Lineage Dynamics in the Absence of Control
One way to identify the control needs of a system, and the strategies that may be used to address those needs, is to build models and explore their behavior. A is a general representation of an unbranched cell lineage that begins with a pool of stem cells, ends with a postmitotic cell type, and possesses any number of transit-amplifying progenitor stages. If cells at each stage are numerous, and divisions asynchronous, then the behavior of such a system over time can be represented by a system of ordinary differential equations (B) with two main classes of parameters. The v-parameters quantify how rapidly cells divide at each lineage stage (in particular, v = ln2/λ, where λ = the duration of a cell cycle). The p-parameters quantify the fraction of the progeny of any lineage stage that remains at the same stage (i.e., 1-p is the fraction that differentiates into cells of the next stage). Thus p may be thought of as an amplification, or replication, probability. As each lineage stage has its own v and p, we use subscripts to distinguish them.
Lineage Behavior in the Absence of Control
Let us refer to the number of terminal-stage cells at any point in time as the output
of a lineage system. From B, we can see that a system is not stable—over time the output increases without bound—if pi
> 0.5 for any i
. In contrast, if pi
< 0.5 for all i
, stem and progenitor cells eventually run out, and the production of new terminal-stage cells stops. Provided terminal-stage cells do not die at an appreciable rate, such a system will reach a final state with a fixed number of terminal-stage cells. Finally, if p0
= 0.5, and pi
< 0.5 for i
> 0, then the system will eventually produce terminal-stage cells at a constant rate. If such cells die or are shed with a constant probability per unit time (represented in B by the rate constant d
), then the output will approach a steady state, the solution for which is given in C (solutions for certain cases of final-state behavior are also given in Protocols S1
, sections 5 and 6).
The result in C describes a steady state that is quite sensitive to the system's parameters. For example, output is proportional to the number of stem cells (χ0, which remains constant at its initial value) and the rate of stem cell division (v0), and inversely proportional to the rate of terminal-stage cell death (d). Output varies even more sensitively with the pi. For example, increasing the value of a pi from 0.45 to 0.4725—a 5% change—necessarily produces a 74% increase in the output of terminally differentiated cells. In engineering, parameter sensitivity is usually quantified as the fold change in output for a given fold change in the parameter (equivalent to the slope of a log-log plot of output vs. parameter). Thus, a linear relationship corresponds to a sensitivity of 1 (directly proportional) or −1 (inversely proportional). From C, we may calculate that the sensitivity of the output to any pi is pi/(1 − 3pi + 2pi2), which for pi < 0.5 is always greater than 1, and grows without bound as pi approaches 0.5.
In well-regulated biological systems, parameter sensitivities ≥ 1 tend to be undesirable, since genetic or environmental variability can easily cause several-fold changes in the biological processes (levels of proteins, cell growth rates, etc.) that underlie parameters [27
]. A system that cannot compensate for such variation is justifiably considered fragile
(the opposite of robust).
Arguably, the most severe fragility of the system in is the constraint placed on the stem cell replication probability: p0
must be exactly
0.5 for a non-zero steady state to exist (effectively, the system's sensitivity to p0
is infinite). This is simply another way of stating that, unless exactly half of all stem cell progeny are stem cells, lineages eventually either go extinct or explode. Meeting this constraint can be achieved by having every stem cell undergo perfect asymmetric divisions, but that does not seem to be what normally happens. Rather, individual stem cells behave stochastically, sometimes giving rise to two, one, or zero stem cells (e.g., [6
]). For the exact condition p0
= 0.5 to arise as a population average, when such behavior is not a cell autonomous imperative, is an extraordinary—and yet poorly understood—feature of stem cell systems.
