Theoretical model
We analyze the proportions of two different GBM cell phenotypes, dormant (D, please refer to Table
1 for symbols used in the equations) and rapidly proliferating (P) cells, in a mathematical model including the influence of different treatment conditions. In the following, we characterize the cells in terms of their fitness, which we define as the growth rate in comparison to cells of the other phenotype. Dormant cells always have a very low or even zero growth rate
ε, which we assume to be independent of the exact composition of the population and the treatment condition. Rapidly proliferating P cells, on the other hand, have a very large fitness advantage compared to dormant cells, which means they proliferate much faster, but they also compete with each other for space and resources. Facing another P cell, a focal P cell has an intermediate fitness, which we assume to be still much larger than the growth rate of D cells
ε. Their fitness therefore depends on the relative fraction of D vs. P cells. Due to the very slow growth of D cells, P cells will represent the vast majority of glioblastoma cells in the absence of treatment.
Table 1
Overview of all used symbols in the model
n
X
| number of cells of type X |
x
X
| ratio of cells of type X in population |
D, P | index for dormant or rapidly proliferating cell type, respectively |
ε
| Fitness of dormant (D) cells |
λ
| treatment cost on normally growing cells |
σ
| probability for spontaneous conversion between types |
\( \overline{f} \)
| total average fitness of all cell types in the population |
Under treatment conditions, however, the population composition changes. Even though D cells still have the same very low (or zero) growth rate
ε, P cells experience a fitness cost
λ due to treatment. The reduction of the fitness due to treatment only applies to P cells, because cytotoxic drugs mostly affect rapidly dividing cells. The fitness cost parameter
λ can be adjusted to account for the strength of the applied treatment. In principle we can continuously vary this parameter. However, for simplicity we focus on two different treatment strategies: In high dosage (HD) chemotherapy the treatment strength parameter
λ is large compared to the growth rate of the P type. Since high dosage chemotherapy has strong side effects for the whole organism (for GBM: [
3]), in reality this treatment strategy cannot be maintained for extended time periods. Therefore, strong treatment needs to be applied in turns with weaker or no treatment. For low dosage (LD) chemotherapy,
λ means only a small reduction of the growth rate of the P cells. As the side-effect stress to the organism should also be lower, this treatment regime could be applied for longer time spans.
Dormant (D) and rapidly proliferating (P) phenotypes in glioblastoma and their aforementioned interactions can be described by the following payoff matrix [
18]:
$$ {\displaystyle \begin{array}{c}\\ {}\mathrm{D}\\ {}\mathrm{P}\end{array}}{\displaystyle \begin{array}{c}\mathrm{D}\kern1.50em \mathrm{P}\\ {}\left(\begin{array}{cc}\varepsilon & \varepsilon \\ {}1-\varepsilon -\lambda & \frac{1}{2}-\lambda \end{array}\right)\end{array}} $$
This matrix gives the fitness for each type if confronted with any of the two other types. Here, we find for example that the fitness of a focal P cell interacting with a D cell is 1 − ε − λ, which includes both the small or zero growth rate of D cells ε and the fitness cost for P cells under treatment λ.
As the phenomenon of dormancy is presumably a reversible process that also occurs without any treatment, we assume that conversion between both phenotypes is possible with a small rate σ. Thus, P cells may enter a dormant phenotype, and D cells may exit from their quiescent state, converting into a P phenotype at any time point.
In the following, we include these fitness effects and phenotypic conversion into a set of ordinary differential equations. In general, the growth of a whole cell population can be explained in terms of a differential equation that describes the change in the number of individuals over time
$$ \frac{dn}{dt}=r\left(n,t\right)n. $$
Here n is the number of individuals, t is the time and r(n, t) is the growth rate, which can itself depend on the number of cells and the time.
