## Abstract

Heterogeneity in the size distribution of cancer cell populations is linked to drug resistance and invasiveness. However, understanding how such heterogeneity arises is still damped by the difficulties of monitoring the proliferation at the typical timescales of mammalian cells. Here, we show how to infer the growth regime and division strategy of leukemia cell populations using live cell fluorescence labeling and flow cytometry in combination with an analytical model where cell growth and division rates depend on powers of the size. We found that the dynamics of the size distribution of Jurkat T-cells is reproduced by (i) a sizer-like division strategy, with (ii) division times following an Erlang distribution and (iii) fluctuations up to ten percent of the inherited fraction of size at division. Overall, our apparatus can be extended to other cell types and environmental conditions allowing for a comprehensive characterization of the growth and division model different cells adopt.

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## Introduction

Heterogeneity in cancer cell phenotypes enhances the population’s capability of resisting fluctuating adverse environments^{1,2,3}, like those produced by the administration of drugs and anticancer treatments^{4}. In particular, recent works suggest that size heterogeneity specifically impacts cancer fitness. However, the exact relationship between size and disease is poorly understood, as much as the strategies cancer cells adopt to create and maintain such heterogeneity^{5}. At a mechanistic level, cell size is determined by the combination of homeostatic processes operating at different timescales. If on shorter timescales, size is set by the capability of cells to regulate processes like water entry, membrane trafficking, and protein synthesis; on longer timescales, size results from the interplay between cell growth and division^{6}. As a result, under steady-state conditions, isogenic populations tend to maintain cell sizes around typical^{7}, type-specific mean values, although significant cell-to-cell variability is often observed^{8}. Indeed, observed size distributions span over an order of magnitude^{9}, usually display lognormal shapes and are characterized by coefficients of variation (CVs) of 0.1–0.3^{5,10}. Such cell-to-cell variability in clonal cell populations is ultimately linked to the inherently stochastic nature of molecular processes, which reflects in fluctuations of the production/degradation of cellular constituents and noisy partition of cell content/volume during cell division^{11,12,13,14}.

Studies on bacteria and yeast populations showed that size distribution homeostasis can be obtained by modulating the amount of growth produced during the cell cycle in such a way that on average larger cells at birth grow less than small ones. The strategies that cellular populations may adopt to reach and maintain such size homeostasis have been roughly classified into three distinct models, depending on whether divisions occur after a certain time (timer), upon reaching a certain size (sizer), or after the size of the cell has increased by a finite volume (adder).

In particular, a phenomenological linear relation (usually referred to as stochastic map) has been observed between the size at birth (*s*_{b}) and the size at division (*s*_{d})^{15}: *s*_{d} = *a**s*_{b} + *η*, where *η* represent noise derived from the specific biological size control model, while the slope, *a*, defines the size control models. Usually, experiments look at the quantity, \({{\Delta }}=\left\langle {s}_{d}\right\rangle -\left\langle {s}_{b}\right\rangle =(a-1)\left\langle {s}_{b}\right\rangle +\left\langle \eta \right\rangle\). So that, for *a* = 2, the size at division is directly proportional to the size at birth. This mechanism is referred to as a timer model, hypothesizing that the cell size is controlled by a cell cycle timer that sets a time limit for the growth phase, and once the time limit is reached, the cell divides. Intuitively, if cell size growth is exponential such a mechanism will end up in big cells to proliferate faster than small cells thus producing a divergent size variance. Only a linear size increase under specific constraints on division symmetry is compatible with this homeostatic strategy^{16}. For *a* = 1, a certain volume < *η* > is added before division, which is uncorrelated to the initial size. This behavior has been proposed for various populations of bacteria, cyanobacteria, and budding yeast populations^{17,18}. Finally, if *a* = 0, the size at division is completely set by the stochastic term, which is called a sizer mechanism. The sizer model suggests that the cell size is determined by a threshold size, and once the cell reaches this threshold size, it triggers the cell division process. Sizer-like mechanism has been found for organisms such as the fission yeast *S.Pombe*^{19,20,21}.

To determine which model is followed by the different kinds of cell types and determine the cell size distribution in both bacterial and eukaryotic cell populations, various experimental techniques have been developed^{22,23}, comprising time-lapse microscopy^{24}, single-cell tracking^{25}, and gene tagging^{26}.

For instance, ref. ^{27} used fluorescence microscopy united with a mathematical framework, showing that it is possible to get dynamical information from fixed cells and revealing feedback linking between cell growth and cell cycle. Sung and coworkers developed a phase microscopy technique for highly accurate measurements of cell mass in adherent mammalian cells^{28}.

Parallel to experimental progress, different models of cell size regulation (with few key parameters) have been proposed. From the pioneering works of Powell^{29} and Anderson^{30}, that compared analytical predictions with cell counts experiments to more recent works that proposed mathematical models to interpret both population and single cell-based data^{17,31,32,33,34,35,36,37,38,39}.

In particular, progress in the quantitative measurement of single-cell features, using, for instance, mother machines, allows for the characterization of the growth and division strategy for fast-proliferating cell populations, like bacteria and yeasts (whose doubling times are on the order of hours). For other cell types, like mammalian ones, it is harder to track several divisions due to lower division rates and more stringent growing conditions. A work by ref. ^{40} investigated a wide spectrum of mammalian cell types combining a microfluidic system with real-time cell imaging to find different adopted homeostatic strategies, ranging from quasi-timers to size-like ones. Varsano et al.^{41} studied the growth of leukemic basophils and macrophages finding that smaller cells follow a size-like division strategy while average cells behave more like adders.

