Death rate during the exponential growth of E. coli

Loïc Brot1, Zengru Di2, Romain Morichon3, Annie Munier-Godebert4, Bertrand
Roehner5, Joëlle Sobczak-Thépot6, Angélique Vinit7

Version of 10 December 2021

Very provisional. Comments are welcome.

Key-words: Escherichia coli, cell division, mortality rate, exponential phase,
stationary phase

1: Centre de Recherche Saint Antoine (CRSA), Sorbonne Université, 75012 Paris,
France. Email<EMAIL_ADDRESS>2: School of Systems Science,
Beijing Normal University, China. Email<EMAIL_ADDRESS>3: Centre Cytométrie
Imagerie Saint-Antoine (CISA). Sorbonne Université, 75012 Paris, France Email:
<EMAIL_ADDRESS>4: Centre Cytométrie Imagerie Saint-
Antoine (CISA). Sorbonne Université, 75012 Paris, France Email:
<EMAIL_ADDRESS>5: Institute for Theoretical and High Energy
Physics (LPTHE), Pierre and Marie Curie campus, Sorbonne University, Paris,
France. Email<EMAIL_ADDRESS>6: Sorbonne Université, National
Institute for Health and Medical Research (INSERM). Email<EMAIL_ADDRESS>7: Centre Cytométrie Imagerie Saint-Antoine (CISA). Sorbonne Université, 75012
Paris, France Email<EMAIL_ADDRESS>
Abstract

So fast is the growth of a culture of E. coli that it led researchers to
overlook a possible death rate. As a matter of fact, the experiments done in
the first half of the 20th century were unable to detect any mortality. It is
only at the beginning of the 21th century that positive evidence emerged which
confirmed the existence of a sizeable (albeit small) mortality. In the present
paper this mortality was measured at successive 45mn time intervals from early
exponential growth into the stationary phase. Done with a flow cytometer, the
successive measurements of the ratio dead/living also provided an estimate of
the standard deviation of the mortality rate. In a forthcoming experiment we
are planning to focus more closely on the stationary phase in order to find
out if its mortality rate is lower or higher than in the exponential phase.

##  1 Introduction

###  1.1 Reasons for measuring the mortality rate of bacteria

In any population, whether human or non-human, the birth and death rates are
the two most basic variables which define its evolution. Historically, these
were the first variables which could be measured annually (i.e. independently
of the decenial censuses). One should not think it was an easy task. For
instance, in the United states, in spite of the fact that the earliest data,
namely those for Massachusetts, go back to the 19th century, it is only around
1930 that the so-called birth and death “Registration Areas” were able to
cover all the states. Basic as they are, these rates nevertheless continue to
raise unsolved questions. Here are two examples. Regarding birth rates,
successive prediction attempts put foward in past decades regularly proved
wrong by a wide margin (of 15% or more).

For death rates the situation is different because the causes of death are
well known which means that, in contrast to birth rates, there are no
unexplained fluctuations. What remains mysterious however is the overall
pattern of death rates as a function of age. Over the age of 15 years, the
death rate follows the well defined empirical law discovered by Benjamin
Gompertz in the 19th century (Gompertz 1825). This law states that the death
rate is an exponential function of age; basically it doubles when age
increases by 10 years. For ages under 15 there is also a well-defined law; it
states that the death rate $\mu$ decreases with age like an hyperbola
function: $\mu\sim 1/t^{\alpha}$, where $t$ is the age and $\alpha$ an
exponent which is close to 1. In previous studies (Berrut et al. 2016, Bois et
al. 2019) we have shown that this law also holds for non-human multicellular
organisms. It is of great interest to see if it holds also for unicellular
organisms for the following reason. In multicellular organisms one can
distinguish separate organs, with death usually occurring when one or several
crucial organ fail. In bacteria the situation is less clear. Thus, this test
may give us information on the internal organization of unicellular organisms.
However, before turning to age-specific death rates we wish to give fairly
accurate estimates of the global death rate.

###  1.2 Outline of the paper

The paper proceeds as follows.

$\bullet$ In the next section we describe the methods and results obtained in
previous studies.

$\bullet$ In section 3 we carefully discuss and define the death rate of a
population of bacteria.

$\bullet$ In section 4 we explain the protocol of our measurements, we present
our results and we discuss how their accuracy can be improved.

$\bullet$ In the last section we discuss related perspectives.

