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Stochastic holin expression can
account for lysis time variation
in the bacteriophage l
rsif.royalsocietypublishing.org
Abhyudai Singh1 and John J. Dennehy2,3
1
Research
Cite this article: Singh A, Dennehy JJ. 2014
Stochastic holin expression can account for
lysis time variation in the bacteriophage l.
J. R. Soc. Interface 11: 20140140.
http://dx.doi.org/10.1098/rsif.2014.0140
Received: 6 February 2014
Accepted: 14 March 2014
Subject Areas:
systems biology, biomathematics, biophysics
Keywords:
stochastic gene expression, lambda phage,
first-passage time, feed-forward circuit,
host lysis, stochastic promoter switching
Author for correspondence:
Abhyudai Singh
e-mail: [email protected]
Electronic supplementary material is available
at http://dx.doi.org/10.1098/rsif.2014.0140 or
via http://rsif.royalsocietypublishing.org.
Department of Electrical and Computer Engineering, Biomedical Engineering, Mathematical Sciences,
University of Delaware, Newark, DE 19716, USA
2
Department of Biology, Queens College, Queens, NY 11367, USA
3
The Graduate Center, City University of New York, New York, NY 10016, USA
The inherent stochastic nature of biochemical processes can drive differences
in gene expression between otherwise identical cells. While cell-to-cell variability in gene expression has received much attention, randomness in timing
of events has been less studied. We investigate event timing at the single-cell
level in a simple system, the lytic pathway of the bacterial virus phage l. In
individual cells, lysis occurs on average at 65 min, with an s.d. of 3.5 min.
Interestingly, mutations in the lysis protein, holin, alter both the lysis time
(LT) mean and variance. In our analysis, LT is formulated as the first-passage
time (FPT) for cellular holin levels to cross a critical threshold. Exact analytical formulae for the FPT moments are derived for stochastic gene expression
models. These formulae reveal how holin transcription and translation efficiencies independently modulate the LT mean and variation. Analytical
expressions for the LT moments are used to evaluate previously published
single-cell LT data for l phages with mutations in the holin sequence or
its promoter. Our results show that stochastic holin expression is sufficient
to account for the intercellular LT differences in both wild-type phages,
and phage variants where holin transcription and the threshold for lysis
have been experimentally altered. Finally, our analysis reveals regulatory
motifs that enhance the robustness of lysis timing to cellular noise.
1. Introduction
Phenotypic variation is an intrinsic feature of biological systems. Most phenotypic
variation stems from genetic or environmental variation, but some phenotypic variation can exist even for identical genotypes in the same environment.
This variation often emerges from the stochastic nature of biochemical reactions
within a cell. Small fluctuations in transcriptional and translational machinery
can lead to large variations in important regulatory molecules, impacting cellular
switches, pathways and, ultimately, cell fates [1–5].
The causes and effects of stochastic variation have been best explored in
microbial systems such as bacteriophages [6–8], bacteria [9–13], yeast [14–19]
and human viruses [20–23]. For example, fluctuations in gene expression make
it possible for populations of isogenic l phages in the same environment to
take one of two distinct developmental fates: immediate lysis of the host cell or
lysogeny as a prophage in the host genome. This developmental switch is controlled by cytoplasmic levels of bacteriophage l protein CII [24]. When CII
levels are high, the lysogenic pathway is favoured, but when CII levels are low,
phages will enter the lytic pathway. While the switch is essentially stochastic,
the probabilities of lysis or lysogeny can vary according to environmental
conditions and the multiplicity of infection [8,25–27].
Bacteriophage l also possesses a simpler regulatory network that determines the
timing of lysis. Lysis is accomplished by a suite of four genes: S (encodes holin and
antiholin), R (encodes endolysin) and Rz/Rz1 (encode spanins). Expression of these
genes, and all l structural and assembly proteins, is under the control of the l late
promoter pR 0 . While expression from promoter pR 0 is constitutive, transcription terminates upstream of the lysis genes [28]. When the ‘decision’ is made to pursue the
& 2014 The Author(s) Published by the Royal Society. All rights reserved.
