Quantitative Analysis of the Correlation between Cell Size and Cellular Uptake of Particles

Component: Cell Model; Description: MDA-MB 231 human breast cancer cells (adherent); cultured in RPMI 1640 with 10% FBS and 1% penicillin/streptomycin

Component: Nanoparticles; Description: Green fluorescent polystyrene nanoparticles (Thermo Scientific FluoroMax): mean diameters of 26, 47, 100, and 200 nm; used without modification

Component: Nanoparticle Concentration; Description: Not specified in text (diluted from stock in RPMI 1640)

Component: Uptake Duration; Description: 5 hours

Component: Cell Size Measurement; Description: Forward scatter (FSC-A) from flow cytometry; calibrated with 2 μm and 3 μm fluorescent beads; confirmed by fluorescence microscopy (Zeiss Apotome 2, 63× objective; ImageJ analysis)

Component: Uptake Quantification; Description: Fluorescence intensity of cell-internalized nanoparticles (533/30 nm bandpass; normalized to 3 μm bead fluorescence)

Component: Model Type; Description: Reaction-diffusion model with spherical cell geometry; incorporates nanoparticle diffusion, cell-surface transporter-mediated uptake, and cellular heterogeneity

Component: Key Model Parameters; Description: • Transporter density (n): ~0.946 μm⁻² (mean)<br>• α (size-transporter correlation): −0.79 to −0.90<br>• kf (forward rate constant): 0.079–0.495 μm³/s (size-dependent)<br>• D (diffusivity): 2.18–16.8 μm²/s (size-dependent)

Component: Number of Events; Description: ≥20,000 per sample; >80% of events were viable cells

Parameter: Cell Culture; Details: MDA-MB 231 cells seeded in 24-well plates (100,000 cells/mL); 24 h attachment; 5 h nanoparticle incubation at 37°C, 5% CO₂

Parameter: Flow Cytometry; Details: BD Accuri C6 plus; 488 nm argon-ion laser; FSC-A and fluorescence (533/30 nm); ≥20,000 events/sample

Parameter: Cell Size Estimation; Details: FSC-A peaks: 2 μm beads, 3 μm beads, MDA-MB 231 cells; linear extrapolation; microscopy image analysis (1,500 cells) for validation

Parameter: Model Implementation; Details: Reaction-diffusion model solved in Python; lognormal distributions for cell size and transporter density; α parameter for size-transporter correlation

Parameter: Model Fitting; Details: Simultaneous fitting to scatter plots (cell size vs. uptake) and histograms (uptake distribution); four particle sizes

Parameter: Controls; Details: Cell autofluorescence (green channel, untransfected cells); bead fluorescence normalization

Parameter: Replicates; Details: Not explicitly stated; single-cell data from ≥20,000 events per sample

Analysis Category: Cell Size Measurement; Methods: • Flow cytometry FSC-A (calibrated with 2 and 3 μm beads)<br>• Fluorescence microscopy with ImageJ analysis (1500 cells)<br>• Lognormal distribution fitting: mean size ~8 μm, σc = 0.20

Analysis Category: Uptake Quantification; Methods: Flow cytometry: fluorescence intensity of internalized nanoparticles normalized to 3 μm bead fluorescence (I/Ib)

Analysis Category: Model Development; Methods: Reaction-diffusion model with spherical cell; Michaelis-Menten kinetics at cell surface; diffusion in extracellular space

Analysis Category: Heterogeneity Modeling; Methods: Lognormal distributions for cell size [ln(r₀) ~ N(μc, σc²)] and transporter density [ln(n) ~ N(μt, σt²)]

Analysis Category: Model Fitting; Methods: Python implementation; fitting to scatter plots and histograms; estimation of n, σt², α, kf

Analysis Category: Parameter Estimation; Methods: α (size-transporter correlation): −0.79 to −0.90 (from fitting)<br> Transporter density (n): 0.946 μm⁻²<br> σt²: 0.071–0.116

Analysis Category: Noise Analysis; Methods: Cellular noise in uptake; diffusion effect on noise suppression

Parameter: Mean cell radius (FSC, uncorrected); Value: ~11 μm

Parameter: Mean cell radius (microscopy, corrected); Value: ~8 μm

Parameter: Distribution (σc); Value: 0.20 (lognormal)

Parameter: FSC vs. microscopy; Value: FSC overestimates size by ~3 μm; distribution shape preserved

Model: Uptake ∝ surface area; Relationship: r₀²; Fit to Data: Poor

Model: Uptake ∝ volume (mass); Relationship: r₀³; Fit to Data: Poor

Model: Linear model; Relationship: r₀; Fit to Data: Excellent (R² high)

Model: Uptake vs. cell radius; Relationship: Linear; Fit to Data: Observed experimentally

Model: Noise (σi); Relationship: 0.263; Fit to Data: Captured in linear model

α (size-transporter correlation): α = 0 (nominal); Reaction-Limited Uptake Dependence: r₀² (nonlinear); Diffusion-Limited Uptake Dependence: r₀ (linear)

