QUANTITATIVE LAWS June 13 -June 24

1. Gene
expression
noise,
regula0on,
and
noise
propaga0on
Erik
van
Nimwegen
Biozentrum,
University
of
Basel,
and
Swiss
Ins8tute
of
Bioinforma8cs
Basel
Our
group

2. Cartoon
of
the
steps
in
gene
expression
Gene
X
RNA
polymerase
Gene
X
RNA
polymerase
mRNA
gene
X
Transcrip0on
rate:
rλ
mRNA
decay
rate
Protein
X
Transla0on
rate
Protein
decay
rate:
pλ
pµ
rµ

3. Gene
expression
differen3al
equa3ons
•
P
=
Amount
of
protein
X.
•
R
=
Amount
of
mRNA
X.
•
P
increases
due
to
transla0on
of
mRNA
and
decreases
due
to
protein
decay.
p p
dP
R P
dt
λ µ= −
•
R
increases
due
to
transcrip0on
and
decreases
due
to
mRNA
decay.
r r
dR
R
dt
λ µ= −
Steady-­‐state:
P =
λr
λp
µr
µp
R =
λr
µr
•
In
reality
there
are
are
really
an
integer
number
p(t)
of
proteins
at
0me
t,
and
r(t)
mRNAs.
•
Numbers
may
be
small,
e.g.
there
is
only
one
copy
of
the
gene
in
the
DNA.
•
The
RNA
polymerases,
ribosomes,
and
mRNAs
are
tumbling
around
in
the
cell,
constantly
bumping
into
other
molecules
(i.e.
following
Brownian
mo0on).
Discreteness
and
Stochas3city:

4. Surprise
surprise:
Gene
expression
is
stochas3c
Low copy
Plasmid
• GFP
fluorescence
per
cell
propor0onal
to
protein
number.
• Not
surprisingly,
fluctua0ons
are
observed
between
cells.
• What
kind
of
fluctua0ons
would
one
expect
in
a
simplest
possible
model?

5. Stochas3c
transcrip3on
and
decay
Gene
X
RNA
polymerase
Gene
X
RNA
polymerase
mRNA
gene
X
Probability
per
unit
0me
to
transcribe
a
new
mRNA.
Differen0al
equa0on
for
the
distribu0on:
1 1
( )
( ) ( 1) ( ) ( ) ( )n
r n r n r r n
dP t
P t n P t n P t
dt
λ µ λ µ− += + + − +
Probability
that
there
are
n
mRNAs
at
0me
t:
rλ rµ
Pn
(t)
Probability
per
mRNA
per
unit
0me
that
it
will
decay.

6. Steady-­‐state
is
Poisson
distribu3on
Probability
to
have
n
mRNAs:
Pn
=
1
n!
λr
µr
⎛
⎝
⎜
⎞
⎠
⎟
n
e
−λr /µr
Mean:
n =
λ
µ
Variance:
var(n) = n =
λ
µ
Standard-­‐devia3on:
σ (n) = n
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
Number of mRNA n
Probability
0 2 4 6 8 10
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Number of mRNA n
Probability
0 5 10 15 20 25 30
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Number of mRNA n
Probability
λr
µr
= 0.1 10r
r
λ
µ
=
λr
µr
=1

7. (Shahrezaei,
Swain
PNAS
2008)
Transla3on
amplifies
mRNA
fluctua3ons
mean
and
variance:
a =
λr
µp
b =
λp
µp
“burst
size”:
transla0ons
per
mRNA
life0me.
n = ab, var(n) = (b+1) n
λr
µr
µp
transcrip0on
mRNA
decay
transla0on
protein
decay
λp
λr
µr
λp
µp
• Proteins
are
oZen
long-­‐lived:
approxima0on
protein-­‐decay
slow
rela0ve
to
mRNA
decay.
• Solu0on
in
terms
of
two
ra0os:
Transcrip0on
events
per
protein
life0me.
Pn
=
Γ(a + n)
Γ(a)n!
b
b +1
⎛
⎝⎜
⎞
⎠⎟
n
1−
b
b +1
⎛
⎝⎜
⎞
⎠⎟
a
noise:
η(n) =
σ(n)
n
=
var(n)
n
2
=
b +1
n

8. mRNAs
per
cell
for
E.
coli
hp://book.bionumbers.org/

9. Typical
genes
have
less
than
1
mRNA
per
cell
in
E
coli
Fluorescently
labeling
single
mRNAs
(Fluorescence
In
Situ
Hybridiza0on).
Coun0ng
mRNAs
per
cell
under
the
microscope.
Mean
mRNAs
per
cell
Taniguchi
et
al,
Science
(2010)
From:
Milo
and
Phillips,
Cell
Biology
by
the
numbers

