Simplicity is Complexity in Masquerade. Michael A. Savageau The University of California, Davis July 2004

Transcription

1 Simplicity is Complexity in Masquerade Michael A. Savageau The University of California, Davis July 2004

2 Complexity is Not Simplicity in Masquerade -- E. Yates

3 Simplicity is Complexity in Masquerade

4 One of John s s Principles of Outside Good Design Transparent to the user Works like magic Simple Inside Inscrutable to the user Rich in robust control structures Complex

5 A Comparative Nonlinear Approach to the Elucidation of Function, Design and Evolution of Biochemical Systems

6 Motivation Importance of comparison Well-controlled comparisons Methodology Modeling strategies Canonical nonlinear formalism Mathematically controlled comparisons Biological design principles Outline Anticipatory control in biosynthetic pathways Coupling of elementary gene circuits Demand for alternative modes of gene control

7 Importance of Comparison Why is there something and not nothing? Why is there something and not something else? Comparison is central to biology Experimental investigation Evolution Optimization (foundations)

8 Difficulties in Making Well-Controlled Comparisons Single changes in isogenic background (complexity) Single changes have multiple consequences (structure) Secondary consequences can mask primary consequences (information) State changes in nonlinear systems can have dramatic consequences (design)

9 Example End-product inhibition Mutant lacking inhibition NA mrna NA mrna AA Enzymes AA Enzymes Substrate Intermediate Product Substrate Intermediate Product v P = V P 1+ [ product] 2 K P 1 V P /2 Stable v P = V P Unstable v P = V P 1+ [ product] 2 K P 1 V P /2 Stable v P = V P /2 Stable

10 Motivation Importance of comparison Well-controlled comparisons Methodology Modeling strategies Canonical nonlinear formalism Mathematically controlled comparisons Biological design principles Outline Anticipatory control in biosynthetic pathways Coupling of elementary gene circuits Demand for alternative modes of gene control

11 Two Modeling Strategies Specific system Identify a specific system of interest Assemble available information and formulate a model Estimate parameter values and simulate known behaviors Successful outcome o o Mimic real system Predict additional behaviors Class of systems Identify class with many members Abstract essential characteristics and formulate a model Symbolic analysis and statistical sampling Successful outcome o o Understand the basis for nearly universal designs Discover rules for distinguishing alternative designs

12 Interlaced Levels of Description for a Chemical Reaction Time/Number Scale Small QM wave function Potential energy function Probability distribution function Rate law function Boolean function Discrete/Stochastic Continuous/Deterministic Discrete/Stochastic Continuous/Deterministic Discrete/Deterministic Large (scale, physics)

13 Power-Law Formalism dx i dt r = α ik k =1 n r g X ijk j β ik j =1 k =1 n j =1 X j h ijk Canonical from Four Different Perspectives Fundamental (physics) Recast (foundations) Local (scale) Piece-wise (design) M. Savageau, Chaos 11: 142 (2001)

14 Methodology Implications of the Canonical Power-Law Formalism Fundamental representation Reference for detailed kinetic descriptions Generalization of mass-action kinetics Local representation Regular mathematical structure Reasonable degree of local accuracy Piece-wise representation Regular mathematical structure Reasonable degree of global accuracy Recast representation Globally equivalent Converts implicit equations into explicit equations Efficient solver for ODEs and algebraic equations Irvine & Savageau, SIAM J. Numerical Anal. 27:704 (1990) Mueller, Burns & Savageau, Appl. Math. Comput. 90:167 (1998)

15 Recast Representation dx / dt = (y )e x x(0) = 3.85 dy / dt = e x (50 + z) y(0) = 7.16 dz / dt = (y 9.75)z z(0) = 7.98 dx 1 / dt = 0.343x 1 x 2 x 1 (0) = dx 2 / dt = x 1 x 3 x 4 x 2 (0) = dx 3 / dt = x x 2 x 4 x 3 (0) = dx 4 / dt = x 2 x x 4 x 4 (0) = 1 X 1 =e X, X 2 =y+17.15, and X 3 X 4 =z+50 Savageau & Voit, Math. Biosci. 87:83 (1987)

16 Global Accuracy of Recast Representations X X X Time X

17 Mathematically Controlled Comparison Two designs are represented in a canonical nonlinear formalism Differences are restricted to a single specific process One design is chosen as the reference Internal equivalence is maintained External equivalence is imposed The systems are characterized by rigorous mathematical and computer analysis Comparisons are made on the basis of quantitative criteria for functional effectiveness (foundations)

