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Conceived and designed the experiments: MDP SDM. Performed the experiments: MDP SDM. Analyzed the data: MDP SDM LR. Contributed reagents/materials/analysis tools: MDP SDM. Wrote the paper: MDP SDM. Conceived the experiments and contributed to manuscript writing: ABL FT.

The authors have declared that no competing interests exist.

The spread of epidemics not only depends on the average number of parasites produced per host, but also on the existence of highly infectious individuals. It is widely accepted that infectiousness depends on genetic and environmental determinants. However, even in clonal populations of host and viruses growing in homogeneous conditions, high variability can exist. Here we show that

In the development of epidemics it is not only the average parasite production that matters, but also the distribution of secondary cases infected by single individuals. The spread of a disease can be deeply affected by heterogeneities in infectiousness, with high viral charge ‘superspreader’ individuals triggering stronger epidemic events

Variability in the outcome of infection has been largely documented and is generally assumed to be due to genetic or environmental variability

Here, we focus on the first stages of a viral infection and examine the evolution of viral charge distribution as large cell-cell differences emerge. We performed experiments with

We combined single-cell measures and a mathematical model to investigate the mechanisms generating variability in clonal populations of virally infected bacteria growing in a homogeneous environment. By using a modified virus that constitutively expresses a fluorescent protein, we were able to quantify the amount of virus produced per cell and to follow the evolution of the viral charge distribution in the course of infection. We explain the variability in virus accumulation within cells as the amplification of initial differences in bacterial growth rate. A mathematical model shows that this simple mechanism allows us to reproduce the kinetics of the variability and the total virus production in different controlled environmental conditions.

We found that, far from being an unusual property of the phage α studied by Max Delbrück, the high heterogeneity in virus production per cell is a common characteristic of several phages, in spite of marked differences in their infection strategy and impact on the host.

Distribution of Plaque Forming Units counts from single cell (or lineages in the case of M13) for the following phages: phage α (reproduced from Delbrück, 1945); the double-stranded RNA phage R17, the double-stranded DNA phage T4 and the filamentous single-stranded DNA phage M13. The phage M13 is not lytic, so the number of virions is measured one hour after infection (1–2 host cells). All phages show a strong heterogeneity in the number of virions produced per cell: all the distributions are wide and asymmetric, the extreme case being the filamentous phage M13. See

We expect the contribution of mutations to be minor in the generation of burst size variability. Phage M13 relatively low replication error rate –a maximum of 7.10^{−7} errors per base measured for phage M13

The estimation of phage production in single cells by PFU counts is very time-consuming and it is not suited for large sample sizes, nor for high time resolution. In order to quantify the distribution of the viral charge per cell and its kinetics, we hence introduced the Yellow Fluorescent Protein (

(

On the contrary to viral proteins, YFP are not extruded from the cell and instead accumulate within the cells. The instantaneous YFP production of a single cell is proportional to the number of dsDNA molecules present inside this cell. Similarly, the different elements composing the virions, namely the capsid proteins and the ssDNA are also produced from the dsDNA and therefore virions production is expected to be proportional to the production of YFP. Assuming that binomial partitioning of molecules at cell division does not significantly decorrelates the number of YFP and the number of virions, YFP intensity of a single cell should hence reflect its viral charge.

This statement was not directly demonstrated because of the difficulty to simultaneously measure on individual cells both the number of virions produced and cell fluorescence. Therefore, we first indirectly verified that fluorescence is a good measure of virions production in whole-population assay. We showed that the distribution of PFU counts in single cells one hour after infection rescales to the fluorescence intensity distribution (

We addressed the dynamics of the emergence of variability among infected cells by following over time cells fluorescence within infected populations. After synchronous infection of the bacteria, we let the population of infected cells grow until they enter the stationary phase. We will call this growth condition S, as saturating. Besides single-cell measurements, traditional population-level quantities have been observed: the absorbance of the culture (Optical Density) and the number of infectious free phage in suspension (PFU). These measures have been repeatedly performed in the course of the infection by the phage, so as to obtain the evolution of the distribution of fluorescence and of the total virus production in the course of time (see

First column: (_{d} = 30, δ = 1.8, τ = 30, f_{thresh} = 10^{4}. The two different growth conditions differ by their growth rates: in E condition g_{0} = 1.5 (measured g_{0} = 1.5±.6), in S condition the system was forced with the slope of a fit (2^{rd} order polynomial) of the measured optical density in the course of the non-diluted growth. A delay of 5 minutes has been introduced for accounting for the maturation time of the fluorescent proteins. The initial growth rate distribution is obtained by rescaling the growth rate distribution measured in non-infected cells so that the average growth rate matches that of the infected culture (standard deviation σ = 0.2).

