PDE model vector field u1,u22. We use u1,u2 to move junctions of cells (ode system)3. We minimize energy functional

C. Dahmann, K. Basler, Cell 2000

Clone of Smoothened gene in wing

M. Rauzi, P. Venant, T. Lecuit, P. Lenne, Nature Cell Biology 2008

Example of cell-sorting

Conclusions

Continuum model + cellular model + energy minimization

1. Continuum model : purse-string model: curvature type flow + other terms

2. Cellular model : ode system (or level set?)

3. Energy functional: line tension -> equilibrium

Aim: to treat experiments with cells expressing different genes

Some applications:

1. Wound healing in Drosophila embryo2. Wound healing in Drosophila pupa3. Dorsal closure in morphogenesis4. Zebrafish epiboly5. C-shaped wound

Thank you for your attention.

Morphogenetic movements in embryogenesis

3h

5h

8h

11h

14h

6

7

8

3

10

14

12

11

15

17

gastrulation

mesoderm

endoderm

germ band extension

germ band retraction

dorsal closure

ventral closure

head involution

Beyond wild-type : Spastin cost function

Spastin: Micro-Tubule severing protein – reduces Filopodia density and activity (Jankovics, Brunner, Dev Cell 2006)

Convexity: Wounds in Ventral Epidermis of Pupa

“beans”

Wounds in Pupae (with Jacinto) versus Wounds in Embryos (with Hutson)=

Fibroctic versus Regenerative Repair

Study positive curvature flows (with A. Chambolle and M.Novaga)

C-shaped wounds

10 days

Convexity: In vitro single layer «wound healing »

Caco2BBE gut cells (Jacinto et al. Curr. Biol., 2000)

Convexity: Zebrafish epiboly

Zebrafish epiboly - C.P. Heisenberg (IST Austria)

(model organism for regenerative repair)

Solnica-Krezel, Curr. Opin. Gen. & Dev. 2006

Conclusions

Purse-string model: curvature type flow + other terms depending on applications

Some applications:

1. Wound healing in Drosophila embryo2. Wound healing in Drosophila pupa3. Dorsal closure in morphogenesis4. Zebrafish epiboly5. C-shaped wound

In morphogenesis we validated model with wild-type and mutants

Work in progress:

6. Validation in wound healing with wild-type mutants7. Purse-string formation before contraction

Thank you for your attention.

Collaborators

Mechanics and Numerical methods : L. Almeida (CNRS) and A. Habbal (Nice)

Curvature Flows: A. Chambolle (Polytechnique) and M. Novaga (Padova)

Gene regulatory networks and multiagent models: J, Demongeot (Grenoble)

Dorsal Closure: F. Serman and S. Noselli (Nice), G. Edwards (Duke)

Wound Healing in Pupa: A. Jacinto (Lisbon)

Wound Healing in Drosophila Embryos: S. Hutson (Vanderbilt)

Zebrafish Tail Fin Regeneration in Larvae: A. Jacinto (Lisbon) and N. Peyriéras (INAF)

Zebrafish Tail Fin Regeneration in Adults: D. Dhouailly and F. Giroud (UJF, Grenoble)

Boundary evolution between 2 equilibria

Harmonic extension inside the woundfor the level set method

Purse-string formation: actio-myosin and deformation waves

In collaboration with Jacinto

Wound healing in Pupa: first minutes

Simple PDE model

+

to have the harmonic extension of ui in the inside

Linearity of the model We solve one PDE for each parameter

Dorsal Closure and fly embryo wounds: a model for vertebrate and adult wound healing and morphogenetic events

- Wounds in chick embryos (Lawson and England 1998)

- Wounds in Drosophila larva and pupae: requirement of the JNK pathway during wing imaginal discs healing (Bosh et al. 2005)

- Wounds in Drosophila adults:expression of the Jnk pathway at the edge of abdomen wounds (Rämet et al. 2002)

- Small wounds in adult rabbit cornea (Danjo and Gibson 1998)

- Epiboly in Zebrafish embryos: actin ring pulls the superficial enveloping layer (Solnica-Krezel 2006)

- Morphogenesis in Mice : involvement of the Jnk pathway in closure of the eyelid, neural tube closure and optic fissure (Xia and Karin 2004)

phalloidin

Conserved mechanisms across species

Hypothesis :

2D

two arcs of circle

three forces on the middle point of each leading edge (cable tension, amnioserosa contraction, ectodherm resistance) Newton’s second law of dynamics - in a condition of high viscosity - at the symmetry point on the Y axis

dW = - a nb of filopodial contact dt = - a wf

= - a / tan(q) ≈ - a W / 2H

h

Previous work: ODE model (Hutson et al., Science, 2003)

filopodia (uniform length l)

wf =lenght with filopodial contact

2lq

Finally...2 parameters... V rate of closure kz rate of zipping

Very restrictive on the geometry (mostly because of the symmetry requirement)

Previous work

Wound Healing :

Math: ODE models (essentially for circular wounds - Murray and collaborators, ‘90)

Fly embryos - influence of small GTPases (Wood et al. Nature Cell Biol. 2003)

Beyond wild-type

Spastin: Micro-Tubule severing protein – reduces Filopodia density and activity (Jankovics, Brunner, Dev Cell 2006)

Our model Hutson & al model

In vitro single layer «wound healing »

Human keratinocytes (G. Ponzio and R. Busca, Fac. Medecine, Nice)