I. Introduction

Asymmetric cell division is referred to as the mechanism by which a mother cell splits into two daughter cells with distinct cellular fates. As first shown by Edwin Conklin [1], this process is usually achieved by asymmetric inheritance of intrinsic cell fate determinants, like specific proteins or RNA, and plays a crucial role during the organisms’ development, favoring cell differentiation and self-renewal [2, 3].

Increasing evidence is vouching for a more ubiquitous presence of fluctuations in the partitioning of cellular elements. Indeed, asymmetric segregation has been observed in non-differentiating cell populations as different as bacteria [4], yeast [5], and tumoral cells [6, 7]. In bacterial populations, differential segregation of cellular components has been linked to antibiotic resistance [8]. Similarly, Katajisto et al. [9], studying the partition of mitochondria in mammalian epithelial stem-like, found that asymmetric segregation of older mitochondria enables cells to protect themselves from aging by preventing the accumulation of misfolded proteins. Asymmetric partition of centrosomes has been observed in human neuroblastoma and colorectal cancer stem cells, controlled by polarity factors and influenced by the segregation of subcellular vesicles during cell division [10].

To quantify the strength of fluctuations at division, the common route goes through fluorescence microscopy measurements of either fixed or live cells, where fluorescent dyes are used as a proxy for the cellular element of interest. The partition statistics are then estimated by looking at the fraction of fluorescence intensity inherited by the two daughter cells [11, 12]. This approach is, in principle, much informative; however, it needs the (often manual) identification of hundreds of division events to be reliable: a time-consuming procedure that dampens a wide characterization of the partition statistics of different cell types.

To cope with these limitations, we propose using high-throughput flow cytometry measurements to quantify the fluctuations in the partitioning of cellular components. In particular, flow cytometry has already been applied to study asymmetric division in specific cellular systems: Yang et al. [5], sorted budding yeast cells to analyze the differential segregation of proteins between daughters cells, while Peruzzi and coworkers first applied multi-color flow cytometry to show that leukemia cells divide mitochondria and membrane proteins asymmetrically [13]. Here, we generalized the protocol to adherent cell types and probed its accuracy by comparing the measured fluctuations to those observed with extensive time-lapse microscopy experiments on a panel of normal and cancerous epithelial cells.

Our work shows that (i) there is an exact, analytical expression linking the inherited cellular components distribution dynamic to the specifics of the partition process. Based on this result, we (ii) proposed an experimental protocol to tag with fluorescent dyes cellular components and measure the dynamics of fluorescent distribution, whose analysis allows for the accurate estimates the degree of asymmetry in the partition process. Applying our method to a panel of both normal and cancerous human colon cells, we found that (iii) different lines of colon adenocarcinoma display very distinct extents of fluctuations in the cytoplasm partition, that reflect in (iv) different degrees of asymmetric division of their size.

II. Results

A. Modeling cells population dynamic in presence of partitioning noise

We look for a theoretical framework able to describe the time evolution of a cell population and its dependence on the partitioning process. In particular, we consider a population of cells characterized by a certain initial distribution, P (m)₀, of element m. Monitoring the time evolution of the abundance of specific cellular elements over multiple divisions in single-cell experiments, it is observed [14] that a component’s counts presents dynamics qualitatively similar to the one reported in Figure 1(b). Indeed, variations in the components number are determined by: (i) the production and degradation processes along the duration of the cell growth phase; and (ii) non identical apportioning of the components between the two daughter cells. Here, we will assume that the components number varies only due to the partition events and look for a quantitative description of the evolution of the element distribution. Such an assumption allows us to neglect the time between divisions and reduces the dynamics to a progressive dilution of the components numbers in subsequent generations. Note that it is possible to model the complete growth and division process (see for instance refs. [15–17]), but it is more difficult to separate the different sources of noise.