##  2 Previous studies of death rates of bacteria in exponential growth

The investigation of the death rate of bacteria in their phase of exponential
growth has been going on for over a century. This long-lasting sustained
interest underlines that the question is of importance. At the same time the
fact that it is not yet completely solved suggests that this measurement still
represents an experimental challenge. That is mostly due to the fact that the
death rate is small. Novadays we know that for E. coli the death rate is of
the order of 1 per 1,000 living cells and per hour. In order to identify the
trend of the death rate across the exponential phase and in the early
stationary phase requires an even higher accuracy because the error bars must
be smaller than the effect of the trend itself.

Actually the very definition of the death rate is not obvious. Even if one
leaves apart the question of the biological meaning of the notion of “death”,
it is not obvious to adapt standard demographic notions to a population which
doubles every 20mn.

In the following subsections it will be seen that there are basically three
methods for distinguishing dead from living cells.

$\bullet$ The CFU (Colony Forming Unit) method. The fact that this traditional
method does not have the required accuracy is clearly shown in Wilson (1922),
a paper which will be discussed in more details below.

$\bullet$ The second method consists in following all individual cells
belonging to one cohort over as many generations as possible. Introduced in
1932, this method was vastly improved in a paper of 2005 by Stewart et al.

$\bullet$ Finally, the third method is based on Flow Cytometry (FC). It is
this method that will be used in the present paper.

###  2.1 The Wilson paper (1922)

In his paper Wilson called into question the dogma that in a young broth
culture (up to 24 hours) all bacteria are living. Here is what he writes at
the beginning of his paper.

“On looking up the literature it was found that of the many observers [the
author cites 9 papers published between 1898 and 1920] who had made a
comparison of the two counts [namely the total number of cells that can be
counted in a culture, whether dead or alive on the one hand and the number of
viable cells defined as cells that are able to fission on the other hand], the
discrepancy was passed over with little comment. Any discrepancy was
attributed to “errors.”

In a 41-page long paper the author examines one by one all successive
operations required in counting procedures. He tries to make them as rigorous
as possible and he estimates the remaining error margin. For instance dilution
before counting is an operation which requires great care but nevertheless
cannot be made very precise. In fact, the author was facing an impossible
task. Such global counting techniques were beset by too many uncertainties.
Therefore, it is not surprising that Wilson arrives at vague and disappointing
conclusions (p.444).

“It seems that in cultures of Bact. suipestifer there is a normal death rate
even during the period of maximum growth. Its extent will vary from culture to
culture. In some it is as high as 43%, in others it is only 20% or 10%, while
finally in a few it is for a short period actually nil.

###  2.2 Paper of Kelly and Rahn (1932)

The first lines of the paper provide a clear statement of why this is a key-
issue.

“It has been assumed by many bacteriologists that during the period of rapid
growth, in a satisfactory culture medium, some bacteria will die in spite of
good food and favorable environment. No doubt this assumption was derived from
an analogy with populations of higher forms of life, of which a number of
individuals are known to die before they reach the reproductive age even with
good care.” (Kelly et al. 1932, p.147)

The observation of individual cells pioneered by Kelly and Rahn represented a
breakthrough. It is true that they were unable to see any death in the
exponential phase but that was only because their sample of 1,766 cells was
too small. Their methodology was sound and opened the door to further
observations with larger samples. Yet, this did not happen until 73 years
later. Indeed, in 2005 this investigation was resumed. Thanks to modern
computerized counting techniques and a sample some 20 times larger namely
35,000 E. coli cells, the experiment revealed some 16 deaths; see Stewart et
al. (2005) and the discussion below. .

Using a method pioneered by J. Orskov (1922), Kelly and Rahn followed the
replication of individual bacteria (and one yeast species) on a solid medium.
They recorded the family trees spanning 4 generations. Altogether they
observed 1,766 divisions: 977 of Bacterium aerogenes, 325 of Bacillus cereus,
464 Saccharomyces ellipsoideus. On average the time intervals between fissions
of Bacterium aerogenes was 30 mn but with great inter-individual fluctuations
(coefficient of variation $\sigma/m$ close to 100%).

The observations led the authors to the following conclusions.

$\bullet$ There was not a single instance where a cell became dormant.

$\bullet$ In the very words used by the author, there was not a single case of
“infant mortality”, that is to say a cell which died after division.

The first conclusion is reassuring because, even today, researchers often
worry about the risk of mistaking dormant cells for dead cells (see the
discussion at the begining of Garvey et al. 2007).

The second conclusion is also interesting for it shows that if such an infant
mortality exists (as is indeed observed in experiments done in the past two
decades as reported below) then its rate is lower than: 1/1766 = 0.56 per
1,000. (for a time interval of 30mn). This is a rough estimate obtained by
bulking together all three species. Estimates detailed by species (at least
for the two largest) are given in Table 1.

