A mathematical model of the role of aggregation in sonic hedgehog signalling

Abstract

Effective regulation of the sonic hedgehog (Shh) signalling pathway is essential for normal development in a wide variety of species. Correct Shh signalling requires the formation of Shh aggregates on the surface of producing cells. Shh aggregates subsequently diffuse away and are recognised in receiving cells located elsewhere in the developing embryo. Various mechanisms have been postulated regarding how these aggregates form and what their precise role is in the overall signalling process. To understand the role of these mechanisms in the overall signalling process, we formulate and analyse a mathematical model of Shh aggregation using nonlinear ordinary differential equations. We consider Shh aggregate formation to comprise of multimerisation, association with heparan sulfate proteoglycans (HSPG) and binding with lipoproteins. We show that the size distribution of the Shh aggregates formed on the producing cell surface resembles an exponential distribution, a result in agreement with experimental data. A detailed sensitivity analysis of our model reveals that this exponential distribution is robust to parameter changes, and subsequently, also to variations in the processes by which Shh is recruited by HSPGs and lipoproteins. The work demonstrates the time taken for different sized Shh aggregates to form and the important role this likely plays in Shh diffusion.

Fig 1. The formation of Shh aggregates.

Multimerisation (a) of Shh occurs by protein monomers aggregating together on the producing cell surface. Alternatively, small Shh multimers may be recruited to (b) heparan sulfate proteoglycans (HSPGs) before disassociating from the surface of the cell. In (c) Lipoprotein transportation is facilitated by the binding of lipoproteins to the producing cell surface whereby Shh monomers can individually associate with the particle before it disassociates from the cell surface.

Fig 2. Cell associated Shh distribution: Shh aggregate formation as a result of multimerisation, HSPG and lipoprotein association.

Simulation shown is at 24 hours with a breakdown of the total aggregate formation given in Table 1.

Fig 3. Dispersed Shh distribution: Shh aggregate formation as a result of multimerisation, HSPG and lipoprotein association.

Simulation shown is at 24 hours with a breakdown of the total aggregate formation given in Table 2.

Fig 4

Cell associated Shh aggregate distributions after: (a) 30 minutes; (b) 1 hour; (c) 3 hours; (d) 6 hours; (e) 12 hours; and (f) 24 hours.

Fig 5

Dispersed Shh aggregate distributions after: (a) 30 minutes; (b) 1 hour; (c) 3 hours; (d) 6 hours; (e) 12 hours; and (f) 24 hours.

Fig 6. The shape of the Shh aggregate distribution is independent of the rate of dispersal.

Here monomer and aggregate dispersal has been removed. Model simulation is at 24 hours.

Fig 7. Shh aggregate formation when lipoproteins recruit multimeric Shh.

Here lipoproteins recruit multimers consisting of up to five monomers. No significant differences in the observed distribution and the underlying competition between mechanisms was observed in comparison with the results of Fig 2. Model simulation is at 24 hours.

Fig 8. Shh aggregate formation when HSPGs recruit monomeric Shh.

Model simulation is at 24 hours.