Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Computational modelling of cell-cell adhesion
in development and disease
Jonathan A. Sherratt
Department of Mathematics
Heriot-Watt University
Microsoft Research Ltd, February 13, 2009
This talk can be downloaded from my web site
www.ma.hw.ac.uk/∼jas
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Collaborators
Nicola Armstrong Formerly Heriot-Watt University
Stephen Gourley University of Surrey
Kevin Painter Heriot-Watt University
Stephen Turner Royal Holloway, University of London
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Outline
1
Introduction and Basic Modelling
2
A New Continuum Model for Cell Adhesion
3
Application I: Cell Sorting and Aggregation
4
Application II: Somite Formation
5
Application III: Cancer Invasion
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
What is Cell-Cell Adhesion?
Theoretical Modelling of Cell Populations
Cell Adhesion in Individual Cell-Based Models
Potts-type Model for Cancer Invasion
Cell Adhesion in Continuum Models
Outline
1
Introduction and Basic Modelling
2
A New Continuum Model for Cell Adhesion
3
Application I: Cell Sorting and Aggregation
4
Application II: Somite Formation
5
Application III: Cancer Invasion
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
What is Cell-Cell Adhesion?
Theoretical Modelling of Cell Populations
Cell Adhesion in Individual Cell-Based Models
Potts-type Model for Cancer Invasion
Cell Adhesion in Continuum Models
What is Cell-Cell Adhesion?
Cells bind to each other
through cell adhesion
molecules.
This is important in a range
of developmental and
pathological contexts:
Embryonic cells adhere selectively, enabling them to sort
into tissues and organs
Altered adhesion properties are thought to be important in
tumour invasion
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
What is Cell-Cell Adhesion?
Theoretical Modelling of Cell Populations
Cell Adhesion in Individual Cell-Based Models
Potts-type Model for Cancer Invasion
Cell Adhesion in Continuum Models
Theoretical Modelling of Cell Populations
Individual cell based models:
explicit representation of each
cell
Jonathan A. Sherratt
lattice-based models
lattice-free models
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
What is Cell-Cell Adhesion?
Theoretical Modelling of Cell Populations
Cell Adhesion in Individual Cell-Based Models
Potts-type Model for Cancer Invasion
Cell Adhesion in Continuum Models
Theoretical Modelling of Cell Populations
Individual cell based models:
explicit representation of each
cell
lattice-based models
lattice-free models
Continuum models: cell density
is a function of space and time
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
What is Cell-Cell Adhesion?
Theoretical Modelling of Cell Populations
Cell Adhesion in Individual Cell-Based Models
Potts-type Model for Cancer Invasion
Cell Adhesion in Continuum Models
Cell Adhesion in Individual Cell-Based Models
Representation of adhesion in individual cell-based models is
relatively straightforward because they include explicit
representations of the cell boundaries.
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
What is Cell-Cell Adhesion?
Theoretical Modelling of Cell Populations
Cell Adhesion in Individual Cell-Based Models
Potts-type Model for Cancer Invasion
Cell Adhesion in Continuum Models
Cell Adhesion in Individual Cell-Based Models
Representation of adhesion in individual cell-based models is
relatively straightforward because they include explicit
representations of the cell boundaries.
Example: Potts-type model for cancer invasion
(S. Turner, J.A. Sherratt (2002) J. Theor. Biol. 216:85).
each cell is represented as a group of squares on a lattice
cell movement occurs via rearrangements that tend to
reduce overall energy
“energy”=adhesion energy + elastic energy
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
What is Cell-Cell Adhesion?
Theoretical Modelling of Cell Populations
Cell Adhesion in Individual Cell-Based Models
Potts-type Model for Cancer Invasion
Cell Adhesion in Continuum Models
Potts-type Model for Cancer Invasion
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
What is Cell-Cell Adhesion?
Theoretical Modelling of Cell Populations
Cell Adhesion in Individual Cell-Based Models
Potts-type Model for Cancer Invasion
Cell Adhesion in Continuum Models
Potts-type Model for Cancer Invasion
1
Randomly choose a lattice point
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
What is Cell-Cell Adhesion?
