We now solve equations (6.2.4) and (6.2.5) in a line of 50 cells, selecting initial conditions such that it represents a line running from a region where β-catenin is not expressed to the dorsal region where β-catenin is expressed. This may represent a line of cells running from the ventral region to the dorsal region, although since gastrulation occurs as the stages when mesendoderm forms, it is not clear where our line of cells corresponds to once this rearrangement of cells has occurred. Recall that in axolotl the only maternal factor involved in mesendoderm induction in our model is β-catenin, since VegT is not involved in the axolotl mesendoderm GRN. Initial conditions are chosen such that the first[n/3]cells represent a region where β-catenin is absent and in the remaining[2n/3]we consider two different cases. In the first case we assume that a gradient of β-catenin runs from cell j=n/3+1 to cell j=n, such that
Ci(0) = 0 for i≤ [n/3] a(i− [n/3]) for[n/3] <i≤ [n], (6.2.6) where a is a positive constant. We chose this gradient of β-catenin based on experimental ev- idence that β-catenin can induce mesoderm and endoderm in a dose dependent manner in axolotl animal caps [16]. However, β-catenin is deposited maternally and localised with cor- tical rotation [98], and there is currently no evidence that a gradient of β-catenin exists in the embryo. Thus we consider a second case where a constant level of β-catenin is found in the last [n/3]cells but is not expressed elsewhere, such that
Ci(0) = 0 for i<[2n/3] b for i≥ [2n/3], (6.2.7)
where b is a positive constant. We now proceed investigate time-dependent solutions to (6.2.4) and (6.2.5) subject to these two sets of initial conditions using the Matlab solver for stiff ODEs, ode15s.
Gradient ofβ-catenin
Plotted in figure 6.4 are solutions to (6.2.4)-(6.2.5) subject to the initial concentration of β-catenin defined by (6.2.6) such that a gradient of β-catenin levels is found in cells 16 to 50 and absent elsewhere. Parameter values (given in tables 6.1 and 6.2) were selected such that (6.2.4) is consistent with experimental observations in Xenopus where Nodal is pinned by Antivin [95].
CHAPTER6: MULTICELLULARMODELS OFMESENDODERMSPECIFICATION
Variable Parameter Value Variable Parameter Value
N λP,N 3 T λP,T 12 θP,N 0.666 θP,T 0.999 λC,N 1 δT 20 θC,N 1 µT 1 λC2,N 1 σT 0.125 θC2,N 1 T0 kT 1 N0 kN 1 k−T 100 k−N 100 µTo 1 µNo 1 P kP 10 T‡ l T 1 µP 0.01 l−T 100 S S 1 µT‡ 0 kS 1 ¯ν 1 k−S 0.01 C µC 0.01
Table 6.1:Dimensionless parameters used to solve (6.2.4). Parameters are selected such that in a single cell case (6.2.4) is bistable.
Variable Parameter Value Variable Parameter Value
M λP,M 12 B λK∗,B 12 θP,M 3 λP,B 20 θB,M 1 θP,B 1 L λP,L 1 G λLI,G 1 θP,L 1 λM,G 8 E λB,E 5 θM,G 1 K∗ kK∗ 10 θG,G 3 µK∗ 0.01 Eo σE 0.125 K kK 1 δE 1 k−K 0.01 kE 1 all other µ 1 k−E 100
Table 6.2:Dimensionless parameters used to solve (6.2.5). Parameters are selected such that in a single cell case (6.2.5) is bistable with steady states corresponding to mesoderm and anterior mesendoderm.
We find that Nodal is restricted from spreading more than one or two cells away from the region where it is induced by β-catenin. There is a gradient of Nodal, Antivin and P-Smad2 which is similar to the gradient of β-catenin. Downstream of Nodal signalling we find that Mix and Goosecoid are co-expressed in regions with high levels of β-catenin and Brachyury and eFGF are expressed in regions with a lower level of β-catenin. These results are qualitatively similar to dose response data in axolotl animal caps [16] where mesoderm forms for a low level of β-catenin and anterior mesendoderm forms for a higher dose of β-catenin. We also observe that, in our model, Brachyury spreads into the region with no active Nodal signalling, this spread being due to the mutual positive regulation of eFGF and Brachyury. Figure 6.5 shows solutions to the model beyond t=100. We find that levels of components of the Nodal signalling pathway decrease as time proceeds. As the levels of P-Smad2 decrease, Mix and Goosecoid become downregulated and Brachyury and FGF become upregulated in these cells. Thus at steady state Nodal is not maintained and eFGF and Brachyury are maintained via their mutual positive regulation.