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the evolution of bacteriophage lysis timing as well as the
implications for the study of variation in gene expression.
mean = 65 min
s.d. = 3.5 min
2. Stochastic gene expression model formulation
frequency
20
We denote by x(t), the number of holin molecules inside the
cell at time t, where t ¼ 0 corresponds to the onset of holin
expression. As holin does not degrade over relevant timescales
[36], x(t) increases over time. The goal is to compute the FPT
(time taken by a random walker to first reach a defined point)
15
10
5
FPT ¼ inf(t 0 : x(t) X)
55
60
65
lysis time (min)
70
75
Figure 1. LT distribution in the bacteriophage l. Single-cell measurements
show considerable variation in lysis timing across a clonal cell population.
Data taken from [34]. Methods and materials from [34] are provided in
the electronic supplementary material as a convenience to the reader.
(Online version in colour.)
lytic life cycle, protein Q is produced and acts to prevent premature termination of late mRNA transcription approximately
10–15 min after lysis induction [29,30]. Following the production
of the late mRNA transcript, the lysis proteins are translated
by host ribosomes. Holin accumulates evenly in the Escherichia
coli plasma membrane until a genetically encoded time when
they spontaneously aggregate to form large holes [31]. These
holes allow endolysin and the spanins to access and disrupt
their targets beyond the periplasmic space [32]. Subsequently,
the cell ruptures and phage progeny are released into the
surrounding medium.
As hole formation and cell rupture are nearly simultaneous, lysis timing depends on the accumulation of holin
in the cell membrane up to a critical threshold [33]. Dennehy
and Wang observed the lysis time (LT) of single l lysogens
using a temperature-controlled perfusion chamber mounted
on a microscope stage (see electronic supplementary material,
figure S1 and movie). Their measurements show that l lysis
occurred on average at 65 min, with an s.d. of 3.5 min
(figure 1). Interestingly, mutations in the holin amino acid
sequence can significantly change the LT mean and variance
[34]. As randomness in the onset of the late promoter
expression cannot account for these variations [34,35], stochastic holin expression and accumulation may be key in
determining LT differences. Ultimately, the expression and
accumulation of holin in the inner membrane depend on several factors: the rate of transcription of late mRNA, the rate of
holin protein translation by host ribosomes, the rate of holin
insertion into the plasma membrane and the threshold holin
concentration required for membrane hole formation.
The impact of these factors on LT variation is investigated
using a stochastic gene expression model. LT is modelled as
the first-passage time (FPT) for cellular holin copy numbers
to cross a critical threshold in the stochastic model (figure 2).
Exact analytical formulae for the FPT statistical moments predict how LT mean and coefficient of variation (CV) change in
response to experimental alterations of the holin transcription
efficiency, translational efficiency and the threshold concentration of holin required for lysis. Some of these predictions
are confirmed through analyses of LT distribution data
obtained by Dennehy & Wang [34] for wild-type phage, and
phages carrying mutations in either the holin sequence or the
pR 0 promoter. We discuss the impact of these findings on
(2:1)
2
to the threshold X 1 assuming x(0) ¼ 0. The CV , defined as
CV2FPT :¼
kFPT2 l kFPTl2
(2:2)
kFPTl2
is used to quantify the randomness in FPT, where the symbol k:l
denotes expected value. Let, Ti be independent and identically
distributed random variables denoting the time interval between
the i 2 1th and ith mRNA transcription event from the promoter.
We consider gene expression in the burst limit: transcribed
mRNAs degrade instantaneously after producing a translational
burst of protein molecules [37–40]. Consistent with measurements [41], we assume protein bursts are geometrically
distributed with a mean translational burst size b ¼ kp/gm.
Here, kp denotes the protein translation rate per mRNA and gm
is the mRNA degradation rate. Thus, protein count
x(t) ¼
n
X
Bi , b :¼ kBi l, 8i 1
(2:3)
i¼1
is an accumulation of independent, identical geometric bursts Bi,
n is the number of transcription events in [0, t] and the inter-burst
arrival time distribution is given by Ti.
3. Analytical calculation of first-passage time
moments
First, the minimum number of transcriptional events N
required for x(t) to reach the threshold X is quantified. Let,
!
9
n
X
>
>
N ¼ inf n 1:
Bi X , n [ {1, 2, . . . } ) >
>
>
=
i¼1
(3:1)
!