At first, we focus on the number of D cells,
n
D
, in the population over time, which have a very small but constant growth rate
ε$$ \frac{d{n}_D}{dt}=\varepsilon\ {n}_D. $$
For P cells on the other hand, the growth rate of
n
P,
given by the average fitness from the payoff matrix (weighted to the cell fractions), changes with the composition of the population
$$ \frac{d{n}_{\mathrm{P}}}{dt}={n}_{\mathrm{P}}\left(\left(1-\varepsilon -\lambda \right)\frac{n_D}{n_D+{n}_{\mathrm{P}}}+\left(\frac{1}{2}-\lambda \right)\frac{n_{\mathrm{P}}}{n_D+{n}_{\mathrm{P}}}\right). $$
Since the system under consideration is constrained, both in terms of nutrients and space, in reality the cell population only grows exponentially as indicated by the growth equations in the very beginning of the process where the constraints regarding space or nutrients are negligible. However, we are mainly interested in the fraction of D cells
\( {x}_D=\frac{n_D}{n_D+{n}_{\mathrm{P}}} \) in the population and vice versa the fraction of P cells
\( {x}_{\mathrm{P}}=1-{x}_D=\frac{n_{\mathrm{P}}}{n_D+{n}_{\mathrm{P}}} \). To obtain the change in fractions for both types, we subtract the average growth rate
\( \overline{f} \) of the population from both individual growth rates,
$$ \overline{f}\kern0.5em =\varepsilon {x}_D+\left[\left(1-\varepsilon -\lambda \right){x}_D+\left(\frac{1}{2}-\lambda \right){x}_{\mathrm{P}}\right]{x}_{\mathrm{P}} $$
From this we obtain two differential equations for the fractions of D and P cells,
$$ {\displaystyle \begin{array}{cl}{\dot{x}}_D& ={x}_D\left(\varepsilon -\overline{f}\right)\\ {}{\dot{x}}_{\mathrm{P}}& ={x}_{\mathrm{P}}\left(\left[\left(1-\varepsilon -\lambda \right){x}_D+\left(\frac{1}{2}-\lambda \right){x}_{\mathrm{P}}\right]-\overline{f}\right)\end{array}} $$
Next, we include the spontaneous conversion between phenotypes with a constant rate
σ, which is independent of the cellular growth. This leads to an additional term to the differential equation of both phenotypes
$$ {\displaystyle \begin{array}{cl}{\dot{x}}_D& =\left[\varepsilon -\overline{f}\right]{x}_D+\sigma \left({x}_{\mathrm{P}}-{x}_D\right)\\ {}{\dot{x}}_{\mathrm{P}}& =\left[\left(1-\varepsilon -\lambda \right){x}_D+\left(\frac{1}{2}-\lambda \right){x}_{\mathrm{P}}-\overline{f}\right]{x}_{\mathrm{P}}+\sigma \left({x}_D-{x}_P\right)\end{array}}. $$
(1)
These equations have the important difference to the usual replicator-mutator equation [
15] that phenotype conversion is a spontaneous process with a constant rate and is independent of the growth in the population. This allows conversion from D to P even if D cells do not grow at all.
Using these equations, we model different therapy schedules combining different treatment strengths in different cycling time plans. Since the equations are nonlinear, we use numerical integration with
Odeint of the Python library Scipy
1 to examine the temporal dynamics of the system under different treatment regimes. Additionally we analytically determine the fixed points of the system and their stability.
Experimental model
Cell culture and cell number determination
The GBM cell line LN229 was purchased from ATCC/LGC Standards (Middlesex, UK, ATCC-CRL 2611) and cultured in Dulbecco’s modified eagle medium (DMEM) plus 10% fetal calf serum (FCS, PAN Biotech, Aidenbach, Germany).
Mycoplasma contaminations were routinely excluded by bisbenzimide staining. The GBM cell line identity was proven routinely by STR (Short Tandem Repeat) profiling at the Department of Forensic Medicine (Kiel, Germany) using the Powerplex HS Genotyping Kit (Promega, Madison, WC). Briefly, DNA was amplified with a STR multiplex PCR, electrophoretic separation was performed with the 3500 Genetic Analyser (Thermo Fisher Scientific, Waltham, MA, USA), and evaluated using the Software GeneMapper ID-X (Thermo Fisher Scientific). For determination of cell numbers after low and high dose chemotherapy treatment, 25,000 cells/well were seeded in 6 well plates (Greiner Bio-one, Frickenhausen, Germany). Cells were grown for 24 h, then washed with phosphate buffered saline (PBS), supplemented with fresh DMEM + 10% FCS and temozolomide concentrations (Sigma-Aldrich, St. Louis, MO, USA; dissolved in dimethyl sulfoxide DMSO) as indicated in Fig.
