INRA, Micalis CNRS-UMR 1319, 78350 Jouy-en-Josas, France

AgroParisTech, Micalis CNRS-UMR 1319, 78350 Jouy-en-Josas, France

Paris Dauphine University, CNRS-UMR 7534, Place du maréchal De Lattre de Tassigny, 75775 Paris, France

University of Rennes 1, CNRS-UMR 6625, Campus de Beaulieu, 35042 Rennes, France

CNRS/UPMC University of Paris 6, FRE 3231, Laboratoire Jean Perrin LJP, 75005 Paris, France

INRIA Paris-Rocquencourt, Domaine de Voluceau, BP 105, 781153 Le Chesnay, France

UPMC University of Paris 6, JL Lions Lab., 4 place Jussieu, 75005 Paris, France

Abstract

Background

Many organisms coordinate cell growth and division through size control mechanisms: cells must reach a critical size to trigger a cell cycle event. Bacterial division is often assumed to be controlled in this way, but experimental evidence to support this assumption is still lacking. Theoretical arguments show that size control is required to maintain size homeostasis in the case of exponential growth of individual cells. Nevertheless, if the growth law deviates slightly from exponential for very small cells, homeostasis can be maintained with a simple ‘timer’ triggering division. Therefore, deciding whether division control in bacteria relies on a ‘timer’ or ‘sizer’ mechanism requires quantitative comparisons between models and data.

Results

The timer and sizer hypotheses find a natural expression in models based on partial differential equations. Here we test these models with recent data on single-cell growth of

Conclusions

Confrontations between cell cycle models and data usually suffer from a lack of high-quality data and suitable statistical estimation techniques. Here we overcome these limitations by using high precision measurements of tens of thousands of single bacterial cells combined with recent statistical inference methods to estimate the division rate within the models. We therefore provide the first precise quantitative assessment of different cell cycle models.

Background

Coordination between cell growth and division is often carried out by ‘size control’ mechanisms, where the cell size has to reach a certain threshold to trigger some event of the cell cycle, such as DNA replication or cell division

Bacterial division is often assumed to be under size control but conclusive experimental evidence is still lacking and the wealth of accumulated data presents a complex picture. In 1968, building on the seminal work of Schaechter

Besides the work of Donachie, the assumption of size control in bacteria originates from a theoretical argument stating that such a control is necessary in exponentially growing cells to ensure cell size homeostasis, i.e. to maintain a constant size distribution through successive cycles. The growth of bacterial populations has long been mathematically described using partial differential equation (PDE) models. These models rely on hypotheses on division control: the division rate of a cell, i.e. the instantaneous probability of its dividing, can be assumed to depend either on cell age (i.e. the time elapsed since birth) or cell size. In the classical ‘sizer’ model, the division rate depends on size and not on age whereas in the ‘timer’ model it depends on age and not on size. Mathematical analysis of these models sheds light on the role of size control in cell size homeostasis. In particular, it has been suggested that for exponentially growing cells, a timer mechanism cannot ensure a stable size distribution

In the present study, we test whether age (i.e. the time elapsed since birth) or size is a determinant of cell division in

Results and discussion

Description of the data

Age and size distribution of the bacterial population

The results reported in this study were obtained from the analysis of two different datasets, obtained through microscopic time-lapse imaging of single _{
i
}: _{
i
}: _{
i
} and _{
i
} lead to different PDE models, and the statistical analysis was adapted to each situation (see below and Additional file _{
i
} and _{
i
}) we extracted the results of three experiments (experiments _{1},_{2} and _{3} and _{1},_{2} and _{3}). Each experiment _{
i
} corresponds to the growth of approximately six microcolonies of up to approximately 600 cells and each experiment _{
i
} to the growth of bacteria in 100 microchannels for approximately 40 generations.

**Supplementary text and figures.**

Click here for file

Given the accuracy of image analysis, we do not take into account variations of cell width within the population, which are negligible compared to cell-cycle-induced length variations. Thus, in the present study we do not distinguish between length, volume and mass and use the term _{
i
} or _{
i
}, by using a simple kernel density estimation method (kernel estimation is closely related to histogram construction but gives smooth estimates of distributions, as shown in Figure _{
i
} or sparse tree _{
i
}) and different experimental conditions, the distributions for the two datasets are not identical. The age distribution is decreasing with a maximum for age zero and the size distribution is wide and positively skewed, in agreement with previous results using various bacterial models

Distributions of cell age and cell size.