α (size-transporter correlation): α = −0.8 to −0.9; Reaction-Limited Uptake Dependence: r₀ (linear); Diffusion-Limited Uptake Dependence: r₀ (linear)

α (size-transporter correlation): α = −2; Reaction-Limited Uptake Dependence: r₀⁰ (size-independent); Diffusion-Limited Uptake Dependence: -

α (size-transporter correlation): α = −5; Reaction-Limited Uptake Dependence: Optimal size for uptake; Diffusion-Limited Uptake Dependence: -

α (size-transporter correlation): Conclusion; Reaction-Limited Uptake Dependence: Negative α explains linear correlation in reaction-limited regime

Particle Size (nm): 26; kf (μm³/s): 0.495; D (μm²/s): 16.80; α: −0.878; σt²: 0.099

Particle Size (nm): 47; kf (μm³/s): 0.079; D (μm²/s): 9.29; α: −0.792; σt²: 0.071

Particle Size (nm): 100; kf (μm³/s): 0.160; D (μm²/s): 4.29; α: −0.896; σt²: 0.077

Particle Size (nm): 200; kf (μm³/s): 0.269; D (μm²/s): 2.18; α: −0.811; σt²: 0.116

Particle Size (nm): Fixed parameters; kf (μm³/s): n = 0.946 μm⁻²; D (μm²/s): kr = 0.1 s⁻¹; α: k₁ = 0.02 s⁻¹; σt²: -

Particle Size (nm): Estimated transporter molecules per cell; kf (μm³/s): ~760 (for 8 μm radius cell); D (μm²/s): -; α: -; σt²: -

Parameter: Transporter density (n); Value: 0.946 μm⁻²; Interpretation: One transporter handles one particle at a time

Parameter: Transporters per cell (8 μm radius); Value: ~760; Interpretation: Equivalent to clathrin-coated pits: 4-15 transporter molecules/pit (if 50-150 pits/cell)

Condition: Cell culture medium; Diffusivity: 1 cP viscosity; Uptake Relationship: Reaction-limited; Noise Level: High noise

Condition: 10× reduced D; Diffusivity: 0.1×; Uptake Relationship: Reaction-limited; Noise Level: High noise

Condition: 30× reduced D; Diffusivity: 0.03×; Uptake Relationship: Transition; Noise Level: Moderate noise

Condition: 100× reduced D; Diffusivity: 0.01×; Uptake Relationship: Diffusion-limited; Noise Level: Low noise

Condition: Reaction-limited; Diffusivity: -; Uptake Relationship: r₀ (with α ≈ −0.8); Noise Level: Significant noise

Condition: Diffusion-limited; Diffusivity: -; Uptake Relationship: r₀; Noise Level: Deterministic

1. Model simplification: The model ignores individual factors associated with distinct endocytic pathways (clathrin-mediated, caveolae-mediated, macropinocytosis). The authors state: "Incorporation of the detailed molecular mechanisms and individual factors associated with various endocytic mechanisms is beyond the scope of this study."

2. In vitro only: All experiments were performed in cell culture medium with water-like viscosity; transport effects in vivo (tumor interstitial matrix, collagen networks, biological barriers) were not experimentally tested.

3. Single cell line: Only MDA-MB 231 breast cancer cells were studied; generalizability to other cell types was not demonstrated.

4. FSC-based cell sizing limitations: The authors acknowledge that "FSC-A may provide an unreliable estimate of cell sizes" due to refractive index, intracellular structures, and device design; they corrected for this using microscopy.

5. Convective transport ignored: The model considers pure diffusion, typical for cell culture, but in vivo tissues may have advective transport; "more complex model and experimental investigation are necessary to determine how reaction, diffusion, and advection together may determine cell-size-dependent nanoparticle uptake in in vivo tissue conditions."

6. Outliers not fully explained: The data showed "considerably larger uptake in a small fraction of cells that fall outside the range of the theoretical values" and "the model does not incorporate a mechanism to account for these outliers."

7. Transporter as hypothetical unit: The transporter molecule is a "hypothetical unit that processes only one particle at a time" and does not correspond to a specific molecular entity; its interpretation as clathrin-coated pits is speculative.

8. No direct measurement of transporter density: Transporter density (n) was estimated from model fitting, not directly measured by imaging or biochemical assays.

9. Particle size range: Only four particle sizes (26-200 nm) were tested; the model's applicability to larger particles (>200 nm) or smaller molecules was not tested.

10. No exploration of surface chemistry: All nanoparticles were polystyrene with identical surface chemistry; the effect of surface charge, functionalization, or protein corona was not investigated.

11. Uptake mechanism not verified: The study assumes transporter-mediated uptake but does not experimentally confirm the involvement of specific endocytic pathways for each particle size.

12. No investigation of uptake kinetics: Only a single time point (5 h) was used; the model assumes steady-state conditions and does not account for time-dependent uptake dynamics.

Report prepared based on the published Biophysical Journal article. For full experimental details, supplementary materials, and complete references, please refer to the original publication.