10. Some
addi3onal
numbers
for
E.
coli
•
RNA
polymerases
per
cell:
1’500-­‐10’000
(depending
on
growth
rate).
•
Ribosomes
per
cell:
14’000
(1
doubling
per
hour)
–
45’000
(2
doublings
per
hour).
•
mRNA
decay
rate:
1-­‐15
minutes
half-­‐life.
• Protein
decay
rate:
typically
a
few
hours.
• Protein
dilu0on
rate:
cell
doubling
0me,
i.e.
30
min
to
2
hours.
Bernstein
et
al,
PNAS
(2002)
Taniguchi
et
al,
Science
(2002)
Distribu3on
mRNA
half-­‐lifes
Distribu3on
mean
proteins
per
cell

11. Measuring
variability
within
and
across
cells
Two
3mes
the
same
promoter
Intrinsic
and
extrinsic
noise
• Total
variance
in
fluorescence
per
cell
can
be
decomposed
into
two
parts:
• Intrinsic
=
variance
within
cell:
• Extrinsic
variance
=
the
rest,
i.e.
variability
across
cells:
vtot
= var(g) + var(r) = vi
+ ve
vi
=
1
2
(g − r)2
ve
= gr − g r
Hey!
That
covariance
could
be
nega8ve!
How
can
a
variance
be
nega8ve?

12. How
to
properly
infer
intrinsic
and
extrinsic
variance
Gives
orthodox
sta0s0cal
es0mators
that
can
give
nega0ve
es0mates.
A
Bayesian
solu3on
is
never
pathological
and
much
more
accurate
when
extrinsic
noise
is
small
Extrinsic:
Gaussian
distribu0on
of
mean
μi
across
cells
i:
Intrinsic:
Gaussian
devia0on
of
green
gi
and
red
ri
from
mean
μi:
P(gi
,ri
| µi
) =
1
2πv
exp −
(gi
− µi
)2
+ (ri
− µi
)2
2v
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
Posterior
for
the
intrinsic
variance
v
and
extrinsic
variance
vμ:
P(v,vµ
| D) = vµ
+ v / 2( )
−(n−1)/2
v−n/2
exp −
n
4v
(g − r)2
−
n
(2vµ
+ v)
var
r + g
2
⎛
⎝⎜
⎞
⎠⎟
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
Example
with
low
extrinsic
noise
Inference
based
on
only.
(g − r)2
Bayesian
result.
Result
assuming
extrinsic
noise
known.

13. Extrinsic
noise
implies
transcrip3on/transla3on/decay
rates
fluctuate
Extrinsic
noise
in
Elowitz
et
al:
Intrinsic
noise
falls
as
the
promoter
is
induced.
Extrinsic
noise
peaks
at
intermediate
induc0on.
R
Phillips
(Annu
Rev
Con
Mat
Phys,
2015)
• Transcrip0on
rate
can
vary
when
the
promoter
switches
between
different
states.
• Switching
rates
depend
on
concentra0ons
of
DNA
binding
proteins
(polymerases,
TFs).
• These
concentra0ons
will
fluctuate
from
cell
to
cell.

14. Noise
propaga3on
• Regulatory
cascade:
Gene
1
induces
gene
2.
Gene
3
cons0tu0ve.
• As
gene
1
is
induced,
its
own
noise
level
drops.
• Gene
2
goes
through
an
intermediate
peak
in
noise
level.
• Gene
3’s
noise
is
unaffected.
Interpreta3on:
At
intermediate
levels
of
gene
1,
the
promoter
of
gene
2
shows
most
switching
between
bound
and
unbound
states
and
most
sensi0vity
to
fluctua0ons
in
the
concentra0on
of
gene
1.

15. Cells
are
not
sta3c:
Inves0ga0ng
stochas0c
regulatory
dynamics
Wish
list
• Follow
growth
and
gene
expression
dynamics
in
single
cells
over
long
0me
scales.
• Accurate
quan0fica0on.
• Follow
different
cell
lineages
separately
to
allow
observa0on
of
rare
events.
• Precise
dynamical
control
over
growth
environment.
Wang
et
al.
Robust
growth
of
Escherichia
coli.
Curr
Biol.
2010
The
mother
machine

16. Our
extension:
The
Dual
Input
Mother
Machine

17. Switching
growth
media
between
glucose
and
lactose
• GFP/lacZ
fusion
reports
lac-­‐operon
expression.
• Switch
glucose/lactose
every
4
hours.
• Immediate
growth
arrest
at
first
switch
to
lactose.
• Stochas0c
induc0on
of
lac-­‐operon
and
restart
of
growth.
• Dilu0on
of
GFP/lacZ
during
glucose
phase.
• No
more
growth
arrests
upon
later
switches.

