of 30
RESEA
RCH
ARTICL
E
Reconciling
kinetic
and
thermodynamic
models
of
bacterial
transcription
Muir
Morrison
ID
1
, Manuel
Razo-Mejia
ID
2
, Rob
Phillips
ID
1,2
*
1
Department
of Physics,
Californi
a Institute
of Technolo
gy,
Pasade
na,
California,
USA,
2
Division
of Biology
and
Biologic
al Engine
ering,
California
Institute
of Technolo
gy,
Pasade
na,
Californi
a, USA
*
phillips
@pboc.caltec
h.edu
Abstract
The
study
of transcription
remains
one
of the
centerpieces
of modern
biology
with
implica-
tions
in settings
from
development
to metabolism
to evolution
to disease.
Precision
mea-
surements
using
a host
of different
techniques
including
fluorescence
and
sequencing
readouts
have
raised
the
bar
for
what
it means
to quantitatively
understand
transcriptional
regulation.
In particular
our
understanding
of the
simplest
genetic
circuit
is sufficiently
refined
both
experimentally
and
theoretically
that
it has
become
possible
to carefully
dis-
criminate
between
different
conceptual
pictures
of how
this
regulatory
system
works.
This
regulatory
motif,
originally
posited
by
Jacob
and
Monod
in the
1960s,
consists
of a single
transcriptional
repressor
binding
to a promoter
site
and
inhibiting
transcription.
In this
paper,
we
show
how
seven
distinct
models
of this
so-called
simple-repression
motif,
based
both
on
thermodynamic
and
kinetic
thinking,
can
be
used
to derive
the
predicted
levels
of gene
expression
and
shed
light
on
the
often
surprising
past
success
of the
thermodynamic
mod-
els.
These
different
models
are
then
invoked
to confront
a variety
of different
data
on
mean,
variance
and
full
gene
expression
distributions,
illustrating
the
extent
to which
such
models
can
and
cannot
be
distinguished,
and
suggesting
a two-state
model
with
a distribution
of
burst
sizes
as
the
most
potent
of the
seven
for
describing
the
simple-repress
ion
motif.
Author
summary
With
the
advent
of
new
technologies
allowing
us
to
query
biological
activity
with
ever
increasing
precision,
the
deluge
of
quantitative
biological
data
demands
quantitative
mod-
els.
Transcriptional
regulation—a
feature
that
lies
at
the
core
of
our
understanding
of
cel-
lular
control
in
myriad
context
ranging
from
development
to
disease—is
no
exception,
with
single-cell
and
single-molecule
techniques
being
routinely
deployed
to
study
cellular
decision
making.
These
data
have
served
as
a fertile
proving
ground
to
test
models
of
tran-
scription
that
mainly
come
in
two
flavors:
thermodynamic
models
(based
on
equilibrium
statistical
mechanics)
and
kinetic
models
(based
on
chemical
kinetics).
In
this
paper
we
study
the
correspondence
between
these
theoretical
frameworks
in
the
context
of
the
sim-
ple
repression
motif,
a common
regulatory
architecture
in
prokaryotes
in
which
a repres-
sor
with
a single
binding
site
regulates
expression.
We
explore
the
consequences
of
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OPEN
ACCESS
Citation:
Morrison
M, Razo-Mejia
M, Phillips
R
(2021)
Reconcilin
g kinetic
and thermodynam
ic
models
of bacterial
transcripti
on. PLoS
Comput
Biol 17(1):
e1008572.
https://doi.
org/10.1371/
journal.pcb
i.1008572
Editor:
James
R. Faeder,
Univers
ity of Pittsburgh
,
UNITED
STATES
Received:
June
14, 2020
Accepted:
November
28, 2020
Published:
January
19, 2021
Peer Review
History:
PLOS
recognize
s the
benefits
of transpar
ency
in the peer review
process;
therefore,
we enable
the publication
of
all of the content
of peer review
and author
response
s alongside
final,
published
articles.
The
editorial
history
of this article
is available
here:
https://doi.o
rg/10.1371/jo
urnal.pcbi
.1008572
Copyright:
©
2021
Morrison
et al. This is an open
access
article
distributed
under
the terms
of the
Creative
Commons
Attribution
License,
which
permits
unrestricte
d use, distribu
tion, and
reproduction
in any medium,
provided
the original
author
and source
are credited.
