The effects of phenotypic plasticity on the fixation probability of mutant cancer stem cells

Abstract

The cancer stem cell hypothesis claims that tumor growth and progression are driven by a (typically) small niche of the total cancer cell population called cancer stem cells (CSCs). These CSCs can go through symmetric or asymmetric divisions to differentiate into specialised, progenitor cells or reproduce new CSCs. While it was once held that this differentiation pathway was unidirectional, recent research has demonstrated that differentiated cells are more plastic than initially considered. In particular, differentiated cells can de-differentiate and recover their stem-like capacity. Two recent papers have considered how this rate of plasticity affects the evolutionary dynamic of an invasive, malignant population of stem cells and differentiated cells into existing tissue (Mahdipour-Shirayeh et al., 2017; Wodarz, 2018). These papers arrive at seemingly opposing conclusions, one claiming that increased plasticity results in increased invasive potential, and the other that increased plasticity decreases invasive potential. Here, we show that what is most important, when determining the effect on invasive potential, is how one distributes this increased plasticity between the compartments of resident and mutant-type cells. We also demonstrate how these results vary, producing non-monotone fixation probability curves, as intercompartmental plasticity changes when differentiated cell compartments are allowed to continue proliferating, highlighting a fundamental difference between the two models. We conclude by demonstrating the stability of these qualitative results over various parameter ranges.

1. Introduction

Cancer invasion is a complex process of the cellular ecosystem and micro-environment. Typically, cancerous cells achieve evolutionary success by mimicking, and subverting, the behaviour of regular, healthy tissues in the host (Kreso and Dick, 2014). In healthy tissue a distinct stratification of tissue structure is observed where the self-renewal capabilities of terminally differentiated cells rely upon a discrete subclass of the cellular population deemed stem cells (Weissman, 2000). In particular, these stem cells can behave invasively through the process of division and specialisation (Borovski et al., 2011). As healthy, specialised (or differentiated) cells die, the stem cells divide producing the required differentiated cell as a byproduct of mitosis. In this way the population of specialised, terminally differentiated cells is maintained (Borovski et al., 2011; Watt and Hogan, 2000). However, in order to maintain tissue homeostasis, this mechanism must be controlled by both positive and negative feedback loops. To that effect there is mounting evidence that differentiated cells can de-differentiate back into adult stem cells in both healthy and non-healthy tissue (Watt and Hogan, 2000; Tata et al., 2013; Chaffer et al., 2011).

This process of cellular invasion in the tissue driven by a particular niche of cells is very similar to the dynamics observed in invasive cancers. This has led to the cancer stem cell hypothesis that posits that the invasion, metastasis, and regulation of solid tumors are driven by a (typically) small sub-population of cells that behave very much like stem cells (O’Brien et al., 2009; Reya et al., 2001; Sprouffske et al., 2013; Marjanovic et al., 2013). In particular, this same behaviour of differentiated cells undergoing some process of de-differentiation and recovering some stem-like properties has been implicated in many forms of cancers experimentally (Gupta et al., 2011; Cabrera et al., 2015; Huels and Sansom, 2015; Philpott and Winton, 2014).

Many models have been proposed to examine the mechanics of the invasion of these plastic cancer cells in tissue for instance in Zhou et al. (2019) the authors discover conditions under which de-differentiation is selected while in Jilkine and Gutenkunst (2014) the authors discuss how de-differentiation influences the time to acquisition of mutations. Another two papers that have mathematically modeled the effect of this stem-cell plasticity and de-differentiation on the evolutionary dynamics of cancer stem cells (Mahdipour-Shirayeh et al., 2017; Wodarz, 2018) arrived at contradictory results given similar starting assumptions. Both papers are concerned with how the invasive efficacy of mutated stem cells depends upon the degree of de-differentiation considered. In particular, both investigate a situation where a resident type of stem cells and differentiated cells have fully saturated within a population and by introducing a single mutant type within the population the fixation probability of these mutants is recovered. In Mahdipour-Shirayeh et al. (2017) the authors conclude that as the rate of plasticity increases in a population of cancer stem cells, the fixation probability of mutant stem cells increases. In Wodarz (2018), however, the author concludes that as the plasticity increases in a population of cancer stem cells, the fixation probability of mutant stem cells decreases. These results are evidently at odds with one another and it is the purpose of this report to reconcile and explain the differing dynamical predictions of these models.

2. Modeling

The structure of the models is the same in both papers (Mahdipour-Shirayeh et al., 2017; Wodarz, 2018). The authors consider a population of resident stem cells in an equilibrium state. They then introduce a mutant, cancerous stem cell into the wild-type population and investigate the probability that the single mutant achieves fixation as a function of the plasticity of the mutant stem cell. That is, the probability ρ_S that as time t tends to infinity the wild type stem cells have all either died or differentiated and the mutant type stem cells remain for a given de-differentiation rate η.

2.1. A discrete moran model

In Mahdipour-Shirayeh et al. (2017) the authors consider modeling the birth-death process by way of a Moran model. In particular, the authors consider two types of cells, stem cells (S_i) and differentiated cells (D_i) where i ∈ {1, 2} denotes whether the cells in question are wild-type (i = 1) or mutant-type (i = 2). Cells can react and transition between compartments according the following rules (summarised in Fig. 1a). Stem cells undergo division at a rate r_i. There is a probability u_i that the stem cell produces both a stem cell S_i and a differentiated cell D_i during division. Similarly, with probability (1 − u_i), the stem cell produces two stem cells as the result of this division. Moreover, differentiated cells can reproduce as well at a rate ${\tilde{r}}_i$. There is a probability η_i that this division is asymmetric, producing one stem cell S_i and one differentiated cell D_i, and a probability (1 − η_i), that the division is symmetric, producing two differentiated cells D_i. Similarly, with probability d_i and ${\tilde{d}}_i$, the stem cells and differentiate cells die.

Fig. 1.

In both models the differentiation, de-differentiation, and death events corresponding to a four-compartment model where S_i and D_i are the stem-cells and differentiated cells for the wild-type (i = 1) and mutant-type (i = 2) respectively.

