Supporting Material
A Reaction-Diffusion Mechanism Influences Cell Lineage Progression as a Basis for
Formation, Regeneration, and Stability of Intestinal Crypts
Lei Zhang1,3,4, Arthur D. Lander2,3,4, Qing Nie1,3,4§
1
Department of Mathematics, University of California, Irvine, CA 92697, USA
Department of Development and Cell Biology, University of California, Irvine, CA
92697, USA
3
Center for Complex Biological Systems, University of California, Irvine, CA 92697,
USA
4
Center for Mathematical and Computational Biology, University of California, Irvine,
CA 92697, USA
2
§
Corresponding author
Email addresses:
Lei Zhang: [email protected]
Arthur D. Lander: [email protected]
Qing Nie: [email protected]
Table of Contents
I. Cell Lineage Model and Numerical Methods
II. Linear Stability Analysis
III. Tables
IV. Figures
V. References
1
I. Cell Lineage Model and Numerical Methods
Cell and morphogen equations
In the cell equation (1), we assume that the volume of substance between cells is
negligible and the overall cell population maintains a uniform density, i.e. C0  C1  1 up
to normalization, in both space and time. Then the velocity equation in (1) can be
obtained by taking a sum of two equations in (1). Since the TD cells may have higher
death rate on the luminal surface, the degradation rate for TD cell d1 is modeled as a
function of C1 :
d1  d1  tanh(
C1  C
1
 1) / 2.
[S1.1]
Here, d1 is the maximal degradation rate of TD cell.  1 is the transition width of decay
and C is the TD cell concentration at the boundary of luminal surface. In the simulations,
we take 1  0.05,C  0.6.
In Eq. (3), the convection timescale is due to the tissue growth is L and the diffusion
V
2
timescale is L D , where L represents the length scale, V the average velocity, and D is
the diffusion rate. The ratio between these two time scales is D . The steady state
VL
crypt length is 400 m , and the stem cell cycle length is between 12 and 32h with an
average of 24h [1]. As a result, the ratio is usually large (e.g., 10.8 in this case) and we
can neglect the convection terms in the equations (3). Moreover, because the typical
timescales of cell cycle lengths and tissue growth are days, whereas the timescale for
molecule interactions is typically hours, the morphogen system quickly reaches steady
state with a time scale of cell cycle lengths [2], therefore, we use the quasi-steady-state
approximation:

 2 [Wnt]
W
D
 W s 2  0C0  (1  ( [Wnt]) nW )(1  ( [In])mW )  dW [Wnt]  0,
W
I

2
W
  [In]
[S1.2]

 d I [In]  0,
 DI
2
s
1  (W [Wnt]) nI

  2 [BMP]
 DB
 1C1  d B [BMP]  0,
s 2

Crypt Dynamics
We use a smooth curve (x(s,t), y(s,t)) to represent the shape of crypts and spatial
dynamics of crypts through gradient of the energy functional [3]. The energy
E(x, y,C0 ,C1 ) depends on progenitor cells based on the following assumptions:
1. Population of progenitor cells is mostly located at the bottom of crypts, which
represents a minimum energy of the crypts.
2
2. The depth and width of crypts depend on the maximum density and total amount
of progenitor cells, respectively.
By using these basic and simple assumptions, we choose two strategies to employ the
crypt mechanics:
Strategy I: We fix two end points of the crypt and choose the initial crypt (luminal
surface) as a flat curve, then one simple form of the energy is
d(C0 ) | x  arg max(C0 (x)) |
x
[S1.3]
E(x, y,C0 )  (y  max(C0 )(tanh(
)  1) / 2  b)2 ,
0
where d(C0 ) is the diameter of crypt which is chosen as a scale of integration of the
progenitor cells along a crypt. The parameter b is the location of the luminal surface and
 0 represents the sharpness of the crypt shape. Two end points are
(x(1,t), y(1,t))  (0.1,0) and (x(1,t), y(1,t))  (0.1,0) where s [1,1] .
This strategy is used for simulations in Fig. 2-5, 6A-C, 7, and 8.
Strategy II: We can apply the polar coordinate (r, ) to the energy functional shown in
Strategy I:
d(C0 ) | r  arg max(C0 (r)) |
x
[S1.4]
E(r,C0 )  (r  length(C0 )(tanh(
)  1) / 2  b)2 .
0
We take x  r  cos( ) and y  r  sin( ) with     2 (  is the ratio of the major
axis and minor axis in the ellipse). We use this strategy for crypt multiplication shown in
Fig. 6D by choosing   3.5 .
Crypt model in growing domain
The dynamics of the crypt length Lmax is governed by the crypt dynamics
b
Lmax (t) 