Feedback Control of Transit-Amplifying Cells: Insights from the Olfactory Epithelium
The idea that negative feedback is used to regulate tissue size and enhance regeneration is an old one. Over 40 y ago, Bullough [31
] introduced the term chalone
to refer to secreted factors that inhibit growth of the tissues and organs that secrete them. When a tissue is injured or partially removed, reduction in chalone levels would thus result in an up-regulation of tissue production. The view that chalones are secreted factors was supported by in vitro experiments, and by experiments with parabiotically joined pairs of animals in which partial hepatectomy in one animal led to liver cell proliferation in the other [32
Although many of the original, in vitro–defined chalones have yet to be fully characterized, genetic studies in the 1990s demonstrated that growth and differentiation factor 8 (GDF8)/myostatin (Mstn1
, MGI:95691), a member of the transforming growth factor β (TGFβ) superfamily of secreted signaling molecules, is specifically expressed by striated muscle cells (the terminal-stage cells of muscle lineages), inhibits the production of muscle, and when genetically eliminated from animals, results in the production of supernumerary muscle cells and an increase in muscle mass [33
]. Subsequently, GDF11 (MGI:1338027)—a close relative of GDF8—was shown to be produced specifically by cells of the neuronal lineage of the mouse OE, and to provide feedback to inhibit the production of neurons (olfactory receptor neurons; ORNs) in that system [34
]. Animals deficient in GDF11 also develop supernumerary ORNs. In recent years, factors that exert negative feedback on growth have been described for many other tissues, including skin, liver, bone, brain, blood cells, retina, and hair (Table S1
). Many of these factors turn out to be members of the TGFβ superfamily, especially the TGFβ/activin branch of that superfamily [35
The OE of the mouse is a particularly useful system for studying lineage progression and feedback: It is continually self-renewing; its lineage stages are well defined; its cells can be studied in tissue culture; and it can be manipulated in vivo through genetic, chemical, or surgical means [36
]. The OE neuronal lineage consists of a stem cell (which expresses Sox2
[MGI: 98364], a gene encoding an SRY-box transcription factor), that gives rise to cells that express the proneural gene Mash1
, MGI: 96919), which in turn give rise to cells that express another proneural gene, Neurogenin1
; MGI: 107754), which in turn give rise to cells that exit the cell cycle and differentiate into ORNs. Recent data have raised the possibility that the Sox2+
stages are not truly distinct, but rather are interchangeable states of the stem cell (K. K. Gokoffski et al., unpublished data). However, the Ngn1+
cell—which is usually referred to as the Immediate Neuronal Precursor, or INP—is clearly a distinct transit-amplifying cell stage (A; [34
Strategies for Feedback Regulation of Transit-Amplifying Cells
The INP appears to give rise solely to ORNs, i.e., it does not represent a lineage branch point [39
]. It is therefore interesting that the feedback actions of GDF11 seem to be directed specifically at INPs [34
]: In vitro, GDF11 completely, but reversibly, arrests INP divisions, yet it has no effect on proliferation of Mash1+
cells. In vivo, the increase in neuronal number observed in Gdf11−/−
mice is accompanied by an increase in INPs, but not in Mash1+
cells. These data imply that GDF11 regulates tissue size by inhibiting the proliferation of a committed transit-amplifying cell.
Because GDF11 can slow and even arrest INP divisions, it is natural to model GDF11-mediated negative feedback as an increase in the cell-cycle length of the INP (B). Indeed, there is abundant literature showing that GDF11, GDF8, and other TGFβ superfamily members slow rates of progression through the cell cycle, at least in part by inducing cyclin-dependent kinase inhibitors [34
]. Increasing the INP cell-cycle length is equivalent to decreasing its v
(B). Unfortunately, the result in C states that the steady state outputs of lineage systems are independent of all v
except for that of the stem cell (v0
). This makes intuitive sense: if one decreases the division rate of an intermediate-stage cell in a lineage, the unchanged influx of cells from the previous lineage stage will cause its numbers to rise proportionately. From the standpoint of the lineage output, the two effects will cancel.
Apparently then, having GDF11 (or any other factor) feed back onto the INP cell division rate can be of no use in controlling the steady state level of ORNs. Could such feedback serve a function related to some other performance objective, such as rate control? As mentioned earlier, without control, lineage systems would be expected to return to steady state after a perturbation (i.e., regenerate) with a time scale similar to that over which terminal-stage cells normally turn over. In principle, feedback onto the cell division rate of a lineage intermediate could improve this. However, as explained below, the utility of this strategy turns out to be very limited:
C shows a simulated regeneration experiment in which output, via GDF11, feeds back onto v1
. At the start of the experiment, all ORNs are synchronously destroyed, and the time course of the return to steady state is followed (this type of perturbation can be produced experimentally by transecting the olfactory nerve or removing one or both olfactory bulbs of the brain [45
]). For comparison, the figure also shows what the time course of the return to steady state would be in the absence of feedback (dashed line). From C, we can see that feedback enables the system to regenerate faster, but we also observe a very high proportion of INPs (they are virtually as numerous, at steady state, as ORNs). It turns out that speeding up regeneration requires a large feedback gain (the parameter h in B), which in turn drives down steady state ORN numbers (relative to other cells). If we define progenitor load
as the percentage of the entire tissue that is composed of progenitors (stem cells plus INPs), we find that requiring the steady state progenitor load to be less than 50% limits any improvement in regeneration speed to about 3.2-fold; restricting progenitor load to 10% drops this value to about 2.6-fold (Figures S16
in Protocols S1
). In fact, experimental data indicate that the progenitor load in the OE is below 10% [46
There is another cost of achieving fast regeneration through feedback on v1
: the lower the progenitor load, the more necessary it becomes to use values of p1
that are perilously close to 0.5 (i.e., nearly half the output of INPs needs to be more INPs; Figures S16
in Protocols S1
). As discussed earlier, when p
-parameters are close to 0.5, system output becomes extremely sensitive to small variations in those parameters (and thus very fragile).
