Cell-based models of morphogenesis in normal and pathogenic development - Continued

Maria Audi ByrneSeptember 21st 2007

Mathbiology and Statistics SeminarUniversity of South Alabama

Presentation Outline

1. Morphogenesis

2. Model of normal development: myxobacteria fruiting body development

3. Model of pathogenic development: tumorigenesis in the prostrate duct

Modeling ofParacrine Signaling

Project 2: Aim 3

Core B: Image Fusion

Gore

Core C:Biomath &

BioinformaticsShyr

Project 3:Bone metastasis

Mundy

Vanderbilt-Ingram Cancer Center

Mou

se M

odels o

f Hum

an

Cance

r Conso

rtium

Van

der b

i lt Int e

gr a

tive

Cance

r B

iol o

gy C

en

t er

Prostate Center

Center for Bone Biology VU In

stitu

te

for I

mag

ing

Scie

nces

Brea

st S

PORE

BioMathem

atics

Small animal

imaging

Proteomics

Core A:Protein

collection & Proteomics

Caprioli

Biostatistics

Project 1: Breast Cancer

Moses

Matrisian

TGFeffectors

Vanderbilt University Tumor Microenvironment Network

VUTMEN

Project 2:Prostate Cancer

Hayward & Bhowmick

Paracrine Signaling

A cell or tissue produces a factor which acts upon an adjacent tissue.

Examples include many growth factors during development, adult homeostasis and in cancer.

ParacrineInteractions

Epithelium

Stroma

TGF as a master regulator of host:tumor interactions

Bierie and Moses, Cytokine Growth Factor Reviews, 2006

Preliminary Model: Static Model of Paracrine

Interactions in Prostate

What is the Prostate?

• From Wikipedia:

• The prostate is an exocrine gland of the male mammalian reproductive system.

• The main function of the prostate is to store and secrete fluid that constitutes 10-30% of the volume of the seminal fluid.

Prostate Cancer

Again from Wikipedia:• Prostate cancer is one of the most common cancers

affecting older men in developed countries and a significant cause of death for elderly men (estimated by some specialists at 3%). Regular rectal exams are recommended for older men to detect prostate cancer early.

• Many men never know they have prostate cancer. Autopsy studies of men who died of other causes have found prostate cancer in thirty percent of men in their 50s, and in eighty percent of men in their 70s. [Breslow et al, 1977]

Modeling Problem: Non-linear effect of stroma on epithelium

• Tissue Recombination Project– Normal stromal cells were mixed with altered stromal cells.– The altered stromal cells were unable to respond to TGF-beta.– Effect on epithelial cells was observed.

• In experiments intermediate levels of altered stroma yield the worst epithelial changes.

– 0% AS: Normal epithelial tissue– 50% AS: Proliferative and invasive epithelial tissue.– 100% AS: Proliferative epithelial tissue.

Drs. Neil Bhowmick and Hal Moses

Mathematical modeling of epithelial-stromal interactions

Modeling GoalHow can we define epithelial and stromal cell rules that

(1) are biologically motivated,(2) model correct proliferative behavior,(3) model correct invasive behavior?

Method: Hypothesize a set of simplified biologically motivated rules and use computer simulations to check if they are sufficient to yield expected cell behaviors.

Warning: If successful, we identify rules that are sufficient to explain experimental observations. Discourse between model predictions and further experiments are needed to further validate/refine the model.

Proliferation is controlled by a diffusing morphogen secreted by the altered stroma that acts on the normal epithelium (e.g., HGF).

Simplified Biological Assumptions

AlteredStroma

NormalEpithelium

ProliferativeEpithelium

InvasiveEpithelium

HGF

Proliferation is controlled by a diffusing morphogen secreted by the altered stroma that acts on the normal epithelium (e.g., HGF).

Migration is controlled by a diffusing morphogen secreted by the normal stroma that acts on proliferative epithelium (e.g., SDF1).

Simplified Biological Assumptions

NormalStroma

NormalEpithelium

ProliferativeEpithelium

InvasiveEpithelium

SDF1

NormalStroma

AlteredStroma

NormalEpithelium

ProliferativeEpithelium

InvasiveEpithelium

HGF SDF

NormalStroma

AlteredStroma

NormalEpithelium

ProliferativeEpithelium

InvasiveEpithelium

HGF SDF

0% Altered Stroma Normal Epithelium

NormalStroma

AlteredStroma

NormalEpithelium

ProliferativeEpithelium

InvasiveEpithelium

HGF SDF

100% Altered Stroma Proliferative Epithelium

NormalStroma

AlteredStroma

NormalEpithelium

ProliferativeEpithelium

InvasiveEpithelium

HGF SDF

50% Altered Stroma Invasive Epithelium

A B C

TGF

Schematic describing how TGF can be simultaneously tumor repressive and promoting.

Developmental Model Hypothesis

LGCA lattice gas cellular automata

• Particles are modeled as points with a defined “interaction neighborhood”.

• Each particle is assigned a “state” corresponding to the particle “channel” (orientation).

• Important exclusion rule: only one cell per channel.• Computationally efficient updating with a 2-step

transition rule at every time step:(1) Interaction step: Cells are assigned new velocities based on

neighborhood interactions.(2) Transport step: All cells are transported simultaneously

along velocity channels.