Data
Availabilit
y Statement:
All data and custom
scripts
were
collected
and stored
using
Git version
control.
Code
for Bayesia
n inference
and figure
generatio
n is available
on the GitHub
repository
different
levels
of
coarse-graining
of
the
molecular
steps
involved
in
transcription,
finding
that,
at
the
level
of
mean
gene
expression,
the
different
models
cannot
be
distinguished.
We
then
study
higher
moments
of
the
gene
expression
distribution
which
allows
us
to
dis-
card
several
of
the
models
that
disagree
with
experimental
data
and
supporting
a minimal
kinetic
model.
Introduction
Gene
expression
presides
over
much
of
the
most
important
dynamism
of
living
organisms.
The
level
of
expression
of
batteries
of
different
genes
is altered
as
a result
of
spatiotemporal
cues
that
integrate
chemical,
mechanical
and
other
types
of
signals.
As
our
ability
to
experi-
mentally
observe
and
measure
the
dynamical
processes
that
constitute
the
central
dogma
improves,
there
is an
opportunity
to
undertake
a theory-experiment
dialogue
in
order
to
sharpen
our
understanding
of
such
a fundamental
biological
process.
One
of
the
remaining
outstanding
challenges
to
have
emerged
in
the
genomic
era
is our
continued
inability
to
pre-
dict
the
regulatory
consequences
of
different
regulatory
architectures,
i.e.
the
arrangement
and
affinity
of
binding
sites
for
transcription
factors
and
RNA
polymerases
on
the
DNA.
This
chal-
lenge
stems
first
and
foremost
from
our
ignorance
about
what
those
architectures
even
are,
with
more
than
60%
of
the
genes
even
in
an
ostensibly
well
understood
organism
such
as
E.
coli
having
no
regulatory
insights
at
all
[1–4].
But
even
once
we
have
established
the
identity
of
key
transcription
factors
and
their
binding
sites
for
a given
promoter
architecture,
there
remains
the
predictive
challenge
of
understanding
its
input-output
properties,
an
objective
that
can
be
met
by
a myriad
of
approaches
using
the
tools
of
statistical
physics
[5–24].
One
route
to
such
predictive
understanding
is to
focus
on
the
simplest
regulatory
architecture
and
to
push
the
theory-experiment
dialogue
to
increase
the
predictive
power
of
our
theoretical
models
[25,
26].
If we
demonstrate
that
we
can
pass
that
test
by
successfully
predicting
both
the
means
and
variance
in
gene
expression
at
the
mRNA
level,
then
that
provides
a more
solid
foundation
upon
which
to
launch
into
more
complex
problems—for
instance,
some
of
the
pre-
viously
unknown
architectures
uncovered
in
[2]
and
[27].
To
that
end,
in
this
paper
we
examine
a wide
variety
of
distinct
models
for
the
simple
repression
regulatory
architecture.
This
genetic
architecture
consists
of
a DNA
promoter
regu-
lated
by
a transcriptional
repressor
that
binds
to
a single
binding
site
as
developed
in
pioneer-
ing
early
work
on
the
quantitative
dissection
of
transcription
[28,
29].
One
of
the
main
features
of
the
models
we
explore
is that,
by
construction,
all
features
related
to
the
microstates
in
which
the
repressor
is bound
to
the
promoter
can
be
separated
from
the
microstates
in
which
the
RNA
polymerase
(RNAP)
is bound.
From
a modeling
perspective,
this
means
that
some
of
the
models
can
be
written
as
effective
two-state
models
for
which
there
is a rich
litera-
ture
[17,
21,
24,
30–36].
Here,
we
systematically
compare
the
predictions
of
several
models
with
different
levels
of
coarse
graining
written
in
terms
of
thermodynamic
and
kinetic
parame-
ters.
One
goal
in
exploring
such
coarse-grainings
is to
build
towards
the
future
models
of
regu-
latory
response
that
will
be
able
to
serve
the
powerful
predictive
role
needed
to
take
synthetic
biology
from
a brilliant
exercise
in
enlightened
empiricism
to
a rational
design
framework
as
in
any
other
branch
of
engineering.
More
precisely,
we
want
phenomenology
in
the
sense
of
coarse-graining
away
atomistic
detail,
but
still
retaining
biophysical
meaning.