Each cell i of the population divides into two daughter cells that we can label unambiguously as 2i and 2i + 1, which inherit respectively m_2i and m_2i+1 components. The process, exactly at division, must conserve the total number of elements, i.e. m_i = m_2i + m_2i+1 and we call partitioning fraction the quantity . To model noise in partitioning, we assume that at each division a random value for f is extracted from a probability distribution function, П(f). Given these assumptions, we can write the probability distribution of the components number of a daughter cell m after g divisions from the mother M as:

where P_k(M) is the components distribution at generation k and it is given by the sum of the distributions of all siblings cells of the k-th generation. After g generations, the cell lineage generated from cell i forms a lineage tree composed of n_g = 2^g cells. Without losing generality, the division probability can be expressed, as:

Therefore, we can write the probability distribution, P (m), of the daughter cells inheriting a fraction, f, of the mother elements:

As is often the case, it is convenient to characterize the distributions in terms of their moments [16]. We identify the cell m_2i (m_2i+1) as the one inheriting a fraction f (1 — f). In particular, the mean of the daughter cell 2i components distribution can be expressed as:

Where E[·] stands for the average. Note that every computation can be symmetrically done for the cell 2i + 1. Thus, we have that in a single partition, the mean number of inherited elements of the daughter cells depends on the asymmetry of the П(f) and on the mean of the P(m_i).

In Figure 1(d), we show the evolution of the components number distribution at different generations. The П(f) chosen for the simulation is a Cauchy distribution with location and scale γ = 0.7.

The total distribution is the sum of 2^g sub-populations which derives from the division process. For example, if we assume , after one division we would have a population distributed around fμ_i and one at (1—f)μ_i. After two division the populations would grow to four sub-populations centered in f²μ_i, f (1 — f)μ_i, (1 − f)fμ_i and (1 — f)²μ_i and so on.

To compute the mean relative to an entire generation, we need to sum over all the different sub-populations:

where is the mean of k-th sub-population at generation g and 2^g is the total number of sub-populations for that generation. This equation [A5] can be simplified into the form (see Appendix for the full computation):

Fig. 1

where μ₀ is the mean of the initial population. It is relevant to notice, that independently of the characteristic of П(f), the mean always halves itself at each generation. This result is simply understood considering that for each cell that inherits a fraction of the compound, its sisters inherit a fraction 1 − . Hence, at every division, the compound counts is diluted by an average factor of 2 in the population, and partition distributions with different properties lead to the same behavior of μ_g (see Figure 1(f)). Therefore, no relevant insights on the П(f) can be obtained from the first moment. We thus moved to consider the second moment, i.e. the variance. Similarly to what is done for the mean, the variance of the inherited fraction of elements can be expressed as:

and in turn, the variance of a generation, starting from the consideration of the variance of a mixture of distributions can be written as (see Supplementary materials for details):

with

It is possible to obtain a recursive equation for A_g (see Appendices), A_g = A_g-1E[f²] = A₀E[f²]^g which leads to a compact form for equation [A8]:

where . Finally, equation 10, links the variance of the components distribution for the entire cells population at generation g, to the properties of the П(f). It is interesting to highlight that there’s no dependence on the symmetric properties of the process. Up to this order, the only relevant feature that shapes the dynamic is the extent of the fluctuations. Indeed, as shown in Figure 1(e), the same П(f), but with a different variance shows a different behavior for . Our analytical results show that the dynamical behavior of the population variance depends non-trivially on the second moment of the underlining partitioning distribution. The next section will be devoted to demonstrating, how this condition can be exploited in experiments to measure the partition noise component.

B. Measuring fluctuations via flow cytometry

As shown by Peruzzi et al. [13] for leukemia cells growing in suspension, it is possible to use flow cytometry to follow the evolution of properly marked cellular elements in time. Here, we generalized the protocol considering a panel of colon cell lines, i.e. Caco2, a human epithelial colorectal adenocarcinoma cell line; HCT116, a human colorectal carcinoma cell line; and CCD-18Co, a normal human colon fibroblast cell line. To measure the degree of asymmetric division, we marked cells’ cytoplasm via a fluorescent dye and followed its dilution through successive generations. As explained in detail in the Materials and Methods section, the experimental procedure we used is based on the following main steps: staining of the selected compound, sorting of the initial population, plating, and acquisition of the distinct populations in the following days. The sorting allows us to have an initial population with a narrow peak at a specific fluorescence intensity.