Whereas the measurement of Stewart et al. (2005) relies on the observation of
individual cells, a paper of 2008 by Fontaine et al. relies on a global (not
individual) observation. It is to the discussion of these modern
investigations that we turn now.

Table 1: Landmark papers about bacterial death rates in the exponential phase

$\matrix{&\hbox{Year}&\hbox{Paper}\hfill&\hbox{Method}\hfill&\hbox{Sample}&\hbox{Number}&\hbox{Death
rate (dr)}\cr&&\hbox{}&\hbox{}&\hbox{size}&\hbox{of deaths}&\hbox{per
1,000}\cr\vrule
height=0.0pt,depth=5.0pt,width=0.0pt&&\hbox{}&\hbox{}&\hbox{}&\hbox{}&\hbox{and
per hour}\cr\hrule\cr\vrule
height=15.0pt,depth=0.0pt,width=0.0pt1&1922&\hbox{Wilson}\hfill&\hbox{Global}\hfill&\hbox{several
sets}&\hbox{unreliable}&\hbox{unreliable}\cr 2&1932&\hbox{Kelly
(1)}\hfill&\hbox{Individual}&\hfill 733&<1&\hbox{dr}<4.1\cr 3&1932&\hbox{Kelly
(2)}&\hbox{Individual}&\hfill 420&<1&\hbox{dr}<1.4\cr
4&2005&\hbox{Stewart}\hfill&\hbox{Individual}&\hfill 35,049&\hfill
16&1.5\cr\vrule
height=0.0pt,depth=5.0pt,width=0.0pt5&2008&\hbox{Fontaine}\hfill&\hbox{Global}\hfill&\hfill
1,000,000&\hfill 700&0.7\cr\hrule\cr}$

Notes: “Global” means measurement performed on a large number of cells in
liquid medium. For this global measurement it is the technique of flow
cytometry which brought about a breakthrough and allowed reliable
measurements. The following bacteria and yeast were investigated (in
parenthesis is the length in minutes of the reproduction cycle). 1: Bact.
suipestifer and other species, 2: Bacterium aerogenes (30mn), 3: Saccharomyces
ellipsoideus (105mn), 4: E. coli (30mn), 5: E. coli (30mn). Sources: Based on
the papers cited in the third column.

###  2.3 The paper by Stewart et al. (2005)

In the Kelly and Rahn (1932) experiment successive divisions were followed
over 4 generations, a process which from each single initial cell produced
$2^{4}=16$ cells. In the Stewart et al (2005) experiment up to 9 generations
were followed, a process through which each initial cell gave rise to
$2^{9}=512$ E. coli cells. Time-lapse images were taken and analyzed
automatically thanks to dedicated software. As 94 colonies were analyzed this
led to a total of 35,049 divisions.

The criterion used for the definition of death was immobility combined with no
division. Some 16 cell death were observed. Unfortunately their sizes were not
included in the publication because the main purpose of the paper was a
different issue.

###  2.4 Paper of 2008 by Fontaine et al. (FC measurement)

In the technique that is used here the cells are not monitored individually.
Instead, the recourse to flow cytometry allows global estimates. Stained dead
cells, or more precisely those cells whose breached membranes allow the stain
to drift into the cytoplasm, are counted thanks to a flow cytometer (FC). In
such a device the light of a laser is diffused by the cells when they move
through the beam, then received by a sensor and amplified by a
photomultiplicator and finally analyzed by a computer software algorithm. Flow
cytometry allows many characteristics of the cells to be identified and
recorded. Here this technique is used to count stained or fluorescent dead
cells.

Flow cytometry began to be used in the 1950s and was really a game changer. It
replaced the successive manipulations needed in the CFU method.

Although FC is the key of the measurement method, a number of additional
verification tests are required to ensure that what is measured by the device
is indeed the appropriate death rate.

##  3 Definition of the global death rate of bacteria

In previously published mortality data for bacteria (for instance in Steward
et al. 2005) their real meaning and definition, for instance with respect to
the time interval, was not clearly indicated. That is why we discuss this
point in some detail.

First, we recall the standard definition of the death rate in human
demography. Then we examine how this definition should be adapted to apply to
organisms whose life duration is of the order of 20 mn.

###  3.1 Global death rates in human populations

In Appendix A we recall the reasons which lead to the definition of the death
rate as accepted in human demography.

Consider a population of size $x$ that one observes during a time interval
$\Delta t$. If $\Delta y$ denotes the number of deaths during $\Delta t$, the
death rate in this time interval will be defined by:

$\mu(t)=1000\left({1\over\Delta t}\right){\Delta y\over x(t)}$ (1)

where: $x(t)$ denotes the size of the population. If $\Delta t$ is expressed
in a time unit $u$, the unit of $\mu(t)$ will be “per 1,000 individuals and
per $u$”.

