Abstract
Many fast renewing tissues are characterized by a hierarchical cellular architecture, with tissue specific stem cells at the root of the cellular hierarchy and differentiating into a whole range of specialized cells. There is increasing evidence that tumors are structured in a very similar way, mirroring the hierarchical structure of the host tissue. In some tissues, differentiated cells can also revert to the stem cell phenotype, which increases the risk that cells that have already acquired mutations lead to long lasting clones in the tissue. Recently, the modelling community has paid special attention to the consequences of de-differentiation on cellular hierarchies. However, the adaptive significance of de-differentiation is still poorly understood and thus it is unclear under which circumstances de-differentiating cells will invade a tissue. To address this, we developed mathematical models to investigate how de-differentiation could be selected as an adaptive mechanism in the context of cellular hierarchies. We consider the cases of stepwise and jumpwise de-differentiation in this study. Our results show that the emergence of de-differentiation is driven by the combination of the properties of the cellular hierarchy and the de-differentiation pattern and derive thresholds for which de-differentiation is expected to emerge.
Introduction
In multicellular organisms, it is important that the inevitable replication errors of cells do not persist and threaten the functioning of the organism as a whole. Many tissues that need to undergo continuous cell turnover are organized in a hierarchical multi-compartment structure, which reduces the risk of the persistence of such mutations [1–13]. Each compartment represents a certain stage of cellular differentiation (Fig 1). At the root of the cellular hierarchy are tissue specific stem cells (SCs) which are capable of self-renewal and differentiation into more mature cells [14]. It is often argued that cancers may have similar hierarchical structures, where cancer stem cells (CSCs) possess characteristics associated with SCs in normal tissues [14, 15]. The CSCs scenario assumes that some cancerous tissues are hierarchically organized, similar to normal tissues [16].
The hierarchical tissue architecture proposes a unidirectional cascade from less differentiated stages to more differentiated stages (Fig 1 a). This would minimize the risk of the accumulation of genomic damage in the long term self renewing stem cells. However, there is significant evidence that the directional relation between different stages of differentiation could be broken in some tissues [17–22]. In other words, cells in later differentiated stages can, under some circumstances, revert to earlier differentiated stages, or even the stem cell stage, in a process known as de-differentiation (Fig 1b and c). De-differentiation could play an important role in regeneration and tumorigenesis [17]. In particular, even though the origin of cancer stem cells is still an open question, growing evidence shows that non-stem cancer cells can reacquire stem-like characteristics in colon cancer [23], breast cancer [20, 21], melanoma [24], leukemia [25–28], glioblastoma [29], and other cancers. For example, expression of the MLL-AF9 gene in committed hematopoietic progenitor cells led to the development of a leukemic stem cell population where only four of these cells were able to result in disease in a mouse model that could be transferred from one mouse to another, confirming the presence of a stem cell population [27].
More recently, special attention has been paid to the effect of de-differentiation on the cellular hierarchy by mathematical modeling of their impact. Previous works have considered how de-differentiation influences the waiting time to carcinogenesis [30], the fixation probability of a mutant [31, 32], the phenotypic equilibrium [33–35], transient overshoots [36, 37], and radiation sensitivity [29]. However, the adaptive significance of de-differentiation is still poorly understood: Under which circumstances would de-differentiation arise in the first place and rise in abundance? It is still unclear whether de-differentiation is a crucial improvement or just an unintended consequence of cellular hierarchy. A related problem is why de-differentiation arises in only some tumors, but not in others. Moreover, the comparison between different patterns of de-differentiation has received little attention.
Here we develop a matrix population model [38] of a stage-structured population for studying the evolution of de-differentiation. Two typical de-differentiation cases are taken into account in our model: One is stepwise de-differentiation which happens from a downstream compartment to an adjacent upstream compartment (Fig 1b), the other is jumpwise de-differentiation which is directly from a highly differentiated compartment to the stem cell compartment without there being in intermediate stages (Fig 1 c). Given hierarchically structured multi-compartment cell population, we are concerned about the selection of stepwise or jumpwise de-differentiating mutant cell population in the competition with non de-differentiating resident cell population. By comparing the growth rates of different cell populations, we analyze the favorable conditions for different de-differentiation patterns to invade a tissue. We hope that our work contributes to the theoretical understanding of the emergence of de-differentiation in multicellular tissues.
