reproduce the gene expression patterns from axolotl?
Recall from chapter 3 that the key difference between the axolotl and Xenopus mesendoderm network downstream of Activin is the action of Mix on Brachyury (compare figures 3.8 and 3.1). In axolotl Mix is required for the expression of Brachyury, but in Xenopus Mix represses Brachyury. We found that both the Xenopus and axolotl in vitro models are bistable with steady states corresponding to mesoderm and anterior mesendoderm but with qualitative differences in the time course expression of genes. In this section we ask if a mathematical model with the Xenopus network topology (i.e. Mix repressing Brachyury) can reproduce experimental data from axolotl animal caps. To do this we use our hybrid parameter estimation algorithm to minimise the error between the model simulation and experimental data. As in the previous section (for axolotl models) we do this in two models; in the first model the level of Activin remains fixed and in the second model Activin is allowed to turn over.
Recall that the Xenopus in vitro model is given by dM dt =λA,MH A θA,M , m1 1− H B θB,M , m2 −µMM, (4.3.1a) dG dt =λM,GH M θM,G, m3 1− H G θG,G, m4 −µGG, (4.3.1b) dB dt = λA,BH A θA,B , m5 +λB,BH B θB,B , m8 1− H G θG,B + M θM,B , m7 −µBB, (4.3.1c) In the model where Activin can turn over we have the rate of change of Activin over time defined in equation (4.2.6).
CHAPTER4: PARAMETERESTIMATION INSINGLE-CELLMODELS OFMESENDODERM SPECIFICATION INAXOLOTL
Figure 4.25: Model simulation for the Xenopus in vitro model. Experimental data (dots) and simulation results (lines) for the in vitro mathematical model of the axolotl mesendoderm network as given in equation (4.3.1). Model parameters were sought using a parameter estimation algorithm as described in the main text, with the re- sulting parameters given in the ‘Xenopus A’ column of table 4.3.
Figure 4.25 gives a fit to the Xenopus in vitro model without the turnover of Activin. The value of the fitness function is 2.16 compared with 2.49 for the equivalent axolotl model, meaning that the parameters found for the Xenopus in vitro model provide a closer fit to the experimental data than the axolotl model. However, the model does not qualitatively reproduce key behaviour of the experimental data.
A solution to the Xenopus in vitro model with Activin degradation is given in figure 4.26. This model qualitatively reproduces the experimental gene expression data from the Activin dose response experimental in axolotl. The behaviour of the model beyond t=60 is shown in figure 4.27, Mix and Goosecoid decay to zero, but in 1pg Activin caps Brachyury reaches a non-trivial steady state. The model prediction for different doses of Activin is given in figure 4.28. As the dose of Activin increases, the quantitative behaviour of Mix and Goosecoid remains the same; the expression of each gene increases to a maximum concentration before decaying, with max- imum level of Mix/Goosecoid increasing with the dose of Activin. The mathematical model is bistable, with either no Brachyury in the system (the trivial steady state) or up regulated Brachyury. At low dose of Activin Brachyury evolves to the upregulated steady state and at high doses Brachyury evolves to the trivial steady state. For intermediate concentrations (such as A=5) Brachyury is expressed at low levels for a time period before being upregulated and evolving to an upregulated steady state. Note that there is no ‘spike’ in Brachyury expression at high levels of Activin in the Xenopus model. In figure 4.29 the model is solved both in the presence and absence of Activin turnover. In the presence of Activin turnover the levels of Mix
CHAPTER4: PARAMETERESTIMATION INSINGLE-CELLMODELS OFMESENDODERM SPECIFICATION INAXOLOTL
Figure 4.26: Model simulation for theXenopus in vitro model with Activin degradation. Ex- perimental data (dots) and simulation results (lines) for the in vitro mathemati- cal model of the axolotl mesendoderm network as given in equations (4.3.1) and (4.2.6). Model parameters were sought using a parameter estimation algorithm as described in the main text, with the resulting parameter set given in the ‘Xenopus B’ column of table 4.3. The fitness function is 1.89.
and Goosecoid decrease until they reach a steady state value, but in the system where Activin does not turnover the levels continue to increase until they reach a steady state value. The in- crease in levels of Mix and Goosecoid causes Brachyury to be down regulated in 1pg Activin caps. A knock out of Mix, Brachyury and Goosecoid is carried out to compare the behaviour of the Xenopus in vitro model with the axolotl in vitro model. A knock out of Mix in the Xenopus in vitro model results in the upregulation of Brachyury at both 1pg and 25pg of Activin, i.e. the caps will become mesoderm. In the absence of Mix, Goosecoid is also not expressed (figure 4.30). This is different to what is seen in the axolotl model simulations where a knockdown of Mix results in no Brachyury or Goosecoid expression, i.e. caps remain as ectoderm (figure 4.11). In the Xenopus model a knock out of Goosecoid does not change the levels of Mix or Brachyury (figure 4.31), but in the axolotl model Goosecoid is required for the repression of Brachyury (figure 4.15). Both in the Xenopus model and the axolotl model (figure 4.32 and fig- ure 4.15, respectively) a knock down of Brachyury results in no change in the levels of Mix and Goosecoid.
CHAPTER4: PARAMETERESTIMATION INSINGLE-CELLMODELS OFMESENDODERM SPECIFICATION INAXOLOTL
Figure 4.27: Model simulation for theXenopus in vitro model as given in equations(4.3.1)
and(4.2.6), including the turnover of Activin. Model parameters are given in the ‘Xenopus B’ column of table 4.3.
Figure 4.28: Investigating the level of gene expression in response to Activin.The Xenopus in vitro model is as given in equations (4.3.1) and (4.2.6) solved subject to parameters given in the ‘Xenopus B’ column of table 4.3.
Figure 4.29: Investigating the role of Activin turnover. The model is solved in the presence (dashed line) and absence (solid line) of Activin turnover. The Xenopus in vitro model is as given in equations (4.3.1) and (4.2.6) solved subject to parameters given in the ‘Xenopus B’ column of table 4.3.
CHAPTER4: PARAMETERESTIMATION INSINGLE-CELLMODELS OFMESENDODERM SPECIFICATION INAXOLOTL
Figure 4.30: Investigating the action of Mix.The model is solved in the presence (dashed line) and absence (solid line) of Mix. The Xenopus in vitro model is as given in equations (4.3.1) and (4.2.6) solved subject to parameters given in the ‘Xenopus B’ column of table 4.3.
Figure 4.31: Investigating the action of Goosecoid. The model is solved in the presence (dashed line) and absence (solid line) of Goosecoid. The Xenopus in vitro model is as given in equations (4.3.1) and (4.2.6) solved subject to parameters given in the ‘Xenopus B’ column of table 4.3.
CHAPTER4: PARAMETERESTIMATION INSINGLE-CELLMODELS OFMESENDODERM SPECIFICATION INAXOLOTL
Figure 4.32: Investigating the action of Brachyury.The model is solved in the presence (dashed line) and absence (solid line) of Brachyury. The Xenopus in vitro model is as given in equations (4.3.1) and (4.2.6) solved subject to parameters given in the ‘Xenopus B’ column of table 4.3.