For a human population $x(t)$ can be measured at the beginning, mid-point or
end of the one-year observation time because it changes little over one year.
The situation is very different when one is dealing with a population whose
size doubles every 20mn. In the next subsection it will be seen that cell
numbers should rather be replaced by life cycles.

As an example of how to use the previous definition let us assume a population
of one million in which 120 deaths are counted in the month of January. If we
take years as unit of time, the interval of observation will be: $\Delta t=$
1/12; thus, the January death rate will be $\mu=1000\times 12\times 120\times
10^{-6}=1.44$ per 1,000 individuals and per year.

###  3.2 Two special requirements for a population of bacteria

For bacteria formula (1) must be applied with special care. Why?

Strictly speaking, $x(t)$ should not be the whole population existing at time
$t$ but only the individuals who are subject to the risk of death during the
time interval $\Delta t$. For instance, if one wishes to compute the death
rate of the French population in year 2000 one should not take for $x(t)$ the
whole population alive on 31 December 2000 for this population would include
individuals born on the last day of 2000 and who are subject to the risk of
dying for only one day. Clearly, the number of such individuals is negligeable
compared to the whole population and one can therefore forget this
restriction. However, with a population which doubles every 20mn this
restriction must be taken into account. We will see below how to do that.

There is another meaningful difference. To explain it, let us consider a
fictitious human population in which all individuals live 100 years. We assume
further that the initial population is 1 million and that it remains constant
with births strictly compensating the deaths. Now, suppose we wish to estimate
the death rate over a period $\Delta t$ of 300 years. This is basically the
situation we are facing with a sample of E. coli, except that here we assumed
the population to be constant for the sake of simplicity. In applying formula
(1) should we take $x(t)$ equal to 1 million? That would clearly be incorrect,
for during the 300 years there were 3 generations successively exposed to the
risk of dying, each producing a number of dead individuals which will be
counted cumulatively. In other words, instead of taking for $x(t)$ 1 million
one should take the number of life cycles which is 3 millions. Therefore, when
applying formula (1) to E. coli one must also take for $x(t)$ the number of
life cycles which occurred during the interval of observation $\Delta t$.

The number of life cycles for a population growing exponentially is computed
in Appendix A.

###  3.3 Global death rate of bacteria: modeling method

The previous method required several assumptions to be made. Can one devise a
more direct argument? The present modeling method relies on two simple
equations which describe a population in exponential growth. Any population
can be fully described by two parameters: (i) the birth rate and (ii) the
death rate $\mu$. In order to estimate them one needs to connect them to two
variables that can be measured in our experiment.

For the birth rate we can start from the fact that the population grows
exponentially: $x(t)=x_{0}\exp(\alpha t)$. The exponent $\alpha$ can be
readily obtained from the data of the spectrophotometer. It is true that this
device makes no difference between the cells which are alive and those that
are dead, but the second are about 1,000 times less numerous than the first
and can therefore be neglected.

The cytometer provides the ratio $y(t)/x(t)$ where $x(t)$ represents the
number of living cells and $y(t)$ the number of dead cells. We have already
written an evolution equation for $x(t)$, can we also write one for $y(t)$?
The evolution equation results from the definition (1) written in differential
form:

$\mu={1\over dt}{dy\over x(t)}\rightarrow{dy\over dt}=\mu x(t)=\mu
x_{0}\exp(\alpha t)$

If one assumes that $\alpha$ is a constant, this equation can be solved easily
and one gets a relationship between $x(t)$ and $y(t)$:

$y(t)-y_{0}={\mu x_{0}\over\alpha}\exp(\alpha t)+K\quad\rightarrow\quad
y(t)-y_{0}={\mu\over\alpha}[x(t)-x_{0}]$

where $x_{0},y_{0}$ are the initial values at the start of the 45mn time
intervals between successive measurements.

