Stochasticity in Signaling Pathways and Gene Regulation:

The NFκB Example and the Principle of Stochastic Robustness

Marek Kimmel

Rice University, Houston, TX, USA

Credits

• Rice University– Pawel Paszek– Roberto Bertolusso

• UTMB – Galveston– Allan Brasier– Bing Tian

• Politechnika Slaska– Jaroslaw Smieja– Krzysztof Fujarewicz

• Baylor College of Medicine– Michael Mancini– Adam Szafran– Elizabeth Jones

• IPPT – Warsaw– Tomasz Lipniacki– Beata Hat

Gene regulation

TNF

TNF Signaling PathwayTNF Signaling Pathway

Apoptosis Signal

NF-B AP-1

Inflammation Proliferation

Nuclear Factor-B (NF-B)

• Inducible (cytoplasmic) transcription factor• Mediator of acute phase phase reactant

transcription (angiotensinogen, SAA)• Mediator of cytokine and chemokine

expression in pulmonary cytokine cascade• Plays role in anti-apoptosis and confering

chemotherapy resistance in drug resistant cancers

IB

Rel A:NF-B1

nucleus

TNF

TRAF2/TRADD/RIP

TAK/TAB1

IKK

Nuclear factor-Nuclear factor-B (NF-B (NF-B) PathwayB) Pathway

Rel A:NF-B1

nucleus

2

Activated IKK

NF-B “Activation”

IKK

nucleus

TNFR1

Rel A:NF-B1

A20

Negative autoregulation of the NF-B pathway

Rel A

IB

IBC-Rel

NF-B1

NF-B2

RelB

Rel ATRAF1

TNF mRNA

TTP/Zf36

Intrinsic sources of stochasticity

• In bacteria, single-cell level stochasticity is quite well-recognized, since the number of mRNA or even protein of given type, per cell, might be small (1 gene, several mRNA, protein ~10)

• Eukaryotic cells are much larger (1-2 genes, mRNA ~100, protein ~100,000), so the source of stochasticity is mainly the regulation of gene activity.

Simplified schematic of gene expression

• Regulatory proteins change gene status.

1)(,0)(

,,

AI

IAAI

GG

genectivegenenactive dc

rK

HG

proteinmRNA

mRNA 1

Discrete Stochastic Model

Time-continuous Markov chain with state space

and transition intensities

ProteinRNAGene}1,0{ ZZ

)()()(

)()(

trytKxdt

tdy

txHGdt

tdx

Continuous Approximationonly gene on/off discrete stochastic

0 2 4 60

2

4

6x 10

4 Free nuclear NF-kB

0 2 4 60

0.5

1

1.5Activity of IkBa gene

0 2 4 60

100

200

300IkBa mRNA transcript

0 2 4 60

5

10x 10

4 Total IkBa

0 2 4 60

2

4

6x 10

4

0 2 4 60

0.5

1

1.5

0 2 4 60

100

200

300

0 2 4 60

5

10x 10

4

0 2 4 60

2

4

6x 10

4

0 2 4 60

0.5

1

1.5

0 2 4 60

100

200

300

0 2 4 60

5

10

15x 10

4

0 2 4 60

2

4

6x 10

4

0 2 4 60

0.5

1

1.5

0 2 4 60

100

200

300

0 2 4 60

5

10x 10

4

Four single cell simulations

Trajectories projected on (IB,NF-Bn,,time) space, red: 3 single cells, blue: cell population

Any single cell trajectory differs from the “averaged” trajectory

White et al. experiments

What happens if the number of active receptors is small?

Low dose responses

How to find out if on/off transcrition stochasticity plays a role?

• If on/off rapid enough, its influence on the system is damped

• Recent photobleaching experiments →

TF turnover ~10 sec

• However, does this quick turnover reflect duration of transcription “bursts”?

FRAP (Mancini Lab)Fluorescence recovery after photobleaching

f

N

B

AR

E

The Model

Nkfkt

N

Bkfkt

B

fkkNkBkfDt

f

dNN

dBB

NBdNdB

)(

zyx

kB

kdB

kdN

kN

The Model

• Fit the model to photobleaching data

• Obtain estimates of binding constants of the factor

• Invert binding constants to obtain mean residence times

• Effect: ~10 seconds

Estimation of mean times of transcription active/ inactive

Estimation of mean times of transcription active/ inactive

Transcription of the gene occurs in bursts, which are asynchronous in different cells.

Estimation of mean times of transcription active/ inactive

hrTE

hrTE

AI

IA

I

A

2.2)(

8.0)(

,

,

1

1

Parameters estimated by fitting the distribution of the level of nuclear message, apparently contradict photobleaching experiments.

A single gene (one copy) using K-E approximation

)()()(

tGtydt

tdy

Amount of protein:

Where:• and are the constitutive activation and deactivation

rates, respectively,• is an inducible activation rate due to the action of protein dimers.

,1)(,0)(

,, 02

20

AI

IAAI

GG

dycc

oc od

2c

Deterministic description

The system has one or two stable equilibrium points depending on the parameters.

,][

),()()(

02

20

220

dycc

yccGE

GEtydt

tdy

Transient probability density functions

Stable deterministic solutions are at 0.07 and 0.63

Transient probability density functions

Stable deterministic solutions are at 0.07 and 0.63

Transient probability density functions

Stable deterministic solutions are at 0.07 and 0.63

Transient probability density functions

Stable deterministic solutions are at 0.07 and 0.63

Conclusions from modeling

• Stochastic event of gene activation results in a burst of mRNA molecules, each serving as a template for numerous protein molecules.

• No single cell behaves like an average cell.• Decreasing magnitude of the signal below a threshold

value lowers the probability of response but not its amplitude.

• “Stochastic robustness” allows individual cells to respond differently to the same stimulus, but makes responses well-defined (proliferation vs. apoptopsis).

References

• Lipniacki T, Paszek P, Brasier AR, Luxon BA, Kimmel M. Stochastic regulation in early immune response. Biophys J. 2006 Feb 1;90(3):725-42.

• Paszek P, Lipniacki T, Brasier AR, Tian B, Nowak DE, Kimmel M. Stochastic effects of multiple regulators on expression profiles in eukaryotes. J Theor Biol. 2005 Apr 7;233(3):423-33.

• Lipniacki T, Paszek P, Brasier AR, Luxon B, Kimmel M. Mathematical model of NF-kappaB regulatory module. J Theor Biol. 2004 May 21;228(2):195-215.