In this paper, we propose an experimental protocol coupled with a minimal mathematical model to determine the homeostatic strategy adopted by populations of leukemia cells that does not rely on cell imaging data. In particular, the experimental protocol, we propose, makes use of flow cytometry measurements which yield information on the cell size (via the collected forward scattering signal^{42}) and permit (via live cell fluorescent tags^{14}) the determination of cell lineage and partition noise. Experimental data are compared with the predictions of a minimal model where the variation of a single parameter allows the exploration of the different size-homeostasis strategies. In particular, we developed a quantitative analytical framework based on three key features: (i) cell growth and division rates depend on powers of the cell size, (i) cell division can take place after the cell accomplishes a certain number of intermediate tasks, and (iii) daughter cells can inherit an uneven portion of the mother cell size.

We show that: (i) using forward scattering as a proxy for cell size^{43} allows us to observe the dynamics of cell size distributions, which are in qualitative agreement with those shown by both numerical simulations of agent-based systems and a minimal analytical model based on a population balance equation. (ii) A simple exponential distribution of division times can not reproduce the observed dynamics, which instead requires an Erlang distribution with at least the sum of three independent exponentially distributed intermediate times. Finally, (iii) stratifying data according to cell generations allows us to fully infer the division strategy adopted by leukemia cells.

## Results

### Experimental protocol

To measure the cell size distribution and follow its dynamics, we developed a protocol based on flow cytometry measurements. Extending the procedure we previously proposed to measure the partition noise of cellular compounds^{14}, we made use of CellTrace-Violet (CTV), a fluorescent dye, to mark cell cytoplasm (see Method section) and follow the proliferation of the population by looking at the dynamics of the fluorescence and forward scattering signals in time via a series of flow cytometry measurements. As depicted in Fig. 1 and explained in more detail in the Methods, marked cells are first sorted (see panel a), so that an initial population with a narrow CTV distribution is selected (Fig. 1b). The sorted population is then collected and cultured in standard growth conditions (see *Methods*). Samples of the population are collected at different times during the cell proliferation, recording CTV, forward FSC, and side scattering SSC, intensities for each analyzed cell (see Fig. 1c). SSC and FSC are used to define the viable population at each time point (see Supplementary Fig. 5 for details). Figure 1d shows the evolution of the distributions of \({\log }_{2}\) CTV fluorescence intensities. The purple curve (low-right corner of the panel) corresponds to the post-sorting distribution, which is considered the time zero of the dynamics. Looking at the CTV intensity at different times during the proliferation of the cell population, one observes a progressive shift of the initial fluorescence and the appearance of multiple peak distributions. As CTV homogeneously binds to cytoplasmic proteins, upon division, the fluorescence of each mother cell is divided into the two daughters as a result of the cell division process. Each division produces two daughters, thus on average the CTV distribution of the daughter cells has half the mean fluorescence of the mother distribution (see inset in Fig. 1d).

Parallel to the evolution of the CTV intensity, we track the evolution of the FSC intensity. As it can be seen from Fig. 1e, the distribution shifts toward higher values than those presented at the initial time point (purple curve), while its variance increases. This behavior can be explained by recalling that the initial population is sorted, consequently, its size distribution is out of equilibrium.

### Forward scattering evolution probes cell size dynamics

To identify the different generations from the CTV intensity profiles, we applied a fitting protocol via a Gaussian Mixture Model of the form: \(P(\ln x)={\sum }_{g}{w}_{g}N(\ln x,{\bar{x}}_{g},{\sigma }_{g})\) combined with an Expectation Maximization algorithm (see *Methods* for details). An example of the result of the fitting procedure is shown in Fig. 2a. From left to right, the distributions of the log2 of the CTV intensity for different time points are shown in gray, while the Gaussian distributions obtained as best fits of the experimental data are reported in different colors corresponding to the different identified generations. Together with the mean and variance of each Gaussian, the fitting procedure yields the probability of each cell to belong to the various identified generations. Since for each cell, both CTV and FSC signals are measured at each time point, we can use the information coming from the GM procedure to identify the subpopulations corresponding to different generations from the FSC distributions. Figure 2b displays the FSC intensity distributions of the same population from which the CTV distributions (shown in panel a) were measured. In this case, gray curves mark the total population, while colored ones correspond to the different generations. Comparing the distributions of the same generation in different snapshots, one notes that newer generations have distributions shifted toward smaller FSC values with respect to older ones. In particular, the distributions (dots) of the grand-daughter cells are reported in Fig. 2c, together with the best fits of a normal distribution for each distinct time point. Mean and variance as a function of the snapshot times are reported in Fig. 2d. Finally, we evaluate the Pearson correlation coefficients between CTV and FSC intensities as a function of time (Fig. 2e). Results show that the correlation is null just after the sorting procedure and then reaches a high positive value (around 0.6) in the next snap times. This behavior appears reasonable since cytoplasm volume scales with the cell size; on the other hand, an almost zero correlation after sorting may indicate that the rates of CTV uptake are moderately dependent on cell size.

### Minimal model for size dynamics

In single-cell size measurements, the division strategy can be directly assessed by looking at the slope of the linear relation between birth and division sizes of the cells. In our framework, to infer the control strategy, we compared data with a theoretical framework that accounts for the time evolution of the cell size distribution under different possible growth and division regimes, i.e., different size-homeostasis strategies.

#### Exponentially distributed division times

To allow for a comparison, we aimed at modeling the growth and division dynamics of a population of cells growing in a controlled environment. We assumed that each cell of the population is thus characterized by a size, *s*, which changes in time as the cell grows and divides, and by its generation, *g*.

Assuming balanced exponential growth of the population and referring to *n*_{g}(*s*(*t*), *t*) as the number of cells in the population having size *s* at time *t* and having divided *g* times, this number will evolve in time according to:

where *g*(*s*) and *γ*(*s*) are the size-dependent growth and division rate, respectively, and *ϕ*(*x*∣*p**y*) quantifies the probability that a daughter cell inherits a fraction p of mother cell size y. See Supplementary Note 1 for the derivation of Eq. (1).

