Supplement: Biomolecular Feedback Systems

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Domitilla Del Vecchio (U. Michigan) and Richard M. Murray (Caltech)

This supplement is intended for researchers interested in the application of feedback and control to biomolecular systems. The material has been designed so that it can be used in parallel with Feedback Systems as part of a course on biomolecular feedback and control systems, or as a standalone reference for readers who have had a basic course in feedback and control theory. The supplement is being written by Domitilla Del Vecchio and Richard Murray based on a variety of presentations, lectures and notes.

This page contains working notes for the material that is being prepared for the supplement. We anticipate that a preliminary version of the notes will be available by July 2009.


The supplement is intended to be useful to three distinct audiences:

  • First year PhD students in biology and bioengineering interested in feedback and control mechanisms in cells
  • Juniors through graduate students in engineering who are interested in biological circuit design
  • More senior researchers in the biological sciences who want to understand the application of principles and tools from feedback systems as applied to biomolecular systems

The pre-requisites for the supplement are a basic understanding of probability, differential equations and some linear algebra at an undergraduate level, along with a first course in feedback and control systems at the level of the main text (can be taken concurrently).


The current plan for the supplement is based on the following draft outline. Comments are welcome on topics that are missing.

  1. Review of Core Processes
    • Modeling of transcription and translation
      • ODE models (coming from chemical reactions and thermodynamics)
      • time constants
      • cell affects (dilution, degradation)
    • Transcriptional Regulation
      • repression
      • activation
      • (Thermodynamics → chemical kinetics →Hill functions, Michaelis-Menten)
    • Intracellular sensing
    • Intracellular communication
    • Intracellular computation
  2. Dynamic behavior
    • Steady state analysis (log/log diagrams)
    • Phase plane analysis
    • Time response
    • Limit cycles
  3. Feedback examples
    • Lactose metabolism
    • Heat shock
    • Chemotaxis
  1. Stochastic behavior
    • Noise modeling (intrinsic/extrinsic), spectrum
    • Stochastic simulation analysis (SSA)
    • Markov modeling and analysis
    • Linearized modeling and analysis: include disturbance attenuation in freq domain (?)
  2. Biological circuit design
    • DNA-protein, vs protein-protein, etc.
    • Feedforward loops
    • PID in biocircuits
    • Activation vs repression (demand theory)
    • Modularity and retroactivity
    • Logical functions: AND, switches, inverters, toggles, etc.
  3. Design examples
  4. Robustness and evolvability
  5. Multicellular systems
    • Additional core processes
    • Quorum Sensing
    • Development


The material above is intended to be useful in analyzing a large variety of natural and engineered biomolecular feedback systems. Some of these examples will be included in the text, while others may be mentioned in exercises or simply left for the reader to explore on their own.

  • Repressilator: eqs, simulations, limit cycle stability using PB+ and Hastings
  • Toggle switch: eqs, nullclines and equilibria, stability, Lyapunov
  • Self repression: frequency response, time response
  • Self activation: dynamics plus Lyapunov
  • Activator-repressor clock (two implementations): eqs, simulations, PB for the 2D model reduction, Hopf Bifurcation for the 4D (singular perturbation for the 2D reduction?)
  • Chemotaxis: integral feedback (Iglesias), modeling (incl methylation)
  • Input adaptation: MAPK with feedback? (Kholodenko?)
  • Biochemical circuits (in vitro)
  • Development/differentiation: signal switching (Tomlin, Ashthagiri)
  • Sporalation, fate choice (Elowitz)
  • MAPK cascades (Sauro, Kholodenko)
  • Quorum Sensing (Weiss, ?), synchronization
  • Methabolic Pathways/engineering (yeast demand analysis, Hoffmeyer)
  • Hemoglobin (Kirshner) for allostery feedback?
  • Glucose/Lactose (Jacob and Monod)
  • Anti-sense switches (Smolke)
  • FFL (Alon)?
  • Heat Shock (El Samad)
  • Calcium Regulation (Khammash)

Additional Information