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. 2017 Oct 30;12(10):e0187000.
doi: 10.1371/journal.pone.0187000. eCollection 2017.

Phenotypic heterogeneity in modeling cancer evolution

Ali Mahdipour-Shirayeh  1 , Kamran Kaveh  2 , Mohammad Kohandel  1   3 , Sivabal Sivaloganathan  1   3
Affiliations

Phenotypic heterogeneity in modeling cancer evolution

Ali Mahdipour-Shirayeh et al. PLoS One. .

Abstract

The unwelcome evolution of malignancy during cancer progression emerges through a selection process in a complex heterogeneous population structure. In the present work, we investigate evolutionary dynamics in a phenotypically heterogeneous population of stem cells (SCs) and their associated progenitors. The fate of a malignant mutation is determined not only by overall stem cell and non-stem cell growth rates but also differentiation and dedifferentiation rates. We investigate the effect of such a complex population structure on the evolution of malignant mutations. We derive exactly calculated results for the fixation probability of a mutant arising in each of the subpopulations. The exactly calculated results are in almost perfect agreement with the numerical simulations. Moreover, a condition for evolutionary advantage of a mutant cell versus the wild type population is given in the present study. We also show that microenvironment-induced plasticity in invading mutants leads to more aggressive mutants with higher fixation probability. Our model predicts that decreasing polarity between stem and non-stem cells' turnover would raise the survivability of non-plastic mutants; while it would suppress the development of malignancy for plastic mutants. The derived results are novel and general with potential applications in nature; we discuss our model in the context of colorectal/intestinal cancer (at the epithelium). However, the model clearly needs to be validated through appropriate experimental data. This novel mathematical framework can be applied more generally to a variety of problems concerning selection in heterogeneous populations, in other contexts such as population genetics, and ecology.

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Conflict of interest statement

Competing Interests: The authors have declared that no competing interests exist.

