of 55
Supporting Information for: Reconciling Kinetic and Equilibrium Models of
1153
Bacterial Transcription
1154
Contents
1155
S1 Derivations for non-bursty promoter models
36
1156
S1.1 Thermodynamic models of gene regulation . . . . . . . . . . . . . . . . . . .
36
1157
S1.1.1 The Two-State Equilibrium Model . . . . . . . . . . . . . . . . . . .
36
1158
S1.1.2 The Three-State Equilibrium Model . . . . . . . . . . . . . . . . . . .
37
1159
S1.2 Derivation of chemical master equation . . . . . . . . . . . . . . . . . . . . .
38
1160
S1.3 Matrix form of the multi-state chemical master equation . . . . . . . . . . .
41
1161
S1.4 General forms for mean mRNA and Fano factor . . . . . . . . . . . . . . . .
43
1162
S1.4.1 Promoter state probabilities
h
~
m
0
i
....................
44
1163
S1.4.2 First moments
h
~
m
i
and
h
m
i
.......................
45
1164
S1.4.3 Second moment
h
m
2
i
and Fano factor
................
46
1165
S1.4.4 Summary of general results . . . . . . . . . . . . . . . . . . . . . . .
47
1166
S1.5 Kinetic Model One - Poisson Promoter . . . . . . . . . . . . . . . . . . . . .
48
1167
S1.5.1 Mean mRNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
1168
S1.5.2 Fold-change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
1169
S1.5.3 Fano factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
1170
S1.6 Kinetic Model Two - RNAP Bound and Unbound States with RNAP escape
50
1171
S1.6.1 Mean mRNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
1172
S1.6.2 Fold-change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
1173
S1.6.3 Fano factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
1174
S1.7 Kinetic Model Three - Multistep Transcription Initiation and Escape . . . .
53
1175
S1.7.1 Mean mRNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
1176
S1.7.2 Fold-change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
1177
S1.7.3 Fano factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
1178
S1.7.4 Generalizing
<
1tomorefine-grainedmodels . . . . . . . . . . . .
56
1179
S1.8 Kinetic Model Four - “Active” and “Inactive” States . . . . . . . . . . . . .
56
1180
S1.8.1 Mean mRNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
1181
S1.8.2 Fold-change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
1182
S1.8.3 Fano factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
1183
S2 Bursty promoter models - generating function solutions and numerics 58
1184
S2.1 Constitutive promoter with bursts . . . . . . . . . . . . . . . . . . . . . . . .
58
1185
S2.1.1 From master equation to generating function . . . . . . . . . . . . . .
58
1186
S2.1.2 Steady-state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
1187
S2.1.3 Recovering the steady-state probability distribution . . . . . . . . . .
64
1188
S2.2 Adding repression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
1189
S2.2.1 Deriving the generating function for mRNA distribution . . . . . . .
66
1190
S2.3 Numerical considerations and recursion formulas . . . . . . . . . . . . . . . .
71
1191
S2.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
1192
S2.3.2 Particulars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
1193
35
S3 Bayesian inference
74
1194
S3.1 The problem of parameter inference . . . . . . . . . . . . . . . . . . . . . . .
74
1195
S3.1.1 Bayes’ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
1196
S3.1.2 The likelihood function . . . . . . . . . . . . . . . . . . . . . . . . . .
75
1197
S3.1.3 Prior selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
1198
S3.1.4 Expectations and marginalizations . . . . . . . . . . . . . . . . . . .
77
1199
S3.1.5 Markov Chain Monte Carlo . . . . . . . . . . . . . . . . . . . . . . .
77
1200
S3.2 Bayesian inference on constitutive promoters . . . . . . . . . . . . . . . . . .
78
1201
S3.2.1 Model 1 - Poisson promoter . . . . . . . . . . . . . . . . . . . . . . .
78
1202
S3.2.2 Model 5 - Bursty promoter . . . . . . . . . . . . . . . . . . . . . . . .
81
1203
S3.3 Bayesian inference on the simple-repression architecture . . . . . . . . . . . .
82
1204
S1 Derivations for non-bursty promoter models
1205
In this section we detail the calculation of mean mRNA levels, fold-changes in expression,
1206
and Fano factors for both thermodynamic and kinetic promoter models in Figure 1. These
1207
are the results that were quoted but not derived in Sections 2 and 3 of the main text. In
1208
each of these models, the natural mathematicization of their cartoons is either an equilibrium
1209
model based on statistical mechanics, or a chemical master equation. These derivations will
1210
go through the specifics of the models in Figure 1. We point the readers to some other
1211
excellent reviews on the general frameworks [1]–[3].