Theoretical Modelling of Cell Populations
Cell Adhesion in Individual Cell-Based Models
Potts-type Model for Cancer Invasion
Cell Adhesion in Continuum Models
Potts-type Model for Cancer Invasion
1
2
Randomly choose a lattice point
Randomly choose a neighbouring point
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
What is Cell-Cell Adhesion?
Theoretical Modelling of Cell Populations
Cell Adhesion in Individual Cell-Based Models
Potts-type Model for Cancer Invasion
Cell Adhesion in Continuum Models
Potts-type Model for Cancer Invasion
Potential
site
copy
1
2
3
Randomly choose a lattice point
Randomly choose a neighbouring point
Calculate the total energy before and after a “site copy”
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
What is Cell-Cell Adhesion?
Theoretical Modelling of Cell Populations
Cell Adhesion in Individual Cell-Based Models
Potts-type Model for Cancer Invasion
Cell Adhesion in Continuum Models
Potts-type Model for Cancer Invasion
Potential
site
copy
1
2
3
4
Randomly choose a lattice point
Randomly choose a neighbouring point
Calculate the total energy before and after a “site copy”
If ∆E < 0, accept the copy; otherwise accept with
probability exp(−∆E/constant) (“Boltzmann weighting”)
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
What is Cell-Cell Adhesion?
Theoretical Modelling of Cell Populations
Cell Adhesion in Individual Cell-Based Models
Potts-type Model for Cancer Invasion
Cell Adhesion in Continuum Models
Potts-type Model for Cancer Invasion
Potential
site
copy
1
2
3
4
5
Randomly choose a lattice point
Randomly choose a neighbouring point
Calculate the total energy before and after a “site copy”
If ∆E < 0, accept the copy; otherwise accept with
probability exp(−∆E/constant) (“Boltzmann weighting”)
Repeat many times
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
What is Cell-Cell Adhesion?
Theoretical Modelling of Cell Populations
Cell Adhesion in Individual Cell-Based Models
Potts-type Model for Cancer Invasion
Cell Adhesion in Continuum Models
Potts-type Model for Cancer Invasion
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
What is Cell-Cell Adhesion?
Theoretical Modelling of Cell Populations
Cell Adhesion in Individual Cell-Based Models
Potts-type Model for Cancer Invasion
Cell Adhesion in Continuum Models
Potts-type Model for Cancer Invasion
Click to play
the movie
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
What is Cell-Cell Adhesion?
Theoretical Modelling of Cell Populations
Cell Adhesion in Individual Cell-Based Models
Potts-type Model for Cancer Invasion
Cell Adhesion in Continuum Models
Potts-type Model for Cancer Invasion
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
What is Cell-Cell Adhesion?
Theoretical Modelling of Cell Populations
Cell Adhesion in Individual Cell-Based Models
Potts-type Model for Cancer Invasion
Cell Adhesion in Continuum Models
Cell Adhesion in Continuum Models
Representation of adhesion in continuum models is much
harder because they do not involve explicit representation of
the cell boundaries.
Importance:
there are many established continuum models for other
aspects of development, cancer, wound healing etc
mathematical analysis is often possible for continuum
models, but is rarely possible for discrete models
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Derivation of the Model
Model Details
Outline
1
Introduction and Basic Modelling
2
A New Continuum Model for Cell Adhesion
3
Application I: Cell Sorting and Aggregation
4
Application II: Somite Formation
5
Application III: Cancer Invasion
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Derivation of the Model
Model Details
Derivation of the Model I
Mass conservation:
Rate of change
of cell density
∂n/∂t
=
Input
−output
+
Birth/
death
−∂J/∂x
Here n(x, t) = cell density, and J = cell flux
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Derivation of the Model
Model Details
Derivation of the Model I
Mass conservation:
Rate of change
of cell density
∂n/∂t
=
Input
−output
+
Birth/
death
−∂J/∂x
Here n(x, t) = cell density, and J = cell flux
Adhesive flux Ja is proportional to the force due to breaking