>
n
X
>
>
>
Probability(N n) ¼ Probability
Bi X : >
;
i¼1
Then, the statistical moments for N are given by (see
electronic supplementary material, appendix C)
kNl ¼
X
þ 1 and
b
kN 2 l kNl2 ¼
X1þb
:
b b
(3:2)
The average number of transcriptional events required for
lysis is the lysis threshold (X ) divided by the mean translational burst size (b) plus one. The plus one arises from the
fact that, even if the threshold is close to zero, at least
one transcription event is required for x(t) X. Given N
(the number of burst events) and Ti (the time interval
between bursts), time taken to reach the threshold is
9
N
P
>
=
FPT ¼ T ) kFPTl ¼ kNlkT l
i
i
i¼1
and kFPT2 l kFPTl2 ¼ kNl(kTi2 l kTi l2 ) þ (kN 2 l kNl2 )kTi l2 :
>
;
(3:3)
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(a)
(b)
decreasing transcription
protein level/cell
decreasing translation
or increasing threshold
time
mean first-passage time
2
Figure 2. Predicted CV versus mean trends for the FPT (arbitrary units) in the constitutive transcription model. LT is captured by the FPT for protein copy numbers
to cross a critical threshold (a). Stochastic expression ensures that the threshold is reached at different times in individual cells, causing cell-to-cell variations in lysis
timing. Decreasing holin transcriptional efficiency increases the mean FPT without altering the CV (b). By contrast, decreasing the holin translational efficiency or
increasing the lysis threshold increases the mean FPT but decreases the CV. (Online version in colour.)
Below, distribution for Ti is determined for two different mRNA production processes: constitutive and bursty
transcription.
Next, a more sophisticated mRNA production process is
considered, where the promoter switches between different
transcriptional states.
3.1. Constitutive transcription model
3.2. Stochastic promoter switching model
Assuming mRNA production is a Poisson process with rate
km, Ti is an exponential random variable with mean 1/km,
and (3.3) yields
b2 þ X þ 2bX
X
1
CV2FPT ¼
þ
1
and
kFPTl
¼
, (3:4)
b
km
(b þ X)2
where CV2FPT is the FPT CV2. For X/b 1 (number of transcriptional events required for lysis is large), (3.4) simplifies to
CV2FPT ¼
1 þ 2b
X
and
kFPTl ¼
X
,
bkm
b¼
kp
:
gm
(3:5)
Note that under similar assumptions, we obtain
CV2FPT ¼
1þb
X
and kFPTl ¼
X
,
bkm
(3:6)
for deterministic arrival of bursts (i.e. kTi2 l kTi l2 ¼ 0) in (3.3).
Thus, for b 1, randomness in the FPT is halved for fixed time
intervals between protein bursts compared with exponentially
distributed intervals. When b 1, we obtain the Poisson limit
CV2FPT ¼ X1 . The results above show that for a given X, CV2FPT
and kFPTl can be independently controlled via holin translation rate (kp) and transcription rate (km). Desired FPT mean
and CV can be set by choosing
b¼
CV2FPT X 1
2
and km ¼
X
:
bkFPTl
(3:7)
Equation (3.5) predicts how FPT CV2 changes with mean as km,
b or X are experimentally altered (figure 2):
— CV2FPT is inversely related to kFPTl for perturbations in the
lysis threshold. More specifically, increasing X will
increase kFPTl but decrease CV2FPT .
— For alterations in translational efficiency kp (and hence,
the mean translational burst size b), increase in kFPTl
comes with a decrease in CV2FPT .
— Changes in the transcriptional efficiency km alter kFPTl
without affecting CV2FPT .
Promoter transitions between active and inactive states have
been well documented in various genes [42 –46]. In the stochastic formulation, the promoter resides in active and
inactive states for exponentially distributed time intervals
with means 1/koff and 1/kon, respectively, where kon and
koff represent transition rates (figure 3, top right). From the
active state, mRNA production is a Poisson process with
rate km. As before, we assume transcribed mRNAs degrade
instantaneously after producing a geometrically distributed
burst of protein molecules of mean size b. Let, Ti be independent and identically distributed random variables denoting
the time interval between the i 2 1th and ith transcriptional
event from the active state. Then, the mean and variance
for the inter-burst time interval is given by
kTi l ¼
koff þ kon
km kon
kTi2 l kTi l2 ¼
and
2km koff þ (kon þ koff )2
,
2 k2
km
on
(3:8)
for all i 1 (see electronic supplementary material, appendix D).
