Computational models of stem cell fates

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Computational models of stem cell fates

Carsten Peterson

Summer workshop

“Mathematical medicine and biology”

Nottingham, July 7-11, 2008 Part 1

Computational Biology & Biological Physics Lund 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

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Computational Biology

Computational models of stem cell fates

Carsten Peterson

Summer workshop

“Mathematical medicine and biology”

Nottingham, July 7-11, 2008

Part 1:

Computational Biology & Biological Physics Lund University, Sweden

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

Part 2:

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Computational models of stem cell fates

Part 1:

Part 2:

Important issues not covered:

-

Interactions between cells - Signaling pathways

- Cell population models

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Stem cells – development and maintenance

Mature stem cells – organ specific

Embryonic stem cell

...

brain blood liver intestinal skin

Development

Maintenance

Self-renewal Differentiation

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Stem cells – development and maintenance

Mature stem cells – organ specific

Embryonic stem cell

...

brain blood liver intestinal skin

Development

Maintenance

Self-renewal Differentiation

(6)

The cancer stem cell connection

100 cancer stemcells 100.000 cancer cells

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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

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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

O_B promoter

O_A g1

AND

A B

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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.

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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

Promoter Operons

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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

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The Shea-Ackers approach

One can write down a “truth table”

A B R weights

1 1 1

1 0 1

0 1 1

. . .

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The Shea-Ackers approach

bound

The transcription rate is proprtional to not bound

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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

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Reaction scheme defining the network

At thermodynamic equilibrium one has

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

Relation to equilibrium calculations

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Relation to equilibrium calculations

• 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

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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

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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

Fast - equilibrium

Approximate to zeroe

Solving the fixed point equation

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The Michaelis-Menten approach

Assume a constant amount of enzyme

One gets

For the production of P as function of S one has

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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

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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

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The Langevin equation

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The Langevin equation

Langevin Gillespie

From D. Gonze

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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

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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

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Bistability

Compute the nullclines

More later ... stem cell switches

Bistability – either or needs to be larger than 1

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Model summary

Reaction kinetics

Michaelis-Menten (Hill function)‏

Shea-Ackers Boolean

Stochastic simulations (e.g. Gillespie)‏

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Model summary

Reaction kinetics

Shea-Ackers Boolean

Deterministic

Stochastic

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Model summary

Reaction kinetics

Shea-Ackers Boolean

Langevin Langevin

Deterministic

Stochastic

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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

(31)

Sneak preview (part 2)‏

The embryonic stem cell switch

Indentify motifs, bistability, missing components etc.

(32)

Sneak preview (part 2)‏

The erythroid - myeloid switch

Indentify motifs, bistability, missing components etc.

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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)