and forming adhesive bonds
(Stokes’ Law: low Reynolds number)
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Derivation of the Model
Model Details
Derivation of the Model II
The force on a cell at x exerted by cells a distance x0 away
depends on
1
cell density at x + x0
2
distance |x0 |
3
sign of x0 (⇒ direction of force)
f (x, x0 ) = α · g (n(x + x0 , t)) · ω(x0 )
Jonathan A. Sherratt
x
x + x0
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Derivation of the Model
Model Details
Derivation of the Model II
The force on a cell at x exerted by cells a distance x0 away
depends on
1
cell density at x + x0
2
distance |x0 |
3
sign of x0 (⇒ direction of force)
f (x, x0 ) = α · g (n(x + x0 , t)) · ω(x0 )
x
x + x0
Total force = sum of all forces acting on cells at x
R +R
F (x) = −R f (x, x0 ) dx0
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Derivation of the Model
Model Details
Model Details: The Sensing Radius, R
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Mathematical Model for One Cell Population
Aggregation and Sorting in an In Vitro Experiment
A Numerical Simulation of Aggregation and Sorting
Computational Issues
Outline
1
Introduction and Basic Modelling
2
A New Continuum Model for Cell Adhesion
3
Application I: Cell Sorting and Aggregation
4
Application II: Somite Formation
5
Application III: Cancer Invasion
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Mathematical Model for One Cell Population
Aggregation and Sorting in an In Vitro Experiment
A Numerical Simulation of Aggregation and Sorting
Computational Issues
Mathematical Model for One Cell Population
To model cell aggregation in vitro, we assume random
(diffusive) and adhesive cell movement, with no birth/death.
This gives the nondimensional model equation
#
" Z
+1
∂
∂2n
∂n
−α
n(x + x0 , t) [1 − n(x + x0 , t)] sign(x0 ) dx0
n
=
∂t
∂x
∂x 2
−1
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Mathematical Model for One Cell Population
Aggregation and Sorting in an In Vitro Experiment
A Numerical Simulation of Aggregation and Sorting
Computational Issues
Mathematical Model for One Cell Population
To model cell aggregation in vitro, we assume random
(diffusive) and adhesive cell movement, with no birth/death.
This gives the nondimensional model equation
#
" Z
+1
∂
∂2n
∂n
−α
n(x + x0 , t) [1 − n(x + x0 , t)] sign(x0 ) dx0
n
=
∂t
∂x
∂x 2
−1
The parameter α reflects the strength of adhesion; we
expect aggregation of disassociated cells when α is large.
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Mathematical Model for One Cell Population
Aggregation and Sorting in an In Vitro Experiment
A Numerical Simulation of Aggregation and Sorting
Computational Issues
Mathematical Model for One Cell Population
To model cell aggregation in vitro, we assume random
(diffusive) and adhesive cell movement, with no birth/death.
This gives the nondimensional model equation
#
" Z
+1
∂
∂2n
∂n
−α
n(x + x0 , t) [1 − n(x + x0 , t)] sign(x0 ) dx0
n
=
∂t
∂x
∂x 2
−1
The parameter α reflects the strength of adhesion; we
expect aggregation of disassociated cells when α is large.
Linear stability analysis implies aggregation when α > αcrit
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Mathematical Model for One Cell Population
Aggregation and Sorting in an In Vitro Experiment
A Numerical Simulation of Aggregation and Sorting
Computational Issues
A Numerical Simulation of Aggregation
Cell Density
t=0
t=5
t = 10
t = 500
1
1
1
1
0.5
0.5
0.5
0.5
0
0
10
x
20
0
0
10
x
Jonathan A. Sherratt
20
0
0
10
x
20
0
0
10
x
Computational modelling of cell-cell adhesion
20
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Mathematical Model for One Cell Population
Aggregation and Sorting in an In Vitro Experiment
A Numerical Simulation of Aggregation and Sorting
Computational Issues
Aggregation and Sorting in an In Vitro Experiment
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Mathematical Model for One Cell Population
Aggregation and Sorting in an In Vitro Experiment
A Numerical Simulation of Aggregation and Sorting
Computational Issues
Extending the Model to Interacting Cell Populations
To consider cell sorting, we extend the model to two