This equation implies that if $x(t)$ growth exponentially, then, so does
$y(t)$; therefore, the initial values can be neglected in comparison to the
values taken by $x(t),y(t)$ some 45mn later. $x_{0}\ll x(t=45$ mn) and
$y_{0}\ll y(t=45$ mn). Thus, the relationship between $x(t)$ and $y(t)$ can be
written in the simpler form:

$\mu={y(t)\over x(t)}\alpha\quad\hbox{where }\alpha=\log[x(t)/x_{0}]/t$ (3)

In practice, the ratio $x(t)/x_{0}$ can be read on the graph of the optical
density. Just to see what is the order of magnitude of the death rate $\mu$,
we take an $\alpha$ corresponding to a doubling time of 25mn which is the
average in our experiment. $\alpha=\log(2)/0.42=1.66\ \hbox{h}^{-1}$ This
gives:

$\mu=1000\times 1.66{y(t)\over x(t)}$

Five measurements of the ratio $y(t)/x(t)$ covering the first 3 hours of the
exponential phase, gave the following average: $<y(t)/x(t)>=0.0011\pm 0.00022$
which in turn leads to the following expression of the death rate:

$\mu=1.83\pm 0.20\hbox{ per 1,000 individuals and per hour}$

##  4 Methodology of the experiment

The following description includes two parts that are often omitted but which
are of great importance for the success of our experiment. The first is the
selection of the micro-organism best suited for our investigation. The second
is the list of preliminary tests performed to check for possible sources of
errors in the successive steps of the measurement process. By listing these
sources of fluctuations our hope is also that it may help to improve the
accuracy of the observation.

###  4.1 Design of the experiment: selection of an appropriate strain

As our objective is well defined we need to select the micro-organisms which
are best suited. There are several requirements that we discuss below and
which are summarized in Table 2.

1 The main requirement is a short division time so that one could get a large
population in a fairly short time.

Table 2 Characteristics of unicellular organisms suitable for mortality
studies

$\matrix{\hbox{Name}&\hbox{Shape}\hfill&\hbox{Fission}\hfill&\hbox{Division}\hfill&\hbox{Size}&\hbox{Chain}&\hbox{Prokaryot}\cr\hbox{}&\hbox{}\hfill&\hbox{type}\hfill&\hbox{time}\hfill&&\hbox{formation}&\hbox{or}\cr\vrule
height=0.0pt,depth=5.0pt,width=0.0pt\hbox{}&\hbox{}\hfill&\hbox{}\hfill&\hbox{(mn,
hour)}\hfill&\hbox{(micro-m)}&\hbox{}&\hbox{Eukaryot}\cr\hrule\cr\vrule
height=15.0pt,depth=0.0pt,width=0.0ptE.\
coli\hfill&\hbox{rod}\hfill&\hbox{transversal}\hfill&\hbox{20mn}\hfill&1&\hbox{little}&\hbox{P}\cr
Bacillus\
subtilis\hfill&\hbox{rod}\hfill&\hbox{transversal}\hfill&\hbox{20mn}\hfill&1&\hbox{yes}&\hbox{P}\cr
Saccharomices\
cerevisae\hfill&\hbox{round}\hfill&\hbox{budding}\hfill&\hbox{2h}\hfill&3&\hbox{yes}&\hbox{E}\cr
Euglena\
gracilis\hfill&\hbox{elastic}\hfill&\hbox{longitudinal}\hfill&\hbox{2h}\hfill&20-100&\hbox{yes}&\hbox{E}\cr\vrule
height=0.0pt,depth=5.0pt,width=0.0ptParamecium\
caudatum\hfill&\hbox{long}\hfill&\hbox{transversal}\hfill&\hbox{20h}\hfill&100&\hbox{no}&\hbox{E}\cr\hrule\cr}$

2 After measuring the overall death rate over a life cycle, in the second part
of this investigation, we are planning to measure the age-specific death rate
which means that we need a way to estimate the individual age of each cell.
For rod-like organisms experiencing transversal fission the bacteria length
provides a possible proxy for its age. The characteristics of diverse model
organisms summarized in Table 1 show that Escherichia coli, Bacillus subtilis
and Paramecium caudatum are three possible candidates. However, if one wishes
to count the dead cells with a cytometer, the last two should be discarded
because they may clog up the thin pipes of the cytometer (Paramecium should
anyway be discarded due to its long generation time). This leaves us with E.
coli.

3 Another consequence of pipe narrowness is the fact that one cannot use
highly concentrated cultures. In addition such cultures would not be suitable
because at the end of their exponential phase. Therefore, if one wishes to
test millions of bacteria (in order to get several thousands of deaths) one
must use a cytometer which accepts a large sample volume.

###  4.2 Organization of the experiment

The successive steps of the experiment follow quite naturally from the end
objective. They are summarized in Fig.1. Every 45mn, a sample was extracted
from the culture which was growing in the incubator. Then, (i) its OD was
measured (ii) the sample was diluted with PBS so as to have in each trial the
same cell concentration (iii) Propidium iodide was added (iv) 1ml of the
sample ws introduced in the cytometer.

In terms of time intervals,

$\bullet$ It took about 10mn from sample extraction to the PI step.