We wondered what is the origin of such variability, and whether its rapid increase at the entry of stationary phase is a response to bacterial density. In order to address these questions, we repeated the experiment and kept the culture in exponential growth (E condition) without nutrient limitations. We did so by re-diluting the culture in fresh medium every hour. We used two different media: rich medium Luria-Bertani (LB) and exhausted medium (supernatant from a two-hours culture) supplemented with yeast extract, the main nutritional component of LB medium. This second medium could contain chemical cues or quorum sensing effectors that carry information on population density in the early stationary phase, but it is not depleted in nutrients. This way, we checked whether variability emerges also if resources are not limiting or it is triggered by signalling processes taking place at the entry in the stationary phase. The culture grown in replenished supernatant behaved as the culture growing in LB medium. This demonstrates that signals of population density have no effect on the dynamics of the fluorescence intensity distribution.

In the E condition, although the nutrient availability is kept almost constant, we still observe (

(

The experimental observations in saturating growth conditions suggest that viral proteins accumulation may be related to bacterial growth rate, higher fluorescence being associated to slower growth of the culture. That stochastic variations in growth rate can shape protein concentration distribution in cells has been proposed in the context of chromosomally encoded genes ^{−16}) was evidenced in the growth rate of mothers and daughters

In the case of saturating growth, we performed simulations with the same parameters as in condition E, but forcing the average growth rate to be the same as the one measured in the course of the experiment. This leads to a fluorescence distribution wider than in the E condition, and whose moments increase at a progressively slower rate.

However, when cell mortality starts to be important, the model cannot be quantitatively compared to the experimental measures of cell fluorescence, since in the simulations dead cells are immediately removed, while in the culture cells that do not divide seem to accumulate fluorescence before lysing, thus increasing the weight of high fluorescence classes. This or other oversimplifications of the model, that does not take into account any variation in physiological properties at the entry in stationary phase other than the growth rate, may be the reason for the model predicting a quantitatively lower fluorescence than that observed in the experiments in S condition.

Total phage production is a quantity typically measured in the course of infections and is currently considered a population-level estimate of virulence. We measured the total virus population in the supernatant (PFU) in the conditions E and S and during 6 hours following infection (

Time evolution of the plaque forming unit (PFU) counts of the whole population in exponential (E, blue symbols) and saturating (S, green symbols) culture conditions following infection by the phage M13yfp. Dots represent the mean +/− the standard error of three independent experiments. In both conditions the increase in PFU is maximal in the first hour following infection. The solid lines represent the total fluorescence calculated by the model (parameters as in

The model points out the processes in the individual-level dynamics and in the distribution kinetics that are important in shaping such a trend in epidemics.

By studying the parameter sensitivity of the model, we can interpret the population-level measures: the initial fast rate of virions production corresponds to the phase of cell-to-cell variability expansion. The cross over to a phase of constant growth rate occurs as the viral concentration within individual cells approaches an equilibrium value, whose magnitude depends on the feedback of virus concentration onto its own production. Such feedback can be due to active regulation of dsDNA replication or to the competition of viruses for cellular resources. In the model, no feedback in virions production is needed for reproducing the infection dynamics, nor is temporal modulation of virions production rate. Instead, cell death seems to be essential in decreasing the virions production rate at the entrance in stationary phase, consistent with the significant percentage of dead cells observed in S growth conditions.

It is commonly observed that differences exist in the individual response to viral infection, from the level of single cells to that of complex organisms endowed with an immune system. This variability in virus production, and in turn in infectiousness, has been mostly attributed to genetic diversity of the host

In the case of the filamentous M13 phage, we explain the emergence of variability as the outcome of two opposing dynamical processes: the autocatalytic amplification of the virus and its dilution as cells grow. We suggest that a key factor determining the kinetics of viral repartition among cells is the difference in growth rate between the infected bacteria and their parasites: the virus accumulates faster in bacteria growing slower than in those having a faster growth rate. At the level of a population, the same mechanism explains why when bacterial growth slows down, as at the entry in stationary phase, virus production increases.

