# The one-message-per-cell-cycle rule:
A conserved minimum transcription level for essential genes

Teresa W. Lo  Han Kyou James Choi  Dean Huang  Department of Physics,
University of Washington, Seattle, Washington 98195, USA  Paul A. Wiggins
<EMAIL_ADDRESS>Department of Physics, University of Washington, Seattle,
Washington 98195, USA  Department of Bioengineering, University of Washington,
Seattle, Washington 98195, USA  Department of Microbiology, University of
Washington, Seattle, Washington 98195, USA

###### Abstract

The inherent stochasticity of cellular processes leads to significant cell-to-
cell variation in protein abundance. Although this noise has already been
characterized and modeled, its broader implications and significance remain
unclear. In this paper, we revisit the noise model and identify the number of
messages transcribed per cell cycle as the critical determinant of noise. In
yeast, we demonstrate that this quantity predicts the non-canonical scaling of
noise with protein abundance, as well as quantitatively predicting its
magnitude. We then hypothesize that growth robustness requires an upper
ceiling on noise for the expression of essential genes, corresponding to a
lower floor on the transcription level. We show that just such a floor exists:
a minimum transcription level of one message per cell cycle is conserved
between three model organisms: Escherichia coli, yeast, and human.
Furthermore, all three organisms transcribe the same number of messages per
gene, per cell cycle. This common transcriptional program reveals that
robustness to noise plays a central role in determining the expression level
of a large fraction of essential genes, and that this fundamental optimal
strategy is conserved from E. coli to human cells.

## Introduction

All molecular processes are inherently stochastic on a cellular scale,
including the processes of the central dogma, responsible for gene expression
[1, 2]. As a result, the expression of every protein is subject to cell-to-
cell variation in abundance [1]. Many interesting proposals have been made to
describe the potential biological significance of this noise, including bet-
hedging strategies, the necessity of feedback in gene regulatory networks, etc
[1, 3, 4]. However, it is less clear to what extent noise plays a central role
in determining the function of the gene expression process more generally. For
instance, Hausser et al. have described how the tradeoff between economy (e.g.
minimizing the number of transcripts) and precision (minimizing the noise)
explains why genes with high transcription rates and low translation rates are
not observed [5]. Although these results suggest that noise may provide some
coarse limits on the function of gene expression, this previous work does not
directly address a central challenge posed by noise: How does the cell ensure
that the lowest expression essential genes, which are subject to the greatest
noise, have sufficient abundance in all cells for robust growth?

To investigate this question, we first focus on noise in Saccharomyces
cerevisiae (yeast), and find that the noise scaling with protein abundance is
not canonical. We re-analyze the canonical stochastic kinetic model for gene
expression, the telegraph model [6, 7, 8], to understand the relationship
between the underlying kinetic parameters and the distribution of protein
abundance in the cell. As previously reported, we find that the protein
abundance for a gene is described by a gamma distribution with two parameters:
the message number, defined as the total gene message number transcribed per
cell cycle, and the translation efficiency, which is the mean protein number
translated per message. Protein expression noise is completely determined by
the message number [3, 9]. Although these results have been previously
reported, the distinction between message number per cell versus per cell
cycle and even between mean protein number and mean message number is often
neglected (e.g. [10]).

To explore the distinction between these parameters and provide clear evidence
of the importance of the message number, we return to the analysis of noise in
yeast. In yeast, the translation efficiency increases with message number
[11]. By fitting an empirical model for the translation efficiency, we
demonstrate that the noise should scale with a half-power of protein
abundance. We demonstrate that this non-canonical scaling is observed and that
our translation model makes a parameter-free prediction for the noise. The
prediction is in close quantitative agreement with observation [12],
confirming that the message number is the key determinant of noise strength.

Figure 1: Robustness hypothesis: The stochasticity in gene expression is
represented by the red shading. We hypothesize that robust growth requires
sufficiently low noise levels for cellular function. We hypothesize that this
critical noise level should be below the level where the signal (mean) equals
noise (standard deviation).

Finally, we use this result to explore the hypothesis that there is a minimum
expression level for essential genes, dictated by noise. The same mean
expression level can be achieved by a wide range of different translation and
transcription rates with different noise levels. We hypothesize that growth
robustness requires that essential genes (but not non-essential genes) are
subject to a floor expression level, below which there is too much cell-to-
cell variation to ensure growth. To test this prediction, we analyze
transcription in three model organisms, Escherichia coli, yeast, and Homo
sapiens (human), with respect to three related gene characteristics:
transcription rate, cellular message number, and message number per cell
cycle. As predicted by the noise-based mechanism, we observe an organism-
independent floor for the number of messages transcribed per cell cycle for
essential genes, but not non-essential genes. We conclude that virtually all
essential genes are transcribed at a rate of at least once per cell cycle.
This analysis strongly supports the hypothesis that the same biological
optimization imperatives, which determine the transcription rates of many low-
expression genes, are conserved from E. coli to human.

## Results

Implications of noise on growth robustness. With the realization of the
stochasticity of central dogma processes, a key question is how cells can grow
robustly in spite of cell-to-cell variations in protein expression. The noise
in protein abundance is defined as the coefficient of variation squared [13,
14, 12]:

${\rm CV}^{2}_{p}\equiv\textstyle\frac{\sigma^{2}_{p}}{\mu^{2}_{p}},$ (1)

where $\sigma^{2}_{p}$ is the variance of protein number and
$\mu_{p}\equiv\overline{N}_{p}$ its mean. It is important to emphasize that
protein abundance must double between birth and cell division in symmetrically
dividing cells during steady state growth. The protein abundance should
therefore be interpreted either as expression per unit volume [15] or the
abundance associated with cells of a defined volume [12].

The coefficient of variation is inversely related to protein abundance and
therefore low-copy proteins have the highest noise [13, 14, 3, 9, 15, 12]. The
challenge faced by the cell is that many essential proteins, strictly required
for cell growth, are relatively low abundance. How does the cell ensure
sufficient protein abundance in spite of cell-to-cell variation in protein
number? It would seem that growth robustness demands that, for essential
proteins, the mean should be greater than the standard deviation:

${\rm CV}^{2}_{p}<1,$ (2)

in order to ensure that protein abundance is sufficiently high enough to avoid
growth arrest. To what extent do essential proteins obey this noise threshold?

What determines the strength of the noise? Usually, noise is argued to be
proportional to inverse protein abundance (e.g. [3, 4, 10]):

${\rm CV}^{2}_{p}\propto\mu_{p}^{-1},$ (3)

for low abundance proteins, motivated both by theoretical and experimental
results [10, 15] and in some cases obeying a low-translation efficiency limit
[15]:

${\rm CV}^{2}_{p}\approx\mu_{p}^{-1}.$ (4)

Can this model be used to make quantitative predictions of the noise? E.g., is
the scaling of Eq. 3 correct? Can the coefficient of proportionality be
predicted? Although Eq. 3 appears to describe E. coli quite well [15], the
situation in yeast is more complicated 111Although there have been claims that
Eq. 3 is consistent with the data [10], these authors did not fit competing
models, nor did they perform a proteome-wide analysis of protein abundance and
noise.. To analyze the statistical significance of the deviation from the
canonical noise model in yeast, we can fit an empirical model to the noise
[13, 14]:

${\rm CV}_{p}^{2}=\textstyle\frac{b}{\mu_{p}^{a}}+c.$ (5)

In the null hypothesis, $a=1$ (canonical scaling), while $b$ and $c$ are
unknown parameters. $c$ corresponds to the noise floor. In the alternative
hypothesis, all three coefficients are unknown. (A detailed description of the
statistical model is given in the Supplemental Material Sec. A.5.)

The canonical model fails to fit the noise data for yeast as reported by
Newman et al. [12]: The null hypothesis is rejected with p-value $p=6\times
10^{-36}$. The model fit to the data is shown in Fig. 2. The estimated scaling
exponent for protein abundance in the alternative hypothesis is $a=0.57\pm
0.02$, and a detailed description of the statistical model and parameter fits
is provided in Supplementary Material Sec. A.5.4. As shown in Fig. 2, even
from a qualitative perspective, the scaling of the yeast noise at low copy
number is much closer to  $\mu_{p}^{-1/2}$ than to canonical assumption
$\mu_{p}^{-1}$ (Eq. 3). In particular, above the detection threshold, the
noise is always larger than the low-translation efficiency limit (Eq. 4).