All told, feeding back onto the rate at which INPs divide does not seem to be a particularly good control strategy. We wondered whether GDF11 might do a better job if it fed back onto a different parameter of INP growth: p1, the replication, or amplification, probability. Analysis of a model of this sort of feedback (D) reveals several remarkable things:
First, with feedback on p1
, the constraint p1
≤ 0.5 goes away: Any INP replication probability allows for establishment of a steady state. Second, the fragility of the steady state output can be substantially reduced. In particular, sensitivity to the number of stem cells, the rate of stem cell division, and the death rate of terminally differentiated cells can be made arbitrarily small for appropriate parameter choices. Sensitivity to p1
can also be greatly reduced (to values <1), even if p1
is large (Figures S1
in Protocols S1
Finally, such a system can mount explosive regeneration after a perturbation. In some cases, the return to steady state can be as much as 100 times faster than in the absence of feedback. Furthermore, this can be accomplished without the need for a high progenitor load. E shows this behavior for a particularly effective set of parameters. Notice how, in response to an acute loss of terminal-stage cells (ORNs), transit-amplifying cells (INPs) undergo a rapid, but transient, increase in number, following which, terminal-stage cells are restored rapidly to values close to steady state. This sort of behavior closely parallels what is seen in the OE following olfactory bulbectomy (in which ORN degeneration is induced by olfactory bulb removal): a transient upsurge in progenitor cell numbers, followed by a wave of neuronal production [20
GDF11 Controls Replication Probabilities
The fact that feedback aimed at p1
can, in theory, produce more useful and realistic behaviors than feedback aimed at v1
, raised the possibility that the actual target of GDF11 might be p1
, and not v1
, as initially thought. To resolve this issue, we carried out tissue culture experiments in which mouse OE progenitor cells were pulse-labeled with 5-bromo-2-deoxyuridine (BrdU; to label cells undergoing division), and evaluated at successive times thereafter to determine when the progeny of dividing cells acquire immunoreactivity for NCAM, a marker for terminally differentiated ORNs. As shown previously, most dividing cells in these cultures are INPs, and their cell cycle length is about 17 h [39
]. If all INP divisions result in production of ORNs, the acquisition of NCAM immunoreactivity by all BrdU-labeled cells should occur after sufficient time to progress through the rest of S-phase, G2-phase, M-phase, and however long it takes for NCAM levels to rise above the threshold of detection. If some INPs replicate, however, then a fraction of labeled cells will not express NCAM until one cell cycle (~17 h) later (if the replicating fraction is high enough, some progeny will go through several cell cycles before acquiring NCAM immunoreactivity; cf. [39
]). Accordingly, delay in the onset of NCAM expression can be used as a measure of the INP replication probability.
shows the effect of GDF11 (added to the culture medium 12 h prior to BrdU labeling) on acquisition of NCAM expression by BrdU pulse-labeled cells. In J, data for two different “chase” periods are graphed. In the absence of GDF11, about 60% of BrdU-labeled cells become NCAM-positive within 18 h. In the presence of low levels of GDF11, this percentage rises as high as 75%, then falls again at high concentrations of GDF11 to less than 10%.