18 Motivation Importance of comparison Well-controlled comparisons Methodology Modeling strategies Canonical nonlinear formalism Mathematically controlled comparisons Biological design principles Outline Anticipatory control in biosynthetic pathways Coupling of elementary gene circuits Demand for alternative modes of gene control

19 Simple Systemic Behavior -- Autocatalytic Growth of Bacteria Cells

20 Steady-State State Growth ln OD Time Numerous models

21 Addition of a Nutrient ln OD Cit Time None predict this behavior!

22 Elucidation of the Underlying Mechanisms Reveals Layers of Complexity -- a Case Study

23 Added Citruline Should Increase Arginine and Promote Growth Citruline Argininosuccinate Arginine Arg-t-RNA Citruline 0

24 Feed-Forward Forward Inhibition Causes Self Starvation! Citruline Argininosuccinate Arginine Arg-t-RNA Citruline 0 But analysis shows that it cannot

25 Alternative Fates for Arginine Might Causes Self Starvation Citruline Argininosuccinate Arginine Arg-t-RNA Citruline 0 Design principle for control of branch points shows that it can, under certain circumstances

26 What Might be the Normal Function of Feed-Forward Forward Inhibition? X 0 X 1 X 2 X 3 X n-1 X n X n+1 A well-controlled comparison shows that there is no significant difference, with or without feed-forward inhibition by the penultimate end product

27 What About Other Feed-Forward Forward Inhibitors? X 0 X 1 X 2 X 3 X n-1 X n X n+1 Analysis shows that stability and temporal responsiveness are optimal when X 1 is the feed-forward inhibitor (design)

28 Enzyme Complexes to Eliminate Diffusion Delays X 0 X 1 X 2 X 3 X n-1 X n X n+1 (structure) X n-2 X 4 Experimental evidence for the isoleucine biosynthetic pathway confirms the predicted design with the first intermediate as feed-forward inhibitor and a complex involving the first and last enzyme of the pathway

29 Motivation Importance of comparison Well-controlled comparisons Methodology Modeling strategies Canonical nonlinear formalism Mathematically controlled comparisons Biological design principles Outline Anticipatory control in biosynthetic pathways Coupling of elementary gene circuits Demand for alternative modes of gene control

30 Two Extreme Forms of Coupling Gene Expression Perfect Coupling Complete Uncoupling NA, X 4 mrna, X 1 NA, X 4 mrna, X 1 AA, X 5 Enzyme/Regulator, X 2/ X 0 Regulator, X 0 AA, X 5 Enzyme, X 2 Substrate, X 6 Inducer, X 3 Substrate, X 6 Inducer, X 3 (information)

31 Equations Perfect Coupling Complete Uncoupling dx 1 dt = α 1B β 1 X 1 X 3 < X 3L dx 1 dt = α 1B β 1 X 1 X 3 < X 3L dx 1 dt = α p g p 1 X 12 g p 2 X 13 3 β 1 X 1 X 3L < X 3 < X 3H dx 1 dt = α 1 u X 3 g 13 u β 1 X 1 X 3L < X 3 < X 3H dx 1 dt = α 1M β 1 X 1 X 3H < X 3 dx 1 dt = α 1M β 1 X 1 X 3H < X 3 dx 2 dt = α 2 X 1 β 2 X 2 dx 2 dt = α 2 X 1 β 2 X 2 dx 3 dt = α 3 X 2 g 32 X 4 g 34 β 3 X 2 h 32 X 3 h 33 dx 3 dt = α 3 X 2 g 32 X 4 g 34 β 3 X 2 h 32 X 3 h 33

32 Log(enzyme concentration) Constraints for External Equivalence Induction characteristic Unique parameters Capacity Gain α u 1 = β 1 α 1 p β 1 α 2 β 2 g 12 p / h 22 h 11 h 22 h 11 h 22 g 12 p g 21 Threshold g 13 u = g 13 p h 11 h 22 h 11 h 22 g 12 p g 21 Log(inducer concentration)

33 Design Space Repressor control Activator control Intermediate Low High 0 g Inducer Stable Unstable g Regulator Low High 0 g Inducer Stable Unstable g Regulator Line of equivalence g 13 = h 11 h 22 h 33 L 24 g 21 g 34 h 33 L 24 g 34 g 12

