^{1}

^{2}

^{1}

^{3}

^{2}

The authors have declared that no competing interests exist.

Many fast renewing tissues are characterized by a hierarchical cellular architecture, with tissue specific stem cells at the root of the cellular hierarchy, 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 mutant cells lead to long lasting clones in the tissue. However, 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 is selected as an adaptive mechanism in the context of cellular hierarchies. We derive thresholds for which de-differentiation is expected to emerge, and it is shown that the selection of de-differentiation is a result of the combination of the properties of cellular hierarchy and de-differentiation patterns. Our results suggest that de-differentiation is most likely to be favored provided stem cells having the largest effective self-renewal rate. Moreover, jumpwise de-differentiation provides a wider range of favorable conditions than stepwise de-differentiation. Finally, the effect of de-differentiation on the redistribution of self-renewal and differentiation probabilities also greatly influences the selection for de-differentiation.

How can a tissue such as the blood system or the skin, which constantly produces a huge number of cells, avoids that errors accumulate in the cells over time? Such tissues are typically organized in cellular hierarchies, which induce a directional relation between different stages of cellular differentiation, minimizing the risk of retention of mutations. However, recent evidence also shows that some differentiated cells can de-differentiate into the stem cell phenotype. Why does de-differentiation arise in some tumors, but not in others? We developed a mathematical model to study the growth competition between de-differentiating mutant cell populations and non de-differentiating resident cell population. Our results suggest that the invasion of de-differentiation is jointly influenced by the cellular hierarchy (e.g. number of cell compartments, inherent cell division pattern) and the de-differentiation pattern, i.e. how exactly cells acquire their stem-cell like properties.

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 [

We illustrate our models by considering a four-compartment hierarchical structure. (_{1}. In each cell division, it can either give birth to two identical stem cells (self-renewal) with probability _{1} or two identical daughter cells in adjacent downstream compartment 2 (differentiation) with probability _{1}. Similar division pattern can also happen to cells in compartments 2 and 3 (with division rates _{2} and _{3} respectively). Compartment 4 represents terminally differentiated cells which cannot divide and are removed from the tissue at rate _{i}) in cell division, the self-renewal probability of each cell in compartment _{i} to _{i} − _{i}, while its differentiation probability is changed from _{i} to _{i} − (1 − _{i}. Here, we have introduced the redistributing factor _{3} to _{3} − _{3}, and its differentiation probability is changed from _{3} to _{3} − (1 − _{3}. (

The hierarchical tissue architecture proposes a unidirectional cascade from less differentiated stages to more differentiated stages (

More recently, special attention has been paid to the effect of de-differentiation on the cellular hierarchy by mathematically modeling its impact [

Here, we develop a matrix population model [

Consider a cell population composed of _{i}. With probability _{i}, it divides symmetrically, giving birth to two identical cells in compartment _{i}, it differentiates symmetrically, generating two identical daughter cells in compartment

We use the vector _{0} is the projection matrix which is given by
_{i}(_{i} − _{i}) represents the effective self-renewal rate of compartment _{i}_{i} represents the influx rate from compartment _{i}) into our model, the effective self-renewal rate of compartment _{i}(_{i} − _{i}), while the influx rate from compartment _{i}_{i} to 2_{i}_{i} + _{i}_{i}. We can see that the characteristics of matrix _{0}, such as essentially non-negativity (all the off-diagonal elements are non-negative [

Let _{0} is an essentially non-negative and lower triangular matrix. According to the standard theory of matrix population models [_{0} is the real largest eigenvalue. The largest eigenvalue hence characterizes the asymptotic growth rate of the whole population, which is often used as a measure of fitness in matrix population models [_{0} > 0, remain in homeostasis if λ_{0} = 0, or shrink if λ_{0} < 0. Here, we are interested in the cases when λ_{0} ≥ 0, i.e. we assess whether a mutant can invade an expanding or steady resident population by comparing their fitness measures. Besides, due to the intense inevitable internal and external noise in cellular dynamics [_{0} to have multiple eigenvalues [_{0} is unique (or simple).

Let us now introduce de-differentiation processes given the non de-differentiating resident cell population

For stepwise de-differentiation, a mutant cell in compartment _{i}. Then, the influx rate from compartment _{i} _{i}. We denote the self-renewal and differentiation probabilities of each mutant cell in compartment _{i}. Due to the current lack of knowledge regarding the effect of de-differentiation on the self-renewal and differentiation probabilities, there is no way to know how much the self-renewal probability or differentiation probability changes individually. In view of this, we introduce a parameter

It has been reported that de-differentiation is generally a rare event [_{i} = 2_{i}_{i} ≪ 1. As the occurrence of de-differentiation for different stages of differentiation is poorly understood, for simplicity we assume that all the _{i} are the same, i.e. they are independent of index

Jumpwise de-differentiation provides an alternative pattern where even highly differentiated cells can directly revert to stem cells without being in intermediate stages (