Experimental Demonstration That GDF11 Regulates p1 and v1
The increase in neuronal differentiation in response to low levels of GDF11 documents that GDF11 indeed suppresses INP replication (i.e., it lowers p1
). The fact that this increase gives way to a large decrease in neuronal differentiation at high GDF11 levels is most likely due to the additional effect of GDF11 on the rate of cell cycle progression: As the INP cell cycle is progressively lengthened, one would expect that an 18-h chase period would cease being long enough to allow BrdU-labeled cells to go on to differentiate. This would lead to a sharp drop-off in the percentage of BrdU-labeled cells that acquire NCAM expression, but with longer chase times (e.g., 36 h), this effect would be overcome. That is indeed what is observed (J). A numerical simulation of the experiment, in which GDF11 negatively regulates both p1
, replicates both qualitative and quantitative features of the experimental data (K; Protocols S1
, section 10).
Having the output of the OE lineage feed back onto p1 seems to be an effective strategy for meeting two control objectives: steady state robustness (low sensitivity to stem cell number χ0, cell division rates v0, and v1, and the death rate constant of the terminal-stage cell d) and rapid regeneration. But the ability to meet each objective separately does not guarantee that both can be met together (i.e., for the same sets of parameters).
As it turns out, the two strategies are largely incompatible. Numerical exploration of the parameter space shows a strong negative correlation between robustness and enhancement of regeneration (A). Cases for which the sensitivity to χ0, v0, or d is less than 0.4 (i.e., a 2-fold change in parameter will cause ≤32% change in output), generally do not exhibit acceleration in regeneration speed exceeding approximately 8-fold. In fact, this result is skewed by cases in which regeneration speed goes from extremely slow (in the absence of feedback) to merely very slow. If one restricts the analysis to cases in which regeneration from complete loss of terminal-stage cells is 80% complete in fewer than 29 transit-amplifying cell cycles (~20 d for INPs), then to achieve parameter sensitivities less than 0.4, the best possible improvement in regeneration speed is less than 2-fold (A and B).
Performance Tradeoffs Associated with Feedback Strategies
Upon closer inspection, other unfortunate tradeoffs can be seen: For the cases in A, improvement in regeneration speed was calculated by simulating a complete loss of terminal-stage cells and then measuring the return to steady state. If we use a milder perturbation (a 75% loss of terminal-stage cells), but otherwise the same parameters, the return to steady state is, unexpectedly, quite slow (C). The need to sustain injury that is massive before regeneration can be rapid hardly seems like a good strategy for an organism in the real world. To define the conditions under which this phenomenon occurs, we calculated, for all the cases in A, the ratio of two regeneration times: the time for regeneration from a 100% perturbation, and the time for regeneration from a 75% perturbation. In D, this value (“speed ratio”) is plotted against fold improvement in regeneration speed (for the 100% perturbation, compared with no feedback). The data show that the speed of regeneration following massive injury cannot be improved by more than about 3-fold, without sacrificing the speed of regeneration following less-than-massive injury.
Altogether, tradeoffs among regeneration speed, sensitivity to parameters, and sensitivity to initial conditions make the control strategy of having GDF11 feed back onto p1
less attractive than it originally seemed. Analysis of cases in which GDF11 inhibits both p1
(which corresponds most closely to what GDF11 does in vitro; J and K) shows some improvement in the tradeoff between regeneration speed and parameter sensitivity, but the effect is not dramatic (Figure S18
in Protocols S1
). Accordingly, we wondered whether additional control elements might still be missing.
Two Loops Are Better Than One
As mentioned in Table S1
, many feedback inhibitors of tissue and organ growth belong to the TGFβ superfamily of growth factors, with those of the TGFβ/activin branch (which signals through the intracellular proteins Smad2 and Smad3) being the most highly represented. Recently, we found that activinβB (Inhbb
; MGI: 96571; hereafter referred to simply as “activin”) is highly expressed in the OE and, like GDF11, has growth-inhibitory effects on the neuronal lineage. Unlike GDF11, however, activin's effects are aimed specifically at the Sox2+
populations, and not at INPs (K. K. Gokoffski et al., unpublished data). This implies that two feedback loops exist in the OE, one aimed at stem cells, and one aimed at transit-amplifying cells (E).
Like GDF11, activin could potentially feed back onto a v-parameter (namely v0, the rate of stem cell division) or a p-parameter (namely p0, the stem cell replication probability), or both. For technical reasons, a pulse-chase experiment similar to that in cannot be performed to sort this out. However, we infer that feedback onto p0 must occur, because Sox2+ and Mash1+ populations are markedly expanded in the OE of ActβB−/− mice (K. K. Gokoffski et al., unpublished data). If activin only regulated v0, loss of activin would result in stem cells that cycle faster, but it could not increase their numbers.