$\bullet$ 15mn were given to the dead cells for absorbing the PI stain. Note
that the absorption is faster when the concentration of the stain is higher.
The PI is said to be toxic but it would require a separate assay to estimate
the concentration level above which the toxicity effect becomes notable.

$\bullet$ The treatment of the 1ml sample in the FC took about 10mn to 15mn.
In fact, the value of the ratio dead/total observed after 2 or 3mn remained
very stable in the remaining time. This means that a smaller volume would have
given the same results.

Fig.1 Organization of the experiment. The two main devices are the
spectrophotometer which measures the optical density (which in turn gives a
measure of the number of cells per cubic cm) and the flow cytometer which
counts the number of dead cells after they have been made fluorescent by
absorption of propidium iodide. Note that the time interval between extraction
of a sample from the culture and its introduction in the cytometer is about
20mn.

###  4.3 Preliminary tests

Our tests concerned mainly three points.

$\bullet$ When does the stationay phase really begin? The transition becomes
clearly visible toward an optical density of 0.3 (for a more elaborate
discussion see Sezonof et al. 2007). However, by starting a culture with a
concentration of cells some 1,000 smaller than the smallest OD that the
spectrophotometer could measure, we have seen that there is in fact a steady
increase of the doubling time.

$\bullet$ In order to test the accuracy of the identification of the dead cell
we prepared a mixture 50%–50% of dead cells (killed by a temperature shock)
and living cells. A proportion of 45% dead cells was detected by the FC. As a
complementary test, the dead cells sorted by the FC were reintroduced into the
FC. A ratio dead/total of 97% was observed.

$\bullet$ In an attempt to better understand the connection between “PI death”
and “immobility death”, dead cells sorted by the FC were put under a
microscope equipped with a light source in the fluorescence range of the PI.
It was verified that there remained no moving cells in the sample. This test
tells us that our “PI dead” would also be counted as dead with the immobility
criterion used in the experiment of Stewart et al. (2005). In other words, PI
death is either equivalent to or “deeper” than “immobility death111During an
observation of the embryonic growth of eggs of Zebra fish, one embryo remained
completely motionless during two days. It was thought to be dead but in fact
resumed a normal development. Compared with normal embryos, the only
difference was that this one hatched two days later.”.

##  5 Results

###  5.1 Absolute versus relative measurements

It must be stated from the outset that we are more interested in relative than
in absolute values of the death rate. There are two main reasons for that.

$\bullet$ The absolute value of the death rate is dependent upon the criterion
used to define “death”. We have already mentioned that PI death seems to be a
sharper criterion than death defined by immobility. In addition, even within
PI death the technical definition of death relies on how one selects the lower
limit of PI positive events. On Fig.2 this limit is represented by the left-
hand side of the rectangle which defines the PI positive events222Also to be
mentioned is the fact that the number of living cells is affected by the
background noise of the cytometer. This is seen very clearly through a test-
run in which only PBS+LB (without cells) is introduced in the FC. Then the
background noise appears in the form of a set of events at the far left of the
domain delimiting the living. This means that the total number of living cells
detected by the FC is an overestimate. The purpose of setting an appropriate
left-hand side limit for the domain delimiting the living cells is precisely
to eliminate these spurious events.. Moreover, one should realize that the
absolute level of the death rate will not really affect our understanding of
the cells’ mechanisms.

$\bullet$ On the contrary relative death rates are of great significance. The
word “relative” can be given different meanings. In the present paper our main
interest is to see how the death rate behaves when the culture moves from the
exponential phase to the stationary phase. Does it increase or does it
fall333Arguments in favor of a predicted fall were given in a recent separate
paper (Di et al. 2021).? In the framework of our investigation a second
possible meaning is to determine whether there is an “infant mortality effect”
in the terms of Kelly and Rahn’s paper. It means comparing the death rates of
newly divided versus mature cells. We expect a death rate peak for freshly
divided cells followed by a steady decline as the size of the cells increases.

Fig.2 PI positive events for 8 samples extracted at 45mn time intervals. The
scale of the x-axis gives the intensity of PI fluorescence as detected by the
cytometer. In spite of the fact that the same procedure was implemented in the
successives assays, the distributions show notable fluctuations. Needless to
say, these fluctuations affect the estimates of the number of dead cells. It
will be the purpose of forthcoming experiments to determine whether these
fluctuations can be reduced or whether they are purely random fluctuations.

###  5.2 Graph of dead/living

Fig.3a,b Optical density and ratio dead/living for 8 samples taken from the
culture at 45mn time intervals. (a) This growth curve serves to define the two
regions labelled as “Exponential phase” and “Stationary phase” in Fig.3b. (b)
This curve gives the ratio dead/living which is denoted $r=y/x$ in the text.
From this ratio it is easy to derive the death rates by using formula (4)
given in the text. Note that whereas $r$ is a number without dimension, for
the death rate one must specify the time unit.