**Distributions of cell age and cell size.** Cell age **(A)** and cell size **(B)** distributions for a representative experiment of the _{i} dataset from Stewart _{i} dataset from Wang

Testing the timer versus sizer models of division

Age-structured (timer) and size-structured (sizer) models

The timer and sizer hypotheses are easily expressed in mathematical terms: two different PDE models are commonly used to describe bacterial growth, using a division rate (i.e. the instantaneous probability of division) depending either on cell age or cell size. In the age-structured model (Age Model) the division rate _{a} is a function only of the age

with the boundary condition

In this model, a cell of age _{a}(

In the size-structured model (Size model), the division rate _{s} is a function only of the size

In the Size Model, a cell of size _{s}(

For simplicity, we focused here on a population evolving along a full genealogical tree, accounting for _{
i
} data. For data _{
i
} observed along a single line of descendants, an appropriate modification is made to Equations (1) and (2) (see Additional file

Testing the Age Model (timer) and the Size Model (sizer) with experimental data

In this study we tested the hypothesis of an age-dependent versus size-dependent division rate by comparing the ability of the Age Model and Size Model to describe experimental data. The PDE given by Equations (1) and (2) can be embedded into a two-dimensional age-and-size-structured equation (Age & Size Model), describing the temporal evolution of the density _{a,s}

with the boundary condition

In this augmented setting, the Age Model governed by the PDE (1) and the Size Model governed by (2) are restrictions to the hypotheses of an age-dependent or size-dependent division rate, respectively (_{a,s}=_{a} or _{a,s}=_{s}).

The density ^{
λ
t
}
_{a,s}=_{a}), the existence of this stable distribution requires that growth is sub-exponential around zero and infinity

We estimate the division rate _{a} of the Age Model using the age measurements of every cell at every time step. Likewise, we estimate the division rate _{s} of the Size Model using the size measurements of every cell at every time step. Our estimation procedure is based on mathematical methods we recently developed. Importantly, our estimation procedure does not impose any particular restrictions on the form of the division rate function _{s}(_{a}(

We measure the goodness-of-fit of a model (timer or sizer) by estimating the distance _{
i
} or _{
i
}, thanks to a simple kernel density estimation method.

Analysis of single-cell growth

As mentioned above, to avoid unrealistic asymptotic behavior of the Age Model and ensure the existence of a stable size distribution, assumptions have to be made on the growth of very small and large cells, which cannot be exactly exponential. To set realistic assumptions, we first studied the growth of individual cells. As expected, we found that during growth, a cell diameter is roughly constant (see inset in Figure ^{2} coefficient of determination, which is classically used to measure how well a regression curve approximates the data (a perfect fit would give ^{2}=1 and lower values indicate a poorer fit). The inset of Figure ^{2} coefficient for all single cells for exponential (red) and linear (green) regressions, demonstrating that the exponential growth model fits the data very well and outperforms the linear growth model. We then investigated whether the growth of cells of particularly small or large size is exponential. If growth is exponential, the increase in length between each measurement should be proportional to the length. Therefore, we averaged the length increase of cells of similar size and tested whether the proportionality was respected for all sizes. As shown in Figure _{
m
i
n
} and _{
m
a
x
} below and over which the growth law may not be exponential (e.g. for the experiment _{1} shown in Figure _{
m
i
n
}=2.3 µm and _{
m
a
x
}=5.3 µm).

Analysis of single-cell growth.

**Analysis of single-cell growth.****(A)** Cell length vs cell age for a representative cell (black dots); exponential fit (red curve) and linear fit (black line). Inset: Cell width vs cell age for the same cell. **(B)** Increase in cell length during one time step (i.e. 1 min) as a function of cell length for _{i} data. During the lifetime of a cell, the cell length is measured at each time step and the increase in cell length between successive time steps is calculated. Black dots are the average length increase for every cell of a given experiment _{1}, as a function of cell length; error bars are the average +/−2 SEM (standard error of the mean). The red line is a linear fit for lengths between 2.5 µm and 4.5 µm. Inset: For each single cell of _{1}, the evolution of cell length with age was fitted with a linear or an exponential function (as shown in panel A). We thus obtain a distribution of ^{2} coefficients corresponding to the linear (green) and exponential (red) fits.