Using (3.3), (3.8) and assuming X/b 1, we obtain
X koff þ kon
(3:9a)
kFPTl ¼
b
km kon
and
kFPT2 l kFPTl2 ¼
X 2km koff þ (kon þ koff )2
2 k2
b
km
on
2
X 1 þ b koff þ kon
þ
:
b b
km kon
(3:9b)
As expected, (3.9) reduces to (3.5) for constitutive
transcription, i.e. kon koff and promoter is always in the
active state.
In the absence of promoter switching, decreasing transcriptional efficiency is predicted to increase kFPTl without affecting
CV2FPT (figure 2). However, inclusion of stochastic promoter
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first-passage time CV2
event threshold
3
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event timing
histogram
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4
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first-passage time CV2
mRNA
km
changes in km
kon
J. R. Soc. Interface 11: 20140140
changes in koff
first-passage time CV2
changes in kon
koff
mean first-passage time
mean first-passage time
2
Figure 3. Predicted CV versus mean trends for the FPT (arbitrary units) in the stochastic promoter switching model. Schematic of the stochastic gene expression
model where promoter randomly toggles between active and inactive states with rates kon and koff (top right). Transcription only occurs from the active state at a
rate km. Qualitative trends for the FPT CV2 as a function of the mean, for alterations in km (top left), koff (bottom right) and kon (bottom left). Trends obtained from
analysis of (3.9). (Online version in colour.)
switching complicates the relationship between kFPTl and
CV2FPT . Qualitative CV2FPT versus kFPTl trends for perturbations
in the transcriptional efficiency (measured by either km, kon or
koff ) are shown in figure 3. Interestingly, modulating the transcriptional efficiency can have different effects on CV2FPT
depending on the underlying parameter that is being altered.
For example, decreasing kon or km will both increase kFPTl,
but the former is predicted to increase CV2FPT , while the latter
decreases CV2FPT (figure 3). As in the previous section, FPT
CV2 changes inversely with the mean for alterations in X or b.
4. Measured lysis time variation consistent with
model predictions
If LT variation is indeed a result of noisy holin expression,
then randomness in event timing should vary with mean as
predicted by (3.5) or (3.9), i.e. CV2FPT should decrease with
increasing kFPTl as the lysis threshold is perturbed. Before
we test this prediction, quantification of CV2FPT and kFPTl
from single-cell LT measurements is discussed.
4.1. Connecting lysis time to first-passage time
LT can be related to the FPT as LT ¼ FPT þ TpR 0 , where TpR 0 is
the time for the onset of transcription from the late promoter.
As TpR 0 is shown to be uncorrelated with FPT [35],
kFPTl ¼ kLTl kTpR0 l
CV2FPT ¼
s2LT
and
s2T pR0
(kLTl kT pR0 l)2
,
(4:1)
where sI denotes the s.d. in I [ {LT, T pR0 }, and kT pR0 l is taken
to be 15 min [29,30]. As TpR 0 variation only accounts for a
small fraction of LT variability [34,35], we assume sT pR0 ¼ 0.
Non-zero values of sT pR0 do not change the CV2FPT versus
kFPTl trend described below. For wild-type l (see electronic
supplementary material, appendix B),
kLTl 65 min and
sLT 3:5 min,
(4:2)
CV2FPT 0:005:
(4:3)
which implies from (4.1),
kFPTl 50 min and
As we ignore TpR 0 variation in these calculations, the reported
values for CV2FPT are an upper bound. Assuming constitutive
transcription from the wild-type pR’ and a lysis threshold X
of 1500 holin molecules [36], we obtain the following estimates for the translational burst size and the transcription
rate from (3.5) and (4.3)
b 3 and
km 10 min1 :
(4:4)
Noisy holin expression with these values is sufficient to
generate intercellular LT differences similar to data (figure 4).
These parameter values are consistent with previous estimates
of holin transcription/translation efficiencies measured using
bulk assays [36].
