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

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.Author summary: The sonic hedgehog (Shh) pathway is vital for normal development in a wide variety of species and its activity is strictly regulated to ensure correct spatiotemporal patterning of numerous developing tissues. Shh signalling requires the formation of Shh aggregates, formed on producing cells via a range of different mechanisms, that then diffuse to receiving cells. We formulate and analyse a mathematical model of the most well described mechanisms, namely monomer multimerisation, and recruitment of Shh by heparan sulfate proteoglycans and lipoproteins. Our results illustrate a distribution of the size and quantities of aggregates formed by these mechanisms. We found that as a consequence of competition between the mechanisms for Shh monomers the shape distribution of Shh aggregates resembles an exponential distribution. We also found the distribution to be robust to both parameter changes and variations to the processes by which mechanisms recruit Shh. We report that our approach and subsequent results demonstrate that these mechanisms act in synergy allowing Shh to aggregate in various quantities with diverse diffusive abilities. We postulate that this regulation contributes significantly to aid precision in signalling for Shh in areas of development.