Exact solution to the reversible association process followed by zeroth-order kinetics

An explicit solution is presented for the two-step consecutive kinetics problem consisting of a reversible association process followed by a zeroth-order decay. This problem models the kinetically reversible combination of two molecules to yield a complex dimer which is then processed by an enzyme provided that we are under substrate saturation conditions. The exact solution, which involves a linear combination of Airy functions of the first and second kind and its derivatives, is valid for general initial conditions. This closed-form expression was obtained via a suitable transformation of the original non-linear differential equation and matches the numerical solution of the latter. A full analysis that includes the discussion of the phase portrait, the asymptotic behavior and the effect of parameters is presented. Both the homo and cross associations between molecules are considered and the passage to the purely irreversible limit is also investigated. The newly developed equations are applied for the dissociation of tubulin heterodimer under various constant injection rates thereof. Other situations where these novel solutions could be useful are discussed.