In figures 6.6 and 6.7 solutions to (6.2.4)-(6.2.5) subject to the initial concentration of β-catenin defined by (6.2.6) are plotted for λP,A = 1 and λP,A = 50, corresponding to the cases where Nodal can spread throughout the line of cells, and where Nodal is pinned by Antivin, respec- tively. When Nodal can spread throughout the line of cells we still observe regions correspond-
CHAPTER6: MULTICELLULARMODELS OFMESENDODERMSPECIFICATION
ing to mesoderm and anterior mesendoderm, but see that components of Nodal signalling spread into the region where β-catenin is not expressed (see figure 6.6). When λP,A = 50, the rate of production is high, which means that the level of P-Smad2 in response to Nodal sig- nalling is low and Antivin pins Nodal to β-catenin expressing regions. Downstream of Nodal we see that a Brachyury region forms, but we do not see a region corresponding to anterior mesendoderm.
Constant level ofβ-catenin
Plotted in figure 6.8 are solutions to (6.2.4)-(6.2.5) subject to the initial concentration of β-catenin defined by (6.2.7) such that a uniform concentration of β-catenin is found in cells 33 to 50 and absent elsewhere. We set λP,A = 10, such that Antivin prevents Nodal from spreading more than one or two cells from its source, and investigate how cell to cell signalling results in the formation of regions corresponding to mesoderm and anterior mesendoderm. The solutions plotted in figure 6.8 show that cells 32 to 50, where β-catenin is expressed, correspond to ante- rior mesendoderm, while mesoderm forms on the boundary of the β-catenin expressing region. The solutions shown in figure 6.9 are for the same model as in figure 6.8, except we set σE=0, such that there is no cell to cell communication via the FGF signalling pathway. We see that in this case only cells 30 and 31 express Brachyury and eFGF and does not spread throughout the line of cells. Thus eFGF signalling is essential for expanding the number of cells corresponding to mesoderm. Figure 6.10 shows solutions where σN = σT = σT‡ = 0, such that there is no
cell to cell communication via the Nodal signalling pathway. As before, we find that anterior mesendoderm forms in cells 32 to 50, but in the absence of Nodal spreading outside of this region.
6.2.5
Discussion
In this section we developed mathematical models of the axolotl mesendoderm GRN in a line of cells. We found that Brachyury expressing (i.e. mesoderm) and Mix and Goosecoid expressing (i.e. anterior mesendoderm) populations of cells form for both an initial gradient of β-catenin and a uniform level of β-catenin in a selected number of cells. While experimental evidence shows that β-catenin can induce mesoderm and anterior mesendoderm in a dose dependent manner in axolotl animal caps [16], it is likely that a gradient of β-catenin does not exist in vivo. Our investigations show that regions corresponding to mesdoderm and anterior mesendoderm can form when β-catenin is uniformly expressed in some cells and absent from others, and that the diffusion of Nodal is essential for the formation of these two distinct regions in this case. To aid further development of multicellular models of the axolotl mesendoderm GRN, experi- mental data is required. A logical first step is to collect data for the expression of components of the Nodal signalling pathway to estimate model parameters, the extend the analysis to com- ponents downstream of Nodal. In the next section we clone Antivin, a gene involved in the regulation of Nodal signalling, which has not previously been identified in axolotl.
CHAPTER6: MULTICELLULARMODELS OFMESENDODERMSPECIFICATION
Figure 6.4:Numerical solutions for (6.2.4)-(6.2.5) subject to the initial condition of β-catenin given by (6.2.6) with a = 1/16 for n = 50 cells. No β-catenin is found in cells i=1 to 16, and a gradient of β-catenin exists in cells i=17 to 50. Nodal (Nio) and its downstream component P-Smad2 (Pi) are restricted to regions expressing β-catenin.
Parameters are selected as in tables 6.1 and 6.2, with λP,A = 10, such that Antivin
pins Nodal expression. Downstream of P-Smad2, we find that Mix is expressed in regions with high β-catenin (cells 32 to 50) and Brachyury is initially expressed in regions with low β-catenin, but spreads into the region where β-catenin is absent.
CHAPTER6: MULTICELLULARMODELS OFMESENDODERMSPECIFICATION
Figure 6.5:Numerical solutions for (6.2.4)-(6.2.5) subject to the initial condition of β-catenin given by (6.2.6) with a = 1/16 for n = 50 cells. No β-catenin is found in cells i=1 to 16, and a gradient of β-catenin exists in cells i=17 to 50. Nodal (Nio) and its downstream component P-Smad2 (Pi) are restricted to regions expressing β-catenin.
Parameters are selected as in tables 6.1 and 6.2, with λP,A = 10, such that Antivin
pins Nodal expression. Downstream of P-Smad2, we find that Mix is expressed in regions with high β-catenin (cells 32 to 50) and Brachyury is initially expressed in regions with low β-catenin, but spreads into the region where β-catenin is absent.