Note that in each of these events the population of stem cells and differentiated cells changes by at most 1 in either direction. Moreover, as births and deaths of cells are assumed to occur instantaneously, the state of the system at any point in time is determined entirely by the state at the previous time. Hence, the particular stochastic process being considered is Markovian. Further, the authors assume that each individual compartment has constant population size across wild and mutant types: i.e. the number of stem cells and the number of differentiated cells remain static, merely what proportion of these stem or differentiated cells are wild or mutant type is the variable.

It is important to note that in Mahdipour-Shirayeh et al. (2017), the model is developed to assume the S_i and D_i compartments correspond to positive and negative biomarker cells. Current biomarkers used do not necessarily correspond directly to stem or non-stem cells. In particular, negative-biomarker (or differentiated-analogue) cells can still be observed to proliferate, as represented in the model. In reality it is often assumed that the only cells capable of driving the growth and invasive potential of a tumor are stem cells and hence differentiated cells cannot proliferate to any meaningful degree. However, since biomarkers are imperfect, some stem-like cells may be classified as differentiated (Zapperi and La Porta, 2012). This is a natural consequence of imperfect biomarkers, but is worth mentioning explicitly to justify why differentiated cells are allowed to proliferate in the model represented in Fig. 1a.

Let N_S denote the size of the stem cell compartment, N_D the size of the differentiated compartment, and N = N_S + N_D the total number of cells considered. Further, we are concerned primarily with invasion of mutant types; that is, when the number of S₂ or D₂ cells is at a maximum (equal to N_S or N_D). Hence we let n_S and n_D represent the number of S₂ or D₂ cells. Therefore, the number of S₁ or D₁ cells is given by N_S − n_S and N_D − n_D, respectively. Let $W_S^{+}\left(n_S,n_D\right)$, $W_S^{-}\left(n_S,n_D\right)$, $W_D^{+}\left(n_S,n_D\right)$, and $W_D^{-}\left(n_S,n_D\right)$ represent the transition probabilities corresponding to an increase or decrease by one (represented by the + or − in the superscript) in the number of stem cells or differentiated cells (represented by the S or the D in the subscript). Then, if 1/N is the duration of each time step in this model, one can represent the following master equation for the probability density function of the stochastic process as follows (Mahdipour-Shirayeh et al., 2017):

$$\begin{gathered} \frac{1}{N}\frac{\partial p\left(n_S,n_D;t\right)}{\partial t}=W_S^{+}\left(n_S-1,n_D\right)p\left(n_S-1,n_D;t\right)+W_S^{-}\left(n_S+1,n_D\right)p\left(n_S+1,n_D;t\right) \\ +W_D^{+}\left(n_S,n_D-1\right)p\left(n_S,n_D-1;t\right)+W_D^{-}\left(n_S,n_D+1\right)p\left(n_S,n_D+1;t\right) \\ -(W_S^{+}\left(n_S,n_D\right)+W_D^{+}\left(n_S,n_D\right)+W_S^{-}\left(n_S,n_D\right) \\ +W_D^{-}\left(n_S,n_D\right))p\left(n_S,n_D;t\right), \end{gathered}$$ (1)

where the transition probabilities are given as follows:

$$W_S^{+}\left(n_S,n_D\right)=\left(r_2\left(1-u_2\right)n_S+{\tilde{r}}_2{\eta }_2{n}_D\right)\left(\frac{N_S-n_S}{N_S}\right)$$ (2)

$$W_S^{-}\left(n_S,n_D\right)=\left(r_1\left(1-u_1\right)\left(N_S-n_S\right)+{\tilde{r}}_1{\eta }_1\left(N_D-n_D\right)\right)\left(\frac{n_S}{N_S}\right)$$ (3)

$$W_D^{+}\left(n_S,n_D\right)=\left({\tilde{r}}_2\left(1-{\eta }_2\right)n_D+r_2{u}_2{n}_S\right)\left(\frac{N_D-n_D}{N_D}\right)$$ (4)

$$W_D^{-}\left(n_S,n_D\right)=\left({\tilde{r}}_1\left(1-{\eta }_1\right)\left(N_D-n_D\right)+r_1{u}_1\left(N_S-n_S\right)\right)\left(\frac{n_D}{N_D}\right).$$ (5)

The results of Fig. 2a clearly demonstrate that fixation probability increases as plasticity of the mutant increases. In particular, when the resident wild type cells were given no plasticity and the mutant type cells plasticity was treated as a variable it was observed that increasing the mutant types plasticity gave these mutant types a greater invasive potential when placed in a population dominated by wild types population.

Fig. 2.

This Figure shows the results of a simulation of both the discrete Moran model from Mahdipour-Shirayeh et al. (2017) and the continuous Gillespie model from Wodarz (2018) where the wild-type differentiated cells are completely non-plastic. The figures show the contrasting results where in 2a the fixation probability increases as mutant-type plasticity increases and in 2b the fixation probability decreases as mutant-type plasticity increases. The figures are recreated from similar figures in Mahdipour-Shirayeh et al. (2017) and Wodarz (2018).

2.2. A continuous gillespie model

On the other hand, in Wodarz (2018) the author aims to describe the dynamics of stem cell invasion not merely by considering births and deaths and the selective pressures therein, but by directly considering the positive and negative feedbacks that control differentiation and de-differentiation in the stem cell hierarchy. To this end, the author constructs multiple systems of differential equations that model the behaviour of stem cells and differentiated cells under various assumptions. The key differences between this paper and Mahdipour-Shirayeh et al. (2017), are in the details of these models. In particular, the author does not limit themselves to constant population size of stem cell and differentiated cell compartments. Moreover, the author does not explicitly allow differentiated cells to proliferate, as the categories S_i and D_i in this model do directly correspond to stem and differentiated cells directly, and not by way of a biomarker analogue as in Mahdipour-Shirayeh et al. (2017). As before, the author considers a four compartment model where S_i represents the stem cells and D_i represents the differentiated cells:

$$\begin{gathered} \frac{{\dot{\text{S}}}_i}{\dot{\text{t}}}={\widehat{r}}_i{S}_i\left(2p_i-1\right)+g_i{D}_i \\ \frac{{\dot{\text{D}}}_i}{\dot{\text{t}}}=2{\widehat{r}}_i{S}_i\left(1-p_i\right)-{\alpha }_i{D}_i-g_i{D}_i \end{gathered}$$ (6)

where i ϵ {1, 2} represents the wild type (i = 1) and mutant type (i = 2), as before.

