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Über dieses Buch

This brief examines a deterministic, ODE-based model for gene regulatory networks (GRN) that incorporates nonlinearities and time-delayed feedback. An introductory chapter provides some insights into molecular biology and GRNs. The mathematical tools necessary for studying the GRN model are then reviewed, in particular Hill functions and Schwarzian derivatives. One chapter is devoted to the analysis of GRNs under negative feedback with time delays and a special case of a homogenous GRN is considered. Asymptotic stability analysis of GRNs under positive feedback is then considered in a separate chapter, in which conditions leading to bi-stability are derived. Graduate and advanced undergraduate students and researchers in control engineering, applied mathematics, systems biology and synthetic biology will find this brief to be a clear and concise introduction to the modeling and analysis of GRNs.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

In this chapter, background material is presented for the gene regulation process and mathematical models of such systems are discussed. In particular, most popular classification and regression methods are briefly mentioned to give an idea on how data collected using microarrays can be used in modeling gene regulatory networks. The chapter ends with the formal definition of the continuous-time ODE-based model with delay to be analyzed in the rest of the book.
Mehmet Eren Ahsen, Hitay Özbay, Silviu-Iulian Niculescu

Chapter 2. Basic Tools from Systems and Control Theory

This chapter sets up the notation for the rest of the book and introduces basic concepts from control theory for the readers who are not familiar with fundamental feedback stability analysis techniques. Delay-differential equations are considered, and the small gain theorem is given for a delay independent stability condition for linear feedback systems. The Nyquist criterion is given in order to derive the necessary and sufficient conditions for delay dependent stability of such systems.
Mehmet Eren Ahsen, Hitay Özbay, Silviu-Iulian Niculescu

Chapter 3. Functions with Negative Schwarzian Derivatives

This chapter is devoted to the analysis on functions with NSD (negative Schwarzian derivatives). First, basic properties of functions with NSD are given and a classification result is proven for such functions. Then, an analysis is made on the fixed points for functions with NSD.
Mehmet Eren Ahsen, Hitay Özbay, Silviu-Iulian Niculescu

Chapter 4. Deterministic ODE-Based Model with Time Delay

This chapter is devoted to the derivation of the ODE-based model with time delay that is to be analyzed in the forthcoming chapters. In particular, an equivalent simplified mathematical model of the GRN model is proposed through some interpretations of the interconnection scheme. Next, the stability conditions for the linearized system around an equilibrium point are discussed. Finally, a specific example (the repressilator) is given to illustrate the motivation behind the model considered.
Mehmet Eren Ahsen, Hitay Özbay, Silviu-Iulian Niculescu

Chapter 5. Gene Regulatory Networks Under Negative Feedback

In this chapter, we consider the simplified GRN model, with the assumption that it is under delayed negative feedback. By analyzing the fixed points of a single function determined from the nonlinear connections, we show that in this case the system has a unique equilibrium point in the positive cone. Then, delay independent global stability, and instability, conditions are derived. For a delay dependent stability condition the secant condition is extended to cover systems with time delays. Special stability conditions are derived for homogenous GRNs where nonlinearities are Hill functions.
Mehmet Eren Ahsen, Hitay Özbay, Silviu-Iulian Niculescu

Chapter 6. Gene Regulatory Networks Under Positive Feedback

In this chapter, we consider the simplified GRN model, with the assumption that it is under delayed positive feedback. By analyzing the fixed points of a single function determined from the nonlinear connections, we show that the system may have three equilibrium points in the positive cone. When the system has a unique equilibrium, generically all solutions converge to this point. When there are three equilibrium points, the system shows a bistable behavior. Homogenous GRNs under delayed positive feedback are analyzed, and their stability and bistability are determined from the parameters of the Hill function used in the nonlinear coupling.
Mehmet Eren Ahsen, Hitay Özbay, Silviu-Iulian Niculescu

Chapter 7. Summary and Concluding Remarks

In this chapter, the analysis results for GRNs under negative and positive feedback are discussed and concluding remarks are made. Some possible future research directions are also pointed out.
Mehmet Eren Ahsen, Hitay Özbay, Silviu-Iulian Niculescu

Backmatter

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