x&2 ( ,t)  y&2 ( ,t)d ,
a
where  is the parametrization of the crypt curve (x(t), y(t)) , and a,b are two end points
of the curve.
To solve the system of cells and morphogens in growing domain, we first transform Eqs.
L
(S1.1-S1.3) by scaling s with max such that the new spatial variable is in a fixed domain
2
[-1, 1] and the dynamics of Lmax is embedded in the coefficients of the transformed
PDEs.
We make the change of variables:
2s
  t, x 
,
Lmax ( )
then the transformed cell and morphogen equations (1) and (3) become
3
 C0 xL 'max ( ) C0
2 (VC0 )


 v0 (2 p0  1)C0 ,



L
(

)
x
L
(

)
x
max
max

 C1 xL 'max ( ) C1
2 (VC1 )


 v0 (2(1  p0 )C0 )  d1C1 ,

Lmax ( ) x Lmax (t) x
 
 2 V
 v0C0  d1C1

 Lmax ( ) x
[S1.5]
and
 [Wnt] xL 'max [Wnt]
2 (V[Wnt])
4DW  2 [Wnt]
 2
   L ( ) x  L ( )
x
Lmax ( ) x 2
max
max


W
  0 C0 
 dW [Wnt],

 nW
mW
(1

(

[Wnt])
)(1

(

[In])
)
W
I

 [In] xL 'max [In]
2 (V[In])
4DI  2 [In]


 2

[S1.6]
Lmax ( ) x
Lmax ( ) x
Lmax ( ) x 2
 

W


 d I [In],
1  (W [Wnt]) nI


2
 [BMP]  xL 'max [BMP]  2 (V[BMP])  4DB  [BMP]
Lmax ( ) x
Lmax ( )
x
L2max ( ) x 2
 

 1C1  d B [BMP],

L
L
The computational domain is chosen as [  max , max ] with Lmax  0.2 for the fixed
2
2
domain. A periodic boundary condition is chosen in order to account for spatial
distribution of cells in periodic multiple crypts. To solve the systems in both fixed
domain and growing domain, we apply Fourier spectral method for the spatial
discretization, and a semi-implicit scheme is carried out for the temporal discretization
[4].
A typical number of spatial grid points used in the simulations are 256 with a time-step
size 10 4 . Numerical tests have been conducted to ensure sufficient spatial and temporal
resolutions for convergence of the numerical solution. The Turing pattern of crypt is very
robust to the noise in the progenitor cells and the initial Wnt distribution if the removal
rate of Wnt is low (Fig. S7). For high removal rate of Wnt, the number of crypts at steady
state may be varied for different initial distribution of Wnt (Fig. 3).
In our one-dimensional models, we focused on cell movement and dynamics of Wnt and
BMP along the crypt direction without incorporating details of the geometry and growth
of the crypt. To incorporate dynamic growth and geometry of crypt, one needs to modify
Eqs (1-4) by replacing the independent variable s by the dynamic arc-length
variable s( ,t) where  is the independent parametrization variable in a fixed domain
(e.g. [0,1]). Now the derivatives with respect to s in Eqs (1-4) become
4

