Analytical solution of coupled nonlinear rate equations. II. Kinetics of positive catalytic feedback loops

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Abstract
Closed positive feedback loops of catalytic reactions between macromolecules provide a kinetic mechanism whereby each species serves to catalyze self-reproduction of its successor in the loop. The dynamics of these catalytic networks, hypercycles as they are known, is described by coupled nonlinear differential equations for which the present study develops approximate analytic solutions. The method involves transformation of the rate equations to suitable generalized coordinates and subsequent iterative solution of derived integral equations expressed in these coordinates. The integral equations and their solutions reveal solution of hypercycles in time as the unfolding of memory functions which reflect at any instant accumulated past dynamical history. Quantitatively, the analytical solutions are sufficiently close to computer solutions of the differential equations to justify their providing a reliable picture of the dynamics. The three species system is shown to be stable and to exhibit oscillatory exponential decay towards its fixed point. The four species hypercyle is quasistable in that the fixed point is approached likewise in an oscillatory manner, but asymptotically at the slow rate of inverse square root of time. The five species hypercycle is unstable and so evolves into a limit cycle characteristic of a biochemical clock whose period and structure is highly nonlinear, yet traceable analytically.