Dissecting Particle Uptake Heterogeneity in a Cell Population Using Bayesian Analysis

Abstract

Individual cells in a solution display variable uptake of nanomaterials, peptides, and nutrients. Such variability reflects their heterogeneity in endocytic capacity. In a recent work, we have shown that the endocytic capacity of a cell depends on its size and surface density of endocytic components (transporters). We also demonstrated that in MDA-MB-231 breast cancer cells, the cell-surface transporter density (n) may decay with cell radius (r) following the power rule n ∼ r^α, where α ≈ −1. In this work, we investigate how n and r may independently contribute to the endocytic heterogeneity of a cell population. Our analysis indicates that the smaller cells display more heterogeneity because of the higher stochastic variations in n. By contrast, the larger cells display a more uniform uptake, reflecting less-stochastic variations in n. We provide analyses of these dependencies by establishing a stochastic model. Our analysis reveals that the exponent α in the above relationship is not a constant; rather, it is a random variable whose distribution depends on cell size r. Using Bayesian analysis, we characterize the cell-size-dependent distributions of α that accurately capture the particle uptake heterogeneity of MDA-MB-231 cells.

Significance

Cells acquire nutrient molecules, peptides, and nanoscale materials from the environment. However, the ability to acquire these external materials varies from cell to cell. Therefore, a population of cells under the same environment displays a distribution in the uptake of these materials. The variables and mechanisms behind such cellular heterogeneity remain unclear. In this work, an integrative theoretical modeling and experimental approach is taken to investigate what cellular attributes may determine this heterogeneity. The study is motivated by the need to better understand cellular responses to a common environment. Moreover, such knowledge of cellular uptake is important for delivering nanoscale materials into cells for many biological, biomedical, and biopharmaceutical applications.

Introduction

Mammalian cells often demonstrate molecular and phenotypic heterogeneity under the same growth condition (1). Such heterogeneity or noise obscures valuable information about the regulatory mechanisms of cell functions at the molecule, cell, and population scale (2, 3, 4, 5, 6, 7). This work focuses on understanding how a population of cells displays heterogeneity in their endocytic capacity even in the same growth environment. By several endocytic routes, cells uptake nanoscale external materials such as nanoparticles, peptides, and nutrients from the environment. When incubated in a solution, cells of the same type display a distribution in the amount of these materials (8, 9, 10). This distribution is a direct reflection of their relative endocytic capacities. The question we ask is what variables at the cellular or molecular level may define such distribution in a cell population.

Cellular uptake of an external particle or molecule can be broadly defined as a three-step process (11,12). The first step involves the transport of a particle from the extracellular medium to the surface of a cell. The second step involves its recognition and capture by an endocytic component or a transporter protein in the cell plasma membrane. The final step involves its internalization and trafficking into the cell cytoplasm. The first step could be the rate-limiting step in a physiological tissue or high-viscosity medium due to slow or hindered diffusion (13,14). On the other hand, the latter two steps could be rate limiting in a regular cell culture medium, where diffusion is relatively fast.

It is well established that many endocytic structures or transporter proteins mediate particle recognition and trafficking in the cell plasma membrane. Some examples of these components or proteins include clathrin-coated pits, caveolae, micropinocytes, and many membrane-anchored transporter or receptor proteins. Nevertheless, depending on the nature of the particle or molecule, only a small subset of these different components may be relevant in an uptake process (15). It is intuitive to think that the relative abundance of these components in a cell will determine its endocytic capacity for a specific molecule or particle. However, in addition to this, the size of a cell could also be a crucial determinant of its endocytic capacity (16,17). Cells are naturally heterogeneous in size and typically follow a lognormal distribution (18,19). A cell having a larger volume should experience a greater demand for external materials. This demand could be met by taking advantage of its greater surface area, which is capable of accommodating more endocytic components and creating a bigger mass transfer interface with the environment. Moreover, a larger cell, because of its greater transcriptional output, may express more endocytic components in the plasma membrane as well. Therefore, the heterogeneity in the endocytic capacity in a cell population may reflect the distribution in cell size, relative abundance of specific transporter proteins, and interdependency of these two variables (19).

To better understand the above roles of cell-surface endocytic components and cell size in determining the uptake heterogeneity, a simple model may be conceptualized. In the model, a spherical cell of radius r contains n molecules of an endocytic transporter per unit area of its plasma membrane. Each of these transporter molecules can form complex with a particle from the solution, and the complex is then internalized in a single-step process. Subsequently, the internalized particle may also be exocytosed in a single step. Here, this transporter can be viewed as a generic hypothetical molecule representing all different endocytic structures contributing to particle uptake. For example, we may substitute a clathrin-coated pit with N_p such transporter molecules, assuming the former mediate uptake at a rate N_p times faster than a single transporter molecule, which handles one particle at a time. Now, ignoring diffusion, the following equation can be used to describe the amount of particle uptake (U) as a function of cell size (r) and cell-surface transporter density (n):

$$\frac{dU}{dt}=\frac{4\pi r^2{k}_{1}nS}{K_M+S}-k_{e}U,$$ (1)

where S is the particle concentration in the extracellular medium. The first term on the right-hand side describes a Michaelis-Menten process for the particle endocytosis, and the second term describes single-step exocytosis. The parameters K_M and k₁ are the Michaelis-Menten constant and the rate constant associated with particle internalization, respectively. Parameter k_e is the rate constant associated with exocytosis. The equation shows how heterogeneity in U may originate from the cell-to-cell variability in r and n. For a sufficiently long incubation time (τ), the cells should reach a steady state, and the uptake can be expressed as

$$U_{ss}=\frac{4\pi r^2{k}_{1}nS}{k_e\left(K_M+S\right)}\sim n{r}^2.$$ (2)

Based on this equation, the endocytic capacity (ψ) of a cell of radius r can be defined as its particle uptake relative to an average cell in a population. Here, the average cell implies a cell representing the characteristic or population-average size $\langle r\rangle$.

$$\psi =\frac{U_{ss}}{\langle U_{ss}\rangle }=\frac{r^2}{{\langle r\rangle }^2}\frac{n}{\langle n\rangle },$$ (3)

where $\langle \cdot \rangle$ stands for a mean (population-average) quantity.

