Transient frailty induced by cell division.

Observation, reasons and implications

Zengru Di1, Eduardo M. Garcia-Roger2, Peter Richmond3,

Bertrand M. Roehner4, Stephane Tronche5

Version of 10 December 2021

Provisional. Comments are welcome.

Key-words: Escherichia coli, cell division, mortality rate, frailty

1: School of Systems Science, Beijing Normal University, China. Email:
<EMAIL_ADDRESS>2: Institut Cavanilles de Biodiversitat I Biologia Evolutiva,
University of València, Spain. Email<EMAIL_ADDRESS>3: School of
Physics, Trinity College Dublin, Ireland. Email<EMAIL_ADDRESS>4:
Institute for Theoretical and High Energy Physics (LPTHE), Pierre and Marie
Curie campus, Sorbonne University, National Center for Scientific Research
(CNRS), Paris, France. Email<EMAIL_ADDRESS>5: Jean Perrin
Laboratory, Pierre and Marie Curie campus, Sorbonne University, Paris, France.
Email<EMAIL_ADDRESS>
Abstract

We know that stress-factors, e.g. X-rays, have an effect on cells that is more
lethal in rapid exponential growth than in stationary phase. It is this effect
which makes radiotherapy effective in cancer treatment. This stress effect can
be explained in two ways: (a) more vulnerability in the growth phase, (b)
improved protection capacity and repair mechanisms in the stationary phase.
Although the two explanations do not exclude each other, they are very
different in the sense that (a) is a general mechanism whereas (b) is strain
and stress-factor dependent. In this paper we explore major facets of (a).
Firstly, we emphasize that (a) can account for known experimental stress-
factor evidence. Secondly, we observe that (a) rightly predicts that slow
exponential growth (meaning with a doubling time of several hours) results in
a lower death rate than fast exponential growth (doubling time of a fraction
of one hour), an effect that cannot be explained in the (b) framework because
both organisms are in the same phase. Thirdly, we conjecture that the stress-
factor effect can be extended to situations of chronic stress due to non-
optimal environmental conditions. If correct, this conjecture would imply that
even in normal culture conditions the natural death rate is lower in the
stationary phase than in the growth phase. Finally, the paper closes with the
description of several open questions and of appropriate test-experiments
meant to address them.

##  1 Introduction

To facilitate reading it can be observed that this paper was written in what
can be called the “spirit of physics”. This means that it focuses on general
causes rather than on the details of specific cases. As a parallel in physics
one can mention the Navier-Stokes equations which describe the basic features
of fluid flows, whether in air, water or any other fluid.

###  1.1 Vital shocks

It is a natural idea to suspect that when a living organism undergoes a major
transition from a state $A$ to a state $B$ it faces elevated risks until the
adjustment is completed successfully. Note that the change may affect the
environment of the organism, or/and its internal state. An example which comes
to mind is the process of moulding, e.g. when a cicada gets rid of its former
exoskeleton . Although the new replacement exoskeleton has taken form under
the old one, it remains soft for several hours (or days in case of larger
species). During this short time the cicada is more vulnerable to predators.

In a series of previous papers (Richmond et al. 2016a,b) we have considered
transient situations in human demography. Several cases were described in
which an abrupt change in living conditions leads to a mortality spike. This
effect was summarized in what we called the “Transient Shock conjecture”. It
provides a qualitative model which leads to testable predictions. For
instance, statistical evidence shows that persons who become widowed
experience within months a mortality spike that is higher than their average
long-term death rate either before or after becoming widowed. This case may
not come as a surprise. More revealing is the marriage transition. Marriage
certainly brings about a major change in personal and social conditions and
according to our conjecture one would expect a mortality spike in the months
following marriage. At first sight this may seem an unlikely proposition but
it was shown that the existence of such a mortality spike is indeed supported
by statistical evidence. (Richmond et al 2016b)

There is a noteworthy difference between widowhood and marriage in the sense
that widowhood occurs usually in old age when there is already a situation of
frailty. A similar case considered in the papers mentioned above concerns
elderly persons who move from their home to a nursing home. The few data that
are available for such cases often display a mortality shock in the months
following admission. Here too, there is certainly a frailty effect. The
biological phenomenon considered in the present paper will also involve these
two components: (i) a shock in the form of an exogenous stress factor and (ii)
a situation of frailty.

###  1.2 Births seen as transient shocks

Coming now to biological cases, in humans and in many other species, birth
constitutes a major shock which, not surprisingly, results in a huge mortality
spike (Berrut et al. 2016). This means that the death rate in the days
following birth is much higher than the death rate in utero in the weeks
preceding birth and also higher than the neonatal death rate (i.e. average
death rate over the first 6 weeks). It can be observed that a large part (over
50%) of these early deaths are due to congenital malformations which were
compatible with life in utero, but become lethal in the new environment; lung
defects are a clear example. The congenital malformations may appear during
embryogenesis but may also be triggered by defects already present in the
initial female and male germ cells111The extent of such defects is attested by
the huge mortality peak in the initial phase of the embryogenesis (see Chen et
al. 2020), i.e. the oocytes and spermatozoons.