18. Automated
Image
Analysis:
The
Mother
Machine
Analyzer
Florian
Jug
Gene
Myers
MPI
Cell
Biology,
Dresden
• Tracking
and
segmenta0on
done
in
parallel
using
a
single
objec0ve
func0on.
• Interac3ve
cura3on:
• User
input
interpreted
as
addi0onal
constraints.
• Automa0c
re-­‐op0miza0on.

19. Cells
expand
exponen3ally
during
their
cell
cycle
2
3
4
2
3
4
2
3
4
2
3
4
0 4 8 12 16 20
time (h)
celllength(µm)
0.970 0.975 0.980 0.985 0.990 0.995 1.000
0.0
0.2
0.4
0.6
0.8
1.0
Pearson correlation exp. growth curve
FractionCellCycles
Cumula3ve
correla3on
coeff.
of
log(size)
vs
3me
Example
growth
dynamics
of
log-­‐size
vs
3me
Roughly
two-­‐fold
variability
in
growth
rates

20. Fluorescence
roughly
tracks
cell
size
but
produc3on
fluctuates
significantly
Approximately
4-­‐fold
varia3on
in
produc3on
rate
Examples
of
total
fluorescence
against
0me
for
single
cells
growing
in
lactose.
Distribu0on
of
GFP
molecules
produced
per
second.

21. Distribu3on
of
total
fluorescence
and
fluorescence
concentra3ons
5000 10000 15000 20000 25000 30000 35000
0.00000
0.00005
0.00010
0.00015
Fluorescence HAUL
Probabilitydensity
Total Fluorescence Distribution
m=10'616, s=2911, sêm=0.274
8.5 9.0 9.5 10.0 10.5
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Log Fluorescence HAUL
Probabilitydensity
Total Log Fluorescence Distribution
m=9.23,s2
=0.07
4000 6000 8000 10000 12000
0.0000
0.0002
0.0004
0.0006
0.0008
Fluorescence concentrationHAUêmicronL
Probabilitydensity
Fluorescence Concentration Distribution
m=4278, s=661, sêm=0.154
8.0 8.5 9.0 9.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Log Fluorescence concentrationHAUêmicronL
Probabilitydensity
Log Fluorescence Concentration Distribution
m=8.35,s2
=0.022
Very
roughly
log-­‐normal
distribu0ons.
Concentra0on
has
significantly
less
varia0on.

22. Measuring
transcrip3on
from
all
E.
coli
promoters
in
single
cells
• GFP
fluorescence
per
cell
propor0onal
to
protein
number.
• GFP
levels
of
single
cells
can
be
measured
in
high-­‐throughput
using
FACS.
• Quan0ta0vely
characterize
the
distribu0on
of
expression
levels
across
single
cells,
for
all
E.
coli
promoters.
ORF1
ORF2
ORF4
E. coli genomeORF3
Plasmid
Zaslaver et al.
2006
Silander
et
al.
PLoS
genet
2012
Wolf
et
al.
eLife
2015

23. FACS:
Measuring
and
selec3ng
single
cells
• Cells
move
one-­‐by-­‐one
in
a
flow
channel.
• Each
cell
passes
in
front
of
a
laser
and
its
fluorescence
is
measured.
• By
selec0vely
charging
par0cles
based
on
their
measured
fluorescence,
one
can
select
cells
whose
fluorescence
lies
in
a
certain
range.