While the author of Wodarz (2018) does not frame the model as a reaction network, it is not hard to recover the set of reactions from the differential equations. These are presented in Fig. 1b.

One should note that if $p_i\ne \frac{{\alpha }_i-g_i}{2{\alpha }_i}$, then the only equilibrium is S_i = D_i = 0. Otherwise, the entire set [S_i, D_i] = [S_i, (r_i/α_i)S_i] is an equilibrium. These results do not coincide with what is observed in experiments. The author argues that feedback on the division rate, self-renewal probability, and de-differentiate rate occurs in order to maintain finite-size cell populations. To model this, it is assumed that ${\widehat{r}}_i$, p_i, and g_i are decreasing functions of the cell population sizes in the following way:

$${\widehat{r}}_i=\frac{{\widehat{r}}_i^{\prime }}{1+h_{i,1}{D}_i^{k_{i,1}}},p_i=\frac{p_i^{\prime }}{1+h_{i,2}{D}_i^{k_{i,2}}},g_i=\frac{g_i^{\prime }}{1+h_{i,3}{S}_i^{k_{i,3}}}$$

In this case, the model permits an equilibrium point internal to the first quadrant that is stable when $p_i>\frac{{\alpha }_i-g_i}{2{\alpha }_i}$.

To numerically establish the fixation probability the following experiments were run. The author begun by placing S₁ and D₁ at the equilibrium values and then introduced a single mutant stem cell, S₂ = 1 and D₂ = 0. Gillespie simulations were then performed (Gillespie, 1977) for many realisations (more than 10⁸). The fraction of simulations in which the mutants fixated was recorded and from this the fixation probability was estimated. This was done for various values of de-differentiation rates g = g₁ = g₂.

In contrast to the results of the discrete Moran model in Fig. 2a, Fig. 2b shows that the fixation probability of a neutral mutant decreases when the de-differentiation rate increases in the continuous Gillespie model.

3. Reconciliation of previous results

In the Wodarz model there are four parameters considered for each cell-type (mutant or wild) (p_i, ${\widehat{r}}_i$, α_i, g_i) while in the Shirayeh model there are six (r_i, ${\tilde{r}}_i$, d_i, ${\tilde{d}}_i$, η_i, u_i). The biggest difference between these parameters is that in Wodarz (2018) the numbers p_i and r_i are both functions of D_i, and g_i is a function of S_i.

Moreover, Wodarz (2018) considers four reactions between stem cell and differentiated cell compartments

$$S_i\to S_i+S_i,S_i\to D_i+D_i,D_i\to S_i,D_i\to Ø.$$

whereas in Mahdipour-Shirayeh et al. (2017) there are six

$$S_i\to S_i+S_i,S_i\to S_i+D_i,D_i\to D_i+D_i,D_i\to D_i+S_i,S_i\to Ø,D_i\to Ø.$$

However, all the reactions from Mahdipour-Shirayeh et al. (2017) can be recovered in the Wodarz model (Wodarz, 2018). In particular, in order for stem cells to die they must first differentiate and then the differentiated cells must both die. In order to obtain asymmetric stem cell division, the stem cells must first divide into two differentiated cells, then one of the differentiated cell must de-differentiate back into a stem cell. Suppose a differentiated cell first differentiates into a stem cell and then the resultant stem cell symmetrically differentiates. Therefore, the differentiated cell has symmetrically divided. Further, suppose one of those resultant differentiated cells de-differentiates back into a stem cell, then the differentiated cell has asymmetrically divided. In this way all the reaction dynamics of the model from Mahdipour-Shirayeh et al. (2017) have been recovered by multiple reactions from the model by Wodarz (2018).

This realisation lends credence to the theory that the models should, hopefully, predict similar qualitative dynamics. To that end, we begin by creating differential equations via the reactions given in Fig. 1a and compare them to the differential equations presented in Wodarz (2018). By making the mass-action assumption, we derive

$$\begin{gathered} \frac{{\dot{\text{S}}}_i}{\dot{\text{t}}}={\tilde{r}}_i{\eta }_i{D}_i+r_i\left(1-u_i\right)S_i-d_i{S}_i \\ \frac{{\dot{\text{D}}}_i}{\dot{\text{t}}}={\tilde{r}}_i\left(1-{\eta }_i\right)D_i+r_i{u}_i{S}_i-{\tilde{d}}_i{D}_i. \end{gathered}$$ (7)

Contrasting (7) with (6) it is not hard to derive the following relations. To frame the Wodarz model (Wodarz, 2018) in terms of parameters from Mahdipour-Shirayeh et al. (2017),

$$p_i=1-\frac{r_i{u}_i}{2\left(r_i-d_i\right)}$$

$${\widehat{r}}_i=r_i-d_i$$

$${\alpha }_i={\tilde{d}}_i-{\tilde{r}}_i$$

$$g_i={\tilde{r}}_i{\eta }_i$$

conversely, framing the model in Mahdipour-Shirayeh et al. (2017) in terms of parameters from the Wodarz model (Wodarz, 2018) we leave d_i and ${\tilde{d}}_i$ as free parameters, then

$$r_i=d_i+{\widehat{r}}_i$$

$${\tilde{r}}_i={\tilde{d}}_i-{\alpha }_i$$

$${\eta }_i=\frac{g_i}{{\tilde{d}}_i-{\alpha }_i}$$

$$u_i=2\frac{{\widehat{r}}_i\left(1-p_i\right)}{d_i+{\widehat{r}}_i}.$$

Hence the qualitative dynamics of (7) are dynamically equivalent to those of (6), under a renaming of parameters. Moreover, these formulae allow us to more carefully compare the two systems predictions under differing notations and parameter choices, as the formulae provide a way to transcribe parameter values and notational choices from one system into the notation of the other. In particular, note that the two plasticity parameters g_i and η_i can be compared directly, as they are just linear, increasing functions of one another. Hence an increase in g_i is analogous to an increase in η_i (and vice versa). Moreover, η_i = 0 is analogous to g_i = 0.

