Maintaining and escaping feedback control in hierarchically1
organised tissue: a case study of the intestinal epithelium2
Matthias M. Fischer,
1, ∗
Hanspeter Herzel,
1, †
and Nils Bl¨uthgen
1, ‡
3
1
Institute for Theoretical Biology, Charit´e and
4
Humboldt Universit¨at zu Berlin, 10115 Berlin, Germany
5
(Dated: June 11, 2021)6
The intestinal epithelium is one of the fastest renewing tissues in mammals with
an average turnover time of only a few days. It shows a remarkable degree of stability
towards external perturbations such as physical injuries or radiation damage. Tissue
renewal is driven by intestinal stem cells, and differentiated cells can de-differentiate if
the stem cell niche is lost after tissue damage. However, self-renewal and regeneration
require a tightly regulated balance to uphold tissue homoeostasis, and failure can lead
to tissue extinction or to unbounded growth and cancerous lesions. Here, we present
a mathematical model of intestinal epithelium population dynamics that is based on
the current mechanistic understanding of the underlying biological processes. We
derive conditions for stability and thereby identify mechanisms that may lead to loss
of homoeostasis. A key results is the existence of specific thresholds in feedbacks after
which unbounded growth occurs, and a subsequent convergence of the system to a
stable ratio of stem to non-stem cells. A biologically interesting property of the model
is that the number of differentiated cells at the steady-state can become invariant
to changes in their apoptosis rate. Moreover, we compare alternative mechanisms
for homeostasis with respect to their recovery dynamics after perturbation from
steady-state. Finally, we show that de-differentiation enables the system to recover
more gracefully after certain external perturbations, which however makes the system
more prone to loosing homoeostasis.
Keywords: Cancer, Colon, Dedifferentiation, Intestinal epithelium, Regeneration, Tissue
7
homoeostasis8
∗
matthias.fischer@charite.de
†
h.herzel@biologie.hu-berlin.de
‡
Corresponding author – nils.bluethgen@charite.de
(which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.
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2
I. INTRODUCTION9
A tissue is said to be hierarchically organised if it consists of different cell types constituting
10
a characteristic hierarchical structure. Generally, two classes of cells can be distinguished:
11
Adult stem cells have an unlimited capacity of indefinite self-renewal, but also differentiate
12
and thus directly or indirectly give rise to differentiated cells which perform the designated
13
function of the tissue [
1
]. Additionally in cases of tissue damage and regeneration the
14
dedifferentiation of differentiated cells back into cycling stem cells has been observed, for
15
instance in case of the intestinal epithelium [
2
,
3
], the airway epithelium [
4
], and the kidney
16
epithelium [
5
]. In order to uphold the homoeostasis of such a tissue in the face of external
17
perturbations, a tight regulation of the stem cell compartment is required. In case of tissue
18
damage, stem cells need to increase proliferation according to tissue requirements; however
19
over-proliferation of the stem cell compartment must be avoided in order to prevent unlimited
20
growth [
6
]. Such a tight control seems to be maintained through specific feedback loops
21
exerted by differentiated cells onto the stem cell compartment regulating the size of the
22
latter [
7
]. In contrast, control of the dedifferentiation of differentiated cells seems to be
23
exerted by the stem cell compartment (see Tata et al.
[4]
, and Beumer and Clevers
[8]24
as well as the references therein). Escaping one or multiple of these stability-conferring
25
control mechanisms may cause the tissue to loose homoeostasis and subsequently switch to a
26
behaviour of unbounded, malignant growth.27
28
The intestinal epithelium and the colon epithelium are prime examples of such hierarchically
29
organised tissues. Despite its single-layered, simple epithelial structure it is able to withstand
30
continuous mechanical, chemical and biological insults due to its specific tissue architecture
31
in combination with a high rate of cellular turnover [
9
]: Stem cells residing at the bottom
32
of the intestinal crypts cycle continuously approximately once per day and give rise to new
33
cells. These cells then mature while migrating upwards, until they terminally differentiate
34
and become part of the villi, eventually committing apoptosis and being shed off into the
35
intestinal lumen [
10
]. Control of the intestinal and the colon stem cell compartment is
36
realised via differentiated epithelial cells releasing Indian Hedgehog (Ihh), which stimulates
37
mesenchymal cells to release Bone Morphogentic Proteins (BMPs). These, in turn, interfere
38
with intracellular effects of WNT signalling and thus stimulate stem cell differentiation [
11
–
15
].
39
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3
40
Previous theoretical research on hierarchically organised tissues has often focussed on the
41
abstract case of arbitrary tissues: Rodriguez-Brenes et al.
[16]
considered arbitrary hierar-
42
chically organised tissues consisting of two compartments: a compartment of cycling and
43
differentiating stem cells, and a compartment of non-cycling differentiated cells committing
44
apoptosis at a fixed rate. They assumed that the differentiated cell compartment may exert
45
feedback onto the stem cells by both decreasing their rate of proliferation and by reducing
46
the probability of a stem cell division resulting in two daughter stem cells compared to the
47
probability of a division yielding two differentiated cells. They then studied the order in
48
which mutations in these feedbacks need to arise in a single new clone to enjoy a selective
49
advantage and spread throughout the system. Limiting their model to sigmoidal Hill-like
50
feedback functions, they then also fitted their model to a number of time-course data of the
51
overall population size of growing tissue from the literature. The same model was used in
52
Rodriguez-Brenes et al.