interacting cell populations
The extended model includes self-population adhesion and
cross-population adhesion
Extension to 2-D is valuable and straightforward
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Mathematical Model for One Cell Population
Aggregation and Sorting in an In Vitro Experiment
A Numerical Simulation of Aggregation and Sorting
Computational Issues
A Numerical Simulation of Aggregation and Sorting
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Mathematical Model for One Cell Population
Aggregation and Sorting in an In Vitro Experiment
A Numerical Simulation of Aggregation and Sorting
Computational Issues
A Numerical Simulation of Aggregation
Click to play
the movie
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Mathematical Model for One Cell Population
Aggregation and Sorting in an In Vitro Experiment
A Numerical Simulation of Aggregation and Sorting
Computational Issues
Computational Issues
Discretise equations in space
and time
At each time step, we must
calculate an integral over a
circular region
Niave computation is very
expensive in computer time
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Mathematical Model for One Cell Population
Aggregation and Sorting in an In Vitro Experiment
A Numerical Simulation of Aggregation and Sorting
Computational Issues
Computational Issues
Discretise equations in space
and time
At each time step, we must
calculate an integral over a
circular region
Niave computation is very
expensive in computer time
Alf Gerisch (Halle) has developed a sophisticated method
based on
1
2
pre-calculating weights for the projection of the spatial grid
onto a radial grid within the circle
using FFT to simplify the resulting system of linear equations
7→ 20-fold speed-up (Gerisch, Chaplain (2008) J. Theor. Biol. 250:684)
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Somite Formation
Adding Adhesive Cells to the Chemical Model
Overview of Somite Application
Outline
1
Introduction and Basic Modelling
2
A New Continuum Model for Cell Adhesion
3
Application I: Cell Sorting and Aggregation
4
Application II: Somite Formation
5
Application III: Cancer Invasion
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Somite Formation
Adding Adhesive Cells to the Chemical Model
Overview of Somite Application
Introduction to Somite Formation
Somites are an initial stage of
segmentation along the
head–tail axis of vertebrates.
Jonathan A. Sherratt
Posterior
Stage 11 chick
Computational modelling of cell-cell adhesion
Anterior
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Somite Formation
Adding Adhesive Cells to the Chemical Model
Overview of Somite Application
Introduction to Somite Formation
Somites are an initial stage of
segmentation along the
head–tail axis of vertebrates.
Posterior
Stage 11 chick
They form in a regular anterior–
posterior sequence, via:
1
Pre-pattern forms in PSM
2
Cells coalesce into somites
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Anterior
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Somite Formation
Adding Adhesive Cells to the Chemical Model
Overview of Somite Application
Mathematical Model for Chemical Signalling
Collier et al (J Theor Biol 207:305, 2000)
proposed a mathematical
model that represents the
chemical signalling that is
thought to underlie somite
formation. Model variables:
u(x, t) conc of a cell
adhesion
molecule
v (x, t) conc of the
signalling
molecule
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Somite Formation
Adding Adhesive Cells to the Chemical Model
Overview of Somite Application
Mathematical Model for Chemical Signalling
Collier et al (J Theor Biol 207:305, 2000)
proposed a mathematical
model that represents the
chemical signalling that is
thought to underlie somite
formation. Model variables:
u(x, t) conc of a cell
adhesion
molecule
∂u
∂t
∂v
∂t
=
=
(u + µv )2
u
Γu (x, t) −
2
κ2
κ1 + κ2 u
2
Γv (x, t)
∂ v
−v +D 2
κ3 + u
∂x
Γu (x, t) = H(ct − x + x1 )
Γv (x, t) = H(ct − x + x2 )
v (x, t) conc of the
signalling
molecule
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Somite Formation
Adding Adhesive Cells to the Chemical Model
