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E-grāmata: Analysis of Deterministic Cyclic Gene Regulatory Network Models with Delays

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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.

1 Introduction
1(12)
1.1 A Brief Glimpse into Biology
1(2)
1.2 Gene Regulatory Networks
3(3)
1.3 Models for Gene Regulatory Networks
6(7)
1.3.1 Boolean Networks
7(1)
1.3.2 Reverse Engineering Gene Regulatory Networks
8(1)
1.3.3 Continuous-Time Models
9(4)
2 Basic Tools from Systems and Control Theory
13(12)
2.1 Preliminary Definitions and Notations
13(2)
2.2 Linear Time Invariant Systems
15(6)
2.3 Functional Differential Equations
21(2)
2.4 Exercises
23(2)
3 Functions with Negative Schwarzian Derivatives
25(18)
3.1 Classification of Functions with Negative Schwarzian Derivatives
25(6)
3.2 Fixed Points
31(11)
3.3 Exercises
42(1)
4 Deterministic ODE-Based Model with Time Delay
43(10)
4.1 Model Transformation and Simplification
43(3)
4.2 Analysis of the Linearized Model
46(2)
4.3 A Synthetic Circuit: The Repressilator
48(2)
4.4 Exercises
50(3)
5 Gene Regulatory Networks Under Negative Feedback
53(20)
5.1 Stability Conditions for GRNs Under Negative Feedback
54(11)
5.2 Homogeneous Gene Regulatory Networks with Hill Functions
65(7)
5.3 Exercises
72(1)
6 Gene Regulatory Networks Under Positive Feedback
73(14)
6.1 General Conditions for Global Stability
74(3)
6.2 Analysis of Homogenous Gene Regulatory Networks
77(7)
6.3 Exercises
84(3)
7 Summary and Concluding Remarks
87(2)
7.1 GRNs Under Negative Feedback
87(1)
7.2 GRNs Under Positive Feedback
88(1)
References 89(4)
Index 93