In
particular
a
key
question
is:
at
this
level
of
coarse-graining,
what
microscopic
details
do
we
need
to
explic-
itly
model,
and
how
do
we
figure
that
out?
For
example,
do
we
need
to
worry
about
all
or
even
any
of
the
steps
that
individual
RNA
polymerases
go
through
each
time
they
make
a transcript?
PLOS COMP
UTATIONAL
BIOLOGY
Reconciling
kinetic
and
thermodyn
amic
models
of bacterial
transcrip
tion
PLOS
Computationa
l Biology
| https:/
/doi.org/10.13
71/journal.p
cbi.1008572
January
19,
2021
2 / 30
(https://git
hub.com/RPG
roup-PBoC
/bursty_
transcripti
on).
Funding:
This material
is based
upon
work
supported
by the National
Science
Foundation
Graduate
Research
Fellowship
under
Grant
No.
DGE-1745
301 (to M.J.M.).
This work
was also
supported
by La Fondation
Pierre-Gilles
de Gennes,
the Rosen
Center
at Caltech,
and the NIH
5R35GM1
18043-05
(MIRA)
to R.P. M.R.M.
was
supported
by the Caldwell
CEMI
fellowship.
The
funders
had no role in study
design,
data collection
and analysis,
decision
to publish,
or preparation
of
the manuscript.
Competing
interests
:
The authors
have declared
that no competing
interests
exist.
Turning
the
question
around,
can
we
see
any
imprint
of
those
processes
in
the
available
data?
If the
answer
is no,
then
those
processes
are
irrelevant
for
our
purposes.
Forward
modeling
and
inverse
(statistical
inferential)
modeling
are
necessary
to
tackle
such
questions.
We
com-
bine
both
approaches
in
order
to
discard
models
that
cannot
empirically
satisfy
the
main
fea-
tures
of
experimental
data.
First
we
apply
forward
modeling
to
demonstrate
that
none
of
the
models
are
distinguishable
at
the
level
of
mean
gene
expression.
We
then
extend
the
modeling
to
look
at
higher
moments
of
the
distribution,
eliminating
models
that
do
not
empirically
sat-
isfy
the
observed
cell-to-cell
variability.
Finally
we
arrive
at
a minimal
model
on
which
we
can
apply
inverse
modeling
in
order
to
infer
the
parameters
that
explain
the
data.
Fig
1A
shows
the
qualitative
picture
of
simple
repression
that
is implicit
in
the
repressor-
operator
model.
An
operator,
i.e.,
the
binding
site
on
the
DNA
for
a repressor
protein,
may
be
found
occupied
by
a repressor,
in
which
case
transcription
is blocked
from
occurring.
Alterna-
tively,
that
binding
site
may
be
found
unoccupied,
in
which
case
RNA
polymerase
(RNAP)
may
bind
and
transcription
can
proceed.
The
key
assumption
we
make
in
this
simplest
incar-
nation
of
the
repressor-operator
model
is that
binding
of
repressor
and
RNAP
in
the
promoter
region
of
interest
is exclusive,
meaning
that
one
or
the
other
may
bind,
but
never
may
both
be
simultaneously
bound.
It is often
imagined
that
when
the
repressor
is bound
to
its
operator,
RNAP
is sterically
blocked
from
binding
to
its
promoter
sequence.
Current
evidence
suggests
this
is sometimes,
but
not
always
the
case,
and
it remains
an
interesting
open
question
pre-
cisely
how
a repressor
bound
far
upstream
is able
to
repress
transcription
[1].
Suggestions
include
“action-at-a-distanc
e”
mediated
by
kinks
in
the
DNA,
formed
when
the
repressor
is
bound,
that
prevent
RNAP
binding.
Nevertheless,
our
modeling
in
this
work
is sufficiently
coarse-grained
that
we
simply
assume
exclusive
binding
and
leave
explicit
accounting
of
these
details
out
of
the
problem.
The
logic
of
the
remainder
of
the
paper
is as
follows.