Figure 2: A non-canonical scaling is observed for gene-expression noise in
yeast. The protein expression noise (${\rm CV}^{2}_{p}$) for yeast scales like

$\mu_{p}^{-1/2}$

(purple) rather than the canonical

$\mu_{p}^{-1}$

(orange) for low-abundance proteins. (Data from Ref. [12].) An empirical noise
model (Eq. 5, green) fit to the essential genes gives an estimate of the
protein-abundance scaling of $\mu_{p}^{-0.57}$.  Figure 3: Panel A: Kinetic
model for the central dogma. The telegraph model is a stochastic kinetic model
for protein synthesis, described by four gene-specific rate constants: the
transcription rate ($\beta_{m}$), the message degradation rate ($\gamma_{m}$),
the translation rate ($\beta_{p}$), and the dilution rate ($\gamma_{p}$).
Panel B: Statistical model for the central dogma. The predicted distribution
in protein abundance is described by a gamma distribution, which is
parameterized by two unitless constants: the shape parameter $\mu_{m}$, the
mean number of messages transcribed per cell cycle, and the scale parameter
$\varepsilon$, the mean number of proteins translated per message. Panel C:
Message number. The message number ($\mu_{m}$) is defined as the mean total
number of messages (dark blue) transcribed per cell cycle. Here, four total
messages are transcribed and translated to protein (light blue); however, due
to message degradation, at time $t^{\prime}$, only one message is present in
the cell. Cellular message number ($\mu_{m/c}$) is defined as the mean number
of messages per cell at time $t$.

Stochastic kinetic model for central dogma. To understand the failure of the
canonical assumptions, we revisit the underlying model. The telegraph model
for the central dogma describes multiple steps in the gene expression process:
Transcription generates mRNA messages [17]. These messages are then translated
to synthesize the protein gene products [17]. Both mRNA and protein are
subject to degradation and dilution [18]. (See Fig. 3A.) At the single cell
level, each of these processes are stochastic. We will model these processes
with the stochastic kinetic scheme [17]:

$\begin{CD}{\rm DNA}@>{\beta_{m}}>{}>{\rm mRNA}&@>{\beta_{p}}>{}>{\rm
Protein}\\\ &&@V{\gamma_{m}}V{}V&@V{\gamma_{p}}V{}V\\\ &&\O&&&\;\O,\end{CD}$
(6)

where $\beta_{m}$ is the transcription rate (s-1), $\beta_{p}$ is the
translation rate (s-1), $\gamma_{m}$ is the message degradation rate (s-1),
and $\gamma_{p}$ is the protein effective degradation rate (s-1). The message
lifetime is $\tau_{m}\equiv\gamma_{m}^{-1}$. For most protein in the context
of rapid growth, dilution is the dominant mechanism of protein depletion and
therefore $\gamma_{p}$ is approximately the growth rate [19, 20, 15]:
$\gamma_{p}=T^{-1}\ln 2$, where $T$ is the doubling time. We will discuss a
more general scenario below.

Statistical model for protein abundance. To study the stochastic dynamics of
gene expression, we used a stochastic Gillespie simulation [21, 22]. (See
Supplemental Material Sec. A.1.) In particular, we were interested in the
explicit relation between the kinetic parameters
$(\beta_{m},\gamma_{m},\beta_{p},\gamma_{p})$ and experimental observables.

Consistent with previous reports [3, 9], we find that the distribution of
protein number per cell (at cell birth) was described by a gamma distribution:
$N_{p}\sim\Gamma(\theta_{\Gamma},k_{\Gamma})$, where $N_{p}$ is the protein
number at cell birth and $\Gamma$ is the gamma distribution which is
parameterized by a scale parameter $\theta_{\Gamma}$ and a shape parameter
$k_{\Gamma}$. (See Supplementary Material Sec. A.1.) The relation between the
four kinetic parameters and these two statistical parameters has already been
reported, and have clear biological interpretations [9]: The scale parameter:

$\displaystyle\theta_{\Gamma}=\varepsilon\ln 2,$ (7)

is proportional to the translation efficiency:

$\displaystyle\varepsilon\equiv\textstyle\frac{\beta_{p}}{\gamma_{m}},$ (8)

where $\beta_{p}$ is the translation rate and $\gamma_{m}$ is the message
degradation rate. $\varepsilon$ is understood as the mean number of proteins
translated from each message transcribed. The shape parameter $k_{\Gamma}$ can
also be expressed in terms of the kinetic parameters [9]:

$k_{\Gamma}=\textstyle\frac{\beta_{m}}{\gamma_{p}};$ (9)

however, we will find it more convenient to express the scale parameter in
terms of the cell-cycle message number:

$\displaystyle\mu_{m}\equiv\beta_{m}T=k_{\Gamma}\ln 2,$ (10)

which can be interpreted as the mean number of messages transcribed per cell
cycle. Forthwith, we will abbreviate this quantity message number in the
interest of brevity.

|  |  |  |  | Total number of | Average
---|---|---|---|---|---|---
Model organism |  Growth condition |  Doubling time: |  Message lifetime: |  Message recycling ratio: |  messages /cell: |  messages /cell-cycle: |  proteins: |  translation efficiency: |  translation rate:
|  |  $T$ |  $\tau_{m}$ |  $T/\tau_{m}$ |  $N^{\rm tot}_{m/c}$ |  $N^{\rm tot}_{m}$ |  $N^{\rm tot}_{p}$ |  $\varepsilon$ |  $\beta_{p}$ (h-1)
Escherichia coli |  LB |  $30$ min |  $2.5$ min |  $12$ |  $7.8\times 10^{3}$ |  $9.4\times 10^{4}$ |  $3\times 10^{6}$ |  $32$ |  $770$
(E. coli) |  M9 |  $90$ min |  $2.5$ min |  $36$ |  $2.4\times 10^{3}$ |  $8.6\times 10^{4}$ |  $3\times 10^{6}$ |  $35$ |  $833$
Sacchromyces cerevisiae (Yeast–haploid) |  YEPD |  $90$ min |  $22$ min |  $4$ |  $2.9\times 10^{4}$ |  $1.2\times 10^{5}$ |  $5\times 10^{7}$ |  $420$ |  $1100$
Homo sapiens (Human) |  Tissue |  $24$ h |  $14$ h |  $1.7$ |  $3.6\times 10^{5}$ |  $6.2\times 10^{5}$ |  $2\times 10^{9}$ |  $3.2\times 10^{3}$ |  $230$

Table 1:  Central dogma parameters for three model organisms. Columns three
through seven hold representative values for measured central-dogma parameters
for the model organisms described in the paper. The sources of the numbers and
estimates are described in the Supplemental Material Sec. A.2.

In terms of two gamma parameters, the mean and the squared coefficient of
variation are:

$\displaystyle\mu_{p}$ $\displaystyle=$ $\displaystyle
k_{\Gamma}\theta_{\Gamma}=\textstyle\mu_{m}\varepsilon\,$ (11)
$\displaystyle{\rm CV}^{2}_{p}$ $\displaystyle=$
$\displaystyle\textstyle\frac{1}{k_{\Gamma}}=\textstyle\frac{\ln 2}{\mu_{m}},$
(12)

where the noise depends on the message number ($\mu_{m}$), not the mean
protein number ($\mu_{p}$). (Eq. 12 only applies when $\varepsilon\gg 0$ [3,
9].) Are these theoretical results consistent with the canonical model (Eq.
3)? We can rewrite the noise in terms of the protein abundance and translation
efficiency:

${\rm CV}_{p}^{2}=\textstyle\frac{\varepsilon\ln 2}{\mu_{p}},$ (13)

which implies that the canonical model only applies when the translation
efficiency ($\varepsilon$) is independent of expression ($\mu_{p}$).

Measuring the message number. The prediction for the noise (Eq. 12) depends on
the message number ($\mu_{m}$). However, mRNA abundance is typically
characterized by a closely related, but distinct quantity: Quantitative RNA-
Seq and methods that visualize fluorescently-labeled mRNA molecules typically
measure the number of messages per cell [6]. We will call the mean of this
number the cellular message number $\mu_{m/c}$. In the kinetic model, these
different message abundances are related:

$\mu_{m}=\textstyle\frac{T}{\tau_{m}}\mu_{m/c},$ (14)

by the message recycling ratio, $T/\tau_{m}$, which can be interpreted as the
average number of times messages are recycled during the cell cycle. To
estimate the message number, we will scale the observed cellular message
number $\mu_{m/c}$ by the message recycling ratio, using the mean message
lifetime. Fig. 3C illustrates the difference between the message number and
the cellular message number. The mean lifetimes, message recycling ratios, as
well as the total message number for three model organisms are shown in Tab.
1.