To achieve this, we used fluorescence-activated cell sorting (FACS) to isolate a specific cell population based on cell morphology and fluorescence intensity. The sorted population is divided into distinct wells, one per acquisition, in equal numbers and kept under identical growth conditions to mitigate inoculum-dependent variability among samples [18]. Figure 2(a) shows an example of the time course dynamics obtained with HCT116 cells (see Methods for details). Each distribution represents a different acquisition of a whole well at increasing times from plating (0 to 84 hours from the bottom to the top). It is possible to neatly observe the succession of peaks associated with different generations that succeed one another and the average decrease of the fluorescence intensity. Note that different generations can coexist at the same time point. One can spot the flow of cells towards higher generations and the behavior of the peak heights, which grow over time, reach dominance and then decay. The shift on the y-axis allows to follow the evolutionary dynamics while being able to see the overlapping of the same generation at different time points.

As in each measurement, the distribution of fluorescent intensity is a mixture of distributions corresponding to different generations:

To obtain the necessary information on the properties of each generations we performed a fitting of the data via a Gaussian Mixture Model (GMM) combined with an Expectation Maximization algorithm (details are explained in “Materials and Methods”). Indeed, assuming that each generation is log-normally distributed with mean μ_g and variance of the corresponding Gaussian distribution, equation 11 can be rewritten as :

An example of the outcome of the GMM algorithm is shown in Figure 2(b) for two acquisitions (48h and 60 h from plating). Blue-shaded distributions represent experimental data, while the best solution of the Gaussian mixture is shown with a solid line. Each generation’s fit is identified with a different color. Upon an overnight, we can identify generations 0, 1, and 2 in both the acquisitions. One can see that their relative fractions changed: generation 3 became dominant overnight and generation 4 appeared.

Figure 2(c) shows the behaviors of μ_g and σ_g as functions of the generation number g for three independent replicas of the time course experiments with CaCo2 cells. Dashed lines represent the best fit of the data against Equation 10, which yields the variance σ_f of the partition distribution probability, П(f). The outcomes of the fitting procedure on the time course dynamic for all the replicas of three studied cell lines are reported in Figure 2(d). In particular, Caco2 and CCD18Co cells show a higher degree of asymmetry with respect to HCT116 as measured by the division asymmetry of the corresponding partition distribution.

Fig. 2

C. Validation of the method via live fluorescence microscopy

The flow cytometry-based protocol combined with the theoretical model provides insights into the properties of the cell partition function (see Figure 2). To validate our approach, we performed extensive live time-lapse microscopy experiments to measure the partition fraction of a labelled compound during proliferation, providing a direct measure of the П(f). Specifically, we aimed to follow a growing cell colony over time, detect divisions, evaluate the fluorescence intensity of mother and daughter cells, and calculate the inherited fraction. The detailed experimental procedure is described in the Materials and Methods. Briefly, cells are stained for the cytoplasm and are plated at low density (approximately 10³ cells/well) on IBIDI cell imaging chambers (μ-Slide 4 and 8 wells). After 24 hours (sufficient time for complete cell adhesion to the slide) the wells are washed, the media is renewed and live imaging acquisition time-lapse is started. Multiple fields are followed in each well, and for each field, we acquire a z-stack bright-field and fluorescence image (a single plane is shown in Figure 3a) at constant intervals (20 min) for 3 days. The fields are manually selected at the beginning of the acquisition based on cell density and fluorescence intensity. The recorded time lapses have been then manually analyzed to identify divisions. A sample outcome of the analysis procedure is shown in Figure 3(b). The total fluorescence of the cells is displayed as a function of time, before and after the division. The mother cell fluorescence decays exponentially over time and at division it halves due to components partitioning between the two daughters. In this specific case, we can observe that the fraction of cytoplasm inherited by the two cells is not equal, indicating an asymmetrical division. To compute the partition fraction f, we fit the fluorescence intensity of the daughters’ cells vs time with:

where I(t) is the fluorescence intensity at time t, m accounts for the decaying process and 2i and 2i + 1 respectively refer to the two daughter cells. We constrain the slope to be the same between the two cells. The fraction of inherited fluorescence, f , is obtained as :