In
the
section
1,
we
show
how
both
thermodynamic
models
(Fig
1B)
and
kinetic
models
based
upon
the
chemical
master
equation
(Fig
1C)
all
culminate
in
the
same
underlying
functional
form
for
the
fold-change
in
the
aver-
age
level
of
gene
expression
with
an
effective
free
energy
Δ
F
R
capturing
the
regulation
given
by
the
transcription
factor,
and
a term
ρ
describing
the
level
of
coarse-graining
of
the
transcrip-
tional
events
as
shown
in
Fig
1D.
Section
2 goes
beyond
an
analysis
of
the
mean
gene
expres-
sion
by
asking
how
the
same
models
presented
in
Fig
1C
can
be
used
to
explore
noise
in
gene
expression.
To
make
contact
with
experiment,
all
of
these
models
must
make
a commitment
to
some
numerical
values
for
the
key
parameters
found
in
each
such
model.
Therefore
in
Sec-
tion
3 we
explore
the
use
of
Bayesian
inference
to
establish
these
parameters
and
to
rigorously
answer
the
question
of
how
to
discriminate
between
the
different
models.
Materials
and
methods
All
data
and
custom
scripts
were
collected
and
stored
using
Git
version
control.
Code
for
Bayesian
inference
and
figure
generation
is available
on
the
GitHub
repository
(https://github.
com/RPGroup-PBoC/bursty_
transcription).
Results
1 Mean
gene
expression
As
noted
in
the
previous
section,
there
are
two
broad
classes
of
models
in
play
for
computing
the
input-output
functions
of
regulatory
architectures
as
shown
in
Fig
1.
In
both
classes
of
model,
the
promoter
is imagined
to
exist
in
a discrete
set
of
states
of
occupancy,
with
each
such
state
of
occupancy
accorded
its
own
rate
of
transcription–including
no
transcription
for
many
of
these
states.
This
discretization
of
a potentially
continuous
number
of
promoter
states
PLOS COMP
UTATIONAL
BIOLOGY
Reconciling
kinetic
and
thermodyn
amic
models
of bacterial
transcrip
tion
PLOS
Computationa
l Biology
| https:/
/doi.org/10.13
71/journal.p
cbi.1008572
January
19,
2021
3 / 30
Fig
1.
An
overview
of
the
simple
repressio
n motif
at
the
level
of
means.
(A)
Schematic
of
the
qualitati
ve
biological
picture
of
the
simple
repression
genetic
architectur
e. (B)
and
(C)
A variety
of
possible
mathemati
cized
cartoons
of
simple
repression,
along
with
the
effective
parame
ter
ρ
which
subsumes
all
regulatory
details
of
the
architectur
e that
do
not
directly
involve
the
repressor.
(B)
Simple
repression
models
from
a thermodyn
amic
perspective
. (C)
Equivalent
models
cast
in
chemical
kinetics
language.
(D)
The
“master
curve”
to
which
all
cartoons
in
(B)
and
(C)
collapse
.
https://
doi.org/10.1371
/journal.pcbi.10
08572.g001
PLOS COMP
UTATIONAL
BIOLOGY
Reconciling
kinetic
and
thermodyn
amic
models
of bacterial
transcrip
tion
PLOS
Computationa
l Biology
| https:/
/doi.org/10.13
71/journal.p
cbi.1008572
January
19,
2021
4 / 30
(due
to
effects
such
as
supercoiling
of
DNA
[37,
38]
or
DNA
looping
[39])
is analogous
to
how
the
Monod-Wyman-Changeux
model
of
allostery
coarse-grains
continuous
molecule
confor-
mations
into
a finite
number
of
states
[40].
The
models
are
probabilistic
with
each
state
assigned
some
probability
and
the
overall
rate
of
transcription
given
by
average rate of transcription
¼
X
i
r
i
p
i
;
ð
1
Þ
where
i
labels
the
distinct
states,
p
i
is the
probability
of
the
i
th
state,
and
r
i
is the
rate
of
tran-
scription
of
that
state.
Ultimately,
the
different
models
differ
along
several
key
aspects:
what
states
to
consider
and
how
to
compute
the
probabilities
of
those
states.
The
first
class
of
models
that
are
the
subject
of
the
present
section
focus
on
predicting
the
mean
level
of
gene
expression.
These
models,
sometimes
known
as
thermodynamic
models,
invoke
the
tools
of
equilibrium
statistical
mechanics
to
compute
the
probabilities
of
the
pro-
moter
microstates
[5–11,
13–15].
