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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@MUSC.edu

       **Department of Microbiology and Immunology
                The University of Michigan
                  Savageau@UMich.edu
Three Ways to Understand Systems

• Bottom-up — molecular biology

• Top-down — global expression data

• Random systems — statistical regularities
        Five-Part Presentation

• From reduction to integration with
  approximate models
• From maps to equations with power-laws
• Typical analyses
• Parameter estimation
• Introduction to PLAS
      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
                   Paradigms
• Dominant Paradigms
  – Guides “normal science”
  – Exclude alternatives
• Paradigm Shifts
  –   Unresolved paradoxes
  –   Crises
  –   Emergence of alternatives
  –   Major shifts are called revolutions
         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
         Inherent Limitations
• Reductionist is also a "reconstructionist"
• Problem: reconstruction is seldom carried
  out
• Paradoxically, at height of success,
  weaknesses are becoming apparent
     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?
              Dynamics

t   X1   X2        X3    X4
0
1
2
3
4
5
6
7
8
.
.
.
     Critical Quantitative Relationships


t   X1       X2         X3        X4


0
1
2
3                                  or   ?

4
.
.
.
              Alternative Designs
     a                       b
X1                                    X1

         X2                      X2

                   X3                  X3
                        c


              X3



                        X1
   Emergent Systems Paradigm
• Focuses on problems of complexity and
  organization
• Program unclear, few documented successes
• On the verge of paradigm shift
       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
     Contrast Complex and Simple

Character              Complex systems   Simple systems



Numbers of variables      Many              Few
Interactions              Strong            Weak

Mode of coupling          Nonlinear         Linear
Processes                 Associative       Additive
  Quantitative Understanding of
      Integrated Behavior
• Focus is global, integrative behavior
• Based on underlying molecular
  determinants
• Understanding shall be relational
             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
                  Rate Law
• Mathematical function
  – Instantaneous rate
  – Explicit function of state variables that
    influence the rate
• Problems
• The general case
                 Examples
•   v = k1 X 1
•   v = k2 X1X2
•   v = k 3 X1 2.6

•   v = VmX1/(Km+X1)
•   v = VhX12/(Kh2+X12)
                      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
             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, Newton’s laws
 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
  Justification for Approximation
• Natural organization of organisms suggests
  simplifications
  – Spatial
  – Temporal
  – Functional
• Simplifications limit range of variables
• In this range, approximation often sufficient
       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
       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
    Functional Simplifications
• Feedback control provides a good example
  – Some pools become effectively constants
  – Rate laws are simplified
• Best shown graphically
Rate Law Without Feedback
                 B
V                •



     A   •

         XA      XB
                      X
    Rate Law With Feedback

V

                          •    C




        A   •    •   A'



            XA            XB
                                   X
  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
 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
Model Design: Maps
             ATP                      Ribose 5-P
            ADP                         2,3-DPG

       PP-Ribose-P
        Synthetase                 NAD
                                   FAD
                                   Other
                                   Nucleotides

                     PP-Ribose-P                     Glutamine




                                                   Amido-
                                                    PRT




                                    P-Ribosyl-NH2



 ATP, GTP          AMP, GMP              IMP
Example from Genetics
                           +


          +           -        -                   mgl
                          ga lS
                                        B            A              C

                  p                p
                                        Ga la ctos e trans port
      +
                                    -
                      -                              ga l
                                         E           T              K   (M)
CRP
                                    p
                                        Ga la ctos e utilizat ion
          ga lR

      p
                                   -?
                                          ga l
              +                            P

                                   p
                                        Ga la ctos e trans port
        Components of Maps
• Variables (Xi, pools, nodes)
• Fluxes of material (heavy arrows)
• Signals (light or dashed arrows)



X4         X1         X2          X3
                       Rules
• Flux arrows point from node to node
• Signal arrows point from node to flux arrow

            X3                      X3

       X1             X2       X1           X2

            Correct                 Incorrect
               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
         Steps of Model Design
            1. Initial Sketch

Homoserine
             -        O-Homoserine-P            Threonine


         Homoserine                Threonine           Threonyl-tRNA
           kinase                  synthetase            synthetase
2. Conversion Table
            Table 2-1. Conversion Table for the Graph in Figures 2-11 and 2-12.