To ensure the highest possible accuracy, we (i) measured the total fluorescence of the mother cell and those of the two daughter cells for at least 2 hours before and after the division event to increase the determination of the fluorescence splitting for each detected mitosis. Moreover, (ii) we removed all dynamics that showed an absolute Pearson correlation value between luminosity and pixel size higher than 0.9 as a function of time (see Supplementary Information). In fact, single cell’s pixel size, which corresponds to the cell size projection on the focal plane, is found to shrink before mitosis and enlarges after division. Fluorescent intensity is instead proportional to the number of stained cytoplasmic proteins, which are expected to remain constant (except for a constant decay of the fluorescence). Therefore, a high correlation between pixel size and fluorescence intensity indicates a high noise-to-signal ratio. Figure 3(c)(bottom) displays the strip plot of the obtained partition fraction, f , for Caco2, HCT116, and CCD18Co cells, randomly spread on the y-axis, and their corresponding fitted П(f) distributions (top). Note that via microscopy imaging, it is possible to measure the whole partition distribution and not just its moments; however, hundreds of events are required to get a reliable estimate. Here, we characterize it by fitting the data with a double Gaussian distribution of the form:

that allows for the measure of both < f > and = .

The fitted distributions clearly show how HCT116 cells are the most symmetric with only one central and narrow peak, while for CCD18Co and CaCO2 cells two asymmetric peaks are visible, associated to higher fluctuations. In Figure 3(d), we compare the outcomes of the flow cytometry vs microscopy measurements. For all three cell lines, the found degrees of asymmetry are statistically consistent between the two approaches.

Fig. 3

D. Size division bias accounts for cytoplasmic fluctuations

To investigate the observed fluctuations in the partition of the cell cytoplasm, we start by recalling that the used dyes bind a-specifically to cytoplasmic amines. Thus, (i) it is expected to be uniformly distributed in the cellular cytoplasmic space and (ii) the number of labeled cytoplasmic components can be considered large. In this framework, the least complex model one can assume is a binomial one with the parameter p, measuring the bias in the process. Via equation 10 we got a direct link between the measurable variance of the population and the second moment of the underlining partition probability distribution, П(f). Henceforth we sought for a relationship between the second moment of П(f) and the parameters of the binomial distribution.

To begin with, we can write the partition distribution for the fraction of inherited component, assuming a level of asymmetry p (q = 1 − p), as:

As we have already observed in the general model, due to symmetry, < f >= 1/2 for any form of the П(f). No information on the system can be obtained by looking at the first moment.

For the second moment, we note that the variance of a single branch of the П(f) is:

where P(N_i) is the probability distribution of the mother’s components number at the moment of division, while , is the variance of the П(f)^p given the number of internal component N_i in the mother cell.

Recalling that m_2i is the number of inherited intracellular components of one of the two daughter cells, the partitioning fraction f can be expressed as f = m_2i/N_i which leads to:

Therefore, by substituting it into equation B2 we obtain:

Which explicitly depends on P (N_i) and coherently returns for P(N_i) = δ(N_i — N).

An equivalent computation can be done for the symmetric branch (σ^2(p)). The variance of the full distribution is given by:

Note that this relationship depends on the integral:

which requires the knowledge of the mother element distribution P(N_i) to be computed. To get an idea of its behavior, we observe that assuming a population of cells with a fixed number of elements before division, i.e. P(N_i) = δ(N — N_i), the integral simply goes as 1/N; in a more realistic scenario, the elements can be considered log-normally distributed, P(N_i) = . as that the log-normal distribution is the limit distribution for the product of random variables, similar to what happen to number of tagged components in a cell as product of the inherited fraction of all the previous divisions. In this case, one gets the following expression for :

where μ =< log(N_i) >.