Equation 3 indicates a simple relationship between the cellular endocytic capacity ψ, cell size r, and cell-surface transporter density n. However, in this relationship, the two variables r and n may not necessarily be independent. Indeed, our recent study (19) indicates that n and r are anticorrelated. Here, we further investigate this previous finding. We use a Bayesian approach to accurately capture the dependency between these two variables and investigate how this dependency dictates heterogeneity in nanoparticle uptake by MDA-MB-231 breast cancer cells. We provide a detailed and quantitative analysis of relative contributions of r and n in determining the distribution of endocytic capacity in a cell population.

Materials and Methods

We performed flow cytometry to characterize the size of MDA-MB-231 breast cancer cells and their uptake of 100 nm (diameter) polystyrene nanoparticles. Individual cell size in a flow cytometer sample was calculated from the forward light scatter (FSC) signal. We also performed confocal imaging to directly evaluate cell size and make the necessary correction in the FSC-derived information. The cell size was then mapped to its normalized fluorescence intensity of internalized nanoparticles. Below, we provide details of our experimental methods.

Nanoparticle preparation

We used green fluorescent polystyrene nanoparticles (Fluoro-Max; Thermo Fisher Scientific, Fremont, CA) without further modification and purification. The particles had a mean size (diameter) of 100 nm. Particle stock solutions were stored per the manufacturer’s instructions. Using dynamic light scattering, we confirmed the particle size. Before the uptake experiment, we vortexed and sonicated the particle solutions per the manufacturer’s recommendations. For use in the dynamic light scatter measurements (Zetasizer Nano ZS; Malvern Panalytical, Westborough, MA), we diluted the sample stock solutions with deionized water to maintain the specific concentrations recommended in the Zetasizer protocol.

We prepared working particle solutions by diluting the stock solution with deionized water at room temperature. The solution was then further diluted in RPMI 1640 media and vortexed to ensure uniform mixing. The RPMI 1640 media were prewarmed to 37°C for better particle dispersion.

Cell culture

We grew MDA-MB-231 cells in RPMI 1640 medium (Roswell Park Memorial Institute 1640 media; Corning cellgro Mediatech, Manassas, VA) supplemented with 10% fetal bovine serum (Gibco Life Technologies, Grand Island, NY) and 1% penicillin/streptomycin (Gibco). The cells were kept in a humidified incubator at 37°C and 5% CO₂ and split at 70–80% confluence using 0.25% trypsin-EDTA solution (Gibco).

Flow cytometry

For flow cytometry, we seeded cells in 24-well plates. Each well had 1 mL media (10⁵ cells per mL of RPMI 1640) added to it and was incubated for 24 h to allow cell attachment. After the incubation, we replaced the media in each well with a solution of nanoparticles in RPMI 1640. Here, we used 20 μg particles/mL. We incubated cells with the nanoparticle solution for 5 h and then conducted flow cytometry. For the particle concentration used, cells reached the steady-state condition in their number of internalized particles within ∼3 h of incubation. We confirmed the steady-state condition by taking the time series measurement of the number of cell-internalized particles.

For time series data, we washed cells three times with phosphate-buffered saline (PBS) and detached them with 0.25% trypsin-EDTA (Gibco). We mixed the cell suspension with 500 μL of fresh RPMI 1640 media solution for flow cytometric measurement. We performed measurement using a BD Accuri C6 Plus flow cytometer (Becton Dickinson, Franklin Lakes, NJ) with a 488 nm argon-ion laser. Fluorescence was collected through a 533/30 nm bandpass filter. At least 10⁵ events per sample were taken for analysis. We noticed two distinct populations in a forward scatter (FSC) versus 90° side scatter (SSC) log-log plot, one with low SSC and FSC and the other with high SSC and FSC. We discarded the former, considering dust or debris, and used the latter, which accounted for over 80% of the data, for analysis.

Confocal imaging

For the confocal experiment, we seeded cells in eight-well glass bottom plates. The cell incubation time and other conditions were identical to those used for flow cytometry analysis. After incubation, we washed the cells three times with PBS and detached them with 0.25% trypsin-EDTA. We mixed the cell suspension with 500 μL of fresh RPMI 1640 media solution. We imaged the MDA-MB-231 cell suspension on a cover glass using a Nikon A1R HD25 confocal microscope (Tokyo, Japan) with a 60× and 100× objective lens. For three-dimensional confocal imaging, cells were fixed after the three-time wash with PBS. The fixed cells were added with actin (red) and nucleus (blue) stains. Subsequently, z-stack images of single cells were taken using a 100× objective.

Cell size estimation

Using the same BD Accuri C6 Plus that we used for the measurement of cell-internalized particle fluorescence, we collected forward scatter (FSC-A) data for a mixture of standard fluorescent beads of mean diameter 2 and 3 μm. The data revealed two distinct peaks corresponding to the two particle sizes in the mixture. Based on the difference in the peak locations for the 2 and 3 μm beads, we calculated the size individual MDA-MB-231 cells from linear extrapolation: cell size (μm) = 3 + (F_c − F₃)/(F₃ − F₂). Here, F_c, F₂, and F₃ represent the FSC-A peak intensities of the MDA-MB-231 cells, 2 μm beads, and 3 μm beads, respectively.