###  1.3 How the process of fast cell replication induces frailty

The question that we address in this contribution is whether or not cells
experience a transient shock (similar to birth) in the process of division. We
will propose reasons valid for both eukaryotic cells and bacteria for a state
of fragility associated with cell division.

For a bacteria like Escherichia coli, the generation time is about 20mn.
During this short time, many complicated processes have to be carried out
(Fig.1). One can keep in mind (Soufi et al. 2015) that the proteome (i.e. the
entire set of proteins expressed in the cell) of E. coli is comprised of some
2,300 different proteins with concentrations ranging from just a few copies to
some 300,000 for the more abundant. Appendix A gives an exemple of the working
of a protein enzyme.

Fig.1 Replication and division of E. coli. The first phase is marked by
intensive protein production and replication of the genome and other
components. It is the analog of the interphase in eukaryote cells. The
replication of all the components of the cell is certainly a major task of
information processing. On the right-hand side of the diagram it was assumed
that, due to a defect in the replication procedure, a key-protein $P_{1}$ was
not produced in appropriate amount in the upper cell. Despite that, the
situation remained manageable as long as exchanges were possilble with the
mother cell. After separation the lack of $P_{1}$ may prove lethal. This
fragility induced by division will be called the “replication frailty”.

In the manufacturing of an airliner accuracy requirements differ from one
component to the other. If millimetric accuracy may be acceptable for the
airframe, there are tighter requirements for the engines. There are good
reasons to think that in the process of Fig.1 the most critical part is the
replication phase.

$\bullet$ Any fault in the production of the proteins which function as the
bricks and tools of the replication process may subsequently lead to fatal
failures of the kind shown in the figure.

$\bullet$ It is true that there are control and reparation mechanisms. For
instance if a gap appears in the replication of the DNA the two ends may be
joined again thanks to the cohesion forces which exist between consecutive
elements. The problem is that control and reparations take time whereas in
fact the whole process is designed to achieve high speed. For instance,
instead of starting from one location of the DNA molecule, replication starts
simultaneously from different locations222More precisely, (i) the replication
starts in two directions and (ii) once the specific starting point has been
duplicated, replication may also start from this point.This speed requirement
is a major cause of fragility. It will be seen shortly that bacteria having a
doubling time of 20mn are much more vulnerable to stress factors than others
whose doubling time is 5 hours.

Another crucial moment is the separation. However, what makes it risky are the
mishaps that occured in the replication. To work well, E. coli cells need
thousands of proteins. Now, suppose that something went wrong in the
replication with the result that an essential enzyme-protein $P_{1}$ is not
produced in appropriate quantity. As long as exchanges remain possible, the
daughter cell can rely on the $P_{1}$ produced by the mother cell. This kind
of sharing becomes impossible after separation. In other words, the death rate
of daughter cells will reflect the replication defects, much in the same way
as post-birth human mortality reflects the manufacturing anomalies that
occurred during embryogenesis. Naturally, not all defects are lethal
immediately after separation. In humans it is well known that some congenital
defects (e.g. heart valve anomalies) become life-threatening only in old age
when compounded with other factors, e.g. increased membrane rigidity. Whenever
they occur, non lethal replication anomalies leave daughter cells in a state
of fragility. If in addition the cells are subjected to adverse environmental
conditions (e.g. X-rays, non optimal temperature, pressure or acidity) this
fragility will bring about a death rate spike.

###  1.4 How the paper proceeds

So far we have explained the mechanisms on which our understanding relies. In
the rest of the paper we adress two questions.

1 How compatible are the data on death rates in bacteria with the existence of
a transient state of frailty during cell division. Many experimental results
are available which describe the response of bacteria to harmful exogenous
factors. It appears that the populations with high growth rates are much more
sensitive to those harmful conditions than those with slow growth rates.
Illustrative evidence will be presented and discussed.

2 Has the transient shock model predictive implications not yet considered and
which may suggest new experimental tests? Can the tests on bacteria be
extended to other systems of growing organisms, e.g. replication of viruses in
cells, replication of eukaryote cells, cleavage of first cell in an embryo?

##  2 Response of bacteria to adverse exogenous factors

Adverse exogenous factors can be of many kinds:

$\bullet$ Ionizing radiations (e.g. X or gamma rays)

$\bullet$ Low or high temperature

$\bullet$ High pressure

$\bullet$ Inadequate acidity

$\bullet$ Disinfectants and antibiotics

$\bullet$ Pulsed electromagnetic fields

$\bullet$ Ultra violet light

Here we wish to focus on parameters which modify the division rate. The
following can be considered.

$\bullet$ Normal cells versus cancer cells will be considered in the next
subsection. As is well known cancer cells are more vulnerable to X-rays than
normal cells.

$\bullet$ Different strains of bacteria have generation times (also referred
to as doubling times) ranging from 20mn to several hours. Those with the
fastest growth are the most vulnerable. An illustrative example will be given.

$\bullet$ Any new cultures of bacteria starts with a period of fast
exponential growth but after a while it enters a stationary phase in which its
birth rate converges toward zero. It will be seen that bacteria taken from the
rapid growth regime are much more vulnerable to stress factors than those from
the stationary phase. An illustrative example will be described.