Overview of Somite Application
Mathematical Model for Chemical Signalling
Collier et al (J Theor Biol 207:305, 2000)
proposed a mathematical
model that represents the
chemical signalling that is
thought to underlie somite
formation. Model variables:
u(x, t) conc of a cell
adhesion
molecule
v (x, t) conc of the
signalling
molecule
Jonathan A. Sherratt
Time
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Somite Formation
Adding Adhesive Cells to the Chemical Model
Overview of Somite Application
Adding Adhesive Cells to the Chemical Model
We add a third equation, for the cell density n(x, t)
∂
∂n(x, t)
∂n(x, t)
=
Dn
+ αu(x, t)n(x, t)·
∂t
∂x
∂x
Z r
u(x + x0 , t) n(x + x0 , t) (nmax − n(x + x0 , t)) ω(x0 ) dx0
−r
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Somite Formation
Adding Adhesive Cells to the Chemical Model
Overview of Somite Application
Adding Adhesive Cells to the Chemical Model
We add a third equation, for the cell density n(x, t)
∂
∂n(x, t)
∂n(x, t)
=
Dn
+ αu(x, t)n(x, t)·
∂t
∂x
∂x
Z r
u(x + x0 , t) n(x + x0 , t) (nmax − n(x + x0 , t)) ω(x0 ) dx0
−r
The parameter α determines the strength of cell-cell
adhesion
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Somite Formation
Adding Adhesive Cells to the Chemical Model
Overview of Somite Application
Adding Adhesive Cells to the Chemical Model
We add a third equation, for the cell density n(x, t)
∂
∂n(x, t)
∂n(x, t)
=
Dn
+ αu(x, t)n(x, t)·
∂t
∂x
∂x
Z r
u(x + x0 , t) n(x + x0 , t) (nmax − n(x + x0 , t)) ω(x0 ) dx0
−r
The parameter α determines the strength of cell-cell
adhesion
Cell adhesion depends on the concentration of the
adhesion molecule u
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Somite Formation
Adding Adhesive Cells to the Chemical Model
Overview of Somite Application
Adding Adhesive Cells to the Chemical Model
We add a third equation, for the cell density n(x, t)
∂
∂n(x, t)
∂n(x, t)
=
Dn
+ αu(x, t)n(x, t)·
∂t
∂x
∂x
Z r
u(x + x0 , t) n(x + x0 , t) (nmax − n(x + x0 , t)) ω(x0 ) dx0
−r
The parameter α determines the strength of cell-cell
adhesion
Cell adhesion depends on the concentration of the
adhesion molecule u
Note that there is no feedback from the cell equation to the
chemical equations
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Somite Formation
Adding Adhesive Cells to the Chemical Model
Overview of Somite Application
Simulation of Somite Formation
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Somite Formation
Adding Adhesive Cells to the Chemical Model
Overview of Somite Application
Simulation of Somite Formation
Click to play
the movie
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Somite Formation
Adding Adhesive Cells to the Chemical Model
Overview of Somite Application
Overview of Somite Application
Conceptual
biological
models
↓
Mathematical model
of chemical signalling
(Collier et al)
↓
Our extended
mathematical
model
→
Jonathan A. Sherratt
Predictions on the
values of α, c and N
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
Outline
1
Introduction and Basic Modelling
2
A New Continuum Model for Cell Adhesion
3
Application I: Cell Sorting and Aggregation
4
Application II: Somite Formation
5
Application III: Cancer Invasion
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
Introduction to Cancer Invasion
Cells in a solid tumour invade
surrounding tissue due to changes in:
migration
protease/anti-protease production
adhesion
Carcinoma of the uterine cervix
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
Introduction to Cancer Invasion
Cells in a solid tumour invade
surrounding tissue due to changes in:
migration
protease/anti-protease production
adhesion: decreased cell-cell
adhesion and increased cell-matrix
adhesion
Carcinoma of the uterine cervix
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
Modelling Adhesion in Cancer
Variables: n(x, t) tumour cell density, m(x, t) matrix density
cell-cell
adhesion
cell-matrix
adhesion
cell
z
}|
{ z
}|
{ proliferation
z }| {
∂
∂
∂n
= −
[n · Knn ] −
[n · Knm ] + n(1 − n)
∂t
∂x
∂x
∂m
2
= − λ
| · n{z· m}
∂t
matrix
degradation
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