We chose to represent the ratio $y/x$ rather than the death rate itself for
two reasons.

$\bullet$ Whereas the ratio $y/x$ is read directly on the cytometer, the
computation of the death rate relies on a set of assumptions described in the
text. It is true that the overall level of $y/x$ depends upon parameter
selection and fine tuning of the cytometer, but the relative variations of
$y/x$ across the 8 measurements performed over a period of five and a half
hours are independent of how the parameters of the cytometer have been
selected.

$\bullet$ Whereas the absolute level of the death rate may be useful for the
purpose of inter-species comparisons, one should realize that anyway such
comparisons are necessarily hazardous due to the fact that even for close
species such as E. coli and B. subtilis there are many ingrained differences.

Is there a fall in the death rate $\mu(t)$ in the course of time as was
predicted in Di et al (2021)? The fluctuations are somewhat too large to
permit a clear conclusion. However, it appears that there is no major
increase.

Remark If the ratio $y/x$ is considered to be approximately steady, formula
(3) shows that $\mu(t)$ is proportional to the growth exponent $\alpha$ of the
population. As $\alpha$ decreases with $t$ it might seem that $\mu$ will also
fall with $t$. Unfortunately, this reasoning is not correct for in solving the
differential equation $\alpha$ was supposed constant.

A note about fluctuations Fig.3b might suggest a fairly high level of
fluctuations. In reality, the absolute level of the fluctuations are of the
order of one per 1,000 which means that, compared to similar observations,
they are in fact of same order. Here is a comparison.

Using a hemocytometer, Jones et al. (1985, p.78, table 1) determined the
proportion of viable spleen cells of mice through a staining method using a
mixture of propidium iodide and fluorescein diacetate. In order to assess the
accuracy of their method they repeated the measurements 8 times. The
coefficient of variation (standard deviation divided by the mean) of the 8
proportions was 40%. For the 8 measurements represented in Fig.3b the
coefficient of variation is 51%. In other words, it is only because the
proportion that we measure is quite small that the fluctuations depicted in
Fig.3b appear to be large.

In addition the graph shows that in order to clearly identify a downward
trend, the fluctuations would have to be further reduced by a factor 2 or 3.

##  6 Conclusion and perspectives

While still open to improvement, our death rate measurements appear more
accurate than those of Stewart et al. (2005) and Fontaine et al. (2008). To be
fair one should add that the death rate measurement was not the main goal of
these papers. The important point in Fig.3b is not so much the absolute level
but rather the magnitude of the fluctuations. Although already faily small,
the fluctuations will have to be reduced further in order to see if a trend
can be identified. In addition it would be appropriate to add two or three
data points in the stationary phase that is to say above the optical density
of 1.

Although the cytometric measurements must be handled with great care they
appear to be the only promising method for this kind of bio-demographic
investigations.

##  Appendix A Appendix A. Death rates in human and bacterial populations

###  A.1 Global death rates in human populations

For a human population the global death rate of a population $A$ is obtained
by first counting the number of deaths in one year (that we denote $\Delta y$)
and by dividing this number by the size, $x(t)$, of the population. The ratio
$r_{d}=\Delta y/x(t)$ represents the relative fall of the population.

However, this definition does not give a satisfactory death rate. The reason
can be seen by considering neonatal mortality data. Usually they are given for
the following successive intervals: (i) less than one day, (ii) from one day
to less than one week (iii) from 8 days to less than 4 weeks. Needless to say,
$\Delta y$ will be heavily dependent upon the respective time intervals: for
(i) it is one day, whereas for (iii) it is 28-7=21 weeks. With a stable number
of deaths on successive post natal days the $r_{d}$ of interval (iii) would
artificially appear 21 times larger than for interval (i). If the number of
deaths on successive days are not the same (as is indeed the case) this
phenomenon will be completely hidden by the change in time intervals. This
difficulty can be overcome easily by normalizing $r_{d}$ by the length of the
time interval, thus instead of $r_{d}$ one should consider: $\mu=r_{d}/\Delta
t$. With this definition identical deaths in successive days will indeed lead
to a constant $\mu$. This $\mu$ can be taken as an appropriate definition of
the death rate. The present normalization argument could appear somewhat
trivial, but actually in many national data for neonatal death rate
statistics, this normalization is not made.

Because, $x(t)$ is usually much larger than $\Delta y$, it is often convenient
to multiply $\mu$ by 1,000 or equivalently to express $x(t)$ in thousands of
individuals.