1212
S1.1 Thermodynamic models of gene regulation
1213
The first class of models we will explore are the so-called thermodynamic models of gene reg-
1214
ulation [4]. The premise for these models is that we imagine the steps leading to transcription
1215
–binding and unbinding of transcription factors and of RNAP– to occur in a timescale much
1216
faster than the downstream processes –open complex formation and RNAP escape [5]. What
1217
this allows us to do is to assume that the steps leading to transcription can be modeled as
1218
being in quasi-equilibrium. This is enormously advantageous since we can use the theoretical
1219
tools of equilibrium statistical mechanics to model such process.
1220
This means that we can enlist the possible microstates in which we can find the promoter
1221
in order to calculate their probabilities based on quantities such as molecular counts and
1222
interaction energies. We then assume that the gene expression level is proportional to the
1223
probability of finding the promoter in the transcriptionally active microstate. As mentioned
1224
in the main text, these models can only deal with the mean gene expression level. This is
1225
because the probability space we consider is that of the state of the repressor, rather than
1226
both the state of the repressor and the mRNA counts. Let’s now work through models 1
1227
and 2 from Figure 1(B).
1228
S1.1.1 The Two-State Equilibrium Model
1229
In this simplest model, depicted as (1) in Figure 1(B), the promoter is idealized as existing
1230
in one of two states, either repressor bound or repressor unbound. The rate of transcription
1231
36
is assumed to be proportional to the fraction of time spent in the repressor unbound state.
1232
From the relative statistical weights listed in Figure 1, the probability
p
U
of being in the
1233
unbound state is
1234
p
U
=
1+
R
N
NS
e
"
R
1
.
(S1)
The mean rate of transcription is then given by
rp
U
,asassumedbyEq.1.Themeannumber
1235
of mRNA is set by the balance of average mRNA transcription and degradation rates, so it
1236
follows that the mean mRNA level is given by
1237
h
m
i
=
r
1+
R
N
NS
e
"
R
1
,
(S2)
where
r
is the transcription rate from the repressor unbound state,
is the mRNA degra-
1238
dation rate,
R
is repressor copy number,
N
NS
is the number of nonspecific binding sites
1239
in the genome where repressors spend most of their time when not bound to the operator,
1240
1
/k
B
T
,and
"
R
is the binding energy of a repressor to its operator site.
1241
Fold-change
The fold-change is found as the ratio of mean mRNA with and without
1242
repressor as introduced in Eq. 2. Invoking that definition results in
1243
fold-change =
1+
R
N
NS
e
"
R
1
,
(S3)
which matches the form of the master curve in Figure 1(D) with
=1and
F
R
=
"
r
1244
log(
R/N
NS
).
1245
In fact it was noted in [6] that this two-state model can be viewed as the coarse-graining of
1246
any equilibrium promoter model in which no transcriptionally active states have transcription
1247
factor bound, or put di
erently, when there is no overlap between transcription factor bound
1248
states and transcriptionally active states. We will see this explicitly in the 3-state equilibrium
1249
model below, but perhaps surprising is that an analogous result carries over even to the
1250
kinetic models we consider later.
1251
S1.1.2 The Three-State Equilibrium Model
1252
Compared to the previous model, here we fine-grain the repressor unbound state into two
1253
separate states: empty, and RNAP bound as shown in (2) in Figure 1(B). This picture was
1254
used in [7] as we use it here, and in [8] and [6] it was generalized to incorporate small-molecule
1255
inducers that bind the repressor. The e
ect of this generalization is, roughly speaking, simply
1256
to rescale
R
from the total number of repressors to a smaller e
ective number of available
1257
repressors which are unbound by inducers. We point out that the same generalization can be
1258
incorporated quite easily into any of our models in Figure 1 by simply rescaling the repressor
1259
copy number
R
in the equilibrium models, or equivalently
k
+
R
in the kinetic models.
1260
The mean mRNA copy number, as derived in Appendix S1 from a similar enumeration of
1261
states and weights as the previous model, is
1262
h
m
i
=
r
P
N
NS
e
"
P
1+
R
N
NS
e
"
R
+
P
N
NS
e
"
P
,
(S4)
37
where the new variables are
"
P
, the di
erence in RNAP binding energy to its specific
1263
site (the promoter) relative to an average nonspecific background site, and the RNAP copy
1264
number,
P
.
1265
Fold-change
The fold-change again follows immediately as
fold-change =
P
N
NS
e
"
P
1+
R
N
NS
e
"
R
+
P
N
NS
e
"
P
1+
P
N
NS
e
"
P
P
N
NS
e
"
P
(S5)
=
1+
R
N
NS
e
"
R
1+
P
N
NS
e
"
P
!
1
(S6)
=(1+exp(
F
R
log
))
1
,
(S7)
with
F
R
=
"
R
log(
R/N
NS
)and
=1+
P
N
NS
e
"
P
as shown in Figure 1(B). Thus far,
1266
we see that the two thermodynamic models, despite making di
erent coarse-graining com-
1267
mitments, result in the same functional form for the fold-change in mean gene expression. We
1268
now explore how kinetic models fare when faced with computing the same observable.