Among the many publications on these effects a good introduction is provided
by a series of three papers which emphasize the role of the growth rate,
namely Ihssen et al. 2004, Berney et al. 2006, Lindqvist et al. 2014. They
consider the cases of temperature jumps, ultra violet light and acidity, The
impact of very high pressures of over 1,000 bars has been studied in Benito et
al. 1999 and Manas et al. 2004.

###  2.1 Early observation of the division effect

One of the first observations of the replication effect was made in the early
20th century when it was discovered that ionizing radiations can destroy
cancer cells without too much affecting surrounding cells. After coming about
through the activation of germ-like stem cells, normal cells multiply
exponentially until the required number is reached. Then, they enter a
stationary phase in which their growth rate is almost zero. On the contrary,
cancer cells can grow exponentially without limitation. As an example, in the
case of small cell lung cancer the doubling time of the cells can be as short
as 30 days.

Why were cancer cells more affected than normal cells? There are (at least)
three possible interpretations.

1 One explanation is to say that the radiations disrupt the DNA to the point
of making any subsequent division impossible. Yet, taken alone, this reason
cannot explain why normal cells are less affected.

2 Another explanation is to say that in the stationary phase the cells develop
a number of protections which shields them from the radiations.

3 The third explanation relies on the fact that, as decribed above, cell
divisions bring about a state of fragility which makes them vulnerable, not
just to radiations but to any adverse factor. In this interpretation the more
divisions cells undergo, the more vulnerable they are.

One great difference between interpretations (2) and (3) is that (3) not only
explains the radiation effect but has also predictive implications for cell
responses to other harmful factors. Actually, (2) and (3) do not exclude one
another but are rather complementary. It is true that in the stationary phase
cells develop means for long term preservation. The transformation of cells
into spores is a clear illustration of this kind of mechanism. Yet,
explanation (2) has no predictive power; it gives a description of the
sporulation process whenever there is one, but it cannot predict what cells
have the ability to form spores or what other preservation mechanisms can be
activated.

Quite surprisingly, despite its early observation, the fragility mechanism has
been all but forgotten in subsequent decades. A 343-page manual on how
bacteria are affected by radiations, published in 1973 by the “International
Atomic Energy Agency” (IAEA) does not even mention this effect. In contrast,
There have been numerous papers describing various protection mechanisms of
type (2). As such mechanisms are species-dependent, it may be a case of trees
concealing the forest. We hope that by highlighting the frailty explanation
the present paper will bring it back to the attention of researchers.

###  2.2 Prokaryotic versus eukaryotic cells

Why should replication and division put bacteria (that is to say prokaryotic
cells) like E. coli in a particularly vulnerable situation? The fact that in
bacteria the replication of the DNA progresses at the impressive speed of
1,000 nucleotides per second, compared to 100 per second for eukaryotes
suggests that such high speed processes are probably hypersensitive even to
small perturbations. In the present paper we will not discuss this point in
more detail; our plan is to accept it as a fact and analyze its implications.

Radiotherapy relies on the replication effect in eukaryotic cells and this
makes us expect an even stronger division effect in bacteria. In the following
subsections we give examples which quantify this effect. It will be shown to
what extent the death rate occasioned by adverse conditions is conditioned by
the frequency of replications.

###  2.3 Effect of ionizing radiations

We start with this case because, as described above, it was historically the
first to attract the attention of researchers333It has the additional
advantage that radiations are always harmful. In contrast, for assessing the
effect of a surge in temperature, one needs to know what is the cells’ optimal
temperature..

The following observation conveys very clearly the idea of the frailty effect.
It has been shown (see Lacassagne 1930 and Lea 1947, p.307) that when a cell
is irradiated death does not occur immediately but at (or following) the next
division. For instance, 10,000 roentgens delivered to yeast allows the cells
to divide but their daughter cells are usually unable to divide further and
eventually die. A much larger dose is required to kill the cells immediately;
thus, a dose of 30,000 roentgens will kill only 50%. The same effect can be
seen for eukaryote cells. When chick tissue growing in culture is irradiated,
a dose of 100 r suffices to cause the death of the cells which attempt to
divide, but 2,500 r are required to cause the death of an appreciable
proportion of resting cells, i.e. those which do not divide.

In an experiment described in the aforementioned report of IAEA (1973, p.41)
different populations of bacteria were subjected to increasing doses of
radiations until only 0.1% survived (Fig.2). This was done successively with
two species. For Micrococcus radiodurans ($G=5$h) this result is obtained with
a dose of 1.20 Megarad (rad means “radiation absorbed dose”, this old unit is
related to the SI unit called Gray by the relation: 1 Gray=100 rad).

Fig.2 Sensitivity of bacteria to X-rays. The sensitivity to X-ray radiations
is strongly conditioned by the generation time G of the bacterial strain. The
faster the bacteria divide, the more sensitive they are to radiation. Source:
The data are from IAEA (1973,p.41)

In contrast, for Salmonella typhimurium ($G=20$mn), only 0.05 Megarad were
required to get the same result. In other words, division of $G$ by 15
resulted in a reduction of the dose by 24.