Modelling Adhesion in Cancer
Variables: n(x, t) tumour cell density, m(x, t) matrix density
cell-cell
adhesion
cell-matrix
adhesion
cell
z
}|
{ z
}|
{ proliferation
z }| {
∂
∂
∂n
= −
[n · Knn ] −
[n · Knm ] + n(1 − n)
∂t
∂x
∂x
∂m
2
= − λ
| · n{z· m}
∂t
matrix
degradation
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
Modelling Adhesion in Cancer
Variables: n(x, t) tumour cell density, m(x, t) matrix density
cell-cell
adhesion
cell-matrix
adhesion
cell
z
}|
{ z
}|
{ proliferation
z }| {
∂
∂
∂n
= −
[n · Knn ] −
[n · Knm ] + n(1 − n)
∂t
∂x
∂x
∂m
2
= − λ
| · n{z· m}
∂t
matrix
degradation
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
Modelling Adhesion in Cancer
Variables: n(x, t) tumour cell density, m(x, t) matrix density
cell-cell
adhesion
cell-matrix
adhesion
cell
}|
{ z
}|
{ proliferation
z
z }| {
∂
∂
∂n
= −
[n · Knn ] −
[n · Knm ] + n(1 − n)
∂t
∂x
∂x
Z 1
Knn = α
n(x + x0 , t) · (2 − n(x + x0 , t) − m(x + x0 , t)) · ω(x0 ) dx0
−1
∂m
2
= − λ
| · n{z· m}
∂t
matrix
degradation
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
Modelling Adhesion in Cancer
Variables: n(x, t) tumour cell density, m(x, t) matrix density
cell-cell
adhesion
cell-matrix
adhesion
cell
z
}|
{ z
}|
{ proliferation
z }| {
∂
∂
∂n
= −
[n · Knn ] −
[n · Knm ] + n(1 − n)
∂t
∂x
∂x
Z 1
Knn = α
n(x + x0 , t) · (2 − n(x + x0 , t) − m(x + x0 , t)) · ω(x0 ) dx0
Knm = β
Z
−1
1
m(x + x0 , t) · (2 − n(x + x0 , t) − m(x + x0 , t)) · ω(x0 ) dx0
−1
∂m
2
= − λ
| · n{z· m}
∂t
matrix
degradation
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
Modelling Adhesion in Cancer
Variables: n(x, t) tumour cell density, m(x, t) matrix density
cell-cell
adhesion
cell-matrix
adhesion
cell
z
}|
{ z
}|
{ proliferation
z }| {
∂
∂
∂n
= −
[n · Knn ] −
[n · Knm ] + n(1 − n)
∂t
∂x
∂x
Z 1
Knn = α
n(x + x0 , t) · (2 − n(x + x0 , t) − m(x + x0 , t)) · ω(x0 ) dx0
Knm = β
Z
−1
1
m(x + x0 , t) · (2 − n(x + x0 , t) − m(x + x0 , t)) · ω(x0 ) dx0
−1
∂m
2
= − λ
| · n{z· m}
∂t
matrix
degradation
Extension to 2-D is straightforward
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
Simulation of a Non-Invasive Tumour
For cell-cell adhesion (α) relatively large and cell-matrix
adhesion (β) relatively small, the model predicts a non-invasive
tumour
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
Simulation of a Non-Invasive Tumour
For cell-cell adhesion (α) relatively large and cell-matrix
adhesion (β) relatively small, the model predicts a non-invasive
tumour
t =0
t = 75
Jonathan A. Sherratt
t = 150
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
Simulation of a Non-Invasive Tumour
For cell-cell adhesion (α) relatively large and cell-matrix
adhesion (β) relatively small, the model predicts a non-invasive
tumour
t =0
t = 75
t = 150
Invasion can be initiated either by decreasing cell-cell adhesion
(α) or by increasing cell-matrix adhesion (β)
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
The Sequential Development of an Invasive Tumour
t =0
Stage 1:
non-invasive
tumour growth
Jonathan A. Sherratt
t = 75
t = 150
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
The Sequential Development of an Invasive Tumour
Stage 2:
mutation,
followed by
tumour invasion
(small increase
in cell-matrix
adhesion)
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
The Sequential Development of an Invasive Tumour
Stage 2:
mutation,
followed by
tumour invasion
(large increase
in cell-matrix
adhesion)
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
The Sequential Development of an Invasive Tumour
Tumour morphology:
Detailed studies of
tumour pathology
reveal a correlation
between the invasive
potential of tumours
and their shape.
(Tumour shape is often
quantified via fractal
dimension.)
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
Explanation of Tumour Fingering
Model solns predict:
invasion of uniform matrix ⇒ flat boundary
invasion of non-uniform matrix ⇒ fingering
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
Explanation of Tumour Fingering
Model solns predict:
invasion of uniform matrix ⇒ flat boundary
invasion of non-uniform matrix ⇒ fingering
Basic explanation: invasion speed varies with matrix density.