4.2. Mutations in the holin sequence
Mutations in the holin amino acid sequence can both increase
and decrease LT mean and s.d. compared with wild-type
[34]. These mutations may alter holin self-affinity or membrane
insertion rate, thus affect the total cellular holin threshold
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simulated lysis
time distribution
first-passage time CV2
holin level/cell
1000
0.06
0.05
0.04
0.03
0.02
0.01
WT
0
10
500
10
20
30
lysis
start of holin
induction expression
40
50
time (min)
60
70
80
mean lysis
time
Figure 4. Stochastic holin expression can account for LT differences in wildtype bacteriophage l. Representative single-cell trajectories of holin copy
numbers obtained from Monte Carlo simulations of the stochastic model
with parameter estimates (4.4). LT is captured by the FPT for holin cellular
level to cross a critical threshold (1500 molecules). Holin expression is
assumed to begin 15 min after lysis induction. LT distribution obtained
from 10 000 trajectories is shown on top. Noisy gene expression is sufficient
to generate cell-to-cell LT differences similar to data in figure 1. The simulated LT distribution has a mean of 65 min and an s.d. of 3.5 min as in
figure 1. Monte Carlo simulations performed using the stochastic simulation
algorithm [47]. (Online version in colour.)
required for lysis. For example, mutant holin with a stronger
membrane binding affinity than the wild-type will have a
larger fraction of the expressed holin present in the plasma
membrane. Consequently, the critical holin concentration
required for hole formation will be reached with less total
holin expression. In our model, this corresponds to having a
lower threshold X, and hence, a shorter LT compared with
the wild-type.
Single-cell LT measurements for six holin amino acid
sequence mutations (see electronic supplementary material,
appendix B) reveal that shorter LTs are associated with
higher a CV2 (figure 5). Thus, as predicted by a stochastic
model of holin expression, results show an inverse relationship between CV2FPT and kFPTl for changing X.
It is important to point out that the inverse correlation
between CV2FPT and kFPTl in figure 5 is inconsistent with a
model where LT differences arise due to cell-to-cell variations
in holin production rate. Consider an alternative model,
where holin molecules are produced at a rate k, in which k
has a certain distribution across the cell population. Cellto-cell differences in k would reflect intercellular variation
in the abundances of host ribosomes and/or RNA polymerases, which arise due to differences in cell volume, cell
cycle or stochastic expression of these enzymes. As holin
levels build up at different rates inside individual cells,
some cells would lyse earlier than others generating a LT
distribution similar to data. However, in this case,
FPT ¼
X
1
) kFPTl ¼ X
k
k
(4:5a)
k1/k2 l k1/kl2
,
k1/k2 l
(4:5b)
kl
and
CV2FPT ¼
20
30
40
50
mean first-passage time (min)
60
70
Figure 5. Mutant l phages exhibit LT variation consistent with model predictions. FPT CV2 as a function of the mean FPT for l phages carrying holin
sequence mutations. Each data point corresponds to a specific l variant with
a holin amino acid sequence mutation and is calculated from the LT distribution using (4.1). Point labelled WT corresponds to wild-type l. These
mutations alter the lysis threshold (X ), and CV2FPT decreases with increasing
kFPTl as predicted by the model (equation (3.5) shown as black line). Error
bars are 95% CIs for CV2FPT obtained using bootstrapping. (Online version
in colour.)
and CV2FPT is independent of X. Hence, CV2FPT is predicted to
be uncorrelated with kFPTl for alterations in X. The inverse
trend between CV2FPT and kFPTl obtained from experimental
measurements (figure 5) suggests that LT differences are
due to stochasticity in holin expression rather than difference
in holin production rates between cells.
4.3. Mutations in the late promoter
Previous work measured LT variation for four l variants carrying mutations in the late promoter pR 0 TATA box region
[34]. Only two of these mutants showed a significant decrease
in promoter strength, which increased both the LT mean and
CV2 (see electronic supplementary material, appendix B).
For example,
kFPTl 95 min and
CV2FPT 0:03,
(4:6)
for promoter mutant SYP208 [34]. Note a twofold increase in
kFPTl, and a sixfold increase in CV2FPT , compared with their
respective values in wild-type l (see (4.3)). The observed
increase in CV2FPT cannot be explained by a reduced transcription rate km in the constitutive model, as in this case CV is
invariant of km (figure 2). However, as discussed below, these
results can be explained by a model that includes random promoter transitions between active and inactive states (figure 3).