Interestingly, when we add the feedback effects of both activin and GDF11 into the equations for the behavior of the ORN lineage, the expression for the steady state value of ORNs becomes very simple: (2p0
, where j
is the feedback gain for activin (Protocols S1
, section 4). This constitutes a dramatic improvement in robustness—the system will, at steady state, always produce the same number of terminal-stage cells regardless of how many stem cells it starts with, how fast stem cells divide, or how quickly terminal-stage cells are lost.
Perhaps even more strikingly, the problematic constraint that the stem cell population must intrinsically “know” to replicate exactly half the time (p0 = 0.5) vanishes. As long as p0 > 0.5, feedback automatically ensures that the stem cell population behaves in the necessary way.
All of these improvements in steady state control come solely from the single feedback loop of system output onto p0. When such a loop is in place, however, feedback onto other p- and v-parameters can have additional useful effects:
Consider, for example, the matter of regeneration speed, which we previously found could be increased through feedback onto p1
, but only by sacrificing robustness, low progenitor loads, or the ability to regenerate quickly from a variety of initial conditions (C and A–D). When feedback is directed solely at stem cells, we also fail to achieve good performance: Feedback onto p0
hardly improves regeneration speed at all (Figure S19
in Protocols S1
), and although feedback onto p0
together can produce fast rates of regeneration (Figure S21
in Protocols S1
), those rates still show a very sensitive dependence on initial conditions (Figure S22
in Protocols S1
In contrast, when feedback is directed at both stem and transit-amplifying cell stages—i.e., the arrangement that actually occurs in the OE—it becomes possible to achieve very rapid regeneration, with low progenitor loads, from almost any starting conditions. This includes conditions in which variable numbers of stem, transit-amplifying, or terminal-stage cells are depleted. F shows an example of such a case.
Not only is such performance possible, it occurs over a substantial fraction of the parameter space (that is, a substantial fraction of randomly chosen sets of parameters meet all of these performance objectives). A shows graphically how, as feedback loops are added one at a time, good control (robustness, stability, low progenitor load, and fast regeneration from a variety of conditions) is found over an increasing fraction of the parameter space (exploring wide ranges on all parameters). In evaluating the magnitude of this effect, it should be noted that fractions of parameter space in the range of 0.1%–1.5% are remarkably high, given the numbers of parameters in each model (cf. [52
]). For example, when there are eight independent parameters (as there are when feedback is directed at p0
, and v1
), good performance over 0.1% of the parameter space means that the average parameter value “works” over 42% (~0.0011/8
) of its range. In , most parameters were explored over three orders of magnitude (i.e., they were randomly selected from a log-uniform distribution with a 1,000-fold range), so for such cases, 42% means that the average parameter can be varied over an 18-fold range (1,0000.42
) without loss of good control.
Effects of Feedback Configuration on Regeneration from Diverse Perturbations
What is the significance of a control system that works over a large portion of its parameter space? It means that the output of the system can be adjusted (through changes to the parameters) without the control strategy itself being jeopardized. From a biological perspective, this means that the system is evolvable, a feature we should expect to observe in most biological control systems [53
Sensitivity and Geometry
So far, we have said much about the cell stages and processes that are targets for feedback in cell lineages, and little about the quantitative details of feedback signals. In and , feedback was modeled using Hill functions; these are natural choices for the actions of secreted growth factors, since saturable binding of ligands to receptors is usually well described by them [54
Hill functions typically employ a parameter n, the Hill coefficient, to fit dose-response relationships that are positively (n > 1) or negatively (n < 1) cooperative. In , , and A, a Hill coefficient of 1 was used, but more detailed exploration of the two-loop feedback system (with feedback on p0, v0, p1, and v1) shows that system performance increases steadily as n goes from 0.5 to 2 (B and C). This makes intuitive sense if we consider that high values of n make Hill functions more switch-like. In the limit of a perfect switch (infinite n), the drive for increased growth would be zero when output is at the desired value, yet maximal when output is even slightly below the desired value. Such a strategy clearly achieves the fastest possible regeneration following a perturbation.