As
seen
in
Fig
1B,
even
within
the
class
of
thermodynamic
models,
we
can
make
different
commitments
about
the
underlying
microscopic
states
of
the
promoter.
Model
1 considers
only
two
states:
a state
in
which
a repressor
(with
copy
number
R
)
binds
to
an
operator
and
a transcriptionally
active
state.
The
free
energy
difference
between
the
repressor
binding
the
operator,
i.e.
a specific
binding
site,
and
one
of
the
N
NS
non-specific
sites
is given
by
Δ
ε
R
(given
in
k
B
T
units
with
β
(
k
B
T
)
1
). Model
2 expands
this
model
to
include
an
empty
promoter
where
no
transcription
occurs,
as
well
as
a state
in
which
one
of
the
P
RNAPs
binds
to
the
promoter
with
binding
energy
Δ
ε
P
. Indeed,
the
list
of
options
con-
sidered
here
does
not
at
all
exhaust
the
suite
of
different
microscopic
states
we
can
assign
to
the
promoter.
The
essence
of
thermodynamic
models
is to
assign
a discrete
set
of
states
and
to
use
equilibrium
statistical
mechanics
to
compute
the
probabilities
of
occupancy
of
those
states.
The
second
class
of
models
that
allow
us
to
access
the
mean
gene
expression
use
chemical
master
equations
to
compute
the
probabilities
of
the
different
microscopic
states
[16–23].
The
main
differences
between
both
modeling
approaches
can
be
summarized
as:
1)
Although
for
both
classes
of
models
the
steps
involving
transcriptional
events
are
assumed
to
be
strictly
irre-
versible,
thermodynamic
models
force
the
regulation,
i.e.,
the
control
over
the
expression
exerted
by
the
repressor,
to
be
in
equilibrium.
This
does
not
need
to
be
the
case
for
kinetic
models.
2)
Thermodynamic
models
ignore
the
mRNA
count
from
the
state
of
the
Markov
pro-
cess,
while
kinetic
models
keep
track
of
both
the
promoter
state
and
the
mRNA
count.
3)
Finally,
thermodynamic
and
kinetic
models
coarse-grain
to
different
degrees
the
molecular
mechanisms
through
which
RNAP
enters
the
transcriptional
event.
As
seen
in
Fig
1C,
we
con-
sider
a host
of
different
kinetic
models,
each
of
which
will
have
its
own
result
for
both
the
mean
(this
section)
and
noise
(next
section)
in
gene
expression.
1.1
Fold-changes
are
indistinguishable
across
models.
As
a first
stop
on
our
search
for
the
“right”
model
of
simple
repression,
let
us
consider
what
we
can
learn
from
theory
and
experimental
measurements
on
the
average
level
of
gene
expression
in
a population
of
cells.
One
experimental
strategy
that
has
been
particularly
useful
(if
incomplete
since
it misses
out
on
gene
expression
dynamics)
is to
measure
the
fold-change
in
mean
expression
[25].
The
fold-change
FC
is defined
as
FC
ð
R
Þ ¼
h
gene
expression with
R
>
0
i
h
gene
expression with
R
¼
0
i
¼
h
m
ð
R
Þi
h
m
ð
0
Þi
¼
h
p
ð
R
Þi
h
p
ð
0
Þi
;
ð
2
Þ
where
angle
brackets
h�i
denote
the
average
over
a population
of
cells
and
mean
mRNA
h
m
i
and
mean
protein
h
p
i
are
viewed
as
a function
of
repressor
copy
number
R
.
What
this
means
is that
the
fold-change
in
gene
expression
is a relative
measurement
of
the
effect
of
the
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transcriptional
repressor
(
R
>
0)
on
the
gene
expression
level
compared
to
an
unregulated
promoter
(
R
= 0).
The
third
equality
in
Eq
2 follows
from
assuming
that
the
translation
effi-
ciency,
i.e.,
the
number
of
proteins
translated
per
mRNA,
is the
same
in
both
conditions.
In
other
words,
we
assume
that
mean
protein
level
is proportional
to
mean
mRNA
level,
and
that
the
proportionality
constant
is the
same
in
both
conditions
and
therefore
cancels
out
in
the
ratio.
This
is reasonable
since
the
cells
in
the
two
conditions
are
identical
except
for
the
pres-
ence
of
the
transcription
factor,
and
the
model
assumes
that
the
transcription
factor
has
no
direct
effect
on
translation.