Figure 4:  Panel A: An empirical model for protein number $\mu_{p}$ in yeast.
The canonical noise model assumes constant translation efficiency, which would
imply that protein number is proportional to the message number (orange);
however, the empirical fit (green) shows that protein number scales close to
the square of message number (purple):

$\mu_{p}\propto\mu_{m}^{2}$

. The protein abundance has a cutoff near $10^{1}$ due to the autofluorescence
cutoff [12]. Panel B: The statistical noise model predicts the observed noise.
The statistical noise model (Eq. 12) and empirical model for protein number
(Eq. 15) make a parameter-free prediction of the noise (green). This
prediction both closely matches the observed scaling (

$\propto\mu^{-1/2}_{p}$

, purple) relative to the canonical scaling (

$\propto\mu^{-1}_{p}$

, orange) and quantitatively estimates magnitude (vertical offset). This
prediction does not include the contribution of noise floor, relevant for
describing high-expression proteins.

Construction of an empirical model for protein number. To model the noise as a
function of protein abundance ($\mu_{p}$), we will determine the empirical
relation between mean protein levels and message abundance by fitting to Eq.
11. Note that the objective here is only to estimate $\mu_{m}$ from $\mu_{p}$,
not to model the process mechanistically (e.g. [23].) The message numbers are
estimated from RNA-Seq measurements, scaled as described above (Eq. 14). The
protein abundance numbers come from fluorescence and mass-spectrometry based
assays [12, 24], with overall normalization chosen to match reported total
cellular protein content. (See Supplemental Material Sec. A.3.4.) The
resulting fit generates our empirical translation model for yeast:

$\displaystyle\mu_{p}$ $\displaystyle=$ $\displaystyle 8.0\ \mu_{m}^{2.1},$
(15)

where both means are in units of molecules. (An error analysis for both model
parameters is described in Supplementary Material Sec. A.4.2.) The data and
model are shown in Fig. 5A.

Prediction of the noise scaling with abundance. Now that we have fit an
empirical model that relates $\mu_{p}$ and $\mu_{m}$, we return to the problem
of predicting the yeast noise. We apply the relation (Eq. 15) to Eq. 12 to
make a parameter-free prediction of the noise as a function of protein
abundance:

${\rm CV}_{p}^{2}=1.9\ \mu_{p}^{-0.48}.$ (16)

An error analysis for both model parameters is described in Supplementary
Material Sec. A.4.5. Our noise model (Eq. 16) makes both a qualitative and
quantitative prediction: (i) From a qualitative perspective, the model
suggests that the $\mu_{p}$ exponent should be roughly $\textstyle\frac{1}{2}$
for yeast, rather than the canonically assumed scaling exponent of 1. (ii)
From a quantitative perspective, the model also predicts the coefficient of
proportionality if the empirical relation between protein and message
abundances is known (Eq. 15).

Observed noise in yeast matches the predictions of the empirical model. Newman
et al. have characterized protein noise by flow cytometry of strains
expressing fluorescent fusions expressed from their endogenous promoters [12].
The comparison of this data to the prediction of the statistical expression
model (Eq. 16) are shown in Fig. 5. From a qualitative perspective, the
predicted scaling exponent of $-0.48$ comes very close to capturing the
scaling of the noise, as determined by the direct fitting of the empirical
noise model (Eq. 5 and Fig. 2). From a quantitative perspective, the predicted
coefficient of Eq. 16 also fits the observed noise.

From both the statistical analysis (Eq. 5) and visual inspection (Fig. 5C), it
is clear that the noise in yeast does not obey the canonical model (Eq. 3).
However, the noise in E. coli does obey the canonical model for low copy
messages [15]. (See Fig. 5C.) Why does the noise scale differently in the two
organisms? The key difference is that the empirical relation between the
protein and message numbers are different. In E. coli,
$\mu_{p}\propto\mu_{m}^{1}$ [25]. Our analysis therefore predicts the
canonical model (Eq. 3) should hold for E. coli, but not for yeast, as
illustrated schematically in Fig. 5. (Additional discussion can be found in
the Supplementary Material Sec. A.5.6.).

Figure 5:  Understanding the distinct central dogma strategies using the
amplifier analogy. Panel A: Yeast. High expression ($\mu_{p}$) is typically
achieved by coordinated small increases in both transcription ($\mu_{m}$) and
translation ($\varepsilon$), relative to low-expression genes. Panel B: E.
coli. High expression ($\mu_{p}$) is typically achieved by a large increase in
transcription ($\mu_{m}$) only, relative to low-expression genes. Translation
($\varepsilon$) is uncorrelated. Panel C: Distinct noise scaling with gene
expression. Due to the coordinated changes in both transcription and
translation in yeast, noise scaling is weaker than in E. coli, where only
transcription changes. The noise of high-expression E. coli genes is
determined by the noise floor.

Implications of growth robustness for translation. Before continuing with the
noise analysis, we to focus on the significance of the empirical relationship
between the protein and message numbers (Eq. 15). How can the cell counteract
noise-induced reductions in robustness? Eq. 11 implies that gene expression
can be thought of as a two-stage amplifier [17]: The first stage corresponds
to transcription with a gain of message number $\mu_{m}$, and the second stage
corresponds to translation with a gain in translation efficiency
$\varepsilon$. (See Fig. 5AB.) The noise is completely determined by the first
stage of amplification, provided that $\varepsilon\gg 0$ [3, 9]. Genes with
low transcription levels are the noisiest. For these genes, the cell can
achieve the same mean gene expression ($\mu_{p}$) with lower noise by
increasing the gain of the first stage (increasing message number) and
decreasing the gain of the second stage (the translation efficiency) by the
same factor. This is most clearly understood by reducing $\varepsilon$ at
fixed $\mu_{p}$ in Eq. 13. Highly transcribed genes have low noise and can
therefore tolerate higher translation efficiency in the interest of economy
(decreasing the total number of messages) [5]. Growth robustness therefore
predicts that the translation efficiency should grow with transcription level.

Translation efficiency increases with expression level in yeast. The
translation efficiency (Eq. 8) can be determined from the empirical
translation model (Eq. 15):

$\displaystyle\varepsilon$ $\displaystyle=$ $\displaystyle 8.0\
\mu_{m}^{1.1},$ (17)

as a function of message number. (An error analysis for both model parameters
is described in Supplementary Material Sec. A.4.4.) In yeast, the translation
efficiency clearly has a strong dependence on message number $\mu_{m}$, and
grows with the expression level, exactly as predicted by robustness arguments.
We note the contrast to the translation efficiency in E. coli, which is
roughly constant [25]. (See Supplementary Material Sec. A.5.6.) We will
speculate about the rationale for these differences in the discussion below.

Implications of growth robustness for transcription. In addition to the
prediction of translation efficiency depending on transcription, a second
qualitative prediction of growth robustness is that essential gene expression
should have a noise ceiling, or maximum noise level (Eq. 2), where noise above
this level would be too great for robust growth. The fit between the
statistical model and the observed noise has an important implication beyond
confirming the predictions of the telegraph and statistical models for noise:
The identification of the message number, $\mu_{m}$, as the key determinant of
noise allows us to use this quantity as a proxy for noise in quantitative
transcriptome analysis.

To identify a putative transcriptional floor, we now broaden our consideration
beyond yeast to characterize the central dogma in two other model organisms:
the bacterium Escherichia coli and Homo sapiens (human). We will also analyze
three different transcriptional statistics for each gene: transcription rate
($\beta_{m}$), cellular message number ($\mu_{m/c}$), and message number
($\mu_{m}$). Analysis of these organisms explores orders-of-magnitude
differences in characteristics of the central dogma, including total message
number, protein number, doubling time, message lifetime, and number of
essential genes. (See Tab. 1.) In particular, as a consequence of these
differences, the three statistics describing transcription: transcription
rate, cellular message number and message number are all distinct. Genes with
matching message numbers in two different organisms will not have matching
transcription rates or cellular message numbers. We hypothesize that cells
must express essential genes above some threshold message number for robust
growth; however, we expect to see that non-essential genes can be expressed at
much lower levels since growth is not strictly dependent on their expression.
The signature of a noise-robustness mechanism would be the absence of
essential genes for low message numbers.

