Mathematical models of DV patterning

Several mathematical models of DV patterning in Zebrafish [156], Drosophila [33, 146] and Xeno- pus [7] are available in the literature. All these models consist of systems of reaction diffusion equations representing the interactions between various extracellular signals. Zhang et al [156] formulate a model of DV patterning using a 3D geometry to capture the shape of a Zebrafish embryo. The interactions of BMP and Chordin are modelled along with BMP-receptor and BMP-Chordin complexes, with the BMP-Chordin complex being degraded via the action of Tolloid. The model reproduces key experimental observations, showing that both Chordin and Tolloid are required to form a sharp BMP gradient. Positive BMP feedback and negative Chordin feedback are also shown to regulate the BMP gradient. Models of DV patterning in Drosophila [33, 146] also use reaction diffusion models to capture key interactions. Eldar et al [33] show that the conditions needed for robustness of DV patterning are restricted diffusion of the free BMP ligand, and that Sog (analogous to Chordin in Xenopus) is only cleaved efficiently when in complex with BMP.

A continuous model of DV patterning inXenopus

Ben-Zvi et al [7] use a model of DV patterning to explore the scaling of the BMP gradient with embryo size. The model includes two BMP ligands, BMP (representing the combined inputs of BMP2, BMP4 and BMP7) and ADMP, Chordin (an inhibitor of BMP signalling) and Xlr (a protease which cleaves Chordin). BMP and ADMP ligands bind to Chordin to give Chordin- ligand complexes Chd+BMP kBmp −−−−−→ChdBMP, (5.1.1a) Chd+ADMP k_Admp −−−−−→ChdADMP. (5.1.1b)

CHAPTER5: DORSAL-VENTRALPATTERNING IN ASINGLE-CELLMODEL OFMESENDODERM SPECIFICATION INXenopus

Chordin can be cleaved by Xlr, either when in complex (releasing BMP/ADMP) or as a single ligand ChdBMP+Xlr λBmp_Chd −−−−−→BMP+Xlr, (5.1.2a) ChdADMP+Xlr λAdmp_Chd −−−−−→ADMP+Xlr, (5.1.2b) Chd+Xlr λChd −−−−−→Xlr. (5.1.2c)

All species in the model can diffuse and the production of BMP/ADMP are defined using Hill functions of the total signalling level (S(x, t) = ADMP(x, t) +BMP(x, t)). ADMP is produced in regions with a low signal, such that αADMP(S(x, t)) = 10−3

T4 ADMP

S(x,t)4_+T4 ADMP

, and BMP is pro- duced in regions with high signalling, such that αBMP(S(x, t)) =10−3 S(x,t)

4

S(x,t)4_+T4 BMP

. ADMP and BMP are assumed to turnover at constant rates, βADMPand βBMP, respectively. The model then consists of the following system of reaction-diffusion equations

∂[Chd] ∂t =DChd∇ 2_{[Chd] − [Chd]}_k Admp[ADMP] +kBmp[BMP] +λChd[Xlr]  , (5.1.3a) ∂[ADMP] ∂t =DAdmp∇ 2_{[ADMP] −k} Admp[ADMP][Chd] +λ admp Chd [Xlr][ChdADMP]

+αADMP(S) −βADMP[ADMP], (5.1.3b)

∂[BMP] ∂t =DBmp∇ 2_{[BMP] −k} Bmp[BMP][Chd] +λBmp_Chd[Xlr][ChdBMP] +αBMP(S) −βBMP[BMP], (5.1.3c) ∂[ChdADMP] ∂t =DChdAdmp∇ 2_{[ChdADMP] +k} Admp[ADMP][Chd] −λ admp Chd [Xlr][ChdADMP], (5.1.3d) ∂[ChdBMP] ∂t DChdBmp∇ 2_{[ChdBMP] −k} Bmp[BMP][Chd] +λBmp_Chd[Xlr][ChdBMP]. (5.1.3e)

Boundary conditions are defined such that all fluxes are zero at the ventral pole. At the dorsal pole the flux of Chordin is given by DChd∇[Chd] =ηChd, where ηChdis a constant and the flux of Admp is given by DAdmp∇[Admp] = αAdmp, where αAdmp is a constant. All proteins and complexes are initially absent, except for BMP, which is uniformly distributed.

A systematical screen of model parameters in [7] reveals that two possible mechanisms in- volved in the formation of BMP gradients emerge: shuttling-based and inhibition-based. In a shuttling based mechanism, the activator (BMP) is physically translocated to ventral regions of the embryo, facilitated by its binding to the inhibitor (Chordin), followed by the activator being released from the inhibitor by a protease (Xlr) that degrades Chordin. In this model, BMP and ADMP diffuse at a faster rate once in complex with Chordin, and cleavage by Xlr is also more effective when Chordin is in a complex. The inhibition based mechanism does not require the physical translocation of the activator; instead the BMP gradient reflects the gradi- ent of Chordin. Here all the proteins diffuse at the same rate and Chordin and its complexes