1269
Before jumping into the derivations of the general computation of the mean mRNA level
1270
and the Fano factor we will work through the derivation of an example master equation. In
1271
particular we will focus on model 1 from Figure 1(C). The general steps are applicable to all
1272
other chemical master equations in this work.
1273
S1.2 Derivation of chemical master equation
1274
(A)
(B)
mRNA count
mRNA count
promoter state
promoter state
0
1
2
...
0
1
2
...
Figure S1. Two-state promoter chemical master equation.
(A) Schematic of the two
state promoter simple repression model. Rates
k
+
R
and
k
R
are the association and dissociation
rates of the transcriptional repressor, respectively,
r
is the transcription initiation rate, and
is
the mRNA degradation rate. (B) Schematic depiction of the mRNA count state transitions. The
model in (A) only allows for jumps in mRNA of size 1. The production of mRNA can only occur
when the promoter is in the transcriptionally active state.
38
The chemical master equation describes the continuous time evolution of a continuous or
1275
discrete probability distribution function. In our specific case we want to describe the time
1276
evolution of the discrete mRNA distribution. What this means is that we want to compute
1277
the probability of having
m
mRNA molecules at time
t
+
t
,where
t
is a su
ciently small
1278
time interval such that only one of the possible reactions take place during that time interval.
1279
For the example that we will work out here in detail we chose the two-state stochastic simple
1280
repression model schematized in Figure S1(A). To derive the master equation we will focus
1281
more on the representation shown in Figure S1(B), where the transitions between di
erent
1282
mRNA counts and promoter states is more explicitly depicted. Given that the DNA promoter
1283
can exist in one of two states– transcriptionally active state, and with repressor bound– we
1284
will keep track not only of the mRNA count, but on which state the promoter is. For this we
1285
will keep track of two probability distributions: The probability of having
m
mRNAs at time
1286
t
when the promoter is in the transcriptionally active state
A
,
p
A
(
m, t
), and the equivalent
1287
probability but when the promoter is in the repressor bound state
R
,
p
R
(
m, t
).
1288
Since mRNA production can only take place in the transcriptionally active state we will
1289
focus on this function for our derivation. The repressor bound state will have an equivalent
1290
equation without terms involving the production of mRNAs. We begin by listing the possible
1291
state transitions that can occur for a particular mRNA count with the promoter in the
1292
active state. For state changes in a small time window
t
that “jump into” state
m
in the
1293
transcriptionally active state we have
1294
Produce an mRNA, jumping from
m
1to
m
.
1295
Degrade an mRNA, jumping from
m
+1to
m
.
1296
Transition from the repressor bound state
R
with
m
mRNAs to the active state
A
with
1297
m
mRNAs.
1298
Likewise, for state transitions that “jump out” of state
m
in the transcriptionally inactive
1299
state we have
1300
Produce an mRNA, jumping from
m
to
m
+1.
1301
Degrade an mRNA, jumping from
m
to
m
1.
1302
Transition from the active state
A
with
m
mRNAs to the repressor bound state
R
with
1303
m
mRNAs.
1304
The mRNA production does not depend on the current number of mRNAs, therefore these
1305
state transitions occur with probability
r
t
.Thesameistrueforthepromoterstatetran-
1306
sitions; each occurs with probability
k
±
R
t
. As for the mRNA degradation events, these
1307
transitions depend on the current number of mRNA molecules since the more molecules of
1308
mRNA there are, the more will decay during a given time interval. Each molecule has a
1309
constant probability
t
of being degraded, so the total probability for an mRNA degrada-
1310
tion event to occur is computed by multiplying this probability by the current number of
1311
mRNAs.
1312
To see these terms in action let us compute the probability of having
m
mRNA at time
1313
39
t
+
t
in the transcriptionally active state. This takes the form
1314
p
A
(
m, t
+
t
)=
p
A
(
m, t
)
+
m
1
!
m
z
}|
{
(
r
t
)
p
A
(
m
1
,t
)
m
!
m
+1
z
}|
{
(
r
t
)
p
A
(
m, t
)
+
m
+1
!
m
z
}|
{
(
m
+1)(
t
)
p
A
(
m
+1
,t
)
m
!
m
1
z
}|
{
m
(
t
)
p
A
(
m, t
)
+
R
!
A
z
}|
{
(
k
R
t
)
p
R
(
m, t
)
A
!
