Figure 1.
A: Scheme of the proposed cell cycle model with three phases
and
The dashed border between the
and the
phase indicates that the
and
phase are pooled into a single phase. The random time
a cell needs to complete the processes associated to each of the phases, follows a delayed exponential distribution with specific parameters
and
for each phase. B: Delayed-exponential completion time distribution density
with parameters
and
C: Best fit of the complementary cumulative distribution
to the fraction of undivided cells after birth obtained by time lapse cinematography [5] of slow and fast dividing cell lines. D: Best fit of
defined by Eq. 4 (solid line) to inter-mitotic time distribution density measured by long-term video tracking of in vitro proliferating B-cells [10]. The data in C and D were read from the graphs in the original publications ([5] and [10] respectively).
Figure 2.
DAPI-BrdU pulse-chase labelling FACS data.
Samples taken at several time points after pulse labelling proliferating U87 human glioblastoma cells with The four gated populations are
and
which are defined precisely in the main text. Briefly, the subscript indicates the phase at the instant of labelling, while the superscripts ‘u’, ‘lu’ and ‘ld’ refers to cells ‘unlabelled’, ‘labelled and undivided’ and ‘labelled and divided’, respectively. The data was generated as described in the Experimental Methods section.
Figure 3.
Model based parameter estimation.
A: Best fit of the model predictions (lines) to experimentally determined cell fractions after BrdU pulse labelling (dots). U87: In vitro cultured U87 human glioblastoma cancer cell line (three replicates). V79: In vitro cultured V79 Chinese hamster cells (single replicate) (courtesy G. Wilson). Best fit parameter values used to compute model predictions (U87: V79:
units are hours). B: Approximate ML regions for the parameters
and
associated to each phase (gray:
red:
green:
). C: Bayesian bi-variate 99%-credibility regions for the parameters
and
for each phase. Arrows indicate point estimates and the dashed lines delineate the information that could have been gained in our thought experiment under noise-free conditions from two support points, one at
and a second at
. The U87 data set was generated as described in the Experimental Methods section. The V79 data set was a kind gift of G. Wilson.
Table 1.
Bayesian summary statistics.
Figure 4.
A: Simplified schematic representations of the protocols corresponding to a conventional single pulse labelling with one nucleoside analog (e.g., BrdU) and a dual pulse labelling experiment with two different nucleoside analogs (e.g., BrdU together IdU or EdU). B: Artificial staining of single-pulse labelling data (for original data see Fig. 2), showing eight of the nine subpopulations that could potentially be identified with double-pulse labelling. Notice that the four population
and
that can be followed by the conventional protocol, have each been subdivided according to the cell cycle phases. The naming convention for the populations is as follows: the superscript (
= ‘labelled undivided’,
= ‘labelled divided’,
= ‘unlabelled’) indicates whether the population is labelled and whether it has divided since the time of the first pulse; the first and the second subscript (
) stand for the phase in which the population was at the time of the first and the second pulse respectively. Double subscripts are used only when necessary.
Figure 5.
Analysis of simulated dual pulse labelling data.
A: Average kinetics of unlabelled (dashed line) and labelled cell cohorts (colored lines) were computed from Eq. 25, using ML parameter estimates from the U87 and the V79 data sets (U87: V79:
units are hours). Support points and repeats were chosen according to the real experiments. Multinomial noise was added, mimicking the residuals found in the original data sets (see the Computational Methods section for more details). Finally, model solutions (lines) were fitted to the synthetic data sets (triangles). Best fit parameters (U87:
V79:
units are hours) B: ML parameter estimates from simulated data. All ML regions converge to point estimates (arrows). Squares indicate parameters used for generating the data (see A). C: Bayesian bi-variate 99%-credibility regions for the parameters
and
for each phase, based on the artificial data.
Figure 6.
Robustness of parameter estimates to empirical phase duration distributions that are not delayed exponential functions.
A-B: Least-squares fitting of histograms predicted from a hypoexponential distribution with two decay and one delay parameter to measurements of phase durations using fluorescent biosensors [35]. The number of cells that were tracked in the original study was around 15 cells. C: Best fit of the cell cycle model with delayed exponential completion time distribution densities to synthetic data generated from a model with hypoexponential completion time distribution densities for the
and
phase with parameters as in A and B. D: Recovery of the initial distribution densities (solid lines) using the delayed exponential model (dashed line). Both the average and the variability in the
phase completion time distribution (original average: 10.70 h, estimated average: 10.88 h; original std: 2.03 h, estimated std: 1.99 h) were estimated accurately. The data shown in A-B was read from the graphs in the original publication ([35]).
Figure 7.
Effect of cell death and completion time distribution on parameter estimates.
A: Comparison of analytical predictions (lines, Eq. 29) with simulated BrdU labelling experiment (squares). Cell death is assumed to occur exclusively during S phase with probability 0 (red) and 0.3 (blue) respectively. Only the population is considered. Parameters:
units are hours. B: Difference between Eq. 29 (accounting for cell death) and Eq. 24 (neglecting cell death) at time
h (see dashed line in A), as a function of
C: BrdU labelling experiments were simulated assuming gamma distributed phase completion times (red curve, graphs on left column) and cell death during S phase with probability
and
(green curve, graphs on left column). The effective completion time
(gray density plot, left column), the population growth (middle column) and the estimation of the mean and the standard deviation of
are shown for both cases. Approximate confidence intervals for the estimates are computed as 1.96 times the standard error. Even though
and the population growth are strongly influenced by the value of
both
and the estimates extracted from
are barely affected. The dashed lines in the middle column indicate the time of the first pulse, which was chosen such that the average population was similar in both scenarios. Parameters for gamma distributed completion time distribution of the three phases: shape:
scale:
delay:
Figure 8.
as a function of
for fixed values of
For
(green circle) the real part of Q takes, depending on
a value in the interval
The values for x are increasing from A-D, while
and
remain unchanged. For relatively low values of
(A-B) the real part
is positive for
After one or several turns, i.e by increasing
the spiral can potentially cross the origin only once (empty circle). In A the spiral misses the origin, while in B the spiral crosses the origin after one turn. Crossing of the origin means that the corresponding complex number
is a root of Q. In C the spiral starts at the origin. This represents the only real positive root of Q. For initially negative values of
(D) the spiral can never cross the origin because the distance to the center point (gray circle) is already in the beginning for
larger than the distance between the latter and the origin. By increasing y this distance will even grow further according to Eq. 33.