Negative feedback on production rates is modeled in the model put forth by Mahdipour-Shirayeh et al. (2017), but not as explicitly as in Wodarz (2018). In particular, the last factor in the transition probabilities of Mahdipour-Shirayeh et al. (2017) provide this bias. If n_S is high, or the number of stem cells is close to maximum, then the transition probability $W_S^{+}$ is near zero, due to the (N_S − n_S)/N_S factor at the end. Hence when the number of mutant stem cells is high, the probability of making more mutant stem cells is lowered (this mimics the behaviour observed in the g₂(S₂) function in the Wodarz model (Wodarz, 2018)). Similarly the n_S/N_S term in the $W_S^{-}$ transition probability creates pressure where creating more wild-type stem cells is less likely in the presence of many wild-type stem cells (similar to the g₁(S₁) function in the Wodarz model (Wodarz, 2018)). Analogous behaviour occurs in the $W_D^{\pm }$ probabilities, mimicking the behaviour of the ${\widehat{r}}_i$ and p_i functions decreasing as D_i increases.

In the finite population size model, negative feedback on (de-) differentiation rates and production rates are governed by the select pressures inherit in finite population sizes, whereas in the unbounded population size model these same feedback mechanisms are modeled by non-constant, non-linear reaction rates.

A more subtle difference between the two systems is due, in part, to a lack of clarity in notational differences. In the original presentation of Wodarz (2018) the i subscripts were not present (though different values of the same parameter between compartments was implied in the figure descriptions). Among other things, this obscures the fact that g (the parameter representing de-differentiation, or plasticity) is being altered at the same rate for both the mutant-type population and for the resident-type population. Hence, the plots in Fig. 2b really plot the fixation probability ρ(g₁, g₂) along the diagonal cross-section of the first quadrant, that is the plots show

$$\frac{\partial }{\partial g}\left({\rho \left(g_1,g_2\right)|}_{g_1=g_2=g}\right)<0$$

whereas the plots in Fig. 2a show

$$\frac{\partial }{\partial {\eta }_2}\left({\rho \left({\eta }_1,{\eta }_2\right)|}_{{\eta }_1=0}\right)>0.$$

The tacit assumption in Mahdipour-Shirayeh et al. (2017) is that part of the phenotypic difference between the mutant stem cells and the resident stem cells is that resident stem cells have no plasticity and mutant stem cells do have plasticity. The analysis then focuses around qualifying the effects of this plasticity. Contrast this, the tacit assumption in Wodarz (2018) is that the mutant and resident stem cells are phenotypically different in some other way and share the same plasticity rate.

To conclude, we wished to consider measuring ρ(η, η) for 0 ⩽ η ⩽ 1 in the discrete Moran model (that is, the fixation probability from Wodarz (2018) in the model by Mahdipour-Shirayeh et al. (2017)) and to measure ρ(0, g) for 0 ⩽ g ⩽ 1 in the continuous Gillespie model (that is, the fixation probability from Mahdipour-Shirayeh et al. (2017) in the model by Wodarz (2018)). In particular, as can be seen in Fig. 3, we have recovered the results of the contrasting models. That is, in Fig. 3a we see that ρ(η, η) is a decreasing function of η as was seen in Wodarz (2018) (Fig. 2b) and in Fig. 3b that ρ(0, g) is an increasing function of g as was seen in Mahdipour-Shirayeh et al. (2017) (Fig. 2a).

Fig. 3.

The figure indicates that when varying both the wild-type and mutant-type plasticity parameters at the same rate in the discrete Moran models, the fixation probability function is negatively sloped. Similarly, when increasing only the mutant-type plasticity parameter and keeping the wild-type plasticity parameter at zero in the continuous Gillespie model, the fixation probability function increases. Note that these results are similar to what was observed in Fig. 2, just with the modelling framework switched.

4. Further results

The results of Section 3 demonstrate that it is vitally important to consider how a change in plasticity in a mutant type affects a change in plasticity of a wild type, if at all. In particular, the authors of Mahdipour-Shirayeh et al. (2017) and Wodarz (2018) were effectively answering different research questions. In Mahdipour-Shirayeh et al. (2017) the authors were concerned with what the effect of increased plasticity in a mutant-type would have on fixation probability assuming the wild-type stayed the same. This corresponds to a case where the mutant-type gains increased plasticity as part of the mutation. Moreover, this assumes that this increased plasticity in the mutant type does not have any effects on the plasticity in the resident type. Contrarily, in Wodarz (2018) the authors assume that the increased plasticity is common to both the mutant and wild type stem cells. This could be the case where an increased mutant-type plasticity affects the wild-type stem cells by some kind of increased selective pressure, or via a change in the micro-environment thus influencing the wild-type to also become more plastic.

A natural extension to the results of Section 3 is to consider decoupling the plasticity parameters all together and contrast the behaviour of the models over this larger parameter range. For computational reasons we proceed by examining the discrete Moran model and derive non-monotone (see Fig. 4) behaviour before concluding in Section 4.2 that this behaviour can not be observed in the continuous Gillespie case.

Fig. 4.

A plot demonstrating the presence of a saddle point in the fixation probability as a function of both de-differentiation parameters. It indicates that increasing the mutant-type de-differentiation parameter does not always result in increased fixation of mutant-type stem cells, even when keeping the wild-type de-differentiation parameter constant.

