summer workshop “mathematical medicine and biology” part 1...

Computational Biology Lund University

Computational models of stem cell fates

Carsten Peterson

Summer workshop“Mathematical medicine and biology”Nottingham, July 7-11, 2008Part 1

Computational Biology & Biological PhysicsLund University, Sweden

http://home.thep.lu.se/~carsten

Part 1:

Stem cell systems

Computational challenges

Modeling transcriptional dynamics

Part 2:

The embryonic stem cell switch

Hematopoietic switches

Bistability, irreversibility, priming



Carsten Peterson

Summer workshop“Mathematical medicine and biology”Nottingham, July 7-11, 2008

Part 1:

Stem cell systems



Computational Biology & Biological PhysicsLund University, Sweden

http://home.thep.lu.se/~carsten

Part 2:






Part 1:

Stem cell systems



Part 2:




Important issues not covered:

- Interactions between cells

- Signaling pathways

- Cell population models


Stem cells – development and maintenance

Mature stem cells – organ specific

Embryonic stem cell

.......brainblood skinintestinalliver

Development

Maintenance

Self-renewal Differentiation


The cancer stem cell connection

100 cancer stemcells 100.000 cancer cells


Stem cell fates

Self-renewal, pluripotency

Differentiation

Apoptosis

What is the program determining this fate

External signals triggers the transcriptional machinery

Provides switch behaviour

- Genetic network architecture

- Dynamics


Transcriptional regulation

Example:

Transcription factors (TF) A and B causes gene g1 to transcribe

With proper interactions e.g. AND logics is obtained

For mammals in general more than 2 TFs

Model small modules of genes:

Verify functionality

Infer additional components/interactions

RNAp

OBpromoterOA

g1

AND

BA


Modeling transcriptional regulation

• Boolean models

High level description; crude approach; does not relate to underlying physics/chemistry

• Michaelis-Menten

Useful phenomenological parametrizations

• Shea-Ackers thermodynamical formalism

Transparent with regard to different interaction components; gateway to stochastic methods

• Stochastic methods

Both Michaelis-Menten and Shea-Ackers assume a “large” number of molecules – one works with concentrations. When this is not the case, one has to treat each molecule stochastically (e.g. Gillespie algorithm).

Computationally tedious.


The Shea-Ackers statistical mechanics approach

Method originally developed for the lysis/lysogeny switch in Lambda phage

Two time scales:

- Slow:

Transcription/Translation/Degradation

- Fast:

Binding/unbinding of TFs to gene – thermal equilibrium makes sense

Possible cases: TF, TF+RNAP, RNAP - probability associated with each

Enumerate all cases, compute probability of bound RNAP

Transcription rate is proportional to promoter occupancy

Gene

TFs RNAP

PromoterOperons


Example: Two transcription factors A and B

Enumerate all possibilities - binding/unbinding of A,B and RNAP The “partition function” Z contain 23 = 8 terms

The Shea-Ackers approach

i,j,k = [1, 0] (binding/unbinding) ‏

[A] and [B]: TF concentrations[R]: RNAP concentration

T: temperature

Free energies (parameters)‏

Low free energy – strong binding and vice versa



One can write down a “truth table”

A B R weights

1 1 1

1 0 1

0 1 1 . . .





bound

The transcription rate is proprtional to

not bound




Relation to equilibrium calculations

Equilibrium calculations, commonly used in enzymatic reactions, yields the same transcription rates as the Shea-Ackers approach (as it should)‏

We illustrate this with a single autoregulatory gene X that binds to itself

The transition rate is given by

There are four possible states subject to the normalization


Reaction scheme defining the network

At thermodynamic equilibrium one has

In this lingo the fractional probability of the gene being bound by RNAP is




• The statistical mechanics (Shea-Ackers) approach is more intuitive

• However: The equilibrium approach allows us to define a reaction scheme in terms of measured kinetic constants

Platform for stochastic simulations (Gillespie)‏

Equilibrium approach

Shea-Ackers


The Michaelis-Menten approach

Yet another way of doing it (deterministically)

In general: An enzyme E binds to a substrate S and turns it into a product P

Rate equations for the concentrations:

1 2



Yet another way of doing it (deterministically)

In general: An enzyme E binds to a substrate S and turns it into a product P

Rate equations for the concentrations:

1 2

Fast - equilibrium

Approximate to zeroe

Solving the fixed point equation



Assume a constant amount of enzyme

One gets

For the production of P as function of S one has


The Michaelis-Menten and Hill equations

With P being the transcribed gene, TF the substrate and DNA the enzyme one has

Similarly for repression, one has

Problem: slow response with [TF]

Improvement: Hill exponents n

Can be deduced from a model where the TFs have multiple binding sites


What about noise? - The Langevin equation

Biochemical reactions take place in noisy environments

Yet, we propose deterministic ordinary differential equations (ODEs)‏

Large number of molecules needed for justification. How large? Problem dependent (10-100)‏

Biochemical reactions take place in noisy environments

Langevin stochastic differential equation for genes X

If the noise is white (uncorrelated), we have

Mean

Variance


The Langevin equation


The Langevin equation

LangevinGillespie

From D. Gonze


Putting things together – bistability etc.

This is where the fun starts .....

So far transcription of single genes

With a set of interacting genes (subnetwork) probe the dynamics e.g. stationary solutions


Bistability

Hill-type equations plus degradation terms

In Gardner et. al. (2000) a genetic switch was constructed with two genes repressing each other by manipulation of DNA in E.Coli

Allows for direct comparisons with models

Compute the nullclines


Bistability

Compute the nullclines

More later ... stem cell switches

Bistability – either or needs to be larger than 1


Model summary

Reaction kinetics

Michaelis-Menten (Hill function)‏

Shea-Ackers

Boolean

Stochastic simulations (e.g. Gillespie) ‏


Model summary

Reaction kinetics


Shea-Ackers

Boolean


Deterministic

Stochastic


Model summary

Reaction kinetics


Shea-Ackers

Boolean


Langevin Langevin

Deterministic

Stochastic


Stem cell fates

Self-renewal, pluripotency

Differentiation

Apoptosis

What is the program determining this fate

External signals triggers the transcriptional machinery

Provides switch behaviour

- Genetic network architecture

- Dynamics


Sneak preview (part 2) ‏


Indentify motifs, bistability, missing components etc.


Sneak preview (part 2) ‏

The erythroid - myeloid switch

Indentify motifs, bistability, missing components etc.


References (minimal/incomplete)

V. Chickarmane, C. Troein, U. Nuber, H.M. Sauro and C. Peterson (2006) PLoS Computational Biology 2, e123 (2006)

N.E. Buchler, C. Gerland and T. Hwa PNAS 100, 5136-5141 (2003)

L. Bintu, N.E. Buchler, H.G. Garcia, C. Gerland and T. Hwa Curr Opin Gen Dev 15, 125-135 (2005)

V. Chickarmane, S.R. Paladagdu, F. Bergmann and H.R. Sauro Bioinformatics 18, 3688-3690 (2005)

summer workshop “mathematical medicine and biology” part 1...

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