Propagation of weakly advantageous mutations in cancer cell population

TL;DR

This work investigates cancer evolution under weakly advantageous passenger mutations by developing deterministic differential-equation models and stochastic Gillespie simulations to track mutation-wave propagation and population growth within a carrying-capacity environment. It introduces explicit birth, death, and mutation processes, derives two-time-scale dynamics, and analyzes quasi-stationary mutation waves with cutoff adjustments to reflect finite populations. The authors validate their framework by confronting predictions with public cancer datasets, including TCGA-OncoVar driver analyses, Casasent et al. DCIS-to-INV breast cancer data, and PET-based tumor growth rates, finding concordance with observed increases in mutation load and with power-law growth patterns. The study suggests that many cancers can be driven by aggregates of weakly advantageous passenger mutations, a scenario compatible with driver-lacking TCGA cases and with clonal progression observed in single-cell sequencing, offering a quantitative scaffold for interpreting cancer evolution beyond classical driver-centric views.

Abstract

Research into somatic mutations in cancer cell DNA and their role in tumour growth and progression between successive stages is crucial for improving our understanding of cancer evolution. Mathematical and computer modelling can provide valuable insights into the scenarios of cancer growth, the roles of somatic mutations, and the types and strengths of evolutionary forces they introduce. Previous studies have developed mathematical models of cancer evolution, incorporating driver and passenger somatic mutations. Driver mutations were assumed to have a strong advantageous effect on the growth of the cancer cell population, while passenger mutations were considered fully neutral or mildly deleterious. However, according to several studies, passenger mutations may have a weakly advantageous effect on tumour growth. In this paper, we develop models of cancer evolution with somatic mutations that introduce a weakly advantageous force to the evolution of cancer cells. The models used in this study can be classified into two categories: deterministic and stochastic. Deterministic models are based on systems of differential equations that balance the average number of cells and mutations during evolution. To verify the results of our deterministic modelling, we use a stochastic model based on the Gillespie algorithm. We compare the predictions of our modelling with some observational data on cancer evolution.

Figures (5)

• Figure 1: Graphical representation of possible events in the analyzed scenario of cancer cells populations evolution.
• Figure 2: Comparison of solutions to deterministic differential equations model Eq (\ref{['eq:master_fit']}), with the solution to model Eq (\ref{['eq:master_fit_FPmod']}) with cutoff modification Eq (\ref{['eq:FP_modifications']}), for modelling evolution with weakly advantageous mutations, for the parameter, set $A=1$, $f=0.0005$, $p_{f}=0.025$, $N_{C}=10~000$. Left panels present plots of mutation waves at three time instants $t=500$, $t=1~000$ and $t=2~500$. The right panels show time plots of mean numbers of mutations, $\chi_{f}(t),$ (upper plots), the variance of mutation numbers, $\sigma_{f}^{2}(t)$, (middle plots) and number of cells in the cancer population $N(t)$ (lower plots). For all plots, bold blue, solid lines show time plots computed by using numerical integration of the system of differential equations without Eq (\ref{['eq:master_fit']}) while analogous black lines represent solutions to modified equations, Eq (\ref{['eq:master_fit_FPmod']}).
• Figure 3: Comparison of deterministic (with cutoff modification) versus stochastic modelling for evolution with weakly advantageous mutations, for the parameter set $A=1$, $f=0.0005$, $p_{f}=0.025$, $N_{C}=10~000$. Left panels present plots of mutation waves at three time instants $t=2~500$, $t=5~000$ and $t=7~500$. The right panels show time plots of mean numbers of mutations, $\chi_{f}(t),$ (upper plots), the variance of mutation numbers, $\sigma_{f}^{2}(t)$, (middle plots) and the number of cells in the cancer population $N(t)$ (lower plots). For all plots, black, bold, solid lines show time plots computed by using numerical integration of the system of differential equations with cutoff modification Eq (\ref{['eq:master_fit_FPmod']}). Red plots present the results of stochastic simulations obtained by using Gillespie algorithm.
• Figure 4: plots of functions $\sigma_{f}^{2}(N)$ for different values of positive selection coefficient ($f=0.0005$, $f=0.001$, $f=0.0015$) and for different values of exponent parameter ($A=0.1$, $A=0.3$, $A=0.5$, $A=1$). Plots corresponding to different values of $A$ are drawn with different colours. $A=0.1$ - with red color, $A=0.3$ - with green color, $A=0.5$ - with blue color and $A=1.0$ - with black color.
• Figure 5: Comparisons of growth patterns of cancer cells population size $N(t)$, for evolution with weakly advantageous mutations, for parameters $f=0.0005$, $p_{f}=0.025$, $N_{C}=10000$ and different values of exponent parameter $A$ in cell death intensity relation (\ref{['eq:death_intensity']}). Values of $A$ used in computations/simulations are $A=0.1$, $A=0.3$, $A=0.5$ and $A=1.0$. (a): Time plots $N(t)$. Time plots $N(t)$ computed by using analytical relation Eq (\ref{['eq:N_analytical']}) are drawn with bold black lines. Time plots of $N(t)$ obtained by using stochastic modelling (Gillespie algorithm) are drawn as red curves. (b): Plots of $log_{10}(\frac{dN}{dt})$ versus $log_{10}(N(t))$. Plots obtained by using relation Eq (\ref{['eq:log_dtN_fit_N']}) are drawn as black bold lines. Growth patterns of $N(t)$ obtained on the basis of stochastic simulations (red plots in Fig \ref{['fig:5a']}) are represented by red asterisks. The coordinates of each asterisk are computed as base $10$ logarithms of averaged values of $N(t)$ (horizontal coordinate) and $\frac{dN}{dt}$ (vertical coordinate). Averaging over bins of the size $1~000$ in the time scale is done for the purpose of reducing the large variation of $\frac{dN}{dt}$ in stochastic simulations.