We consider varying both η₁ and η₂ (the plasticity of the wild-type and mutant-type cells) independently of one another. Since η₁ and η₂ are probabilities, we varied them over the closed unit square [0, 1] × [0, 1] in a 36 × 36 grid. At each point in the grid we calculated the fixation probability of the mutant-type stem cells. Further details of the numerical simulation can be found in Appendix A. Note that allowing η₁ > 0 corresponds to de-differentiation of healthy, wild-type tissue. Mathematically, this is motivated by the original model strategy of Wodarz (2018) but biologically this choice is reasonable in light of experimental evidence that de-differentiation is present not only in malignant tissue, such as was studied in Zhou et al. (2019), but in healthy tissue as well (Tata et al., 2013; Chaffer et al., 2011). Moreover, especially in the case of the discrete Moran model, we wish to reemphasise that the compartment labels of stem and differentiated cells can be thought of as positive and negative stem biomarker cells, in which case the assumption that η₁ > 0 could be understood to represent imperfect sorting of such compartments by modern biomarkers (Zapperi and La Porta, 2012).

The purpose of the experiment was to further investigate the exact interplay between plasticity in the two compartments on the fixation probability. However, there are 6 other parameters to consider, r_i, ${\tilde{r}}_i$, and u_i. In order to investigate the direct effect of a change in plasticity on the fixation probability in isolation, we considered the case where relative fitness was the same in the stem and differentiated cells (that is, $r_i={\tilde{r}}_i$).

We further considered experiments where the relative fitness of the mutant-type cells $\left(r_2={\tilde{r}}_2\right)$ were varied along with the de-differentiation probabilities (η₁ and η₂). Moreover, to further analyse the stability, we considered varying the differentiation probability (u₂) of the mutant-type cells along with the de-differentiation probabilities.

In the case of the neutral mutant (where $r_2={\tilde{r}}_2=r_1={\tilde{r}}_1$ and u₂ = u₁), the results are summarised in Figs. 4, 5, and 7. The plot in Fig. 4 indicates the fixation probability as a function of both de-differentiation parameters. It indicates that in general, increasing η₂, the de-differentiation probability of the mutant cells, increases the fixation probability. Moreover, this fixation probability is maximal for (η₁, η₂) = (0, 1). However, it also shows that for large enough η₁, increasing η₂ will not always result in an increased fixation probability. In the extreme case, where η₁ = 1, increasing η₂ initially increases the fixation probability, but eventually this fixation probability decreases as η₂ is closer to 1.

Fig. 5.

A stackplot demonstrating the fixation probability as a function of the de-differentiation probability η₂ for various choices of η₁. The solid line is the calculated fixation probability and the dashed lines indicate one deviation by the standard error of the mean.

Fig. 7.

The mean fixation probability as a function of η₂ with the mean taken over the parameter η₁ assuming a uniform distribution. The dashed bars indicate a single deviation by standard error of the mean.

This result can be seen more clearly in Fig. 5, where the fixation probability is plotted for constant values of η₁ as a function of varying η₂. When η₁ = 0 the fixation probability is a concave down, increasing, saturating function of η₂. However, for η₁ = 0.2 the fixation probability is no longer strictly increasing and undergoes a change in concavity early on. As η₁ is increased further the fixation probability is no longer a strictly increasing function of η₂ but initially increases and eventually decreases as η₂ becomes larger. For larger η₁ the fixation probability function decreases over a larger range of η₂ and decreases more drastically. Conversely, as η₁ is increased, the average fixation probability of the mutant-type cells is increased. That is, for η₁ = 0.8, the maximum of ρ(0.6, η₂) is around 0.045 (or 4.5%), but the maximum of ρ(1, η₂) is closer to 0.075 (or 7.5%). This can be recovered by recognising the saddle-like behaviour of Fig. 4. This symmetry can be explained similarly, as η₁ increases, more wild-type differentiated cells are de-differentiating into stem-cells. This removes selective pressure from mutant-type differentiated cells, allowing these mutant-type cells to grow larger and, for appropriate values of η₂, de-differentiate back into mutant-type stem cells.

This non-monotonic fixation probability function is an interesting phenomenon. This implies that for particular parameter sets, it is actually worse for mutant type stem cell fixation to increase the rate at which mutant differentiated cells de-differentiate into mutant stem cells. This is non-obvious, as one would assume increasing the pool of mutant stem cells, by decreasing the pool of mutant differentiated cells, is vital to the fixation of the mutant stem cells. However, by decreasing the pool of mutant differentiated cells, the wild type differentiated cells have a selective pressure removed and can see an increased response in growth. For large enough η₁, these wild type differentiated cells can de-differentiate back into wild type stem cells and thus provide a selective disadvantage to the mutant stem cells. This is an important phenomenon to fully understand. Much of the results of Section 4.1 is to indicate that this phenomenon persists over a wide range of parameter space and is not the result of a particular, lucky set of parameters.

The original paper by Mahdipour-Shirayeh et al. (2017) considered only a particular constant η₁ value while allowing η₂ to vary. Thus far we have considered various constant η₁ values and noted interesting dynamics occurring. In the paper by Wodarz (2018), they considered varying η₁ as a linear function of η₂ (in their particular case, η₁ was identically η₂). In the next set of experiments we considered varying η₁ as other linear functions of η₂.

In the first plot in Fig. 6, we note that, compared to the η₁ = η₂ case, in general the effect is a decrease in fixation probability for η₂ smaller than, approximately, 0.55, and an increase in fixation probability as η₂ reaches its maximum. In this scenario, the initial decrease in fixation probability is intuitive. A more plastic wild-type compartment would allow more wild-type stem cells to be created and hence result in a selective pressure on mutant-type stem cells, resulting in less probable fixation for the mutant cells. However, when η₂ is sufficiently large, than the difference between η₂ and η₁ is less pronounced. This is similar to the theme observed in Fig. 5. For large enough values of η₂ the selective pressures from the more plastic wild-type stem cells are reduced. Moreover, in these scenarios the smaller de-differentiation value compartment has a selective advantage in production of differentiated cells, and both differentiated and stem cell compartments are necessary for fixation.