1 

,
s s 
2
1 2
s 
 2
 3
2
2
s
s 
s 
with s( ,t)  x2 ( ,t)  y2 ( ,t) being updated in time through Eq. (4). The new overall
system for one-dimensional growth model then can be solved by using the similar
computational approach in this paper.
II. Linear Stability Analysis
To investigate the conditions of Turing instability for molecules Wnt and Wnt inhibitor,
we apply the method in [5] to study the first two equations in [S1.4]:
[Wnt]
2 [Wnt]
W
 DW
  0 C0 
 dW [Wnt],
2
nW
t
x
(1  (W [Wnt]) )(1  ( I [In])mW )
[S2.1]
[In]
2 [In]
W
 DI

 dI [In].
2
t
x
1  (W [Wnt]) nI
[S2.2]
Assuming W = 1 ,  I ? 1 , W = 1 , which greatly simplifies the calculation, the system
is reduced to
[Wnt]
2 [Wnt]
[Wnt]nw
 Dw
 0C0  dW [Wnt]  
,
t
x 2
[In]mw
[S2.3]
[In]
2 [In]
nI
 DI
 d I [In]  [Wnt] ,
t
x 2
where    W  I (W )nw / ( I )nI ,    w (w )nI .
In order to simplify the equation further, we take nw  nI  2, mw  1 , which are the
numbers used for all simulations and figures, and we normalize the system with a new
scale:
d 
d2
W  w [Wnt],
I  w 2 [In],
dI
dI
t  dwt,
x
dw
Dw  x,
Then Eq. [S2.3] in the new variables becomes
W
W2
 W    W 
,
t
I
I
 DI  d(W 2  I ),
t
2
D
d
where D  I D ,
d Id ,

,
x 2
w
w
[S2.4]
  0C0 d  .
I
First, the homogeneous steady-state solution is
5
W0  1  ,
I 0  W02 .
The perturbation of this solution takes the form:
W  W0  W ,
I  I 0   I with
W  W0 et eikx ,
 I   I 0 et eikx
Following the method in [5], for the general system
ut  u   f (u, v),

vt  Dv   g(u, v),
the conditions for the generation of spatial Turing patterns at the steady state are
fu  gv  0,
fu gv  fv gu  0,
Dfu  gv  0,
(Dfu  gv )2  4D( fu gv  fv gu )  0.
The derivatives fu and gv must be of opposite sign. Thus, we have
1 
1
fw 
,
fI  
 0,
1 
(1   )2
gw  2d(1   )  0,
gI  d  0.
Since fu and gv must have opposite signs, it requires   1 . With these expressions, the
conditions of instability require
1  
1    d