The age-size joint distribution of

We used both the Age Model and Size Model to fit the experimental age-size distributions, following the approach described above. The growth law below _{
m
i
n
} and above _{
m
a
x
} is unknown. Therefore, to test the Age Model, growth was assumed to be exponential between _{
m
i
n
} and _{
m
a
x
} and we tested several growth functions _{
m
i
n
} and _{
m
a
x
}, such as constant (i.e. linear growth) and polynomial functions. Figure _{1} shown in Figure _{1} data) with the reconstructed distribution shown in Figure _{1} data) we can see that the Age Model fails to reconstruct the experimental age-size distribution and produces a distribution with a different shape. In particular, its localization along the _{1} data (panels A and C), the red area corresponding to the maximum of the experimental distribution is around 2.4 on the

Experimental and reconstructed age-size distributions for representative experiments from Stewart _{1}) and Wang _{1}).

**Experimental and reconstructed age-size distributions for representative experiments from Stewart****(**_{1}**) and Wang****(**_{1}**).****(A,B)** Experimental age-size distributions for representative experiments _{1} (A) and _{1} (B). The frequency of cells of age **(C,D)** Reconstruction of the distributions using the Age Model (C: reconstruction of the data _{1} shown in panel A; D: reconstruction of the data _{1} shown in panel B). These reconstructed distributions were obtained from simulations with the Age Model using a division rate estimated from the data (C: from _{1}, D: from _{1}). The growth functions used for the simulations are detailed in the Methods section. **(E,F)** Reconstruction of the distributions using the Size Model (E: reconstruction of the data _{1} shown in panel A; F: reconstruction of the data _{1} shown in panel B). These distributions were obtained from simulations with the Size Model using a division rate estimated from the data (E: from _{1}, F: from _{1}) with an exponential growth function (see Methods).

As an additional analysis to strengthen our conclusion, we calculated the correlation between the age at division and the size at birth using the experimental data. If division is triggered by a timer mechanism, these two variables should not be correlated, whereas we found a significant correlation of −0.5 both for _{
i
} and _{
i
} data (^{−16}; see Additional file

We used various growth functions for _{
m
i
n
} and _{
m
a
x
} but a satisfying fit could not be obtained with the Age Model. In addition, we found that the results of the Age Model are very sensitive to the assumptions made for the growth law of rare cells of very small and large size (see Additional file

In contrast, the Size Model is in good agreement with the data (Figure

The quantitative measure of goodness-of-fit defined above is coherent with the curves’ visual aspects: for the Size Model the distance _{
i
} data (16% to 26% for _{
i
} data) whereas for the Age Model it ranges from 51% to 93% for _{
i
} data (45% to 125% for _{
i
}).

The experimental data has a limited precision. In particular, the division time is difficult to determine precisely by image analysis and the resolution is limited by the time step of image acquisition (for _{
i
} and _{
i
} data, the time step represents respectively 5% and 8% of the average division time). By performing stochastic simulations of the Size Model (detailed in Additional file

Size control is robust to phenotypic noise

Noise in the biochemical processes underlying growth and division, such as that created by stochastic gene expression, may perturb the control of size and affect the distribution of cell size. We therefore investigated the robustness of size control to such phenotypic noise. The Size Model describes the growth of a population of cells with variable age and size at division. Nevertheless, it does not take into account potential variability in individual growth rate or the difference in size at birth between two sister cells, i.e. the variability in septum positioning. To do so, we derived two PDE models, which are revised Size Models with either growth rate or septum positioning variability (see Additional file

Variability in individual growth rate has a negligible effect on the size distribution

For each single cell, a growth rate can be defined as the rate of the exponential increase of cell length with time

We recently extended the Size Model to describe the growth of a population with single-cell growth rate variability (the equation is given in Additional file _{
i
} data. The resulting size distribution is virtually identical to the one obtained without growth rate variability (Figure

Influence of the variability in individual growth rate and septum positioning on the cell size distribution.