We analyzed confocal images using ImageJ software (20) to quantify the size of individual MDA-MB-231 cells. Images of 1500 MDA-MB-231 cells were analyzed to collect their size statistics. These actual sizes obtained from the confocal images were used to correct the FSC-based estimate for the entire cell population in the flow cytometer sample following the procedure in (19).

Results

Particle uptake heterogeneity in MDA-MB-231 cells

A population of cells incubated with particles, molecules, or peptides displays a distribution of uptake, indicating heterogeneity in the cellular endocytic capacities. In our earlier work (19), we demonstrated that the endocytic capacity of a cell is determined by its size (radius r, assuming the cell is spherical) and endocytic transporter density in the plasma membrane (n). Here, we focus on dissecting how these two variables independently and jointly dictate the heterogeneity in particle uptake by cells.

To demonstrate the heterogeneity in particle uptake in a cell population, we characterized cell-size-dependent uptake of nanoparticles in MDA-MB-231 cells, which are nearly spherical in suspension. Fig. 1 A shows a snapshot of such suspended spherical cells and the fluorescence of their internalized particles. Contrary to the suspended cells, attached MDA-MB-231 cells take irregular shapes. Fig. 1 B shows a high-resolution image of an attached MDA-MB-231 cell and its internalized nanoparticle (green). In our analysis, we used the suspended cells, which permitted more convenient and accurate estimation of cell sizes.

Figure 1.

Confocal images of MDA-MB-231 cells and internalized nanoparticles. (A) MDA-MB-231 cells in suspension are shown. (B) A high-resolution image of surface-attached MDA-MB-231 cells is shown. Green, red, and blue represent nanoparticles, plasma membrane, and nucleus, respectively. To see this figure in color, go online.

As described in the Materials and Methods, we analyzed the flow cytometer FSC-A signal to obtain the size of individual cells in a flow cytometer sample of 10⁵ cells. We also analyzed confocal images to collect cell-size statistics from a relatively small sample (1500 cells). Following (19), the confocal image-derived cell-size statistics were used to correct the FSC-based calculation, which otherwise yielded an inaccurate estimate for cell size. The size of each cell was mapped to its amount (fluorescence intensity) of internalized nanoparticles.

The scatter plot in Fig. 2 A shows the population-scale variability in cell size (radius r) and the heterogeneity in relative particle uptake ψ (Eq. 3) in our flow cytometer sample. Consistent with our earlier work (19), the data reveals a nearly linear correlation between r and ψ despite the high degree of noise. For a particular cell size, this noise represents cell-to-cell variability in n (Eq. 3). The gray peak in Fig. 2 B shows the distribution of the log-transformed values of ψ. The plot indicates ψ is lognormally distributed with standard deviation (SD) σ_ψ ≈ 0.35.

Figure 2.

Particle uptake heterogeneity in MDA-MB-231 cells. (A) A colormap (probability density) is given showing distribution of relative particle uptake ψ (Eq. 3) and cell radius r in a flow cytometer sample of 10⁵ cells. Corresponding colorbar is shown on the right. (B) The probability distribution of log-transformed relative uptake ψ in the sample is shown. The peak fits to a normal distribution (solid curve) with SD 0.35. The root mean-square error for the fitting is 0.0084. To see this figure in color, go online.

The distribution above obscures the joint contributions from cell-to-cell variability in cell size r and cell-surface transporter density n, considering the remaining parameters in Eq. 3 are identical among the cells. The question we are interested in is whether the independent contributions of r or n in this distribution can be quantified and captured in a model.

Relating cell size and transporter density

In Khetan et al. (19), we proposed that particle flux into a cell directly reflects the density of endocytic transporters in the plasma membrane. The underlying theoretical basis was as follows. Consider Eq. 1, which describes the time-dependent accumulation of particles in a cell. If the incubation time τ $\gg$ t_ss, where t_ss is the time needed to reach steady state, the time-averaged uptake rate is the net amount of particle uptake divided by τ:

$${\dot{U}}_{\tau }=\frac{1}{\tau }{\int }_{t=0}^{t=\tau }U\left(t\right)dt\approx \frac{1}{\tau }{\int }_{t=0}^{t=\tau }{U}_{ss}dt=\frac{4\pi r^2{k}_{1}nS}{k_e\left(K_M+S\right)}\sim r^{2}n.$$ (4)

In our experiments, τ = 5 h, whereas the steady state was reached within 50 min of incubation. By definition, the time-averaged particle flux into a cell is the rate of uptake divided by its surface area: ${\dot{U}}_{\tau }$/4πr². Following the same convention as ψ, we can define ϕ to be the particle flux into a cell of size r relative to an average cell with size $\langle r\rangle$:

$$\varphi =\frac{{\dot{U}}_{\tau }}{\langle {\dot{U}}_{\tau }\rangle }\times \frac{r^{-2}}{{\langle r\rangle }^{-2}}=\frac{n}{\langle n\rangle },$$ (5)

where $\langle \cdot \rangle$ represents a population-average quantity. Because n ∼ ϕ, a plot of ϕ against r should uncover any potential dependence of n on r as well. Our analysis in (19) showed that n might decay in a growing cell following a power rule n ∼ r^α, i.e.,

$$\phi =\frac{n}{⟨n⟩}=\frac{r^{\alpha }}{{⟨r⟩}^{\alpha }}$$ (6)

In (19), we also showed that α ≈ −1, which led to the linear relationship between uptake and cell radius.

