SUPPLEMENTARY NOTES FOR "DESIGN PRINCIPLES UNDERLYING CIRCADIAN CLOCKS

The linear mapping M relates the parameter change k to the change ( ˜, ) in the reparameterised limit cycle and period. It is therefore is from s-dimensional space to an infinite-dimensional space. To estimate the singular spectrum of M we must approximate it by a finite-dimensional operator i.e. by a matrix. We do this by approximating the curve ˜ (t) by a vector. We fix a large integer N and approximate ˜ = ˜ (t) by the vector whose jth entry is ˜ (j/N) and approximate M by MN : k ! (, ) which is given by = P ii · k i where ¯ i is the vector whose jth entry is i(j/N) and where i(t) = @ @ki k=k0 ˜ k(t). This gives a matrix representation for M(N) in terms of the basis vectorsi. We have developed a software tool that rapidly calculates the quantities i(t). Using the above results this enables us to compute M and its singular value decomposition to arbitrary accuracy.