Figure 6: Transcription in three model organisms. We characterized different
gene transcriptional statistics in three model organisms. In E. coli, two
growth conditions were analyzed. Panel A: The distribution of gene
transcription rate. The transcription rate varies by two orders-of-magnitude
between organisms. Panel B: The distribution of gene cellular message number.
There is also a two-order-of-magnitude variation between cellular message
numbers.  Figure 7: Panel A: Transcription in three model organisms. The
distribution of gene message number. All organisms have roughly similar
distributions of message number for essential genes, which are not observed
for message numbers below a couple per cell cycle. However, non-essential
genes can be expressed at much lower levels. Panel B: Nonessential genes
tolerate higher noise levels than essential genes. The floor of message number
is consistent with a noise ceiling of ${\rm CV}_{p}^{2}=0.7$ for essential
genes (green). Nonessential genes (red) are observed with lower transcription
levels. Panel C: Conserved transcriptional program for essential genes. The
message number per gene (number of messages transcribed per cell cycle) is
roughly identical in E. coli, yeast, and human. We show this schematically.

| Estimated minimum essential gene
---|---
Model organism |  Maximum essential gene noise: |  messages /cell-cycle: |  messages /cell: |  transcription rate: |  proteins:
|  $\max{\rm CV}_{p}^{2}$ |  $\mu_{m}^{\rm min}$ |  $\mu_{m/c}^{\rm min}$ |  $\beta_{m}^{\rm min}$ (h-1) |  $\mu_{p}^{\rm min}$
E. coli (LB) |  $0.7$ |  $1$ |  $0.08$ |  $2$ |  $30$
(M9) |  $0.7$ |  $1$ |  $0.03$ |  $0.7$ |  $30$
Yeast |  $0.7$ |  $1$ |  $0.2$ |  $0.7$ |  $400$
Human |  $0.7$ |  $1$ |  $0.6$ |  $0.04$ |  $3000$

Table 2:  Estimates of threshold levels for the central dogma in three model
organisms. Estimates for the lower thresholds of transcription statistics as
inferred from our analysis based on the one-message-per-cell-cycle rule.

No organism-independent threshold is observed for transcription rate or
cellular message number. Histograms of the per-gene transcription rate and
cellular message number are shown in Fig. 6 for E. coli, yeast, and human.
Consistent with existing reports, essential genes have higher expression than
non-essential genes on average; however, there does not appear to be any
consistent threshold in E. coli (even between growth conditions), yeast, or
human transcription, either as characterized by the transcription rate
($\beta_{m}$) or the cellular message number ($\mu_{m/c}$). For instance, the
per gene rate of transcription is much lower in human cells than E. coli under
rapid growth conditions, with yeast falling in between.

An organism-independent threshold is observed for message number for essential
genes. In contrast to the other two transcriptional statistics, there is a
consistent lower limit, or floor, on message number ($\mu_{m}$) of somewhere
between 1 and 10 messages per cell cycle for essential genes. (See Fig. 7.)
Non-essential genes can be expressed at a much lower level. This floor is
consistent not only between E. coli, growing under two different conditions,
but also between the three highly-divergent organisms: E. coli, yeast and
human. We will conservatively define the minimum message number as

$\mu_{m}^{\rm min}\equiv 1,$ (18)

and summarize this observation as the one-message-per-cell-cycle rule for
essential gene expression.

In addition to the common floor for essential genes, there is a common gene
expression distribution shape shared between organisms dependent on the
message numbers, especially for low-expression essential genes. This is
observed in spite of the significantly larger number of essential genes in
human relative to E. coli. (See Fig. 7.) Interestingly, there is also a
similarity between the non-essential gene distributions for E. coli and human,
but not for yeast, which appears to have a much lower fraction of genes
expressed at the lowest message numbers.

What genes fall below-threshold? We have hypothesized that essential genes
should be expressed above a threshold value for robustness. It is therefore
interesting to consider the function of genes that fall below this proposed
threshold. Do functions of these genes give us any insight into essential
processes that do not require robust gene expression?

Since our own preferred model system is E. coli, we focus here. Our essential
gene classification was based on the construction of the Keio knockout library
[26]. By this classification, 10 essential genes were below threshold. (See
Supplementary Material Tab. 4.) Our first step was to determine what fraction
of these genes were also classified as essential using transposon-based
mutagenesis [27, 28]. Of the 10 initial candidates, only one gene, ymfK, was
consistently classified as an essential gene in all three studies, and we
estimate that its message number is just below the threshold ($\mu_{m}=0.4$).
ymfK is located in the lambdoid prophage element e14 and is annotated as a CI-
like repressor which regulates lysis-lysogeny decision [29]. In $\lambda$
phase, the CI repressor represses lytic genes to maintain the lysogenic state.
A conserved function for ymfK is consistent with it being classified as
essential, since its regulation would prevent cell lysis. However, since ymfK
is a prophage gene, not a host gene, it is not clear that its expression
should optimize host fitness, potentially at the expense of phage fitness. In
summary, closer inspection of below-threshold essential genes supports the
threshold hypothesis.

Maximum noise for essential genes. The motivation for hypothesizing a minimum
threshold for message number was noise-robustness, or the existence of a
hypothesized noise ceiling above which essential gene expression is too noisy
to allow robust cellular proliferation. With the one-message-per-cell-cycle
rule, $\mu_{m}^{min}\equiv 1$, we can estimate the essential gene noise
ceiling using Eq. 12:

${\rm CV}_{p}^{2}\leq 0.7,$ (19)

for essential genes. Since noise depends only on the message number, we expect
to observe the same limit in all organisms if the message number floor is
conserved.

Estimating the floor on central-dogma parameters. If message number floor is
conserved, a limit can be estimated for the floor value on other
transcriptional parameters. Using Eq. 14, we can estimate the floor on the
cellular message number (as measured in RNA-Seq measurements):

$\mu_{m/c}^{\rm min}=\textstyle\frac{\tau_{m}}{T},$ (20)

for essential genes. Similarly, we can use Eq. 9 to estimate the minimum
transcription rate:

$\beta_{m}^{\rm min}=\textstyle\frac{1}{T},$ (21)

for essential genes. Again, this result has an intuitive interpretation as the
one-message-per-cell-cycle rule. Finally, we can estimate a floor on essential
protein abundance, assuming a constant translation efficiency using Eq. 11:

$\mu_{p}^{\rm min}={\varepsilon},$ (22)

for essential genes, where $\varepsilon$ is the translation efficiency (which
we will assume is well approximated by the mean in the context of the
estimate). All four floor estimates for each model organism are shown in Tab.
2.

## Discussion

Noise by the numbers. Although there has already been significant discussion
of the scaling of biological noise with protein abundance [3, 9, 10, 12, 15],
our study is arguably the first to test the predictions of the telegraph and
statistical noise models against absolute measurements of protein and message
abundances. This approach is particularly important for the message number
($\mu_{m}$), which determines the magnitude of the noise in protein
expression, and facilitates direct comparisons of noise between organisms as
well as identifying the common distributions of message number for genes, that
are conserved from bacteria to human.

Noise scaling in E. coli versus yeast. A key piece of evidence for the
significance of the message number was the observation of the non-canonical
scaling of the yeast noise with protein abundance (Fig. 5); however, the
canonical model (Eq. 3) does accurately describe the noise in E. coli (see
Fig. 11). Why does the noise scale differently? In E. coli, the translation
efficiency is only weakly correlated with the gene expression [25], and
therefore the canonical model is a reasonable approximation (Supplementary
Material Sec. A.5.4). However, we also argued that translation efficiency
should grow with expression level. Why is this not observed in E. coli? Due to
the high noise floor in E. coli, nearly all essential genes are expressed at a
sufficiently high expression level such that the noise is dominated by the
noise floor [15]. As a consequence, increasing the message number, while
decreasing translation efficiency, does not decrease the noise even as it
increases the metabolic load as a result of increased transcription. (A
closely related point has recently been made in Bacillus subtilis [30], where
Deloupy et al. report that the noise cannot be tuned by adjusting the message
number due to the noise floor.) Our expectation is therefore that other
bacterial cells will look similar to E. coli: They will have a higher noise
floor and a similar scaling of noise with protein abundance.

In contrast, due to the lower noise floor, we expect eukaryotic cells to
optimize the central dogma processes like yeast and as a result will have a
similar non-canonical scaling of noise with protein abundance. Although this
non-canonical scaling is clear from the abundance data (Fig. 5B), there is an
important qualification to emphasize: the mechanism that gives rise to the
non-canonical scaling is due to the correlation between translation efficiency
and transcription. Regulatory changes that effect only transcription (i.e.
increase $\mu_{m}$) and not translation ($\varepsilon$) should obey the
canonical noise model (Eq. 3). This scenario may help explain why Bar-Even et
al. claim to observe canonical noise scaling in yeast [10], studying a subset
of genes under a range of conditions resulting in differential expression
levels. The failure of the canonical noise model (Eq. 3) at the proteome level
in yeast (Eq. 16) is a consequence of genome-wide optimization of the relative
transcription and translation rates.