However, in this work, we propose that α is not a single-valued parameter; rather, it is a random variable whose distribution depends on cell size r. To demonstrate this, we divided the 10⁵ cells in Fig. 3 into 17 distinct subpopulations. Each subpopulation contained cells with radius r_i ± 0.25 μm, where r_i ∈ {4, 4.5, 5, …, 12} μm. Thus, two cells in such a subpopulation could differ at most by 0.5 μm. Considering this small gap in size, the effect from the size heterogeneity within each subpopulation can be ignored. The distribution in ϕ within each subpopulation should arise primarily from the cell-to-cell variability in the surface transporter density n. However, if we compare the mean ϕ of two such subpopulations i and j, we can tell how ϕ (or n) may change because of the difference between cell sizes r_i and r_j. In Fig. 3, we plot the mean ϕ for each of the 17 subpopulations in our sample. The error bar associated with each point represents the subpopulation noise (SD of the peak). The data shows an exponential decay in ϕ (or n), indicating a negative value for the exponent α in Eq. 6. By fitting Eq. 6 to this data, we obtained α ≈ −0.8. The solid line in Fig. 3 represents Eq. 6 with this best-fit value.

Figure 3.

Dependence of the cell-surface particle influx ϕ on cell radius r. The 10⁵ cells in the entire sample (Fig. 2) were binned into 17 distinct subpopulations. The cells in subpopulation i had radius r_i ± 0.25 μm, where r_i ∈ {4, 4.5, 5, …, 12} μm. Each data point (filled circle) and associated error bar represent the mean and SD of ϕ in corresponding subpopulation.

It is important to note that the error bar (the SD of ϕ in each subpopulation) also decays with r. This indicates that the larger cells may be subject to less-stochastic variations in n. To account for this cell-size dependence of n and its noise, it is more appropriate to treat α as a stochastic variable, with its SD being a function of r. Thus, a more general form of Eq. 6 to account for the heterogeneity in r and n is

$$\varphi =\frac{n}{\langle n\rangle }=\frac{r^{\alpha \left(r\right)}}{{\langle r\rangle }^{\langle \alpha \rangle }},$$ (7)

where $\langle \cdot \rangle$ indicates average over the entire sample of 10⁵ cells. Here, $\langle \alpha \rangle$ is expected to be the same as the deterministic best-fit value −0.8. Based on Fig. 2 B, if we consider a lognormal distribution for ϕ (or n) for each subpopulation, then α(r) is expected to follow a normal distribution. However, we were interested in identifying the actual distribution for α(r) so that it can be used to describe heterogeneity in ϕ or n based on Eq. 7. Moreover, we wanted to investigate the mean (subpopulation-average) value of α(r) for each of the 17 subpopulations to see if this quantity indeed remains uniform at the best-fit value −0.8 obtained from the deterministic relationship in Eq. 6.

Bayesian framework for predicting the uptake heterogeneity

Based on Eq. 7, the random variate α(r) obscures heterogeneity associated with both r and n. The identification of this parameter should enable the prediction of the population-scale distributions of ϕ and ψ. However, because the distribution of α(r) is a function of r, its Bayesian estimation poses a problem quite different from standard model parameter estimation. Bayesian parameter estimation (21) yields the joint posterior distribution of a set of model parameters. By contrast, here our interest is to identify the distributions of the same model parameter α at different r. Fig. 3 indicates that the SD of α(r) takes different values at each of the 17 data points. To identify these distributions of α(r), we took an approach explained below.

According to the Bayesian theory,

$$P\left(\Theta |\mathcal{D}\right)=\frac{P\left(\mathcal{D}|\Theta \right)P\left(\Theta \right)}{P\left(\mathcal{D}\right)},$$ (8)

where $\mathcal{D}$ is an experimental outcome (data or an observation) and Θ is a set of parameters underlying $\mathcal{D}$. $P\left(\Theta |\mathcal{D}\right)$ and P(Θ)are the posterior and prior probability of Θ, respectively. $P\left(\mathcal{D}|\Theta \right)$ is the likelihood of $\mathcal{D}$ for a given Θ. $P\left(\mathcal{D}\right)$ is the evidence, i.e., the probability of $\mathcal{D}$ integrated over the entire parameter space.

To explain Eq. 8 in the context of our problem, let us conceptualize a hypothetical experiment in which we have s = 17 distinct cell culture wells. Each well i ∈ {1, 2, …, s} contains cells of identical sizes with radius r_i ∈ {4, 4.5, …, 12}. In the wells, cells are incubated with nanoparticle solutions of identical concentration for the same period. This incubation time τ is longer than the time needed to reach steady state, i.e., τ > t_ss. After incubation, one cell from each well is sampled randomly and analyzed for its time-averaged particle influx ϕ (Eq. 5). Thus, a tuple of s distinct particle flux data, {ϕ_i}, for i ∈ {1, 2, …, s} is generated. This tuple {ϕ_i} is a random outcome in the hypothetical experiment. From this tuple, a plot ϕ_i vs. r_i can be created, although the curve is expected to be very stochastic. It may deviate considerably from the subpopulation-average curve in Fig. 3. However, many such tuples can be created by random sampling. The mean values and associated SDs of these samples should coincide with the 17 data points and associated error bars in Fig. 3.

We can use Eq. 7 to simulate above hypothetical experiment. We consider α(r) to be distinct and independent at each of the s = 17 data points and accordingly define an s-dimensional joint prior: P(Θ) = P({α_i}) = Π_iP(α_i). To mimic the random sampling of cells from the hypothetical wells, we sample {α_i} with P(Θ) and create a tuple of synthetic data $\left\{M\left({\alpha }_i\right)\right\}=\left\{r_i^{{\alpha }_i}/{\langle r\rangle }^{\langle \alpha \rangle }\right\}$. Here, {M(α_i)} implies a prediction as contrary to its experimental counterpart {ϕ_i}. We calculate the likelihood of this sample against the (mean) data points in Fig. 3: $P\left(\mathcal{D}|\Theta \right)=\text{exp}\left\{-{\sum }_{i=1}^c\left(1/{\sigma }_i^2\right){\left[\text{log}{\varphi }_i-\text{log}M\left({\alpha }_i\right)\right]}^2\right\}$. Here, logϕ_i stands for the log (10-based)-transformed value of the subpopulation-averaged experimental ϕ_i (a data point in Fig. 3), and σ_i stands for the corresponding SD (error bar in Fig. 3). We then calculate the posterior probability based on Eq. 8: $P(\Theta |\mathcal{D})=P(\mathcal{D}|\Theta )P\left(\Theta \right)$ by setting $P\left(\mathcal{D}\right)$ = 1. Evidence $P\left(\mathcal{D}\right)$ can be set to 1 because this term gets eliminated in the Metropolis-Hasting algorithm (22) while implementing the Markov chain Monte Carlo (MCMC) (23), which we explain below.