Fold-change
has
proven
a very
convenient
observable
in
past
work
[41–44].
Part
of
its
util-
ity
in
dissecting
transcriptional
regulation
is its
ratiometric
nature,
which
removes
many
sec-
ondary
effects
that
are
present
when
making
an
absolute
gene
expression
measurement.
Also,
by
measuring
otherwise
identical
cells
with
and
without
a transcription
factor
present,
any
bio-
logical
noise
common
to
both
conditions
can
be
made
to
cancel
out.
Fig
1B
and
1C
depicts
a
smorgasbord
of
mathematicized
cartoons
for
simple
repression
using
both
thermodynamic
and
kinetic
models,
respectively,
that
have
appeared
in
previous
literature.
For
each
cartoon,
we
calculate
the
fold-change
in
mean
gene
expression
as
predicted
by
that
model,
deferring
most
algebraic
details
to
the
S1
Supporting
Information.
What
we
will
find
is that
for
all
car-
toons
the
fold-change
can
be
written
as
a Fermi
function
of
the
form
FC
ð
R
Þ ¼ð
1
þ
exp
ð
D
F
R
ð
R
Þ þ
log
ð
r
ÞÞÞ
1
;
ð
3
Þ
where
the
effective
free
energy
contains
two
terms:
the
parameters
Δ
F
R
, an
effective
free
energy
parametrizing
the
repressor-DNA
interaction,
and
ρ
,
a term
derived
from
the
level
of
coarse-
graining
used
to
model
all
repressor-free
states.
In
other
words,
the
effective
free
energy
of
the
Fermi
function
can
be
written
as
the
additive
effect
of
the
regulation
given
by
the
repressor
via
Δ
F
R
, and
the
kinetic
scheme
used
to
describe
the
steps
that
lead
to
a transcriptional
event
via
log(
ρ
)
(See
Fig
1D,
left
panel).
This
implies
all
models
collapse
to
a single
master
curve
as
shown
in
Fig
1D.
We
will
offer
some
intuition
for
why
this
master
curve
exists
and
discuss
why
at
the
level
of
the
mean
expression,
we
are
unable
to
discriminate
“right”
from
“wrong”
cartoons
given
only
measurements
of
fold-changes
in
expression.
1.1.1 Two- and three-state
thermodynamic
models
We
begin
our
analysis
with
models
1 and
2 in
Fig
1B.
In
each
of
these
models
the
promoter
is idealized
as
existing
in
a set
of
discrete
states;
the
difference
being
whether
or
not
the
RNAP
bound
state
is included
or
not.
Gene
expression
is then
assumed
to
be
proportional
to
the
probability
of
the
promoter
being
in
either
the
empty
state
(model
1)
or
the
RNAP-bound
state
(model
(2)).
We
direct
the
reader
to
the
S1
Supporting
Information
for
details
on
the
der-
ivation
of
the
fold-change.
For
our
purposes
here,
it suffices
to
state
that
the
functional
form
of
the
fold-change
for
model
1 is
FC
ð
R
Þ ¼
1
þ
R
N
NS
e
b
D
ε
R
1
;
ð
4
Þ
where
R
is the
number
of
repressors
per
cell,
N
NS
is the
number
of
non-specific
binding
sites
where
the
repressor
can
bind,
Δ
ε
R
is the
repressor-operator
binding
energy,
and
β
(
k
B
T
)
1
.
This
equation
matches
the
form
of
the
master
curve
in
Fig
1D
with
ρ
= 1 and
Δ
F
R
=
β
Δ
ε
R
log
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(
R
/
N
NS
). For
model
2 we
have
a similar
situation.
The
fold-change
takes
the
form
FC
ð
R
Þ ¼
1
þ
R
N
NS
e
b
D
ε
R
1
þ
P
N
NS
e
b
D
ε
P
!
1
ð
5
Þ
¼ ð
1
þ
exp
ð
D
F
R
þ
log
r
ÞÞ
1
;
ð
6
Þ
where
P
is the
number
of
RNAP
per
cell,
and
Δ
ε
P
is the
RNAP-promoter
binding
energy.