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 _{0}, λ_{S} and λ_{J} of _{0}, _{S} and _{J}, respectively. Note that _{S} and _{J} can be seen as matrix perturbations to _{0}. According to the eigenvalue perturbation theory (see e.g. Theorem 4.4 in [_{S} and λ_{J} are differentiable with respect to _{0} is simple. In this way, we have
_{S} and Δλ_{J} are given by
_{0} respectively (see

For a given parameter set (_{i}, _{i}, _{i}, _{S} characterizes the selective difference between an _{S} > 0, for example, the _{S} corresponds to a selection gradient and acts as a comparative fitness measure of the _{J}. We thus term Δλ_{S} and Δλ_{J} as selection gradients of the

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 _{0} is a lower triangular matrix whose eigenvalues are just the diagonal elements. Note that the resident cell population in _{0}. In this way, the largest eigenvalue λ_{0} is the largest among all the non-negative diagonal elements of _{0}. Note that −_{0} is always in the form of _{0} is the compartment that maximizes this quantity.

Next, we turn to stepwise de-differentiation, _{0} is unique, which implies that _{j}(_{j} − _{j}) for _{0}. Thus, all the Γ_{j,k,l} in _{0} = 1, Δλ_{S} = Γ_{1,1,2} is positive. In other words, an _{0} = _{1}(_{1} − _{1}) is positive. The other is for the populations at steady state (homeostasis), i.e. when λ_{0} is zero. We can see that the selection gradient Δλ_{S} is always positive, even though different patterns of function relation are present for left and right panels. That is, the stepwise de-differentiation always provides a fitness advantage, regardless of whether the resident cell populations are expanding or at steady state. Actually, this result is quite in line with biological intuition. Given that stem cells have the highest self-renewal potential, i.e. the self-renewal potential is gradually lost in the process of differentiation, de-differentiation effectively leads to a faster growth rate of the population.

Illustration of the selection gradient (comparative fitness) of the _{S} as a function of division rates and symmetric division probabilities, provided that the stem cell compartment has the largest effective self-renewal rate, i.e. λ_{0} = _{1}(_{1} − _{1}). In both panels, colored lines represent analytical approximations from _{1} = 0.99, _{3} = 0.3. (_{0} > 0). De-differentiation provides a fitness advantage for all values of _{1} and _{2}. Here _{2} = 0.55, _{3} = 0.6 and the range of _{1} (0.55 < _{1} < 1.0) ensures that _{1}(_{1} − _{1}) is the largest eigenvalue. (_{0} = 0). De-differentiation also provides a fitness advantage for all values of _{2} and _{2}. Here _{1} = 0.5, _{3} = 0 and the range of _{2} (0 < _{1} < 0.3) ensures that λ_{0} = 0 is the largest eigenvalue.

In general, stem cells are defined as having the greatest potential for long term self-renewal. There is also evidence that stem cells replicate slowly and therefore in many tissues it is the progenitor cells that lead to amplification and maintenance of tissues due to a process of replication, self-renewal and differentiation [_{0} > 1 in our model.

From _{S} is a linear combination of Γ_{j,k,l} and _{0} > 1. It is interesting to see that Δλ_{S} is negatively correlated with _{S} is surely positive. With an increase of _{S} could become negative. Hence, there are typically two scenarios of Δλ_{S}: either it is always larger than zero for any _{S} changes with _{0} = 2). In the expanding case (left panel) both of these two scenarios are present, whereas in the homeostasis case (right panel) Δλ_{S} is always larger than zero. Actually, when the population is at homeostasis, i.e. λ_{0} = _{2}(_{2} − _{2}) = 0, we can show that Γ_{1,2,1} is larger than 1 and note that Γ_{2,2,3} is always positive, then Δλ_{S} is shown to be positive for any 0 ≤ _{S} as a function of both _{2} in the scenario that Δλ_{S} can change from positive to negative. It is shown that with the increase of _{2}, the critical value _{j,k,l} represents the effect of cellular hierarchy on de-differentiation, and

Illustration of the selection gradient (comparative fitness) of the _{S} as a function of redistributing factor and division rates provided that λ_{0} = _{2}(_{2} − _{2}). In both panels, colored lines represent the eigenvalue perturbation results from _{0} > 0). In this case, there are two different scenarios: For _{S} is always positive (blue color); For _{1} > 0.1950, Δλ_{S} changes from positive to negative with the increase of _{1} = 0.5, _{2} = 0.95, _{3} = 0.55, _{2} = 0.44, and _{3} = 0.17. (_{0} = 0). In this case, Δλ_{S} is always positive. Here _{1} = 0.001, _{2} = 0.5, _{3} = 0.001, _{1} = 0.99, and _{3} = 0.8.