Essential versus non-essential genes. What genes are defined as essential is
highly context specific [31]. It is therefore important to consider whether
the comparison between these two classes of genes is informative in the
context of our analysis. We believe the example of lac operon in E. coli is
particularly informative in this respect. The genes lacZYA are conditionally
essential: they are required when lactose is the carbon source; however, these
genes are repressed when glucose is the carbon source. Our expectation is that
these conditionally essential genes will obey the one-message-per-cell-cycle
rule when these genes are required; however, they need not obey this rule when
the genes are repressed. By analyzing essential genes, we are limiting the
analysis to transcriptionally-active genes, whereas the non-essential category
contains both transcriptionally-active and silenced genes.

Protein degradation and transcriptional bursting. Two important mechanisms can
act to significantly increase the noise above the levels we predict: protein
degradation and transcriptional bursting. Although the dominant mechanism of
protein depletion is dilution in E. coli, protein degradation plays an
important role in many organisms, especially in eukaryotic cells [32, 33]. If
protein degradation depletes proteins faster than dilution, the shape
parameter decreases below our estimate (Eq. 9), increasing the noise.
Likewise, the existence of transcriptional bursting, in which the chromatin
switches between transcriptionally active and quiescent periods, can also act
to increase the noise [1, 34, 7]. Since the presence of both these mechanisms
increases the noise beyond what is predicted by the message number, they do
not affect our estimate of the minimum threshold for $\mu_{m}$.

The biological implications of noise. What are the biological implications of
gene expression noise? Many important proposals have been made, including bet-
hedging strategies, the necessity of feedback in gene regulatory networks, etc
[1]. Our analysis suggests that noise influences the optimal function of the
central dogma process generically. Hausser et al. have already discussed some
aspects of this problem and use this approach to place coarse limits on
transcription versus translation rates [5]. The transcriptional floor for
essential genes that we have proposed places much stronger limits on the
function of the central dogma.

Although we describe our observations as a floor, a more nuanced description
of the phenomenon is a common distribution of gene message numbers, peaked at
roughly 15 messages per cell cycle and cutting off close to one message per
cell cycle. Does this correspond to a hard limit? We expect that this does not
since there are a small fraction of genes, classified as essential, just below
this limit; however, it does appear that virtually all essential genes have
optimal expression levels above this threshold. The common distribution of
message number clearly suggests that noise considerations shape the function
of the central dogma for virtually all genes. Exploring this hypothesis will
require quantitative models that explicitly realize the high cost of noise-
induced low essential-protein abundance. We will present such an analysis
elsewhere.

Adapting the central dogma to increased cell size and complexity. Although
core components of the central dogma machinery are highly-conserved, there has
been significant complexification of both the transcriptional and
translational processes in eukaryotic cells [35]. Given this increased
regulatory complexity, it is unclear how the central dogma processes should be
adapted in larger and more complex cells. An important clue to this adaptation
comes from E. coli proliferating with different growth rates. Although there
are very significant differences between the cellular message number as well
as the overall transcription rate under the two growth conditions, there is
very little difference in message number. In short, roughly the same number of
messages are made during the cell cycle, but they are made more slowly under
slow growth conditions.

How does this picture generalize in eukaryotic cells? Although both the total
number of messages and the number of essential and non-essential genes are
larger in both yeast and human cells, the distribution of the message number
per gene is essentially the same as E. coli (Fig. 7). The conservation of the
message number between organisms is consistent with all of these organisms
being optimized with respect to the same trade-off between economy and
robustness to noise.

Data availability. We include a source data file which includes the estimated
message numbers as well as essential/nonessential classifications for each
organism.

Acknowledgments. The authors would like to thank B. Traxler, A. Nourmohammad,
J. Mougous, K. Cutler, M. Cosentino-Lagomarsino, S. van Teeffelen, and S.
Murray. This work was supported by NIH grant R01-GM128191.

Author contributions: T.W.L., H.K.J.C., D.H. and P.A.W. conceived the
research. T.W.L. and P.A.W. performed the analysis. H.K.J.C. and D.H.
performed experiments and analysis. T.W.L., H.K.J.C., D.H. and P.A.W. wrote
the paper.

Competing interests: The authors declare no competing interests.

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###### Contents

  1. A Supplemental analysis
    1. A.1 Gillespie Simulation of the telegraph model
    2. A.2  Selection of central dogma parameter estimates
      1. A.2.1 E. coli data
      2. A.2.2 Yeast data
      3. A.2.3 Human data
    3. A.3 Quantitative estimates of central dogma parameters
      1. A.3.1 Estimating the cellular message number: $\mu_{m/c}$
      2. A.3.2 Estimating the transcription rate: $\beta_{m}$
      3. A.3.3 Estimating the message number: $\mu_{m}$
      4. A.3.4 Estimating the protein number: $\mu_{p}$
    4. A.4 Empirical models for yeast gene expression
      1. A.4.1 The meaning of the error estimates
      2. A.4.2 Empirical model for protein number
      3. A.4.3 Empirical model for message number
      4. A.4.4 Empirical model for translation efficiency
      5. A.4.5 Empirical model for the coefficient of variation
    5. A.5 Supplemental analysis of gene expression noise
      1. A.5.1 Inclusion of noise floor in the yeast analysis
      2. A.5.2 Hypothesis test I
      3. A.5.3 Hypothesis test II
      4. A.5.4 Maximum likelihood estimates of the parameters
      5. A.5.5 Statistical details MLE estimate of the variance
      6. A.5.6 Detailed discussion of noise in E. coli

##  Appendix A Supplemental analysis

###  A.1 Gillespie Simulation of the telegraph model

Protein distributions of the telegraph model for E. coli were simulated with a
Gillespie algorithm. Assuming the lifetime of the cell cycle ($T_{cc}=30$ min)
[36], mRNA lifetime ($\tau_{m}=2.5$ min) [37], and translation rate
($\beta_{p}\approx 500$ hr-1), the protein distributions for several mean
expression levels were numerically generated for exponential growth with
100,000 stochastic cell divisions, with protein partitioned at division
following the binomial distribution.

The gamma distributions for each mean message number with scale and shape
parameters determined by the corresponding translation efficiency and message
number ($\theta=\varepsilon\ln 2$, $k=\frac{\mu_{m}}{\ln 2}$) as used for the
Gillespie simulation were also plotted with the protein distributions.

$p(n|\theta,k)=\frac{1}{\Gamma(k)\theta^{k}}n^{k-1}e^{-\frac{n}{\theta}}$ (23)

Figure 8: The protein abundance is approximately gamma distributed. Protein
abundance was modeled for eight different transcription rates using a
Gillespie simulation, including the stochastic partitioning of the proteins
between daughter cells at cell division. The range in abundance matches the
observed range of expression levels in the cell. We observed that the
simulated protein abundances were well fit by gamma distributions.

|  |  |  |  | Total number of | Average
---|---|---|---|---|---|---
Model organism |  Growth condition |  Doubling time: |  Message lifetime: |  Message recycling ratio: |  messages /cell: |  messages /cell-cycle: |  proteins: |  translation efficiency: |  translation rate:
|  |  $T$ |  $\tau_{m}=\gamma_{m}^{-1}$ |  $m=T/\tau_{m}$ |  $N^{\rm tot}_{m/c}$ |  $N^{\rm tot}_{m}$ |  $N^{\rm tot}_{p}$ |  $\varepsilon$ |  $\beta_{p}$ (h-1)
Escherichia coli |  LB |  30 min [36] |  $2.5$ min [37] |  12 |  $7.8\times 10^{3}$ [38] |  $9.4\times 10^{4}$ |  $3\times 10^{6}$ [39] |  $22$ |  $530$
(E. coli) |  M9 |  90 min [36] |  $2.5$ min [37] |  36 |  $2.4\times 10^{3}$ [38] |  $8.6\times 10^{4}$ |  $3\times 10^{6}$ [39] |  $24$ |  $580$
Sacchromyces cerevisiae (Yeast–haploid) |  YEPD |  90 min [40] |  22 min [41] |  4 |  $2.9\times 10^{4}$ [42] |  $1\times 10^{5}$ |  $5\times 10^{7}$ [43] |  $4\times 10^{2}$ |  $410$
Homo sapiens (Human) |  Tissue |  24 h [35] |  14 h [44] |  1.7 |  $3.6\times 10^{5}$ [40] |  $5\times 10^{5}$ |  $2\times 10^{9}$ [39] |  $4\times 10^{3}$ |  120

Table 3:  Central dogma parameters for three model organisms with detailed
references. Columns three through seven hold representative values for
measured central-dogma parameters for the model organisms described in the
paper. Each value is followed by a reference for its source. The last three
columns hold estimates for the lower threshold on transcription inferred from
our analysis. The sources of the numbers and estimates are described in the
Supplemental Material Sec. A.2.