For
this
model
we
have
Δ
F
R
=
β
Δ
ε
R
log(
R
/
N
NS
) and
r
¼
1
þ
P
N
NS
e
b
D
ε
P
. Thus
far,
we
see
that
the
two
thermodynamic
models,
despite
making
different
coarse-graining
commitments,
result
in
the
same
functional
form
for
the
fold-change
in
mean
gene
expression.
We
now
explore
how
kinetic
models
fare
when
faced
with
computing
the
same
observable.
1.1.2 Kinetic
models
One
of
the
main
difference
between
models
shown
in
Fig
1C,
cast
in
the
language
of
chemi-
cal
master
equations,
compared
with
the
thermodynamic
models
discussed
in
the
previous
section
is the
probability
space
over
which
they
are
built.
Rather
than
keeping
track
only
of
the
microstate
of
the
promoter,
and
assuming
that
gene
expression
is proportional
to
the
probabil-
ity
of
the
promoter
being
in
a certain
microstate,
chemical
master
equation
models
are
built
on
the
entire
probability
state
of
both
the
promoter
microstate,
and
the
current
mRNA
count.
Therefore,
in
order
to
compute
the
fold-change,
we
must
compute
the
mean
mRNA
count
on
each
of
the
promoter
microstates,
and
add
them
all
together
[32].
Again,
we
consign
all
details
of
the
derivation
to
the
S1
Supporting
Information.
Here
we
just
highlight
the
general
findings
for
all
five
kinetic
models.
As
already
shown
in
Fig
1C
and
1D,
all
the
kinetic
models
explored
can
be
collapsed
onto
the
master
curve.
Given
that
the
repressor-bound
state
only
connects
to
the
rest
of
the
promoter
dynamics
via
its
binding
and
unbinding
rates,
k
þ
R
and
k
R
respectively,
all
models
can
effectively
be
separated
into
two
catego-
ries:
a single
repressor-bound
state,
and
all
other
promoter
states
with
different
levels
of
coarse
graining.
This
structure
then
guarantees
that,
at
steady-state,
detailed
balance
between
these
two
groups
is satisfied.
What
this
implies
is that
the
steady-state
distribution
of
each
of
the
non-repressor
states
has
the
same
functional
form
with
or
without
the
repressor,
allowing
us
to
write
the
fold-change
as
a product
of
the
ratio
of
the
binding
and
unbinding
rates
of
the
pro-
moter,
and
the
promoter
details.
This
results
in
a fold-change
of
the
form
FC
¼
1
þ
k
þ
R
k
R
r
1
;
ð
7
Þ
¼ ð
1
þ
exp
ð
D
F
R
þ
log
ð
r
ÞÞÞ
1
;
ð
8
Þ
where
D
F
R
log
ð
k
þ
R
=
k
R
Þ
, and
the
functional
forms
of
ρ
for
each
model
change
as
shown
in
Fig
1C.
Another
intuitive
way
to
think
about
these
two
terms
is as
follows:
in
all
kinetic
models
shown
in
Fig
1C
the
repressor-bound
state
can
only
be
reached
from
a single
repressor-free
state.
The
ratio
of
these
two
states
--repressor-bound
and
adjacent
repressor-free
state-
- must
remain
the
same
for
all
models,
regardless
of
the
details
included
in
other
promoter
states
if
Δ
F
R
represents
an
effective
free
energy
of
the
repressor
binding
the
DNA
operator.
The
pres-
ence
of
other
states
then
draws
probability
density
from
the
promoter
being
in
either
of
these
two
states,
making
the
ratio
between
the
repressor-bound
state
and
all
repressor-free
states
dif-
ferent.
The
log
difference
in
this
ratio
is given
by
log(
ρ
).
Since
model
1 and
model
5 of
Fig
1C
consist
of
a single
repressor-free
state,
ρ
is then
necessarily
1 (See
the
S1
Supporting
Informa-
tion
for
further
details).
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The
key
outcome
of
our
analysis
of
the
models
in
Fig
1 is the
existence
of
a master
curve
shown
in
Fig
1D
to
which
the
fold-change
predictions
of
all
the
models
collapse.
This
master
curve
is parametrized
by
only
two
effective
parameters:
Δ
F
R
, which
characterizes
the
number
of
repressors
and
their
binding
strength
to
the
DNA,
and
ρ
,
which
characterizes
all
other
fea-
tures
of
the
promoter
architecture.