R
z
}|
{
(
k
+
R
t
)
p
A
(
m, t
)
,
(S8)
where the overbrace indicates the corresponding state transitions. Notice that the second to
1315
last term on the right-hand side is multiplied by
p
R
(
m, t
)sincethetransitionfromstate
R
1316
to state
A
depends on the probability of being in state
R
to begin with. It is through this
1317
term that the dynamics of the two probability distribution functions (
p
R
(
m, t
)and
p
A
(
m, t
))
1318
are coupled. An equivalent equation can be written for the probability of having
m
mRNA
1319
at time
t
+
t
while in the repressor bound state, the only di
erence being that the mRNA
1320
production rates are removed, and the sign for the promoter state transitions are inverted.
1321
This is
1322
p
R
(
m, t
+
t
)=
p
R
(
m, t
)
+
m
+1
!
m
z
}|
{
(
m
+1)(
t
)
p
R
(
m
+1
,t
)
m
!
m
1
z
}|
{
m
(
t
)
p
R
(
m, t
)
R
!
A
z
}|
{
(
k
R
t
)
p
R
(
m, t
)+
A
!
R
z
}|
{
(
k
+
R
t
)
p
A
(
m, t
)
.
(S9)
All we have to do now are simple algebraic steps in order to simplify the equations. Let
1323
us focus on the transcriptionally active state
A
. First we will send the term
p
A
(
m, t
)to
1324
the right-hand side, and then we will divide both sides of the equation by
t
.Thisresults
1325
in
1326
p
A
(
m, t
+
t
)
p
A
(
m, t
)
t
=
rp
A
(
m
1
,t
)
rp
A
(
m, t
)
+(
m
+1)
p
A
(
m
+1
,t
)
m
p
A
(
m, t
)
+
k
R
p
R
(
m, t
)
k
+
R
p
A
(
m, t
)
.
(S10)
Upon taking the limit when
t
!
0 we can transform the left-hand side into a derivative,
1327
obtaining the chemical master equation
1328
dp
A
(
m, t
)
dt
=
rp
A
(
m
1
,t
)
rp
A
(
m, t
)
+(
m
+1)
p
A
(
m
+1
,t
)
m
p
A
(
m, t
)
+
k
R
p
R
(
m, t
)
k
+
R
p
A
(
m, t
)
.
(S11)
Doing equivalent manipulations for the repressor bound state gives an ODE of the form
1329
dp
R
(
m, t
)
dt
=(
m
+1)
p
R
(
m
+1
,t
)
m
p
R
(
m, t
)
k
R
p
R
(
m, t
)+
k
+
R
p
A
(
m, t
)
.
(S12)
40
In the next section we will write these coupled ODEs in a more compact form using matrix
1330
notation.
1331
S1.3 Matrix form of the multi-state chemical master equation
1332
Having derived an example chemical master equation we now focus on writing a general
1333
matrix form for the kinetic models 1-4 shown in Figure 1(C) in the main text. In each of
1334
these four models, the natural mathematicization of their cartoons is as a chemical master
1335
equation. For model 1 we have the master equation
1336
d
dt
p
R
(
m, t
)=
R
!
U
z
}|
{
k
R
p
R
(
m, t
)+
U
!
R
z
}|
{
k
+
R
p
U
(
m, t
)+
m
+1
!
m
z
}|
{
(
m
+1)
p
R
(
m
+1
,t
)
m
!
m
1
z
}|
{
mp
R
(
m, t
)
d
dt
p
U
(
m, t
)=
R
!
U
z
}|
{
k
R
p
R
(
m, t
)
U
!
R
z
}|
{
k
+
R
p
U
(
m, t
)+
m
1
!
m
z
}|
{
rp
U
(
m
1
,t
)
m
!
m
+1
z
}|
{
rp
U
(
m, t
)
+
m
+1
!
m
z
}|
{
(
m
+1)
p
U
(
m
+1
,t
)
m
!
m
1
z
}|
{
mp
U
(
m, t
)
.
(S13)
Here
p
R
(
m, t
)and
p
U
(
m, t
) are the probabilities of finding the system with
m
mRNA
1337
molecules at time
t
either in the repressor bound or unbound states, respectively.
r
is
1338
the probability per unit time that a transcript will be initiated when the repressor is un-
1339
bound, and
is the probability per unit time for a given mRNA to be degraded.
k
R
is the
1340
probability per unit time that a bound repressor will unbind, while
k
+
R
is the probability
1341
per unit time that an unbound operator will become bound by a repressor. Assuming mass
1342
action kinetics,
k
+
R
is proportional to repressor copy number
R
.
1343
Next consider the cartoon for kinetic model 2 in Figure 1(C). Now we must track probabilities
1344
p
R
,
p
P
,and
p
E
for the repressor bound, empty, and polymerase bound states, respectively.
1345
By analogy to Eq. S13, the master equation for model 2 can be written
1346
d
dt
p
R
(
m, t
)=
R
!
U
z
}|
{
k
R
p
R
(
m, t
)+
U
!
