Supporting Information for: Reconciling Kinetic and Equilibrium Models of
1153
Bacterial Transcription
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Contents
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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
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S1.3 Matrix form of the multi-state chemical master equation . . . . . . . . . . .
41
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S1.4 General forms for mean mRNA and Fano factor . . . . . . . . . . . . . . . .
43
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S1.4.1 Promoter state probabilities
h
~
m
0
i
....................
44
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S1.4.2 First moments
h
~
m
i
and
h
m
i
.......................
45
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S1.4.3 Second moment
h
m
2
i
and Fano factor
⌫
................
46
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S1.4.4 Summary of general results . . . . . . . . . . . . . . . . . . . . . . .
47
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S1.5 Kinetic Model One - Poisson Promoter . . . . . . . . . . . . . . . . . . . . .
48
1167
S1.5.1 Mean mRNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
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S1.5.2 Fold-change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
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S1.5.3 Fano factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
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S1.6 Kinetic Model Two - RNAP Bound and Unbound States with RNAP escape
50
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S1.6.1 Mean mRNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
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S1.6.2 Fold-change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
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S1.6.3 Fano factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
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S1.7 Kinetic Model Three - Multistep Transcription Initiation and Escape . . . .
53
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S1.7.1 Mean mRNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
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S1.7.2 Fold-change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
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S1.7.3 Fano factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
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S1.7.4 Generalizing
⌫
<
1tomorefine-grainedmodels . . . . . . . . . . . .
56
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S1.8 Kinetic Model Four - “Active” and “Inactive” States . . . . . . . . . . . . .
56
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S1.8.1 Mean mRNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
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S1.8.2 Fold-change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
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S1.8.3 Fano factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
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S2 Bursty promoter models - generating function solutions and numerics 58
1184
S2.1 Constitutive promoter with bursts . . . . . . . . . . . . . . . . . . . . . . . .
58
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S2.1.1 From master equation to generating function . . . . . . . . . . . . . .
58
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S2.1.2 Steady-state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
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S2.1.3 Recovering the steady-state probability distribution . . . . . . . . . .
64
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S2.2 Adding repression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
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S2.2.1 Deriving the generating function for mRNA distribution . . . . . . .
66
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S2.3 Numerical considerations and recursion formulas . . . . . . . . . . . . . . . .
71
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S2.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
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S2.3.2 Particulars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
1193
35
S3 Bayesian inference
74
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S3.1 The problem of parameter inference . . . . . . . . . . . . . . . . . . . . . . .
74
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S3.1.1 Bayes’ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
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S3.1.2 The likelihood function . . . . . . . . . . . . . . . . . . . . . . . . . .
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S3.1.3 Prior selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
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S3.1.4 Expectations and marginalizations . . . . . . . . . . . . . . . . . . .
77
1199
S3.1.5 Markov Chain Monte Carlo . . . . . . . . . . . . . . . . . . . . . . .
77
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S3.2 Bayesian inference on constitutive promoters . . . . . . . . . . . . . . . . . .
78
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S3.2.1 Model 1 - Poisson promoter . . . . . . . . . . . . . . . . . . . . . . .
78
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S3.2.2 Model 5 - Bursty promoter . . . . . . . . . . . . . . . . . . . . . . . .
81
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S3.3 Bayesian inference on the simple-repression architecture . . . . . . . . . . . .
82
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S1 Derivations for non-bursty promoter models
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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].
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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
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both the state of the repressor and the mRNA counts. Let’s now work through models 1
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and 2 from Figure 1(B).
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S1.1.1 The Two-State Equilibrium Model
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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
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unbound state is
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p
U
=
✓
1+
R
N
NS
e