Transcription in E.coli by the numbers
•Size of typical gene = 1kb
•Typical number of mRNAs = 10
•Typical rate of mRNA production = 10 min-1
•Typical mRNA degradation rate = 1 min-1
mRNA production in E.coli
Golding et al. ‘05
hmi
r=1/7min
°=ln2/50min
dhm i
dt
= r ¡ ° hmi
Master equation for mRNA production
Master equation for p(m,t):
@p(m ;t )
@t
= r p(m ¡ 1; t) + ° (m + 1)p(m + 1; t) ¡ r p(m; t) ¡ ° mp(m; t)
Steady state:
³
@p( m ;t )
@t
´
= 0
p(m) =
m
¡ ¹ ¹
e m!
Poisson distribution
with mean ¹ = r
°
Variance of the mRNA number
var(m) = 4 <m>
Poisson: var(m) = <m>
Poisson
Inconsistent with unregulated transcription model
Single cell trajectories
¢n
Transcription occurs in bursts!
Genetic switch
Gardner, Cantor and Collins ‘00
Flipping the genetic switch
Bistability!
Nullclines and flow field
Phase portrait of the genetic switch
Repressilator
Elowitz and Leibler ‘00
The Poisson distribution
Probability
What is the distribution
of people per square?
Histogram
Poisson
people/unit area