[17]
in order to reveal that during recovery from an injury significant
53
damped oscillations in the path back to the steady-state may occur, and that this oscillatory
54
behaviour is more pronounced when the stem cell load represents only a small fraction of the
55
entire cell population. Nonetheless, oscillations may still be avoided, however at the price of
56
slowing down the speed at which the system is able to recover after an injury. The same
57
model topology has also been studied by Sun and Komarova
[18]
using the framework of a
58
two-dimensional Markov process in order to obtain analytical solutions for the mean and
59
variance of the cell compartment sizes. Recently, Wodarz
[19]
has extended the model by also
60
taking into account the possibility of differentiated cells dedifferentiating into cycling stem
61
cells again. Assuming sigmoidal Hill-like feedback onto stem cell cycling rate and self-renewal
62
probability, he studied the effect of a linear and a sigmoidal dedifferentiation term, showing
63
how unbounded, cancerous tissue growth may arise as a consequence of escaping this feedback.
64
By means of numerical simulations, he also demonstrated how dedifferentiation may allow
65
for speedier regeneration dynamics after perturbations.66
67
More concrete theoretical studies have for example been carried out on the hematopoietic sys-
68
tem [
20
–
24
], the mammalian olfactory epithelium [
25
,
26
], or on the development, treatment
69
and recurrence of breast cancer [
27
]. In contrast, however, theoretical examinations on the
70
homoeostasis and dynamics of the intestinal and colon epithelium have been rare. A note-
71
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4
worthy exception is Johnston et al.
[28]
, who have derived and studied a three-compartment
72
ODE model consisting of stem, transit-amplifying and terminally differentiated cells. They
73
assumed that the first two of these three compartments limit themself either via a negative
74
quadratic term (reminiscent of classical single-species population dynamic models which
75
assume logistic growth [
29
]) or via a negative saturating term. However, no mechanistic
76
justification of these models has been presented in the paper, possible also owing to the fact
77
that our mechanistic understanding of the biology of intestinal stem cells has been rather
78
limited until very recently [9].79
80
In this work, we set out to derive a model of intestinal and colon epithelial population
81
dynamics based on our current understanding of the involved underlying biological processes.
82
We use this model in order to answer some fundamental questions about maintaining and
83
losing the homoeostatic stability of this system: Both for the case without dedifferentiation
84
and for the case with dedifferentiation, we derive all possible ways the system can loose
85
stability and exhibit unbounded malignant growth. We prove analytically how allowing for
86
dedifferentiation opens up an additional way of losing stability and switching to unbounded
87
growth. For all cases of unbounded growth, we prove that – under the biologically reasonable
88
assumption of saturating rate functions – after some period of transient behaviour the system
89
will always converge to a stable ratio of cell types which we can calculate analytically. A
90
special focus is given to the study of the transient behaviour of the system while recovering
91
from different kinds of external perturbations. We examine how the shape of the feedback
92
functions shapes the system behaviour during recovery, and will compare how graceful and
93
efficient the colon model is able to recover from different kinds of external perturbations
94
compared to other imaginable model topologies throughout the entire model parameter
95
space. Finally, we show analytically and illustrate with numerical simulations how adding
96
dedifferentiation can tremendously speed up recovery and reduce frequency and duration of
97
oscillations, which especially applies to the case of perturbations of the stem cell compartment.
98
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5
II. MATERIALS AND METHODS99
A. Colon epithelium model100
Our main model consists of two cell compartments
S
and
D
, denoting stem cells, and
101
differentiated cells, respectively, following earlier approaches such as Rodriguez-Brenes et al.
102
[16]
. Stem cells cycle at a constant rate
β >
0, whereas differentiated cells are cell-cycle
103
arrested, but die with an apoptosis rate
ω >
0. Finally, stem cells differentiate with a rate
104
δ
(
D
) that is a function of the size of the differentiated cell compartment. Overall, the model
105
is described by the following set of two coupled ordinary differential equations:106
dS(t)
dt
= βS(t) − δ(D)S(t)
dD(t)
dt
= δ(D)S(t) − ωD(t),
(1)
where
β, ω ∈ R
+
and
δ
is a continuously differentiable and monotonically increasing function
107
R
+
→ R
+
. A sketch of the model is shown in Figure 1a. Note that at this model does not
108
include the possibility of dedifferentiation, and we will extend this later.109
110
At this point, the structural similarity between our model and the classical Lotka-Volterra
111
model of predator-prey dynamics [
30
,
31
] may also be pointed out. In particular, if
δ
is a
112
linear function with an intercept of zero, i.e. if the basal stem cell differentiation rate in
113
the absence of differentiated cells is zero, then the models are mathematically identical. As
114
remarked by Peschel and Mende
[32]
, however, even such minor qualitative changes to the
115
Lotka-Volterra model will affect its qualitative behaviour and cause the loss of its typical
116
harmonic oscillations and for instance the emergence of a limit cycle or a stable steady-state
117
instead.118
B. Comparison of different model topologies119
Next, we generalise the previous model to the family of all models containing exactly one
120
explicit feedback loop from one compartment onto one rate parameter. Since we have
121
two compartments which could potentially be able to exert a feedback (stem cells
S
, and
122
differentiated cells
D
), and three rates which could potentially be affected by such a feedback
123
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