R
z
}|
{
k
+
R
p
E
(
m, t
)+
m
+1
!
m
z
}|
{
(
m
+1)
p
R
(
m
+1
,t
)
m
!
m
1
z
}|
{
mp
R
(
m, t
)
d
dt
p
E
(
m, t
)=
R
!
U
z
}|
{
k
R
p
R
(
m, t
)
U
!
R
z
}|
{
k
+
R
p
E
(
m, t
)+
m
+1
!
m
z
}|
{
(
m
+1)
p
E
(
m
+1
,t
)
m
!
m
1
z
}|
{
mp
E
(
m, t
)
.
+
A
!
U
z
}|
{
k
P
p
P
(
m, t
)
U
!
A
z
}|
{
k
+
P
p
E
(
m, t
)+
m
1
!
m, A
!
U
z
}|
{
rp
P
(
m
1
,t
)
d
dt
p
P
(
m, t
)=
A
!
U
z
}|
{
k
P
p
P
(
m, t
)+
U
!
A
z
}|
{
k
+
P
p
E
(
m, t
)+
m
+1
!
m
z
}|
{
(
m
+1)
p
P
(
m
+1
,t
)
m
!
m
1
z
}|
{
mp
P
(
m, t
)
.
m
!
m
+1
,A
!
U
z
}|
{
rp
P
(
m, t
)
.
(S14)
k
+
P
and
k
P
are defined in close analogy to
k
+
R
and
k
R
, except for RNAP binding and unbinding
1347
instead of repressor. Similarly
p
P
(
m, t
) is defined for the active (RNAP-bound) state exactly
1348
as are
p
R
(
m, t
)and
p
E
(
m, t
)fortherepressorboundandunboundstates,respectively.The
1349
41
new subtlety Eq. S14 introduces compared to Eq. S13 is that when mRNAs are produced,
1350
the promoter state also changes. This is encoded by the terms involving
r
,thelasttermin
1351
each of the equations for
p
E
and
p
P
.Theformeraccountsforarrivalsintheunboundstate
1352
and the latter accounts for departures from the RNAP-bound state.
1353
To condense and clarify the unwieldy notation of Eq. S14, it can be rewritten in matrix form.
1354
We define the column vector
~
p
(
m, t
)as
1355
~
p
(
m, t
)=
0
@
p
R
(
m, t
)
p
E
(
m, t
)
p
P
(
m, t
)
1
A
(S15)
to gather, for a given
m
, the probabilities of finding the system in the three possible promoter
1356
states. Then all the transition rates may be condensed into matrices which multiply this
1357
vector. The first matrix is
1358
K
=
0
@
k
R
k
+
R
0
k
R
k
+
R
k
+
P
k
P
0
k
+
P
k
P
1
A
,
(S16)
which tracks all promoter state changes in Eq. S14 that are
not
accompanied by a change
1359
in the mRNA copy number. The two terms accounting for transcription, the only transition
1360
that increases mRNA copy number, must be handled by two separate matrices given by
1361
R
A
=
0
@
000
00
r
000
1
A
,
R
D
=
0
@
000
000
00
r
1
A
.
(S17)
R
A
accounts for transitions
arriving
in a given state while
R
D
tracks
departing
transitions.
1362
With these definitions, we can condense Eq. S14 into the single equation
1363
d
dt
~
p
(
m, t
)=(
K
R
D
m
I
)
~
p
(
m, t
)+
R
A
~
p
(
m
1
,t
)+
(
m
+1)
I
~
p
(
m
+1
,t
)
,
(S18)
Straightforward albeit tedious algebra verifies that Eqs. S14 and S18 are in fact equiva-
1364
lent.
1365
Although we derived Eq. S18 for the particular case of kinetic model 2 in Figure 1, in fact
1366
the chemical master equations for all of the kinetic models in Figure 1 except for model 5 can
1367
be cast in this form. (We treat model 5 separately in Appendix S2.) Model 3 introduces no
1368
new subtleties beyond model 2 and Eq. S18 applies equally well to it, simply with di
erent
1369
matrices of dimension four instead of three. Models 1 and 4 are both Eq. S18 except that
1370
R
D
=
R
A
R
,sinceinthesetwomodelstranscriptioninitiationeventsdonotchange
1371
promoter state.
1372
Recalling that our goal in this section is to derive expressions for mean mRNA and Fano
1373
factor for kinetic models 1 through four in Figure 1, and since all four of these models
1374
are described by Eq. S18, we can save substantial e
ort by deriving general formulas for
1375
mean mRNA and Fano factor from Eq. S18 once and for all. Then for each model we can
1376
42
simply plug in the appropriate matrices for
K
,
R
D
,and
R
A
and carry out the remaining
1377
algebra.