The curve represents the boundary with Δλ_{S} = 0, which is generated by the eigenvalue perturbation approximation from _{S} = 0. The parameters are _{1} = 0.0885, _{2} = 0.4145, _{3} = 0.5555, _{1} = 0.4723, _{3} = 0.0727,

We now turn our attention to the selection gradient (comparative fitness) of the _{j,k,l} in _{0} < _{J} is always positive, i.e. the _{0} = 1 and _{0} = 2. For each case, it is shown that Δλ_{J} is positive, regardless of whether the resident cell populations are expanding or maintaining homeostasis.

Illustrations of the selection gradient (comparative fitness) of the _{J} for the cases _{0} = 1 and _{0} = 2. In all panels, colored lines represent analytical approximations from _{J} as a function of _{1} provided an expanding population in which compartment 1 has the largest effective self renewal rate, i.e. λ_{0} = _{1}(_{1} − _{1}) > 0. Here _{2} = 0.55, _{3} = 0.6, _{1} = 0.2, and _{3} = 0.3. (_{J} as a function of _{2} provided an expanding population in which compartment 2 has the largest effective self renewal rate, i.e. λ_{0} = _{2}(_{2} − _{2}) > 0. Here, _{1} = 0.55 _{3} = 0.6, _{1} = 0.2, _{3} = 0.3. (_{J} as a function of _{2} provided a steady population in which compartment 1 has the largest effective self renewal rate, i.e. λ_{0} = _{1}(_{1} − _{1}) = 0. Here _{1} = 0.5, _{3} = 0.1, _{2} = 0.4, and _{3} = 0.6. (_{J} as a function of _{1}, provided a steady population in which compartment 2 has the largest effective self renewal rate, i.e. λ_{0} = _{2}(_{2} − _{2}) = 0. Here, _{2} = 0.5, _{3} = 0.1, _{1} = 0.4, and _{3} = 0.6.

On the other hand, for _{0} = _{J} is negatively correlated with the redistributing factor _{J} changes with _{0} = _{J} is always positive (blue line), or it changes from positive to negative at some critical point 0 < _{J} is always positive for all _{i,n − 1,i} is larger than 1, and then the product

Illustration of the selection gradient Δλ_{J} as a function of the redistributing factor _{0} = _{3}(_{3} − _{3}). In both panels, colored lines represent eigenvalue perturbation results in _{0} > 0). In this case, there are two different scenarios: For _{J} is always positive (blue color). For _{1} < 0.45, Δλ_{S} is changed from positive to negative with the increase of _{1} = 0.5, _{2} = 0.65, _{3} = 0.85, _{2} = 0.4, _{3} = 0.6. (_{0} = 0). In this case, Δλ_{J} is always positive. Here _{1} = 0.01, _{3} = 0.5, _{1} = 0.8, _{2} = 0.7, and _{3} = 0.2.

A comparison between Eqs (_{J} is always positive for any 1 ≤ _{0} < _{S} is always positive only for _{0} = 1. Secondly, Δλ_{S} only depends on the parameters related to the neighborhood compartments of _{0}, but Δλ_{J} 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 compartments 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.

In this study, we have explored the adaptive significance of de-differentiation in hierarchical multi-compartment structured cell populations. Favorable conditions for de-differentiation have been presented by comparing the fitness measures 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, and the cell division rate. These correspond to the parameter landscape of (_{i}, _{i}, _{i}) 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 _{S} and Δλ_{J}) 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. In general the stem cells are the cells with the highest potential for long term self-renewal. There is also agreement that stem cells replicate slowly and therefore in many tissues it is the progenitor cells that lead to amplification and maintenance of tissues. There is evidence that cells downstream of the stem cells can undergo self-renewal, albeit not long term or indefinite. In hematopoiesis, for example, erythroid progenitors that are committed to produce red blood cells undergo self-renewal that is regulated by Bm1-1 and PU-1 [

According to our results, 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 stem cells have the largest effective self-renewal rate. This result is quite intuitive. Stem cells are normally considered to have the greatest self-renewal potential, and due to de-differentiation the stem cells compartment receives the influx from differentiated cells. In this way, de-differentiation contributes to a faster growth rate of the whole population. 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. Interestingly, these results apply in both expanding and steady cell populations. For the expanding case, advantageous de-differentiation can speed up the growth rate of the whole population. For the steady case, de-differentiating mutant cell populations with fitness advantage can escape from the homeostasis and expand with time. A significant biological implication of this result is that de-differentiation could play a very important role in tumor initiation [

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 de-differentiation on self-renewal and differentiation probabilities greatly influences the selection conditions. Our results suggest that de-differentiation is more likely to be favored if there is less effect on self-renewal than on differentiation. That is, the smaller the redistributing factor

Note that the presented study is based on matrix population models with constant elements, which in principle do not take any non-linearity into account. 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) [

(PDF)

We would like to thank the Department for Evolutionary Theory at the MPI Plön for feedback.