An alternative approach (see Appendix B) could be conducted by comparing the
required dose in the exponential phase of M. radiodurans to the dose for the
same bacteria but in the stationary phase (Serianni et al. 1968, p.198). One
finds that a dose about 3 times larger is required in the stationary phase. In
short, the faster the cells divide the more sensitive they are to radiation.

###  2.4 Effect of cold

Among numerous papers devoted to studying the effect of stress factors, the
experiments done by James Sherman and William Albus (1923) are of particular
interest for (at least) two reasons.

$\bullet$ Whereas most papers study the effect of temperature rises, this one
investigates the effect of cold.

$\bullet$ In contrast with many papers in which the protocol is neither well
designed nor well explained (see Appendix B), here the methodology is
carefully designed and well explained.

The authors start with the observation that it is well known that a brief
exposure to cold (above the freezing point of the medium) causes almost no
reduction in the number of viable E. coli bacteria as counted by the CFU
(Colony Forming Unit, see Appendix B) method, provided that the population is
in the stationary phase. Then, the authors describe the following experiment.

First experiment. A culture of E. coli in 1% pepton was kept for 8 days at
laboratory temperature (25 degree Celsius) before being diluted in distilled
water at a temperature of 2C. The numbers of viable cells were counted on a
small sample, immediatly after dilution ( $N_{0}$) and again after one hour
exposure at this temperature ($N_{1}$). They found (average of two trials):

$\hbox{Old culture (8 days)}\Longrightarrow N_{1}/N_{0}=0.99\pm 0.014$

Then, they repeated the same experiment with a culture which was only 4 hours
old. This time they found the following numbers (again average of two
experiments):

$\hbox{Young culture (4 hours)}\Longrightarrow N_{1}/N_{0}=0.47\pm 0.26$

They were tempted to conclude that there was an over-mortality among the young
cells due to sudden cooling. However, the authors observe quite judiciously
that it was not possible to conclude that it was the cooling which caused the
mortality for in fact there were two changes: (i) temperature and (ii)
transfer from a nutrient medium to distilled water. At this point the authors
could have performed two tests:

1 Replace the water at 2C by water at 25C. This would show the water effect
alone.

2 Replace the water at 2C by pepton medium at 2C. This would show the
temperature effect alone.

The authors decided to do the second test. In a sense this was a logical
decision because it is in the temperature effect that they were interested.
However, just as a confirmation, it would also have been interesting to try
the first test (see Appendix C).

Second experiment. A culture of E. coli in 1% pepton was kept for 12 days at
laboratory temperature (25C) before being diluted in pepton medium at a
temperature of 2C. The numbers of viable cells were counted immediatly after
dilution ( $N_{0}$), and after 1 hour exposure at this temperature ($N_{1}$).
They found (each measurement was done only once):

$\hbox{Old culture (12 days)}\Longrightarrow N_{1}/N_{0}=0.96$ $\hbox{Young
culture (3 hours)}\Longrightarrow N_{1}/N_{0}=0.42$

The fact that the $N_{1}/N_{0}$ ratio for the young culture was almost the
same as in the first experiment suggests that the pepton to water (both at
25C) transition had almost no effect in terms of mortality. Naturally, in
distilled water a cell cannot produce any daughter cell because the basic
materials (e.g. carbon) are unavailable.

So, is this experiment the ultimate test? Not entirely. From the data given by
the authors we observe that in the old culture, $N_{0}=970$ per cubic cm,
whereas in the young culture: $N_{0}=3,420$ per cubic cm. In other words, the
density of bacteria in the young solution is over 3 times higher than the
density in the old culture. Can that be of importance? Probably not, but to be
on the safe side it would have been easy to dilute the young culture so as to
reach the same density as for the old culture.

The authors measured also the number of cells after 2 and 3 hours at 2C. There
were only slight decreases:

$N_{2}/N_{1}=0.80\quad N_{3}/N_{2}=0.85$

These numbers are consistent with the interpretation that the effect is mostly
due to the cells which were in the process of division shortly before cooling.
Such cells were very numerous in the young culture at the moment of its
dilution in the medium at 2C. What happened to them? Their growth was
certainly stopped. Were they killed or only inactivated in the sense of
becoming unable to form colonies? We do not really know.

Novadays, measurements with a spectrophotometer would allow us to know whether
or not some cells could successfully complete their division in the minutes
after cooling. One would not be surprised that cells in the final phase might
be able to terminate their division.

###  2.5 Effect of other stress factors

It can be noted that in the previous experiment by Sherman et al. (1923) the
replication effect was rather weak in the sense that the number of viable
cells was divided by only two whereas in the X-ray experiment we have seen
much greater reductions. This is understandable because the cooling froze all
new divisions; only the few cells whose division process was well advanced
were affected. This interpretation is confirmed by the next experiment in the
same paper. In this case, the stress factor consisted in exposure in a
solution of 2% NaCl during 1 hour. This factor stressed the cells but did not
stop the divisions. As a result, the number of young cells was divided by 30.
In another experiment (in the same paper) the stress factor was exposure to a
temperature of 53C during 20mn. In this case, the excess reduction of the
young cells with respect to the old cells was a factor 75.