CHAPTER5: DORSAL-VENTRALPATTERNING IN ASINGLE-CELLMODEL OFMESENDODERM SPECIFICATION INXenopus

model parameter value model parameter value both [Xlr] 10−2 both λ_ChdAdmp 1

both λ_ChdBmp 1 both kAdmp 10−2

both kBmp 1 both TAdmp 10−4

both DChd 10 both DComp 10

Shuttling DLig 10−1 Shuttling λChd 10−2

Shuttling ηChd 1 Inhibition DLig 10

Inhibition λChd 1 Inhibition ηChd 103

Table 5.1:Parameters used to solve (5.1.3)

are all cleaved by Xlr at the same rate. Ben-Zvi et al [7] propose that a shuttling mechanism is required for the scaling of a dorsal half Xenopus embryos, by showing that the scaling of dorsal half embryos only occurs when parameters corresponding to a shuttling mechanism are used. Ben-Zvi et al [7] then explore the scaling of the BMP gradient in a dorsal half embryos, built around the assumption that Xenopus embryos scale with size. However the evidence used to make this assumption is misinterpreted [35]. In particular, Spemann’s experiments are quoted as being evidence that scaling occurs in dorsal half embryos. Spemann [29] divided cleaving salamander eggs (i.e. a urodele amphibian) into two halves: the half containing the future dor- sal lip produces a well proportioned embryo, and a belly piece is formed from a ventral half. Ben-Zvi et al [7] take this to be evidence that Xenopus (an anuran amphibian) also exhibit scal- ing in dorsal halves. Ben-Zvi et al [7] also quote a paper by Cooke [23] to be evidence that dorsal half embryos produce well proportioned embryos, when in fact the quoted paper only stated that mesoderm patterning scales in transverse sections of tailbud embryos. There is ev- idence that dorsal halves from the 8-cell Xenopus blastula develop into tadpoles with normal heads and a small body and ventral halves develop into belly pieces [67]. Taken together, these papers suggest that, while dorsal halves from urodeles scale with embryo size, dorsal halves from anurans (such as Xenopus) do not.

A signalling profile is said to scale with embryo size if for the activation thresholds S= 10−2 and S = 10−1, the relative position, scaled by embryo length, shifts by less than 20%. This is quite a large shift in the position of an activation theshold. To produce an in-proportion embryo, the shift in position may need to be much less than 20%. We now reproduce the numerical results given in [7] using a modified version of MATLAB’s PDE solver, as provided by Danny Ben-Zvi, which runs faster than the standard MATLAB PDE solver. Unless otherwise stated the parameter values from table 5.1 are used. We attempt to reproduce the results of the model for both shuttling and inhibition parameters, but setting the ADMP and BMP degradation terms such that they are non-zero. For a wild type embryo, we set L=1000µm and for a dorsal half L=500µm; we also plot results for L=250µm. All results then are scaled by setting X=x/L In the shuttling mechanism (see figure 5.2(a)) BMP ligands are translocated to ventral regions of the embryo. Our results, like those given in Ben-Zvi et al [7], show that the BMP gradient scales with embryo length for a dorsal half embryo. In the inhibition based model, the BMP gradient does not scale with embryo length. Instead, the profile for a dorsal half embryo is the same as the profile for the dorsal half of a wild type embryo.

CHAPTER5: DORSAL-VENTRALPATTERNING IN ASINGLE-CELLMODEL OFMESENDODERM SPECIFICATION INXenopus

(a)Shuttling Model (Scaled Axis) (b)Inhibition Model (Scaled Axis)

Figure 5.2:Solutions to (5.1.3), using the parameters given in table 5.1. (a) The BMP signalling profile for the shuttling based mechanism, for a wild type embryo (black line) and a dorsal half (grey line), with the dorsal half scaled to full length. (b) The BMP signalling profile for the inhibition based mechanism, for a wild type embryo (black line) and a dorsal half (grey line), with the dorsal half scaled to full length. βAdmp=

0.01

Summary

In this subsection the mathematical model of BMP gradient formation given in Xenopus was reviewed. To reproduce the numerical results of the model given in [7], several modifications need to be made. Firstly, Ben-Zvi et al [7] state that the BMP activation profiles are plotted at their steady state. However, they set the ADMP degradation term to zero, meaning that due to the mathematical properties of the Hill function used for the rate of ADMP production, ADMP never reaches a steady state concentration. Either a non-trivial rate of ADMP degradation or using a Heaviside function, whereby for concentrations of BMP above a threshold level ADMP stops accumulating, need to be used to allow ADMP to reach a steady state value and to re- produce plots similar to those in [7]. Secondly, in the parameter screen, ADMP production only enters the model via a flux term on the dorsal side of the embryo. However, to reproduce the model results of [7] the ADMP production term is included across the whole embryo. An- other issue was experienced whilst attempting to reproduce the BMP activation profiles in the inhibition-based model. The profiles plotted for a wild-type embryo (L =1000) and a dorsal- half embryo (L = 500) are not the same as those given in [7], while plotting for a wild-type embryo with L=500 and L=250 for a dorsal-half produces profiles similar to those in [7]. The model of [7] considers the extracellular protein interactions of BMP, ADMP and Chordin, showing that a gradient of BMP can form by two different mechanisms. In this chapter we are concerned with adding Vent (a downstream target of BMP signaling), rather than on the mech- anisms of BMP gradient formation. As such we procced to formulate a single cell model of DV patterning, noting that this model could eventually be combined with a model of extracellular components of DV patterning as introduced in this section.

CHAPTER5: DORSAL-VENTRALPATTERNING IN ASINGLE-CELLMODEL OFMESENDODERM SPECIFICATION INXenopus