This mechanism of amplification of small growth rate differences is general and may occur whenever a parasite undergoes autocatalytic replication, that is its production rate is proportional to its cellular concentration. This typically occurs at the very beginning of an infection, and it is such initial stage of an epidemic that the present model aims at describing. Within a short time from the infection, variability in virus production can increase enormously and give rise to highly skewed distributions, as those observed for lytic phages (

Another factor that is likely to influence the distribution of viruses is the imperfect heritability of growth rate within cell lineages. In the case of

Besides accurately accounting for the observed distributions of fluorescence and phage productions, the model predicts that cells belonging to classes with growth rate higher than the maximal phage production rate should eventually get rid of the virus, and thus generate a fast-growing subpopulation of ‘healed’ cells, susceptible of being re-infected as soon as the virus stops repressing the production of the F pilus. This scenario is compatible with the observation that after few hours from the infection, non-infected bacteria appear

The proposed mechanism provides a simple explanation for the trade-off between horizontal and vertical transmission commonly observed in parasites that do not induce the death of their host

We model the population of infected bacteria as composed of a large number of classes, corresponding to different growth rates. These classes can be thought to as being lineages with epigenetic (heritable) differences in doubling time.

Each class is characterized by three variables: the biomass B, the intracellular concentration D of double-stranded viral genome (dsDNA) and the intracellular concentration F of fluorescent proteins. These state variables evolve deterministically based on four parameters: growth rate g of the bacteria, viral DNA duplication rate δ, viral DNA expression rate (or fluorescent protein production rate) τ, carrying capacity K_{d} for viral ds-DNA, and are defined by the following equations:_{d} and taken for simplicity identical in all cells), and to an initial absence of fluorescent proteins. The growth rates are assigned according to a given distribution of average g_{0}. The distribution of fluorescence is computed taking into account the different biomass of classes with different growth rates and with cells dying stochastically with a probability that is proportional to the fluorescence intensity, up to a threshold value f_{thresh} above which death is certain. Such an assumption is confirmed by the observation that highly fluorescent cells often fail to divide and eventually die.

The parameter g_{0} is assigned according to the measured growth rates of infected cultures: for the E condition it is simply the rate of exponential growth, while in the S condition the system is forced by progressively changing g_{0} so as to match the measured values of declining velocity. The growth rate distribution is that measured for non-infected bacteria (Fig. S4c in the SI), and translated so that its average is g_{0} and its standard deviation is maintained to σ = 0.2.

The carrying capacity for the double-stranded DNA (K_{d} = 30) has been taken from independent measures of an average dsDNA content per cell (13).

The parameters δ_{LB} = 1.8 h^{−1} and τ = 30 h^{−1} of the model have been chosen so as to reproduce the experimentally measured shape of the fluorescence distribution and dynamics of its first three moments in E conditions.

The viral DNA duplication rate has to be chosen of the same order of magnitude of the bacterial duplication rate, otherwise either the viral content would explode and the host-parasite couple disappears, or the virus would be diluted out (15).

A fluorescence concentration threshold f_{thresh} = 10^{4}, above which the accumulation of fluorescent particles kills the bacterial cells, has been introduced in order to reproduce the large mortality observed in the S condition. This threshold is so high that its effect in the case of exponential growth is minor, in accordance with the fact that a lower number of dead cells is observed in E condition.

The expansion in the fluorescence concentration reflects the initial exponential growth of the virus. Later, this saturates and a quasi-steady state sets in, where internal concentrations of viruses and fluorescence are stable, but the repartition changes according to the differential demography: classes with higher growth rate tend to be proportionally more represented. The shift of the distribution due to such process is however much slower than the infection dynamics and is almost negligible on the time scale considered here.

Sensitivity analysis has been performed by numerical simulation, starting from an initial Gaussian distribution of mean g_{0} and standard deviation σ. Although the expansion of the initial distribution always occur when parameters are varied, the specific shape of the distribution and the kinetics of its moments depend rather sensitively on the dsDNA duplication rate δ. This reflects the importance of the difference in growth rate between host and parasite, and is confirmed by the fact that similar population dynamics are obtained if the viral growth rate -instead of the bacterial one- is distributed. Increasing δ to values higher than the maximal bacterial growth rate causes the distribution to displace visibly towards higher fluorescence values, before reaching a quasi-steady state, which does not correspond to our observations. Neither does the qualitative behaviour of simulations at low δ values, where after an initial burst, the distribution collapses towards small fluorescence values.

The maximal number of viral dsDNAs sustainable by one cell, K_{d}, controls the extent of the distribution expansion: the smaller it is, the narrower the quasi-stationary distribution and the shorter the transient before reaching it.

Finally, the parameter τ measures the fluorescence production rate, but has no quantitative meaning, since fluorescence is measured in relative units.

Modifications in the initial growth rate distribution affect the distribution kinetics in two ways: a decrease/increase in average growth rate is qualitatively equivalent to increasing/decreasing δ, corresponding to the fact that what matters is the growth rate differentials rather than the absolute replication rate of parasite and host.

By changing the variance of the initial growth rate distribution, the main effect is a change in the transient to the quasi-stationary distribution, which is faster for wider distributions.