Results in §3.2 show that in the stochastic promoter
switching model (figure 3, top right), reducing transcription
from the active state (decreasing km) increases kFPTl but
decreases CV2FPT . By contrast, enhancing the stability of the
inactive state (decreasing kon) increases both kFPTl and
CV2FPT . Finally, for decreasing active-state stability (increasing koff ), CV2FPT exhibits an inverted U-shaped profile with
increasing kFPTl. Thus, an increase in both kFPTl and CV2FPT
in response to a reduced holin transcription is consistent
with promoter mutations decreasing kon or increasing koff in
the promoter switching model (figure 3). A recent study
has shown that promoter mutations in the TATA box region
primarily modulate the transcription burst frequency kon [48].
Assuming constitutive transcription from the wild-type pR 0
(kon koff ), mutations that significantly decrease kon will
induce additional promoter switching noise in holin
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5. Discussion
(i) LT was determined by the FPT for cellular holin levels
to reach a critical threshold. Exact analytical formulae
for the FPT mean and CV2 were derived for different
stochastic gene expression models. These formulae are
given by (3.4) and (3.9) for the constitutive transcription model and the stochastic promoter switching
model, respectively.
(ii) FPT derivations reveal that for a fixed lysis threshold,
LT CV2 is minimized by reducing the translation burst
size b. As discussed later, mechanisms for reducing
b are embedded in l’s lytic pathway in order to
ensure precision in lysis timing.
(iii) Using (3.4) and (3.9), predictions on how biologically
meaningful parameters (transcription rate, translation
rate, lysis threshold, etc.) impact LT statistics were generated. These predictions are summarized in figure 2
(for the constitutive transcription model) and figure 3
(for the stochastic promoter switching model).
(iv) Analysis of data from [34] confirms model predictions.
In particular, when the threshold for lysis is altered
through holin sequence mutations, increases in mean
LT are associated with decreases in LT CV2. By contrast, when holin transcription is altered through
holin promoter mutations, increases in mean LT are
associated with increases in CV2.
Below, we discuss the implications of some of these findings
in further detail.
5.1. Independent control of lysis time mean
and variation
Formulae for the FPT moments reveal how transcriptional/
translational efficiencies independently modulate the mean
and variation in lysis timing. For constitutive transcription,
FPT CV2 increases linearly with the translational burst size b,
but is independent of the transcription efficiency (figure 2).
Precision in the FPT can be achieved by setting b sufficiently
small. When b 1, the limit of noise suppression for a given
lysis threshold X is attained: CV2FPT ¼ X1 . This limit corresponds to a Poisson arrival process for holin production.
Once b is chosen for a desired CV2FPT , holin transcription
efficiency can be adjusted to have any mean FPT.
5.2. An incoherent feed-forward circuit minimizes lysis
time stochasticity
The l S gene encodes two proteins with opposing function:
holin and antiholin [50–52]. These proteins are produced in
2 : 1 ratio, i.e. for every two holins, there is one antiholin molecule. These two proteins differ only in that antiholin has an
extra two residues, methionine and lysine, at the N-terminus.
The positively charged lysine prevents antiholin from accessing holin’s characteristic three transmembrane domain
configuration in the E. coli inner membrane [53]. Experimental evidence suggests that antiholin binds holin and
sequesters it in the cytoplasm, preventing it from participating in hole formation [54,55]. This creates an incoherent
feed-forward circuit in l’s lytic pathway [56].
Preliminary analysis provides some clues on the functioning of this feed-forward circuit. Let, holin and antiholin
translation from the mRNA be Poisson processes with rates
ka and kah, respectively. Then, the probability of producing i
holin and j antiholin molecules from the mRNA in time interval [0, t] will be
(ka t)i (kah t)j
exp ( (kah þ ka )t),
i!
j!
i, j [ {0, 1, 2, . . . }:
(5:1)
We denote by x and y, the total number of holin and antiholin
molecules synthesized by an individual mRNA before it
decays. Random variables x and y would be correlated,
because an mRNA that lives longer than average by random
chance would create more of both holin and antiholin. Assuming each mRNA lives for an exponentially distributed time
interval with mean 1/gm, the joint probability distribution of
x and y is
Probability (x ¼ i, y ¼ j)
ð1
(ka t)i (kah t)j
exp ( (kah þ ka þ gm )t)dt
¼ gm
i!
j!