24. Gene
expression
distribu3ons
for
two
example
promoters
µ1 µ2
σ1
σ2
Promoter 1 Promoter 2

25. Means
and
variances
of
na3ve
E.
coli
promoters
• Variance
in
log-­‐expression
in
shows
a
trend
of
decreasing
with
mean
expression.
• Different
promoters
with
same
mean
can
show
significantly
different
variance.
• There
seems
to
be
a
clear
lower
bound
on
variance
as
a
func0on
of
mean.
5 6 7 8 9 10 11
0.0
0.2
0.4
0.6
0.8
Mean Log@GFP IntensityD
VarianceLog@GFPpercellD
background
2
*
background

26. 7 8 9 10 11 12 13
0.0
0.2
0.4
0.6
0.8
Mean Log@proteins per cellD
VarianceLog@proteinspercellD
7 8
0.0
0.2
0.4
0.6
0.8
Excessnoise
Means
and
variances
of
na3ve
E.
coli
promoters
Red
curve:
σab
2
= 0.025, b = 450
n = ab, var(n) = (b+1) nAt
constant
transcrip0on/transla0on/decay
rates:
Assume
a
and
b
both
fluctuate:
var(n) = (b +1) n +σab
2
n
2
nmeas
= nbg
+ n + ε var(n) var log(nmeas
)⎡⎣ ⎤⎦ = σab
2
1−
nbg
nmeas
⎛
⎝
⎜
⎞
⎠
⎟
2
+
(b +1)
nmeas
1−
nbg
nmeas
⎛
⎝
⎜
⎞
⎠
⎟

27. Noise
levels
vary
across
na3ve
E.
coli
promoters
7 8 9 10 11 12 13
0.0
0.2
0.4
0.6
0.8
Mean Log@ proteins per cellsD
Excessnoise
Excess
noise
(variance
–
lower
bound
as
func.
mean)
Selec3on
on
noise
levels
High
noise
DriZ?
Selected
for
noise?
Low
noise.
Selec0on
to
minimize
noise?
What
noise
would
one
get
without
selec3on?
Evolve
synthe8c
promoters
in
a
precisely
controlled
selec0ve
environment.

28. Directed
evolu0on
of
promoters
that
express
at
a
desired
level
*
*
*
28

29. Evolu0on
of
popula0on
expression
levels
Selec0ng
for
Medium
expression
29
Selec0ng
for
High
expression

30. Expression
distribu0ons
of
individual
synthe0c
promoters
• We
isolated
~400
clones
from
evolu0onary
runs
for
both
medium
and
high
expression.
• Measured
each
clone’s
expression
distribu0on.
How
do
noise
levels
of
synthe3c
promoters
compare
with
those
of
na3ve
promoters?

31. Na0ve
promoters
Synthe0c
promoters
• Synthe0c
promoters
were
not
selected
on
their
noise
proper0es.
• Low
noise
is
the
default
behavior
of
E.
coli
promoters.
• Selec0on
must
have
acted
so
as
to
increase
the
noise
levels
of
some
na0ve
promoters.
Iden0cal
distribu0ons
at
the
low
noise
end.
High
noise
enriched
in
na0ve
promoters.
Selec0on
caused
increased
noise
in
a
substan0al
frac0on
na0ve
promoters
What
is
`special’
about
na3ve
promoters
that
show
high
noise?

32. Noisy
genes
have
more
regulatory
inputs
• 185
E.
coli
transcrip0on
factors
(TFs).
•
4123
known
regulatory
interac0ons
TF
→
promoter.
Genes
with
higher
noise
have
(on
average)
higher
numbers
of
known
regulatory
inputs.
2
or
more
inputs
1
known
input
no
known
inputs
synthe0c
proms.
Why
is
there
a
general
associa3on
between
noise
and
regula3on?
Why
did
selec3on
cause
noise
to
increase?

33. Noise-­‐propaga3on:
nuisance
or
opportunity?
Noise
as
an
unavoidable
side-­‐effect
of
regula3on
• Explains
the
general
associa0on
of
noise
and
regula0on.
• `Fluctua0on-­‐dissipa0on
rela0on’:
Genes
that
need
complex
regula0on
unavoidably
couple
to
the
noise
in
their
regulators.
• Generally
assumed
to
be
detrimental:
reduces
the
accuracy
of
regula0on.
Stochas3city
as
a
bet-­‐hedging
strategy
• Phenotypic
diversity
can
generally
be
selected
for
in
fluctua0ng
environments.
• Maybe
noise-­‐propaga0on
can
be
beneficial
in
some
circumstances?
Let’s
do
some
theory
on
how
gene
expression
noise
affects
fitness

34. Fitness
func0on
in
a
single
environment
f (x | µ*,τ ) = exp −
(x −µ* )2
2τ 2
"
#
$
%
&
'
p(x | µ,σ ) =
1
2πσ
exp −
(x −µ)2
2σ 2
"
#
$
%
&
'
f (µ,σ | µ*,τ ) = dxp(x | µ,σ ) f (x | µ*,τ ) =∫
τ 2
τ 2
+σ 2
exp −
(µ −µ* )2
2(τ 2
+σ 2
)
#
$
%
&
'
(
The
fitness
of
a
promoter
`genotype’
(frac0on
of
its
cells
selected)
is
a
convolu0on
of
these
two
func0ons
(approx.
area
on
the
intersec0on):
Fitness
(probability
to
be
selected):
Promoter
expression
distribu0on:
σ = 0.1
µ µ*
τ