###  A.2  Selection of central dogma parameter estimates

The estimates for central dogma model parameters come from two types of data:
(i) quantitative measurement of cellular-scale parameters for each organism
(total number of messages in the cell, cell cycle duration, etc) and (ii)
genome-wide studies quantitative of mRNA and protein abundance.

For the cellular-scale central dogma parameters, we relied heavily on an
online compilation of biological numbers: BioNumbers [45]. This resource
provides a collection of curated quantitative estimates for biological
numbers, as well as their original source. In the interest of conciseness, we
have cited only the original source in the Tab. 1, although we are extremely
grateful and supportive of the creators of the BioNumbers website for helping
us very efficiently identify consensus estimates for the parameters of the
central dogma parameters.

For the selection of genome-wide studies on abundance, we used many of the
same resources cited in BioNumbers as well as studies selected by a previous
study of a quantitative analysis of the central dogma: Hausser et al. [5].

####  A.2.1 E. coli data

Message lifetimes: The message lifetimes (and median lifetime) were taken from
a recent transcriptome-wide study by Chen et al. [37]. These investigators
measured the lifetime in both rapid (LB) and slow growth (M9).

Noise: Taniguchi et al. have performed a beautiful simultaneous study of the
proteome and transcriptome with single-molecule sensitivity [15]. Although we
use the noise analysis data from this study for our supplemental analysis of
E. coli noise, it is not the source for our E. coli transcriptome data due to
the extremely slow growth of the cells in this study (150 minute doubling
time), which is not consistent with the growth conditions for the other
sources of data.

mRNA abundance: Instead, we used data from the more recent Bartholomaus et al.
study [38], which characterizes the transcriptome in both rapid (LB) and slow
growth (M9).

Total cellular message number. This study was chosen since it was the source
of the BioNumbers estimates of cellular message number in E. coli (BNID 112795
[45]).

Doubling time: The source of the doubling times for rapid (LB) and slow (M9)
growth of E. coli comes from Bernstein [36].

Essential gene classification. The classification of essential genes in yeast
comes from the construction of the Keio knockout collection from Baba et al.
[26].

Protein number. The total protein number in E. coli came from Milo’s recent
review of this subject [39].

####  A.2.2 Yeast data

Message lifetimes: The message lifetimes (and median lifetime) were taken from
Chia et al. [41].

Noise: The noise data was taken from the Newman et al. study, which used flow
cytometry of a library of fluorescent fusions to characterize protein
abundance with single-cell resolution [12].

mRNA abundance: The transcriptome data comes from the very recent Blevins et
al. study [46].

Total cellular message number. There are a wide-range of estimates for the
total cellular message number in yeast: $1.5\times 10^{4}$ [47] (BNID 104312
[45]), $1.2\times 10^{4}$ [48] (BNID 102988 [45]), $6.0\times 10^{4}$ [49]
(BNID 103023 [45]), $2.6\times 10^{4}$ [42] (BNID 106763 [45]) and $3.0\times
10^{4}$ [50]. We used the compromise value of $2.9\times 10^{4}$.

Doubling time: The doubling time was taken from [40].

Protein number. The total protein number in yeast comes from Futcher et al.
[43].

Essential gene classification. The classification of essential genes in yeast
comes from van Leeuwen et al. [51].

Proteome abundance data: The proteome abundance data came from two sources:
flow cytometry of fluorescent fusions from Newman et al. [12] as well as mass-
spec data from de Godoy et al. [24].

####  A.2.3 Human data

Message lifetimes: The message lifetimes (and median lifetime) were taken from
Yang et al. [44] who reported a median half life of 10 h which corresponds to
a lifetime of 14 h.

mRNA abundance: The transcriptome data comes from the data compiled by the
Human Protein Atlas [52], which we averaged over tissue types.

Total cellular message number. The total cellular message number in human
comes from Velculescu et al. [53] (BNID 104330 [45]).

Doubling time: The doubling time was taken from [35].

Protein number. The total protein number in human came from Milo’s recent
review of this subject [39].

Essential gene classification. The classification of essential genes in human
comes from Wang et al. [54].

###  A.3 Quantitative estimates of central dogma parameters

####  A.3.1 Estimating the cellular message number: $\mu_{m/c}$

For each model organism (and condition), we found a consensus estimate from
the literature for the total number of mRNA messages per cell $N_{m/c}^{\rm
tot}$. This number and its source are provided in Tab. 1. To estimate the
number of messages corresponding to gene $i$, we re-scaled the un-normalized
abundance level $r_{i}$:

$N_{m/c,i}=N_{m/c}^{\rm tot}\textstyle\frac{r_{i}}{\sum_{j}r_{j}},$ (24)

where the sum over gene index $j$ runs over all genes.

####  A.3.2 Estimating the transcription rate: $\beta_{m}$

To estimate the transcription rate for gene $i$, we start from the estimated
cellular message number $N_{m/c,i}$ and use the telegraph model prediction for
the cellular message number:

$N_{m/c,i}=\beta_{m,i}/\gamma_{m,i},$ (25)

where $\gamma_{m,i}$ is the message decay rate. Since gene-to-gene variation
in message number is dominated by the transcription rate (e.g [37]), we
estimate the decay rate as the inverse gene-median message life time:

$\gamma_{m,i}=\tau_{m}^{-1},$ (26)

for which a consensus value was found from the literature. This number and its
source are provided in Tab. 1. We then estimate the gene-specific
transcription rate:

$\beta_{m,i}=N_{m/c,i}/\tau_{m}.$ (27)

####  A.3.3 Estimating the message number: $\mu_{m}$

To estimate the message number of gene $i$, we use the predicted value from
the telegraph model:

$N_{m,i}=T\beta_{m,i}=\textstyle\frac{T}{\tau_{m}}N_{m/c,i},$ (28)

where $T$ is the doubling time and $N_{m/c,i}$ is the cellular message number
(Eq. 24).

####  A.3.4 Estimating the protein number: $\mu_{p}$

The protein abundance data for yeast grown in YEPD media and measured with
flow cytometry fluorescence [12] were given in arbitrary units (AU). In order
to convert from AU to protein number, the fluorescence values were rescaled by
comparing with mass-spectrometry protein abundance data for yeast grown in YNB
media [24]. Since the protein abundance from mass-spectrometry was given in
terms of Intensity, the Intensity values were first rescaled by the total
number of proteins in yeast, $5\times 10^{7}$. The mass-spectrometry protein
data was thresholded at 10 proteins, based on the assumption that the noise of
the data for 10 and fewer proteins makes the data unreliable. Next, the log of
the fluorescence protein abundance in AU as a function of the log of
thresholded mass-spectrometry protein abundance was fit as a linear function
with an assumed slope of 1 to find the offset, 3.9, (Fig. 9) which corresponds
to a multiplicative scaling factor (Eqn. 30). We then used that offset value
to rescale the fluorescence data from AU to protein number. We also compared
to yeast grown in SD media [12] and found a similar offset result.

$\displaystyle\log\mu_{P}^{\text{F}}=m\log\mu_{P}^{\text{MS}}+b$ (29)
$\displaystyle\mu_{P}^{\text{F}}=b(\mu_{P}^{\text{MS}})^{m}$ (30)

Figure 9:  Fit to rescale fluorescence intensity to protein number.  Protein
abundance from flow cytometry fluorescence [12] as a function of mass-
spectrometry scaled abundance [24]. The mass-spectrometry data was thresholded
at 10 proteins, and then a linear fit was performed to find the offset of 3.9,
which was used to convert protein fluorescence AU to number.

###  A.4 Empirical models for yeast gene expression

To generate the empirical model for protein number as a function of message
number, we used protein abundance data from Newman et al. [12], re-scaled to
estimate protein number (Sec. A.3.4) and transcriptome data from Lahtvee et
al. [55], re-scaled to estimate message number (Sec. A.3.3).

####  A.4.1 The meaning of the error estimates

Before providing a detailed error analysis, it is important to place our error
estimates in perspective. The error that we will be estimating is the
statistical error associated with the finite sample size; however, this is not
the dominant source of error. A far more important consideration are
systematic problems with our analysis and the underlying experiments. For
instance, since we do not have a detailed model for the error of the
experiments analyzed, there are multiple distinct analyses (i.e. assumptions
about the error model) that could be implemented for the data fitting, each
giving slightly different model parameters. These model to model differences
still give rise to predictions consistent with our qualitative conclusions;
however, they are likely larger than the statistical uncertainty we compute
(while assuming a particular model).