Along the lines of previous works^{31,33,37,44}, we assumed that both growth and division rates are given by power of the cell size, i.e., we assumed \(g(s)=\frac{ds}{dt}=\lambda {s}^{\alpha }\) and *γ*(*s*) = *κ**s*^{β}. As shown by ref. ^{45}, with this minimal assumption it is possible to recover all main size-homeostatic strategies tuning the parameter *ω* = *β* − *α*.

Note that we refer to linear/exponential growth considering the single-cell growth rate, *d**s*/*d**t*, which is linear for *α* = 0 and exponential for *α* = 1. We make the approximation that the growth rate expression remains the same independently from the cell cycle state (see for example in ref. ^{46}). For what concerns the growth of the population in terms of the number of cells as a function of time, we assume that the population is in balanced exponential growth as previously stated.

If we now look at the variation of the total number of cells at generation *g*, we have:

which can be recast as

The first term of the integral goes to zero thanks to the fact that either the size or the number of cells is zero in the integration extrema; thus the above equation becomes:

where < *x* > _{g} = ∫ *d**s**x**ρ*_{g} and we introduced the probability of finding a cell with size *s* at time *t* and generation *g* as

The fraction *N*_{g−1}/*N*_{g} is difficult to be experimentally measured. Thus, we want to recast it in a more handy form. To do so, we define the fraction of cells belonging to a certain population at each time as:

Eq. (4) can be used to compute the dynamics of the fractions, *P*_{g}. In fact, we have that^{3}:

which, after some calculation, can be expressed as:

It remains to obtain expressions for the mean and variance dynamics. Again, we can start from Eq. (4). In fact,

and

Reordering and dividing by *N*, we get

Without loss of generality, one can express *ϕ* as

where *π*(*p*) is a general probability function of the fraction of inherited cell size.

Thanks to Eqs. (11) and (12), we can easily compute the distribution moments evolution equations as:

where < *p*^{i} > _{π} refers to the i-th moment of *π*(*p*) (see Supplementary Note 7 for details on how the last term is obtained).

#### Erlang distributed division times

Assuming size-dependent growth and division rates end up producing exponentially-distributed division times^{33}, while it has been previously shown how intergeneration division time statistics is better captured by Erlang distributions^{33,47}. To retrieve an Erlang-like distribution, we introduce a series of intermediate states, the cell has to go through, before starting the division as done for instance by ref. ^{45} to describe E. coli size dynamics. Each of the state’s duration has an exponential distribution of times. Introducing another index accounting for the intermediate states the cell transit in before division, and repeating all calculations (see Supplementary Notes 4, 5), one ends up with:

where \({\left\langle \cdot \right\rangle }_{g,q}\) stands for the statistical average over *ρ*_{g,q}, the density of cells that divided *g* times and passed *q*-th out of Q intermediate states and *δ*_{i,j} is the Kronecker delta. Similarly, the fraction of cells being at generation *g* and state *q* evolves according to

for *q* > 0, and

otherwise.

Equations (14) and (15), (16) fully describe the dynamics of the cell population, however except for some specific sets of parameters, this set of equations is not closed; in fact, the time derivative of the i-th moment may contain higher moments depending on the values of *α* and *β*. Indeed, the set is closed only in the case of a division rate that does not depend on the cell size (i.e., *β* = 0). To solve the system in the general case, we must choose a moment closure strategy. To do so, we exploit our findings on the FSC distributions stratified by generations^{44}. In fact, the latter are well fitted by normal distributions, showing that our experimental resolution on the real generation size distribution can at best be on the second moment. Considering additional moments, one would risk inserting experimental noise in place of real signal, while restraining only to the evolution of the first two moments assures considering the less unbiased statistics from a maximum entropy viewpoint^{48,49}.

Thus, we assume that the single generation size distributions have normal moments (note that this is not the case of the total population size distribution, which shows a log-normal distribution instead) and opt for a normal moment closure (see Supplementary Note 6 for details).

To validate the obtained relations and test the adopted moment closure, we compare the solution of the differential equations with the results of stochastic simulations of an agent-based model, where an initial population of cells grow and divide following the same grow and division rates functional form used in Eq. (1). To associate at each cell a proper division time, a Gillespie procedure has been adopted. See the Method section for a detailed description of the stochastic simulation protocol. The outcomes of the simulations are recapitulated in Fig. 3. In particular, Fig. 3a provides a schematic representation of the life cycle of a single agent, i.e., cell in the population: a cell is born with initial size, *s*_{b}; it grows according to a certain growth rate, *g*(*s*), for a certain set of times, \({{\tau }_{q}}_{0}^{Q}\), which encode a series of independent intermediate states the cell has visited before actual division (see Supplementary Note 5). From a biological point of view, such states can be linked to the phases of the cell cycle. Upon reaching the division size, *s*_{d}, the mother cell splits into the two daughter cells, one inheriting a fraction p of the mother volume and the other keeping the remaining 1 − *p* fraction.

To begin with, we verified that such a framework reproduces the expected sizer, timer, and adder (see Fig. 3c, d) behavior upon varying the *ω* = *β* − *α* parameter. Indeed, simulations with *α* = 1 and *β* taking values 2, 0, or 1 produced the expected trend for the sizer, timer, and adder, respectively. Next, we compared the solution of the model, in the normal closure approximation, with the results of the agent-based stochastic simulations. In Fig. 3e, we show the results for the fractions of cells found in different generations as a function of time. The good agreement, measured by the *R*^{2} of ~ 0.9, suggests the choice of the moment closure provides a good approximation for such kind of dynamical process.

### Model parameters govern distinct and measurable aspects of cell dynamics

The derived framework depends on several parameters, thus, we next proceeded to characterize the role of the different model parameters and their effects on the quantities we can measure experimentally, i.e., the population size distribution moments, their per-generation stratifications, and the relative abundances of cells in different generations during the dynamics.

