Some applications of control theory to the stability analysis of biological and biochemical control systems

The differential equations describing a large class of biological and biochemical control systems are examined from the point of view of stability using the following methods: The stability properties of the equations transformed to the Lur´e canonical form are examined by methods based on the second Lyapunov method. These results illustrate that sustained concentration oscillations cannot arise in a second order chemical systems involving negative feedback and otherwise first order rate laws. For such systems involving feedback that is not purely negative, and for third and higher order chemical systems involving negative feedback, it is possible for limit cycles to arise. Phase-plane methods indicate that limit cycle behavior cannot arise in chemical control Systems involving negative feedback and two components if the rate law for the uninhibited step is described by first order kinetics, a rectangular hyperbola, or a sigmoidal relationship. Linear methods indicate that the potential for instabilities is always present in the case of chemical systems involving positive feedback, the potential for instability and limit cycle behavior increases each time the order of the differential equations describing the system is increased. Digital simulation of the equations illustrate that limit cycles can arise in multi-component chemical systems involving negative feedback even if the rate laws for the uninhibited steps are linear. A mapping of the range of values of the constants in the equations indicates that there is a definite range for which sustained concentration oscillations arise. Finally, the possibility of multiple stationary states arising in biochemical and biological control systems is discussed. Eleven different cases are considered, and examples illustrating the temporal and phase-plane behavior of actual systems involving multiple stationary states are presented. It is illustrated, for example, that the addition of a little positive feedback to an other- ise purely negative chemical feedback system imparts stability to stationary states are presented. It is illustrated, for example, that the addition of a little positive feedback to an otherwise purely negative chemical feedback system imparts stability to stationary states for the system.