View
20
Download
1
Category
Preview:
DESCRIPTION
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 - PowerPoint PPT Presentation
Citation preview
Introduction to the Analysis of Biochemical and Genetic Systems
Eberhard O. Voit* and Michael A. Savageau**
*Department of Biometry and EpidemiologyMedical University of South Carolina
VoitEO@MUSC.edu
**Department of Microbiology and ImmunologyThe 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?
X1 X3 X4X2t
012345678...
Dynamics
X1 X3 X4X2t
01234...
or ?
Critical Quantitative Relationships
X1
X3
X2
a
X1
X3
X2
b
c
X3
X1
Alternative Designs
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 X1
• v = k2 X1X2
• v = k3 X12.6
• v = VmX1/(Km+X1)
• v = VhX12/(Kh
2+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
V
X
•
•
A
B
XBXA
Rate Law With Feedback
V
X
• •
•
A A'
C
XBXA
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: MapsRibose 5-PATP
PP-Ribose-P Glutamine
P-Ribosyl-NH2
IMP
Amido-PRT
PP-Ribose-PSynthetase
ADP 2,3-DPG
Other Nucleotides
NADFAD
AMP, GMPATP, GTP
Example from Genetics
CRP
galS
galR
mgl
Galactose transport
gal
Galactose utilizat ion
Galactose transport
B A C
E T K (M)
P
gal
pp
p
p
p
--
+
+
-+
-
- ?
+
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
X1 X2
X3
X1 X2
X3
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 Design1. Initial Sketch
Homoserine O-Homoserine-P Threonine-
Homoserinekinase
Threoninesynthetase
Threonyl-tRNAsynthetase
2. Conversion TableTable 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 X2Flux through O-homoserine-P V1Flux through threonine V2
Independent Homoserine concentration X3Homoserine kinase concentration X4Threonine synthetase concentration X5Threonyl-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
X 5X 4 X 6
-X 3 X 1 X 2
V1 V2
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)
Act. Xmas F.
Hageman F(XII) Act. Hageman F.
P.T.A. (XI)
Xmas F. (IX)
Act. P.T.A.
Ca++
Failure to Distinguish Types of Multireactants
X 1
X 2
X 3
X 1
X 2
X 3
X 2
X 3
X 1
X 2
X 3
X 1
Failure to Account for Molecularity (Stoichiometry)
X 1 X 2 X 1 X 2
X 1 X 22 X 1 X 22
Confusion Between Material and Information Flow
X 3X 1 X 2X 4
-X 1
Confusion of States, Processes, and Logical Implication
Hunger
Neuron Neuron
Pepsinogen
Pepsin
Food
Neutralizationof acid
Digestionproducts
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
X1X5 X4
X3
X2
X1X5 X4
X3
X2
Variables Outside the System
X 1
X 2
X 3
X 4
X 1
X 2
X 3
X 4
X 5
X 6
A
B
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
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
X4 X1 X2 X3
Example: Gene Circuitry
A
/0/+ /0/+
Regulator gene /0/+
Effector gene
g45
g43
g15
g13
B
X1 mRNA
X2 Enzyme
X4 mRNA
X5 Regulator
X6 NA
X7 AA
X8 Substrate
X6 NA
X7 AA
X3 Inducer
g45g43 g13g15
/+
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
Vi+ i X1
gi1X2
gi2… Xn
gin
Vi– i X1
hi1X2
hi2… Xn
hin
“Power-Law Representation”
Parameters
gij: kinetic orders (positive, negative, or zero)
hij: kinetic orders (positive, negative, or zero)
i: rate constants (positive or zero)
i: 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”:
dXi/dt = i X1
gi1X2
gi2… Xn
gin
– i X1
hi1X2
hi2… Xn
hin
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
Recommended