 Variable                          Variable                              Variable
 type                              name                                  symbol


 Dependent             O-homoserine-P                                       X1
                       Threonine                                            X2
                       Flux through O-homoserine-P                          V1
                       Flux through threonine                               V2


 Independent           Homoserine concentration                             X3
                       Homoserine kinase concentration                      X4
                       Threonine synthetase concentration                   X5
                       Threonyl-tRNA synthetase concentration               X6


 Aggregate             None explicit                                        --


 Constrained           None explicit                                        --


 Implicit              ATP                                                  --
                       ADP                                                  --
                       Mg                                                   --
                       Inorganic phosphate                                  --
                       threonyl-tRNA                                        --
                       tRNAthr                                              --
                       temperature                                          --
                       pressure                                             --
                       pH                                                   --
                       salt concentration                                   --
                       geometry of reaction space                           --
3. Redraw Graph in Symbolic
           Terms
           V1        V2



 X3
      -    X1        X2


      X4        X5        X6
      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
Failure to Account for Removal
           (Dilution)


Hageman F(XII)                  Ac t. Hageman F.




                 P.T.A . (XI)                      Ac t. P.T.A .

                                                    Ca++

                                  Xmas F. (IX)                     Ac t. Xmas F.
Failure to Distinguish Types of
         Multireactants
X1                  X1

          X3                  X3

X2                  X2




               X2                  X2

     X1                  X1

               X3                  X3
   Failure to Account for
 Molecularity (Stoichiometry)

X1      X2       X1      X2




 2X1   X2         X1   2X2
     Confusion Between Material and
            Information Flow

       -
X4         X1   X2   X3          X1
Confusion of States, Processes,
   and Logical Implication
             Neuron            Neuron

 Hunger
                                          Pepsinogen




                                               Pepsin


 Digestion
 products
                                        Food
              Neutralization
                of acid
    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
Open versus Closed Systems

               X2
     X5   X1        X4
               X3


               X2
     X5   X1        X4
               X3
Variables Outside the System
            A
                     X2
                X1        X3
                     X4




            B
                     X2
       X6       X1        X3
                     X4


                     X5
     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)
      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
     Example: Metabolic Pathway


X4        X1          X2          X3

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
        Example: Gene Circuitry
A                 g45                              g15
           /0/+    Regulator gene           /+       Effector gene
           /0/+                           /0/+
                  g43                              g13

B
    g43     g45                           g15       g13

X6 NA           X4 mRNA               X6 NA              X1 mRNA

        X7 AA       X5 Regulator              X7 AA           X2 Enzyme

                                    X8 Substrate           X3 Inducer
      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
                      Result
• No matter what Vi+ and Vi– , and even if Vi+
  and Vi– are not known, the result in
  symbolic form is always

             ai X1 X2 …
                  gi1 gi2      gin
     Vi+                    Xn

             bi X1 X2 … Xn
                hi1    hi2     hin
     Vi– 

  “Power-Law Representation”
              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)
     Meaning of Kinetic Orders
  0 < g, h < 1     --   Saturating functions
      g, h > 1     --   Cooperative functions
– 1 < g, h < 0     --   Partial inhibition
      g, h < – 1   --   Strong inhibition
– 2 < g, h < 2     --   Typical values (higher for
                        fractal kinetics)
        System Description
 dXi/dt = Vi+(X1, …, Xn) – Vi–(X1, …, Xn)

becomes “S-system”:

            ai X1 X2 …
                 gi1 gi2        gin
 dXi/dt =                  Xn

                     bi X1 X2 …
                          hi1 hi2       hin
                 –                    Xn
      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
 Components of a Typical Analysis
• Steady state
  –   Numerical characterization
  –   Stability
  –   Signal propagation
  –   Sensitivities
• Dynamics
  – Time plots
  – Bolus experiments
  – Persistent changes

				
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posted:4/6/2013
language:English
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