Mees A I, Rapp P E
J Math Biol. 1978 Mar 3;5(2):99-114. doi: 10.1007/BF00275893.
For a general multiple loop feedback inhibition system in which the end product can inhibit any or all of the intermediate reactions it is shown that biologically significant behaviour is always confined to a bounded region of reaction space containing a unique equilibrium. By explicit construction of a Liapunov function for the general n dimensional differential equation it is shown that some values of reaction parameters cause the concentration vector to approach the equilibrium asymptotically for all physically realizable initial conditions. As the parameter values change, periodic solutions can appear within the bounded region. Some information about these periodic solutions can be obtained from the Hopf bifurcation theorem. Alternatively, if specific parameter values are known a numerical method can be used to find periodic solutions and determine their stability by locating a zero of the displacement map. The single loop Goodwin oscillator is analysed in detail. The methods are then used to treat an oscillator with two feedback loops and it is found that oscillations are possible even if both Hill coefficients are equal to one.
对于一个通用的多环反馈抑制系统，其中终产物可以抑制任何或所有中间反应，结果表明，具有生物学意义的行为总是局限于包含唯一平衡点的反应空间的有界区域。通过为一般的n维微分方程显式构造一个李雅普诺夫函数，结果表明，对于所有物理上可实现的初始条件，反应参数的某些值会使浓度向量渐近地趋近于平衡点。随着参数值的变化，有界区域内可能会出现周期解。关于这些周期解的一些信息可以从霍普夫分岔定理中获得。或者，如果已知特定的参数值，可以使用数值方法来找到周期解，并通过定位位移映射的零点来确定它们的稳定性。对单环古德温振荡器进行了详细分析。然后使用这些方法来处理具有两个反馈环的振荡器，结果发现，即使两个希尔系数都等于1，振荡也是可能的。