We implement MCMC to determine the joint posterior distribution of {α_i}. In any MCMC step k, we have a current Θ^k = $\left\{{\alpha }_i^k\right\}$, which is obtained from the previous step. We create a new (proposal) Θ for the next step by introducing a small random perturbation: Θ^{k + 1} = Θ^k + Δ, where each element of Δ is Δ_i = γN(0, 1)/||Δ||. Here, γ is a tunable parameter that determines the rate of acceptance of the roposal. We use γ = 0.1, which yields a 33% acceptance rate on average. We calculate P(Θ^{k + 1}) = $\text{exp}\left\{-{\sum }_{i=1}^s{\left[{\alpha }_i^{k+1}-{\alpha }_i^k\right]}^2\right\}$ and $P\left(\mathcal{D}|{\Theta }^{k+1}\right)=\text{exp}\left\{-{\sum }_{i=1}^s\left(1/{\sigma }_i^2\right){\left[\text{log}{\varphi }_i-\text{log}M\left({\alpha }_i^{k+1}\right)\right]}^2\right\}$. We then calculate posterior $P\left({\Theta }^{k+1}|\mathcal{D}\right)$ by plugging $P\left(\mathcal{D}|{\Theta }^{k+1}\right)$ and P(Θ^{k + 1}) into Eq. 8:

$$P\left({\Theta }^{k+1}|\mathcal{D}\right)=\text{exp}\left\{-\sum _{i=1}^s\left(1/{\sigma }_i^2\right){\left[\text{log}{\varphi }_i-\text{log}M\left({\alpha }_i^{k+1}\right)\right]}^2-\sum _{i=1}^s{\left[{\alpha }_i^{k+1}-{\alpha }_i^k\right]}^2\right\}.$$ (9)

The Metropolis-Hasting algorithm (22) is then used to either reject or accept the proposal Θ^{k + 1}. We draw a uniform random number between 0 and 1, ρ ∼ U(0, 1), and accept Θ^{k + 1} if

$$\rho \le min\left\{1,\frac{P({\Theta }^{k+1}|\mathcal{D})}{P({\Theta }^k|\mathcal{D})}\right\}$$ (10)

or reject otherwise. The process is repeated for a large number of steps. We collect samples of Θ at every 100th step after the MCMC chain equilibrates at the end of the so-called burn-in period (21).

It should be noted that at the start of the MCMC chain, there is no existing Θ from a previous step. Therefore, at the start, we initiate Θ = {μ_i + N(0, 1)}, where μ_i = −0.8, the best-fit value obtained from the deterministic fitting in Fig. 3, and N(0, 1) is a random value sampled from the standard normal distribution.

Cell-size dependence of α(r) and its distributions

Fig. 4 shows our Bayesian estimation of α(r) and its distribution for each of the 17 subpopulations. In Fig. 4 A, the red curve shows the Bayesian fit between Eq. 7 (red) and the experimental data in Fig. 3 (black). The red curve is created by randomly sampling 1000 distinct {α_i} from the 17-dimensional joint posterior distribution. These sample α-values were plugged into Eq. 7 to create 1000 trajectories. The red curve shows the mean and SD of these trajectories at each r_i. The plot reveals accurate predictions of the subpopulation-average flux and error bar for each of the 17 data points. Therefore, the estimated joint posterior distribution of α_i should correctly account for the heterogeneity in n. These discrete subpopulation-average α_i-values and their error bars can be interpolated for other intermediate cell sizes.

Figure 4.

Bayesian estimation of parameter α(r). (A) Black represents the experimental data in Fig. 3, and red represents the Bayesian fit of Eq. 7. The experimental and computed SDs associated with each point is shown by the black and red error bars, respectively. (B) Marginal probability density of α for different cell sizes is shown. The widest peak represents the smallest cells (4 μm), and the narrowest peak represents the largest cells (12 μm). The peaks corresponding to other intermediate cell sizes are also shown. (C) The mean values of α and corresponding SDs are plotted as functions of cell size. To see this figure in color, go online.

Fig. 4 B shows the marginal distribution of each α_i. For smaller cells, these distributions are wider and skewed to the left. However, for larger cells, they closely resemble a normal distribution. Fig. 4 C shows the mean $\left({\mu }_{{\alpha }_i}\right)$ and SD $\left({\sigma }_{{\alpha }_i}\right)$ of each marginal distribution. An important thing to note is that these mean values are insensitive to cell size, though the SDs show a systematic decay with increasing cell size. By taking average of the 17 mean values, we get the entire population-average value of α: $\sum _i{\mu }_{{\alpha }_i}/17=\langle \alpha \rangle =-0.82\approx {\mu }_{{\alpha }_i}\in \left\{{\mu }_{{\alpha }_1},{\mu }_{{\alpha }_2},\cdots ,{\mu }_{{\alpha }_{17}}\right\}$. This is close to our deterministic best-fit α = −0.80. This result is important because it indicates that μ_α(r) is independent of r, and it can be substituted by $\langle \alpha \rangle$ = −0.82 to make deterministic predictions for any arbitrary cell size r. However, to make a single-cell prediction or to account for the heterogeneity in a population, we must sample α(r) from its appropriate cell-size-dependent distribution.