At first, we focused on the mean size of the whole population. Measuring the mean FSC as a function of time for populations sorted with three different values of initial mean size, we found that they eventually reached comparable values of the mean size within the first 24 h of the experiment (see Fig. 4a). Comparing the trend shown by experimental data with those of the model (Fig. 4b1, b2), we found that the asymptotic mean size is not a function of the starting mean sizes (see Fig. 4b1) but it is modulated by the ratio of the rate coefficients, *λ*/*κ*. In particular, Fig. 4b2 clearly shows that the higher the ratio *λ*/*κ*, the higher the population mean size in the long time limit. Looking at the trend of the Coefficient of Variation, CV, instead, we found that the size of the Jurkat population displays a CV with an oscillating behavior around a value of about twenty-three percent as shown in Fig. 4c. A comparison with the model trends again shows that this behavior is qualitatively reproduced by the model. Moreover, the key parameter modulating the variance of the cell size distribution is the exponent of the division rate, *β*. In particular, the higher the exponent the lower the fluctuations of the cell sizes (see Fig. 4d2).

To explore the role of the remaining model parameters, i.e., the number of intermediate states and the division noise, we moved to consider the time evolution of the fractions of cells per generation. Figure 4e shows the results for three timecourses as dots. As one can see, experimental data exhibit a trend qualitatively similar to those obtained by solving Eqs. (8) and shown in Fig. 3e. Notably, the maximum fraction of observed first-generation cells in the population is 0.85 ± 0.05. This value is not compatible with a single-task model of cell growth and division but requires a cell division time given by the sum of at least three independent exponentially distributed times. Indeed, this can be seen comparing the predicted maximum fraction of daughter cells obtained by solving Eqs. (8). The inset in Fig. 4e displays the maximum fraction values obtained by changing the number of states, Q, from one to 6. Finally, we quantified the role of division noise in terms of the shapes of the generation fractions curves. As discussed in the previous sections, given the high correlation between FSC and CTV intensity, we assume that the size at division follows the same statistics of the cytoplasmic components. In previous work, we found that Jurkat cells partition their cytoplasm symmetrically^{14}. Thus, we assumed that even the fraction of inherited volume is a random variable with a normal distribution, centered in 1/2 and having a certain variance, \({\sigma }_{p}^{2}\) (see inset in Fig. 4f). Solving the model equations with different values of *σ*_{p} while keeping all other parameters fixed gives the trend reported in Fig. 4f. It can be seen that the higher the level of division noise, the more the maximum fraction of cells per generation decreases, while the same generation tends to endure for longer times, i.e., smaller cells can be produced that require longer times to divide. Note that this reflects an increase in the total population variance.

### Size dynamics behaves according to a size-like homeostatic strategy

Finally, we compared the prediction of the model against the collected data. To begin with, we checked that the proliferation of the population of Jurkat cells was in a balanced exponential regime. In particular, we studied the behavior of the logarithm of the number N of live cells per ml (normalized by the initial density) versus time, which defines the growth curve of the population^{23}. As one can see from Supplementary Fig. 1, data are compatible with an exponential trend after an initial lag phase of approx. 7.5 h (green dotted line in Figure). The obtained doubling time of 19 ± 3 h is in accordance with literature data on the same cell type^{50}.

Next, we proceeded to find the best values of the model parameter able to reproduce the observed experimental trends. As discussed in the previous section, Eqs. (13) and Eqs. (8) depend on four parameters governing the growth and division rates, two parameters fixing the first two moments of the initial size distribution, the number of intermediate tasks, Q, and the variance of the inherited size fraction. In particular, the mean and variance of the initial size distribution are directly measurable from the starting post-sorting forward scattering distribution, while the exponent of the growth rate is fixed to 1, as required to reproduce the exponential growth dynamics compatible with the trend of the variance evolution of the mother cell size variance. Note that we re-scaled forward scattering intensities by the mean of the starting population to work with smaller numbers. Figure 5 shows the results of the best fit between the experimental data and the model. In particular, we minimized the cost function defined as the squared sum of the residues of the fractions of cells in the different generations (Fig. 5a) via Approximate Bayesian Computation Sequential Monte Carlo (ABC SMC) for parameter estimation^{51}. As the model depends on 5 free parameters, we opted not to include size distribution moments information in the cost function to use them as an independent validation in the selection of the best growth model. Best fit curves are obtained with the parameters: *κ* = 0.016 ± 0.007, *λ* = 1.29 ± 0.03,*β* = 6 ± 1, *Q* = 5 ± 1, and \({\sigma }_{p}^{2}=0.003\pm 0.001\).

As one can see from Fig. 5, the parameters of the fit best reproduce the dynamics of mean and variance of the size distribution for both the total population and the single generations (as also testified by the values of the *R*^{2}). Supplementary Fig. 3 shows the outcome of the ABC SMC for a parameter set having *β* = 1, i.e., compatible with a perfect adder strategy, which, while providing good trends for the population fractions, has a poorer agreement on the size moments behaviors.

Notably, the optimal value of the *β* exponent of the cell division rate is equal to 6, which indicates a near-adder strategy for size homeostasis. Note that in the proposed model, a *β* of 1 would have indicated a perfect adder strategy, while *β* → *∞* is a perfect size sensing.

## Discussions

Cell size is a phenotype that exhibits a huge variability across different kinds of cells. Typical sizes of bacteria span a range of 1–10 μ*m*, eukaryotic cells have linear sizes of 5–100 μ*m*, to end at neuronal cells whose size is up to some meters. Besides such inter-kind size heterogeneity, cells of isogenic populations have a well-defined typical size^{52}, that influences and is influenced in return by processes such as transcription, translation, and metabolism^{53,54}, as cell volume and surface area affects molecule reactions and nutrient exchanges^{55}. How this typical size is preserved despite the complex and noisy machinery^{56,57} of cellular processes that are at play in proliferating cells, is a question that remains still largely unanswered^{7}, especially for cancer cells that have ten-fold slower proliferation timescales than bacteria or yeast cells.