Fig. 6.

The effects on fixation probability by varying η₁ as a function of η₂.

In the second plot in Fig. 6, we note that in general for η₁ smaller than η₂ we, mostly, observe that the fixation probability is increased (compared to the η₁ = η₂ case). This is intuitive, when η₁ is smaller, wild-type stem cells are being produced less quickly than mutant-type stem cells. Hence mutant-type stem cells have a selective advantage for invading. However, when η₂ is near 1.0 we observe a very slight dip in fixation probability. In these scenarios, the mutant cells are almost always de-differentiating. Hence the S₂ compartment is filling faster while the D₂ compartment is draining faster. Even though η₁ < η₂, η₁ is still appreciably non-zero, hence the D₁ compartment has a selective advantage due to the small size of the D₂ compartment. Therefore, both wild types perform well and fixate with slightly more frequency.

In the third and fourth plots of Fig. 6, the situation behaves similarly to the first two plots. When η₁ > η₂ there is an initial advantage for the wild-type cells that disappears as η₂ becomes sufficiently large. Contrarily, for η₁ < η₂ there is an initial advantage for the mutant-type cells that disappears only when η₂ becomes sufficiently large. What is unique about these two cases is the loss of monotonicity. In each experiment in the third plot of Fig. 6, increasing the plasticity parameter η₂ also increases η₁, but in this experiment η₁ initially increases fast enough that the selective disadvantage for wild-type cells imposed by larger η₂ is offset by the selective advantage for wild-type cells granted by larger η₁. However, past a certain point the benefit in fixation success for the wild-type cells as a function of η₁ begins to saturate and become outpaced by the selective advantage imposed by larger η₂. The fourth plot behaves in a similar fashion to the third plot. For smaller η₁ initially the fixation probability increases. As η₂ first begins to increase, η₁ increases too but at a slower rate. The effect of this initial slow increase in η₁ is offset by the growing of η₂, hence the fixation probability function is positively sloped. As η₂ becomes larger yet, the increase in η₁ starts to effect the evolutionary dynamics and the fixation probability starts to decrease.

When examining Fig. 4, it appears that in general the fixation probability increases. Moreover, when the fixation probability is increasing, it obtains a larger value. That is, for increasing η₁ the fixation probability is, in general, decreasing. This effect raises the question that in general, in a heterogeneous population of wild-type stem cells where deviations in de-differentiation probability η₁ can be observed, what might be the expected effect on the fixation probability. To that end, we modeled the average fixation probability as a function of η₂ averaged over all values of η₁. That is,

$$\langle \rho \left({\eta }_2\right)\rangle ={\int }_0^1\rho \left({\eta }_1,{\eta }_2\right)f\left({\eta }_1\right){\dot{\eta }}_1$$

where f(η₁) is the probability density function for η₁ in this case taken to be uniform. The result, as presented in Fig. 7, is a non-monotone function of η₂ with a unique maximum, suggesting that in general, increasing the plasticity rate of the mutant-type stem cell will not always result in increased fixation probability, but may be harmful to the mutant-type cells.

4.1. Stability analysis

The previous results were all obtained for the neutral mutant. To demonstrate the generality of these results and their independence on particular parameter choices, we repeated the analysis for varying parameters. In particular, we varied the relative fitness of the mutant-types, $r_2={\tilde{r}}_2$, and the probability of differentiation of the mutant types, u₂. The technical details can be found in Appendix A.

The results of Fig. 8 show that for general mutant increasing the relative fitness increases the fixation probability. For cases where the relative fitness is lower, the de-differentiation parameter η₂ is more important. It is only for relatively larger relative fitness levels that these fixation probability curves begin to demonstrate saturation and diminishing returns as a response to increased mutant cell plasticity. In these cases, saturation is reached quicker for larger $r_2={\tilde{r}}_2$ values. Thus even small values of η₂ are sufficient to maximise the chances of fixation.

Fig. 8.

Average fixation probability for various relative fitness levels of the mutant-type cells, $r_2={\tilde{r}}_2$. The dashed lines indicate one deviation of the standard error of the mean. The average was taken over wild-type de-differentiation probability η₁.

However, for relative fitness values that are closer to neutral (i.e. $r_2={\tilde{r}}_2=0.75$) we recover the non-monotone behaviour. That is, when there is no substantial advantage between the wild-type and mutant-type stem cells, fixation probability is more complicated than just maximising the plasticity. In these cases, as seen in Section 4, the behaviour of the stem cells is governed by competing evolutionary pressures and results in a greater need to balance parameters than to maximise parameters.

In Fig. 9 the probability parameter u₂ was varied. This parameter corresponds to the probability a mutant stem cell will differentiate. These results corroborate the claim that the non-monotonicity of the fixation probability function is a general phenomenon and not a result of careful parameter choice. For smaller u₂ values the fixation probability function is more right-skewed and for larger u₂ values the function is more left-skewed. In fact, for sufficiently large enough u₂ values the resulting function is monotone increasing.

Fig. 9.

Average fixation probability for various values of u₂, the probability the mutant type stem cell differentiates. The dashed lines indicate one deviation of the standard error of the mean. The average was taken over wild-type de-differentiation probability η₁.

The left-skew of the small u₂ functions indicates that when the mutant cells do not differentiate often, increasing the de-differentiation parameter is worse for fixation. This could be because in order for the wild-type cells to fixate they must selectively eliminate all mutant stem cells and differentiated cells. Hence, if there are few mutant differentiated cells to begin with, removing them from the micro-environment provides a selective advantage to the wild-type cells. When u₂ is larger, there are enough differentiated cells in the micro-environment that allowing some to de-differentiate (equivalent to 0 < η₂ < 1) provides a selective advantage for the mutant-type cells. However, allowing too many mutant differentiated cells to de-differentiate (equivalent to η₂ ≈ 1) results in the differentiated cell pool depleting too quickly and the resultant selective advantage for the wild-type cells provides a selective disadvantage to the mutant-type cells. For sufficiently large u₂ as in the plots for u₂ = 0.8 and u₂ = 1, mutant stem cells are almost always de-differentiating, hence the only way to refill the mutant stem cell pool is to increase the rate of de-differentiation. For these scenarios the bottle-neck for fixation is recovering enough stem cells to create a selective disadvantage on the wild-type cells. Hence, increasing η₂ is always the most evolutionary effective means of increasing fixation.

