Analysis and Control of Polynomial Dynamic Models with Biological Applications

1. Introduction

1. Dynamic models for describing biological phenomena

2. Kinetic systems

2.1. Chemical reaction networks with mass action law

2.2. Chemical reaction networks with rational functions as reaction rates

3. QP models

3.1. The original Lotka-Volterra equations

3.2. Generalized Lotka-Volterra equations

2. Basic Notions

1. General nonlinear system representation in the form of ODEs

1.1. Autonomous polynomial and quasi-polynomial systems

1.2. Positive polynomial systems

2. Formal introduction of the QP model form

2.1. The QP model form

2.2. LV systems

2.3. Extension with input term

3. Introduction of kinetic models with mass action and rational reaction rates

3.1. General notions for reaction networks

3.2. Reaction networks with mass action kinetics

3.3. Kinetic realizability and structural non-uniqueness of mass action type reaction networks

3.4. Reaction networks with rational function kinetics

3.5. Extension with input term

4. Basic relations between kinetic and QP models

4.1. Representing kinetic models with mass action reaction rates as QP models

4.2. LV models as kinetic systems

3. Model Transformations and Equivalence Classes

1. Affine and linear positive diagonal transformations

1.1. Affine transformations and their special cases for positive polynomial systems

1.2. Positive diagonal transformation of QP systems

1.3. Positive diagonal transformation of CRNs: linear conjugacy

2. Nonlinear diagonal transformations

2.1. X-factorable transformation

2.2. State-dependent time-rescaling

3. Quasi-monomial transformation and the corresponding equivalence classes of QP systems

3.1. The quasi-monomial transformation (QM-transformation)

3.2. The Lotka-Volterra (LV) form and the invariants

4. Embedding transformations and the relationship between classes of positive polynomial systems

4.1. Embedding smooth nonlinear models into QP form (QP-embedding)

4.2. Embedding rational functions into polynomial form (CRN-embedding)

4.3. The generality and relationship between classes of positive polynomial systems

4. Model analysis

1. Stability analysis of QP models

1.1. Local stability analysis of QP and LV models

1.2. Global stability analysis through the solution of linear matrix inequalities

1.3. Extension with time-reparametrization

2. Stability of kinetic systems

2.1. The deficiency zero and deficiency one theorems

3. Invariants (first integrals) for QP and kinetic systems

3.1. Linear first integrals of kinetic systems and their relations to conservation

3.2. Invariants of QP systems and their computation

4. Relations between the Lyapunov functions of QP and kinetic models

4.1. The physical-chemical origin of the natural Lyapunov function of kinetic models

4.2. Relationship with the logarithmic Lyapunov function of QP models

5. Computational analysis of the structure of kinetic systems

5.1. Computational model for determining linearly conjugate realizations of kinetic systems

5.2. Dense and sparse realizations and their properties

5.3. Computing linearly conjugate realizations with preferred properties

5.4. Computing all possible graph structures of a kinetic system

5.5. Computation of linearly conjugate bio-CRNs

5.6. Computation-oriented representation of uncertain kinetic models and their realizations

5. Stabilizing Feedback Control Design 931. Stabilizing control of QP systems by using optimization

1.1. LQ control of QP systems based on their locally linearized dynamics

1.2. Stabilizing control of QP systems by solving bilinear matrix inequalities

2. Stabilizing state feedback control of nonnegative polynomial systems using special CRN realizations of the closed loop system

2.1. The controller design problem

2.2. Feedback computation in the complex balanced closed loop case

2.3. Feedback computation in the weakly reversible closed loop with zero deficiency case

3. Robustness issues and robust design for the stabilizing control of polynomial systems

3.1. Handling the parametric uncertainty of stabilizing control of polynomial systems in the complex balanced closed loop case

6. Case studies

1. Optimization-based structural analysis and design of reaction networks

1.1. Computing all mathematically possible structures of a G1=S transition model in budding yeast

1.2. Analysis of a 5-node-repressilator with auto activation

1.3. Different realizations of an oscillating rational system

1.4. Linearly conjugate structures of the model

2. Computational distinguishability analysis of an uncertain kinetic model

2.1. Biological background

2.2. Kinetic representation

2.3. Analysis of the uncertain model

3. Stability analysis and stabilizing control of fermentation processes in QP form

3.1. Zero dynamics of the simple fermentation process

3.2. Partially actuated fermentation example in QP-form

3.3. Fully actuated fermentation example in QP-form

3.4. Feedback design for a simple fermentation process

3.5. Model parameters 139
A. Notations and abbreviations
A.1. Notations
B. Mathematical tools
B.1. Directed graphs
B.2. Matrices of key importance
B.3. Basics of the applied computational tools
B.3.1.Linear programming
B.3.2.Mixed integer linear programming and propositional logic
B.3.3.Linear and bilinear matrix inequalities
B.4. Basic notions from systems and control theory
B.4.1.Input-affine nonlinear state space models and their realizations
B.4.2.Stability analysis
B.4.3.Stabilizing feedback controllers
B.4.4.Input-output linearization via state feedback
B.4.5.Uncertainty description in dynamic models