In addition to the time length of an observation $\Delta t$ just introduced,
one must also specify which time unit ($u$) is used. This did not play any
role in the previous argument because implicitly it was the day which was
selected as time unit. In order to see the role of $u$ consider the following
case. One wishes to measure the death rate between two censuses. In this case,
$\Delta t$ is 10 years; if for $u$ we take also 10 years, $\mu$ will give the
death rate per 1,000 and per 10 years. Usually, this is not what one wants
because one wishes to compare $\mu$ to the annual death rate. To obtain an
annual death rate, one should take $u=$ one year. This will give a death rate
that is one tenth of the previous one. It represents the average annual death
rate over the 10-year period.

###  A.2 Estimates of the number of life cycles

In our experiment the number of dead E. coli cells were measured every 45 mn.
However for the sake of simplicity of the present argument we assume that the
measurements were performed every 60 mn. Assuming in addition a generation
time of 20 mn, there will be 3 doublings. If we assume that they were 10,000
bacteria at initial time $t_{0}=0$, at the end of the 60mn there will be
80,000.

At time $t_{1}=40$mn the population will number $40,000$. All these 40,000
cells were born before $t_{1}$ and all of them had been exposed to the risk of
death during their whole life cycle.

If we assume that all cells are synchronized they will all divide at time
$t_{1}$ and the resulting 80,000 cells will be exposed to the risk of dying
during their whole life cycle. In this case the number of life cycles would
be:

$N_{l}=20,000+40,000+80,000=80,000(1/4+1/2+1)=1.75x(t)$

where $x(t)$ is the end population.

However, in reality the cells are not synchronized. Hence the previous formula
should be changed according to the following discussion. It turns out that the
formula $N_{l}=1.75x(t)$ should be replaced by: $N_{l}=1.37x(t)$.

Cells born at $t_{2}=50$mn, will be exposed to the risk of death for only
10mn, that is to say one half of the life cycle; those born after $t_{2}$ will
be exposed to the risk for an even shorter time. Clearly, to continue this
argument one needs to know the distribution by age of the third generation.

One can get an idea of the distribution of E. coli cells by age at a given
moment through the distribution of their lengths. Examples of distributions by
size are given in the literature and were found in agreement with our own
measurements. They show a substantial peak in young age followed by a slow and
fairly steady fall. For the sake of simplicity this kind of distribution will
be schematized as having two separate peaks: one at $t=45$mn corresponding to
3/4 of the generation and a second at $t=55$mn which corresponds to the
remaining 1/4. Those of the first peak will be exposed to the risk of death
for 15mn, i.e. 0.75 life cycle, whereas those of the second peak will be
exposed for only 5mn, i.e. 0.25 life cycle. Thus, the total number of life
cycles subject to the risk of dying can be written:

$N_{l}=20,000+40,000+80,000[(3/4)\times 0.75+(1/4)\times 0.25]$

which gives:

$N_{l}=80,000\times(0.25+0.5+0.62)=1.37x(t)$

Now, if one assumes that $n_{d}$ dead cells have been counted by the cytometer
we can apply definition (1) with:

$x(t)=N_{l}\quad\Delta y=n_{d}\quad\Delta t=1\hbox{hour}\quad u=\hbox{hour}$

Replacement leads to the following death rate:

$\mu=1000{1\over lc}{n_{d}\over 1.37x(t)}=1000\times 0.82r$

where $lc$ denotes the length of a life cycle and $r$ denotes the ratio
dead/living given by the cytometer.

Note that the cytometer will also count a number of dead cells whose death
occured before the 45mn-long observation time; but due to the exponential
growth this number can be considered negligeable compared to the deaths which
occurred during the subsequent interval of observation.

If in the previous argument we replace the time interval of 60mn by the one of
45mn which was actually used in our experiment the same argument leads to the
following expression for total number of life cycles:

$N_{l}=23,784+47,570[(3/4)\times 0.75+(1/4)\times 0.25]=1.12x(t)$

In writing this expression we have separated the time interval of 45mn into
one of 25mn and a second of 20mn. In 25mn the population will reach
$2^{25/20}=23,784$ and all these cells will be able to finish their live
cycle. In contrast, the cells born during the last 20mn will not be able to
finish their life cycle and will give rise to a number of life cycles that can
be estimated by the same argument as above. This leads to the following
expressions for the number of life cycles and for the death rate:

$N_{l}=1.12x(t)\ \longrightarrow\ \mu={1\over lc}{n_{d}\over 1.12x}=1000\times
1.19r$ (2)

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