####  A.4.2 Empirical model for protein number

We initially fit the empirical model for protein number,

$\mu_{p}=C_{0}\mu_{m}^{\alpha_{0}},$ (31)

to the data using a standard least-squares approach; however, the algorithm
led to a very poor fit since it does not account for uncertainty in both
independent and dependent variables. We therefore used an alternative approach
[56], which assumes comparable error in both variables. The model parameters
are:

$\displaystyle\alpha_{0}$ $\displaystyle=$ $\displaystyle 2.1\pm 0.04,$ (32)
$\displaystyle C_{0}$ $\displaystyle=$ $\displaystyle 8.0\pm 1.0,$ (33)

where the uncertainties are the estimated standard errors.

####  A.4.3 Empirical model for message number

For the prediction of the coefficient of variation, it is useful to invert Eq.
31 to generate a model for message number as a function of protein number:

$\displaystyle\mu_{m}$ $\displaystyle=$ $\displaystyle
C_{0}^{-1/\alpha_{0}}\mu_{p}^{1/\alpha{0}},$ (34) $\displaystyle=$
$\displaystyle C_{1}\mu_{p}^{\alpha_{1}},$ (35)

where the last line defines two new parameters: a coefficient $C_{1}$ and an
exponent $\alpha_{1}$. The resulting parameters and uncertainties are:

$\displaystyle\alpha_{1}$ $\displaystyle\equiv$ $\displaystyle 1/\alpha_{0},$
(36) $\displaystyle=$ $\displaystyle 0.48\pm 0.01,$ (37) $\displaystyle C_{1}$
$\displaystyle\equiv$ $\displaystyle C_{0}^{-1/\alpha_{0}},$ (38)
$\displaystyle=$ $\displaystyle 0.37\pm 0.02,$ (39)

where the uncertainties are the estimated standard errors.

####  A.4.4 Empirical model for translation efficiency

To generate an empirical model for translation efficiency, we started from the
empirical model for protein number (Eq. 31), and then use Eq. 11 to relate
protein number, message number, and translation efficiency:

$\displaystyle\varepsilon$ $\displaystyle=$
$\displaystyle\textstyle\frac{\mu_{p}}{\mu_{m}},$ (40) $\displaystyle=$
$\displaystyle C_{0}\mu_{m}^{\alpha_{0}-1},$ (41) $\displaystyle=$
$\displaystyle C_{2}\mu_{m}^{\alpha_{2}},$ (42)

where the last line defines two new parameters: a coefficient $C_{2}$ and an
exponent $\alpha_{2}$. The resulting parameters and uncertainties are:

$\displaystyle\alpha_{2}$ $\displaystyle=$ $\displaystyle\alpha_{0}-1,$ (43)
$\displaystyle=$ $\displaystyle 1.07\pm 0.04,$ (44) $\displaystyle C_{2}$
$\displaystyle=$ $\displaystyle C_{0},$ (45) $\displaystyle=$ $\displaystyle
8.0\pm 1.0,$ (46)

where the uncertainties are the estimated standard errors.

####  A.4.5 Empirical model for the coefficient of variation

To generate an empirical model for the coefficient of variation, we started
from the empirical model for message number (Eq. 35), and then substitute this
into the statistical model prediction for CV${}_{p}^{2}$ (Eq. 12):

$\displaystyle{\rm CV}_{p}^{2}$ $\displaystyle=$
$\displaystyle\textstyle\frac{\log 2}{\mu_{m}},$ (47) $\displaystyle=$
$\displaystyle C_{0}^{1/\alpha_{0}}\log 2\cdot\mu_{p}^{-1/\alpha_{0}},$ (48)
$\displaystyle=$ $\displaystyle C_{3}\mu_{p}^{\alpha_{3}},$ (49)

where the last line defines two new parameters: a coefficient $C_{3}$ and an
exponent $\alpha_{3}$. The resulting parameters and uncertainties are:

$\displaystyle\alpha_{3}$ $\displaystyle\equiv$ $\displaystyle-1/\alpha_{0},$
(50) $\displaystyle=$ $\displaystyle-0.48\pm 0.01,$ (51) $\displaystyle C_{3}$
$\displaystyle\equiv$ $\displaystyle C^{1/\alpha_{0}}_{0}\log 2,$ (52)
$\displaystyle=$ $\displaystyle 1.9\pm 0.1,$ (53)

where the uncertainties are the estimated standard errors.

###  A.5 Supplemental analysis of gene expression noise

The quantitative model for gene expression noise includes multiple
contributions:

${\rm CV}^{2}_{p}\approx\textstyle\frac{1}{\mu_{p}}+\textstyle\frac{\log
2}{\mu_{m}}+c_{0},$ (54)

where the first term can be understood to represent the Poisson noise from
translation, the second term the Poisson noise from transcription, and the
last term, $c_{0}$, is called the noise floor and is believed to be caused by
the cell-to-cell variation in metabolites, ribosomes, and polymerases etc [13,
14].

Figure 10:  Yeast noise fit against canonical noise model, with a noise floor.
Yeast noise data fit with the 2- (null hypothesis with $\mu_{p}^{-1}$
dependence) and 3- parameter ($\mu_{p}^{a}$) models.

####  A.5.1 Inclusion of noise floor in the yeast analysis

In the main text of the paper, we have ignored the role of the noise floor in
the analysis of noise in yeast. Unlike E. coli, where the noise floor is high
(${\rm CV}^{2}_{p}=0.1$) and is determinative of the noise associated with
almost all essential genes [15, 13, 14], in yeast the noise floor is much
lower (${\rm CV}^{2}_{p}=0.01$) and therefore affects only genes with the
highest expression.

In this section, we will consider models that include the noise floor, since
its presence can make the noise scaling more difficult to interpret. To
determine if the scaling of the noise is consistent with the canonical
assumption that the noise is proportional to $\mu_{p}^{-1}$ for low
expression, we will consider two competing empirical models for the noise
(Fig. 10). In the null hypothesis, we will consider a model:

$\eta_{0}(\mu_{p};b,c)=\textstyle\frac{b}{\mu_{p}}+c,$ (55)

and an alternative hypothesis with an extra exponent parameter $a$:

$\eta_{1}(\mu_{p};a,b,c)=\textstyle\frac{b}{\mu_{p}^{a}}+c.$ (56)

We will assume that ${\rm CV}^{2}_{p}$ is normally distributed about $\eta$
with unknown variance $\sigma^{2}_{\eta}$.

In this context, a maximum likelihood analysis is equivalent to least-squares
analysis. Let the sum of the squares be defined:

$S_{I}({\bm{\theta}})\equiv\sum_{i}[{\rm
CV}_{p,i}^{2}-\eta_{I}(\mu_{p,i};{\bm{\theta}})]^{2}$ (57)

for model $I$ where $\bm{\theta}$ represents the parameter vector. The maximum
likelihood parameters are

$\hat{\bm{\theta}}=\arg\max_{\bm{\theta}}S_{I}({\bm{\theta}}),$ (58)

with residual norm:

$\hat{S}_{I}=S_{I}(\hat{\bm{\theta}}).$ (59)

To test the null hypothesis, we will use the canonical likelihood ratio test
with the test statistic:

$\Lambda\equiv 2\log\frac{q_{1}}{q_{0}},$ (60)

where $q_{0}$ and $q_{1}$ are the likelihoods of the null and alternative
hypotheses, respectively. Wilks’ theorem states that $\Lambda$ has a chi-
squared distribution of dimension equal to the difference of the dimension of
the alternative and null hypotheses ($3-2=1$).

####  A.5.2 Hypothesis test I

In our first analysis, we will estimate the variance directly. We computed the
mean-squared difference for successive ${\rm CV}^{2}_{p}$ values, sorted by
mean protein number $\mu_{p}$. The variance estimator is

$\hat{\sigma}^{2}_{\eta}=\textstyle\frac{1}{2}\left<({\rm CV}^{2}_{p,i}-{\rm
CV}^{2}_{p,i+1})^{2}\right>_{i}=6.3\times 10^{-4},$ (61)

where the brackets represent a standard empirical average over gene $i$ for
the $\mu_{p}$-ordered gene ${\rm CV}_{p}^{2}$ values. The test statistic can
now be expressed in terms of the residual norms:

$\displaystyle\Lambda$ $\displaystyle=$
$\displaystyle(\hat{S}_{1}-\hat{S}_{2})/\hat{\sigma}^{2}_{\eta},$ (62)
$\displaystyle=$ $\displaystyle 3.3\times 10^{4},$ (63)

which corresponds to a p-value far below machine precision. We can therefore
reject the null hypothesis.