The

A mid–log phase (OD600 between 0.2 and 0.3)

Modifications of the phage genome leading to an even modest perturbation of the phage life cycle can lead to the eventual death of infected cells and generate non-productive infections. For enhancing the probability of obtaining a viable virus, we started from a M13 derivative, the cloning vector M13mp19 (New England Biolabs, Ipswitch, Massachusset, United States), which possesses the LacZα gene in its origin of replication. To construct the reporter p2

In order to quantify cell-cell differences in fluorescence due to variations in the activity of the promoter, we incorporated the distinguishable cyan (cfp) allele of the yfp gene in the the chromosome of the bacterial host (see bacterial strains section) and under the same promoter p2

Samples were taken at 20 or 30 minutes intervals. In order to obtain a monolayer of cells, samples of growing cultures of M13yfp infected cells were concentrated by centrifugation (1.5 min at maximum speed in a bench centrifuge) and spread on LBagarose (Qbiogene) supplemented with chloramphenicol (Sigma, 30 µg/ml) and propidium iodide (Molecular Probes, 0.5 µg/ml). A series of images of cell monolayer was taken with a CoolSNAP HQ (Princeton Instruments, Trenton, New Jersey, USA) at 100× magnification by an automated microscope (Zeiss 200M; Zeiss, Jena, Germany), in phase contrast and in fluorescence at wavelength 514 nm (YFP), 540nm (propidium iodide) and 420nm (CFP) during 1s exposure time. Excitation light was limited to 50% of the output of the 100-W Hg vapor lamp.

Images were treated with the Metamorph software (Universal Imaging, Downingtown, Pennsylvania, USA). Image analysis procedure identified cells and then quantified their mean fluorescent intensities with YFP, CFP and IP filter sets. We analysed more than 10,000 cells at each time point. Fluorescent background of the LBagarose media was subtracted from each value of fluorescence. Cells were considered dead if the IP fluorescence level was above 1.5 times the medium value of the IP fluorescence of all cells.

The model simulations and statistical analysis were performed with MATLAB (Mathworks, Natick, MA). The same code has been used for the statistical analysis of both the experimental data and the simulated distributions.

Scheme of M13 life cycle. Upon entering the cell, the phage genome is duplicated. Subsequently, transcription, replication, and generation of single-strand genomes occur. Phage proteins assemble around single-stranded genomes to produce virions that are subsequently extruded from the cell. The intracellular dynamics of phage replication in individual cells was tracked by means of a fluorescent gene reporter introduced in the genome of the filamentous phage M13mp19, a derivative of phage M13.

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Effects of yfp insertion on growth rates of the phage and its host. A Effect of yfp insertion on the host. As previously reported, infection reduces the culture growth rate, and the same bacterial growth rate reduction was obtained with the WT M13 phage, the phage M13mp19 possessing a polycloning site in its genome and the YFP encoding phage M13 Yfp. This indicates that the cost of infection does not increase with the size of the phage. The same bacterial population has been infected at time 0 at a multiplicity of infection of 0.1. B effect of gene insertion on phage doubling time (mean +/− standard deviation). The phage doubling time is the time necessary for the number of free virion to double, and is calculated from the exponential increase in PFU measured from 0.5 to 2 hours after infection.

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Relation between the number of phage produced and phage-encoded fluorescence. Number of total PFU counts versus total fluorescence of the culture. The measures have been made between 3 and 6 hours after the beginning of the infection in 166 distinct M13Yfp infected bacterial cultures. Solid line: exponential regression, R2 = 0.88. During conditions of sustained growth of the infected cells, the number of phage particles produced is proportional to the total yfp fluorescence of the culture, indicating a correlation between yfp intensity and number of virions per cell. The use of fluorescence as a measure of virions production is also validated by the comparison of single-cell fluorescence distribution and PFU counts on single-cells (

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Distribution of physiological bacterial parameters. A and B: At each time point, the distribution of CFP fluorescence intensity, which reflects chromosomal expression of the reporter gene, show only relatively low cell-cell variation (A) and follows a distribution whose characteristics are stable with time (B). This is consistent with our construction that places CFP under a constitutive promoter. The relative error is about 10%. C: Distribution of the log2 transformation of the growth rate of non-infected bacterial cells. Growth rates are obtained be time-lapse microscopy of cells plated on agar in nutrient-rich conditions as described in (Stewart, 2005). The bacterial growth rate follows an almost normal distribution of average 2.1 hours-1 and standard deviation 0.2 hours-1. This is used, after rescaling to the measured growth rate of infected cells, as the initial condition for the model simulations.

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We thank G. Paul and E. Stewart for valuable discussions and advices and A. Demarez, E. Sabuncu and M. Mariadassou for help with the experiments.