Over the years, several models have been proposed to explain how cells regulate their size. These models provide insights into the molecular mechanisms that govern cell growth and division and help researchers identify key regulatory pathways that could be targeted for therapeutic purposes. In particular, (i) cells that divide after a certain time from birth are said to follow a timer process; (ii) if division takes place when the cell reaches a certain size one speaks of the sizer model; while (iii) an adder mechanism consists in adding a certain volume which does not depend on the birth size.

The ‘canonical’ way to assess the size homeostatic strategy adopted by a certain cell type is based on the trend shown by the size at birth vs that at division. To obtain such a relation, one has to follow the proliferation of single cells and measure the size of the same cell at its birth and just after its entry into the mitotic phase. While such a procedure provides a reliable way to determine the homeostatic behavior and a sure way to track cell lineages, it also has the limitation of measuring the birth and division sizes, i.e., following the dynamics of individual cells, which in turn limits our understanding of size determination mechanisms in cancer^{5,58}. To address such problems, we sought an alternative/complementary procedure able to provide high statistics while preserving cell growth conditions. We propose an experimental protocol that does not explicitly consider birth and division size but instead utilizes flow cytometry data in combination with a minimal mathematical model to determine the growth and division mechanism of Jurkat cancer T-cells.

Our main finding is that a model based on power functions of the size for division and growth rates successfully reproduces the key features of Jurkat cell size dynamics, including an Erlang distribution of division times, a size-like strategy, and the presence of fluctuations in the inherited size fraction between daughter cells.

In particular, analyzing the maximum fraction of cells found in the first generations, we found that for the model to correctly reproduce observations, cell division time has to be given by the sum of a minimum of three independent and exponentially distributed intermediate times. This is in accordance with the evidence Chao and coworkers provide of the human cell cycle as a series of uncoupled, memory-less phases^{59}. Our protocol proved effective in detecting the dynamics of the size also among the different cell cycle phases, thus it would be possible to further investigate these aspects in the future.

Notably, our prediction of a size dependence in the division rate of leukemic T-lymphoblast is in accordance with the findings of ref. ^{60} who showed that growth rate is size-dependent throughout the cell cycle in lymphoblasts.

Other studies considering leukemia cells concluded that small leukemia cells behave as a sizer, while average-size cells follow an adder model^{41}. Using a combination of microfluid and cell imaging, ref. ^{40} assessed the division model for a wide range of cell types, finding that Raji cells, a suspension lymphoblast-like cell line follow a timer-like division model while mouse leukemic L1210 cells, a line exhibiting lymphoblast morphology set in the adder-sizer region. We note that the partial accordance with previous investigations may rely on biological differences between the considered cells or in the different used methods.

Finally, we measure the division noise as CTV noise through correlation analysis, finding that the shapes of the generation fraction curves were compatible with a symmetrical division with fluctuations around the mean up to ten percent, which is in agreement with the fluctuations observed in another line of leukemic lymphoblast by ref. ^{28} via accurate phase microscopy experiments.

The model we present is not limited to exponential growth, a major assumption in most of the analytical modelizations present in literature^{37}. In fact, while this common assumption holds for various cell types, it is not universal. For example, Schizosaccharomyces pombe (fission yeast) is a case where the increase of cell size with time after birth is non-exponential^{61,62}.

We note that our results depend on both the generation fractions and the FSC signal. While fraction signals are reasonably solid, the forward scattering can only be considered as a proxy for cell size, thus future works should focus on finding a better descriptor. In this respect, possible replacement may come from novel Neural Network-based approaches that use full scattered light to get a more precise estimate of the cell size see for instance ref. ^{63} or fixed-cells dyes, like succinimidyl ester dye, SE-A647, that provides a good measure of cell mass^{27}.

From the analytical point of view, the derived exact expressions for the size moments and the population fractions evolution could be further expanded to account for fluctuations in the key parameters (e.g., *κ* and *λ*), and to combine different strategies in different tasks should be performed.

In conclusion, we proposed an experimental and theoretical apparatus to characterize the growth and division of leukemia cells. We found that (i) while following a size-like homeostatic strategy, Jurkat cells (ii) need to pass a certain number of intermediate states before dividing, which are independent and exponentially distributed. (iii) Experimental data are well reproduced by a minimal model that depends on relatively few, physically meaningful parameters.

## Methods

### Cell culture

E6.1 Jurkat cells (kindly provided by Dr. Nadia Peragine, Department of Cellular Biotechnologies and Hematology, Sapienza University of Rome) were used as a cell model for proliferation study and maintained in RPMI-1640 complete culture media containing 10% FBS, penicillin/streptomycin plus glutamine at 37 °C in 5% CO_{2}. Upon thawing, cells were passaged once prior to amplification for the experiment. Cells were then harvested, counted, and washed twice in serum-free solutions and re-suspended in PBS for further staining.

### Cells fluorescent dye labeling

To track cell proliferation by dye dilution establishing the daughter progeny of a completed cell cycle, cells were stained with CellTrace™ Violet stain (CTV, C34557, Life Technologies, Paisley, UK), typically used to monitor multiple cell generations. To determine cell viability, prior to dye staining, the collected cells were counted with the hemocytometer using the dye exclusion test of Trypan Blue solution, an impermeable dye not taken up by viable cells. For the dyes staining, highly viable 20 × 10^{6} cells were incubated in a 2 ml solution of PBS containing CTV (1/1000 dilution according to the manufacturer’s instruction) for 25 min at room temperature (RT) mixing every 10 min to ensure homogeneous cell labeling. Afterward, complete media was added to the cell suspension for an additional 5 min incubation before the final washing in PBS.

