In this chapter we investigate how the inclusion of time delay alters the dynamic properties of (a) delayed protein cross talk model, (b) time delay model of RNA silencing (also known as RNA interference), and (c) time delay in ERK and STAT interaction. The consequences of a time delay on the dynamics of those systems are analysed using Hopf’s theorem and Lyapunov-Andronov theory. Our analytical calculations predict that time delay acts as a key bifurcation parameter. This is confirmed by numerical simulations.
The aim of this review is to give an extended analytical consideration of the role of time delay in the behaviour associated with dynamical models: (i) delayed protein cross talk model; (ii) time delay model of RNA silencing and (iii) time delay in ERK and STAT interaction.
Some of the results presented here are obtained and published in the papers (Nikolov, Kotev, Georgiev, & Petrov, 2006; Nikolov, Kotev, & Petrov, 2006a; Nikolov, Kotev, & Petrov, 2006b; Nikolov, Vera, Wolkenhauer, Yankulova, & Petrov, 2007; Nikolov & Petrov, 2007; Nikolov, Vera, Kotev, Wolkenhauer, & Petrov, 2008), but new considerations and improvements are also made. The investigations, conducted on time-delay mathematical models, examine how the time-delay influences the processes of protein synthesis, the RNA silencing and the interaction of the ERK and STAT proteins. Using the Lyapunov-Andronov theory and the Hopf theorem, the bifurcation values of the time delay are discovered, the zones of stability and instability are determined, and from there – the zones of norm and pathology (cancer) for each process. Thus, the greatest advantage of such an approach is revealed, namely – the theoretical forecast (prediction) of various diseases, including cancer.
Key Terms in this Chapter
RNA Silencing: Or also known as RNA interference similar to immune system guards against exploitive parasitic elements by (i) identifying non-self-elements; (ii) generating target-specific responses against these foreign elements, and (iii) rapidly amplifying these responses to clear or otherwise inactive the threat.
Andronov-Hopf Bifurcation: From a mathematical point of view, the onset of sustained oscillations generally correspond to the passage through an Andronov-Hopf bifurcation point. Obviously, for a critical value of a control parameter (named bifurcation), the system displays damped oscillations and eventually reaches the steady state- stable focus. Beyond the bifurcation point, a stable solution arises in the form of a small-amplitude limit cycle surrounding the unstable steady state [Golbeter, A., Nature, 420:238-245, 2002]. This bifurcation is very typical for biological systems.
Time delay: Past memory (history).
DDEs: Delay differential equations. Those equations contain in addition derivatives which depend on the solution at previous times. Also they are infinite-dimensional systems which find application in control systems, biology, chemical kinetics, and other areas.
STAT: Signal Transducer and Activator of Transcription is a family of latent cytoplasmic proteins that are activated to participate in gene control when cells encounter various extracellular polypeptides.
NFB: Negative feedback. In this case a signal is caused by the expression of its inhibitor.
PFB: Positive feedback (or autocatalysis). Now, more inhibitors, or other molecules amplify the initial signal and lead to the stabilization of the amplitude, or the increase in the signal’s duration.
Chaotic Dynamics: Chaotic motions are based on homoclinic (heteroclinic) structures which instability accompanied by local divergence and global contraction. Meanwhile, the transition from stability to instability requires the vanishing of stable equilibrium states and of stable periodic motions or sufficiently large increase in the periodic ones.
ERK: Extracelullar Signal-Regulated Kinase. Cell signalling protein activated STAT through serine phosporilation.