Introduction to the Analysis of

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Introduction to the Analysis of Biochemical and
Genetic Systems
Eberhard O. Voit and Michael A. Savageau
Department of Biometry and Epidemiology Medical
University of South Carolina VoitEO_at_MUSC.edu
Department of Microbiology and Immunology The
University of Michigan Savageau_at_UMich.edu
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Three Ways to Understand Systems

• Bottom-up molecular biology
• Top-down global expression data
• Random systems statistical regularities

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Five-Part Presentation

• From reduction to integration with approximate
  models
• From maps to equations with power-laws
• Typical analyses
• Parameter estimation
• Introduction to PLAS

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Module 1 Need for Models

• Scientific World View
• What is of interest
• What is important
• What is legitimate
• What will be rewarded
• Thomas Kuhn
• Applied this analysis to science itself
• Key role of paradigms

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Paradigms

• Dominant Paradigms
• Guides normal science
• Exclude alternatives
• Paradigm Shifts
• Unresolved paradoxes
• Crises
• Emergence of alternatives
• Major shifts are called revolutions

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Reductionist Paradigm

• Other themes no doubt exist
• Dominant in most established sciences
• Physics - elementary particles
• Genetics - genes
• Biochemistry - proteins
• Immunology - combining sites/idiotypes
• Development - morphogens
• Neurobiology - neurons/transmitters

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Inherent Limitations

• Reductionist is also a "reconstructionist"
• Problem reconstruction is seldom carried out
• Paradoxically, at height of success, weaknesses
  are becoming apparent

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Indications of Weaknesses

• Complete parts catalog
• 10,000 parts of E. coli
• But still we know relatively little about
  integrated system
• Response to novel environments?
• Response to specific changes in molecular
  constitution?

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Dynamics
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Critical Quantitative Relationships
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Emergent Systems Paradigm

• Focuses on problems of complexity and
  organization
• Program unclear, few documented successes
• On the verge of paradigm shift

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Definition of a System

• Collection of interacting parts, which
  constitutes a whole
• Subsystems imply natural hierarchies
• Example ... cells-tissues-organs-organism ...
• Two conflicting demands
• Wholeness
• Limits

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Contrast Complex and Simple
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Quantitative Understanding of Integrated Behavior

• Focus is global, integrative behavior
• Based on underlying molecular determinants
• Understanding shall be relational

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Mathematics

• For bookkeeping
• Uncovering critical quantitative relationships
• Adoption of methods from other fields
• Development of novel methods
• Need for an appropriate mathematical description
  of the components

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Rate Law

• Mathematical function
• Instantaneous rate
• Explicit function of state variables that
  influence the rate
• Problems
• The general case

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Examples

• v k1 X1
• v k2 X1X2
• v k3 X12.6
• v VmX1/(KmX1)
• v VhX12/(Kh2X12)

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Problems

• Networks of rate laws too complex
• Algebraic analysis difficult or impossible
• Computer-aided analyses problematic
• Parameter Estimation
• Glutamate synthetase
• 8 Modulators
• 100 million assays required

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Approximation

• Replace complicated functions with simpler
  functions
• Need generic representation for streamlined
  analysis of realistically big systems
• Need to accept inaccuracies
• Laws are approximations
• e.g., gas laws, Newtons laws

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Criteria of a Good Approximation

• Capture essence of system under realistic
  conditions
• Be qualitatively and quantitatively consistent
  with key observations
• In principle, allow arbitrary system size
• Be generally applicable in area of interest
• Be characterized by measurable quantities
• Facilitate correspondence between model and
  reality
• Have mathematically/computationally tractable form

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Justification for Approximation

• Natural organization of organisms suggests
  simplifications
• Spatial
• Temporal
• Functional
• Simplifications limit range of variables
• In this range, approximation often sufficient

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Spatial Simplifications

• Abundant in natural systems
• Compartmentation is common in eukaryotes (e.g.
  mitochondria)
• Specificity of enzymes limits interactions
• Multi-enzyme complexes, channels, scaffolds,
  reactions on surfaces
• Implies ordinary rather than partial differential
  equations

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Temporal Simplifications

• Vast differences in relaxation times
• Evolutionary -- generations
• Developmental -- lifetime
• Biochemical -- minutes
• Biomolecular -- milliseconds
• Simplifications
• Fast processes in steady state
• Slow processes essentially constant

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Functional Simplifications

• Feedback control provides a good example
• Some pools become effectively constants
• Rate laws are simplified
• Best shown graphically

