Stochasticity in Signaling Pathways and Gene Regulation:
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Stochasticity in Signaling Pathways and Gene Regulation: The NF κ B Example and the Principle of Stochastic Robustness. Marek Kimmel Rice University, Houston, TX, USA. Rice University Pawel Paszek Roberto Bertolusso UTMB – Galveston Allan Brasier Bing Tian Politechnika Slaska

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Stochasticity in Signaling Pathways and Gene Regulation: The NFκB Example and the Principle of Stochastic Robustness

Marek Kimmel

Rice University, Houston, TX, USA


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

Credits


Gene regulation


TNF Signaling Pathway

Apoptosis

Signal

NF-kB AP-1

Inflammation

Proliferation

TNF


Nuclear Factor-kB (NF-kB)

  • 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


Nuclear factor-kB (NF-kB) Pathway

TAK/TAB1

TRAF2/TRADD/RIP

IKK

TNF

IkBa

Rel A:NF-kB1

nucleus


Rel A:NF-kB1

NF-kB “Activation”

Activated

IKK

2

nucleus


TRAF1

RelB

NF-kB1

A20

NF-kB2

IkBa

IkBe

TTP/Zf36

Negative autoregulation of the NF-kB pathway

TNFR1

Rel A

IKK

Rel A

C-Rel

Rel A:NF-kB1

TNF mRNA

nucleus


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.


Simplifiedschematic of gene expression

  • Regulatory proteins change gene status.


Discrete Stochastic Model

Time-continuous Markov chain with state space

and transition intensities


Continuous Approximation

only gene on/off discrete stochastic


Four single cell simulations


Trajectories projected on (IB,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


The Model

kB

B

ARE

kdB

f

kN

kdN

N


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

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

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.


Deterministic description

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


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.


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