Stochastische Genexpression

Literatur

Kaern et al. Nature Reviews Genetics Vol.6 p.451 (2005)

Ozbudak, Oudenaarden et al (2004) Multistability in the lactose utilization network of Escherichia coli, Nature 427, p737

Vorlesung System-Biophysik 18. Dez. 2007

Genetische Schalter und Multistabilität

Das Operon-ModellFrancois Jacob und Jaques Monod, 1961

operon

Operon: Genetische Funktionseinheit, die aus regulierten Genen mit verwandter Funktion besteht und enthält:- Promotor: Bindungsstelle für RNA-Polymerase- Operator: kontrolliert Zugang der RNA-Polymerase zu Strukturgen- Strukturgene: Polypeptide codierende Genezusätzlich:Regulatorgen: codiert Repressor Campbell, N.A., Biology

A Transkription-Aktivator and a Transkription-Repressor control the lac-Operon

Genregulation and boolean networks

from Weiss, 2000

Boolean expression of the Lac-Operon

Genetische Netze

transcriptiontranscription

translationtranslation

Genregulatorisches ProteinGenregulatorisches Protein

Transkription factors show cooperativity (e.g. by dimer-formation)

CI D O CIO

KD CI M 2

CI D

CIO Ototal

CI M 2

K KD CI M 2

Cooperative binding

CI M CI M

CI D

Wiederholung

Genregulation and boolean Network

from Weiss, 2000

(Nature, Dec 99)

the genetic Toggle Switch (Flip-Flop)

Weiss et al.

the repressilator (genetic oscillator)

Genetically controlled oscillation of about 24 hours, adapted to the day-night rythm.

Der 24h Rythm is robust, the phase is coupled to the light/dark cycle.

A single gen-mutation is responsible for the familial advanced sleep phase syndrome, FASPS.

[Latein. circa about + dies a day]

Circadian Rhythm – the biological clock

Die circadiane Uhr at Drosophila

The complex fromation Per/Tim supports the entry in the nucleus and

stays stable there for 8-10h. This slows down the feedback loop.

Two proteins Per (Period) and Tim (timeless) regulate each other and form a dimere complex. Monomeres of Per in the nucleus supresses expression. The kinase DBT (double time) phosphorylates und degrades Per.

Decelerated negative feedback via dclk (dclock) (degradation) anddbt (doubletime) (Transport)Synchronization (Entrainment) due to sunlight dependent TIM degradation rate

A couple of phosporilation steps are part of deceleration mechanism

What is life ?

1. How can „biological order“ (life) be explaind by the basic laws of physics?

2. How does life deal with the statistic nature of molecular interactions?

„... wenn wir so empfindliche Organismen wären, daß ein einzelnes Atom oder meinet-wegen ein paar Atome einen wahrnehmbaren Eindruck auf unsere Sinnesorgane machen könnten - du lieber Himmel, wie sähe das Leben dann aus!“

Schrödinger considered 1943 the consequences of the molecular nature of the genetic code in a lecture about „Physics and biology“

The importance of statistical fluctuations in biology

• In a fluctuating environment, heterogeneous cell populations have better chances to grow. (e.g. control of lac.operon, immune system, lysis-networks of lambda-phage)

• Diversification in isogene phenotypes und celltypes (e.g. stem cell diversification)

• Efficiency increase in signal transduction (e.g. chemotaxis regulation oder stochastic resonance (ears))

Noise can be increased with „positive feedback loops“ with advandtages:

• Stabilisation of metabolics / homeostasis

Noise can be decreased via „negative feedback loops“

Biochemical noise: fluctuation of protein concentration

Small numbers of copies of many components e.g. Polymerases, regolatory proteins, Stochastic effects in gene expression play an important role for variations of protein concentrations of bacteria wit identical genes

Asymetries emerge, which are amplified by feedback loops and influence the development of the cell.

Noise in the expression:

Deterministic model of gene expression

from JJ Collins, Nature Reviews 2005

Definitions for noise

2 A2 A

2

z 1

noise

Distribution

Noise amplitude decreases with increasing number of particles!

z: number of data points

Rao, Wolf,Arkin, Nature 2002

Variance

p j jn k1

jk2n j

k1 k2 n

t N

A t 2 A t 2

A t 2

1 2

finite size effect

x

x (noise)

x : mean value

x : standard deviation

1 N

0.1µM corresponds to 30 molecules/bacterium

Decrease of the transcription rate and cell volume with equal factors keeps the protein level constant, but increases noise

beschreibt den Effekt, dass ein Heraufsetzen der translationsrate auch dieFluktuationen verstärkt.

Herabsetzenvon Transkriptionsrate undZellvolumen

Proteinlevel konstantFluktuationen erhöht

„Translational bursting“

Slow promotors increase noise

Transcriptional bursting

low promotor rate

High transcription rate

Noise models

Set of differntial equations (deterministic):

Set of differential-equations (stochastic)Langevin equations:C: concentrations, t: time, v: stoichiometric matrix, r: rates, x(t): white noise

Probability density function

example isomerisation with

k1 = k2 = 1s-1

k1

k2

Simulation for isomerisation :

state A state B

Experiment: stochastisc Gen-Expression

Distinguish between „intrinsic noise“ (e.g. gene expression) and „extrinsic noise“(e.g. other cell components as RNA polymerase)Idea for an exeriment:Gene for CFP (green fluorescent Protein) und YFP (rot fluorecent Protein) are controlled by the same promotor, hence the mean concentration of CFP and YFP is equal => Expression probability should differ only due to intrinsic noise

A: no intrinsic noise => noise is correlated red+green=yellow

B: intrinsic noise => noise not correlated, different colors

Elowitz, M. et al, Science 2002

Two distinguishable genes (CFP and YFP)controlled by the same promotor

Low induction:(low fluorescence)high noise

High induction :(high fluorescene)Low noise

Elowitz, M. et al, Science 2002

Stochastische Genexpressionin einer einzelnen Zelle

x

x

tot2 int

2 ext2

(noise)

Intrinsic noise: inherent stochasticity at identical external conditions

Extrinsic noise: cell to cell variance of expression

Stochastic gene expression

Elowitz et al. 2002

hte „intrinsisc noise“ decreases with increasing protein concentration

Elowitz, M. et al, Science 2002

22int

2exttot