1378
For our purposes it will su
ce to derive the first and second moments of the mRNA distri-
1379
bution from this master equation, similar to the treatment in [9], but we refer the interested
1380
reader to [10] for an analogous treatment demonstrating an analytical solution for arbitrary
1381
moments.
1382
S1.4 General forms for mean mRNA and Fano factor
1383
Our task now is to derive expressions for the first two moments of the steady-state mRNA
1384
distribution from Eq. S18. Our treatment of this is analogous to that given in Refs. [9]
1385
and [10]. We first obtain the steady-state limit of Eq. S18 in which the time derivative
1386
vanishes, giving
1387
0=(
K
R
D
m
I
)
~
p
(
m
)+
R
A
~
p
(
m
1) +
(
m
+1)
I
~
p
(
m
+1)
,
(S19)
From this, we want to compute
1388
h
m
i
=
X
S
1
X
m
=0
mp
S
(
m
)(S20)
and
1389
h
m
2
i
=
X
S
1
X
m
=0
m
2
p
S
(
m
)(S21)
which define the average values of
m
and
m
2
at steady state, where the averaging is over
1390
all possible mRNA copy numbers and promoter states
S
. For example, for model 1 in
1391
Figure 1(C), the sum on
S
would cover repressor bound and unbound states (
R
and
U
1392
respectively), for model 2, the sum would cover repressor bound, polymerase bound, and
1393
empty states (
R
,
P
,and
E
), and so on for the other models.
1394
Along the way it will be convenient to define the following
conditional
moments as
1395
h
~
m
i
=
1
X
m
=0
m
~
p
(
m
)
,
(S22)
and
1396
h
~
m
2
i
=
1
X
m
=0
m
2
~
p
(
m
)
.
(S23)
These objects are vectors of the same size as
~
p
(
m
), and each component can be thought of as
1397
the expected value of the mRNA copy number, or copy number squared, conditional on the
1398
promoter being in a certain state. For example, for model 1 in Figure 1 which has repressor
1399
bound and unbound states labeled
R
and
U
,
h
~
m
2
i
would be
1400
h
~
m
2
i
=
P
1
m
=0
m
2
p
R
(
m
)
P
1
m
=0
m
2
p
U
(
m
)
.
(S24)
43
Analogously to
h
~
m
i
and
h
~
m
2
i
,itisconvenienttodefinethevector
1401
h
~
m
0
i
=
1
X
m
=0
~
p
(
m
)
,
(S25)
whose elements are simply the probabilities of finding the system in each of the possible
1402
promoter states. It will be convenient to denote by
~
1
arowvectorofthesamelengthas
~
p
1403
whose elements are all 1, such that, for instance,
~
1
·
h
~
m
0
i
=1,
~
1
·
h
~
m
i
=
h
m
i
,etc.
1404
S1.4.1 Promoter state probabilities
h
~
m
0
i
1405
To begin, we will find the promoter state probabilities
h
~
m
0
i
from Eq. S19 by summing over
1406
all mRNA copy numbers
m
, producing
1407
0=
1
X
m
=0
[(
K
R
D
m
I
)
~
p
(
m
)+
R
A
~
p
(
m
1) +
(
m
+1)
I
~
p
(
m
+1)]
(S26)
Using the definitions of
h
~
m
0
i
and
h
~
m
i
,andnotingthematrices
K
,
R
D
,and
R
A
are all
1408
independent of
m
and can be moved outside the sum, this simplifies to
1409
0=(
K
R
D
)
h
~
m
0
i
h
~
m
i
+
R
A
1
X
m
=0
~
p
(
m
1) +
1
X
m
=0
(
m
+1)
~
p
(
m
+1)
.
(S27)
The last two terms can be handled by reindexing the summations, transforming them to
1410
match the definitions of
h
~
m
0
i
and
h
~
m
i
.Forthefirst,noting
~
p
(
1) = 0 since negative
1411
numbers of mRNA are nonsensical, we have
1412
1
X
m
=0
~
p
(
m
1) =
1
X
m
=
1
~
p
(
m
)=
1
X
m
=0
~
p
(
m
)=
h
~
m
0
i
.
(S28)
Similarly for the second,
1413
1
X
m
=0
(
m
+1)
~
p
(
m
+1)=
1
X
m
=1
m
~
p
(
m
)=
1
X
m
=0
m
~
p
(
m
)=
h
~
m
i
,
(S29)
which holds since in extending the lower limit from
m
=1to
m
=0,theextratermwe
1414
added to the sum is zero. Substituting these back in Eq. S27, we have
1415
0=(
K
R
D
)
h
~
m
0
i
h
~
m
i
+
R
A
h
~
m
0
i
+
h
~
m
i
,
(S30)
or simply
1416
0=(
K
R
D
+
R
A
)
h
~
m
0
i
.