34 Example of Analytical Comparison Robustness measured by parameter sensitivities Parameter sensitivities defined as S(V i, p j ) = V i p j p j V i External equivalence implies g 13 u = g 13 p h 11 h 22 h 11 h 22 g 12 p g 21 Ratio for comparison ( ) p = ( ) u SV 3,β 2 SV 3,β 2 h 11 h 22 < 1 for g p h 11 h 22 g p 12 < 0 12 g 21 Conclusion: Perfectly coupled circuit with repressor control is more robust than the equivalent completely uncoupled circuit Savageau, Nature 229: 542 (1971) Becskei & Serrano, Nature 405: 590 (2000)

35 Response Time Flux to product V g Time (min) Savageau, Nature 252: 546 (1974) Rosenfeld, et al., J. Mol. Biol. 323: 785 (2002)

36 Coupling of Gene Expression in Elementary Circuits A NA mrna NA mrna AA Regulator AA Enzyme Substrate Inducer B C Log [Regulator] Directly Coupled Uncoupled Inversely Coupled Log [Enzyme] Induction Log [Substrate] Log [Substrate]

37 Predicted Coupling of Gene Expression in Elementary Circuits Mode Capacity Predicted coupling Positive Small Inversely coupled Positive Large Directly coupled Negative Small Directly coupled Negative Large Inversely coupled

38 Experimental Evidence for Coupling of Gene Expression in Elementary Circuits 3 Log (Expression capacity of regulator gene) hutigc-hutuh metr-mete dsdc-dsda laci-laczya arac-arabad D U I Log (Expression capacity of effector gene) Hlavacek & Savageau, J. Mol. Biol. 255: 121 (1996) Wall, et al., Nature Rev. Genetics 5: 34 (2004)

39 Motivation Importance of comparison Well-controlled comparisons Methodology Modeling strategies Canonical nonlinear formalism Mathematically controlled comparisons Biological design principles Outline Anticipatory control in biosynthetic pathways Coupling of elementary gene circuits Demand for alternative modes of gene control

40 Dual Modes of Gene Control (structure, design,information, physics, foundations)

41 Demand Theory of Gene Control A positive mode of control is predicted when there is a high demand for expression of a gene A negative mode of control is predicted when there is a low demand for expression of a gene Savageau, PNAS 71:2453 (1974)

42 Life Cycle of Escherichia coli H H L H L DC (1- DC ) Time (hrs) C Lactase L H L

43 Region of Realizability Modulator Promoter Log [C] Log [D] (design )

44 Rate and Extent of Selection A B H L D 1E+10 1E+8 1E+6 Modulator Promoter H C (hrs) 1E+4 1E+2 E 1E+0 1E+8 1E+7 1E+6 C Lactase H L H L DC (1- DC ) Time (hrs) C L F 1E+5 1E+4 1E+3 1E+5 1E+4 1E+3 1E+2 Response time (hrs) ent of selection mutant 1E-6 fraction) 1E-7 1E-5 1E-4 D 1E-3 1E-2 1E-1 Savageau (1998)

45 Predictions Cycling without colonization 26 hours Colonization without cycling 66 years Rate of re-colonization 4 months Evolutionary response time 3 years M. Savageau, Genetics 149:1677 (1998)

46 LYTIC PHAGE λ Lytic + N CI CRO + Lytic LYSOGENIC PHAGE λ Lytic N CI + CRO Lytic (design )

47 Sample of Design Principles for Gene Circuits Molecular mode control Savageau, Genetics 149: 1677 (1998) Switching characteristics Savageau, Math. Biosci. 180:237 (2002) Signaling Cross-talk Alves & Savageau, Mol. Microbiol. 48:25 (2003) Coupling of expression Wall, et al., Nature Rev. Genetics 5: 34 (2004)

48 Summary Comparisons are critical Need for well-controlled comparisons Example illustrating the difficulties Methodology Distinguishing between two modeling strategies Canonical nonlinear formalism has fundamental implications Mathematically controlled comparisons are nearly ideal Biological design principles Feed-forward inhibition stabilizes long biosynthetic pathways Rules for the coupling of expression in elementary gene circuits Demand for gene expression provides the natural selection for alternative molecular modes of gene control

49 Acknowledgements Eberhard Voit Douglas Irvine Rui Alves Gerald Jacknow William Hlavacek Michael Wall NSF, NIH, ONR, DOE Sloan Foundation Pfizer