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
Conclusions and Challenges
Conclusions:
Our model framework successfully reproduces
experimental results on cell aggregation and sorting,
predicts somite formation when added to the chemical cell
cycle model, and is consistent with traditional thinking on
cancer invasion.
For both somite formation and cancer invasion, the model
makes a number of predictions, some of which are
experimentally testable.
The model raises many computational challenges, in
particular concerning extension to 3-D.
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
References
S. Turner, J.A. Sherratt: Intercellular adhesion and cancer
invasion: a discrete simulation using the extended Potts model.
J. Theor. Biol. 216, 85-100 (2002).
N.J. Armstrong, K. Painter, J.A. Sherratt: A continuum approach
to modelling cell adhesion. J. Theor. Biol. 243, 98-113 (2006).
N.J. Armstrong, K.J. Painter, J.A. Sherratt: Adding adhesion to
a chemical signalling model for somite formation. Bull. Math.
Biol. 71, 1-24 (2009).
J.A. Sherratt, S.A. Gourley, N.J. Armstrong, K.J. Painter:
Boundedness of solutions of a nonlocal reaction-diffusion
model for adhesion in cell aggregation and cancer invasion.
Eur. J. Appl. Math. 20, 123-144 (2009).
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
List of Frames
1
Introduction and Basic Modelling
What is Cell-Cell Adhesion?
Theoretical Modelling of Cell Populations
Cell Adhesion in Individual Cell-Based Models
Potts-type Model for Cancer Invasion
Cell Adhesion in Continuum Models
2
A New Continuum Model for Cell Adhesion
Derivation of the Model
Model Details
3
Application I: Cell Sorting and Aggregation
Mathematical Model for One Cell Population
Extending the Model to Interacting Cell Populations
Aggregation and Sorting in an In Vitro Experiment
A Numerical Simulation of Aggregation and Sorting
Computational Issues
Jonathan A. Sherratt
4
Application II: Somite Formation
Introduction to Somite Formation
Adding Adhesive Cells to the Chemical Model
Overview of Somite Application
5
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
Model Details: The Function ω(x0 )
•
x −R
•
x1
←
Force due to
cells at x1
•
x
→
•
x2
Force due to
cells at x2
ω(x0 ) is an odd function. For simplicity we usually take
−1 if −R < x0 < 0
ω(x0 ) =
+1 if 0 < x0 < +R
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
•
x +R
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
Model Details: The Function g(n)
At low cell densities, the force f (x, x0 ) will increase with cell
density at x + x0 when this is small.
However, there will be a density limit beyond which cells
will no longer aggregate.
We account for this via
a nonlinear g(n); we
take g(n) = n(nmax − n).
Here nmax corresponds
to close-packed cells.
Jonathan A. Sherratt
Computational modelling of cell-cell adhesion
Introduction and Basic Modelling
A New Continuum Model for Cell Adhesion
Application I: Cell Sorting and Aggregation
Application II: Somite Formation
Application III: Cancer Invasion
Introduction to Cancer Invasion
Modelling Adhesion in Cancer
Simulation of a Non-Invasive Tumour
The Sequential Development of an Invasive Tumour
Conclusions and Challenges
Further Model Extension:Two Cell Populations
t = 40
1
0.5
2
4
6
x
t = 80
8
10
Cell Density
Cell Density
2
cell type 1
cell type 2
somitic factor
1.5
0
0
Cell Density
1
0.5
2
4
6
x
t = 160
8
10
4
6
x
t = 120
8
10
12
2
4
6
x
t = 200
8
10
12
2
4
6
x
8
10
12
1
0.5
Cell Density
2
1.5
1
0.5
Jonathan A. Sherratt
2
1.5
0
0
12
2
0
0
1
0.5
2
1.5
0
0
1.5
0
0
12
2
Cell Density
Provided that cross-adhesion
is weak (0 < β < α1 , α2 ), the
model does predict
anterior/posterior
differentiation.
t=0
2
Cell Density
We investigate differentiation
of somites into anterior and
posterior halves by
considering two cell
populations n1 (x, t), n2 (x, t).
2
4
6
x
8
10
12
1.5
1
0.5
0
0
Computational modelling of cell-cell adhesion