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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