In biology, dose-response relationships that are fit by Hill coefficients other than 1 arise for a variety of reasons besides biochemical cooperativity; these include buffering, competition, feedback, and distributed multistep reactions [55
]. Generally speaking, Hill coefficients quantify the sensitivity of output to input (in the limit of high input, the Hill coefficient and the engineering definition of sensitivity are equivalent). Thus, in our models of feedback in the OE, Hill coefficients near 1 mean that the amount of activin and GDF11 signaling in stem cells and INPs (respectively) is roughly proportional (over some range) to the number of cells producing activin and GDF11 (i.e., the size of the tissue).
It occurred to us that this situation—feedback proportional to tissue size—might not be so easy for tissues to achieve. As a tissue grows in size, one can certainly envision the total amount of material it produces increasing proportionally, but it is the concentrations—not the amounts—of factors like GDF11 and activin to which cells respond. How the concentrations of secreted ligands change as tissues grow turns out to depend both on issues of geometry (tissue shape and boundary properties), and issues of cell biology (rates of ligand capture and turnover).
For example, consider a hypothetical tissue surrounded by a boundary across which macromolecules cannot diffuse. In this case, a secreted protein produced everywhere in the tissue should reach a steady state concentration determined by the balance between production and local degradation. If the tissue doubles in size, it will make twice as much of the protein, but distribute it over twice the volume. The result will be no change in concentration. In a truly “closed” tissue, secreted molecules cannot be used as part of a strategy for growth control.
Fortunately, epithelia, such as the OE, are not closed systems. Although tight junctions between epithelial cells prevent escape of molecules from the apical surface, there appears to be little or no impediment to diffusion across a basal lamina into the underlying connective tissue stroma [58
]. Within such a geometry, we may use approaches developed for the analysis of morphogen and signaling gradients [59
] to calculate expected intraepithelial distributions of secreted molecules (Protocols S1
, section 11).
The results of these calculations () show that when an epithelium is very thin, concentrations of secreted molecules in the intercellular space initially go up linearly with tissue size, but soon level off. Does the normal size range of the OE (adult thickness ~80 μm) lie in the linear region, or on the plateau? The answer depends on two factors: The first is the decay length of the molecule of interest. This is the average distance a molecule travels in tissue before being captured and degraded by cells, and is a function of its diffusion coefficient and rate of receptor-binding and degradation.
Effects of Geometry and Degradation on Levels of Secreted Molecules within Epithelia
The second factor is the ratio of decay length within the epithelium to decay length in the adjacent stroma (which, in most cases, simply reflects how much faster or slower degradation proceeds in one location versus the other). If that ratio is low—i.e., if molecules that diffuse into the stroma are not quickly degraded—then intraepithelial concentrations will be poorly sensitive to tissue size long before the epithelium reaches even a single decay length in thickness (A; Figure S27 in Protocols S1
In contrast, if the ratio of decay lengths between epithelium and stroma is high—i.e., if the stroma acts as a sink, quickly eliminating molecules that enter it—then average intraepithelial concentrations will rise more gradually, and not plateau until the epithelium has reached a size of several decay lengths (B). This effect is more pronounced if the concentration that matters is the concentration close to the basal surface of the epithelium, and not the average concentration over the entire epithelial thickness. At this basal location, concentration varies linearly with tissue size for many decay lengths (B; Figure S28
in Protocols S1
Estimates of intraepithelial decay lengths of TGFβ superfamily polypeptides, obtained both from measurements of morphogen gradients and from first-principles calculations, tend to be in the range of tens of micrometers [59
], i.e., on the order of, or less than, the normal thickness of the OE. This suggests that it would be difficult to use activin and GDF11 as “reporters” of OE size, if these molecules merely leaked into the stroma and were not rapidly degraded there (as in A): once the OE grew beyond 0.2 decay lengths in thickness, the poor sensitivity of activin and GDF11 concentrations to OE size would be functionally equivalent to feedback described by Hill coefficients less than 0.5. As already demonstrated (B), such low Hill coefficients undermine good control.
Accordingly, we infer that it would be strategically advantageous for the OE to possess a mechanism that rapidly removes activin and GDF11 in the underlying stroma, as well as a mechanism for restricting the location at which cells measure the level of activin and GDF11, to the basal surface of the tissue. Remarkably, the OE seems to have both:
First, the OE contains large amounts of the protein follistatin (FST; MGI: 95586) in its basement membrane and stroma (C; [34
]). FST not only binds and inhibits both activins and GDF11, it does so irreversibly, effectively eliminating them [67
]. That FST plays a central role in regulating GDF11 and activin function in the OE has recently been demonstrated genetically ([34
] and K. K. Gokoffski et al., unpublished data); what the analysis here provides is an explanation for why FST is used by the OE, and why it should be found primarily beneath the epithelium.