The quantity is obtainable from flow cytometry/microscopy measurements. If the total cell fluorescence is I and F is the fluorescence intensity of each component, then I = N_i F and . The values of μ are instead not directly measurable. However, form Figure 4(a), showing the value of Σ(N) as a function of N for fixed values of , one can see that already for N = 100, the role of Σ(N) is negligible. In general, for the partitioning of components characterized by a great components number this factor can be neglected. In this regime, the expression for becomes:

Fig. 4

By inverting this relationship, we obtained a general mapping between the variance and the level of asymmetry of the binomial distribution.

In Figure 4(c), we compared the analytical expression to the values obtained via microscopy measurements, showing that also for the binomial approximation the two methods are compatible. CaCO2 cells confirm to be as the most asymmetric among the cell lines and since they are known for their heterogeneity in cell morphology [19], we explored the hypothesis of shape heterogeneity and cytoplasmic asymmetry to be linked. Figure 4(d) displays three sample cases of division examined through time-lapse fluorescent microscopy following cytoplasm.

Different time frames are overlaid to depict the entire dynamic at once, with time progressing from the bottom left to the top right of each image.

We computed the pixel size ratio between the daughters cells in the last studied frame and compared it with the fraction of inherited cytoplasm from the mother cell. Although pixel size is a rough proxy for cell volume, we observe that the cell inheriting a larger fraction also displays a greater size. This finding leads us to conclude that asymmetry in cytoplasm partition is linked to size fluctuations, where size dictates the observed bias in the segregation processes.

III. Discussions

Cell division is orchestrated by hundreds of molecular interactions, which are intrinsically stochastic processes [20]. The presence of such stochasticity results in the insurgence of variability among the cells phenotypes sharing the same genome, as it is the case for cancer cells. In this respect, gene expression noise has been extensively characterized [21] as well as the different strategies that cells evolved to regulate the extent of the noise in these channels [22, 23]. Beside such noise sources, proliferating cells are subject to the fluctuations originated by the partition of cellular elements at division. Analysis of this partition noise highlighted that under some circumstances, its contribution exceeds the others in creating heterogeneity [14, 24]. In fact, asymmetrically dividing cells can produce daughters that differ in size, cellular components, and, in turn, fate [25]. In this framework, the origin of such asymmetry may be ascribed to the basal stochasticity of molecular processes taking place at different levels during the division and/or to specific, evolutionary-conserved mechanisms used by cells to control cell fate and generate diversity [3, 26]. In fact, asymmetric partitioning of components alters the initial counts of molecules in the following cell cycle, which can lead the system to a different phenotypic state [27, 28]

Here, we show that an accurate determination of partition fluctuations can be obtained via standard flow cytometry measurements of properly marked cell populations. Our approach is easier and faster with respect to the use of microscopy imaging, which requires the acquisition and analysis of extensive recording of the cells, ultimately limiting the reachable statistics in contrast to the high-throughput readouts of flow cytometry. Moreover, most of the microscopy-based approaches use fixed cells and measure the inherited fraction of fluorescence from a single snapshot of dividing cells. Notably, comparing the degree of asymmetry in the division of the cytoplasm in the three different analysed cell types, we found that microscopy-derived fluctuations are systematically higher than those measured by flow cytometry. While this could be linked to the different used protocols, it seems more probable that the difference is due to a systematic overestimation of the fluctuations measured by microscopy due to its lower statistics.

From a biological point of view, the segregation statistics among the analysed colon cell types are neatly distinct. CaCo2 cells exhibits the highest degree of cytoplasmic partition fluctuations, while HCT116 division is the one with lowest asymmetry. It is interesting to note that normal colon fibroblasts sit in between, suggesting that the cancer-associated dis-regulation of the normal cell activities leads to the reactivation of asymmetric cell division in a cancer-specific manner. Indeed, CaCo2 cells are known to display a markedly heterogeneous distribution of phenotypes [19].

To get insights on the possible mechanisms behind the observed asymmetries, we looked for specific shape of the partition distribution. Following a maximum entropy approach [20, 29, 30], we started by considering the distribution that employs the fewest assumptions about the system, i.e. we consider independent segregation, which directly maps into binomial partitioning. Indeed, this is a common assumption in segregation models that traces back to the pioneering works of Berk [31] and Rigney [32]. Notably, our results indicated biased segregations, with colon cancer cells displaying different biases.