Methods
The matrix population model for cellular hierarchy
Consider a cell population composed of n compartments, each of which represents a certain stage of differentiation [10, 13] (Fig. 1). For example, compartment 1 represents stem cells, and compartment n represents terminally differentiated cells. Each cell in compartment i (1 ≤ i ≤ n − 1) divides at rate ri. With probability pi, it divides symmetrically, giving birth to two identical cells in compartment i (Fig. 1 d). With probability qi, it differentiates symmetrically, generating two identical daughter cells in compartment i + 1. It should be pointed out that, for simplicity, asymmetric division [39, 40] (giving birth to one daughter cell in compartment i and the other in compartment i + 1) is not taken into account here, but our approaches and results are still applicable for the models with asymmetric division. The terminally differentiated cells in compartment n cannot divide and are removed from the tissue at rate d.
We use the vector to denote the cell numbers in different compartments. Then, the hierarchically structured population dynamics composed of non de-differentiating cells can be described as a matrix population model [38] where A0 is the projection matrix which is given by
Here ri(pi − qi) represents the effective self-renewal rate of compartment i, and 2riqi represents the influx rate from compartment i to compartment i + 1 due to differentiation. Let be the total cell number of the population. Note that A0 is an essentially non-negative (all the off-diagonal elements are non-negative [41]) and lower triangular matrix. According to the standard theory of matrix population models [38], the population grows exponentially in the long run, i.e. where λ0 is the real leading eigenvalue. The leading eigenvalue hence characterizes the asymptotic growth rate of the whole population, which is often used as a measure of fitness in matrix population models [42, 43]. Here, we use this fitness measure to assess whether a mutant can invade a resident population.
Stepwise and jumpwise de-differentiation
Let us now introduce de-differentiation processes given the non de-differentiating resident cell population Eq. (1). Since it is biologically unclear how a non de-differentiating resident cell acquires the ability for de-differentiation, here we consider de-differentiation as a result of certain genetic or epigenetic alterations (jointly referred to as mutations). It is assumed that the mutant cells are provided with the additional ability of de-differentiation. More specifically, when these mutant cells divide, besides symmetric division and symmetric differentiation, they can also perform symmetric de-differentiation (Fig. 1 d) with a small probability. In principle, there are two different ways to do this: (i) stepwise de-differentiation, where cells de-differentiate to the previous compartment, and (ii) jumpwise de-differentiation, where de-differentiation happens across multiple compartments at a time. These are the most extreme cases and a mixture between them is possible.
For stepwise de-differentiation, a mutant cell in compartment i gives rise to two daughter cells in its adjacent upstream compartment i − 1 (Fig. 1 b) when de-differentiation happens. Suppose that the de-differentiation probability from compartment i to i − 1 is δi. Then the influx rate from compartment i to i − 1 due to de-differentiation is given by 2riδi. When de-differentiation is taken into account, the self-renewal probability of each mutant cell in compartment i becomes , and its differentiation probability becomes , such that we have . Here, the parameter κ (0 ≤ κ ≤ 1) governs how mutant cell redistributes the probabilities for self-renewal and differentiation when taking de-differentiation into account. We call κ the redistributing factor. De-differentiation is generally a rare event [21], we thus assume that ρi = 2riδi ≪ 1. As the occurrence of de-differentiation for different stages of differentiation is poorly understood, for simplicity we assume that all the ρi are the same, i.e. they are independent of index i and denoted as ρ for short. In this way, the population dynamics of the stepwise de-differentiating mutant population can be modeled with a projection matrix given by
Jumpwise de-differentiation provides an alternative pattern where even highly differentiated cells can directly revert to stem cells without being in intermediate stages (Fig 1 c). Formally, it is assumed that the jumpwise de-differentiating mutant cell in compartment n − 1 give birth to two daughter stem cells in compartment 1 (Fig 1d). Therefore, the projection matrix is given by
Selection gradient for de-differentiation
In the following, we consider the competition between a non de-differentiating resident cell population and a stepwise de-differentiating mutant cell population (which is called S mutant cell population for short), as well as between a non de-differentiating resident cell population and a jumpwise de-differentiating mutant cell population (which is called J mutant cell population for short) by comparing their fitness measures, i.e. the leading eigenvalues λ0, λS and λJ of A0, AS and AJ, respectively. Note that ρ is very small, such that both AS and AJ can be seen as matrix perturbations to A0. According to the eigenvalue perturbation method [44], we have
Here, ΔλS and ΔλJ are given by where and are the left and right eigenvectors associated with λ0 respectively (see Supplementary Information).