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Rate Law Without Feedback
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Rate Law With Feedback
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Consequence of Simplification

• Approximation needed and justified
• Engineering
• Successful use of linear approximation
• Biology
• Processes are not linear
• Need nonlinear approximation
• Second-order Taylor approximation
• Power-law approximation

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Module 2 Maps and Equations

• Transition from real world to mathematical model
• Decide which components are important
• Construct a map, showing how components relate to
  each other
• Translate map into equations

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Model Design Maps
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Example from Genetics
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Components of Maps

• Variables (Xi, pools, nodes)
• Fluxes of material (heavy arrows)
• Signals (light or dashed arrows)

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Rules

• Flux arrows point from node to node
• Signal arrows point from node to flux arrow

Correct
Incorrect
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Terminology

• Dependent Variable
• Variable that is affected by the system
  typically changes in value over time
• Independent Variable
• Variable that is not affected by the system
  typically is constant in value over time
• Parameter
• constant system property e.g., rate constant

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Steps of Model Design1. Initial Sketch
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2. Conversion Table
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3. Redraw Graph in Symbolic Terms
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Examples of Ambiguity

• Failure to account for removal (dilution)
• Failure to distinguish types of reactants
• Failure to account for molecularity
• Confusion between material and information flow
• Confusion of states, processes, and logical
  implication
• Unknown variables and interactions

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Failure to Account for Removal (Dilution)
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Failure to Distinguish Types of Multireactants
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Failure to Account for Molecularity
(Stoichiometry)
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Confusion Between Material and Information Flow
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Confusion of States, Processes, and Logical
Implication
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Analyze and Refine Model

• There is lack of agreement in general
• Discrepancies suggest changes
• Add or subtract arrows
• Add or subtract Xs
• Renumber variables
• Repeat the entire procedure
• Cyclic procedure
• Familiar scientific method made explicit

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Open versus Closed Systems
X
2
X
X
X
1
5
4
X
3
X
2
X
X
X
1
4
5
X
3
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Variables Outside the System
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General System Description

• Variables Xi, i 1, , n
• Study change in variables over time
• Change influxes effluxes
• Change dXi/dt
• Influxes, effluxes functions of (X1, , Xn)
• dXi/dt Vi(X1, , Xn) Vi(X1, , Xn)

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Translation of Maps into Equations

• Define a differential equation for each dependent
  variable
• dXi/dt Vi(X1, , Xn) Vi(X1, , Xn)
• Include in Vi and Vi those and only those
  (dependent and independent) variables that
  directly affect influx or efflux, respectively

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Example Metabolic Pathway

• dX1/dt V1(X3, X4) V1(X1)
• dX2/dt V2(X1) V2(X1, X2)
• dX3/dt V3(X1, X2) V3(X3)
• No equation for independent variable X4

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Example Gene Circuitry
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Power-Law Approximation

• Represent X1, , Xn, Vi and Vi in logarithmic
  coordinates
• yn ln Xn Wi ln Vi Wi ln Vi
• Compute linear approximation of Wi and Wi
• Translate results back to Cartesian coordinates

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Result

• No matter what Vi and Vi , and even if Vi and
  Vi are not known, the result in symbolic form
  is always
• Vi ? ai X1gi1X2gi2 Xngin
• Vi ? bi X1hi1X2hi2 Xnhin
• Power-Law Representation

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Parameters

• gij kinetic orders (positive, negative, or
  zero)
• hij kinetic orders (positive, negative, or
  zero)
• ai rate constants (positive or zero)
• bi rate constants (positive or zero)

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Meaning of Kinetic Orders

• 0 lt g, h lt 1 -- Saturating functions
• g, h gt 1 -- Cooperative functions
• 1 lt g, h lt 0 -- Partial inhibition
• g, h lt 1 -- Strong inhibition
• 2 lt g, h lt 2 -- Typical values (higher for
  fractal kinetics)

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System Description

• dXi/dt Vi(X1, , Xn) Vi(X1, , Xn)
• becomes S-system
• dXi/dt ai X1gi1X2gi2 Xngin
• 
  bi X1hi1X2hi2 Xnhin

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Summary of Power-Law Representation, S-systems

• Taylor series in logarithmic space
• Truncated to linear terms
• Interpretation of power-law function
• Estimation of parameter values
• Supporting evidence in biology

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Components of a Typical Analysis

• Steady state
• Numerical characterization
• Stability
• Signal propagation
• Sensitivities
• Dynamics
• Time plots
• Bolus experiments
• Persistent changes