####  A.5.3 Hypothesis test II

In a more conservative approach, we can use maximum likelihood estimation to
estimate the variance of each model independently as a model parameter. In
this case, the test statistic can again be expressed in terms of the residual
norms:

$\displaystyle\Lambda$ $\displaystyle=$ $\displaystyle
N\log\textstyle\frac{\hat{S}_{1}}{\hat{S}_{2}},$ (64) $\displaystyle=$
$\displaystyle 1.6\times 10^{2},$ (65)

where $N$ is the number of data points. In this case, the p-value can be
computed assuming the Wilks’ theorem (i.e. the chi-squared test):

$p=6\times 10^{-36},$ (66)

again, strongly rejecting the null hypothesis.

####  A.5.4 Maximum likelihood estimates of the parameters

In the alternative hypothesis, the maximum likelihood estimate (MLE) of the
empirical noise model (Eq. 5) parameters are (Fig. 10):

$\displaystyle a$ $\displaystyle=$ $\displaystyle 0.57\pm 0.02,$ (67)
$\displaystyle b$ $\displaystyle=$ $\displaystyle 3.0\pm 0.5,$ (68)
$\displaystyle c$ $\displaystyle=$ $\displaystyle 0.013\pm 0.001,$ (69)

where the parameter uncertainty has been estimated using the Fisher
Information in the usual way using the MLE estimate of the variance.

The noise model parameters were also determined for E. coli:

$\displaystyle a$ $\displaystyle=$ $\displaystyle 1.22\pm 0.01,$ (70)
$\displaystyle b$ $\displaystyle=$ $\displaystyle 1.27\pm 0.02,$ (71)
$\displaystyle c$ $\displaystyle=$ $\displaystyle 0.154\pm 0.002,$ (72)

with the corresponding fit shown in Fig. 11. Since a is close to 1, the
canonical model with $a=1$ (Eqn. 3) is a somewhat reasonable approximation for
the noise in E. coli.

Figure 11: Three-parameter fit to E. coli noise.  The noise as a function of
protein abundance from Taniguichi et al. was fit to the 3 parameter noise
model (Eqn. 5). From the fit, protein noise scales proportionally with
$\mu_{p}^{-1.22}$, which is a close result to the canonical model with
$\mu_{p}^{-1}$.

####  A.5.5 Statistical details MLE estimate of the variance

The minus-log-likelihood for the normal model $I$ is:

$h_{I}(\hat{\bm{\theta}},\sigma^{2})=\textstyle\frac{N}{2}\log
2\pi\sigma^{2}+\textstyle\frac{1}{2\sigma^{2}}\hat{S}_{I},$ (73)

where $\hat{S}_{I}$ is the least-square residual. We then minimize $h_{I}$
with respect to the variance $\sigma^{2}$:

$\partial_{\sigma^{2}}h|_{\hat{\sigma}^{2}}=0,$ (74)

to solve for the MLE $\hat{\sigma}^{2}$:

$\hat{\sigma}^{2}=\textstyle\frac{1}{N}\hat{S}_{I}.$ (75)

Next we evaluate $h$ at the variance estimator:

$h_{I}(\hat{\bm{\theta}},\hat{\sigma}^{2})=\textstyle\frac{N}{2}\left[\log
2\pi\frac{\hat{S}_{I}}{N}+1\right].$ (76)

The test statistics can be written in terms of the $h$’s:

$\displaystyle\Lambda$ $\displaystyle=$ $\displaystyle
2h_{0}(h_{1}(\hat{\bm{\theta}},\hat{\sigma}^{2})-2h_{2}(\hat{\bm{\theta}},\hat{\sigma}^{2}),$
(77) $\displaystyle=$ $\displaystyle
N\log\textstyle\frac{\hat{S}_{1}}{\hat{S}_{1}},$ (78)

which can be evaluated directly in terms of the residual norms for the null
and alternative hypotheses.

Gene | Message |  Annotated function | Essential (E)/
---|---|---|---
name | number: |  from Ecocyc | Nonessential (N)
| $\mu_{m}$ |  | Ref. [26], [57], [28]
alsK | 0.3 |  The alsK gene encodes a D-allose kinase. Its role in the degradation of D-allose is unclear; AlsK is not required for utilization of a D-allose carbon source; this effect may be due to the presence of other ambiguous sugar kinases within E. coli K-12. | E, N, N
bcsB | 0.4 |  BcsB is encoded in a predicted operon together with bcsA, bcsZ and bcsC. In other organisms, these genes are involved in cellulose biosynthesis, a characteristic of the rdar (red, dry and rough) morphotype. However, the K-12 laboratory strain of E. coli does not show a rdar morphotype and does not produce cellulose. | E, N, N
entD | 0.4 |  AcpS is the founding member of a 4’-phosphopantetheinyl (P-pant) transferase protein family that includes E. coli EntD, E. coli o195 protein, and Bacillus subtilis Sfp; family members share two conserved motifs but relatively low sequence identity overall. | E, N, N
yafF | 0.4 |  No information about this protein was found by a literature search conducted on April 19, 2017. | E,-, N
yagG | 0.6 |  yagGH is predicted to be a member of the XylR regulon; its products may mediate transport (YagG) and hydrolysis (YagH) of xylooligosaccharides; putative XylR and CRP binding sites are identified upstream of yagGH. | E,-, N
yceQ | 0.2 |  No information about this protein was found by a literature search conducted on July 12, 2017. | E, E, N
ydiL | 0.2 |  No information about this protein was found by a literature search conducted on April 7, 2017. | E, N, N
yhhQ | 0.4 |  YhhQ is an inner membrane protein implicated in the uptake of queuosine (Q) precursors - 7-cyano-7-deazaguanine (preQ0) and 7-aminomethyl-7-deazaguanine (preQ1) - for Q salvage. Q-modified tRNA is absent in $\Delta$queD and $\Delta$queD $\Delta$yhhQ strains grown in minimal media with glycerol; Q-modified tRNA is detected when a $\Delta$queD strain is grown in minimal media plus 10 nM preQ0 or preQ1 but is absent when a $\Delta$queD $\Delta$yhhQ strain is grown under these conditions. yhhQ expressed from a plasmid restores the presence of Q-modified tRNA in a $\Delta$queD $\Delta$yhhQ strain. | E,-, N
yibJ | 0.3 |  No information about this protein was found by a literature search conducted on July 9, 2018. | E, N, N
ymfK | 0.4 |  YmfK is a component of the relic lambdoid prophage e14 and is likely the SOS-sensitive repressor. It is similar to the P34 gene of the Shigella flexneri bacteriophage SfV and belongs to the LexA group of SOS-response transcriptional repressors. | E, E, E

Table 4:  Below-threshold essential genes identified in E. coli. This table
describes the message numbers and annotations for essential genes that we
estimated to have expression below the threshold of one message per cell
cycle. However, in the final column, we show classifications from three
different studies. Only one of the identified genes, ymfK, was consistently
defined as essential.

####  A.5.6 Detailed discussion of noise in E. coli

In general, the Telegraph model predicts that the noise will have a
coefficient of variation [3, 9]:

${\rm CV}_{p}^{2}\approx\frac{1}{\mu_{p}}+\frac{\varepsilon\ln 2}{\mu_{p}},$
(79)

where the first term is significant whenever the translation efficiency isn’t
$\varepsilon\gg 1$. In both E. coli ($\varepsilon\approx 30$) and yeast
($\varepsilon\approx 420$), this would seem naively to be the case. However,
since translation efficiency in yeast is not uniform, we must consider its
variation for low-expression proteins. We estimate that the detection
efficiency in yeast is roughly $10^{3}$ molecules. Using Eq. 15, we estimate
that $\varepsilon\approx 100$ and our approximation holds at the low-
expression detection limit.

In E. coli, the situation is somewhat more complicated. Unlike yeast, the
translation efficiency is roughly constant (at high to intermediate expression
levels) with respect to expression level [25], and therefore both terms in Eq.
79 are expected to scale like the canonical model ($\propto\mu_{p}^{-1}$).
However, it is clear that the translation efficiency must significantly
decrease for the lowest abundance proteins. This is visible even in Ref. [25]
Fig. 1B, where the data falls below the predicted protein abundance at low
message number. Note that these mass-spec measurements are not as sensitive as
fluorescence-based measurements (e.g. only 64% proteome could be detected
[58]). Furthermore, fits to the E. coli noise (Eq. 71) are consistent only
with low values of $\varepsilon$. At sufficiently high expression levels such
that we are confident about the translation efficiency, the noise is already
very close to the noise floor.