### Cell sorting

Jurkat cells labeled with dyes were sorted using a FACSAriaIII (Becton Dickinson, BD Biosciences, USA) equipped with Near UV 375 nm, 488 nm, 561 nm, and 633 nm lasers and FACSDiva software (BD Biosciences version 6.1.3). Data were analyzed using FlowJo software (Tree Star, version 9.3.2 and 10.7.1). Briefly, cells were first gated on single cells, by doublets exclusion with morphology parameters, both side and forward scatter, area versus width (A versus W). The unstained sample was used to set the background fluorescence for each channel. For each fluorochrome, a sorting gate was set around the max peak of fluorescence of the dye distribution^{64}. In this way, the collected cells were enriched for the highest fluorescence intensity for the markers used. 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 > 99 % for each sample. See Supplementary Figures for more details.

### Time course kinetic for dye dilution assessment

The sorted cell population was seeded into a single well of a 6-well plate (BD Falcon) at 1 × 10^{6} cells/well and kept in culture for up to 72 h. To monitor multiple cell division, an aliquot of the cells in culture was analyzed every 18, 24, 36, 48, 60, and 72 h for the fluorescence intensity of CTV dye by the LSRFortessa flow cytometer. 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. Every time that an aliquot of cells was collected for analysis, the same volume of fresh media was replaced in the culture.

### Expectation-Maximization and the Gaussian Mixture Model

We used the Expectation-Maximization (EM) algorithm to detect the clusters in Gaussian Mixture Models^{65}. The EM algorithm is composed of two steps the Expectation (E) step and the Maximization (M) step. In the E-step, for each data point **f**, we used our current guess of *π*_{g}, *μ*_{g}, and *σ*_{g}, to estimate the posterior probability that each cell belongs to generation *g* given that its fluorescence intensity measure as **f,** *γ*_{g} = P(*g*∣**f**). In the M-step, we use the fact that the gradient of the log-likelihood of *p*(**f**_{i}) for *π*_{g}, *μ*_{g}, and *σ*_{g} can be computed. Consequently, the expression of the optimal value of *π*_{g}, *μ*_{g}, and *σ*_{g} is dependent on *γ*_{g}. It is shown that under, certain smoothness conditions the iterative computation of E-stem and M-step leads us to the locally optimal estimate of the parameters *π*_{g}, *μ*_{g}, and *σ*_{g}, and returns the posterior probability *γ*_{g} which weights how much each point belongs to one of the clusters. Here, we used this model to perform cluster analysis and detect the peaks which correspond to different generations. Then, we estimated *π*_{g}, E[*f*_{g}], and Var[*f*_{g}] from these clusters.

### Gillespie simulation

To validate the mathematical model that was formulated, stochastic simulations of the growth and dividing cell population were carried out. Note that, through simulation, we can also know the birth and division sizes and can therefore compare the trends of Δ Vs < *s*_{b} > .

In particular, simulations were performed starting from *N* = 1000 initial cells, having initial size randomly sampled from a normal distribution of mean *μ*_{s} and variance \({\sigma }_{s}^{2}\).

For each cell, a division time is extracted from the probability distribution *P*(*t*_{d}) via inverse transform sampling. For the considered system, *P*(*t*_{d}) is given by^{33}:

Upon division, each cell is split into two new daughter cells, each inheriting a fraction *p* and (1 − *p*) of the mother size, respectively.

### Used parameters

Simulation results shown in Fig. 2d, e and Fig. 4b, d, e, f are obtained running Gillespie simulations and/or solving Eq. (14) with the following default parameters *κ* = 5, *λ* = 1.6, *α* = 1, *β* = 2, *Q* = 3, < *s* > _{0} = 1, *C**V*_{0} = 0.15, and \(\pi (f)={{{{{{\mathcal{N}}}}}}}(0.5,0.001)\). Curves of panel (b1) are obtained with all default parameters but < *s* > _{0}, that assumed values of 0.8, 0.9, 1.0, and 1.1. Curves of panel (b2) are obtained with all default parameters but *λ*, which assumed values of 0.8, 1.6, and 3.2. Graphs in panel (d1) are obtained with all default parameters but *C**V*_{0}, which assumed values of 0.15, 0.17, and 0.2. Graphs in panel (d2) are obtained with all default parameters but *β*, that assumed values of 1, 2, 3, and 4.

### Approximate Bayesian Computation Sequential Monte Carlo

Approximate Bayesian Computation Sequential Monte Carlo (ABC SMC) for parameter estimation has been performed via the astroABC python library^{51}. The sampler has been initialized with uniform priors and the following default settings: ’dfunc’:dist_metric, ’adapt_t’: True, ’pert_kernel’:2. The cost function has been defined as the squared root of the sum of the squared residual between experimental and predicted fractions of cells per generations.

## 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.

## References

Kussell, E. & Leibler, S. Phenotypic diversity, population growth, and information in fluctuating environments.

*Science***309**, 2075–2078 (2005).McGranahan, N. & Swanton, C. Clonal heterogeneity and tumor evolution: past, present, and the future.

*Cell***168**, 613–628 (2017).De Martino, A., Gueudré, T. & Miotto, M. Exploration-exploitation tradeoffs dictate the optimal distributions of phenotypes for populations subject to fitness fluctuations.

*Phys. Rev. E***99**, 012417 (2019).Korolev, K. S., Xavier, J. B. & Gore, J. Turning ecology and evolution against cancer.

*Nat. Rev. Cancer***14**, 371–380 (2014).Jones, I. et al. Characterization of proteome-size scaling by integrative omics reveals mechanisms of proliferation control in cancer.