Second, the progenitor cells of the OE that respond to activin and GDF11 become increasingly polarized, during early development, to the basal side of the epithelium; eventually they lie within a few cell diameters of the basement membrane. This is shown in D and E, using in situ hybridization for Ngn1
to visualize INPs. Thus, the only concentrations of GDF11 and activin that progenitor cells sense are likely to be those near the basal surface of the epithelium. Interestingly, in many other types of epithelia, stem/progenitor cells also localize near the basement membrane, an observation that has long suggested the existence of a specialized microenvironment, or “niche,” in this region [70
The OE, a self-renewing tissue, maintains its size by continuous replacement of dying cells [51
]. Some organs—such as the mammalian brain—achieve a final size during development and largely cease proliferating [72
]. Such final-state (as opposed to steady state) systems also may be modeled using the equations in , by setting the terminal cell death rate constant, d
, to zero, and allowing replication probabilities to be below 0.5. Like steady state systems, they can be quite fragile.
This point is well illustrated by the mouse brain, which is composed of approximately 108
cells of neural lineage (neurons and glia; [16
]). Although brain cell number varies from mouse to mouse, within a given strain, the coefficient of variation is small, about 5% [16
]. If we hypothesize that the brain is “founded” by a pool of 105
progenitors (probably an overestimate), and we make the simplifying assumption that no cells die during development, then a 1,000-fold expansion in cell numbers is needed (). One way to accomplish this would be to have all progenitors replicate for a time equal to ten cell-cycle lengths (210
= 1,024), and then stop. With this strategy, final cell number will be linearly sensitive (i.e., proportional) to the initial size of the progenitor pool (A), and much more than linearly sensitive to the average length of the cell cycle, or the length of time allowed for proliferation (a mere 5% change in either parameter would produce a 30% change in output). If the brain is founded by fewer progenitors, this fragility only becomes more severe.
Behaviors of Final-State Systems
Now, let us consider a slightly more sophisticated strategy: a progenitor pool that undergoes a mixture of replicative and differentiative divisions, with a replication probability p
set below 0.5. Because proliferating cells replicate less than half the time, the progenitor pool runs out, and the tissue approaches a final state gradually, without need to count cell cycles or time. In this case, the final state is still linearly sensitive to the initial size of the progenitor pool, and although no longer sensitive to time or cell-cycle parameters, it is extremely sensitive to the value of p
itself, which must be very close to 0.5 to produce a 1,000-fold expansion in cell numbers (Protocols S1
, section 5).
One way to circumvent this extreme fragility is to allow p
to change over time, starting above 0.5 (promoting progenitor expansion), then falling below 0.5 (driving progenitor cell extinction). In fact, this very mechanism, illustrated in B, was introduced by Nowakowski et al. [75
] to explain the biphasic expansion and contraction of progenitor pools in the cerebral cortex, and it is supported by considerable experimental data (e.g., [76
]). Mathematical analysis (Protocols S1
, section 5) shows that sensitivity to p
is reduced by this strategy, but it still remains very high (Figure S5
in Protocols S1
). Moreover, the system now becomes quite sensitive to the rate at which p
declines (relative to the cell-cycle length; Figure S4
in Protocols S1
). In addition, such a system is still linearly sensitive to the initial size of the progenitor pool (B).
Given how difficult it seems to be to achieve even modestly robust final states, it is striking how much can be accomplished with the addition of just a single feedback loop. C illustrates a case much like the one in B, in which the p
-value of a progenitor pool declines over time, but this time, the decline is caused by feedback from terminal-stage cells. Superficially (that is, when not perturbed), it behaves just like the Nowakowski-Caviness model [75
], displaying expansion, contraction, and disappearance of a stem cell pool. Yet in this case, a 2-fold change in the initial number of stem cells produces only a minute (0.14%) change in the final state! Even sensitivity to the initial value of p
can be much lower (<5) than in the case without feedback (Figures S6
in Protocols S1
). Just as with our analysis of steady state systems, this sort of behavior arises only when feedback regulates replication probabilities (p
-parameters), and not when it regulates cell cycle lengths (v