###  2.6 Internal changes when exponential growth slows down

We have already observed that there are internal changes. How important are
they? In this respect the comparison between M. radiodurans and S.
typhimurium, both of them considered in their exponential phase, has a clearer
significance than experiments of type (2). defined in Appendix B. Why? When a
culture gradually moves from the exponential phase to the stationary phase,
many internal transformations take place. One of these, which is mentioned in
the literature in connection with cell response to various stress factors, is
the increase in the RpoS factor444Although irrelevant for our argument, the
origin of this cryptic name can be explained as follows. R stands for the
first letter of RNA; po stands for polymerase, a catalyst which produces
chained molecules and is used in DNA to RNA transcription; finally S refers to
the fact that this factor is only active in the initiation of this
transcription. Another often used protective technique against radiation and
other stress factors is clustering .. The production of this factor increases
as the population enters the stationary phase and we are told that it may also
be activated by a situation of stress. However, this can by no means explain
the difference observed between M. radiodurans and S. typhimurium for we are
talking here of bacteria in their exponential phase. In contrast, the
explanation by vulnerability during divisions remains valid. In short,
although we do not deny that there may be some specific defense mechanisms, we
think that, because of its broad validity, the replication-frailty effect
should be considered first.

##  3 Conjecture and predictive implications

###  3.1 Statement of the conjecture

We have shown that instead of relying on a multitude of protective mechanisms
which are strain- and stress-dependent, all the experimental evidence about
responses to stress can be explained by assuming that cell divisions make the
cells particularly vulnerable to any pre-existing stress factor. Is it
possible to extend this mechanism to situations where there is no specific
stress factor? We believe so. This conjecture can be stated as follows.

Frailty conjecture: The death rate of a culture of bacteria grown under nearly
ideal conditions should be correlated to its growth rate. Consequently, it
should be lower in the stationary phase than in the exponential phase.

As all conjectures, this one relies on several assumptions. For instance, it
is assumed that the natural stress factor due to non-perfect conditions does
not increase as the growth rate of the culture decreases.

###  3.2 Supportive argument

A simple argument goes as follows. A cell culture is never in a situation that
is 100% ideal. Nutrients, pH, supply of dissolved oxygen and so many other
parameters are never all completely optimum for the simple reason that in fact
we do not know exactly what the most perfect conditions are. The discrepancy
between existing and perfect conditions are seen by the bacteria as a light
form of stress. Thus, what we have said about stress factors leads to the
prediction that the natural death rate (usually denoted by: $\mu_{n}$) will be
lower in the stationary phase than in the exponential phase.

It can be observed that the previous prediction is rather counter-intuitive.
Why? Usually, the stationary phase is seen as resulting from adverse
conditions in the form of insufficient nutrients and accumulation of residues.
There is no doubt that such unfavorable conditions do exist. This means that
in a batch experiment the two effects occur simultaneously and are in
competition with one another. In the early stationary phase, the growth rate
effect will probably dominate but it is certain that after a while the effect
of bad conditions will prevail.

A graph published in Wilson (1921, p.430) suggests that for a population
without external stress factor, if the stationary phase starts around time
$T_{s}$ (in the case presented in the paper $T_{s}=5$h), then the death rate
begins to swell substantially after time $2T_{s}$ and the number of viable
cells falls back to its initial level toward time $4T_{s}$.

One can get rid of these unwanted competing effects by using a chemostat in
which nutrients are renewed and residues eliminated (see Appendix B).

###  3.3 Experimental tests of the conjecture

Although the previous argument seems reasonable we do not put too much faith
in it. The real test will come from experiments.

However, one must realize that the required measurements will not be easy.
Why? When there is a stress factor, the population falls very substantially,
for instance it may be divided by two or three within one hour. Such big
changes can easily be assessed even with fairly crude techniques such as the
CFU counting technique. On the contrary, for the natural death rate of E. coli
in the exponential phase, one expects a death rate of the order of 1 per 1,000
living cells and per hour. If our conjecture is correct the death rate in the
early stationary phase will be even lower. Needless to say, to get reasonable
estimates for such low death rates one must be able to measure the number of
dead cells with high accuracy.

###  3.4 Amplification of small differences in growth rates

Usually, growth rates are not easy to measure. The replication effect opens
the possibility of replacing growth rate measurements by death rate
measurements.

Translation of growth rate differences into death rate differences can be done
by applying a standardized stress factor (e.g. X-rays or ultra-violet light).
This may be useful in fields (e.g. oncology) where differences in growth rates
are important.

##  4 Extensions to embryo and viruses

“Is it possible to express our conclusions concerning the effects that we have
described in such a way that some general principles emerge from them?”

This is the question raised in one of his papers on this topic by Nobel prize
winner André Lwoff (1959, p.120). To this end we state the replication effect
in more precise form and then we will ask if its presumed applicability can be
extended. This will make the testing game more risky but also more rewarding.
It would be pointless to propose an explanation which cannot be disproved.

###  4.1 Enlarging and focusing

Whether for prokaryotes or eukaryotes, the division process goes through
similar successives steps. Given that the overall objective is the same, it is
hardly surprising that there are common features: production of a great number
of proteins, replication of the DNA and transcription of the RNA, spatial
separation of mother and daughter components at the two ends of the cell and
finally separation555Although the broad mechanisms are fairly clear many
questions regarding their real implementation remain unanswered.Actually, such
steps are indispensable in all cases where a mother cell gives rise to a
daughter cell. Apart from the two cases already considered, one can add to
them two others not yet considered.

i The initial cleavage of a zygote, that is to say the first step in the
process that will lead to an embryo and eventually to the birth of a new
individual.

ii The replication of viruses in their host cell.