For a given parameter set (ri, pi, qi, d, κ), ΔλS characterizes the selective difference between an S mutant cell population and a non de-differentiating cell population. If ΔλS > 0, for example, the S mutant population is favored in this competition — a non de-differentiating resident cell population is invaded by an S mutant cell population. Therefore, ΔλS corresponds to a selection gradient and acts as a comparative fitness measure of the S mutant cell population relative to the non de-differentiating resident cell population. A similar argument also applies for ΔλJ. We thus term ΔλS and ΔλJ as selection gradients of the S mutant cell population and the J mutant cell population, respectively. Based on these quantities, we will analyze the favorable condition for de-differentiation.
Results
We infer whether de-differentiation leads to an increased fitness in the different cases (stepwise and jumpwise), both analytically and numerically.
Let us first focus on the null model without de-differentiation. In this case, the projection matrix A0 is a lower triangular matrix whose eigenvalues are just the diagonal elements. Suppose that the resident cell population in Eq. (1) is not shrinking, which implies that there exists at least one non-negative diagonal element in A0. In this way, the leading eigenvalue λ0 is the largest among all the non-negative diagonal elements of A0. Note that −d is always negative, such that λ0 is always in the form of rj0 (pj0 − qj0), where j0 is the compartment that maximizes this quantity.
Next, we turn to stepwise de-differentiation, cf. Eq. (4). Given λ0 = rj0 (pj0 − qj0), the selection gradient (comparative fitness) of an S mutant cell population is given by (see Supplementary Information) where . In reality, cells in different compartments do not have exactly the same effective self-renewal rate, because of internal or external noise in cellular dynamics [36]. It is thus reasonable to assume that the leading eigenvalue λ0 is unique, which implies that rj0 (pj0 − qj0) is strictly larger than any other rj (pj − qj) for j ≠ j0
Thus, all the гj,k,l in Eq. (8) are positive. In particular, for j0 = 1, ΔλS = г1,1,2 is positive. In other words, an S mutant cell population is always favored in the competition with non de-differentiating resident cell population provided that stem cells have the largest effective self-renewal rate among all cell compartments. We performed exact numerical solutions to verify our theoretical approximation and find a very good agreement, see Fig. 2. Furthermore, with the increase of symmetric division probability of stem cells (p1), the selection gradient ΔλS gradually tends to zero (Fig. 2).
Let us now consider the cases with j0 > 1. In reality, stem cells may have slower cycling time than more differentiated cells [45], or stem cell differentiation and self-renewal could be balanced (p1 = q1 = 0.5), maintaining a stable stem cell population [10]. Thus, sometimes it is more likely for other cell compartments (j0 > 1) to have the largest effective self-renewal rate. From Eq. (8) we can see that ΔλS is a linear combination of гj,k,l and κ. It is interesting to see that ΔλS is negatively correlated with κ. Note that κ is the redistributing factor that characterizes how the introduction of de-differentiation reshapes the probabilities for self-renewal and differentiation. For κ = 0, ΔλS is positive. With an increase of κ, ΔλS could become negative. Fig. 3 typically shows two scenarios of ΔλS: either it is always larger than zero for any κ, or it changes from positive to negative at some critical point 0 < κ* < 1. For the latter scenario, Fig 4 illustrates how ΔλS changes with both the redistributing factor k and the symmetric division probability p2 provided that compartment 2 has the largest effective self-renewal rate (j0 =2). It is shown that with the increase of p2, the critical value κ* decreases, which means it is getting less likely for the S mutant cell population to be favored. Note that гj,k,l represents the effect of cellular hierarchy on de-differentiation, and κ represents how de-differentiation reshapes the cellular division patterns. Therefore, the selection of de-differentiation is a combined result of cellular hierarchy and de-differentiation pattern.
We now turn our attention to the selection gradient (comparative fitness) of the J mutant cell population, which is given by (see Supplementary Information for mathematical details)
Similar to Eq. (8), here all the гj,k,l in Eq. (9) are positive. For the cases with 1 ≤ j0 < n − 1, in particular, ΔλJ is always positive (Fig. 5), i.e. the J mutant cell population is favored in all these cases. On the other hand, for j0 = n − 1, ΔλJ is negatively correlated with the redistributing factor κ which is similar to ΔλS (Fig. 6).