4.2. Monotone response curves

The non-monotonicity observed in Fig. 4, among others, can be seen as arising from the competing reactions $D_i{\to }^{{\tilde{r}}_i{\eta }_i}{S}_i+D_i$ and $D_i{\to }^{{\tilde{r}}_i\left(1-{\eta }_i\right)}{D}_i+D_i$. Both reactions are beneficial to the species, as increasing both the differentiated and stem cell compartment is necessary to achieve fixation. As such, one could imagine the non-monotonicity as arising out of an implicit choice being made to further fixate by either creating a new stem cell, with the expectation that this will result in later increases in the differentiated cell compartment, or, instead, by creating a new differentiated cell immediately. This, evidently, is a process unique to the Moran model outlined in Fig. 1a that has no obvious analogue in the continuous Gillespie model outlined in Fig. 1b. In particular, in the continuous Gillespie model differentiated cells are not permitted to proliferate, and while the two models produce equivalent dynamical systems, up to a change of parameters, the particulars of the implementation of both are not equivalent due to the differing process by which the models enforce selective pressure (in particular, the selective pressure from the Moran model is a result of fixed population sizes whereas in the Gillespie model selective pressure is governed by feedback on reaction rates). As a result, the continuous Gillespie model cannot achieve non-monotone fixation probability curves.

To illustrate how this behaviour is extended by the discrete Moran model, consider setting η_i = 1 and using ${\tilde{r}}_i$ as the plasticity parameter. In this context the reaction that allows differentiated cells to proliferate is removed, and de-differentiation is governed by the parameter ${\tilde{r}}_i$. Fig. 11 demonstrates that this process results in a fixation probability curve $\rho \left({\tilde{r}}_1,{\tilde{r}}_2\right)$ that is strictly increasing in ${\tilde{r}}_2$, strictly decreasing in ${\tilde{r}}_1$, and strictly decreasing for ${\tilde{r}}_1={\tilde{r}}_2=\tilde{r}$ (as was observed in Wodarz (2018), seen more clearly in Fig. 12).

Fig. 11.

A contour plot demonstrating the fixation probability as a function of both plasticity parameters when differentiated cell propagation has been removed. The plot indicates that increasing the mutant-type de-differentiation parameter always increases the fixation probability of mutant-type cells. Likewise, decreasing the wild-type de-differentiation parameter increases the fixation probability of mutant-type cells. In contrast with Figure c 4, no saddle point is observed.

Fig. 12.

A plot indicating that when mutant and wild-type plasticity parameters are kept coupled increasing this plasticity parameter decreases the fixation probability of the mutant-type cells. Dashed lines indicate a difference of one standard error of the mean.

Similarly, in Wodarz (2018), the author introduces a model using transit amplifying cells (Fig. 10). In this model, stem cells either proliferate on their own, or differentiate into an intermediary state called transit amplifying cells. Transit amplifying cells then either de-differentiate back into stem cells, or fully differentiate into differentiated cells. The only reaction differentiated cells can undergo is death. In particular, note that this model includes similar competition, based on the plasticity parameter q_i, between reactions $T_i{\to }^{r_{2,i}\left(1-p_{2,i}\right)q_i}{S}_i+S_i$ and $T_i{\to }^{r_{2,i}\left(1-p_{2,i}\right)\left(1-q_i\right)}{D}_i+D_i$. However, since differentiated cells, in this model, do not provide a selective advantage to the species, only one of these reactions directly increases the evolutionary fitness of the species. As a result, varying the plasticity parameter q results in strictly monotone response curves.

Fig. 10.

A model where stem cells first differentiate into transit amplifying cells before fully differentiating. Transit amplifying cells can then govern the de-differentiation process.

In effect, this demonstrates that when differentiated cells are not allowed to proliferate increasing the plasticity of mutant-type cells (or, equivalently, decreasing the plasticity of wild-type cells) results in an increase in fixation probability of the mutant-type cells. This is analogous to the results observed in Mahdipour-Shirayeh et al. (2017) when η₁ = 0.

5. Conclusion

In conclusion, two, apparently contradictory, stochastic models of cancer stem cell behaviour were considered. Both models were questioning the invasive capacity of mutant stem cells in the presence of a pre-existing stem cell and differentiated cell strata. The models were concerned with how the phenotypic plasticity, or de-differentiation rate, affected the invasive potential of these mutant stem cells. A Moran birth–death model was considered and Gillespie simulations of a reaction network were considered. The apparent discrepancy between these models was discovered to be hidden within assumptions determining how both the resident type and mutant type plasticity rates alter. This apparent contradiction was resolved and the behaviour of the Gillespie simulation model was recovered in the Moran context, and vice versa.

It was also observed that increasing plasticity in the mutant type differentiated cells can result in both positive and negative effects for the fixation probability of the mutant stem cells, as long as differentiated cells are allowed to proliferate at some nonzero rate. This seeming contradiction was demonstrated to be stable across differing parameter sets indicating that this phenomenon may be observed in general. Further, when differentiated cells are not permitted to proliferate, then the system is entirely dependent on increasing the fixation efforts of the stem cells. Hence, when differentiated cells are not permitted to proliferate, increasing mutant plasticity results in increased mutant fixation probability. The continuous Gillespie model of (Wodarz, 2018) is an example of a stem cell model where non-monotone response curves can not be realised as a result of this inability for differentiated cells to proliferate. This elucidates a fundamental difference between the models of Mahdipour-Shirayeh et al. (2017) and Wodarz (2018); namely that since the former allows differentiated cells to proliferate, competitive behaviour between stem and differentiated compartments is observed that the latter model does not experience. These results indicate that while the initial contradiction between the two models has been resolved in this report further differences between the two have been revealed, and explained, as a result of differing assumptions regarding the modeling of differentiated cell populations. Moreover the discrete Moran model was shown to be able to recover the non-monotone response curve behaviour of the continuous Gillespie model under a suitable change of parameters.