Effect of transporter density on uptake heterogeneity

From the analysis above, it is apparent that both the cell-surface transporter density n and associated noise are functions of cell size. We next wanted to analyze how n and its noise could contribute to the heterogeneity in the cell-surface flux ϕ and endocytic capacity ψ if the cells are homogeneous in size (or narrowly distributed in size so that they could be treated as homogeneous). The purpose of this analysis is to dissect the sole contribution of n in determining the observed heterogeneity in uptake and endocytic capacity.

Before providing the experimental data-derived analysis, we provide a theoretical basis for speculating how the noise in n might dictate the heterogeneity in ϕ and ψ. Based on our results in Fig. 4 B, let us consider that the posterior (marginal) distribution of α(r) for any arbitrary r can be approximately described by a normal distribution. Also, based on Fig. 4 C, let us consider μ_α(r) = $\langle \alpha \rangle$ at any r. With these assumptions, taking log on both sides of Eq. 7,

$$\text{ln}\varphi \left(r\right)=\text{ln}\frac{r^{\langle \alpha \rangle }}{{\langle r\rangle }^{\langle \alpha \rangle }}+\left({\sigma }_{\alpha \left(r\right)}\text{ln}r\right)N\left(\mathrm{0,1}\right)$$

The equation above indicates that ϕ should follow a lognormal distribution in the absence of any heterogeneity in cell size. The mean of such a distribution is μ_ϕ(r) = $\text{ln}\left(r^{\langle \alpha \rangle }/{\langle r\rangle }^{\langle \alpha \rangle }\right)$, and the corresponding SD is σ_ϕ(r) = σ_α(r)lnr. This SD qualitatively tells how the heterogeneity in ϕ will scale with a change in the cell size of the population. Fig. 5 A shows distributions of ϕ for five such distinct cell populations. Within each population, cells are homogeneous in size, but between two populations, the cells are of distinct sizes, as indicated. These distributions are created by sampling α(r) from the corresponding marginal distributions obtained from our Bayesian estimation. Let us consider the leftmost peak, which represents cells with r = 12 μm (largest cells in our measurement). Because in our experimental sample, $\langle r\rangle$ = 7.5 μm, the mean of the leftmost peak, μ_ϕ(12) ≈ ln(12/7.5)^−0.82 = −0.39. Taking antilog, ϕ = 0.67. That is, particle influx across the plasma membrane of a cell of radius 12 μm should, on average, be 67% of that of an average cell in our sample. Again, from our Bayesian calculation, we obtained σ_α(12) = 0.068. Therefore, σ_ϕ(12) ≈ 0.068 × ln(12) = 0.17.

Figure 5.

Predicted cell-surface particle flux (ϕ) and relative uptake (ψ) for cells of five different sizes. (A) The probability density for lnϕ(r_i) at r_i ∈ {4, 6, 8, 10, 12} μm, from right to left, is shown. (B) The probability density for lnψ(r_i) for r_i ∈ {4, 6, 8, 10, 12} μm, from left to right, is shown.

The deterministic form of the endocytic capacity ψ in Eq. 3 can be redefined to account for its cell-to-cell variations:

$$\psi \left(r\right)=\frac{r^{2+\alpha \left(r\right)}}{{\langle r\rangle }^{2+\langle \alpha \rangle }}.$$ (11)

Again, assuming normal distribution for α(r), we can show that ψ(r) follows a lognormal distribution with mean μ_ψ(r) = $\text{ln}\left(r^{2+\langle \alpha \rangle }/{\langle r\rangle }^{2+\langle \alpha \rangle }\right)$ and SD σ_ψ(r) = σ_α(r)lnr. Therefore, lnψ(r) and lnϕ(r) have identical distribution peaks separated by a distance |μ_ϕ(r) − μ_ψ(r)|. Nevertheless, there is a remarkable difference in the way ϕ and ψ may change with cell size. Whereas ϕ(r) becomes smaller in larger cells (Fig. 5 A), ψ(r) changes in the opposite direction (Fig. 5 B). Note that this shift in the opposite directions is determined by the power terms associated with corresponding distribution means. Whereas for ϕ(r), this power term is negative (α = −0.82), for ψ(r), it is positive (2 + $\langle \alpha \rangle$ = 2 − 0.82 = 1.18). An interesting study would be to investigate whether this is true for all different cell types. For certain other cell types, if $\langle \alpha \rangle$ < −2, both flux and uptake will shift in the same direction, i.e., these quantities will be smaller in larger cells and vice versa. On the other hand, if $\langle \alpha \rangle$ > 0, both will increase with cell size. Otherwise, they will shift in the opposite directions, as in Fig. 5.

Effect of cell size on uptake heterogeneity

In the previous section, we discussed how the noise in cell-surface transporter density n alone contributes to the heterogeneity in the uptake flux ϕ and endocytic capacity ψ. Here, we dissect the contribution from cell size alone in determining the heterogeneity of these two quantities. To provide a theoretical basis for this analysis, we consider a hypothetical cell population with a size distribution identical to our experimental sample of 10⁵ cells. However, in this hypothetical population, we consider the noise associated with n is zero, meaning n is uniform across the population. Therefore, for this hypothetical population, the error bars in Fig. 3 should all become zero, i.e., σ_α(r) = 0 and α(r) = μ_α(r) = $\langle \alpha \rangle$ = −0.82. Therefore, from Eq. 11,

$$\text{ln}{\psi |}_{{\sigma }_{\alpha \left(r\right)=0}}=\left(2+\langle \alpha \rangle \right)\text{ln}\frac{r}{\langle r\rangle }.$$ (12)