(S31)
So given matrices
K
,
R
D
,and
R
A
describing a promoter, finding
h
~
m
0
i
simply amounts to
1417
solving this set of linear equations, subject to the normalization constraint
~
1
·
h
~
m
0
i
=1.
1418
It will turn out to be the case that, if the matrix
K
R
D
+
R
A
is
n
dimensional, it will
1419
always have only
n
1linearlyindependentequations. Includingthenormalizationcondition
1420
44
provides the
n
-th linearly independent equation, ensuring a unique solution. So when using
1421
this equation to solve for
h
~
m
0
i
,wemayalwaysdroponerowofthematrixequationat
1422
our convenience and supplement the system with the normalization condition instead. The
1423
reader may find it illuminating to skip ahead and see Eq. S31 in use with concrete examples,
1424
e.g., Eq. S59 and what follows it, before continuing on through the general formulas for
1425
moments.
1426
S1.4.2 First moments
h
~
m
i
and
h
m
i
1427
By analogy to the above procedure for finding
h
~
m
0
i
,wemayfind
h
~
m
i
by first multiplying
1428
Eq. S19 by
m
and then summing over
m
.Carryingoutthisprocedurewehave
1429
0=
1
X
m
=0
m
[(
K
R
D
m
I
)
~
p
(
m
)+
R
A
~
p
(
m
1) +
(
m
+1)
I
~
p
(
m
+1)]
,
(S32)
and now identifying
h
~
m
i
and
h
~
m
2
i
gives
1430
0=(
K
R
D
)
h
~
m
i
h
~
m
2
i
+
R
A
1
X
m
=0
m
~
p
(
m
1) +
1
X
m
=0
m
(
m
+1)
~
p
(
m
+1)
.
(S33)
The summations in the last two terms can be reindexed just as we did for
h
~
m
0
i
,freelyadding
1431
or removing terms from the sum which are zero. For the first term we find
1432
1
X
m
=0
m
~
p
(
m
1) =
1
X
m
=
1
(
m
+1)
~
p
(
m
)=
1
X
m
=0
(
m
+1)
~
p
(
m
)=
h
~
m
i
+
h
~
m
0
i
,
(S34)
and similarly for the second,
1433
1
X
m
=0
m
(
m
+1)
~
p
(
m
+1)=
1
X
m
=1
(
m
1)
m
~
p
(
m
)=
1
X
m
=0
(
m
1)
m
~
p
(
m
)=
h
~
m
2
ih
~
m
i
.
(S35)
Substituting back in Eq. S33 then produces
1434
0=(
K
R
D
)
h
~
m
i
h
~
m
2
i
+
R
A
(
h
~
m
i
+
h
~
m
0
i
)+
(
h
~
m
2
ih
~
m
i
)
,
(S36)
or after simplifying
1435
0=(
K
R
D
+
R
A
)
h
~
m
i
+
R
A
h
~
m
0
i
.
(S37)
So like
h
~
m
0
i
,
h
~
m
i
is also found by simply solving a set of linear equations after first solving
1436
for
h
~
m
0
i
from Eq. S31.
1437
Next we can find the mean mRNA copy number
h
m
i
from
h
~
m
i
according to
1438
h
m
i
=
~
1
·
h
~
m
i
,
(S38)
where
~
1
is a row vector whose elements are all 1. Eq. S38 holds since the
i
th
element of the
1439
column vector
h
~
m
i
is the mean mRNA value conditional on the system occupying the
i
th
1440
promoter state, so the dot product with
~
1
amounts to simply summing the elements of
h
~
m
i
.
1441
45
Rather than solving Eq. S37 for
h
~
m
i
and then computing
~
1
·
h
~
m
i
,wemaytakeashortcut
1442
by multiplying Eq. S37 by
~
1
first. The key observation that makes this useful is that
1443
~
1
·
(
K
R
D
+
R
A
)=0
.
(S39)
Intuitively, this equality holds because each column of this matrix represents transitions to
1444
and from a given promoter state. In any given column, the diagonal encodes all departing
1445
transitions and o
-diagonals encode all arriving transitions. Conservation of probability
1446
means that each column sums to zero, and summing columns is exactly the operation that
1447
multiplying by
~
1
carries out.
1448
Proceeding then in multiplying Eq. S37 by
~
1
produces
1449
0=
~
1
·
h
~
m
i
+
~
1
·
R
A
h
~
m
0
i
,
(S40)
or simply
1450
h
m
i
=
1
~
1
·
R
A
h
~
m
0
i
.
(S41)
We note that the in equilibrium models of transcription such as in Figure 1, it is usually
1451
assumed
that the mean mRNA level is given by the ratio of initiation rate
r
to degradation
1452
rate
multiplied by the probability of finding the system in the RNAP-bound state. Reas-
1453
suringly, we have recovered exactly this result from the master equation picture: the product
1454
~
1
·
R
A
h
~
m
0
i
picks out the probability of the active promoter state from
h
~
m
0
i
and multiplies
1455
it by the initiation rate
r
.