To explain the origin of those biases, we looked at relative sizes between the daughter cells right after division. The found correlation between daughter sizes and the fractions of inherited fluorescence vouches size as the origin of the observed biased segregation.

In conclusion, by combining experimental data with statistical modelling, we show how it is possible to use flow cytometry data to extract reliable estimates of the strength of fluctuations during cell division. Our approach has the potential to be applied to different cell types where a quantification of the level of division asymmetries daughter cells can experience may provide new insights into the mechanisms of asymmetric cell division and its role in cancer heterogeneity and plasticity.

IV. Materials and Methods

A. Cell Culture

Caco-2, colorectal adenocarcinoma cell line, was purchased from ATCC (Manassas, VA, USA, HBT-37) and maintained in complete culture media DMEM (D6046) containing 20% FBS, penicillin/streptomycin plus glutamine, nonessential amino acids (NEAA) and sodium pyruvate, all 1/100 dilution. HCT116 VIM RFP, colorectal carcinoma cell line, was purchased from ATCC (Manassas, VA, USA, CCL-247EMT) and maintained in complete culture media McCoy’s 5A (M9309) containing 10% FBS, penicillin/streptomycin plus glutamin.

CCD-18Co, colon fibroblast cell line, were purchased from ATCC (Manassas, VA, USA, CRL-1459) and maintained in complete culture media DMEM containing 10% FBS, penicillin/streptomycin plus glutamin, nonessential amino acids (NEAA) and sodium pyruvate, all 1/100 dilution.

All cells were kept in culture at 37°C in 5% CO2 and passaged according to the experimental protocol. For all experiments cells were harvested, counted, and washed twice in serum-free solutions and resuspended in room temperature (RT) PBS w/o salts for further staining.

In detail, to detach cells from culture flasks we used Trypsin EDTA 0.25%. Culture media was removed and the flasks were washed once with PBS, then Trypsin EDTA was added and allowed to work for 1 min in an incubator at 37°C. Once the cells appeared detached the flasks were mechanically agitated to facilitate cells’ loss of adhesion. Trypsin was then inactivated with one volumes of complete media and cells were collected, pelleted, and washed a second time in PBS. To determine cell viability, prior to dye staining, the collected cells were counted with the hemocytometer using Trypan Blue, an impermeable dye not taken up by viable cells.

B. Flow Cytometry and Cell Sorting

To track cell proliferation by dye dilution for establishing the progeny of a mother cell, cells were stained with CellTrace Violet^TM (CTV). The CTV dye staining (C34557, Life Technologies, Paisley, UK), used to monitor multiple cell generations, was performed according to the manufacturer’s instruction, diluting the CTV 1/1000 in 0.5-1 ml of PBS for 20 min in a water bath at 37°C, mixing every 10 min. Afterward, 5x complete media was added to the cell suspension for an additional 5 min incubation before the final washing in PBS.

Labeled cells were sorted using a FACSAriaIII (Becton Dickinson, BD Biosciences, USA) equipped with Near UV 375nm, 488nm, 561nm, and 633nm lasers and FACSDiva software (BD Biosciences version 6.1.3). Data were analyzed using FlowJo software (Tree Star, version 10.7.1). Briefly, cells were first gated on single cells, by doublets exclusion with morphology parameters area versus width (A versus W), both side and forward scatter. The unstained sample was used to set the background fluorescence. The sorting gate was set around the max peak of fluorescence of the dye distribution. In this way, the collected cells were enriched for the highest fluorescence intensity. Following isolation, an aliquot of the sorted cells was analyzed with the same instrument to determine the post-sorting purity and population width, resulting in an enrichment > 97 % for each sample.