Having thus enlarged the question, we wish at the same time to make it more
focused by asking what is the most crucial part of the division process as far
as manufactoring mishaps are concerned. In fact, this question was already
addressed earlier (see also Appendix A) In the framework of our paper, it is
clearly the early phase of the division which is considered to be the most
critical. Called interphase in the case of eukaryote cells, this phase (more
precisely the G1-S phases of the interphase) comprises the synthesis of many
proteins and the replication of the DNA. Our conjecture is that it is during
this phase that the cell is the most vulnerable to external perturbations. The
main question is whether the mechanism brought to light for bacteria and
eukaryote cells remain valid for viruses and embryonic development. It will be
addressed in the following subsections. However, our purpose is only to open
new perspectives that can be considered more fully in subsequent studies.

###  4.2 Frailty mechanism for viruses

Broadly speaking the existence of such a mechanism is both self-evident and
yet still mysterious. It is self-evident because viruses are living entities
only during the time when they get replicated inside an host cell. Naturally
that does not mean that they cannot be destroyed outside of their host-cells.
Any molecule, and particularly macromolecules, can be broken up into pieces by
radiation or excessive temperature666Lwoff addresses this point by saying that
mere “thermal inactivation of viral particles [i.e. outside of the host cell]
in few hours in temperatures ranging from 37C to 41C is negligeable.”.

Once inside their host-cells the replication of the viruses happens to be
highly temperature dependent (see Lwoff 1959, p. 111). Unlike chemical
reactions which, according to Arrhenius’ law, become faster when the
temperature is (moderately) increased, for viruses it is the opposite. As soon
as the temperature becomes higher than their optimum temperature, their
replication is reduced by several orders of magnitude. Lwoff gives
measurements for the poliovirus which show that at 40C their development is
reduced by a factor of almost 1,000 with respect to what it is at 37C. Such a
high vulnerability to a fairly small temperature change is indeed coherent
with the high growth rates achieved by some viruses. Here again, the fastest
the division, the higher the vulnerability to stress factors.

###  4.3 Frailty mechanism in embryogenesis

The early steps of embryogenesis consist in a replication process. Here we
have a limited goal. We wish to draw attention on just one point which may
help us to better understand the bacteria case.

For most species of fish the fertilization of the eggs produced by the female
occurs outside of the body which means that it can easily be observed from the
very first moment on. In this way one can measure the death rate of the
embryos during their whole development. It appears that the bulk of the deaths
is concentrated toward the beginning of the development and that it can reach
very substantial rates of the order of 10% (see Chen et al. 2020). This is 100
times more than the death rate for E. coli cells during the phase of
exponential growth. Why do we have such high rates here? We believe there is a
simple reason which is related to how the female and male germ cells are
produced.

In the case of E.coli any cell can be considered as an initial cell that has
been produced by a mother cell through a standard process of division that is
known to work fairly well with about only one defect in 1,000 divisions. For
the oocytes produced by the female it is a very different matter. They are
produced through an intricate process which comprises many steps (including
two divisions). Although there may be some control mechanisms along this
production process there is no screening process that would eliminate the
defective eggs. In fact, the actual screening takes place in the first
divisions of the embryogenesis777We did note mention possible defects of the
sperm cell because it is about 1,000 times smaller than the egg, nevertheless
defects cannot be excluded altogether..

If a parallel can be helpful it would be with the launch of a rocket compared
to a departing train in a station. The departure procedure (e.g. announcing
imminent departure on the train’s loud speakers, closing the door, starting
the engine) has been tested multiple times in previous stations. On the
contrary, although the individual components (e.g. the engine, the navigation
system) may have been controlled in many ways, the launch will be the first
time for all components to work together. It is well known that despite
multifold tests, the probability of failure remains usually of the order of
5%. Similarly, the division of the zygote is for the female and male
components the first time to work together.

##  5 Perspectives: complementary experimental tests

In this section we describe a number of open questions. Experiments are
proposed which should help to answer them.

Does cell concentration constitute a stress factor for a sample of cells? This
is an important question that we did not consider so far. Two interrogations
come to mind immediately.

$\bullet$ The term “concentration” can be understood in two ways: (i) with
respect to the total volume of the liquid, $c_{v}$ (ii) with respect to the
amount of nutritive elements, $c_{f}$, contained in the liquid. For the sake
of brevity we call them “volume-concentration” and “food-concentration”,
respectively. When a culture is growing, both $c_{v}$ and $c_{f}$ decrease. It
would be of interest to know the effect of each factor separately. Clearly, if
$c_{f}\rightarrow 0$, division will stop but we do not really know what
happens when $c_{v}\rightarrow 0$ while $c_{f}$ is kept constant. The
experiment summarized in Fig.3 should answer this question.

