Introduction to the Analysis of Biochemical and Genetic Systems

1. 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

2. Three Ways to Understand Systems • Bottom-up — molecular biology • Top-down — global expression data • Random systems — statistical regularities

3. Five-Part Presentation • From reduction to integration with approximate models • From maps to equations with power-laws • Typical analyses • Parameter estimation • Introduction to PLAS

4. 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

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

6. 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

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

8. 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?

9. t X X X X 1 3 4 2 0 1 2 3 4 5 6 7 8 . . . Dynamics

10. t X X X X 1 3 4 2 0 1 2 3 4 . . . or ? Critical Quantitative Relationships

11. Alternative Designs a b X X X X X X X X 2 1 1 3 2 3 3 1 c

12. Emergent Systems Paradigm • Focuses on problems of complexity and organization • Program unclear, few documented successes • On the verge of paradigm shift

13. 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

14. Contrast Complex and Simple

15. Quantitative Understanding of Integrated Behavior • Focus is global, integrative behavior • Based on underlying molecular determinants • Understanding shall be relational

16. 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

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

18. Examples • v = k1 X1 • v = k2 X1X2 • v = k3 X12.6 • v = VmX1/(Km+X1) • v = VhX12/(Kh2+X12)

19. 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

20. 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

21. 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

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

23. 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

24. 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

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

26. Rate Law Without Feedback

27. Rate Law With Feedback

28. 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

29. 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

30. 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

31. Example from Genetics

32. 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)

33. 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

34. 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

35. Steps of Model Design1. Initial Sketch

36. 2. Conversion Table

37. 3. Redraw Graph in Symbolic Terms

38. 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

39. Failure to Account for Removal (Dilution)

40. Failure to Distinguish Types of Multireactants

41. Failure to Account for Molecularity (Stoichiometry)

42. Confusion Between Material and Information Flow

43. Confusion of States, Processes, and Logical Implication

44. 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

45. Open versus Closed Systems X 2 X X X 1 4 5 X 3 X 2 X X X 1 4 5 X 3

46. Variables Outside the System

47. 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)

48. 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

49. 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

50. Example: Gene Circuitry