To monitor multiple cell division, the sorted cell population was seeded in 12 wells plates (Corning, Kennebunk, ME, USA) at cell density bewteen [30–70]·10³ cells/well, according to the experiment, and kept in culture for up to 84 h. Each well corresponds to a time point of the acquisition and cells in culture were analyzed every 24, 36, 48, 60, 72, 84 h by the LSRFortessa flow cytometer. In order to set the time zero of the kinetic, prior culturing, a tiny aliquot of the collected cells was analyzed immediately after sorting at the flow cytometer. The unstained sample was used to set the background fluorescence as described above.

C. Time-lapse Microscopy

To better investigate the cell proliferation dynamics, we performed time-lapse experiments, for up to 3 days. In our previous paper [13] we verified that cell growth is not affected by different dyes combination. Therefore, in the present work, we used both CellTrace Yellow^TM (C34573 A, Life Technologies, Paisley, UK) and CellTrace Far Red^TM (C34572, Life Technologies, Paisley, UK), to stain the cytoplasm for microscopy analysis. Low passage cells at around 60 % confluency were counted and stained with CellTrace dyes 1/500 and plated on IBIDI cell imaging chambers (μ-Slide 4 and 8 wells) at low cell density of 10³ cells/well. After overnight incubation, the chamber is transferred to the inverted microscope adapted with an incubator to keep cells in appropriate growing conditions. Brightfield and fluorescent confocal image stacks were acquired with a 20x air objective (Olympus, Shinjuku, Japan) and Zen Microscopy Software (Zeiss, Oberkochen, Germany), every 20 min. Time-lapse images were analyzed using ImageJ and inhouse Python programs.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Code availability

All codes used to produce the findings of this study are available from the corresponding author upon request. The code for the Gaussian Mixture algorithm is available at https://github.com/ggosti/fcGMM.

Acknowledgements

This research was partially funded by grants from ERC-2019-Synergy Grant (ASTRA, n. 855923); EIC-2022-PathfinderOpen (ivBM-4PAP, n. 101098989); Project ‘National Center for Gene Therapy and Drugs based on RNA Technology’ (CN00000041) financed by NextGeneration EU PNRR MUR—M4C2—Action 1.4—Call ‘Potenziamento strutture di ricerca e creazione di campioni nazionali di R&S’ (CUP J33C22001130001).

Additional information

Author contributions statement

M.M. and G.G. conceived research; D.C., G.P., and S.S. performed cell biology experiment and flow cytometry measurements. D.C., C.G., V.d.T., and M.M. performed timelapse microscopy experiments; F.G. and G.R. contributed additional ideas and helped interpret results; D.C. analyzed data; M.M. performed analytical calculations; D.C. and G.G. performed numerical simulations and statistical analysis; all authors analyzed results, and wrote and revised the paper.

Appendix A: General model for the partitioning noise

We assume that at time zero each cell has m_i marked compounds, with a certain distribution in the whole cell population of mean μ and variance σ². Each cell in the population, if sufficiently healthy, will divide in two cells and we would like to follow the evolution of the marker intensity distribution. The splitting ratio is one of the parameter we want to characterize.

Therefore, by calling mi the number of compounds of the mother cell, m_2i and m_2i+1 the compound in the two daughter cell. The probability of finding a cell with m_2i marked compounds at generation g is:

where П(f) is the distribution probability of the fraction of compound that goes into a daughter cell.

Since we do not study single cells, but populations, we must deal with distributions. We will omit the computation for m_2i+1 since it is equal to m_2i if f is substituted with 1 − f.

Which according to equation A1 becomes:

Hence, the expectation value for m_2i is :

while for m_2i+1:

For the variance we can do a similar computation:

Therefore, both variance and mean of the daughters subpopulations are linked to the variance and mean of the population of mother cells and to the distribution probability of f.

To be able to compare the model with experimental data we need to compute the mean and variance of the whole population, which is the sum of coexisting subsequent generations.

Where we are assuming that a mother cell always give rise to only two daughters. We can split equation A5 into two groups, the one that always inherits a fraction E[f] of the mother’s compounds and the symmetrical one which inherits a fraction E[1 − f].

