A morphogen gradient model for pattern regulation. II. Time description of global morphogen formation and field compartmentalization

In a model for pattern regulation, use was made of local and global morphogens S and Sigma. Sigma is produced from the S-degradation and it is decomposed by first order kinetics while it diffuses along the field. We solve exactly the partial differential equation for the distribution of Sigma in one spatial dimension when its source S is monotonie (for simplicity, linear or generally a power function). Assuming that S and Sigma react reversibly with an allosteric protein P according to a sequential scheme, we derive the evolution in time of the field separation into compartments. At equilibrium the relative extent of each compartment is constant (for variable field size) and so pattern regulation is achieved.