Abstract
Protein ubiquitination and degradation play important roles in many biological functions and are associated with many human diseases. It is well known that for biochemical oscillations to occur, proper degradation rates of the participating proteins are needed. In most mathematical models of biochemical reactions, linear degradation kinetics has been used. However, the degradation kinetics in real systems may be nonlinear, and how nonlinear degradation kinetics affects biological oscillations are not well understood. In this study, we first develop a biochemical reaction model of protein ubiquitination and degradation and calculate the degradation rate against the concentration of the free substrate. We show that the protein degradation kinetics mainly follows the Michaelis-Menten formulation with a time delay caused by ubiquitination and deubiquitination. We then study analytically how the Michaelis-Menten degradation kinetics affects the instabilities that lead to oscillations using three generic oscillation models: 1) a positive feedback mediated oscillator; 2) a positive-plus-negative feedback mediated oscillator; and 3) a negative feedback mediated oscillator. In all three cases, nonlinear degradation kinetics promotes oscillations, especially for the negative feedback mediated oscillator, resulting in much larger oscillation amplitudes and slower frequencies than those observed with linear kinetics. However, the time delay due to protein ubiquitination and deubiquitination generally suppresses oscillations, reducing the amplitude and increasing the frequency of the oscillations. These theoretical analyses provide mechanistic insights into the effects of specific proteins in the ubiquitination-proteasome system on biological oscillations.
Highlights
Protein ubiquitination and degradation, regulated by the ubiquitin-proteasome system (UPS), play important roles in many fundamental biological functions and are associated with many human diseases [1,2,3,4]
Protein degradation is known to be important for many biological functions and the major effect is to maintain a proper protein level for a normal biological function
The study by Wong et al [18] shows that the degradation kinetics may play an important role in promoting oscillations and the study by Buchler et al [42] shows that the nonlinearity in protein degradation can be important for bistability of biological systems
Summary
Protein ubiquitination and degradation, regulated by the ubiquitin-proteasome system (UPS), play important roles in many fundamental biological functions and are associated with many human diseases [1,2,3,4]. In many mathematical models of biochemical reactions [11,12,13,14,15], the degradation rate of a substrate protein S has been modeled as being linearly proportional to its concentration [S], i.e., d1⁄2S=dt~{kd 1⁄2S. This implies that the protein content decays exponentially (i.e., 1⁄2S(t)~1⁄2S(0)e{kdt), which has been shown in experimental measurements [2,16]. The MM kinetics for protein degradation was used in mathematical modeling studies [14,19,20,21], mainly following the Goldbeter-Koshland formulation [22]. Coli that the MM degradation kinetics significantly enlarges the parameter space for oscillations, the underlying mechanisms remain unclear
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