Mathematical Determination of Cell Population Doubling Times for Multiple Cell Lines

Abstract

Cell cycle times are vital parameters in cancer research, and short cell cycle times are often related to poor survival of cancer patients. A method for experimental estimation of cell cycle times, or doubling times of cultured cancer cell populations, based on addition of paclitaxel (an inhibitor of cell division) has been proposed in literature. We use a mathematical model to investigate relationships between essential parameters of the cell division cycle following inhibition of cell division. The reduction in the number of cells engaged in DNA replication reaches a plateau as the concentration of paclitaxel is increased; this can be determined experimentally. From our model we have derived a plateau log reduction formula for proliferating cells and established that there are linear relationships between the plateau log reduction values and the reciprocal of doubling times (i.e. growth rates of the populations). We have therefore provided theoretical justification of an important experimental technique to determine cell doubling times. Furthermore, we have applied Monte Carlo experiments to justify the suggested linear relationships used to estimate doubling time from 5-day cell culture assays. We show that our results are applicable to cancer cell populations with cell loss present.

Appendices

Appendix A: Phase Solutions with no Division

We consider the solution when the system is not exhibiting BEG. Solving (1a)–(1c), when the transition rates between compartments are assumed to be positive constants and $r_{G_{2}M\rightarrow G_{1}}$ is set to zero, gives us analytical formulas for the number of cells N _p , p∈{G ₁,S,G ₂ M}, in each of the phases. We can subdivide the solution of the ODE system into two cases. Firstly, let ${r_{G_{1}\rightarrow S}}\neq r_{S\rightarrow G_{2}M}$, then the system of differential equations (1a)–(1c) can be solved analytically as follows:

(29a)

(29b)

(29c)

(29d)

(29e)

Secondly, when ${r_{G_{1}\rightarrow S}}=r_{S\rightarrow G_{2}M}$, the analytical solution of the system (1a)–(1c) is:

(30a)

(30b)

(30c)

But solution (30a)–(30c) will occur with probability of zero when running a Monte Carlo simulation such as in Sect. 3.3 so it is not considered further.

Appendix B: Proof of Theorem 2.1—Nonlinear Mapping Properties

As discussed in Sect. 2.1 for an established cell line, we have a relationship between proportions in each phase, the rate transitions between phases, the population doubling time and this can be reduced to an implicit relationship involving $\mathbf{r}=\{ {r_{G_{1}\rightarrow S}},r_{S\rightarrow G_{2}M},r_{G_{2}M\rightarrow G_{1}}\}$, and $\boldsymbol{\sigma}=\{ \varPi_{G_{1}},\varPi_{S}, \lambda\}$. So it can be shown that the system can be written

$$ \mathbf{G}(\mathbf{r},\boldsymbol{\sigma})= \mathbf {G}({r_{G_1\rightarrow S}},r_{S\rightarrow G_2M} ,r_{G_2M\rightarrow G_1},\varPi_{G_1},\varPi_{S}, \lambda)=0. $$

(31)

The existence of functional relationships between the variables will be determined by the implicit function theorem and this is conditional on certain characterisations of the Jacobian matrix of G, which is given by

Here

(32a)

(32b)

(32c)

(32d)

with F given by (7). We have partitioned J into the first three columns, and called this square matrix J _σ , with the remaining three columns called J _r , and then the implicit function theorem assures us that the Jacobian $J_{\mathbf {r}}=\frac{d{\boldsymbol{\sigma}}}{{d\mathbf{r}}}$ exists locally provided the Jacobian J _σ is non-singular, i.e.

$$ \det J_{\sigma} =\partial_{\lambda} F \biggl( \frac {\varPi_{G_2M}}{\varPi_S\varPi_{G_1}} + \frac{1}{\varPi_S\varPi_{G_1}} \biggr) >0. $$

(33)

This is true if ∂ _λ F>0 as the proportional of cells in each phase are positive. To see that the condition on F is true for the domain under consideration here, we note for exponential growth that λ>0, and all the rates in (32a) are also positive, so that ∂ _λ F>0. Hence the map σ=R(r) is locally unique and determined.

To invert this mapping, we must look into the singularity of J _r and this can be determined from

(34)

so that provided detJ _r ≠0 anywhere, a local map r(σ) will exist almost everywhere.

To prove this, we first look at (32b) and see this can be written

(35)

and in S′, F(λ)=1, with λ>0 so

(36)

A similar argument applies to the other two derivatives of F in (32a)–(32d). As in S′ all the transition probabilities and λ are positive, it follows that detJ _r <0 and we have the result.

Appendix C: Approximate Solution of $\mathcal{P}(\lambda)=0$

To enable this, we look first at the dependence of F(λ)=1 on λ. So with the understanding of approximating the graph of F(λ), we consider a quadratic that crosses the x-axis at −β, −α and the y-axis at γ and that is positive for large x. We also assume that β<α. Then this quadratic is

$$ y(x)=\frac{\gamma}{\alpha\beta}(x+\beta) (x+\alpha ). $$

(37)

Again with consideration of (7), we consider the roots of the equation y(x)=1, and these are given by

$$ x=\frac{1}{2} \biggl[ -(\beta+\alpha)\pm\sqrt{(\beta +\alpha )^2+ \frac{4\alpha\beta}{(\beta+\alpha)^2}\biggl(\frac{1}{\gamma }-1\biggr)} \biggr]. $$

(38)

With the assumption $\frac{4\alpha\beta}{(\beta+\alpha)^{2}}(\frac{1}{\gamma}-1)\ll 1$, the positive root is given by

$$ \frac{\alpha\beta}{(\beta+\alpha)}\biggl(\frac{1}{\gamma}-1\biggr). $$

(39)

When $\frac{\alpha}{(\beta+\alpha)}\approx1$, which is true for |α| large. We then find an approximation to the positive root is

$$ x\approx\beta\biggl(\frac{1}{\gamma}-1\biggr)-\mathcal{O}\biggl(\frac {\beta}{\alpha } \biggr),\quad\beta< \alpha. $$

(40)

Fig. 7

Function F(λ)=1 is plotted for three random sets of transition rates, where $r_{G_{1}S}<r_{SG_{2}M}$

We observe that the root found to y(x)=1 if a linear polynomial is fitted through the x-axis at −β, and the y-axis at γ, i.e. y(0)=γ is (40) with no error term. So the degree of approximation is determined by how close $\frac{\alpha}{(\beta+\alpha)}\approx1$. In conclusion, the linear approximation of λ in (7), when α,β are chosen from $r_{S\rightarrow G_{2}M}$, $r_{S\rightarrow G_{2}M}$ depending on which is larger, and the y-axis crossing is 1/2, is given by (26). This is because

$$ \alpha=\max \bigl(|{r_{G_1\rightarrow S}}|,|r_{S\rightarrow G_2M}| \bigr),\qquad \beta=\min \bigl(|{r_{G_1\rightarrow S}}|,|r_{S\rightarrow G_2M}| \bigr) $$

(41)

and γ=2.