Agent-based modelling of epithelial cells

An example of rule formulation and extension

Dr Dawn Walker, University of Sheffield, UK

What determines cell behaviour?

Other cells•Intercellular bonds•Intercellular signalling

Environmental factors•Extracellular matrix•Calcium concentration•Growth medium

Genetic ‘rules’•Cell cycle•Differentiation

Modelling strategy

CLOCK

FOR EVERY CELL IN TURNExecute cell behaviour rules

Adjust position of all cells to ensure no overlap

AGENTMODEL

PHYSICALMODEL

Iterative coupled ‘agent – physics’ model

Model Implementation

CELLCOMMUNICATION

APOPTOSISRULES

MOTILITYRULES

BONDINGRULES

SPREADINGRULES

CELL CYCLERULES

For each cell in turn….

For all cells together…. EQUILIBRIATE CELL POSITIONSDUE TO GROWTH, MICRATION ETC.

Cell cycle control – the model

M

G2

S

G1

G0

G1 GROWTH PHASE

Ref- general biological knowledgePublications of urothelial cell proliferation time

Cell cycle control – the model

M

G2

S

G1

G0

G1 GROWTH PHASE

Ref- general biological knowledge

Cell cycle control – the model

M

G2

S

G1

G0

CONTACT INHIBITION?(4 or more bonds)

CELL SPREAD?

G1-G0 checkpoint

GROWTH FACTORS?

General biological knowledgeRef: Nelson & Chen 2002,FEBS Letters 514 pp 238-242

Cell cycle control – the model

M

G2

S

G1

G0

CONTACT INHIBITION?(4 or more bonds)

CELL SPREAD? XQUIESCENCE

G0 QUIESCENT PHASE

GROWTH FACTORS?

Cell cycle control – the model

M

G2

S

G1

G0

CONTACT INHIBITION?

(4 or more bonds) x

CELL SPREAD?

G1 GROWTH PHASE

GROWTH FACTORS?

Cell cycle control – the model

M

G2

S

G1

G0

S PHASE – (CHROMOSOMEREPLICATION)

Cell cycle control – the model

M

G2

S

G1

G0

G2 PHASE – (HOUSEKEEPING)

Cell cycle control – the model

M

G2

S

G1

G0

M PHASE - DIVISION

Bonding Rules

Stochastic process governed by•Cell edge separation•Calcium ion concentration

0.10.20.30.40.50.60.70.80.91

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2.00

Bin

din

g P

rob

abili

tyCa2+ ion concentration

Sep. = 0mSep. = 1mSep. = 5m

Sep

[Ca2+]

Ref: Baumgartner et al, 2000, Cadherin interaction probed by atomic force microscopy PNAS 97(8) 4005-4010.

Migration parameters

Urothelial cells in low Ca2+ (0.09mM)

Physical model

l

F=ma l

Ca2+ dependent behaviour - In Vitro vs. In Virtuo

• Intercellular bonds require the presence of Ca2+ ions

• In Ca2+ conc.> 1mM many bonds are formed

• Cells with several intercellular bonds become contact-inhibited (stop cycling)

• WHAT IS THE EFFECT OF Ca2+ ON GROWTH AND PROLIFERATION?

= STEM CELL = TRANSIT =MITOTIC CELL =QUIESCENT AMPLIFYING CELL (G0) CELL

Model Simulations – urothelial monolayer growth

Physiological Ca2+ (2mM)

Low Ca2+ (0.09mM)

ITERATION NUMBER

NO

. CE

LL

S

Ca2+ = 2mM Ca2+ = 0.09mM

Model Simulations – urothelial monolayer growth

Physiological Ca2+ (2mM)

Low Ca2+ (0.09mM)

ITERATION NUMBER

NO

. CE

LL

S

Ca2+ = 2mM Ca2+ = 0.09mM

= STEM CELL = TRANSIT =MITOTIC CELL =QUIESCENT AMPLIFYING CELL (G0) CELL

In virtuo wound healing (urothelium)

Physiological Ca2+ (2mM) Low Ca2+ (0.09mM)

In virtuo wound healing (urothelium)

Physiological Ca2+ (2mM) Low Ca2+ (0.09mM)

In Vitro wound healing (urothelium)

Low Ca2+Physiological Ca2+

In vitro vs. in virtuo population growth (urothelium)

In vitro model Computational model

0

500

1000

1500

2000

2500

0 10 20 30 40 50 60 70 80

Simulation time in hours

To

tal

ce

ll n

um

be

r

Ca2+ conc.=2.0mMCa2+ conc.=0.09mM

0

20

40

60

80

100

120

Day 1 Day 3 Day 5 Day 7 Day 9

Cel

l n

um

ber

/ x

10E

4 p

er m

L Low Calcium

Physiological Calcium

Rule extension – cell contact and proliferation

Hypotheses:• 1. Short range growth

factor diffusive signal

• 2 Juxtacrine growth factor signal

• 3 E-Cadherin - Catenin related signal

Hypothesis (1) autocrine GF-mediated signalling

Cell{x}……Ligand released=Ls

Free receptors=Rs

Ratio of RT:CT

Determines change in cell behaviour e.g.

cell cycle progression, migration

Internalised complexes, Ce

and receptors Re

Activated surface receptors= Cs

Testing Hypothesis (1)

[Ca2+]=0.05mM[Ca2+]=2.5mM

Initial cell agent seeding density and distribution

Conclusion: Diffusive growth factors – population growth is seeding density, but NOT distribution related

Assembling rules to test hypothesis (2)

EC_high Ca2+EC_low Ca2+

Work in Progress!Thanks to Nik Georgopolous

Summary

• Initial rule formulation can be based on simplifications and abstractions of known biological behaviour

• Iterative comparison with experimental data can improve the accuracy of the model and direct experimental investigation

• The rule set can be extended to model additional aspects of cell behaviour (e.g. differentiation, stratification)

• Rules can be replaced by more complex models (e.g. inter- and intra- cellular signalling)

Thank you for listening

Any Questions?