Chemical kinetics and nonlinear dynamics

Lecture 6: Kinetics and Catalysis

Outline

1. First order
2. Second order
3. Reaction symmetry
4. Single molecule view of degradation*
5. Exponential decay through entropy maximization*

*Notes only (see below).

Reading

PBOC2, Chapter 15.1-15.2

Notes

Kinetics and catalysis

Addendum (points 6 and 7)

Lecture 7: Introduction to enzyme catalysis

Outline

1. Reversible kinetics and approach to equilbrium
2. Enzyme catalysis
   1. Michaelis-Menten mechanism
   2. Solution to Michaelis-Menten by timescale separation
   3. Limiting behavior
3. Some binding problems revisited
   1. Ligand depletion

Reading

(optional) Chapters form Cornish-Bowden, Fundaments of Enzyme Kinetics. (pdf)

Notes

Continued from last time

Lecture 8: Bistability; stability analysis

Outline

1. Mixed inhibition*
   1. Solution by assuming rapid equilibrium
   2. Limiting behaviors: Competitive, uncompetitive, noncompetitive.
2. Phosphorylation cascades
   1. Single site
   2. Multisite
3. Zeroth order ultrasensitivity
   1. Solution by rate balance
   2. Goldbeter-Koshland function
4. Bistable autophosphorylating kinase

Reading

PBOC2, Ch. 15.2.6.

Notes

Phosphorylation cascades

Some binding problems

More enzyme kinetics (old notes)

Lecture 9: Bifurcation analysis in One Dimension

Outline

1. Autophosphorylating kinase
2. Autoregulated gene

Notes

Bifurcation analysis in 1D

Bistability example (older notes)

Analysis of 1D ODE's (older notes)

Reading

(Optional) J.E. Lisman, PNAS, 82,3055-57 (1985).

Lecture 13: Analysis of linear ODEs in Two Dimensions

Outline

1. Review of 1D ODE's
   1. Solution of 1D linear by Taylor expansion
   2. Stability analysis
   3. Solution of inhomogenous equation by changing variables
   4. General solution to linear equations
   5. Expansion of nonlinear equations about a fixed point
2. Linear equations in 2D
   1. Solution in terms of eigenvalues and eigenvectors
   2. Examples
      1. Unstable
      2. Center
      3. Stable spiral
   3. General stability properties

Notes

Analysis of ODE's in 2D (AO2D)

Analysis of 2D ODE's (older notes)

Reading

(Optional) PBOC2, Ch. 15.2.6, 19.3.6.

Lecture 14: Analysis of nonlinear ODE's in Two Dimensions

Outline

1. Stability analysis of nonlinear equations
2. Example: the Brusselator
3. Hopf bifurcations
4. Genetic toggle switch
5. Genetic oscillator

Notes

AO2D-part2

Reading

PBOC2, Ch. 19.3.6.

J.E. Ferell, T. Y. Tsai, and Q. Yang. "Modeling the Cell Cycle: Why Do Certain Circuits Oscillate?" Cell, 144, 874-885. (link).