35. Moving
the
mean
toward
the
desired
level
always
increases
fitness
f (µ,σ | µ*,τ ) =
τ 2
τ 2
+σ 2
exp −
(µ −µ* )2
2(τ 2
+σ 2
)
"
#
$
%
&
'
7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4
0.0
0.2
0.4
0.6
0.8
1.0
Log expression
ExpressionêSelectionprobability
7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4
0.0
0.2
0.4
0.6
0.8
1.0
Log expression
ExpressionêSelectionprobability
f (µ = 8.0,σ = 0.1) = 0.066 f (µ = 8.1,σ = 0.1) = 0.174

36. 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4
0.0
0.2
0.4
0.6
0.8
1.0
Log expression
ExpressionêSelectionprobability
7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4
0.0
0.2
0.4
0.6
0.8
1.0
Log expression
ExpressionêSelectionprobability
At
op0mal
mean
minimal
noise
is
preferred
f (µ,σ | µ*,τ ) =
τ 2
τ 2
+σ 2
exp −
(µ −µ* )2
2(τ 2
+σ 2
)
"
#
$
%
&
'
f (µ = 8.15,σ = 0.1) = 0.196 f (µ = 8.15,σ = 0.025) = 0.625

37. As
mean
moves
away
from
the
op0mum
there
is
a
bifurca0on
to
nonzero
op0mal
noise
f (µ,σ | µ*,τ ) =
τ 2
τ 2
+σ 2
exp −
(µ −µ* )2
2(τ 2
+σ 2
)
"
#
$
%
&
'
f (µ = 8.0,σ = 0.05) = 0.0077
7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4
0.0
0.2
0.4
0.6
0.8
1.0
Log expression
ExpressionêSelectionprobability
f (µ = 8.0,σ = 0.1) = 0.066
7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4
0.0
0.2
0.4
0.6
0.8
1.0
Log expression
ExpressionêSelectionprobability
`Bifurca3on’
in
op3mal
σ
When
,
the
op0mal
noise
level
is
non-­‐zero:
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Expression deviation »mu-mu*»
Optimalsigma
Op3mal
σ
σ* = (µ −µ* )2
−τ 2
τ = 0.05
τ = 0.2µ −µ*
≥ τ

38. Variable
environment:
Fitness
of
an
unregulated
gene
log f (µ,σ )[ ]= −
(µ −µe )2
2(τ 2
+σ 2
)
+
1
2
log
τ 2
τ 2
+σ 2
"
#
$
%
&
'Log-­‐fitness
in
a
variable
environment:
Assuming
no
regula0on,
op0mal
mean
equals
Log-­‐fitness
becomes:
Op3mal
noise
matches
the
varia3on
in
desired
expression
levels:
log f (µ,σ )[ ]= −
var(µe )
2(τ 2
+σ 2
)
+
1
2
log
τ 2
τ 2
+σ 2
"
#
$
%
&
'
This
is
the
bet
hedging
scenario.
But:
Wouldn’t
it
be
beer
to
evolve
gene
regula0on?
σopt
2
= var(µe )−τ 2
µ = µe

39. Effects
of
coupling
a
gene
to
a
regulator
Regulator’s
ac0vity
Gene
coupled
to
the
regulator.
Gene
without
regula0on
TF
TF
Two
main
effects
on
the
gene’s
expression:
1. Condi3on-­‐response:
Mean
depends
on
regulator’s
(condi0on-­‐dependent)
ac0vity.
2. Noise-­‐propaga3on:
Noise
increases
due
to
propaga0on
of
the
regulator’s
noise.
We
developed
a
general
theory
to
calculate
how
these
effects
conspire
to
affect
fitness.

40. Fitness
depends
on
only
4
effec3ve
parameters
Varia0on
in
desired
levels:
V
στ
1.
Expression
mismatch:
Y 2
=
V
σ 2
+τ 2
Varia0on
in
regulator
levels:
Vr
σr
2.
Signal-­‐to-­‐noise
of
the
regulator:
S2
=
Vr
σr
2
3.
Correla3on
regulator/desired
levels:
R
Fitness
effect
of
the
regulatory
interac3on:
4.
Coupling
strength:
X
log[ f ]= −
1
2
Y 2
(1− R2
)+ SX − RY( )
2
(1+ X 2
)
−
1
2
log 1+ X 2"
#
$
%
Scenario:
Start
with
unregulated
promoter.
What
fitness
can
be
obtained
by
coupling
to
regulator
with
signal-­‐to-­‐noise
S
and
correla0on
R?