**Influence of the variability in individual growth rate and septum positioning on the cell size distribution.****(A)** Size distributions simulated using the Size Model with the division rate _{s} estimated from _{1} data and an exponential growth (**(B)** Simulated size distributions using the Size Model with the same division rate _{s} as in A and a constant growth rate

Variability in septum positioning has a negligible effect on size distribution

The cells divide into two daughter cells of almost identical length. Nevertheless, a slight asymmetry can arise as an effect of noise during septum positioning. We found a 4% variation in the position of the septum (Additional file

Conclusions

In the present study, we present statistical evidence to support the hypothesis that a size-dependent division rate can be used to reconstruct the experimental age-size distribution of

Noise in biochemical processes, in particular gene expression, can have a significant effect on the precision of biological circuits. In particular, it can generate a substantial variability in the cell cycle

Our approach is based on comparisons between PDE models and single-cell data for the cell cycle. Such comparisons were attempted a few decades ago using data from yeasts (e.g.

Methods

Data analysis

The data of Stewart _{
i
} data, cell segmentation was based on the localization of brightness minima along the channel direction (see _{
i
} data, local minima of fluorescence intensity were used to outline the cells, following by an erosion and dilation step to separate adjacent cells (see

For both datasets we extracted data from three experiments done on different days. We did not pool the data together to avoid statistical biases arising from day-to-day differences in experimental conditions. Each analysis was performed in parallel on the data corresponding to each experiment.

Numerical simulations and estimation procedures

All the estimation procedures and simulations were performed using MATLAB. Experimental age-size distributions, such as those shown in Figure ^{7} equally spaced points on [ 0,_{
m
a
x
}] and 2^{7} equally spaced points on [ 0,_{
m
a
x
}], where _{
m
a
x
} is the maximal cell age in the data and _{
m
a
x
} the maximal cell size (for instance _{
m
a
x
}=60 min and _{
m
a
x
}=10 µm for the experiment _{1}, as shown in Figure _{s} for each experiment, the distribution of size at division was first estimated for the cell size grid [ 0,_{
m
a
x
}] using the ksdensity function. This estimated distribution was then used to estimate _{s} for the size grid using Equation (20) (for _{
i
} data) or (22) (for _{
i
} data) of Additional file _{s} and an exponential growth function (^{−1} for the _{1} experiment and ^{−1} for _{1}). For the Age & Size Model, we discretized the equation along the grid [ 0,_{
m
a
x
}] and [ 0,_{
m
a
x
}], using an upwind finite volume method described in detail in

meeting the CFL: Courant-Friedrichs-Lewy stability criterion. We simulated ^{−8}). To eliminate the Malthusian parameter, the solution

The age-dependent division rate _{a} for each experiment was estimated for the cell age grid [0,_{
m
a
x
}] using Equation (14) and (16) of Additional file _{
m
i
n
} and _{
m
a
x
}; between _{
m
i
n
} and _{
m
a
x
} growth is exponential with the same rate as for the Size Model). For instance for the fit of the experiment _{1} shown in Figure ^{3}+0.036^{2}−0.094_{1} shown in Figure ^{3}+0.063^{2}−0.33_{1} data).

Simulations of the extended size models with variability in growth rates or septum positioning (Equations (23) and (24) in Additional file ^{7} equally spaced points on [ 0,_{
m
a
x
}] and 100 equally spaced points on [ 0.9_{
m
i
n
},1.1_{
m
a
x
}], where _{
m
i
n
} and _{
m
a
x
} are the minimal and maximal individual growth rates in the data.

Abbreviations

PDE: partial differential equation.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

LR and MD conceived and designed the study, performed the analysis and drafted the manuscript. MH participated in the design of the study and helped to draft the manuscript. NK, SA and JR provided analytical tools. All authors read and approved the final manuscript.

Acknowledgments

We thank S Jun and E Stewart for sharing their data, D Chatenay, E Stewart, J Hoffmann, M Elez and G Batt for critical reading of the manuscript and Richard James for English editing. The research of M Doumic was supported by the ERC Starting Grant SKIPPER ^{
A
D
}.