Eberhard O. Voit and Michael A. Savageau

Recent genome research has begun to produce unprecedented amounts of data that await analysis and interpretation. Clustering genes that are over- or under-expressed in response to a stimulus is an excellent strategy for identifying known and unknown genes involved in the response. However, clustering alone is not sufficient for explaining why the rates of some processes are increased whereas others are decreased, and why the degree of change may vary widely among processes. A true understanding of the well-orchestrated overall response with which an organism reacts to perturbations requires effective mathematical modeling approaches that integrate function on the genomic, biochemical, and physiological levels. These approaches must be sophisticated enough to capture with some validity the complexity observed in gene regulation and metabolism, but simple enough to be tractable when applied to phenomena of relevant magnitude.

This tutorial discusses *Canonical Modeling*, a biomathematical set of techniques that have proven very effective in the analysis of metabolic pathways and regulatory gene circuits. The tutorial consists of five modules. The introductory module discusses the need for mathematical models beyond simple kinetic rate laws and the concept of approximations. The second module introduces simple, effective rules for translating the diagram of a pathway or network into canonical model equations. The third module describes a typical model analysis, including assessments of steady states and explorations of dynamic features. The fourth module discusses methods of parameter estimation in the context of actual examples. In the final module, the attendee will have the opportunity to perform hands-on computer simulations of small systems in an interactive fashion, using the user-friendly freeware *PLAS*.

*Module 1: Need for Models*

- Approaches to representing and analyzing biochemical and genetic systems.
- Desirable features of "good" models.
- Notions of
*exact representations*and*approximations*. - Issues of scales and hierarchies.
- Combination and complementation of algebraic analyses and computer simulations.
- Concepts of change and of differential equations.
- General system equations.

*Module 2: Maps and Equations;*

- Essential components of systems models.
- Basic terminology and notation; dependent variables, independent variables, parameters; flow of material versus flow of information.
- Guidelines for translating biochemical and genetic systems into convenient graphical representations (
*maps*). - Correct maps, faulty maps. Didactic and actual examples.
- Power-law representation and canonical S-system models. Meaning of parameters.
- Guidelines for designing basic models.
- Open, closed systems; conserved quantities and constraints; precursor-product relationships.

*Module 3: A Typical Analysis*

- Concept of a steady state.
- Stability.
- Sensitivities, gains, robustness.
- Dynamics.
- Bolus experiments.
- Persistent changes in system components.

*Module 4: Parameter Estimation*

- Survey of methods for estimating parameter values.
- Data needs for the various approaches. Advantages and limitations of these approaches.
- Estimation of kinetic orders from steady-state data.
- Estimation of kinetic orders and rate constants from traditional rate laws.
- Estimation of parameters from dynamic data.
- Representative didactic and actual examples.

*Module 5: Computer Simulation with PLAS*

- Implementing systems in
*PLAS*. *PLAS*analyses of steady states, stability, gains, sensitivities.- Dynamic solutions.
- Temporary versus persistent modifications of the system.
- Typical computer experiments: bolus experiments, changes in dependent or independent variables, changes in parameter values.
- Representation of results: tables, time courses, phase-plane plots, and sensitivity profiles.
- Emergence of unexpected behaviors from seemingly simple pathways.
- "Free time" for independent exploration.

*General Discussion*

- Philosophical and technical questions from the audience.
- Strengths and limitations of canonical modeling.
- What’s the future?