1   d


[S2.5]
1   D
d  0

(D 1    d)2  4Dd
 1 

 C
Notice that 0    0 0
dI  1 , we have
[S2.6]
0   d I C
0
By setting   0.1 , the phase diagram of Turing instability region (Fig. S1) shows that, in
addition to the requirement that Wnt inhibitor diffuses faster than Wnt, Wnt inhibitor has
to adapt rapidly to any change of the Wnt, which corresponds to the case that the ratio
d
between removal rate d I and dW satisfies a constraint, i.e., I
dW  cons . In addition, the
ratio of the two diffusion coefficients between Wnt inhibitor and Wnt determines the
ratio of the corresponding removal rates for Turing instability that is necessary for
formation of heterogeneous pattern.
6
III. Tables
Table S1: Parameters used for Fig. 2-8 and Fig. S1-S6 unless otherwise specified ‘--’
means “not applicable”.
Parameters
Dw , DI , D B
 0 , 1
p0
n, m, nw , mw , nl ,
 W , B
v0
d1
 w , w
w ,  I , w
C
0
1
Value
10 7 , 10 6 , 10 7
2  10 4 , 2  10 4
1
2, 2, 2, 1, 2
4, 10
0.25
0.05
120, 0.1
0.2, 10 4 , 0.1
0.6
0.002
0.1
Units
cm 2 s 1
s 1 M
-- M 1
Per cell cycle
Per cell cycle
s 1 M
 M 1
cm1
cm
cm
Table S2: Parameters used in Fig. 2-8 and Fig. S1-S6 ‘--’ means “not applicable”.
Parameters
dW
dI
dB
1
1
(s )
(s )
( s 1 )
Fig. 2
2  103
2  103
1.5  103
Fig. 3
Fig. 4A
Fig. 4B-C
Fig. 5
Fig. 6A-D
Fig. 6E
Fig. 7, 8
Fig. S2
Fig. S3
Fig. S4
Fig. S5
Fig. S6
-2  103
2  103
2  103
2  103
2  103
1  10 3
2  10 2
2  103
2  103
1  10 3
2  103
-2  103
2  103
2  103
2  103
2  103
1  10 3
2  10 2
2  103
2  103
1  10 3
2  103
1.5  103
-1.5  103
1.5  103
1.5  103
1.5  103
1.5  103
1.5  103
1.5  103
1.5  103
1.5  103
1.5  103
7
IV. Figures
Figure S1. Phase diagram of Turing instability. For generating Turing instability, the
ratio of the two diffusion coefficients between Wnt inhibitor and Wnt determines the
ratio of the corresponding removal rates. “Unstable” means the region satisfying the
conditions of Turing instability; “Stable” represents the region not satisfying the
D
dI
conditions of Turing instability. x-axis: D  I
Dw , y-axis: d  dw .
8
Figure S2. Initial conditions of Wnt at the removal rate of Wnt dW  0.02s 1 for
Figure 3 in main text.
(A) Initial condition for five crypts at the steady state;
(B) Initial condition for six crypts at the steady state.
9
Figure S3: Dynamics of crypt regeneration when Crypt I was removed.
(A) At T  0 , all progenitor cells in Crypt I are removed from the wild-type steady crypt
( T  100 in Fig. 2A). Progenitor cells traveling from Crypt II start to accumulate and
self-renew, leading to formation of new Crypt I ( T  200 ). All progenitor cells in both
crypts are ultimately regenerated ( T  300 ). (B) The maximal density of progenitor cells
in Crypt I (blue dash line) and Crypt II (red solid line) as functions of time. Cell velocity,
V , (C) and progenitor cell flux, VC0 , (D) are plotted at the time T=0 (blue dash curve),
T=200 (black dot curve), and T=300 (red solid curve).
10
Figure S4. For a system with BMP, different initial distributions of progenitor cells
result in different multiple steady state crypts. However, for the system without
BMP, the same five-crypt steady state is always generated at the steady state for all
the three initial conditions.
(A) Two localized spots of the progenitor cells at T  0 , leading to two steady crypts
with BMP and five steady crypts without BMP.
(B) Three localized spots of the progenitor cells at T  0 , leading to three steady crypts
with BMP and five steady crypts without BMP.
(C) Five localized spots of the progenitor cells at T  0 , leading to five steady crypts
with BMP and five steady crypts without BMP.
11
Figure S5: Temporal dynamics of the progenitor cells by adding exogenous Wnt.
(A) Temporal dynamics of Fig. 10 (B) at   4  104 s 1 M ,
(B) Temporal dynamics of Fig. 10 (D) at   4  10 3 s 1 M ,
(C) Temporal dynamics of Fig. 10 (E) at   0s 1 M .
12
Figure S6: Noise effect on the formation of crypt pattern. Taking different noises in
the initial Wnt at dW  2  103 (A) and (B), it results in the same periodic two-crypt
pattern (C) and (D), respectively. The crypt formation is also very robust to the noise in
the progenitor cells. Two samples of progenitor cells noise are plotted in (E) and (F).
13
Figure S7: Replication probabilities of progenitor cells with the maximal death rate of
TD cell d1  0.02 (red solid curve) and d1  0.2 (blue dash curve).
14
V. References
1.
2.
3.
4.
5.
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Chou CS, Lo WC, Gokoffski KK, Zhang YT, Wan FY, Lander AD, Calof AL,
Nie Q: Spatial dynamics of multistage cell lineages in tissue stratification.
Biophys J 2010, 99:3145-3154.
Evans LC: Partial Differential Equations. American Mathematical Society; 1998.
Chen L-Q, Shen J: Applications of semi-implicit Fourier-spectral method to phase
field equations. Computer Physics Communications 1998, 108:147-158.
Murray JD: Mathematical Biology. Springer-Verlag, Berlin; 1993.
15