## ABSTRACT

Adult tissues in multicellular organisms typically contain a variety of stem, progenitor and differentiated cell types arranged in a lineage hierarchy that regulates healthy tissue turnover. Lineage hierarchies in disparate tissues often exhibit common features, yet the general principles regulating their architecture are not known. Here, we provide a formal framework for understanding the relationship between cell molecular ‘states’ and cell ‘types’, based on the topology of admissible cell state trajectories. We show that a self-renewing cell type – if defined as suggested by this framework – must reside at the top of any homeostatic renewing lineage hierarchy, and only there. This architecture arises as a natural consequence of homeostasis, and indeed is the only possible way that lineage architectures can be constructed to support homeostasis in renewing tissues. Furthermore, under suitable feedback regulation, for example from the stem cell niche, we show that the property of ‘stemness’ is entirely determined by the cell environment, in accordance with the notion that stem cell identities are contextual and not determined by hard-wired, cell-intrinsic characteristics.

This article has an associated ‘The people behind the papers’ interview.

## INTRODUCTION

Adult stem cells were first identified in the 1960s in the context of haematopoiesis (Siminovitch et al., 1963; Becker et al., 1963) and are now believed to reside in all renewing tissues. Because of their ubiquity and regenerative importance, the quest to find a complete set of defining characteristics of adult stem cells has been a longstanding aim of stem cell and developmental biology. However, despite numerous attempts, we still lack an unambiguous characterisation, and the stem cell concept is still widely disputed (Blau et al., 2001; Seaberg and Van Der Kooy, 2003; Shostak, 2006; Clevers, 2015; Laplane, 2015; Vorotelyak et al., 2020).

Historically, the stem cell identity has been viewed as a hard-wired phenotype, which is encoded in distinct patterns of gene and protein expression that confer to the cell the ability to maintain its own state while regenerating the cells of a whole tissue. Modern experimental methods have, however, cast doubt on this interpretation. For example, single-cell transcriptomics experiments have revealed a greater degree of molecular heterogeneity within apparently functionally ‘pure’ cell populations (including stem cell populations) than previously envisaged, and the search for unique stem cell markers has often, accordingly, proven challenging (Ghadially, 2012; Alcolea, 2017). Perhaps more significantly, genetic cell lineage-tracing studies have shown that many cells can exhibit a high degree of functional plasticity. For instance, cells that under homeostatic conditions are committed to differentiation are, in other contexts, such as regeneration or cancer, apparently able to acquire stem cell-associated features, and can re-populate the stem cell pool via de-differentiation (Donati and Watt, 2015; Donati et al., 2017; Tetteh et al., 2015, 2016; Koren and Bentires-Alj, 2015; Van Keymeulen et al., 2015). In response to these studies it has been suggested that rather than being a hard-wired cell-intrinsic property, ‘stemness’ – i.e. the attribute of being a stem cell – is a functional, dynamical cellular feature that is influenced by the cellular environment (Blau et al., 2001; Clevers, 2015; Clevers and Watt, 2018).

In practice, the most commonly accepted characterisation of stem cells is purely pragmatic: an adult stem cell is a cell that has the capability to (1) maintain its own population via self-renewing cell divisions and (2) produce all cells of a specific tissue lineage via differentiation. According to this functional characterisation, ‘stemness’ is assessed by a cell's lineage potential – i.e. what a stem cell and its progeny can become in the future, rather than by its molecular characteristics at present.

Here, we will take this basic functional definition as a starting point and seek to examine the relationships between intracellular molecular ‘states’ and cell ‘types’ in order to clarify, precisely, what we mean when we refer to a stem cell. We start by acknowledging that a cell may be characterised by its internal molecular state (which we will define precisely shortly), and the possible changes in cell states that confer changes in cell function, for example when differentiating. It is reasonable, therefore, to define a cell both in terms of its current state, and the states that it may reach in the future under appropriate stimulus: that is, by the set of admissible cell state trajectories.

Notably, such cell state trajectories may not just be simple paths: they can have a complex topology, including branching points at which cells make fate ‘decisions’ and closed loops or cycles (corresponding to recurrent dynamics, such as the cell cycle). Our essential insight is that we can represent all possible cell states and changes between them as a directed network. Cell state trajectories are then directed paths on this network. Importantly, the structure of this network defines which trajectories are allowed and which are not and so is of central importance.

Here, we analyse the general structural and dynamical properties of the network of cell state trajectories, and use this analysis to define precisely stem cell identities and cell lineages in adult renewing tissues. Typically, we have very limited experimental information available about this potentially very complex network of cell states and transitions. However, we will show, using notions from dynamical systems and network theory, that homeostasis imposes dynamical constraints that ensure that the lineage hierarchy of adult renewing tissues is necessarily very constrained.