*Sci. Adv.***9**, eadd0636 (2023).Cadart, C., Venkova, L., Recho, P., Lagomarsino, MarcoCosentino & Piel, M. The physics of cell-size regulation across timescales.

*Nat. Phys.***15**, 993–1004 (2019).Ginzberg, M. B., Kafri, R. & Kirschner, M. On being the right (cell) size.

*Science***348**, 1245075–1245075 (2015).Modi, S., Vargas-Garcia, CesarAugusto, Ghusinga, KhemRaj & Singh, A. Analysis of noise mechanisms in cell-size control.

*Biophys. J.***112**, 2408–2418 (2017).Hatton, I. A. et al. The human cell count and size distribution.

*Proc. Natl Acad. Sci.***120**, e2303077120 (2023).Scotchman, E., Kume, K., Navarro, F. J. & Nurse, P. Identification of mutants with increased variation in cell size at onset of mitosis in fission yeast.

*J. Cell Sci.***134**, jcs251769 (2021).Huh, D. & Paulsson, J. Non-genetic heterogeneity from stochastic partitioning at cell division.

*Nat. Genet.***43**, 95–100 (2010).Huh, D. & Paulsson, J. Random partitioning of molecules at cell division.

*Proc. Natl Acad. Sci.***108**, 15004–15009 (2011).Miotto, M., Marinari, E. & De Martino, A. Competing endogenous RNA crosstalk at system level.

*PLOS Comput. Biol.***15**, e1007474 (2019).Peruzzi, G., Miotto, M., Maggio, R., Ruocco, G. & Gosti, G. Asymmetric binomial statistics explains organelle partitioning variance in cancer cell proliferation.

*Commun. Phys.***4**, 188 (2021).Phillips, N. E., Mandic, A., Omidi, S., Naef, F. & Suter, D. M. Memory and relatedness of transcriptional activity in mammalian cell lineages.

*Nat. Commun.***10**, 1208 (2019).Thomas, P. Analysis of cell size homeostasis at the single-cell and population level.

*Front. Phys.***6**, 64 (2018).Amir, A. Cell size regulation in bacteria.

*Phys. Rev. Lett.***112**, 208102 (2014).Campos, M. et al. A constant size extension drives bacterial cell size homeostasis.

*Cell***159**, 1433–1446 (2014).Fantes, P. A. Control of cell size and cycle time in schizosaccharomyces pombe.

*J. Cell Sci.***24**, 51–67 (1977).Pan, K. Z., Saunders, T. E., Flor-Parra, I., Howard, M. & Chang, F. Cortical regulation of cell size by a sizer cdr2p.

*eLife***3**, e02040 (2014).Facchetti, G., Knapp, B., Flor-Parra, I., Chang, F. & Howard, M. Reprogramming cdr2-dependent geometry-based cell size control in fission yeast.

*Curr. Biol.***29**, 350–358.e4 (2019).Cadart, C., Zlotek-Zlotkiewicz, E., Berre, MaëlLe, Piel, M. & Matthews, H. K. Exploring the function of cell shape and size during mitosis.

*Dev. Cell***29**, 159–169 (2014).Enrico Bena, C. et al. Initial cell density encodes proliferative potential in cancer cell populations.

*Sci. Rep.***11**, 6101 (2021).Wallden, M., Fange, D., Lundius, EbbaGregorsson, Baltekin, Özden & Elf, J. The synchronization of replication and division cycles in individual

*E. coli*cells.*Cell***166**, 729–739 (2016).Taheri-Araghi, S. et al. Cell-size control and homeostasis in bacteria.

*Curr. Biol.***25**, 385–391 (2015).Andersen, JensBo et al. New unstable variants of green fluorescent protein for studies of transient gene expression in bacteria.

*Appl. Environ. Microbiol.***64**, 2240–2246 (1998).Kafri, R. et al. Dynamics extracted from fixed cells reveal feedback linking cell growth to cell cycle.

*Nature***494**, 480–483 (2013).Sung, Y. et al. Size homeostasis in adherent cells studied by synthetic phase microscopy.

*Proc. Natl Acad. Sci.***110**, 16687–16692 (2013).Powell, E. O. A note on koch & schaechter’s hypothesis about growth and fission of bacteria.

*J. Gen. Microbiol.***37**, 231–249 (1964).Anderson, E. C., Bell, G. I., Petersen, D. F. & Tobey, R. A. Cell growth and division.

*Biophys. J.***9**, 246–263 (1969).Osella, M., Nugent, E. & Lagomarsino, MarcoCosentino Concerted control of escherichia coli cell division.

*Proc. Natl Acad. Sci.***111**, 3431–3435 (2014).Vargas-García, C. A, & Singh, A. Elucidating cell size control mechanisms with stochastic hybrid systems. In

*2018 IEEE Conference on Decision and Control (CDC)*, pages 4366–4371 (IEEE, 2018).Nieto-Acuna, CesarAugusto, Vargas-Garcia, CesarAugusto, Singh, A. & Pedraza, JuanManuel Efficient computation of stochastic cell-size transient dynamics.

*BMC Bioinform.***20**, 647 (2019).García-García, R., Genthon, A. & Lacoste, D. Linking lineage and population observables in biological branching processes.

*Phys. Rev. E***99**, 042413 (2019).Miotto, M. & Monacelli, L. Genome heterogeneity drives the evolution of species.

*Phys. Rev. Res.***2**, 043026 (2020).Miotto, M. & Monacelli, L. TOLOMEO, a novel machine learning algorithm to measure information and order in correlated networks and predict their state.

*Entropy***23**, 1138 (2021).Jia, C., Singh, A. & Grima, R. Cell size distribution of lineage data: analytic results and parameter inference.

*iScience***24**, 102220 (2021).Lin, J. & Amir, A. From single-cell variability to population growth.