$\bullet$ The term “stress factor” can be understood in two ways: (i) a factor
limiting growth (ii) A factor inducing higher mortality. For the sake of
brevity these meanings will be referred to as “growth stress factor” and “life
stress factor” respectively. In the papers reviewed above the reduction in the
number of viable cells was well documented but there was little information
about birth rate changes. Probably, it was assumed that a fall in viable cells
would necessarily be paralleled by an interruption of the division process.
However, that is by no means evident. Cases of human populations having high
death rates coupled with high birth rates are well documented in developing
countries.

###  5.1 Is the volume-concentration of cells a growth-limiting stress factor?

The experiment consists in three steps.

1 After the two samples have been extracted from the culture their optical
density and their their doubling time are measured.

2 Dead cells are added to sample $A$ and the same volume of medium but without
cells is added to sample $B$.

Fig.3 Sensitivity of bacteria to volume concentration. In order to change the
volume concentration without changing the food concentration dead cells are
added to the culture.

3 The doubling time is measured again in order to see if (and by what amount)
the growth of sample $A$ was slowed down with respect to the control sample
$B$.

###  5.2 Effect of: medium $\rightarrow$ water transition.

In the analysis of the experiments conducted by Sherman and Albus (1923), we
came across the question of whether it is possible to define what can be
called the momentum of inertia of a culture.

Fig.4: What amount of building materials are stored in the cells during
exponential growth? The graph at the top would be observed in case of an
important stock of advance material. On the contrary, the bottom graph
corresponds to the assumption of “just in time” production that is to say with
the minimum of advance storage.

Simply stated, this notion gives an estimate of how long a culture can
continue to grow once it is put in distilled water. Here, we propose a simple
experiment .

Two different growth surges can be expected.

1 The growth based on “endogenous” resources may occur shortly after the
transfer into water. One knows that in the phase of rapid growth of E. coli
the replication of the DNA destined to the grand-daughter starts before the
separation of the daughter has occurred. This means that the cells certainly
contain more materials than the amount strictly needed to produce a single
daughter.

2 An exogenous resource is provided by dead cells which release some of their
components. This is a fairly obvious effect in which we are not really
interested. One hopes that the time lag between the first and second effects
is large enough to allow a clear distinction.

This simple experiment should be able to tell us what is the amount of
“building materials” that are stored in the cell at different moments of the
growth cycle. One would expect that this amount dwindles when the speed of
growth decreases.

This effect can also be used to probe the age distribution of the population.
To make this point clearer let us assume that the population is synchronized.
Then, if all cells are freshly divided they do not possess the materials
necessary for replication and therefore will be unable to divide. Thus, one
expects the bottom graph in Fig.4. On the contrary if all cells are mature and
about to divide one expects the top graph in Fig.4. In a mixed case, the
length of the inertia phase will give an estimate of the proportion of mature
cells.

##  Appendix A Appendix A: Manufacturing defects versus mutations

In this appendix we wish to explain the implications of a defective blue-print
(as a result of mutations in the genome) as contrasted with mishaps in how the
prescribed tasks are carried out. It is true that mutations of the genome may
be increased by external factors such as X-rays or ultra violet light.
However, even if the blue-print is correct there will be defects in the
implementation process. This question is outside the scope of our paper but it
is important to realize that mutations of the genome are not the only
mechanism which leads to anomalies.

###  A.1 Defects and mishaps

Instead of the expression “manufacturing defects” we were at first tempted to
use more specifically biological expressions such as “error of metabolism” or
“enzymatic defects” but then we realized that in fact we have in mind few
clear cases of enzymatic defects that would illustrate our argument. In
contrast, examples of manufacturing defects can be cited easily. Consider for
instance the construction of a building. Naturally it will proceed in
accordance with the blueprint established by an architect. But in its
implementation there may be many mishaps. In hot weather concrete may dry too
fast thus impairing the mechanical properties of the walls. Freezing weather
may also cause problems. In other words, at each step, external non-optimal
conditions may exist. In some cases the effect may be light while in others
they may lead to structural flaws.

In biology inappropriate temperature or pH conditions may lead to inordinate
protein production: too little, not at the right location or not in the
appropriate 3-d configuration (for instance a cis- instead of a trans-
molecular structure), any of these defects may possibly lead to life
threatening abnormalities. Naturally external parameters are never 100%
optimum, but may nevertheless be “acceptable”. For a process which does not
require high accuracy (e.g. the production of tissue) the acceptable bounds
will be fairly wide. In contrast, for processes which require high accuracy
they may be rather narrow.

Although we know of few specific examples of defect forming here is one. There
is a mechanism called the “induced-fit process” but that we prefer to call the
“hand-glove adjustment” in which a substrate must do more than simply fit into
the already preformed shape of an enzyme’s active site. When the substrate
approaches the enzyme surface, it induces a change in its shape that optimizes
a correct placement. In short, there is a two-sided adjustment, hence the
hand-glove expression. A case in point is the digestive enzyme
carboxypeptidase in which the binding of the substrate causes a tyrosine
molecule (a standard hydrophobic amino acid used by cells to synthesize
proteins) at the active site to move by as much as 15 angstroms which is about
twice the size of the molecule itself. In fact, depending on the temperature,
the real distance will differ. A large change of that kind may even disrupt
the adjustment and therefore the production of the protein.

