1. A Stochastic Model of Cell Differentiation
B. Laforge
LPNHE Paris, Université Paris VI
Presentation based on:
Modeling embryogenesis and cancer:
An approach based on an equilibrium between the autostabilization of
stochastic gene expression and the interdependence of cells for proliferation
Prog. Biophys. Mol. Biol Sep;89(1):93-120
B.L., D. Guez,M. Martinez M, JJ. Kupiec
Simulation de modèles sélectifs de la différenciation cellulaire.
PhD Thesis of J. Glisse, Université Pierre et Marie Curie–Paris 6, janvier 2009
Directed by JJ. Kupiec and B.L.

2. Why a physicist doing simulation in Biology ?
We all understand the organisation of the universe and its components through a representation : biology needs a theory !
We have to understand if and how we could have a unified representation of physical and biological processes.
It is not coherent to assume that physics is governed by stochastics processes while biology is based on information theory
I am convinced that organisation is really the result of different dynamics occuring inside the system and at its vicinity.
Simulation is the only way to adress this question on a general basis.

3. Emergence also occurs in physics
I don’t think we need non linear processes or chaotic systems
to have not trivial organisation, competition between
Internal dynamics and outside constraints are enough

4. The Darwinian Model of Cell differentiation
Darwinian Model of Cell Identification
Lot of experimental data demon-strate a stochastic component in gene expression
 e.g. see review in Nat. Rev. Genet Jun;6(6):
Need to build a model of molecular biology processes incorporating that phenomenology
random
event A B
No specific signal
Stochastic gene expression
- Homogeneity postulated
Cellular interactions stabilise
cell phenotype
- Predicts intrinsic variability
By simulation : test intrinsic properties of the Darwinian model to create organised tissues

5. Description of the model
Two cell types A and B grow in a 2D matrix
A and B change their phenotype randomly
A cells synthesize a molecules whereas
B cells synthesize b molecules.
a and b molecules diffuse following Fick’s Laws
3 matrices are used to hold cells and (a,b) densities

6. In each cell, at each simulation step
1) a and b molecules are synthesized
2) a and b are degraded
3) a and b diffuse in the environment (Fick’s laws)
4) Cell Identification (A or B) is stochastically determined.
The switching probabilities
depend on a or b molecules densities
Stochastic to
Deterministic
behaviour
Autostabilisation :
the cell is stabilised by its own product
Interstabilisation :
the cell is stabilised by the molecules produced by the other cell type

7. Inter and auto stabilisation effects
Interstabilisation
(small clusters)
Autostabilisation
(large areas)

8. 2nd model + 1st model in autostabilisation mode
Interdependance for proliferation :
A (resp. B) cells metabolise a certain quantity
of b (resp. a) molecules to survive and proliferate
a and b molecules can be seen as pleiotropic
growth factors

9. Results - An invariant bi-layer structure can be obtained with lateral
finite growth :
 no stop signal but equilibrium !
-Two symmetric gradients of a
and b molecules are also created
transversally to the bi-layer.
- ‘‘embryo mortality’’ is bound to the intrinsic stochastic nature of the model
cell death increases the bi-layer formation probability
 could explain an evolutionary
origin of apoptosis !

10. Di-layer formation is conserved
What happens in 3D ?
Di-layer formation is conserved

11. Some details on few important properties of the model
- Molecular Gradient structuration
- Cell death function in living organisms
Impact of the stochasticity on organism
Development
What happens when the equilibrium is
disrupted ?

12. Gradients and bilayer are formed at the same time

13. We could have systems where gradients do not
preexist before the structure
as stated by position information theory

14. Simulation Parameters

15. Stochasticity and Structure Formation kinetics
Limited stochasticity
Very stochastic
Deterministic step
b = 0.1
b = 0.5
b = 4.0
Limited stochasticity provide early bilayer formation with lower formation time fluctuation :
☺ selective advantage for organism ruled by such dynamics

16. Stochasticity and Structure Formation efficiency
29 * 75 * 1000 = simulations

17. Stochasticity and Structure Formation efficiency

18. Number of simulations

19. % of success vs stochasticity

20. Effect of cell death on the probability to produce a structure

21. Stochasticity and Structure Formation efficiency

22. Change of a parameter disrupt the equilibrium
Suggest a new mechanism for
control of cell proliferation

23. Change of a parameter (here L) disrupt the Equilibrium
Suggest a new mecanism for cancer

24. Conclusion
The selective model has to be modified :
Differentiation and tissues organisation could result from an equilibrium between stochastic gene expression, autostabilisation of phenotypes and interdependence for proliferation.
Genetic program could be replaced by an Dynamic Equilibrium Principle
Stochasticity gives a better structuration in our model
Cell apoptosis can be seen as a consequence of double level of selection of organisms better structured with “apoptosis” cells
This model suggest a new mechanism for the control of cell proliferation

25. Backup Slides

26. Darwinian Model of Cell Identification
Cell differentiation Models
Deterministic Cell
differentiation
Darwinian Model of Cell Identification
Kupiec, 1981
random
event
Information
A signal triggers differentiation
Cell stability is assumed
Heterogeneity is postulated
No prediction of intrinsic variability A B
No specific signal
Stochastic gene expression
- Homogeneity postulated
Cellular interactions stabilise
cell phenotype
- Predicts intrinsic variability
By simulation : test intrinsic properties of the Darwinian model to create organised tissues

27. Remove autostabilisation
organisation properties and finite growth lost

28. Remove interdependence
for proliferation
organisation properties and finite growth lost

29. Bi-layer structure production
A-E : Growth phase
E-F : Stabilisation

30. Results - An invariant bi-layer structure can be obtained with lateral
finite growth :
 no stop signal but equilibrium !
-Two symmetric gradients of a
and b molecules are also created
transversally to the bi-layer.
- Simulations produce individuals with common property (bi-layer)
 Definition of a species !
- cell death increases the bi-layer formation probability (selective origin of apoptosis !)
- ‘‘embryo mortality’’ is bound to the intrinsic stochastic nature of the modèle