1456
S1.4.3 Second moment
h
m
2
i
and Fano factor
1457
Continuing the pattern of the zeroth and first moments, we now find
h
~
m
2
i
by multiplying
1458
Eq. S19 by
m
2
and then summing over
m
,whichexplicitlyis
1459
0=
1
X
m
=0
m
2
[(
K
R
D
m
I
)
~
p
(
m
)+
R
A
~
p
(
m
1) +
(
m
+1)
I
~
p
(
m
+1)]
.
(S42)
Identifying the moments
h
~
m
2
i
and
h
~
m
3
i
in the first term simplifies this to
1460
0=(
K
R
D
)
h
~
m
2
i
h
~
m
3
i
+
R
A
1
X
m
=0
m
2
~
p
(
m
1) +
1
X
m
=0
m
2
(
m
+1)
~
p
(
m
+1)
.
(S43)
Reindexing the sums of the last two terms proceeds just as it did for the zeroth and first
1461
moments. Explicitly, we have
1462
1
X
m
=0
m
2
~
p
(
m
1) =
1
X
m
=
1
(
m
+1)
2
~
p
(
m
)=
1
X
m
=0
(
m
+1)
2
~
p
(
m
)=
h
~
m
2
i
+2
h
~
m
i
+
h
~
m
0
i
,
(S44)
for the first sum and
1463
1
X
m
=0
m
2
(
m
+1)
~
p
(
m
+1)=
1
X
m
=1
(
m
1)
2
m
~
p
(
m
)=
1
X
m
=0
(
m
1)
2
m
~
p
(
m
)=
h
~
m
3
i
2
h
~
m
2
i
+
h
~
m
i
(S45)
46
for the second. Substituting the results of the sums back in Eq. S43 gives
1464
0=(
K
R
D
)
h
~
m
2
i
h
~
m
3
i
+
R
A
(
h
~
m
2
i
+2
h
~
m
i
+
h
~
m
0
i
)+
(
h
~
m
3
i
2
h
~
m
2
i
+
h
~
m
i
)
,
(S46)
and after grouping like powers of
m
we have
1465
0=(
K
R
D
+
R
A
2
)
h
~
m
2
i
+(2
R
A
+
)
h
~
m
i
+
R
A
h
~
m
0
i
.
(S47)
As we found when computing
h
m
i
from
h
~
m
i
,wecanspareourselvessomealgebrabymulti-
1466
plying Eq. S47 by
~
1
,whichthenreducesto
1467
0=
2
h
m
2
i
+
~
1
·
(2
R
A
+
)
h
~
m
i
+
~
1
·
R
A
h
~
m
0
i
,
(S48)
and noting from Eq. S41 that
~
1
·
R
A
h
~
m
0
i
=
h
m
i
,wehavethetidyresult
1468
h
m
2
i
=
h
m
i
+
1
~
1
·
R
A
h
~
m
i
.
(S49)
Finally we have all the preliminary results needed to write a general expression for the Fano
1469
factor
. The Fano factor is defined as the ratio of variance to mean, which can be written
1470
as
1471
=
h
m
2
ih
m
i
2
h
m
i
=
h
m
i
+
1
~
1
·
R
A
h
~
m
ih
m
i
2
h
m
i
(S50)
and simplified to
1472
=1
h
m
i
+
~
1
·
R
A
h
~
m
i
h
m
i
.
(S51)
Note a subtle notational trap here:
h
m
i
=
1
~
1
·
R
A
h
~
m
0
i
rather than the by-eye similar
1473
but wrong expression
h
m
i6
=
1
~
1
·
R
A
h
~
m
i
,sothelastterminEq.S51isingeneralquite
1474
nontrivial. For a generic promoter, Eq. S51 may be greater than, less than, or equal to one,
1475
as asserted in Section 3. We have not found the general form Eq. S51 terribly intuitive and
1476
instead defer discussion to specific examples.
1477
S1.4.4 Summary of general results
1478
For ease of reference, we collect and reprint here the key results derived in this section that
1479
are used in the main text and subsequent subsections. Mean mRNA copy number and Fano
1480
factor are given by Eqs. S41 and S51, which are
1481
h
m
i
=
1
~
1
·
R
A
h
~
m
0
i
(S52)
and
1482
=1
h
m
i
+
~
1
·
R
A
h
~
m
i
h
m
i
,
(S53)
respectively. To compute these two quantities, we need the expressions for
h
~
m
0
i
and
h
~
m
i
1483
given by solving Eqs. S31 and S37, respectively, which are
1484
(
K
R
D
+
R
A
)
h
~
m
0
i
=0
(S54)
47