We see that although examples of mishaps can be given, we have little
intuitive understanding of them because these phenomena occur in the
microscopic world with which we are not familiar

###  A.2 Mutations

In contrast, we have a fairly clear view of the effects of DNA mutations
because these are illustrated in daily experiments in which one can describe
side by side the mutation and its phenotypic consequences888Whereas the
connection mutation $\rightarrow$ abnormality is easy, the converse connection
is much more difficult. In fact the genetic basis of most abnormalities is
unknown. . In other words it is not surprising that we are much more familiar
with mutations than with mishaps. By the way, the same holds in our building
parallel. Whereas it is very easy to identify a flaw with respect to the
blueprint, a defect in the structural properties of the walls is not easily
detected.

###  A.3 Randomness versus semi-randomness: the case of strabism

Finally, one must keep in mind one major difference between mishaps and
mutations: while mutations are completely random, mishaps are rather semi-
random. In which sense do we mean that? Depending upon the degree of accuracy
that is required, mishaps will be more or less frequent. Here is an
illustration.

The control of the pupils of the eyes requires an excellent coordination of
the surrounding muscles; this means that only small fluctuations are
acceptable, which in turn imposes narrow bounds on the values of external
parameters. Thus, one will not be surprised that this congenital malformation
(called strabism) occurs with a fairly high frequency of the order of 3%
according to US statistics. This is one of the highest malformation
frequencies. Similarly, the making of heart valves requires strict
manufacturing constraints; as a result heart valve defects have a relatively
high likelihood.

##  Appendix B Appendix B. Comparison of experimental protocols

In reviewing the literature we found that the experimental conditions are
often loosely chosen and described. As a particularly striking case one can
mention the following (IAEA 1973, p. 55, Fig.10). The graph shows two curves
(a) and (b) of surviving bacteria fractions as functions of the radiation
dose. The caption says: “S. typhimurium irradiated in meat at -15C. (a)
Following growth in the meat at 37C for two days. (b) Without pre-irradiation
growth in the meat”. The growth rates before irradiatation are not given. In
(a) the culture was brutally brought from a temperature of 37C to -15C. For
(b) the pre-irradiation temperature is not indicated. In short, this graph
mixes up temperature and radiation effects. Due to an unclear design and a
sketchy description it is impossible to draw any conclusion.

This example shows that great care should be given to both the design and the
description of the experimental conditions. It is the purpose of this appendix
to emphasize some important distinctions in this respect.

###  B.1 Three types of measurements

Before we come to the data we wish to describe how these experiments can be
done. There are basically three methods.

Type 1 The simplest way is to use bacteria which have different generation
times $G$. This is the time interval between successive divisions as long as
one is in the exponential phase. In this phase, the population doubles in time
$G$ which for that reason is also called the doubling time. As examples, for
the bacteria Salmonella typhimurium $G$ is equal to 20mn, whereas for the
bacteria Micrococcus radiodurans $G$ is equal to 5 hours (i.e. 15 times more).

Type 2 If one wishes to focus on a specific bacterium one can make a
comparison between $G$ in the exponential phase and the very large value of
$G$ observed in the stationary phase. Although many papers use the comparison
exponential phase versus stationary phase one should observe that it is a
methodology which is not very accurate for at least two reasons. (i) The
comparison relies on only two values. It is true that in addition $G$ can be
changed by controlling the temperature, however we did not find any paper in
which this possibility is used. (ii) Actually $G$ is not really constant in
the exponential phase; it increases slowly but steadily. In addition, for
bacteria like E. coli which have a small $G$ the whole exponential phase lasts
less than 3 hours which may be too short to fully observe the impact of the
exogenous factor.

Type 3 The best method is to use chemostat-cultivated bacteria. A chemostat is
a device in which a flow is maintained so that the bacteria remain in the
exponential phase indefinitely. In addition, it is possible to control the
growth rate by changing the dilution rate. Unfortunately, the chemostat
methodology requires an appropriate device and an initiation to learn how to
control the various parameters. In the paper by Linqvist et al. (2014) there
is a useful comparison, for the same stress factor, between a batch and a
chemostat experiment.

###  B.2 Two key-variables and how to measure them

These key-variables can be described as follows:

(a) The overall number of cells, whether dead or alive, is measured with a
spectrophotometer which tests the turbidity of the medium. Increases in the
optical density give estimates of the number of cell divisions, and therefore
of the growth rate in the form $\alpha=(1/\Delta t)\log[y(t)/y_{0}]$ which is
constant for an exponential growth. Note that the doubling time is given by:
$\tau=\log 2/\alpha$.

(b) The second key-variable is the number of viable cells. It is estimated
either with a cytometer or through the CFU (colony forming unit) method. When
a cell can give rise to a colony on a Petri dish it is considered viable. Note
that some living cells may become unable to develop into colonies. For this
and other reasons the method is fairly inaccurate. Wilson (1921) gives many
examples of possible pitfalls.

Acknowledgements We are grateful to Nicolas Biais (Sorbonne University) for a
discussion which was at the origin of this paper. One of us (BR) had the
privilege of many stimulating exchanges with Arnaud Chastanet (INRAE) and he
wishes to extend many thanks to him.

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