0
j
¼
kai kah gm (i þ j)!
i!j!(ka þ kah þ gm )1þiþk
,
i, j [ {0, 1, 2, . . . }:
(5:2)
6
J. R. Soc. Interface 11: 20140140
The timing of cellular events, such as mitosis, apoptosis,
developmental differentiation and reversion from latency,
frequently depends on the concentrations of essential regulator molecules. Stochastic gene expression causes variation in
regulator molecule concentrations even for identical cells
induced at the same time. Consequently, molecular threshold
concentrations will be reached at different times in different
cells, giving rise to significant variation in the timing of critical cellular events across a homogeneous cell population. The
holin lysis system of bacteriophage l provides a simple
system to study how event timing is regulated at the singlecell level. The main contributions of this work are as follows:
Note that b ¼ kp/gm is proportional to the translation efficiency kp, where gm is the mRNA degradation rate. In l, all
lysis and structural proteins are translated from the same polycistronic mRNA transcript. Hence, reducing b by making kp
small is a more viable option than increasing gm, as the latter
would affect the translational burst sizes of structural proteins.
S gene, which encodes holin, is reported to have a low translational efficiency [36]. As discussed later, holin translation bursts
are further reduced through production of a second protein,
antiholin, from the S mRNA. Given the direct relationship
between CV2FPT and b, mechanisms in place for minimizing b
are essential for buffering randomness in lysis timing.
Our results on the FPT are complementary to previous
work quantifying protein copy number variations in stochastic gene expression models. These studies have also shown
that a reduced translation efficiency is critical for minimizing
stochastic fluctuations in protein levels [13,38]. However,
there are qualitative differences in the formulae for the FPT
and protein copy number statistical moments. For example,
the FPT CV2 is proportional to the translation efficiency kp.
By contrast, for b 1, the steady-state protein copy number
CV2 is independent of kp [13,38,49].
rsif.royalsocietypublishing.org
expression. Consequently, weaker non-constitutive late promoter expression makes cell lysis longer, and more unpredictable
compared with wild-type l.
Downloaded from http://rsif.royalsocietypublishing.org/ on June 16, 2017
b¼
1 X
i
X
i¼0
j
(i j)
j¼0
kai kah gm (i þ j)!
i!j!(ka þ kah þ gm )1þiþk
(5:3a)
and
ka kah
gm
if ka kah ,
(5:3b)
which decreases with increasing antiholin production rate kah.
Recall from (3.5), that for a fixed X and kFPTl, CV2FPT is minimized by reducing b (see equation (3.5)). Thus, antiholin may
enhance precision in lysis timing by lowering b, and making the
active holin production process less bursty. Removing antiholin
(kah ¼ 0) would increase b, which is predicted to decrease
kFPTl but increase CV2FPT (equation (3.5)). Indeed, l variant
IN62 (no antiholin expression) lyses 10 min faster but with a
twofold higher CV2FPT compared with wild-type [34]. Our
results show that as in other systems [57–59], antiholinmediated incoherent feed-forward loop is a key regulatory
motif in l’s lytic pathway for buffering LT from stochastic
gene expression.
5.3. Evolution of lysis timing
In life-history theory, the timing of lysis is viewed as a trade-off
between present and future reproduction. Assuming a linear
rate of progeny assembly [60], postponing lysis increases progeny number, but delays infection of new hosts. In other
words, increasing fecundity also increases generation time.
Several studies have theoretically and experimentally examined the optimal timing of lysis [60–66]. These studies reach
similar conclusions: when host density and quality are high,
early lysing genotypes will be selectively favoured as they
will have more grandchildren than competing genotypes.
Conversely, poor host quality and low host densities should
favour longer LTs.