The key finding of this article is that functional definitions of ‘stem’ and committed cells arise naturally within this framework. In particular, we show that there is an inherent relationship between the classification of self-renewing versus committed cell types and their position in the cell lineage hierarchy of homeostatic tissues: any self-renewing cell type, if defined as suggested, must reside at an apex of a homeostatic cell lineage. In consequence, self-renewing cells always have maximal lineage potential in homeostatic tissues and can therefore be classified as stem cells. Thus, self-renewal and lineage potential – the two defining characteristics of stem cells – are inherently connected.

Although this is commonly experimentally observed, there is currently no rationale for why this is the case. We demonstrate that this commonly observed structural hierarchy is not a mere coincidence of evolution, but instead is the only possible lineage architecture that is able to maintain homeostasis in renewing tissues, in any complex, multicellular life form. Within this framework, it also becomes apparent why stemness is not necessarily a cell-intrinsic property, but rather is an emergent property of populations of cells interacting with their cellular environment, for example via crowding regulation or instruction from a niche.

## RESULTS

Many adult tissues, such as the epidermis, intestinal epithelium and other epithelia, are constantly renewing: old cells are steadily removed and new cells are produced by cell division. In undisturbed conditions (i.e. in the absence of disease or trauma), adult tissue renewal is in homeostasis – the state of a tissue that is fully developed, unstressed, and behaving physiologically. In particular, during homeostatic tissue renewal, the tissue cell composition does not change significantly over time and this necessarily requires cell division, differentiation and death to be finely balanced. Our aim is to understand the constraints imposed by homeostasis on the structure of possible cell lineage architectures.

Typically, not all cells in a tissue can divide. Instead, only a fraction have the ability to keep dividing in the long term, with their progeny differentiating to provide a constant turnover of tissue-specific cells. It has been disputed whether such self-renewing progenitor cells generally qualify as ‘adult stem cells’ (Seaberg and Van Der Kooy, 2003; Shostak, 2006; Vorotelyak et al., 2020). Here, we show that constraints on tissue renewal imposed by homeostasis are sufficient to define unambiguously the stem cell notion in a mathematically robust and coherent way in any homeostatic renewing tissue.

### Cell types and cell lineages

#### Cell state trajectories

To start, we assume that a cell's phenotype can be characterised by its internal molecular composition, including expression of RNAs, proteins and metabolic components, etc. We will be interested not only in the current state of the cell, but also its potential, including its propensity to divide, change its phenotypic state, die or emigrate out of the tissue of interest. Rather than restrict to any specific technology (e.g. single-cell RNA sequencing) we define a cell's state in terms of the molecular features (measurable or not) that determine these propensities.

The molecular composition of a cell will naturally change over time, but not all molecular changes will be possible at any instance in time. We will refer to any admissible transition between cell states – a directed path from an initial state to a final state – as a ‘cell state trajectory’. The set of all admissible cell state trajectories determines a cell's lineage potential. Notably, the set of admissible cell state trajectories can have a complex topology: it can contain branching points, for example, at which trajectories diverge to different cell fates. Moreover, trajectories may be reversible and form cycles (for example, the cell cycle). Crucially, we can use the topology of the set of cell state trajectories as a basis to define cell types, lineages, and stem cell identity.

#### Cell types

Cell types are typically defined by molecular characteristics, for example expression of surface markers or clustering from single-cell transcriptome data. Although pragmatic, such taxonomies do not provide a rationale for why certain molecular states are associated with specific cell functions. However, we can assess a cell's lineage potential by analysing the set of possible cell state trajectories to which it belongs and are thus able to classify cell states according to the role they play in tissue development and maintenance via their position in the lineage hierarchy. To define a cell type in this context, we propose that cells of the same type should have the same lineage potential, and vice versa. Thus, any two states, *i* and *j*, belonging to the same cell type should share the same outgoing cell state trajectories. This is the case if, and only if, states *i* and *j* are able to interconvert, i.e. if they are mutually reachable via cell state trajectories. In this case, any state reachable from *i* is also reachable from *j*. According to this rationale, we can formulate a definition of a ‘cell type’:

A cell type is a maximal set of cell states that are mutually reachable by cell state trajectories.

Informally, this means that we can decompose the set of all possible cell state trajectories into disjointed groups, such that all cell states in the same group are connected via cyclic cell state trajectories (through which they are mutually reachable), whereas cells in different groups are not connected by cyclic trajectories. In this formulation, each such group constitutes a cell type.

We note that this definition of cell type is not always congruent with informal yet commonly used definitions of a cell type. Typically, cell types are associated with distinct patterns of expression, or distinct morphologies or functions. In our alternative view, cells of the same type may have substantially different molecular profiles or morphologies, so long as they are interchangeable (i.e. spontaneous conversion from one state or morphology to the other is possible in either direction). In this view, molecular or morphological similarity at a particular time point is not essential, rather it is the lineage potential of the cell that matters, allowing for the possibility that cell types can be inherently dynamic. To illustrate this notion, consider the cell cycle: during the cell cycle, a cell may change its molecular state and morphology dramatically, yet we would not consider each stage of the cell cycle as a different cell type. Consistent with this view, our definition groups all cell states within the cell cycle to one cell type.

Furthermore, we note that cell state trajectories, and therefore cell types, may vary with environmental context. For example, two distinct cell types may be present in one environment (e.g. *in vitro* or in grafts), yet in another environment (e.g. *in vivo*) a reversible transition between the states of those types may become possible, and thus, in our framework, only one cell type is present.

#### Cell state networks and lineages

In the following sections, we formalise these arguments, using notions from network theory, and use this reasoning to propose universal features of lineage architectures under homeostatic conditions. To that end, we first represent cell trajectories as transitions between a discrete set of cell states. Although we take a discrete formalism, our results are also valid in a continuous conceptualisation because continuous trajectories can always be discretised in a topology-preserving way (Milnor, 2016).

The set of discrete cell states and admissible cell state transitions forms a network, which we will call the ‘cell state network’. The nodes of this network are the cell states, and there is a directed link from node *i* to node *j* whenever a transition from cell state *i* to cell state *j* is possible without intermediate states (according to the chosen discretisation). Directed paths (i.e. sequences of directed links) in this network represent allowed cell state trajectories. An example of such a cell state network is shown in Fig. 1, left.

*n*

_{i}(

*t*) denote the expected number of cells in state

*i*at time

*t*. Furthermore, let us write the rate of cell division in state

*i*as

*λ*

_{i}, the rate of direct transition from state

*i*to state

*j*as

*ω*

_{i→j}, the rate of cell loss in state

*i*as

*d*

_{i}, and the probability that a cell in state

*i*will produce daughter cells in state

*j*and

*k*when it divides as . The time evolution of the expected number of cells of type

*i*is given by:

*j*produced upon division of a cell in state

*i*. We can write this system compactly in matrix form as follows. First, we define the total transition rate from

*i*to

*j*as , which combines all transitions from state

*i*to state

*j*, via cell division (Eqn 1) and direct state transitions (Eqn 2). Note that

*κ*

_{i→j}≥0 for all

*i*,

*j*. Also, we define the effective loss rate at

*i*as , which combines transitions out of state

*i*and cell loss. Using this notation, we can express the dynamical system (Eqn 4) in matrix form as:

where *A* is the *m*×*m* matrix with (*i*, *j*)-entry *κ*_{i→j} if *i*≠*j* and *κ*_{i→i}−*δ*_{i} if *i*=*j*, and * n*(

*t*)=(

*n*

_{1}(

*t*),

*n*

_{2}(

*t*),…,

*n*

_{m}(

*t*)), a vector.

The matrix *A* encodes the weighted cell state network described in the main text as its adjacency matrix. This network contains all the information about the cell dynamics: the nodes represent the cell states *i*=1, 2,…, *m* and any link from node (i.e. cell state) *i* to node *j* has weight *κ*_{i→j}, representing how frequent this transition is. In particular, a link from cell state *i* to cell state *j* only exists if the transition is possible, i.e. if *κ*_{i→j} > 0. Note that self-loops (links from a node to itself) may occur, and that cell loss is not a node on the network, although we have added it to the graphical representation of our networks for clarity.

In general, the parameters *λ*_{i}, , *ω*_{i→j}, *d*_{i} (*i*, *j*=1,…, *m*) may depend on *n*_{1}, *n*_{2}, …. If so, Eqn 4 will be non-linear. This case is discussed further below, with details provided in Box 3 (a complete mathematical treatment is given in section 2 of the supplementary Materials and Methods). However, when the tissue is at homeostasis, the expected cell numbers do not change under the dynamics (that is, mathematically, ). In this case, the dynamics in both the linear and nonlinear regimes are determined by the properties of the matrix *A*, as we show in section 1 of the supplementary Materials and Methods.

*T*

_{k}consisting of states in the cell state network. The intrinsic dynamics of the cell type

*T*

_{k}, i.e. when neglecting transitions from other cell types, are determined by the equation

where is the vector of expected cell numbers at time *t* in the cell states comprising the cell type *T*_{k}. The matrix *A*_{k} is the *m*_{k}×*m*_{k} submatrix of *A* (see Box 1) corresponding to the rows and columns . Since *A*_{k} is the adjacency matrix of a strongly connected network (the SCC corresponding to *T*_{k}), it is irreducible, and, furthermore, all off-diagonal elements of *A* are non-negative. Therefore, the Perron–Frobenius theorem guarantees that *A*_{k} has a simple real maximal eigenvalue *μ*_{k} (MacCluer, 2000; Arrow, 1989). It is a basic result of dynamical systems theory that *μ*_{k} determines the asymptotic long-term dynamics of the system, that is, asymptotically, where *n*_{k} is the total number of cells in type *T*_{k}. This means that the sign of *μ*_{k} determines whether the population will grow or decline in the long term (Strogatz, 1994) (see section 1 of the supplementary Materials and Methods for further details). We call *μ*_{k} the ‘growth parameter’ of cell type *T*_{k}.

Our intuitive definition of a cell type has a well-characterised counterpart in the terminology of network theory. Two nodes, *i*, *j*, that are mutually reachable via directed paths in a network are said to be ‘strongly connected’. Furthermore, any directed network may be uniquely decomposed into strongly connected components (SCCs) by grouping together all strongly connected nodes (Bollobás, 2013) (see Fig. 1 for an example). From a biological viewpoint, it is reasonable to consider only non-trivial SCCs, i.e. those that contain at least one link (that is, either more than one node or one node with a self-link). Thus, our intuitive definition of a cell type may be naturally phrased in the language of network theory: a cell type is a non-trivial strongly connected component of the underlying cell state network.

Using this definition, we can construct a second network, with cell types (that is, SCCs of the cell state network) as nodes, and a directed link between pairs of nodes (*A* and *B*, say) whenever there is at least one link from a state of type *A* to a state of type *B* in the cell state network (see Fig. 1 for an example). An elementary result of graph theory states that the resulting network, which is known as the condensation, does not contain cycles (directed paths from a node to itself) and is, therefore, hierarchical (Cormen, 2009). More precisely, we can order the cell types *T*_{1}, *T*_{2}, …, such that if there is a trajectory (i.e. a directed path) from *T*_{k} to *T*_{l}, then *k*≤*l* (Fig. 1). In this case, we will say that *T*_{k} is upstream of *T*_{l} and *T*_{l} is downstream of *T*_{k}. Biologically, this means that cell types are necessarily ordered in a hierarchy, as is commonly observed.

### Homeostasis constrains possible cell lineage hierarchies

So far, we have considered the structure of the cell state and cell type networks without imposing any restriction on tissue proliferative dynamics. In the following, we consider tissues at homeostasis. Importantly, cell proliferation and removal must be finely balanced in renewing tissues to ensure homeostasis, and so homeostasis imposes strong constraints on proliferative dynamics. From a mathematical perspective, homeostasis represents a steady state of the cell population dynamics in which the number of cells of each type in the tissue stays, on average, constant over time. Our guiding question is: how does the dynamical constraint of homeostasis restrict the possible hierarchy of cell lineages?

#### Cell population dynamics

*i*=1,2,…,

*m*(discretised as discussed in the previous section). We can represent these three cell processes schematically as:where the empty set symbol indicates ‘cell loss’ and

*i*,

*j*,

*k*denote cell states of which any two or all may be equal or not. The processes, Eqns 1-3, determine the general dynamics of cells in a tissue. In this description, transitions between cell states can occur through cell divisions coupled to cell state changes (Eqn 1, e.g. asymmetric cell division), and direct transitions (Eqn 2). Both transition types constitute the links of the cell state network. Equations describing the dynamics of the expected number of cells in each state can be easily derived from Eqns 1-3, and are provided in Box 1.

*T*

_{a}at the apex of a cell lineage, which is not

*a priori*a self-renewing type. Assume that the kinetic parameters of the cell dynamics,

*λ*

_{i},

*ω*

_{i→j}and

*d*

_{i}(see Box 1), may depend on the density of cells in the ‘niche’, which we assume to be constituted by the cells of type

*T*

_{a}. For constant volume, the density is proportional to the number of cells of type

*T*

_{a}, , and thus these parameters are functions of

*N*

_{a}:

*λ*

_{i}(

*N*

_{a}),

*ω*

_{i→j}(

*N*

_{a}) and

*d*

_{i}(

*N*

_{a}). The time evolution for cell numbers per state in

*T*

_{a}(Eqn 6) then becomes non-linear: . We note that, since the kinetic parameters are functions of

*N*

_{a}, so are the entries of the matrix

*A*

_{a}and thus the growth parameter

*μ*

_{a}=

*μ*

_{a}(

*N*

_{a}) as well. Now, for notational convenience, let us rename all kinetic parameters by

*α*

_{j}, numbered by , in the order

*λ*

_{1},

*λ*

_{2},…,

*ω*

_{1→2},

*ω*

_{1→3},…,

*d*

_{1},

*d*

_{2},…. In section 2 of the supplementary Materials and Methods, we show that a sufficient condition for the number of cells of type

*T*

_{a}to neither increase nor decrease monotonically in the long term (i.e. to obtain either a stable self-renewing state or exhibit finite oscillations/fluctuations around it) is:

for all *N*_{a}_{.} [Furthermore, we make the biologically plausible assumption that for large *N*_{a} (*N*_{a}→∞) each parameter attains a limiting value.] Recall that the crowding dependence of a parameter *α* is positive if *dα*/*dN*_{a}>0 and negative if *dα*/*dN*_{a}<0. For a cell type at the apex of a non-trivial lineage hierarchy the following sufficient criterion holds, which we term ‘sign criterion for regulation of self-renewal’:

If for all kinetic parameters *α*_{j}, the crowding dependence is positive (respectively negative), and the growth parameter dependence on *α*_{j} is negative (respectively positive), then a cell type at the apex of a lineage hierarchy acquires and maintains a long-term self-renewing state (i.e. the state of self-renewal is stable, or cell numbers fluctuate/oscillate around that state).

Note that the sign criterion is sufficient but not necessary, and not all parameters must fulfil the sign criterion. Eqn 10 can also be satisfied if some parameters do not meet this requirement, as long as they do not outweigh those that do. In particular, for any parameter with *dμ*_{a}/*dα*_{j}=0, the crowding dependence of *α*_{j} is not relevant. Illustrative examples of this result are shown in Fig. S1.

#### Dynamic classification of cell types

In order to assess conditions for tissue homeostasis, we classify cell types according to their intrinsic long-term proliferative properties, i.e. their intrinsic ability to perpetuate their own numbers, neglecting influx from other cell types. In Box 2, we show that the intrinsic long-term dynamics of the *k*th cell type population, *T*_{k}, can be uniquely characterised by a single number, its ‘growth parameter’ *μ*_{k}, which is fully defined by the rates and probabilities of the processes depicted in Eqns 1-3. In particular, if *μ*_{k}>0 then the expected number of cells of type *T*_{k} increases; if *μ*_{k}<0 then the expected number of cells of type *T*_{k} decreases until the pool vanishes; and if *μ*_{k}=0 then the expected number of cells of type *T*_{k} stays constant, i.e. cells of that type maintain themselves at homeostasis (Strogatz, 1994). In general, *μ*_{k} may depend on the cellular environment and change over time (see discussion in the section ‘Environmental regulation of the stem cell identity’), but in this section we refer to its steady state value only. This classification of cell types according to their intrinsic dynamics motivates the following definition:

We call a cell type T with growth parameter μ ‘transient’ if μ<0, ‘hyperproliferating’ if μ>0, or ‘self-renewing’ if μ=0.

These three classes represent three distinctly different kinds of dynamics. Firstly, transient cell types will eventually vanish, unless they are replenished through incoming trajectories from other upstream cell types. This means that all transient cells and their progeny will eventually differentiate into another type or will be lost from the tissue (e.g. via cell death or emigration). Secondly, hyperproliferating cell types will continually increase in number in the tissue. Although possible, this situation is clearly not physiological, yet may be encountered in certain pathologies, such as cancer. Thirdly, self-renewing cell types will maintain their numbers, on average, over time. In principle, this definition allows for inert cells that do not divide, differentiate or die. However, this situation only occurs in cell lineages that are not renewing, which we do not consider here. In renewing cell lineages, such cells must precisely balance proliferation and loss, for example by asymmetric cell divisions or neutral competition of equipotent cells for limited niche space (Clayton et al., 2007; Lopez-Garcia et al., 2010; Kitadate et al., 2019), and so exhibit the biological property of self-renewal that is characteristic of stem cells. Such a fine-tuning at a critical point (an exact balance between differentiation and self-renewal) may seem at first glance implausible. However, in the section ‘Environmental regulation of the stem cell identity’ we give an example of a feedback-based regulation mechanism that can achieve such precise balance without intrinsic fine-tuning.

#### Homeostasis imposes fundamental constraints on cell lineage architectures

There are a number of ways that homeostasis can be maintained. In principle, a homogeneous cell population that consists of a self-renewing population of a single type, as defined above, would, by definition, be homeostatic. However, this is a particular situation that is not reflective of the complexity of most tissues. Typically, a tissue will consist of a range of different cells, each with different proliferative capacities, related in a complex hierarchy. In these circumstances, the number of cells in each type are affected by incoming transitions from upstream cell types. Thus, for the cell population as a whole to be homeostatic, further conditions on the architecture of the lineage hierarchy need to be satisfied, which are discussed next.

The cell state dynamics described in Box 1 have the particular feature that the coupling between different states is non-negative, because transition rates *κ*_{i→j}≥0. Furthermore, when considering the conditions for homeostasis – a steady state of the dynamics – the kinetic parameters can be treated as constants. Such a system is a ‘linear cooperative system’, i.e. a system in which all couplings between distinct states are non-negative and constant. We have recently derived general mathematical conditions for a steady state in linear cooperative systems (Greulich et al., 2019), and in section 1 of the supplementary Materials and Methods we show how this can be generalised to the non-linear dynamics described in Box 1. In particular, we show that homeostasis can only prevail in a renewing tissue if:

- (1)
no hyper-proliferating cell type is present;

- (2)
at least one self-renewing cell type is present; and

- (3)
there are no cell state trajectories from one self-renewing cell type to another.

For now, we refrain from addressing the question of how these conditions are maintained in order to achieve homeostasis. We will discuss this later, in the section ‘Environmental regulation of the stem cell identity’, which outlines a potential mechanism that controls homeostasis.

Conditions (1-3) are argued in a mathematically rigorous way in section 1 of the supplementary Materials and Methods, yet they can also be understood intuitively. Condition (1) is obvious, as the presence of hyper-proliferating cells necessarily contradicts homeostasis. Condition (2) follows from the fact that, in the absence of both hyperproliferating and self-renewing cell types, all cell types must be transient and thus would steadily decline in numbers. Condition (3), however, is less intuitive. To understand why a lineage connection between two self-renewing cell types disrupts homeostasis, let us consider a self-renewing type *S*_{1} that is upstream of another one, *S*_{2}. If so, cells of type *S*_{1} must be able to transition to type *S*_{2}. Given that the mean cell number of *S*_{2} cells is constant without contribution from other cell types (by definition of a self-renewing type), this additional steady inflow of cells into type *S*_{2} would mean that the number of *S*_{2} cells would steadily increase, and thus be non-homeostatic. Therefore, self-renewing cell types cannot be connected by cell state trajectories, in a homeostatic state.

From conditions (1) and (3) it follows that all cell types upstream of a self-renewing type must be transient. Therefore, they vanish in the long term from the tissue: that is, their cell numbers will tend to zero (Greulich et al., 2019; supplementary Materials and Methods, section 1). Because at least one self-renewing cell type must exist [condition (2)], this implies – using our definition of a self-renewing cell type – that:

In homeostatic renewing tissues, every self-renewing cell type resides at an apex of a cell lineage hierarchy, and every such lineage has a self-renewing type at its apex.

Because self-renewing cell types reside at the apexes of lineage hierarchies, they can generate all cells of their lineage. Thus, self-renewal and lineage potential are not independent cellular properties of cells: they are fundamentally coupled. In particular, self-renewal implies lineage potential and lineage potential implies self-renewal. According to the commonly used characterisation of stem cells, we can therefore associate any self-renewing cell type in a homeostatic renewing tissue (with a non-trivial lineage) as a homeostatic adult stem cell. Moreover, stem cell identity can be established by either property.

The implications of conditions (1-3) and the lineage hierarchy principle are illustrated in Fig. 2, which shows a typical cell lineage hierarchy. Two features are notable.

First, the lineage hierarchy principle allows cell types upstream of a self-renewing type to be transiently part of a lineage hierarchy in non-homeostatic situations – for example, during development or when a tissue is damaged – but they vanish from the tissue as homeostasis is approached or become dormant. Such dormant stem cells can reside above other self-renewing cells in the lineage hierarchy, but they cannot participate in tissue renewal under homeostatic conditions and are therefore not considered part of the homeostatic lineage (see Fig. 2). Thus, our framework makes a clear distinction between stem cells that maintain homeostasis and those that regulate tissue response to conditions of stress.

Second, multiple apexes of a lineage can in principle exist, with different stem cells at each apex, as is conjectured for mouse mammary epithelium lineage, for instance (Pal et al., 2017).

With these insights, we can classify each cell type according to its growth parameter and relative position in the hierarchy into one of the following three classes: ‘developing cells’ (D), ‘committed cells’ (C) and ‘adult stem cells’ (S). Developing cells are transient cells (growth parameter *μ*<0) upstream of a self-renewing cell type in the lineage hierarchy. Developing cells may transiently divide and differentiate but, because they are not replenished by a self-renewing cell type, they disappear as homeostasis is approached and are thus not present in adult, homeostatic tissues. We therefore associate these cells with developmental populations that do not survive into adulthood (e.g. embryonic stem cells). Committed cells are transient cells (growth parameter *μ*<0) downstream of a self-renewing cell type. They may divide transiently, but all their progeny are eventually lost (e.g. to death or emigration). However, this population does not disappear as it is maintained at homeostasis by replenishment from an upstream self-renewing population. We associate committed cells with adult somatic cells that are not stem cells. Adult stem cells are self-renewing cells (growth parameter *μ*=0, maintaining their population) that can differentiate into at least one committed cell type downstream. According to the lineage hierarchy principle, they must reside at the top of a homeostatic lineage hierarchy.

*S*

_{k}is a homeostatic adult stem cell type of type

*k*, and

*C*

_{l},

*C*

_{m}are (possibly different) committed cell types. These three division types correspond precisely to what is known experimentally about stem cell divisions in homeostasis. In particular, stem cells may divide in the following three ways: symmetric self-renewal divisions (

*S*→

*S*+

*S*), asymmetric self-renewal divisions (

*S*→

*S*+

*C*) and differentiation division (

*S*→

*C*+

*C*). However, following the conditions for homeostasis above, we can say more. For instance, a stem cell division producing one or more stem cells of different types, i.e. a division of the form

*S*

_{k}→

*S*

_{l}+

*S*

_{m}or

*S*

_{k}→

*S*

_{l}+

*C*in which

*l*≠

*k*or

*m*≠

*k*, is not possible, as it would contradict condition (3) above. Furthermore, stem cell divisions that produce developing cells, and divisions of a committed cell to produce a stem cell, are also not possible, according to our lineage hierarchy principle. To make these ideas more concrete, examples of lineage hierarchies that do/do not fulfil these principles are shown in Figs 2 and 3.

Collectively, these results demonstrate that characteristic properties of lineage architectures may be derived from first principles by consideration of the relationship between cell states and cell types, and from dynamical constraints imposed on lineage hierarchies by homeostasis.

### Environmental regulation of the stem cell identity

In order to be self-renewing, the kinetic parameters of a stem cell type (rates of division, cell state transitions, and cell loss) need to be finely tuned to achieve a growth parameter of exactly zero (*μ*=0, see Box 2), and to ensure homeostasis. Any, even slight, deviation from this value may, if sustained over an extended period, lead to a loss of the self-renewing capacity and thus of the stem cell phenotype. In the absence of a homeostatic control mechanism such fine tuning is biologically implausible. In reality, cells are embedded in a tissue environment and may respond to signalling cues from other cells. A simple example of such regulation is crowding feedback, in which cells sense the density of cells in their local microenvironment and respond by adjusting their propensity to divide or differentiate. Such feedback mechanisms have been experimentally observed in various epithelia, such as in developing *Drosophila* (Marinari et al., 2012), human colon and zebrafish epidermis (Eisenhoffer et al., 2012; Eisenhoffer and Rosenblatt, 2013), where increased cell density is known to accelerate cell differentiation. Potential mechanisms to mediate this feedback are by mechanosensing (Eisenhoffer and Rosenblatt, 2013; Puliafito et al., 2012; Shraiman, 2005) or by the limited availability of growth signalling factors (Kitadate et al., 2019), for example.

We can incorporate such regulation in our framework by assuming that the kinetic parameters associated with the dynamics in Eqns 1-3, as defined in Box 1, are not constant, but depend on the cell density. A kinetic parameter may increase with the cell density (i.e. exhibit ‘positive crowding dependence’) or decrease with it (exhibit ‘negative crowding dependence’), or neither. In Box 3, and in section 2 of the supplementary Materials and Methods, we show that a simple sign criterion for the crowding dependence of the kinetic parameters guarantees that the cell type at the apex of the cell lineage hierarchy will acquire and maintain self-renewal potential, on average, in the long term. In short, the sign criterion ensures that the growth parameter *μ* decreases with cell density, and thus with cell number *N* (i.e. *dμ*/*dN*<0).

Recall that the growth parameter *μ* determines the long-term growth rate of the cell population (i.e. asymptotically, ). Therefore, the sign criterion ensures that if the cell number is below the critical value for which *μ*=0, then *μ*>0 and thus the cell number eventually increases toward . Similarly, if the cell number is above , then *μ*<0, and thus the cell number eventually decreases toward . Hence, the tissue cell numbers will be confined around the self-renewing state defined by *μ*=0 and are either stable, or fluctuate or oscillate around the homeostatic state (for more details, see the discussion in section 2 of the supplementary Materials and Methods). Notably, this could include a dynamic state, in which cells persistently switch between a hyperproliferating state (*μ*>0) and a declining state (*μ*<0), in a way, however, that on average cell numbers remain in the long term constant. Either way, the cell population is on average long-term self-renewing, resulting in a (potentially dynamic) homeostatic state for the tissue, assuming all other cell types are transient.

This feedback-based mechanism for regulating stemness is simple and robust against noise, as it does not depend on the exact values of the dynamic parameters, but only on the sign of their dependencies. Crucially, this means that self-renewal need not be a property intrinsically determined by a cell, but it may be acquired by any cell type that is at the apex of a lineage hierarchy, through interaction with the cellular environment.

## DISCUSSION

Stem cells are fundamental to healthy tissue turnover and repair and have accordingly been the subject of concerted research. However, despite more than 60 years of study, our understanding of the general principles that underpin stem cell dynamics remains incomplete. Here, we have proposed a new way to approach understanding stem cell identities from first principles based upon consideration of the relationships between cell states and cell types and general properties of proliferation in hierarchically structured populations. The resulting organisational principles are consistent with common conceptualisations of adult stem cells and lineage architectures, even though they emerge from consideration of the dynamical properties of cell lineage architectures at homeostasis rather than from explicitly biological considerations.

Furthermore, our analysis indicates that these common properties are not coincidental but rather reflect fundamental and universal constraints on tissue dynamics at homeostasis, defined here as a steady state of the tissue cell population. The common observation that self-renewing cells sit at the apex of lineage hierarchies, for example, is not simply due to biological contingency: it is the only possible way that lineage architectures can be constructed to support homeostasis in renewing tissues. Crucially, this means that lineage potential and self-renewal potential are not independent characteristics of adult stem cells in homeostasis, but they imply each other. Thus, in homeostasis, either self-renewal or lineage potential alone is enough to define a stem cell.

It is important to note that our results only hold at homeostasis. In challenged conditions, such as wounding and cancer, lineage architectures may temporarily change to allow de-differentiation, for example. In such situations, properties of the lineage hierarchy not accounted for in our formulation may also be important. For example, our lineage hierarchy principle does not exclude an additional pool of dormant stem cells residing in the lineage hierarchy above an adult stem cell population, such as a dormant population of epidermal stem cells residing in hair follicles (Taylor et al., 2000). However, our analysis shows that such dormant stem cells cannot participate in homeostatic tissue renewal if another self-renewing cell type is present, and so they play no role in healthy adult tissue turnover. They may nevertheless become activated under conditions of stress.

Our results accord well with current experimental evidence in numerous tissues. For example, in many mammalian renewing epithelia, such as mouse interfollicular epidermis (Clayton et al., 2007; Alonso and Fuchs, 2003), oesophagus (Seery and Watt, 2000; Seery, 2002) and intestine (Barker et al., 2007), a single self-renewing cell type, at the apex of the homeostatic lineage hierarchy, can be identified. In some tissues, however, multiple stem cell populations that contribute to the same pool of cells have been identified, such as in the haematopoietic lineage (Cheng et al., 2019), human epidermis (Taylor et al., 2000), sebaceous glands (Feldman et al., 2019), or as conjectured for the mammary gland (Pal et al., 2017). Although not immediately apparent, these cases are generally consistent with our predictions. In the mouse mammary gland, for example, the two stem cell populations each reside on a separate apex of the lineage hierarchy (Pal et al., 2017), and thus are allowed in our framework. In other cases, additional stem cell populations represent dormant stem cells (as discussed above for epidermis; Taylor et al., 2000), or stem cells that are most likely active only in development or under conditions of stress [e.g. Blimp1^{+} sebaceous stem cells, which are distinct from self-renewing progenitor cells (Andersen et al., 2019) and which can recreate sebaceous organoids (Feldman et al., 2019)], or stem cells that transdifferentiate between compartments (in epidermis; Jones et al., 2007). In each of these examples, the activity of multiple stem cells only occurs in a non-homeostatic context – a situation not covered by our theory. An intriguing potential counter-example is the proposition of two distinct types of stem cells in the homeostatic haematopoietic lineage – long-term (LT) and short-term (ST) repopulating haematopoietic stem cells (HSCs) – as it has been suggested that they stand in a hierarchical relationship to one another (Cheng et al., 2019). However, much of our understanding of HSC function comes from bone marrow transplantation experiments, which are fundamentally out of homeostasis, and the functional distinction between LT-HSCs and ST-HSCs has accordingly not yet been fully characterised *in vivo*. Indeed, there is increasing evidence that the mechanism of HSC proliferation in times of stress is fundamentally different to that under steady state. Our analysis could be a guide to future experiments as our lineage hierarchy principle predicts that either: (1) some identified HSCs constitute a dormant population that does not participate in normal homeostasis, but are activated when homeostasis is disturbed, e.g. after blood loss; (2) one of the putative stem cell populations is not genuinely self-renewing, but gradually loses its proliferative potential on a long time scale, or (3) LT-HSCs and ST-HSCs can inter-convert. If so, they would, according to our definition, effectively be a single stem cell type. These issues are hard to resolve experimentally, yet theoretical considerations can be a guide. It is likely that similar issues will arise as we gain a deeper understanding of other systems. If so, then theoretical notions such as those we propose could be invaluable. We anticipate that future developments in stem cell biology will benefit from closer integration of experiment and theory, which will yield a deeper understanding of the organisational principles that regulate stem cell dynamics.

## MATERIALS AND METHODS

The illustrative cell dynamics shown in Fig. 3 and in Fig. S1 are obtained by numerical integration based on a Runge–Kutta 4-5 method of the system of ordinary differential equations for the corresponding networks. The code is implemented in MATLAB and available at https://github.com/cp4u17/stemCellIdentity.git.

## Footnotes

**Author contributions**

Conceptualization: P.G., B.D.M., R.J.S.-G.; Methodology: P.G., C.P., R.J.S.-G.; Software: C.P.; Investigation: P.G., C.P., R.J.S.-G.; Writing - original draft: P.G.; Writing - review & editing: P.G., B.D.M., C.P., R.J.S.-G.; Visualization: C.P.; Supervision: P.G.; Project administration: P.G.

**Funding**

P.G. is supported by a Medical Research Council New Investigator Research Grant (MR/R026610/1), C.P. is supported by a Studentship of the Institute for Life Sciences (University of Southampton), and P.G, B.D.M. and R.J.S.-G. are partially supported by The Alan Turing Institute (EPSRC grant EP/N510129/1).

**Data availability**

The programme code for numerical computations and plotting of its results is available at https://github.com/cp4u17/stemCellIdentity.git.

## Peer review history

The peer review history is available online at https://journals.biologists.com/dev/article-lookup/doi/10.1242/dev.194399

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**Competing interests**

The authors declare no competing or financial interests.

Stem Cell Microenvironments and Beyond