A comparison between Eqs. (8) and (9) reveals some important differences between stepwise and jumpwise de-differentiation patterns. First of all, jumpwise de-differentiation provides a much wider range of favorable condition for de-differentiation than stepwise de-differentiation in the sense that ΔλJ is always positive for any 1 ≤ j0 < n − 1, but ΔλS is always positive only for j0 = 1. Secondly, ΔλS only depends on the parameters related to the neighborhood compartments of j0, but ΔλJ depends on the parameters related to all compartments, ranging from the stem cell stage to the stage where de-differentiation occurs. This implies that, the total number of compartment does matter in the jumpwise case, but not in the stepwise case. In other words, stepwise de-differentiation utilizes the local structure around the compartment with the largest effective self-renewal rate, whereas jumpwise de-differentiation utilizes the global structure throughout the multi-compartment hierarchy.
Discussion
In this study, we have explored the adaptive significance of de-differentiation in hierarchical multi-compartment structured cell populations. By establishing a matrix population model, we study the competition between resident hierarchical structured cell populations without de-differentiation and mutant cell populations with different modes of de-differentiation.
In principle, there are two main factors that could influence the selection of de-differentiation: cellular hierarchy and the de-differentiation pattern. Cellular hierarchy refers e.g. to the number of cell compartments, the inherent cell division pattern, or the cell division rate. These correspond to the parameter landscape of (n, pi, qi, ri) in our model. The de-differentiation pattern refers to different modes of de-differentiation (stepwise or jumpwise), as well as how de-differentiation reshapes the division pattern in the cellular hierarchy (corresponding to κ in our model). Interestingly, our results show that the selection gradients for de-differentiation (ΔλS and ΔλJ) can generally be decomposed into a sum of a cellular hierarchy part and a de-differentiation part, showing that the selection of de-differentiation is a result of the linear combinations of these two factors.
Among all factors in the cellular hierarchy, the most important one is which of the cell compartments has the largest effective self-renewal rate. This is because de-differentiation is more likely to be favored when earlier compartments have the largest effective self-renewal rate. For example, in the stepwise case, de-differentiation is favored provided that the stem cell has the largest effective self-renewal rate. In the jumpwise case, de-differentiation is favored in all cases except when the latest divisible cell compartment has the largest effective self-renewal rate.
Given all the factors in the cellular hierarchy, we are most concerned about how different de-differentiation patterns shape the evolution of de-differentiation. In particular the redistributing factor, i.e. the effect of dedifferentiation on self-renewal and differentiation probabilities greatly influences the selection condition. Our results suggest that de-differentiation is more likely to be favored if there is less effect on self-renewal than on differentiation. In addition, the de-differentiation mode (stepwise or jumpwise) has enormous implications for the selection condition. Our results suggest that de-differentiation is more likely to be favored in the jumpwise case than in the stepwise case. However, jumpwise de-differentiation seems to be biologically much more difficult to achieve, the overall incidence of it would still be very low. Perhaps an example of the differences between stepwise and jumpwise de-differentiation and the implications of the subsequent disease behavior can be illustrated by various types of leukemia. As already mentioned, MLL-AF9 expression in a committed progenitor cells can lead to the development of leukemic stem cells that can result in disease transmission across mice [27, 46]. In general MLL expression is associated with a poor prognosis in acute myeloid leukemia [47, 48]. This may be an example of jumpwise de-differentiation. In contrast, acute promyelocytic leukemia (APL) is an example of acute leukemia that is highly curable [49]. It is therefore possible that in this disease, stepwise de-differentiation - or a situation where a mutant cell can stick in a compartment without differentiating, similar to a stem cell - is occurring that in part makes the disease still potentially curable.
Note that the presented study is based on a matrix population model with constant elements, which in principle captures an exponential growing population. Even though there are still uncertainties regarding the growth patterns of cell populations in different contexts (cancer or normal, solid or hematologic tumor, in vivo or in vitro) [7, 50] and exponential growth is often considered to be unable to capture the biological processes in reality, exponential-like growth models are widely used as default models to describe growing cell populations, especially in early cancer development [21, 51-53]. We followed this idea and used it as a starting point to explore the adaptive significance of de-differentiation. In the future, more complex biological mechanisms such as density-dependent population growth could be taken into account in models for de-differentiation. Moreover, while the hierarchical architecture of tissues is considered to have been selected to minimize the risk of retention of mutations, the risk of acquisition of stem cell line properties by the large population of progenitor cells introduces new dynamics - perhaps in such a scenario two additional considerations could reduce the risk of cancer - namely the low probability that specific mutations lead to acquisition of stem cell like behavior or the average survival of progenitor cells may be low enough to prevent the acquisition of the additional mutations needed to reach the full cancer phenotype. This could be an extension of this work in future.
Footnotes
↵* zhouda{at}xmu.edu.cn (DZ); traulsen{at}evolbio.mpg.de(AT)