We expect phage fitness will be maximized if the LT CV is
minimized around the optimum LT for any particular environment. Indeed, we find that wild-type l has a lower LT CV and a
higher fitness than all laboratory-generated mutant strains
[34,60]. Furthermore, l’s lytic pathway exhibits the ingredients
required for precision in timing: low S gene translational efficiency [36] and antiholin synthesis to further reduce the
active holin translational burst size. At first, production of
antiholin seems energy inefficient as it inactivates holin, translated from the same mRNA. Our analysis and data suggest that
antiholin may reduce LT variations and is reminiscent of
the classical trade-off between energy efficiency and noise
buffering in cellular systems [67,68].
Our results predict how the mean LT should evolve
under selection, while minimizing LT CV. In principle,
mean LT can be modulated through mutations in the following regions: the pR 0 promoter sequence, the holin leader
sequence and the holin coding sequence. Because of the polycistronic nature of the pR 0 promoter sequence, we expect it
can be ruled out. Presumably, mutations in pR 0 will have undesirable pleotropic effects on the expression of l’s structural and
regulatory proteins. We predict that selection for shorter LTs
5.4. Burst size implications of lysis timing variation
Given that progeny are assembled at a linear rate, we expect
that differences in LT will be reflected by differences in progeny burst size (total progeny produced in a single
infection). Consequently, we should observe a direct relationship between LT stochasticity and burst size stochasticity.
Seminal work by Delbrück showed that phage burst size
can vary significantly from cell-to-cell, even after controlling
for cell size [69]. Interestingly, measured viral burst size CV
is an order of magnitude larger than the LT CV (see electronic
supplementary material, appendix E). Recent single-cell
measurements also show a broad viral burst size distribution
for vesicular stomatitis virus (VSV), an animal virus that can
occasionally infect humans. Data show a 300-fold cell-to-cell
difference in the number of VSV progeny produced per cell,
and these differences cannot be attributed to viral genetic
variation [70,71]. Phage l provides an ideal system to study
how viral burst size variations arise from LT variations and
noisy expression of structural genes. Understanding how
non-genetic factors affect viral fitness at the single-cell level
has important implications for health and disease.
6. Summary
In summary, cell-to-cell LT differences arise due to the stochastic transcription and translation of the lysis timekeeper, holin.
We derive analytical formulae for the mean and CV2 of the
time it takes for holin to reach a critical concentration in
the inner membrane. These formulae are used to generate
predictions regarding the impact of biologically relevant parameters, such as transcription and translation rates, on LT
mean and CV2. We evaluate these predictions using previously
published data on single-cell variation in LT [34]. Our analysis
confirms that stochastic holin expression can account for variation in LT. Furthermore, we find that the holin lysis system
contains several features designed to minimize noise.
Acknowledgements. We thank Andrew Rutenberg, Gillian Ryan, IngNang Wang, Pak-Wing Fok, Pavol Bokes, Ryan Zurakowski, Khem
Ghusinga and Ryland Young for discussion on some of the ideas
contained herein.
Funding statement. J.J.D. was supported by grants from the Professional
Staff Congress of the City University of New York and the National
Science Foundation (Division of Environmental Biology Awards
0804039 and 1148879, and Division of Molecular and Cellular
Biosciences Award 0918199). A.S. was supported by the National
Science Foundation grant no. DMS-1312926, University of Delaware
Research Foundation (UDRF) and Oak Ridge Associated Universities
(ORAU).
7
J. R. Soc. Interface 11: 20140140
(i.e. when host quality and density is high) will favour
mutations that increase translational efficiency, and not those
that lower the threshold concentration of holin required for
lysis (X ). From (3.5), we see that, when translational burst
size (b) is small, CV2 1/X, and more sensitive to changes in
X rather than b. So decreasing mean LT by lowering X will
increase CV2. By contrast, selection for longer LTs will favour
mutations that increase X because this will decrease LT CV2,
while mutations decreasing translational efficiency will not
substantially change CV2. These predictions can be tested
by genetic sequencing of l phage following experimental
evolution under high or low host densities.
rsif.royalsocietypublishing.org
Consider holin–antiholin interaction in a limiting case: antiholin rapidly binds to holin making holin inactive in an
irreversible reaction. Then the active holin translational burst
size (number of active holin made per mRNA) would be x 2 y
(if x . y) or 0 (if x y). Equation (5.2) yields the following
mean translational burst size b for active holin
Downloaded from http://rsif.royalsocietypublishing.org/ on June 16, 2017
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