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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
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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?
t X X X X 1 3 4 2 0 1 2 3 4 5 6 7 8 . . . Dynamics
t X X X X 1 3 4 2 0 1 2 3 4 . . . or ? Critical Quantitative Relationships
Alternative Designs a b X X X X X X X X 2 1 1 3 2 3 3 1 c
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
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/(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
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
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 Model Design: Maps
X X X X 4 1 2 3 Components of Maps • Variables (Xi, pools, nodes) • Fluxes of material (heavy arrows) • Signals (light or dashed arrows)
X X X X X X 1 2 3 1 2 3 Rules • Flux arrows point from node to node • Signal arrows point from node to flux arrow 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
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
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 X 2 X X X 1 4 5 X 3 X 2 X X X 1 4 5 X 3
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
X X X X 4 1 2 3 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