Clearly, the distribution of $\text{ln}{\psi |}_{{\sigma }_{\alpha \left(r\right)=0}}$ is determined by the distribution of r. Our experimental data indicated that the size of MDA-MB-231 cells follows a lognormal distribution. The mean cell radius is $\langle r\rangle$ = 7.5 μm, and the SD of the lognormal distribution is σ_c ≈ 0.20. Therefore, for the entire population of 10⁵ cells in our sample, $\text{ln}{\psi |}_{{\sigma }_{\alpha \left(r\right)=0}}$ is expected to follow a lognormal distribution with SD $\left(2+\langle \alpha \rangle \right){\sigma }_c$ = (2 − 0.82) × 0.20 = 0.23. Fig. 6 compares this peak with a second peak with SD 0.35. The second peak represents the measured distribution in Fig. 2 B, which obscures combined effects of the cell-to-cell variability in r and n.

Figure 6.

Two normal peaks comparing heterogeneity in endocytic capacities considering zero or nonzero noise in the cell-surface transporter density n. The dashed line indicates what the distribution of particle uptake (lnψ) would look like were there no cell-to-cell variability n. This peak represents heterogeneity arising from r only. The solid line indicates the combined effects from cell-to-cell variability in r and n. This latter peak represents the measured distribution of our sample in Fig. 2B, which also has SD σ_ψ = 0.35.

Combined contributions from cell size and transporter density

We next investigated the joint contributions of n and r in determining the overall uptake distribution ψ in our sample (gray peak in Fig. 2 B).

To account for the combined effects from both r and n, we must consider cell-size-dependent variations in α(r). Taking log on both sides of Eq. 11,

$$\text{ln}\psi =-\left(2+\langle \alpha \rangle \right)\text{ln}\langle r\rangle +\left[2+\alpha \left(r\right)\right]\text{ln}r.$$ (13)

Therefore, lnψ is a function of two random variables α(r) and lnr, both of which approximately follow normal distributions. We evaluate the probability distribution of this function by applying simulation and using the marginal distributions of α(r) obtained from our Bayesian analysis. First, we sampled lnr ∼ N$\left({\mu }_c,{\sigma }_c^2\right)$ and mapped this sampled r to its bin (subpopulation) i ∈ {1, 2, …, 17}. We then sampled α(r) = α(r_i) ∼ N$\left({\mu }_{\alpha \left(r_i\right)},{\sigma }_{\alpha \left(r_i\right)}^2\right)$. We evaluated lnψ(r) by plugging the sampled r and α(r) into Eq. 13. A total of 10⁵ samples were generated to create the probability distribution for lnψ(r). In Fig. 7, the red curve represents σ_c = 0.20, which is the SD of our measured cell size distribution. This peak approximately recovers our observed distribution in Fig. 2 B. We then compare this distribution with two other cases: σ_c = 0.1 and 0. Again, σ_c = 0 implies that cells are homogeneous in size in the population and that the heterogeneity is only due to the noise in n. As seen in Fig. 7, there is no remarkable difference between this peak and the peak representing σ_c = 0.2. Overall, it indicates that the cell size heterogeneity did not play a major role, given the level of noise in n in our sample.

Figure 7.

Joint contribution of n and r in the heterogeneity of particle uptake (ψ). Each of the three distribution peaks corresponds to a distinct cell size heterogeneity, as indicated by the SD values in the figure legend. The distribution peaks are generated by sampling both r and α(r) and then plugging these sampled values into Eq. 11. To see this figure in color, go online.

Discussion

We have provided a detailed analysis of how cells in a population may differ in their ability to uptake nanoscale materials. Our analysis focused on two specific cell attributes that may allow individual cells in a population to display heterogeneity in their endocytic capacities. One attribute is cell size, and the other is the level of expression of endocytic components (transporters) in the plasma membrane. We dissected cell-to-cell variability of these two attributes and their relative contributions in determining the population-scale heterogeneity in particle uptake.

In our earlier work (19), we showed that the density of transporters in the plasma membrane of MDA-MB-231 cells may decay following the rule n ∼ r^α (Eq. 7), where α ≈ −1. The negative value of α implies that the endocytic transporters in the cell plasma membrane may gradually become sparse in a growing cell, thus reducing the influx of extracellular materials. As a result, the amount of extracellular materials taken by a cell may not grow in proportion to the volume or biomass synthesis of a growing cell. The biological significance might be that it serves as a self-regulatory mechanism for a cell’s growth and maintenance of size homeostasis by avoiding uncontrolled inflow of nutrients. In this study, we have presented a simple stochastic version of the above relationship in which α is a random variable. The mean value of α is a constant, but it takes different distributions based on the cell size. This stochastic relationship accurately captured the heterogeneity in particle uptake by cells. Using this stochastic relationship, we have explained how cell size and cell-surface transporter density independently and jointly contribute to the heterogeneity in particle uptake in a population.

The analysis we have presented is based on experiments with 100 nm spherical nanoparticles in MDA-MD-231 breast cancer cells. These quantitative data and results may be more specific to this particular combination of particle and cell type. Cellular endocytosis of nanoparticles of this size is mediated primarily by the clathrin-coated pits (24, 25, 26). Therefore, the transporter molecule in our study may be more pertinent to clathrin-coated pits, and the results may reflect the way coated pits per cell scale with cell size. For a particle or molecule having a significantly different size or surface property, other endocytic component(s) could be more relevant (24,25,27). For example, particles larger than 500 nm are primarily internalized by the caveolae-mediated pathway (24), whereas particles smaller than 20 nm are internalized by phagocytosis (25). An interesting study would be to employ variable particle sizes and evaluate α to investigate how particle size may alter the stochastic relationship in Eq. 3.