2a (5, 50 or 100 μg/ml for 10 days). Temozolomide (TMZ) is a DNA alkylating drug causing apoptotic cell death and the most commonly used chemotherapeutic in GBM therapy. Control cells were supplemented with 0.5% DMSO, which corresponds to the solvent concentrations of each TMZ stimulated sample. Cells were stimulated for 10 days with TMZ, while media were changed every 2–3 days. After 10 days, cells were detached by trypsination and total cell numbers per well counted using trypan blue exclusion and a Neubauer chamber (Brand, Wertheim, Germany). DMSO stimulated control cells were already detached after 6 days of stimulation, split 1:10 and seeded again to exclude limitations of growth due to space and nutrient limitations. This splitting factor (1:10) was considered when relative cell numbers of TMZ treated samples in comparison to DMSO controls were determined for
n = 5–6 independent experiments.
Immunocytochemistry
For immunocytochemistry, 50,000 cells were seeded onto poly-D-lysine coated glass cover slips, grown for 24 h and supplemented with indicated TMZ or DMSO concentrations as described above. From day 6, growth media were additionally supplemented with 10 μM 5-bromo-2′-deoxyuridine (BrdU, Sigma-Aldrich, St. Louis, MO) to allow for incorporation in the DNA in the S phase of the cell cycle. After 10 days, cover slips were fixed with an ice-cold mixture of methanol and acetone (1:1) for 10 min, rinsed with 0.1% Tween / PBS (3 × 5 min), incubated with 1 M HCl for 30 min, neutralized with 0.1 M sodium borate buffer (pH 8.5), and rinsed again with 0.1% Tween/PBS. Afterwards, cells were blocked for unspecific bindings with 0.5% bovine serum albumin (BSA) / 0.5% glycine in PBS (1 h) and incubated over night with the primary antibody against H2BK (1:300, Biorbyt, Cambridge, UK), a marker of glioma dormancy [
24,
25] and the primary antibody against BrdU (1:200, Abcam, Cambridge, UK). Then cover slips were incubated with the secondary antibodies (donkey anti-rabbit IgG, labelled with Alexa Fluor 488, and donkey anti-sheep labelled with Alexa Fluor 555, both Invitrogen, Carlsbad, CA, USA) for 1 h at 37°, and 4′, 6-diamidino-2-phenylindole (DAPI; Sigma Aldrich, St. Louis, MO, USA; 1 mg/ml, 1:30,000, 30 min at room temperature) to stain nuclei. Cover slips were embedded using Immu-Mount (Thermo Fisher Scientific, Rockford, IL, USA), and analysed with equal exposure times using an Axiovert microscope and digital camera (Zeiss, Jena, Germany). H2BK-immunopositive, BrdU-positive and double positive cells were counted and normalized to total cell numbers in 6 (DMSO controls) to 10 (TMZ samples) fields of view for
n = 4 independent experiments.
DiO retention and cell countings on phenotype conversion
To monitor the conversion to and from dormancy we used the green fluorescent vital dye DiO (Invitrogen), as rapidly proliferating cells lose the dye due to repeated divisions, while resting, dormant (or very slowly cycling) cells retain the dye and can be detected by fluorescence microscopy. Investigating the conversion to dormancy, 150,000 LN229 cells were seeded into 6-well-plates, stained with Vybrant® DiO Cell-Labeling Solution (Thermo Fisher Scientific, Waltham, MA, USA) following the manufacturer’s instructions and stimulated with 100 μg/ml TMZ (or equal volume of the solvent DMSO) for 10–12 days. Cells were photographed combining transmitted-light microscopy and fluorescence microscopy with equal exposure times for TMZ and control treated cells, and green fluorescent cell portions were determined in comparison to total cell counts. To determine the influence of different cell densities on the incidence of conversion, 50,000 and 150,000 cells were seeded, respectively, into 6-well-plates and treated with 100 μg/ml TMZ (or equal volumes of the solvent DMSO) for 10 days. As the DMSO control treated cells rapidly proliferate, cells were detached at day 6 (50,000) or day 3 and 6 (150,000), cell numbers counted using a Neubauer chamber to determine the growth rate over this time period, and seeded again at initial density, to allow for cell growth without limitation of space and nutrients. After 10 days, TMZ and control treated cells were detached and counted. To extrapolate the total cell numbers of control cells, growth rates determined at day 3, 6 and 10 were used, and TMZ surviving cells were calculated as percentage of extrapolated total cells.
Statistical analysis
Statistical analysis and graphical presentation of experimental data were performed with Graph Pad Prism using a two-tailed t-test (*** p < 0.001).