*Phys. Rev. E***101**, 012401 (2020).Genthon, A. Analytical cell size distribution: lineage-population bias and parameter inference.

*J. R. Soc. Interface***19**, 20220405 (2022).Cadart, C. et al. Size control in mammalian cells involves modulation of both growth rate and cell cycle duration.

*Nat. Commun.***9**, 3275 (2018).Varsano, G., Wang, Y. & Wu, M. Probing mammalian cell size homeostasis by channel-assisted cell reshaping.

*Cell Rep.***20**, 397–410 (2017).Vargas-Garcia, C. A., Björklund, M. & Singh, A. Modeling homeostasis mechanisms that set the target cell size.

*Sci. Rep.***10**, 13963 (2020).Li, Q., Rycaj, K., Chen, X. & Tang, D. G. Cancer stem cells and cell size: a causal link?

*Semin. Cancer Biol.***35**, 191–199 (2015).Totis, N. et al. A population-based approach to study the effects of growth and division rates on the dynamics of cell size statistics.

*IEEE Control Syst. Lett.***5**, 725–730 (2021).Nieto, C. ésar, Arias-Castro, J., Sánchez, C., Vargas-García, C. ésar & Pedraza, JuanManuel Unification of cell division control strategies through continuous rate models.

*Phys. Rev. E***101**, 022401 (2020).Conlon, I. & Raff, M. Control and maintenance of mammalian cell size: response.

*BMC Cell Biol.***5**, 1–2 (2004).Yates, C. A., Ford, M. J. & Mort, R. L. A multi-stage representation of cell proliferation as a markov process.

*Bull. Math. Biol.***79**, 2905–2928 (2017).Bialek, W.S.

*Biophysics: Searching for Principles*(Princeton University Press, 2012).Miotto, M. & Monacelli, L. Entropy evaluation sheds light on ecosystem complexity.

*Phys. Rev. E***98**, 042402 (2018).Schoene, N. W. & Kamara, K. S. Population doubling time, phosphatase activity, and hydrogen peroxide generation in jurkat cells.

*Free Radic. Biol. Med.***27**, 364–369 (1999).Jennings, E. & Madigan, M. astroabc: an approximate bayesian computation sequential monte carlo sampler for cosmological parameter estimation.

*Astron. Comput.***19**, 16–22 (2017).Levin, PetraAnne & Angert, E. R. Small but mighty: cell size and bacteria.

*Cold Spring Harbor Perspect. Biol.***7**, a019216 (2015).De Paiva, C. S., Pflugfelder, S. C. & Li, De-Quan Cell size correlates with phenotype and proliferative capacity in human corneal epithelial cells.

*Stem Cells***24**, 368–375 (2005).Miettinen, T. P. et al. Identification of transcriptional and metabolic programs related to mammalian cell size.

*Curr. Biol.***24**, 598–608 (2014).Marshall, W. F. et al. What determines cell size?

*BMC Biol.***10**, 1–22 (2012).Pancaldi, V. Biological noise to get a sense of direction: an analogy between chemotaxis and stress response.

*Front. Genet.***5**, 52 (2014).Miotto, M. et al. Collective behavior and self-organization in neural rosette morphogenesis.

*Front. Cell Dev. Biol.***11**, 1134091 (2023).Ho, Po-Yi, Lin, J. & Amir, A. Modeling cell size regulation: from single-cell-level statistics to molecular mechanisms and population-level effects.

*Annu. Rev. Biophys.***47**, 251–271 (2018).Chao, HuiXiao et al. Evidence that the human cell cycle is a series of uncoupled, memoryless phases.

*Mol. Syst. Biol.***15**, e8604 (2019).Tzur, A., Kafri, R., LeBleu, V. S., Lahav, G. & Kirschner, M. W. Cell growth and size homeostasis in proliferating animal cells.

*Science***325**, 167–171 (2009).Nobs, Jean-Bernard & Maerkl, S. J. Long-term single cell analysis of s. pombe on a microfluidic microchemostat array.

*PLoS ONE***9**, e93466 (2014).Nakaoka, H. & Wakamoto, Y. Aging, mortality, and the fast growth trade-off of schizosaccharomyces pombe.

*PLOS Biol.***15**, e2001109 (2017).Reale, R. et al. A low-cost, label-free microfluidic scanning flow cytometer for high-accuracy quantification of size and refractive index of particles.

*Lab on a Chip***23**, 2039–2047 (2023).Filby, A., Begum, J., Jalal, M. & Day, W. Appraising the suitability of succinimidyl and lipophilic fluorescent dyes to track proliferation in non-quiescent cells by dye dilution.

*Methods***82**, 29–37 (2015).Dempster, A. P., Laird, N. M. & Rubin, D. B. Maximum likelihood from incomplete data via the em algorithm.

*J. R. Stat. Soc. Ser. B***39**, 1–38 (1977).

## Acknowledgements

G.R. thanks the European Research Council Synergy grant ASTRA (n. 855923), the European Innovation Council through its Pathfinder Open Programme, project ivBM-4PAP (grant agreement No 101098989), and 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) for support. M.L. and G.G. thank Project LOCALSCENT, Grant PROT. A0375-2020- 36549, Call POR-FESR “Gruppi di Ricerca 2020”.

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M.M., G.G., and G.P. conceived research; M.L. and G.R. contributed additional ideas; G.P. and S.S. performed experiments; M.M., S.S., and G.G. analyzed data; M.M. performed analytical calculations, numerical simulations, and statistical analysis; all authors analyzed results; M.M. wrote the paper; all authors revised the paper.

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Miotto, M., Scalise, S., Leonetti, M. *et al.* A size-dependent division strategy accounts for leukemia cell size heterogeneity.
*Commun Phys* **7**, 248 (2024). https://doi.org/10.1038/s42005-024-01743-1

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DOI: https://doi.org/10.1038/s42005-024-01743-1

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