Which can be rewritten as:

Therefore, knowing the initial mean value of the population, μ₀ we have:

Equation A7 shows that the mean value of the number of compounds for each generation is independent of the underlying process and does not depend on the distribution П(f). Interesting, although expected, but useless for comparing expectations to experiment.

To obtain a dependence on the partitioning process we need to compute the variance. We want the variance of a distribution which is given by the mixture of the distribution of all the sub-populations of daughter cells. It can be demonstrated that the total variance is:

As we did in A6 we can rewrite equation A8 by taking into account the two daughters sub-populations:

We can rewrite this formula by recalling the relationships A4 and A3.

We can notice that the last two elements of every sum can be rewritten as:

Which leads to:

and considering that μ_f = 1/2.

Therefore, the final form of equation A6 is :

Appendix B: Modeling partition as a binomial process

The model we have developed and tested by comparing two sets of independent experiments allows for a robust characterisation of the partitioning noise. The measures are done via flowcytometry which allows to obtain a great statistics. The key point of what we have proved up to now, is that the variance of the partitioning distribution is the main actor in shaping the population dynamic. In this section we want to propose an interesting case of study, were we make an assumption on the shape of the partitioning distribution. The reasonable assumption is to consider the binomial distribution for the partitioning of components. In particular, for the cytoplasm the high number of tagged components.

Therefore, we want to link the measure of σ of the partitionig distribution done with the general method, to the value of p of the binomial distribution which would lead to the same observed value.

The binomial model consists in assuming that each cell’s component has a certain probability p of ending up in one of the two daughter cell, and 1 − p to end up in the other one.

Therefore, the total partition distribution can be written as:

As we have already observed, due to symmetry, the < f >= 1/2. But, we are interested in how the general noise, map into the binomial one.

The variance of a single branch , can be computed in the following way:

Consider initially one of the two daughter cell distributions for example, P(f)^(p). Its average value will be given by:

and similarly < f >^(q)= q. For the variance instead, :

where the notation indicates the variance of the distribution p as a function of f given a certain N_i. Recalling that f, the fraction of intracellular components and/or fluorescence, is defined as f = , it follows that:

Plugging Eq. B3 in B2 we get:

which, consistently, in the case of a delta distribution (i.e., for a single value of N_i), results in .

Once the main quantities of the individual distribution have been determined, we consider the sum distribution (eq. B1).

The mean value of this distribution is:

Therefore, to determine p, it is necessary to estimate the variance . By definition, it is:

Substituting Eq. B5 one gets:

from which, adding and subtracting < f >^2(p), < f >^2(q) , one gets:

and via Eq. B4

a. Delta Distribution of mother’s components

In principle, P(N_i) can have any functional form, even a very complex one, which is not known. Therefore, let’s start with the simplest case: all mother cells possess the same number of components N₀, so the distribution P(N_i) is a Dirac delta.

In this case, the theoretical variance (eq. B12) will be:

Since the values of the initial distribution have been considered randomly, each of them will have the same probability of being selected: P(N_i) = . Consequently, the integral term in the variance expression (eq. B12) can be simplified as:

and one gets to the final expression:

Neglecting the integral term, the equation reduces to :

b. Lognormal distribution of mothers’ components

In an even more realistic case, the distribution of the number of internal components of the mother cells will be given by a log-normal distribution:

where μ is the average value: μ =< log(N) >.

The integral term in this case is given by:

In general, the distribution of internal components of a cell is not known. However, in microscopy experiments, it is possible to determine fluorescence distributions. Additionally, a direct proportionality between the two variables has been assumed: I = N_i · F, where F is a constant value representing the fluorescence of an internal component.

Now, consider the variance of the distribution as a function of the number of internal components:

The variance of the distribution as a function of N_i, therefore, coincides with that of the fluorescence, which can be determined from experimental measurements.

Consequently, for a log-normal distribution, the contribution of the integral term to the variance, integrated over the interval [1, ∞), will be:

where the variance coincides with that of the fluorescence distribution (eq. B18) and μ = .

Substituting the integral term (eq. B19) into the complete expression for the variance (B10) for a log-normal distribution of the mother cells, we obtain: