Modeling lineage feedback

Cell lineages may be described as pathways of differentiation that begin with a stem cell (SC), progress through some number of self-renewing progenitor stages, and end with one or more terminally differentiated (TD) cells. Here we consider only the simplest of lineages—ones that are un-branched uni-directional, and produce TD cells that are non-dividing (postmitotic)—because, as we will see, the spontaneous emergence of morphogenetic behaviors requires no more complexity than that.

Following [25], the dynamics of such lineages may be written as a system of ordinary differential equations (ODEs), shown in section 1 of S1 Text, in which cell number (or concentration) at any lineage stage i is represented by the variable χ_i, cell cycle speed at stage i by rate parameter v_i, and the probability of self-renewal at stage i by parameter p_i (0 ≤ p_i ≤ 1). In tissues undergoing cell turnover, a rate constant, d, for cell loss or death must also be introduced (minimally for the TD cell). Within this framework, feedback or feedforward effects may be represented as dependencies of parameters upon the χ_i.

When seeking to capture the effects of diffusible molecules, modeling with ODEs is equivalent to assuming that cells are homogenously distributed (“well-stirred”) in space, so that diffusion gradients may be ignored. Although this is not generally a valid assumption for solid tissues, we may still understand the ODE representation of lineage dynamics as applying to sufficiently small spatial domains within tissues. In addition, because solutions to deterministic ODEs do not reveal the effects of stochastic variation that one might expect to encounter at low cell numbers, their analysis must often be supplemented with independent stochastic simulations,. In spite of these limitations, ODE formulations can serve as a good starting point for developing general insights, which can then be extended to regimes in which stochasticity, space and spatial organization matter explicitly. This is the approach we take here.

Mixed feedback in continually-renewing, well-stirred lineages

Fig 1A depicts the simplest possible, continually-renewing unidirectional lineage: a single type of SC that gives rise directly to a single type of TD cell that then turns over with a constant half-life. The ODEs for this situation may be written as (1) in which χ₀ and χ₁ represent SC (cell type “0”) and TD (cell type “1”) cells, respectively; P stands for the SC renewal probability; τ refers to time in units of cell cycles; and δ is the death rate of TD cells relative to the cell cycle rate, i.e. δ = d/v. To represent the possibility that TD cells may be a source of both negative and positive feedback on P, we replace P with the expression , where γ and ϕ quantify the strength of the negative and positive feedback, respectively, while p, a constant between 0 and 1, represents the maximum possible renewal probability. Expressing feedback as a product of Hill functions is convenient insofar as it guarantees a non-negative value of P, and exhibits saturation at large values of χ₁, as would be expected for any biological process. However, the precise form of the feedback function has only minor effects on the qualitative results described below. Likewise, the interaction between negative and positive feedback terms can be represented in a variety of other ways without altering the qualitative behaviors that are observed (see Tables A and B in section 5 of S1 Text, which summarize outcomes of different forms of the feedback functions).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Bistability of a two-stage lineage.

(A) Illustration of a lineage consisting of stem/progenitor cells (χ₀) and terminally differentiated (TD) cells (χ₁), in which feedback promotes self-renewal (p) with strength ϕ and inhibits it with strength γ. The parameter τ = 1/v denotes the cell division time (v is the rate of cell division) and τ/δ represents the lifetime of TD cells, in units of τ (δ = d/v where d is the TD death rate). (B-E) This feedback configuration yields bifurcations and bistability in both the SC/CP and TD cell equilibrium populations. Parameter values for these curves were δ = 0.2, p = 0.8. (F) Steady states are determined by the ϕ/γ ratio, for which the threshold for non-zero growth is approximately 5.8 when p = 1. Here, parameter values were δ = 1, p = 1, and γ = 1; ϕ was varied as shown. The initial conditions were χ₀(0) = 0.3 and χ₁(0) = 0. (G) Small changes in initial conditions can determine which steady state is reached. A larger number of initial progenitors χ₀(0), or an initial χ₁(0) that is neither too large nor too small, favors attainment of the non-zero steady state. (H-J) The probability distributions of χ₁ after 200 CP cell cycles, obtained from simulations in which both the self-renewal parameter (p), and the initial conditions for χ₀ were taken to be normally-distributed random variables with means p = 0.9 and χ₀(0) = 0.3. The variances in both cases were 0.1; p was truncated to ensure it was between 0 and 1 and χ₀ was truncated to ensure it was positive. These variables were sampled every CP cycle and 200 realizations were performed. The results are shown for three different values of ϕ/γ, as labeled. Other parameters were as in panel F.

https://doi.org/10.1371/journal.pcbi.1004814.g001

For non-zero values of ϕ, system 1, with P is defined as above, has three equilibrium states. In these states χ₁ is equal to either and χ₀ is simply δ times the value of χ₁ in each state, which is a consequence of the non-zero χ₀ equilibrium solutions necessitating that P = 0.5. For trivial reasons the solution in which both χ₀ and χ₁ are zero (i.e. both cell types become extinct) is necessarily stable. For the non-zero solutions, a saddle-node bifurcation occurs so that the second solution is unstable, while the third solution locally stabilizes as shown in Fig 1B–1E. The criterion for bistability—in which either of the two stable states is available for a single set of parameters—is the condition that p>0.5 (stem cells must be capable of renewing more than 50% of the time) and: Using the maximum feasible value of 1 for p, we find the minimum of this ratio to be ≈5.8. In other words, bistability requires that positive feedback on progenitor self-renewal be at least 5.8 times stronger than negative feedback.

Fig 1F plots the trajectories of solutions with different ϕ/γ ratios, starting from an initial condition of zero TD cells. In every case the system starts to generate TD cells, but when ϕ/γ is not large enough, TD production collapses and both cell types vanish. The switch between extinction and maintaining a non-zero steady state occurs, as predicted, at ϕ/γ ≈5.8, and this holds regardless of the individual values of ϕ and γ (see S1 Fig).

The fact that extinction occurs when positive feedback is too weak may seem surprising, given that spontaneous extinction was not observed in prior modeling studies that considered only negative feedback (e.g. [25]), but this is simply because such studies had to proceed from the assumption that SC are intrinsically specified to renew with high probability when there is no feedback. Here, we abandon that assumption, allowing the renewal and differentiation behaviors of SC to be largely determined by extrinsic factors, as is likely to be the case in vivo.

Although the existence of a non-zero steady state depends strictly on ϕ/γ, whether that state is reached is a function of initial conditions, as shown in Fig 1G. This is to be expected for a bistable system. If SC numbers are initially too low, only extinction is possible. If initial SC numbers exceed a threshold, the non-zero steady state can be reached only if the initial number of TD cells is neither too high nor too low. This behavior can be rationalized: if TD numbers are too low, there is insufficient positive feedback to keep SC numbers from contracting to the point where SCs are incapable of sustaining themselves; if TD numbers are too high, the effect of positive feedback approaches saturation and negative feedback dominates, driving SC self-renewal below the threshold for sustainability.

To explore the effects of stochastic fluctuations, we allow p, the maximum possible self-renewal fraction to be a random variable, while maintaining the same form of P given above. The reason for focusing on the self-renewal parameter is that the greatest source of stochasticity is likely to come from randomness in cell-division outcomes, whereby cells choose probabilistically between whether to generate zero, one or two differentiated progeny. In Fig 1H–1J, we take p = 0.9+ζ, where ζ a random variable sampled every CP cell cycle from a normal distribution with mean zero and variance 0.2 (p is truncated to ensure it lies between 0 and 1). The results, in terms of the distributions of the TD cell numbers (χ₁) after 100 CP cell cycles, measured over a large number of independent trials, are presented in Fig 1H–1J for different values of ϕ/γ. In these cases, the initial conditions were also similarly distributed random variables centered around the values from Fig 1F. The results show that, when ϕ/γ is small, the most likely state is the zero steady-state (Fig 1H). When ϕ/γ is sufficiently large, the non-zero steady state is the most likely, although the distribution of populations is fairly broad (Fig 1J). For intermediate values, the distribution is bimodal with the zero and non-zero steady states being nearly equally likely (Fig 1I) due to noise-driven switching between the steady states. Because of the possibility of stochastic extinction of small populations, the values of ϕ/γ needed to achieve non-zero steady states in the stochastic case are larger than those required for the deterministic model. Nevertheless, the behaviors of the stochastic and deterministic models are qualitatively similar.

Final-state systems

The system of ODE in (1) models lineages in which TD cells constantly turn over, as occurs in hematopoietic, epidermal and many epithelial lineages. In such a system, steady state solutions can be calculated simply by setting the time derivatives in (1) to zero. Not all tissues reach steady states, however. In some, TD cells last for the lifetime of the organism (e.g. in the nervous system), with SC either disappearing, or becoming quiescent. Such tissues may be modeled by setting the turnover rate δ in (1) to zero, as illustrated in Fig 2A (see also [25]). In this case, the governing equations become (2) again allowing feedback to be captured by a dependence of P on the levels of cell types (e.g., we again take P = ). The lack of TD cell turnover in (2) means that TD cells are now free to accumulate to a level that, through negative feedback, drives SC to extinction. A stationary state is thereby reached, but it is not “steady” in the usual sense of the word, i.e. there is no dynamic equilibrium; we therefore refer to it as a “final state”. Note that TD values at the final state cannot be calculated simply by setting the time derivatives in (2) to zero, because the time derivatives and SC value (χ₀) both go to zero together, leaving P undetermined.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Final-state analysis of the two-stage lineage.

When cell death is negligible, lineages with negative feedback on progenitor self-renewal (A) eventually reach a final state for χ₁ (B) that must lie either below or above a “growth leap” region. The bounds on this region (denoted χ₁^critlow and χ₁^crithigh), are the values of χ₁ at which χ₀' changes sign. As shown in (C), for initial values χ₁(0) below χ₁^critlow (blue) or above χ₁^crithigh (red), SC numbers (χ₀) necessarily shrink monotonically; otherwise χ₀ grows transiently (green line) and then shrinks to zero. (D) Whether a growth leap occurs is determined by the initial values of both χ₀ and χ₁. Here the presence of a growth leap is manifested as a large net increase in total cell number of cells. For small enough initial values of χ₀, such a leap occurs as in panel B, when initial χ₁ values lie between χ₁^critlow and χ₁^crithigh (denoted by red planes). For larger initial values of χ₀ however, the growth leap regime can be accessed at initial values of χ₁ below χ₁^critlow. Parameter values for panels B—D are p = 0.8, ϕ = 0.05, and γ = 0.002. The initial conditions in B were χ₁(0) = 20, 120, and 260, with χ₀(0) = 1, as shown in C.

https://doi.org/10.1371/journal.pcbi.1004814.g002

Strictly speaking, final-state systems require TD cells that never turn over. However, if TD cells turn over slowly, the final state formulation will provide a better description of lineage dynamics on time scales that are short relative to the TD cell half-life. As tissue morphogenesis often occurs rapidly, relative to TD cell lifespans, final state models may thus be inherently better representations of lineage dynamics during morphogenesis, even in self-renewing tissues. We therefore explored the effects of mixed feedback in final-state systems.

As with steady-state systems, we observed behaviors of final-state systems that depended strongly on the value of the ratio ϕ/γ, and exhibited abrupt switching between distinct states in response to small changes in the initial conditions for either χ₀ or χ₁ (see plots in Fig 2). As described in Box 1, the dynamics of TD production can be understood in relation to two critical TD population sizes: χ₁^critlow and χ₁^crithigh. When χ₁ lies between these two values, χ₀ grows; otherwise, χ₀ shrinks to zero. Understanding the dynamics that drive TD cell numbers across these critical values allows us to understand why initial conditions, seen in Fig 2D, cause leaps in growth to coincide with neither critical values nor equilibrium solutions. There are essentially three regimes of behavior:

1. Regime 1 (too many starting TD cells): When χ₁ begins above χ₁^crithigh, χ₀ shrinks monotonically to zero, increasing the value of χ₁ as it does so, until χ₀ = 0 (i.e. SC are exhausted). See the red curves in Fig 2B and 2C.
2. Regime 2 (not enough starting TD cells): If χ₁ begins below χ₁^critlow, χ₀ necessarily falls and χ₁ necessarily rises. If χ₀ runs out before χ₁ reaches χ₁^critlow, χ₁’s final state remains below χ₁^critlow. See the blue curves in Fig 2B and 2C. If not, the system enters regime 3.
3. Regime 3 (the “growth-leap” case): If initial χ₀ is large enough to carry χ₁ past χ₁^critlow, positive feedback then drives a rapid increase in net growth rate that pushes χ₁ above χ₁^crithigh (see the green curves in Fig 2B and 2C, and S2 Fig). The threshold for this “growth-leap” actually requires fewer initial χ₁ cells when there are more χ₀, (Fig 2D). It is only at the limit as initial χ₀ approaches zero that the initial value χ₁ required for a growth leap approaches χ₁^critlow (indicated by the red vertical plane in Fig 2D).

Box 1. Final State Analysis of a Two-Stage Lineage with Feedback

Taking the ODE representation of a two-stage lineage with negligible cell death Eq (2) and representing feedback by defining P as presented in the main text, we obtain the following dynamical system.

It is easy to see that the accumulation of TD cells (χ₁) is just the time integral of the expansion and contraction of the SC pool, i.e.

As t →∞, this integral converges only if χ₀ →0, which is guaranteed by the fact that, for sufficiently large χ₁, χ₀’(τ) necessarily falls below 0. There are then two interesting regimes to consider: when χ₀’(τ) < 0 at all times—this is what occurs if there is only negative feedback on P—and when χ₀’(τ) spends some time in positive territory, which can occur if positive feedback is sufficiently strong relative to negative.

Although the solution for χ₁(∞) in the above integral equation can only be expressed implicitly (see section 3 of S1 Text), we can still find the values of χ₁ at which χ₀’(τ) changes sign. There are two such values, which we refer to as χ₁^critlow and χ₁^crithigh: Note that these are identical to the non-zero steady states of the two-stage lineage system with TD cell turnover Eq (1); accordingly, for χ₁^critlow and χ₁^crithigh to be positive and real, the same lower bound on ϕ/γ exists as is required for bistability of the steady state system.

Essentially, the qualitative dynamics of the final system can be understood by following the trajectories of χ₁ relative to χ₁^critlow and χ₁^crithigh (see section 4 of S1 Text). When either χ₁ > χ₁^crithigh, or 0 < χ₁ < χ₁^critlow, χ₀ proceeds monotonically to extinction (Fig 2C). When χ₁ lies between χ₁^critlow and χ₁^crithigh, both χ₀ and χ₁ grow (Fig 2B and 2C); this persists until χ₁ breaches χ₁^crithigh, at which point χ₀’(τ) switches signs and χ₀ again monotonically shrinks. We refer to the region between χ₁^critlow and χ₁^crithigh as the “growth leap” regime, because within it both SC and TD cells undergo transient expansion, whereas outside of it SC pools always contract.

Mechanistically, the abrupt transitions between regimes, as a function of initial conditions, are analogous to the abrupt transitions between steady states in Fig 1, although the precise relationships between initial conditions and final state are not identical. Final-state systems may not properly be called bistable, since they are not attracted to fixed, dynamic stable states. We therefore refer to them as bimodal, in the sense that they may be switched between slow and fast growth modes as function of the sizes of cell populations. Like bistability, bimodality creates an opportunity for ultrasensitive switching. Not only can small differences in initial conditions cause abrupt changes in the growth trajectories of feedback-controlled tissues, as shown in Fig 2D, so can small changes in the strength of either positive or negative feedback (see S3 Fig).

Similarities and differences between the possible behaviors of steady state and final state systems, analyzed over all possible parameter values, and in the face of different kinds of TD-derived feedback, are summarized in Table 1, and Tables A and B in section 5 of S1 Text (see also S4 Fig in which positive feedback arises from SCs). These results show that the presence of positive feedback is required for bimodality, the presence of negative feedback is required for stability, and to achieve either bistability, or bimodality with a stable final state, both positive and negative feedback are required.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Summary of possible behaviors with feedback when p > 0.5.

This table summarizes possible steady state and final state behaviors when maximal progenitor self-renewal is greater than 0.5. Without feedback, the open loop system’s output is largely dependent on p, the probability that a progenitor cell will divide into an identical cell type instead of a differentiated cell type. Negative feedback eliminates this dependence on p, but such systems can only reach a single fixed point in both the steady and final state scenarios. Positive feedback, on the other hand, permits unbounded growth, which would be detrimental to achieving precision in tissue and organ growth. Together, however, lineages can growth to one of two distinct fixed points, which are either bistable (in homeostatic equilibrium) or bi-modal (i.e. stationary and non-dynamic).

https://doi.org/10.1371/journal.pcbi.1004814.t001

More complex lineages

Next, we consider a three-stage, steady-state lineage, in which we also alter the source and target of positive feedback. In this particular scenario (Fig 3A), an intermediate, committed progenitor (CP) is the target of all feedback, and a constant influx of progenitor cells from a stem cell source guarantees non-extinction of the CP pool (we assume that the SCs are in equilibrium). Negative feedback from TD cells is now mixed with positive feedback from CP cells themselves. The ODEs for this system are (3) in which ζ is the rate of proliferation of SCs, relative to that of CPs, and P = . Such an arrangement models, for example, the apparent roles of GDF11 and FGFs in the mouse olfactory epithelium [25].

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Bistability and spatial bi-modality in a three-stage lineage.

(A) Here, stem cells (SC) are assumed to maintain a constant output of committed progenitor (CP) cells (the mechanism for which is not modeled explicitly). The self-renewal probability of CP cells is taken to be regulated by positive feedback from CP cells themselves, and negative feedback from terminally differentiated (TD) cells. The parameter ζ denotes the division rate of SCs relative to that of the CPs. (B) As with two-stage lineages (Fig 1), the equilibrium state is determined by the positive-to-negative feedback ratio, ϕ/γ (here ϕ is varied while γ is held fixed; initial conditions are χ₀(0) = 1, χ₁(0) = 4, and χ₂(0) = 10, and τ is in units of CP cell cycles). (C) CP and TD steady states display bistability as a function of ϕ/γ. Parameters for B-C are ζ = 0.09, δ = 0.2, p = 0.75, and γ = 0.02. (D-E) The probability distributions of χ₂ after 200 CP cell cycles, obtained from simulations in which both the self-renewal parameter (p), and the initial conditions for χ₀, χ₁, χ₂ were taken to be normally-distributed random variables with means given by their values in panel C and variances 0.001. These variables were sampled every CP cycle and 200 realizations were performed. The results are shown for three different values of ϕ/γ, as labeled. Other parameters were as in panel C. (G-J) The behavior of the same lineage as in A was simulated in a spatial domain (G), in which the initial geometry simulates that of an epithelium, with SCs confined to a basement membrane (BM). The total areas of the epithelium (H), CPs (I) and TDs (J) are plotted as functions of time (measured in CP cell cycles), for different positive feedback strengths, ϕ. λ represents the intrinsic decay length of feedback factor G, which is here taken to be about 50 μm. The spatial distributions of CPs and TDs at the start of the simulation, and at the 12^th and 24^th CP cell cycles, are shown as insets in panels I and J, respectively. As the results show, with positive feedback gain ϕ = 3.0, the initial tissue grows exponentially at early times, but growth is not sustained. In contrast, with ϕ = 4.0, growth occurs at a similar rate at early times, but continues to expand exponentially at later times. A complete list of parameter values can be found in S1 Table.

https://doi.org/10.1371/journal.pcbi.1004814.g003

As observed in the analysis of the two-stage lineage, we see here that the ratio ϕ/γ, which captures the relative strength of positive versus negative feedback, plays a determining role in the emergence of bistability and ultrasensitivity (Fig 3B), although now both stable states are non-zero (Fig 3B and 3C). Further, when p and the initial data are taken to be random variables of the type used in Fig 1H–1J, the distributions of the TD cell population at a given time are similarly dependent on ϕ/γ (Fig 3D–3F). When ϕ/γ is small, the system is subject to stochastic extinction events even though the stable steady states are non-zero (Fig 3D). When ϕ/γ is sufficiently large, the most likely state of the system is the largest stable population, although the distribution of the populations is again fairly broad (Fig 3F). When ϕ/γ is an intermediate value, the distribution tends to be bimodal with peaks near the two steady states; the probability of stochastic extinctions decreases with increasing ϕ/γ.

Although the three-stage lineage model is not as analytically tractable as the two-stage model, numerical solutions clearly demonstrate that the system exhibits bistability with respect to feedback parameters and initial conditions (see S5A–S5F Fig). Similarly, growth bimodality is also observed if cell turnover is set to zero, or made very slow (see S5G and S5H Fig).

Taking account of space

Real tissues are not “well-stirred”, but grow outward into physical space. This potentially influences the feedback regulation of cell lineages in three ways. First, diffusible molecules that mediate feedback will naturally form spatial gradients, so that the concentrations of feedback factors—and thus the effects of feedback—will differ at different locations. Second, growth may displace (advect) both cells and diffusible molecules to potentially different degrees at different locations. Third, boundary conditions (i.e. the disposition of molecules and cells when they reach natural tissue boundaries) may influence both cell behaviors and molecular gradients.

To account for spatial variation, we re-interpret the variables in Eq (2) as representing a local measure of cell concentration, i.e. the volume fraction occupied by each cell type, and add additional equations for the spatial distributions of negative and positive feedback factors, which we refer to as G and F, respectively. Because the time-evolution of the distributions of these molecules is expected to be much faster than time scales of growth, we assume a quasi-steady state distribution for them. We allow G to be produced by TD cells; F to be produced by CP cells, TD cells or both; and in some cases also consider the potential effects of external sources of G or F (e.g., stromal cells). The distribution of F is governed by Eq (7) in Methods; the equation for G is analogous. The forms of the external sources and sinks are given in Eqs (13)–(17) in Methods. The effects of F and G on CP self-renewal are modeled exactly as in Figs 1–3, with parameters γ and ϕ representing the strengths of negative and positive feedback, respectively (see Eq (6) in Methods).

For simplicity, growth and diffusion are simulated here in two-dimensional domains, (extension to three dimensions is straightforward). As an initial condition we typically begin with tissues represented as thin, rectangular domains, as might be encountered in growing epithelia; see the schematic in Fig 3G. We denote one long side as apical [AP], and impose there a reflective (no-flux) boundary condition on molecular diffusion (representative of the tight junctions that exist at the apical surface of most epithelia). On the other side, where one would normally find a basement membrane [BM] that is not itself a barrier to diffusion, we impose no boundary condition, allowing molecules to move unimpeded into the adjacent space, which may be taken to represent the stroma upon which most epithelia sit. In some cases, we also model the presence of molecules in the stroma that act as absorbers, or sinks, for diffusible factors (e.g. follistatin in the stroma beneath GDF11- and activin-expressing epithelia [25, 38]), see Eqs (15) and (16) in Methods.

Diffusion, proliferation, differentiation and cell turnover are then modeled together with soft-tissue mechanics (as described in [39, 40]). The dynamics of the epithelial cells are governed by Eqs (4), (5) and (8)–(12) in Methods. In addition, the SC population is restricted to the basement membrane (BM) and set to self-renew with probability 1/2. Temporal and spatial trajectories of tissue growth in such models were explored over a wide range of parameter values to identify the sorts of qualitative behaviors that emerge. A complete list of dimensionless parameters can be found in S1 Table in Supplementary Materials. The nondimensionalization is given in section 6 of S1 Text. See Methods for a discussion of the numerical implementation of the governing Eqs (4)–(12) and section 6 of S1 Text for additional model and numerical details.

Feedback-driven bimodality of growth

As in non-spatial models, spatial systems may be initiated from small size and grow to include large numbers of CP or TD cells. Interestingly, in spatial models the presence of negative feedback control no longer guarantees that a stable steady state, or final-state, will be reached for all parameter values. If the strength of negative feedback is not sufficient, tissue size will increase indefinitely (see Fig 3H, blue curve). This reflects the fact that diffusible signals have limited ranges of action, so that the incremental increase in feedback, averaged over a tissue, that accompanies any incremental increase in tissue size, gets progressively smaller the larger a tissue gets.

This effect notwithstanding, we clearly observe that the mixture of positive and negative feedback can produce growth that is bimodal, switching ultrasensitively between fast and slow (or zero) growth rates as a function of the ϕ/γ ratio. For example, Fig 3H–3J compare the epithelia dynamics using positive feedback strengths ϕ = 3.0 and ϕ = 4.0 (holding other parameters fixed; see S1 Table). Note that in this example, initial growth is similar for both feedback strengths, but in the ϕ = 3.0 case (green curves and insets), the lack of sufficient positive feedback causes increased differentiation of CP cells into TD cells, elevating the concentration of G, which causes CP numbers to decline even further (Fig 3I). The decline in CPs eventually causes TD production to fall to a low level that is insufficient even to balance TD turnover (Fig 3J). In contrast, when ϕ = 4.0, the exact opposite sequence of events occurs, and CP levels rise to a value that sustains continual growth (see blue curves and insets in Fig 3H–3J). The distributions of the feedback regulators, the self-renewal fractions and the cells can be found in S6 Fig for ϕ = 3.0 and S7 Fig for ϕ = 4.0 in Supplementary Materials. Similar results are found when CP cells also produce G (S8 Fig), TD cells also produce F (S9 Fig), the ratio D_F/D_G is decreased (S10 Fig), and the sink of G in the stroma is removed (S11 Fig).

Although such systems exhibit a sharp threshold in behavior at a critical value of ϕ/γ, the value itself is not easily calculated, as it depends not only on initial conditions, but also on the spatial gradients on G and F, and thereby on the tissue geometry, as well as on the relative locations of cell types (see S12A Fig for a plot of the feedback ratios for the simulations in Fig 3H–3J).

We also explored the effects of stochasticity on the results in this spatial model. In particular, we considered two sources of random fluctuations—the self-renewal parameter p and the relative proliferation rate ζ of SCs. In S12B Fig, both are taken to be normally-distributed random variables in space and time with means given by their deterministic values. The results indicate that, when the variance is relatively small, simulations evolve similarly to what is seen in the deterministic case. When the variance is large, self-sustaining growth may occur spontaneously even when it would not, under deterministic conditions. When ϕ is large enough (e.g. ϕ = 4.0), we found that growth may self-sustain even for small variances.

Exogenous triggering of growth and growth-arrest

One consequence of the dependence of growth trajectories on the spatial distributions of feedback molecules is that localized, exogenous sources of the same molecules can be used to create localized, persistent differences in growth rate. In Fig 4 we consider an epithelial tissue in which the ϕ/γ ratio would not normally be sufficient to sustain continued growth, but an exogenous point source of F is placed near the basement membrane (such a source might represent the production of a growth factor by a group of stromal cells, for example). In response, a local outgrowth appears and continues to grow, producing a stratified arrangement of cells, with CP cell densities highest near the F-source and TD cells highest farther away. Notice that the outgrowth becomes sharply demarcated, i.e. there is a distinct notch, or “neck” that separates it from the adjacent epithelium (see arrows in Fig 4B), despite the fact that the exogenous gradient of F declines smoothly and gradually across this zone. Because of this demarcation, we refer to the structure as a “bud”.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Effect of exogenous application of a graded positive regulator.

The spatial development of a three-stage lineage was simulated as in Fig 3, under conditions (ϕ = 3.0) that would normally not lead to sustained growth. An exogenous source of “F”, the same factor that mediates positive feedback on CP self-renewal is placed near the lower left edge of the initial tissue, leading to sustained growth near the source, and a sharp boundary between rapidly growing and non-growing domains. Colors and shading track the concentrations of F and G (the negative feedback factor) over time and space (A-B), as well as the value of P, the CP self renewal fraction (C), and the volume fractions occupied by SC, CP and TD cells (D-F, respectively). The arrows in B mark the neck of the growing bud. The effect of the exogenous factor is to induce a local increase in the self-renewal fraction of CPs (e.g., see panel C, after the 12^th CP cell cycle), which drives local growth of the epithelium, formation of a bud due to the presence of a sharp border between high and low growth states, and a spatial stratification of the CPs and TDs populations and the feedback regulators they produce. The scale bar is in units of the diffusional decay length of the negative feedback factor G (λ), i.e. approximately 50 μm. See S1 Table for a complete list of parameter values.

https://doi.org/10.1371/journal.pcbi.1004814.g004

Interestingly, once such a bud is induced, it may sustain its growth independently after the F-source is removed. This is shown in Fig 5A, in which the exogenous source is shut off after the 12^th cell cycle (see S13 Fig for the corresponding evolution of the cell distributions). In this case, bud growth continues unabated, with an outcome almost identical to that in Fig 4A (see S21 Fig for a comparison). The reason is that the growing bud itself becomes an effective source of F, and it is this endogenous feedback that drives most of the growth (see S14 Fig, which shows the distributions of P). If exogenous feedback is removed too soon, however, before the bud is large enough, endogenous feedback is insufficient to sustain growth (see S15 and S16 Figs). In particular, CPs differentiate into TD cells, and growth stops.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. Effect of transient application of positive and negative regulators.

(A) The spatial development of a three-stage lineage is modeled as in Fig 4, except that the exogenous source of F is removed after the 12^th CP cell cycle. The spatiotemporal distributions of the positive regulator F (exogenous: purple; endogenous: green), the negative regulator G, and the CP self-renewal fraction P are shown in the three panels. The results show that the transient stimulus is sufficient to drive the CP self-renewal fraction above a critical threshold, beyond which bud growth becomes self-sustaining. (B) In this example, the simulation proceeds identically to panel A for the first 15 CP cell cycles, after a localized exogenous source of G is added for three cell cycles, and then removed. The result is an abrupt and sustained arrest of growth. Spatiotemporal distributions of F, G and P are plotted as in panel A, with an additional dark green scale added for exogenous G. Scale bars in A and B are in units of the diffusional length of feedback factor G (λ), i.e. approximately 50 μm. See S1 Table for a complete list of parameter values.

https://doi.org/10.1371/journal.pcbi.1004814.g005

Just as a transient exogenous positive signal can initiate a self-sustaining process of local epithelial growth, an exogenous source of G can also shut off growth. The simulation in Fig 5B begins in the same way as Fig 5A, with a transient source of F applied for 12 cell cycles, but following its removal a transient source of G is applied from the 15^th to 18^th cell cycle. Growth of the bud gradually arrests, as CPs differentiate into TDs, and the level of endogenous feedback needed to keep P above ½ is no longer sustained. See S17–S20 Figs for plots of the cell distributions, the distributions of P, and an analysis of strength of the exogenous source of G on the growth arrest. We also tested the effects of stochasticity on p and ζ as above, and found it more difficult to trigger growth arrest using a transient exogenous source of negative feedback in the stochastic case, i.e. larger amounts of G were required to stop growth.

These sorts of results (which are summarized in S21 Fig) are reminiscent of the observation in non-spatial systems that a single system can be moved in or out of the “growth-leap” region simply through an appropriate change in initial conditions (Fig 2D). In Fig 5A, the exogenous source of F drives growth of a localized region of the tissue to a point where it has sufficient endogenous F to enter a growth-leap regime. Indeed, as shown in Fig 6, short pulses of exogenous sources of F or G can also be used in non-spatial simulations to toggle, in an analogous manner, between fast- and slow-growing states. In both cases, it is the inherent bistability/bimodality of the systems that ensures an ultrasensitive dependence of such toggling on the levels of F and G applied. In spatial systems (Figs 4 and 5), that ultrasensitivity plays out as a sharp demarcation in the region of the tissue that responds to an exogenous cue, even if that cue forms a shallow spatial gradient over a larger region. Spatially demarcated behavior persists even after removal of an exogenous source because endogenous factors also form gradients, and the responses to them are similarly ultrasensitive.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 6. Ultrasensitive switching using exogenous regulators of renewal.

To explore the mechanism by which transient exogenous signals trigger or suppress high growth states (Fig 5), the effect of exogenous inputs was studied in the simpler, non-spatial, two-stage lineages of Figs 1 and 2. For a steady state system (A), with parameter values that would ordinarily lead to lineage extinction, panel B shows the effects of transient exogenous application, for two cell cycles, of a high level (ϕ_ex = 200) of the same renewal-promoting factor that is normally produced by TD cells, followed after the 100^th cell cycle by the application, for two cell cycles of a high level (γ_ex = 200) of the same renewal-suppressing feedback factor that is normally produced by TD cells. The first treatment shifts the system to a sustained high growth state, while the second treatment shifts it to a zero-growth state. Parameter values are δ = 0.2, p = 0.8, ϕ = 0.25 and γ = 0.015. For a final state system (C), in which cell death is negligible, panel D displays the effects on the final state of providing a pulse of various durations (numbers of cell cycles, τ) of an exogenous, renewal-promoting factor. Although the minimum level of exogenous factor required to trigger the high growth state is a function of pulse duration, the value of the final state that results once a high growth state is triggered is remarkably similar for pulses of all magnitudes and durations. Parameter values are p = 0.8, ϕ = 0.05, and γ = 0.002.

https://doi.org/10.1371/journal.pcbi.1004814.g006

From buds to self-organizing fingers

The sorts of buds that appear in the simulations in Figs 4 and 5 are simple, expanding balloon-like structures, with relatively little spatial organization. One reason for this is that the domains being simulated are relatively small compared with the ranges over which diffusible factors spread. To explore whether more intricate structures can be formed, it is necessary to “zoom out”, either by increasing the size of the simulated domain or—equivalently—reducing the spread of the diffusing molecules. In Figs 7 and 8 we do the latter, decreasing the intrinsic decay lengths of F and G, and adding uptake of the negative feedback factor G in the stroma (equivalent to including stromal expression of an irreversible inhibitor, such as follistatin [25, 38]). In Fig 7, we start with initial conditions from which growth does not spontaneously occur, and initiate growth through a point source of F (Fig 7A and 7E). In panels A-C the ϕ/γ ratio is set at a level that does not allow for self-sustaining growth, and therefore the exogenous source is left on throughout the simulation (as in Fig 4). In panels E-G, the ϕ/γ ratio is slightly higher, so that growth, once initiated, will self-sustain, and thus the exogenous source is removed after 15 cell cycles (for further details, see S22 and S23 Figs, which show the distributions of the cells and F and G). In both cases, buds initiate, but they continue to grow outward as fingers of relatively constant width (Fig 7D and 7H). Interestingly, this width is at least 10 times the characteristic decay lengths of either F or G, implying that it is determined by characteristics of the system other than the spatial ranges of action of diffusible signals.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 7. Formation of growing, constant-width fingers.

Spatial simulations similar to those in Figs 3–5 were carried out, using a larger domain size, slower diffusivity of feedback factors F and G, and incorporating strong uptake of G on the stromal side of the epithelium (for parameters see S1 Table; the characteristic decay lengths of F and G, denoted λ, were identical, and can be appreciated from the 10λ scale bar in the upper right corner). Boundary conditions were also taken to be periodic. Simulations all began from the same initial flat geometry, with an exogenous source of F in the center of the domain (F_ex, arrows), but using different values of the endogenous feedback ratio (ϕ/γ), as shown. A-C. At the lower ϕ/γ ratio, a fingerlike structure elongates as long as the exogenous signal is present. Values of P are high primarily near the source of F. D. The length of the finger in A-C increases linearly in time, but after initial growth, its width remains constant. E-G. At the higher ϕ/γ ratio, a similar-looking structure is generated, but it continues to grow even after the exogenous source of F is removed after the 15^th CP cell cycle. Values of P remain high primarily near the tip of the growing finger. H. Although the finger elongates at an approximately exponential rate, its width also stabilizes at a constant value. Distances in D and H are plotted in units of the intrinsic decay length of both F and G within the tissue (which is the square root of the ratio of diffusivity to the rate constant of uptake).

https://doi.org/10.1371/journal.pcbi.1004814.g007

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 8. Self-organizing pattern.

A-U. Simulations were carried out as in Fig 7, except that no exogenous source of F was provided, higher values of the ϕ/γ ratio were used, and the initial shape of the BM was perturbed by the addition of a small amount of “noise”, creating a slightly rough contour (see section 7 of S1 Text for details). Under such conditions, small inhomogeneities in levels of endogenous F and G near the BM lead to spontaneous and sustained finger growth. In the first two rows (panels A-G and H-N), ϕ = 3.0 and γ = 5.0 (thus ϕ/γ = 0.6), but different noisy initial conditions were used. The third row (O-U) starts from the same initial conditions as H-N, but with ϕ = 3.5 and γ = 5.0 (thus ϕ/γ = 0.7). Panels A-C, H-K and O-Q show the progress of finger growth, shaded as in Fig 7, at the indicated times (numbers of CP cell cycles). The scale bar shows five times the characteristic decay lengths of F and G. To facilitate quantitative analysis of these simulations, BM heights were measured as a function of relative position along the BM contour arc length. In panels D, K and R, these measurements are plotted for every other cell cycle from the start to the end of the simulation. Fourier transforms were used to identify the power spectra (power as a function of spatial frequency) of these graphs, and the evolution of such spectra over time was summarized in the form of kymographs, shown in E, L and S, in which intensity at different frequencies is displayed via a heat map, and time is on the ordinate axis. Panels F,G,M,N,T, and U are periodograms—essentially excerpts from these kymographs at single time points—at the end (F, M, T) or beginning (G, N, U) of the simulations, with power displayed on the ordinate axis (units of dB). In all cases, the results show that dominant frequencies present in the noisy initial conditions (arrowheads at the bottoms of kymographs E, L and S) quickly become replaced by lower frequency components (arrowheads at the top of kymographs E, L and S; dominant frequencies are also shown as dashed red lines in F-G, M-N and T-U). V-W. Results from the three cases presented in A-U and 21 additional independent simulations, initiated from a variety of noisy initial conditions with a range of initial frequency components and ϕ/γ values (see section 7 of S1 Text), were analyzed by plotting spectral moment (V) and average power across all frequencies (W), as a function of time (spectral moment is an aggregate measurement that captures the frequency at which power is “centered”). All simulations show a similar pattern: during the first two CP cell cycles, average power declines as most of the noise in the initial conditions is averaged away (in the inset in V, a logarithmic axis is used to show the pruning of both low and high frequencies; ϕ/γ = 0.6 for these cases). Afterwards, power centered on a low frequency band increases continually, reflecting the elongation of fingers of relatively constant width. The subset of cases in panel W in which power grows more rapidly are those for which ϕ/γ = 0.7, as opposed to 0.6. See S1 Table for a complete list of parameter values.

https://doi.org/10.1371/journal.pcbi.1004814.g008

The ability of a self-sustaining finger to grow in one direction, while maintaining a constant width in the other, suggests some sort of spontaneous self-organizing ability. One way to test for spatial self-organization is to see whether noise (random small inputs) gets transformed into quasi-regular patterns that are independent of the details of the noise. For example, reaction-diffusion systems that generate Turing patterns, when initiated from any of a variety of noisy initial conditions, evolve to produce patterns of stripes or spots with characteristics that directly reflect the underlying parameters of reaction and diffusion.

In our simulations, a simple way to mimic noisy initial conditions is to make many small perturbations to the initial shape of the modeled tissue. In Fig 8, small, sinusoidal perturbations of a variety of arbitrary spatial frequencies were introduced along the basal surface. Values of the ϕ/γ ratio were chosen so that self-sustaining growth would not normally occur. However, the shape irregularities introduced by the initial conditions create tiny locales in which endogenous F can rise above or fall below the average. Some of these reach the threshold for locally self-sustaining growth, with the result that a series of buds begin to emerge. Eventually, buds fuse to produce wide fingers with a relatively regular spacing. Fingers may continue to fuse, or branch to create new fingers, depending upon parameter values. Fig 8A–8C, 8H–8J and 8O–8Q show the epithelia morphologies together with the distributions of P for different ratios of ϕ/γ and initial conditions. The corresponding distributions of F (from Fig 8A–8C and 8H–8J) are shown in S24A and S24B Fig.

As in Fig 7, fingers are many times wider than the characteristic decay lengths of factors F and G (see scale bar, upper right). We find that finger widths are most readily influenced by changes to the diffusion coefficients D_F and D_G, with the widths increasing as a function of D_F and decreasing as a function of D_G (see S24C Fig). If D_F gets too large, however, growth fails to self-sustain, because peak levels of F become too small.

To show that these patterns are truly self-organized, and do not merely represent the expansion of pattern already present in the initial conditions, we used Fourier transforms of contours extracted from the images in Fig 8D, 8K and 8R to follow spatial frequency components over time. As shown in the kymographs in Fig 8E, 8L and 8S, as well as in individual periodograms excerpted from them (Fig 8F, 8G, 8M, 8N, 8T and 8U), simulations initially display power at frequencies corresponding to those used in the (sinusoidal) initial conditions, but quickly redistribute power to frequencies in the range of 2–4 (i.e. periods of 1/2 to 1/4 of the width of the modeled domain, which in these cases is ~12.6 times the characteristic decay length of diffusible factors F and G). This occurs regardless of whether the dominant frequencies in the initial conditions are high or low (Fig 8V and 8W; see also S24D Fig), and are independent of both the exact value of ϕ/γ and the details of the boundary conditions. What these data indicate is that the role of initial conditions in these simulations is merely to get growth started; afterward, spatial interactions organize the behavior of the system so that fingers of similar size are produced. Such self-organization, through spatial interaction, is reminiscent of the evolution of Turing patterns. [21, 22, 41, 42]. As with Turing patterns, this behavior depends upon diffusible, antagonistic factors (“activators” and “inhibitors”). Here, however, the patterns emerge as a spatial manifestation of bistability and bimodality generated by the diffusible factors and cell displacements. Thus, unlike Turing patterns, there is no need for the diffusivities of the factors to be different—indeed the diffusivities of the positive and negative factors are equal in Figs 7 and 8, and the characteristic scales of the patterns are much larger than the characteristic decay lengths of these factors.

Steady state, final state, and dynamics

Analyses of the spatially homogeneous, ordinary differential equation models were performed using Wolfram Mathematica. Simulations of these models were performed using NDSolve. Stochastic simulations were carried out using MATLAB.

Multispecies interactions

The volume fractions of CP (χ₁) and TD (χ₂) satisfy the mass conservation equation (4) where i = 1,2. The flux J_χi accounts for interactions among cell species, and u is the mass-averaged cell velocity. Here we have assumed that the cell densities are matched, that the cells are tightly packed and that the cells move with the same velocity following [39].

The source term Src_χi quantifies the net effects of the proliferation, differentiation and death of the lineage components: (5) The self-renewal fraction of the CPs, p₁, is regulated by a soluble negative feedback factor G (e.g., GDF) and a positive feedback factor F (e.g., FGF) instead of directly by the CP and TD populations in the spatially homogeneous model. Defining [G] and [F] as the concentrations of G and F respectively, we take (6) which is analogous to the feedback in the ODE model. Here P_1,min,P_1,max are the minimum and maximum levels of self-renewal, and ϕ, γ are the positive and negative feedback gains respectively.

The concentration [F] satisfies the quasi-steady reaction-diffusion equation (7) The equation for [G] is identical, but with different coefficients. We use a quasi-steady equation here because the diffusion time scales of G and F (e.g. minutes) are much faster than cell proliferation time scale (e.g. hours). Since it is not known which cell types produceF, the production of F may come from CPs or TDs or both. In particular, and are the production rates from CPs (χ₁) and TDs (χ₂) respectively, d_F is the natural decay rate, u_F is the uptake rate by CPs and D_F is the diffusion coefficient. The last term represents the effects of an exogenous source. In the equation for [G] EXTSrc_G also includes a sink that models the binding of G by Follistatin (FST) [65]. A complete list of dimensionless parameters and their values may be found in S1 Table.

Soft tissue mechanics

Following [39, 66, 67], we use a simplified model of soft tissue mechanics to obtain the cell velocity and flux. In particular, we use a generalized Darcy law to describe the passive motion of cells due to proliferation-induced pressure. This assumes that the tissue can be treated as a viscous, inertia-less fluid and also models flow through a porous media. Other constitutive laws and model formulations may be found in [39, 66–72] and are expected to yield similar results. We take the cell velocity to be (8) where κ is a non-negative cell motility, p is the pressure generated by cell proliferation, μ is the chemical potential defined as (9) where ε measures the thickness of the basement membrane and apical surfaces (both assumed to have the same thickness for simplicity). The parameter λ models the surface tension (stiffness) of the epithelia interfaces. The surface tension may be different for the BM and AP surfaces. The function χ_T denotes the epithelium volume fraction:χ_T = χ₁ + χ₂, and is a double-well potential. Note that the term where λ_BM,λ_AP are the surface tensions of the BM and AP surfaces, H_BM,H_AP are the corresponding mean curvatures of the surfaces and are surface delta functions localized on the basement membrane and apical surfaces, e.g., see [39]. Both terms in Eq (8) model passive cell motion with the first due to differential proliferation and the second due to cell-cell adhesion. Finally, the flux J_χ is modeled as a generalized Fick’s law [39] (10) where M represents the magnitude of the flux. Together, these constitutive relations for the cell velocity and fluxes guarantee that the model is thermodynamically consistent. That is, the adhesion energy , where L_BM,L_AP are the lengths of the BM and AP surfaces, is non-increasing in the absence of net sources, where the integration is taken over a domain containing the tissue, see [39].

To solve for the pressure, we assume that the mass conservation equation also holds for the volume fractions of the stroma (χ_s) and the fluid domain (χ_L) exterior to the AP surface. Defining χ_NE = χ_s+χ_L, we assume 1 = χ_T+χ_NE and correspondingly that . We also assume that there are no net sources in the stroma or fluid domains, i.e. Src_χs = Src_χL = 0. Then, summing the volume fraction equations for the lineage components, the stroma and the fluid, we obtain (11) from which it follows that (12)

Numerical implementation

We solve the system of equations in a rectangular computational domain Ω = {(x,y)|−20 ≤ x 20,−10 ≤ y ≤ 10} that contains the epithelial tissue. We use homogeneous Neumann boundary condition for cell types and feedback factors [F],[G]. In order to allow the tissue to leave the computational domain smoothly, we apply homogeneous Dirichlet boundary conditions for the pressure p and chemical potential μ, following [39]. The no-flux boundary condition at the AP is implemented by the diffuse domain approach (see section 6 of S1 Text). The stem cell population , where A(t) is chosen to ensure that numbers of SCs are maintained, even if the BM changes shape. The details about this approach are also found in section 6 of S1 Text.

To solve the governing equations efficiently, we follow previous work in [66] and use an adaptive finite-difference method with an implicit time discretization. The equations at the implicit time step are solved using the nonlinear multigrid method, where the equations are reformulated as a system of second order equations. Additional details may be found in section 6 of S1 Text. The spatial distributions of cells and feedback factors are visualized using Matlab.

Exogenous source for feedback factors G and F

Let T represent the current time in the simulation. Suppose that the exogenous source for F is introduced from time T_F1 and T_F2. Let (13) for T_F1≤T≤T_F2. Here is the volume fraction of the stroma, is a prescribed Gaussian source centered at the lower left corner of the computational region. F is produced in the stroma at a rate proportional to its difference from a target concentration A_F with a proportionality constant α_F. Given the computational domain {(x,y)|−20 ≤ x ≤ 20,−10 ≤ y ≤ 10}, we choose (14) where M_F measures the magnitude of the exogenous concentration, is the leftmost position of the BM. The feedback factor G has an uptake term in the stroma. Denoting u_G,s as the uptake rate, we define (15) The definition of exogenous source of is analogous. Suppose that the exogenous source for is introduced from to , we define (16) for T_G1≤T≤T_G2. Similar to F,G is produced in the stroma at a rate proportional to its difference from a target concentration (17) with proportionality constant α_G Here M_G measures the magnitude of the exogenous concentration of G, y_c is the leftmost position of the BM. The values for these dimensionless parameters are listed in S1 Table.