Appendix A. Numerics

The Moran simulations were run until fixation occurred or until 15,000 iterations were achieved – whichever came first. A total of 500,000 such simulations were run in 50 batches of 10,000. For each batch of 10,000 simulations, fixation probability was estimated by logging what percentage of the 10,000 iterations achieved fixation. The error of these probabilities was estimated as the standard error of the mean calculated between the 50 different batches. This process was completed once for every unique set of parameter values.

Similarly, the two Gillespie simulations (in Figs. 2 and 3) were run until fixation of or until 10⁸ iterations were achieved. For each parameter set, these simulations were run in 10 batches of 3000 simulations with the fixation probability being calculated for each individual batch of 3000 simulations. The error was calculated from these 10 different fixation probabilities as the standard error of the mean.

A.1. Figure information

In Fig. 2a the plot was generated by Moran simulations with the following parameters: N_S = N_D = 10, u₁ = u₂ = 0.5, $r_1=r_2={\tilde{r}}_1={\tilde{r}}_2=1$, η₁ = 0, $d_1=d_2={\tilde{d}}_1={\tilde{d}}_2=1$, and η₂ varying over 36 discrete values evenly placed between 0 and 1, inclusive.

Similarly, for Fig. 2b was generated by Gillespie simulations (as in Wodarz (2018)) with the following parameters: $r_1^{\prime }=r_2^{\prime }=0.5$, $p_1^{\prime }=p_2^{\prime }=0.8$, h_1,1 = h_2,1 = h_1,2 = h_2,2 = h_1,3 = h_2,3 = 0.01, k₁ = k₂ = k₃ = 1, α₁ = α₂ = 0.5, and $g_1^{\prime }=g_2^{\prime }=g$. The initial condition for these simulations was $\left(S_1,D_1,S_2,D_2\right)=\left(S_1^{*},D_1^{*},1,0\right)$ where $S_1^{*}$ and $D_1^{*}$ are defined as (within rounding to nearest integer) the equilibrium point of the deterministic differential equations in S₁ and D₁ (where S₂ and D₂ are assumed to be zero for the purpose of obtaining the equilibrium values) and vary depending on the value of g. The stochastic simulation was then repeated for each g value in {0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3} corresponding to initial conditions $\left(S_1^{*},D_1^{*}\right)\in \{(114,68),(132,75),(101,62),(166,88),(183,94),(200,100)\}$. Note that while the Moran simulations in Figure c 2a use finite, discrete population sizes, the simulations in Figure c 2b make no such assumptions, allowing continuous, unbounded population sizes.

In Fig. 3 the leftmost plot was generated by Moran simulations with the following parameters: N_S = N_D = 10, u₁ = u₂ = 0.5, $r_1=r_2={\tilde{r}}_1={\tilde{r}}_2=1$, $d_1=d_2={\tilde{d}}_1={\tilde{d}}_2=1$, and η₁ = η₂ = η where η varies over 36 discrete values evenly placed between 0 and 1, inclusive.

Similarly, the rightmost plot was generated by Gillespie simulations with the following parameters: $r_1^{\prime }=r_2^{\prime }=0.5$, $p_1^{\prime }=p_2^{\prime }=0.8$, h_1,1 = h_2,1 = h_1,2 = h_2,2 = h_1,3 = h_2,3 = 0.01, k₁ = k₂ = k₃ = 1, α₁ = α₂ = 0.5, $g_1^{\prime }=0$, and $g_2^{\prime }$ taking values in {0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3}. As in Fig. 2, the initial condition was taken to be $\left(S_1,D_1,S_2,D_2\right)=\left(S_1^{*},D_1^{*},1,0\right)$ with $S_1^{*}$ and $D_1^{*}$ defined as the equilibria points of the (S₁, D₁) system calculated similarly to the above.

In Fig. 4 the data was generated with Moran simulations with the following parameters: N_S = N_D = 10, u₁ = u₂ = 0.5, $r_1=r_2={\tilde{r}}_1={\tilde{r}}_2=1$, $d_1=d_2={\tilde{d}}_1={\tilde{d}}_2=1$, and both η₁ and η₂ vary, independently, from 0 to 1 in 0.1 increments. A cubic spline was generated from the resulting data points (and their standard errors) and used to create the plots in Figs. 5 and 6.

In Fig. 7 the data was generated, for each fixed η₂ value, by averaging over all 36 η₁ values.

Figs. 8 and 9 were generated similarly. In both figures Moran simulations were run with parameters N_S = N_D = 10, $d_1=d_2={\tilde{d}}_1={\tilde{d}}_2=1$, and η₁ and η₂ independently varying over 36 discrete values evenly placed between 0 and 1, inclusive. In Fig. 8 $r_2={\tilde{r}}_2=r$ where r takes on values in {0.25, 0.50, 0.75, 1, 2, 3, 4, 5} while u₁ = u₂ = 0.5. Similarly, in Fig. 9 u₂ was allowed to vary between 0.1 and 0.9 inclusive in increments of 0.1, while $r_2={\tilde{r}}_2=1$ and u₁ = 0.5 were kept constant. The averages were calculated from the resulting data points as in Fig. 7.

Figs. 11 and 12 were generated with Moran simulations run with N_S = N_D = 10 where η₁ = η₂ = 1 and ${\tilde{r}}_1$ and ${\tilde{r}}_2$ were varied over 36 discrete values evenly placed between 0 and 1. For Fig. 11, the values of ${\tilde{r}}_1$ and ${\tilde{r}}_2$ were varied independently, for Fig. 12 they were varied together. The data were generated by averaging the results of 100 simulations 5000 times. The remaining parameters were set as r₁ = r₂ = 1, u₁ = u₂ = 0.5.