41. Fitness
with
op0mal
coupling
to
a
regulator
of
given
correla0on
R
and
signal-­‐to-­‐noise
S
Fitness
of
the
unregulated
promoter.
Y=4
Perfect
correla0on
No
correla0on
Noisy
regulator
Precise
regulator

42. Coupling
to
a
near
op3mal
regulator:
condi3on-­‐response
effect
Y=4
TF
TF
σtot = 0.16
R = 0.95
S = 3.3
Fitness
of
the
unregulated
promoter.

43. Coupling
to
a
noisy
uncorrelated
regulator:
noise-­‐propaga3on
implements
bet
hedging
strategy
Y=4
TF
TF
σtot = 0.55
R = 0
S = 0.19
Fitness
of
the
unregulated
promoter.

44. Intermediate
case:
a
moderately
correlated
regulator
Y=4
TF
TF
σtot = 0.23
R = 0.64
S = 2.45
Fitness
of
the
unregulated
promoter.

45. Op0mal
S
at
a
given
R.
Y=4
Condi3on-­‐response
and
noise-­‐propaga3on
typically
act
in
concert
Regulator
too
noisy.
Regulator
not
noisy
enough.
• Noise-­‐propaga0on
is
oZen
func8onal,
ac0ng
as
a
rudimentary
form
of
regula0on.
• De
novo
evolu0on
of
regula0on:
Star0ng
from
pure
noise-­‐propaga0on
(R=0,S=0)
there
is
a
con0nuum
of
solu0ons
of
increasing
accuracy
along
which
condi0on-­‐
response
and
noise-­‐propaga0on
op0mally
complement
each
other.
• Regulated
genes
are
noisy
because,
whenever
the
condi0on-­‐response
is
imperfect,
maximal
fitness
requires
noisy
regulators.
Summary
Theory:

46. 0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
1.0
Y: Expression mismatch
R:Correlationofregulator'sexpressionwithdesired-levels
σtot
2
=σ 2
Low
noise
regime:
Promoters
with
low
expression
mismatch
Y<1
`do
not
bother’
to
be
regulated.
For
extremely
correlated
regulators,
zero
noise-­‐propaga0on
is
the
op0mum.
Phase
diagram
of
final
noise
aZer
coupling
to
regulators
with
op0mal
noise
levels.

47. 0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
1.0
Y: Expression mismatch
R:Correlationofregulator'sexpressionwithdesired-levels
σtot
2
=σ 2
Noise-­‐propaga3on
regime:
The
final
noise
level
matches
the
frac0on
of
variance
in
desired
levels
not
tracked
by
the
condi0on-­‐response.
σtot
2
= (1− R2
)var(µe )−τ 2
Phase
diagram
of
final
noise
aZer
coupling
to
regulators
with
op0mal
noise
levels.
Amount
of
regula3on
required.
Variance
in
desired
levels
Selec3on
tolerance
Limited
accuracy
of
the
condi3on-­‐response.
Frac3on
variance
not
tracked
by
regula3on.

48. Conclusions
signal
regulator
• We
evolved
synthe0c
promoters
de
novo
in
E.
coli
under
carefully-­‐
controlled
selec0ve
condi0ons.
• No
evidence
E.
coli
promoters
have
been
selected
to
lower
noise.
• Regulated
genes
have
been
selected
to
increase
noise.
Experimental
observa3ons
Theory
• Coupling
a
regulator
to
a
target
promoter
has
two
effects:
1. Condi0on-­‐response.
2. Noise-­‐propaga0on.
• Noise-­‐propaga0on
alone
can
act
as
a
rudimentary
form
of
regula0on.
• Accurate
regula0on
can
evolve
smoothly
along
a
con0nuum
in
which
noise-­‐propaga0on
and
condi0on-­‐response
act
in
concert.
• Whenever
the
condi0on-­‐response
has
limited
accuracy,
noisy
regula0on
is
preferred.
• Explains
the
general
associa0on
between
noise
and
regula0on.

49. Thank
you!
Luise
Wolf
Olin
Silander
Theory/computa3on
PhD
and
post-­‐doc
posi3ons
available!
This
work:
Our
group