Supplementary Discussion
“Role of cell cycle on the cellular uptake and dilution of
nanoparticles in a cell population”
Jong Ah Kim, Christoffer Åberg, Anna Salvati, Kenneth A. Dawson
Numerical Simulations of Nanoparticle Uptake by A Cell
Population Undergoing Exponential Growth
At all times, cells in the G2/M phase have had the longest time to take up nanoparticles,
followed by cells in the S phase; in contrast, cells that have just divided are in the G1 phase.
This fact is certainly consistent with the experimentally observed ranking (Fig. 1c), but in
order to assess if this effect is quantitatively large enough to explain the experimental
observation, simulations were performed with an ensemble of cells under somewhat idealised
circumstances.
In the simulations, all cells were assumed to cycle at the same rate. Different cycle
lengths can be included, but would likely not lead to quantitatively different results. The
positions of individual cells along the cell cycle, their ‘ages’, were randomly chosen
according to an exponentially decaying distribution, as illustrated in the figure below. This
distribution is well-known in cell proliferation analysis1,2,3, and leads to an exponentially
growing population of cells.
Exponentially decaying distribution along cell cycle with the individual phases indicated. Based on the
distribution, the traversal time for a given phase can be related to the fraction of cells in that phase, as
illustrated graphically. The shaded bins show a population of 106 cells randomly chosen according to the
distribution and used as the initial population for the simulations.
In the simulations, cells traversed the cell cycle with time. Since the exponentially
decaying distribution (normalised for the total number of cells) is time-independent, the
fraction of cells in each phase remains constant throughout the simulations. The time spent in
each phase can then be related to the fraction of cells in a given phase2,3, as illustrated in the
figure above. The fact that the fraction of cells in each phase remained constant in the
simulations implicitly assumes that the cell population is in a state of exponential growth and
that effects due to confluence have not been taken into account. We can therefore only
compare the simulations with experiments for a limited time course, during which these
assumptions remain approximately true.
The simulations can be validated to data by using the nucleoside analogue EdU (5ethynyl-2’-deoxyuridine) together with DNA staining (7-AAD) to follow experimentally the
division of those cells that were in S phase at the time of labelling. Since the labelled (S
phase) cells have to traverse G2/M prior to division, a comparison with the simulations can
be performed by counting the number of cells that have divided in the simulations, after a
delay corresponding to the time spent in G2/M. The prediction, using only parameters
independently measured (fraction of cells in the different phases) or acquired (cell population
doubling time), is compared with the experimental data in Fig. 2b to excellent agreement.
As each simulated cell traversed the cell cycle, it was also assumed to take up
nanoparticles with a given rate. In the simplest case (Fig. 1e-f), the rate was assumed
independent of the cell’s instantaneous position along the cell cycle. However, each cell was
allowed to have its own rate, simulating a heterogeneity in the cell population when it comes
to nanoparticle uptake. The rate with which an individual cell takes up nanoparticles was,
furthermore, assumed to be inherited by both daughter cells upon cell division. A log-normal
distribution of rates was used since it reproduces the appearance of the experimentally
observed distributions measured by flow cytometry, though we imply no deeper significance
than that.
To investigate the effect of different rates of uptake during the different phases
(Supplementary Fig. S17), simulations were performed where the uptake during a given
phase was assumed constant, but different among the phases. Again each cell took up
nanoparticles with its own rate; this rate being inherited by both daughter cells upon cell
division. A 'fictitious rate', , was assigned to each cell from the same log-normal
distribution as for the simpler case of constant rate during the cell cycle (previous paragraph).
The rates of uptake during the different phases for a given cell,
,
and
,
respectively, were scaled from the 'fictitious rate' according to
where
denotes the measured rate and
the fraction of cells in phase , respectively.
The respective numerators in the expressions above imply that the ratios of the rates of
uptake during the different phases of any given cell exactly match the ratios of the measured
rates. In other words, a cell that takes up nanoparticles faster than average during the G0/G1
phase, also takes up nanoparticles faster than average during the S and G2/M phases.
Furthermore, the rate of uptake for any given cell during G2/M phase is somewhat larger than
during the other phases. The denominator in the above expressions was used for simplicity of
comparison, since this normalisation ensures that the mean of the full population exactly
matches that for the simpler case of constant rate of uptake during the different phases
(previous paragraph).
No export of nanoparticles was taken into account, and therefore the only diluting effect
on the average number of internalised nanoparticles is cell division. Upon cell division, each
simulated cell was in the simplest case (Fig. 1e-f and Supplementary Fig. S17) assumed to
give exactly half of its intracellular load of nanoparticles to each daughter cell. To investigate
the effect of asymmetrical partitioning in qualitative terms (Supplementary Fig. S9),
simulations were also performed where one of the daughter cells received 70% and the other
30% of the intracellular load of the parent cell.
In the numerical evaluation, we used a cell cycle length of 22 hours (the cell population
doubling time given by the supplier), and fractions of cells in the different phases of 52%
G0/G1, 40% S and 8% G2/M, respectively, as determined experimentally from the EdU and
7-ADD double staining (Supplementary Fig. S5). The initial population consisted of 106 cells
(growing with time), but results were re-normalised to correspond to the 15,000 cells
measured experimentally. The means corresponding to the different phases were calculated
by averaging over the cells in a given phase, as defined by their position along the cell cycle
(illustrated in the figure above).
Results for the simplest case of equal rates of uptake during all phases and symmetrical
split upon cell division are reported in Fig. 1: Figure 1f shows the average accumulations
corresponding to the experiment shown in Fig. 1c, and Fig. 1e shows distributions of ‘cell
fluorescence intensity’ corresponding to the experiment shown in Fig. 1d.
Supplementary Fig. S9 shows the effect of asymmetrical partitioning, keeping the rates
equal during the different phases, on the ‘cell fluorescence intensity’ distributions. Note that
the average nanoparticle accumulation by cells instantaneously in a given phase is not
affected by an asymmetrical partitioning, since the average partitioning is still half (for
example, if one daughter receives 70% of the intracellular load and the other 30%, the
average is still 50%). The averages corresponding to Fig. S9 are therefore the same as shown
in Fig. 1f. In particular, this implies that the ranking of the phases is completely independent
of eventual asymmetry of the partitioning.
Supplementary Fig. S17 compares the average uptake with different rates of uptake in
the different phases. Notably, the somewhat larger uptake rate measured during the G2/M
phase (Fig. 3) does not affect the nanoparticle accumulation significantly, simply due to the
fact that the time spent during G2/M is comparatively short. The observed ranking in
nanoparticle accumulation for cells in the different phases is therefore most likely not due to
the measured (minor) differences in uptake rates.
In the absence of a nanoparticle source, the simulation predicts an exponential decay of
the average number of nanoparticles per cell with a decay constant given by
where
is the cell population doubling time. Figure 2c shows the decay of the average fluorescence
after loading cells with nanoparticles for some time, and subsequent removal of the
nanoparticle source. A one-parameter fit (with initial fluorescence as a free parameter, and
the decay constant fixed) shows that the simulation gives a good description of the
experimental data (solid line in Fig. 2c).
References
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Cytochem. 22, 642-650 (1974).
2. Steel, G. G. Growth Kinetics of Tumors: Cell Population Kinetics in Relation to the Growth and Treatment
of Cancer (Oxford Univ. Press, Oxford, 1977).
3. Montalenti, F., Sena, G., Cappella, P. & Ubezio, P. Simulating cancer-cell kinetics after drug treatment:
Application to cisplatin on ovarian carcinoma. Phys. Rev. E 57, 5877-5887 (1998).