Biological LGCA

• Coined by Edelstein-Keshet and Ermentrout, 1990

• May relax exclusion principle or include novel modeling elements

Lattice-based Computational Model

•Cell-Based•Each lattice node may be occupied by a cell•No more than 1 cell per node•A lattice node is about 5 x 5 m

•Each cell has a “state”•Different cell states correspond to different cell types•Each cell state has its own set of rules that define the behavior of that cell type

•Simulation-based•The lattice at time step t+1 is a function of the lattice at time step t•A “simulation” is an arbitrary number of time steps

Mathematical modeling of epithelial-stromal interactions

5 cell types in model:

•Normal Stroma•Altered Stroma•Normal Epithelia•Transformed Epithelia

Type 1: proliferative phenotype Type 2: invasive phenotype

Mathematical modeling of epithelial-stromal interactions

Cell Rules for Each State

Normal Stroma secretes SDF: SDF(i,j,t+1) = SDF(i,j,t+1) + 1

Altered Stroma secretes Wnt: Wnt(i,j,t+1) = Wnt(i,j,t+1) + 1

Normal Epithelial PhenotypeTransforms at threshold levels of Wntif Wnt(i,j)>5, state changes from normal to proliferative

Proliferative Epithelial PhenotypeTransforms at threshold levels of SDF:if SDF(i,j)>3, state changes from proliferative to invasive

Invasive Epithelial Phenotype

for each cell located at the node (i,j)

•Molecules / morphogen units are discrete.

•Molecules are secreted by cells.

•Molecules diffuse over the lattice by random walk (at a diffusion rate specified for each molecule.

•Model keeps tracks of how many of each type of molecule are at each node at each time-step. Interactions can occur between cells and molecules and between molecules.

Model of diffusing molecules (paracrine interaction)

Cell Rules for Each State

Normal Stroma secretes SDF: SDF(i,j,t+1) = SDF(i,j,t+1) + 1

Altered Stroma secretes Wnt: Wnt(i,j,t+1) = Wnt(i,j,t+1) + 1

Normal Epithelial PhenotypeTransforms at threshold levels of Wntif Wnt(i,j)>5, state changes from normal to proliferative

Proliferative Epithelial PhenotypeTransforms at threshold levels of SDF:if SDF(i,j)>3, state changes from proliferative to invasive

Invasive Epithelial Phenotype

for each cell located at the node (i,j)

Diffusion

• At each time-step, cells, activator molecules and inhibitor molecules diffuse by either:

• resting at their current node with probability p

s

(or )• moving right, up, left or

down with probability (1-p

s)/4.

As the probability of resting ps increases, the diffusion rate of the particle decreases.

Model Particles:Cells, Activator, Inhibitor, and Fibronectin

Initial cell positions are taken from an experimental image.

Blue: Normal fibroblastsCyan: Altered fibroblastsBlack: Normal epithelia

• Altered stroma secretes Wnt that diffuses to the epithelium and causes epithelial cells to transform.

Stage 1: Transformation of Epithelium from Normal to Proliferative

Wnt Production and Diffusion after 10 time steps

- from altered stromal cells only -

Wnt Production and Diffusion Over 100 Time Steps

Blue: Normal fibroblastsCyan: Altered fibroblastsBlack: Normal epitheliaRed: Transformed epithelia

Blue: Normal fibroblastsCyan: Altered fibroblastsBlack: Normal epitheliaRed: Transformed epithelia

Stage 1: Transformation of Epithelium from Normal to Proliferative

• Altered stroma secretes Wnt that diffuses to the epithelium and causes epithelial cells to transform.

– The amount of Wnt increases with time.

– The amount of Wnt increases with the fraction of altered stroma.

– The number of transformed epithelial cells increases with time, fraction of altered stroma.

Stage 2: Transformation of Epithelium from Proliferative to Invasive

• Normal stroma secretes SDF that diffuses to the epithelium and causes transformed epithelial cells to transform further.

– The amount of SDF increases with the fraction of normal stroma.

– Recall, the number of transformed epithelial cells (stage 1) increases with the fraction of altered stroma.

– What fraction of altered stroma will yield the greatest number of invasive epithelial cells? What can we learn about SDF dynamics from the dependence of the number of invasive cells upon the fraction of altered stroma?

Model Results For Different Parameters

Error bars show average and standard deviation of five simulations.

Model Results For Different Parameters

Error bars show average and standard deviation of five simulations.

Future Directions

• Incorporating experimental data (quantitative data will specify some parameters).

• Investigating what limits the total number of invasive cells in the case of 50% altered stroma. (What limits available levels of SDF?)

- spontaneous exponential decay of SDF? (current model)- limited cell receptors for SDF?- ECM reservoirs of SDF?

• Developing a dynamic model based on normal prostate duct development that allows cell division (proliferation) and cell movement (migration).

• Systematically explore parameter space and form simulation-based predictions that will inform us about the consequences of the model assumptions.

Thanks!

The End

Proliferation is controlled by a diffusing morphogen secreted by the altered stroma that acts on the normal epithelium (e.g., HGF).

1. Normal stroma does not produce HGF because HGF is suppressed by TGF-beta.

2. Altered stroma are resistant to TGF-beta and do produce HGF.

3. HGF diffuses from the stroma to the epithelia.4. Threshold levels of HGF transform normal

epithelia to a proliferative phenotype.

Simplified Biological Assumptions for Proliferation

FIGURE 1. Cellular relationships. From the following article: Cancer:  Dangerous liaisons Allan Balmain and Rosemary J. Akhurst Nature 428, 271-272(18 March 2004) doi:10.1038/428271a

Migration is controlled by a diffusing morphogen that is secreted by normal stromal cells and acts upon proliferative epithelial cells (e.g. SDF-1/CXCL12 via AKt pathway).

1. Normal fibroblasts produce SDF1.2. Altered fibroblasts do not produce SDF1.3. Normal epithelia cells do not have SDF receptor CXCR4.4. Proliferative epithelial cells up-regulate CXCR 4 (TGFB dependent).5. Epithelial cells with CXCR4 respond to SDF1 by becoming invasive.

Simplified Biological Assumptions for Migration