The
key
assumption
underpinning
this
result
is that
no
transcription
occurs
when
a repressor
is bound
to
its
operator.
Given
this
outcome,
i.e.,
the
degeneracy
of
the
different
models
at
the
level
of
fold-change,
a mean-based
metric
such
as
the
fold-change
that
can
be
readily
measured
experimentally
is insufficient
to
discern
between
these
different
levels
of
coarse-graining.
The
natural
extension
that
the
field
has
followed
for
the
most
part
is to
explore
higher
moments
of
the
gene
expression
distribution
in
order
to
establish
if those
contain
the
key
insights
into
the
mechanistic
nature
of
the
gene
transcription
process
[24,
35].
Following
a similar
trend,
in
the
next
section
we
extend
the
analysis
of
the
models
to
higher
moments
of
the
mRNA
distribution
as
we
continue
to
examine
the
discrimi-
natory
power
of
these
different
models.
2 Beyond
means
in
gene
expression
In
this
section,
our
objective
is to
explore
the
same
models
considered
in
the
previous
section,
but
now
with
reference
to
the
question
of
how
well
they
describe
the
distribution
of
gene
expression
levels,
with
special
reference
to
the
variance
in
these
distributions.
To
that
end,
we
repeat
the
same
pattern
as
in
the
previous
section
by
examining
the
models
one
by
one.
In
par-
ticular
we
will
focus
on
the
Fano
factor,
defined
as
the
variance/mean.
This
metric
serves
as
a
powerful
discriminatory
tool
to
compare
our
different
models
to
the
null
model
that
the
steady-state
mRNA
distribution
must
be
Poisson,
resulting
a Fano
factor
of
one.
2.1
Kinetic
models
for
unregulated
promoter
noise.
Before
we
can
tackle
simple
repres-
sion,
we
need
an
adequate
phenomenological
model
of
constitutive
expression.
The
literature
abounds
with
options
from
which
we
can
choose,
and
we
show
several
potential
kinetic
mod-
els
for
constitutive
promoters
in
Fig
2A.
Let
us
consider
the
suitability
of
each
model
for
our
purposes
in
turn.
2.1.1 Poisson
noise promoter
The
simplest
model
of
constitutive
expression
that
we
can
imagine
is shown
as
model
1 in
Fig
2A
and
assumes
that
transcripts
are
produced
as
a Poisson
process
from
a single
promoter
state.
This
is the
picture
from
Jones
et.
al.
[33]
that
was
used
to
interpret
a systematic
study
of
gene
expression
noise
over
a series
of
promoters
designed
to
have
different
strengths,
but
no
regula-
tion.
This
model
insists
that
the
“true”
steady-state
mRNA
distribution
is Poisson,
implying
the
Fano
factor
ν
must
be
1. In
[33],
the
authors
carefully
attribute
measured
deviations
from
Fano
= 1 to
intensity
variability
in
fluorescence
measurements,
gene
copy
number
variation,
and
copy
number
fluctuations
of
the
transcription
machinery,
e.g.,
RNAP
itself.
In
this
picture,
all
the
corrections
to
Poisson
behavior
are
derived
as
additive
corrections
to
the
Fano
factor.
This
picture
is appealing
in
its
simplicity,
with
only
two
parameters,
the
initiation
rate
r
and
degradation
rate
γ
. In
other
words,
the
model
is not
excessively
complex
for
the
data
at
hand.
But
for
many
inter-
esting
questions,
for
instance
in
the
recent
work
[47],
attributing
all
deviations
from
the
model
to
extrinsic
noise
sources,
limits
the
kinds
of
predictions
that
can
be
done.
To
make
progress
then
we
need
a (slightly)
more
complex
model
than
model
1 that
would
allow
us
to
incorporate
the
non-Poissonian
features
of
constitutive
promoters
directly
into
a master
equation
formulation.
2.1.2 Sub-Poissoninan
noise promoters
with RNAP
escape
A natural
extension
of
the
one-state
promoter
studied
in
the
previous
section
is to
explicitly
include
an
empty
promoter
state.
This
state
allows
for
single
RNAP
to
bind
and
unbind
from
the
promoter
with
rates
k
þ
P
and
k
P
, respectively,
before
engaging
in
a transcriptional
event.
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