Figures

Fig 1
Fig 1. Phenotypic—Genotypic changes in individuals within a four–compartmental structure.
We consider constant population sizes NS and ND for SCs and DCs respectively. With respect to the finite Markov chain, we consider a generalized model to take into account the competition between normal and malignant individuals in each of the SC and DC subpopulations. Differentiation and dedifferentiation events connect the selection dynamics between the two niches. In (a), all possible differentiation, dedifferentiation, and death events with their corresponding rates are represented. The SC-DC compartmental structure is depicted in (b) with the associated self—renewal and differentiation/plasticity possibilities.
Fig 2
Fig 2. The fixation probability of mutants in the absence of plasticity.
We assume NS = ND = 10, r1=r˜1=1, and η1 = η2 = 0 in simulations (points) and exact calculations (solid lines) for an initial mutant in the SC compartment. Each error bar shown at each point is the standard error of the mean. In (a) changing parameters u1, u2, that are the differentiation rates of normal and tumor SCs respectively, the trends for the fixation probability of mutants is given as a function of relative fitness of mutants, referred to as r2=r˜2=r. In (b) and (c) the fixation probability variation is given in terms of asymmetric differentiation rates u1 = u2 = u and various values of r and the ratio of the differentiation rates of normal SCs ϵ = u2/u1. In (b) ϵ = 0.5 and in (c) ϵ = 1.5.
Fig 3
Fig 3. The fixation probability of mutants in the presence of phenotypic plasticity.
We suppose that NS = ND = 10, r1=r˜1=1, r2 = αr, and r˜2=βr in the exactly calculated results (solid curves) and stochastic simulation (points with bars as the standard error of the mean) starting with an initial mutant within SCs. Changing α from 0.5 to 1 and then to 2 while β remains fixed, the behavior of the system is shown in terms of the fixation probability with respect to the relative fitness r. In subfigure (a) plastic potential has only been considered for malignant individuals: η1 = 0, η2 = 0.5 while in (b) both WT and mutant cells can dedifferentiate to the stemness state (η1 = η2 = 0.5). Straightforward calculations reveal that for the given parameter values in (b), ρ = ρS = ρD.
Fig 4
Fig 4. Effect of change in asymmetric differentiation and plasticity rates on survivability of mutants.
Subfigures show the variation of ρS with respect to the relative fitness of mutants r, in which we start with an initial mutant within SCs. We assume that NS = ND = 10, r1=r˜1=1, and r2=r˜2=r. In subfigure (a), the fixation probability of SCs as a function of η is given, where η = 0.01, 0.1, 0.5 while u1 = u2 = 0.5, and η1 = 0. In (b) η1 = 0 and η = 0.1. Changing parameters u1 and u2, which are the asymmetric division rates of normal and tumor SCs respectively, the fixation probability as a function of u1, u2 is shown. Solid lines represent the exact calculation and points correspond to simulation results (error bars are based on the standard error of the mean).
Fig 5
Fig 5. Effect of change in the rate of phenotypic plasticity for mutant DCs on the survival probability of malignant cells.
In subfigures (a), (b), and (c) we have respectively considered the fixation probability of mutants while the initial malignant mutation respectively occurs in SC, DC, and SC+DC compartments at random. In these plots, we assumed that NS = ND = 10, r1=r˜1=1, r2=r˜2=1.1,u1=0.5 and η1 = 0. Different asymmetric division rates of normal and malignant individuals have also taken into account when the fixation probability is given as a function of η. Solid curves and points are respectively the results of exact calculations and simulation analyses. At each given discrete point, the error bar depicts the standard error of the mean.
Fig 6
Fig 6. Phase diagram of plastic mutant SCs.
The phase boundary for advantageous and disadvantageous mutant populations are given as differentiation and plasticity rates change. We assume that r1=r˜1=1, r2=r˜2=r, u1 = u2 = u, and η1 = 0. Different regions for advantageous and disadvantageous mutant SCs are given in (a) as u changes. A similar analysis has been carried out in (b) as η varies. In (a) η = 0.1, 0.3, 0.7, here the alteration in the plasticity rate of DCs results in a tendency to approach various regions of fixation for mutant SCs, while the extinction domain shrinks with increasing η. In (b) u = 0.1, 0.3, 0.7. Increasing the asymmetric division rate u, the region for advantageous mutants expands to provide a higher survival chance for mutant SCs. In both cases, advantageous criteria relate to either fixation of mutants or coexistence of mutants and WT individuals.
Fig 7
Fig 7. Cellular interactions in the colonic crypt as a newborn mutant arises within the stem or non-stem cell compartments.
Within this schematic cylindrical model, we represent how our model is structured through the four compartments of host and mutant stem and non-stem cells. In contrast to the circular model of five SCs considered in [63], we assume a cylindrical model of two circles, one on the top of the other. SCs are located at the bottom circle while the circle on the top is full of partially DCs.
Fig 8
Fig 8. Expectations for the enhance in the fixation probability of P53 mutants under inflammation.
We propose that when microenvironmental-induced plasticity can occur as a result of inflammatory injury, the increase in the survival chance of mutants reported in [18] may occur as a result of dedifferentiation rather than an enhance in the fitness of P53 mutants. The blue strand show the range of change for the fixation probability represented in [18] with an average shown by dashed green line and error bars shown by green lines. We assume that NS=ND=10,r1=r˜1=1,r2=r˜2=0.96, and η1 = 0. The blue, pink, and red lines are drawn for diverse values for the asymmetric differentiation and fitness of normal SC and non-stem cells are normalized to one while that of mutant SCs/non-SCs are equal to the reproduction rate reported in [18]. Then the fixation probability ρS is given over a range of different plasticity rates. When u1 = 0.5, u2 = 0.25, and η = 0.12 then ρS = 0.343 as has been reported in [18] with an increases in the reproduction rate of mutants from 0.96 to 1.16.
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