CHAPTER6: MULTICELLULARMODELS OFMESENDODERMSPECIFICATION
Figure 6.6:Numerical solutions for (6.2.4)-(6.2.5) subject to the initial condition of β-catenin given by (6.2.6) with a = 1/16 for n = 50 cells. No β-catenin is found in cells i=1 to 16, and a gradient of β-catenin exists in cells i=17 to 50. Nodal (No
i) and its
downstream component P-Smad2 (Pi) are restricted to regions expressing β-catenin.
Parameters are selected as in tables 6.1 and 6.2, with λP,A = 1, such that Nodal
spreads throughout the line of cells. Downstream of P-Smad2, we find that Mix is expressed in regions with high β-catenin (cells 32 to 50) and Brachyury is initially ex- pressed in regions with low β-catenin, but spreads into the region where β-catenin is absent.
CHAPTER6: MULTICELLULARMODELS OFMESENDODERMSPECIFICATION
Figure 6.7:Numerical solutions for (6.2.4)-(6.2.5) subject to the initial condition of β-catenin given by (6.2.6) with a = 1/16 for n = 50 cells. No β-catenin is found in cells i=1 to 16, and a gradient of β-catenin exists in cells i=17 to 50. Nodal (Nio) and its downstream component P-Smad2 (Pi) are restricted to regions expressing β-catenin.
Parameters are selected as in tables 6.1 and 6.2, with λP,A = 50, such that Nodal
is pinned by Antivin. Downstream of P-Smad2, we find that Mix is expressed in regions with high β-catenin (cells 32 to 50) and Brachyury is initially expressed in regions with low β-catenin, but spreads into the region where β-catenin is absent.
CHAPTER6: MULTICELLULARMODELS OFMESENDODERMSPECIFICATION
Figure 6.8:Numerical solutions for (6.2.4)-(6.2.5) subject to the initial condition of β-catenin given by (6.2.7) with b=50/16 for n=50 cells. No β-catenin is found in cells i=1 to 32, and a uniform level β-catenin exists in cells i=33 to 50. Nodal (Nio) and its downstream component P-Smad2 (Pi) are restricted to regions expressing β-catenin.
Parameters are selected as in tables 6.1 and 6.2, with λP,A = 10, such that Antivin
pins Nodal expression. Downstream of P-Smad2, we find that Mix is expressed in regions where β-catenin is expressed (cells 33 to 50) and Brachyury is initially ex- pressed at the boundary of the β-catenin positive region, and then spreads into the region where β-catenin is absent.
CHAPTER6: MULTICELLULARMODELS OFMESENDODERMSPECIFICATION
Figure 6.9:Numerical solutions for (6.2.5) subject to the initial condition of β-catenin given by (6.2.7) with b = 50/16 for n = 50 cells. No β-catenin is found in cells i = 1 to 32, and a uniform level β-catenin exists in cells i = 33 to 50. Nodal (No
i) and its
downstream component P-Smad2 (Pi) are restricted to regions expressing β-catenin.
Parameters are selected as in tables 6.1 and 6.2, with λP,A = 10, such that Antivin
pins Nodal expression and σE=0 such that extracellular FGF does not spread from
cell to cell. Downstream of P-Smad2, we find that Mix is expressed in regions where
β-catenin is expressed (cells 33 to 50) and Brachyury is expressed at the boundary of
CHAPTER6: MULTICELLULARMODELS OFMESENDODERMSPECIFICATION
Figure 6.10:Numerical solutions for (6.2.5) subject to the initial condition of β-catenin given by (6.2.7) with b = 50/16 for n = 50 cells. No β-catenin is found in cells i =
1 to 32, and a uniform level β-catenin exists in cells i = 33 to 50. Nodal (Nio) and its downstream component P-Smad2 (Pi) are restricted to regions expressing β-catenin. Parameters are selected as in tables 6.1 and 6.2, with λP,A = 10, such
that Antivin pins Nodal expression and terms of the form σX=0 such that there is
no cell to cell communication via Nodal signalling. Downstream of P-Smad2, we find that Mix is expressed in regions where β-catenin is expressed (cells 33 to 50) and Brachyury is expressed at the boundary of the β-catenin positive region.
CHAPTER6: MULTICELLULARMODELS OFMESENDODERMSPECIFICATION
hit species gene Accession number identities e value
1 Paralichthys olivaceus lefty AB232902.1 73% 2e-163
2 Oryzias latipes lefty NM_001163090.1 73% 1e-146
3 Xenopus (Silurana) lefty NM_001130253.1 73% 4e-141
tropicalis
4 Oreochromis niloticus left-right determination XM_003440707.1 71% 4e-134 factor 2-like
5 Xenopus laevis Xantivin AF209744.1 70% 8e-131
Table 6.3:NCBI BLAST results for the axolotl Antivin coding sequence. BLASTN 2.2.26+ is a nucleotide blast.