Further study is needed to investigate whether the proposed stochastic relationship in Eq. 7 is valid in other cell lines. In particular, we are interested in taking distinct combinations of particles and cells and studying what relationship may best describe particle uptake in these combinations. In the Results, we have provided the rationale behind the relationship in Eq. 7, which accurately captures our experimental data as well. Note that this equation takes several different forms simply based on whether α takes a positive, negative, or zero value. As explained in the Results, considering various rationally plausible relationships between a cell’s size and its demand for the extracellular materials, α indeed may be positive or negative. Here, we obtained its mean value $\langle \alpha \rangle$ ≈ −0.82. As we have explained before, this negative value implies mean particle flux across the plasma membrane is smaller in larger cells, whereas the overall uptake rate is higher in the larger cells (Fig. 5). Therefore, an anticorrelation between particle flux and uptake is seen in our system. However, as we have noted, a considerably different $\langle \alpha \rangle$ (e.g., α < 2 or α > 0) might lead to a positive correlation between the flux and uptake as well. Such a possibility cannot be ruled out for a different cell line or a different particle transporter depending on the particle size.

It should be noted that our study involved a sample of cells in which the cell size distribution is not expected to change or evolve with time. However, if an individual cell is monitored for a long time, its change in size should be accompanied by a change in the cell-surface transporter density, particle flux, and particle uptake rate. In other words, the mean value of α and its distribution may be time-evolving functions. It might be challenging to monitor a single cell for a long period and observe its time-evolution of size and particle uptake. Nevertheless, our analysis may be extended to model and study such time-evolving behaviors that are difficult to investigate by an experiment. We have already quantified cell-size-dependent α and its distributions, which can be applied to a growing cell based on its instantaneous size. A model can be created considering a discrete change in cell size r over time and corresponding α(r) sampled from appropriate distributions. Cell division can also be incorporated in such a model. If a cell divides, the existing transporters of the dividing cell could be redistributed between the newly created smaller cells. However, more complicated behaviors may also arise if the transcriptional output of a cell also changes upon division.

Recently, Lin et al. (28) have provided an insightful theoretical analysis of how cells maintain mRNA and protein homeostasis. In their model, the authors considered that cellular mRNA and protein content grow exponentially with time and in proportion to cell volume (r³). Nevertheless, homeostasis is maintained because of the limiting RNA polymerase and ribosomes in a cell. Based on the analysis we have provided here and in our earlier work (19), the expression of transporter proteins in a cell scale linearly with cell radius r. The simplistic model provided by Lin et al. (28) could apply to many proteins, but there could be exceptions as well. It would be interesting to experimentally validate whether different proteins in a growing cell follow different rules or is there a general rule for all or most of them. It might be possible that a cell employs active mechanisms to control or adapt the rate of expression or downregulation of specific proteins. Therefore, an aberration from the exponential growth over time is plausible depending on the cellular need.

We have used Bayesian analysis to quantify cellular heterogeneity in a population. The error bars in our data represent the cellular noise of particle intake in a population. It should be noted that in typical Bayesian parameter estimation, the error bars represent measurement errors (21,29), which are usually generated from the duplicates or triplicates of an experiment. Posterior predictions from a Bayesian parameter estimation are not intended to reproduce the error bars in a data set. By contrast, the calculation procedure we followed aimed at capturing the mean as well as the error bars (SDs) of α so that it accurately reflects the heterogeneity in cell size r and cell-surface transporter density n. This Bayesian framework we used may be applied to analyze other types of cellular heterogeneities. For example, a similar study may be employed to study how the cell-surface density of GLUT (a specific glucose transporter) scales with the cell size. Another example might be to study how the copy number of a specific signaling protein scales with cell size. Protein copy numbers usually follow lognormal distribution in cells (30,31). The level of expression of a protein should correlate with cell size because of the differential transcription rates (32,33). The mean transcription rate of a protein may change with cell size following a deterministic rule. Nevertheless, the noise in the transcription rate may lead to heterogeneity in protein expression in cells having identical size (34).

It is important to note that our analysis excluded diffusion effects. The stochastic relationship between ψ, r, and α(r) may be extended to account for diffusion following our earlier model (19). Nevertheless, in our experiments, cells were incubated in a regular cell culture medium, and particle concentration was sufficiently high that we do not expect diffusion to play any significant role. An early theoretical work by Berg and Purcell (35) analyzed the diffusion effect on particle interaction with cells. However, assumptions in this study were that the solutions were dilute and the cells (or cell-surface transporters and/or reactive patches) were perfect sinks. In reality, cells hardly behave as perfect sinks in the context of nanoparticle uptake from its environment. Many of the transporters in the cell plasma membrane have lifetimes ranging from seconds to minutes, and their interactions with a particle could be reversible as well. Moreover, a very small fraction of the cell membrane is occupied by these transporters, and hence, many of the particle-cell encounters are unproductive collisions. Not surprisingly, a recent study by Lindemann et al. (36) found that the diffusion effect on particle uptake in water is insignificant and that the uptake rate is primarily dictated by the particle recognition and endocytic trafficking in the cell membrane.

Conclusions

Our work shows how cell-to-cell variability in cell size (r) and cell-surface transporter density (n) determine the heterogeneity in the endocytic capacity of a cell population. Our analyses indicate that these two variables are not independent. Rather, the mean value of n, as well as its noise, has an inverse relationship with r. We show that n decays with r following a power rule. We use a Bayesian framework to characterize the parameter associated with this relationship and establish a model that accurately captures the endocytic heterogeneity of MDA-MB 231 breast cancer cells.

Author Contributions

Both M.S. and D.B. conceptualized the work. M.S. carried out the experiments and data analysis. D.B. wrote the program for the Bayesian framework. Both authors contributed to manuscript preparation and revision.

Editor: Jochen Mueller.