1. 1 Signal Transduction, Cellerator, and The Computable Plant Bruce E Shapiro, PhD bshapiro@caltech.edu http://www.bruce-shapiro.com/cssb

2. 2 Overview Cellerator Chemical Kinetics Signal Transduction Networks –Modular outlook –Switches, oscillators, cascades, amplifiers, etc. –Deterministic vs. Stochastic simulations Multicellular systems –Synchrony, pattern formation –The Computable Plant project Model Inference

3. 3 Cellerator

4. 4 Short Cellerator Demonstration

5. 5 Canonical form of a chemical reaction: Ri,P i : Reactants, Products s Pi,s Ri : Stoichiometry k : Rate Constant Example: Law of Mass Action: The rate of the reaction is proportional to the product of the concentrations of the reactants. Law of Mass Action

6. 6 Law of Mass Action (2) Formal statement (for a single reaction): Interpretation of

7. 7 Law of Mass Action (3) Add rates for multiple reactions “Oregonator”

8. 8 Cellerator Input for Oregonator stn={{BrO3+Br  HBrO2+HOBr, k1}, {HBrO2+Br  2*HOBr, k2}, {BrO3+HBrO2  HBrO2+2*Ce, k3}, {2*HBrO2  BrO3+HOBr, k4}, {Ce  Br, k5}}; interpret[stn, frozen  {BrO3}]; Hold BrO3 concentration Fixed Stoichiometry Rate Constants

9. 9 Cellerator Output for Oregonator { {Br’[t]==-k1*Br[t]*BrO3[t]+k5*Ce[t]- k2*Br[t]*HBrO2[t], Ce’[t]==-k5*Ce[t]+2*k3*BrO3[t]*HBrO2[t], HBrO2’[t]==k1*Br[t]*BrO3[t]- k2*Br[t]*HBrO2[t] +k3*BrO3[t]*HBrO2[t] -k4*HBrO2[t]^2, HOBr’[t]==2*k2*Br[t]*HBrO2[t] +k4*HBrO2[t]2+k1*Br[t]*BrO3[t]}, {Br, Ce, HBrO2, HOBr} } List of Differential Equations and Variables

10. 10 Cellerator Simulation s= predictTimeCourse[stn, frozen  {BrO3}, timeSpan  1500, rates  {k1  1.3, k2  2*10 6, k3  34, k4  3000., k5  0.02, BrO3[t] .1}, initialConditions  {HBrO2 .001, Br .003,Ce .05,BrO3 .1}; {{0, 1500, {{Br  InterpolatingFunction[{{0., 1500.}}, <>], Ce  InterpolatingFunction[{{0., 1500.}}, <>], HBrO2  InterpolatingFunction[{{0., 1500.}}, <>], HOBr  InterpolatingFunction[{{0., 1500.}}, <>]}}}} Input Output

11. 11 Plot Results of Simulation runPlot[s, plotVariables  {Br}, PlotRange  { {400, 1400}, {0, 0.002}}, TextStyle  {FontFamily -> Times, FontSize -> 24}, PlotLabel  "Br Concentration"]; Optional Input

12. 12 Basic Syntax Format of rate constants varies for different arrows Modifiers are optional Different rate laws for different arrow/modifier combinations We will focus on reaction Generate differential equation by entering interpret[network]

13. 13 Basic Mass Action Reactions Cellerator Syntax: We will generally omit explicitly writing the rate constants in the remainder of this presentation.

14. 14 Catalytic Mass Action Reactions becomes

15. 15 Cascades

16. 16 Michaelis-Menten Kinetics Catalytic Reaction: Mass action Steady-state assumption: where E 0 is total catalyst (bound + unbound)

17. 17 Michaelis-Menten Kinetics (2) Solve for where Therefore where v=kE 0 If then hence

18. 18 Michaelis-Menten in Cellerator

19. 19 Comparison of models

20. 20 GTP: A molecular switch

21. 21 GTP: A reaction schema

22. 22 GTP: Cellerator schema

23. 23 GTP: Cellerator Simulation

24. 24 RASGTP Switch

25. 25 Cascades

26. 26 MAPK: Mitogen Activate Protein Kinase Cell Growth and Survival Heat Shock, Radiation, Chemical, Inflamatory Stress lab of Jim Woodget, http://kinase.uhnres.utoronto.ca/

27. 27 MAPK Cascade Reactions in solution (no scaffold)

28. 28 MAPK in Solution Kinase Reactions 1 st Stage KKK+S  KKK-S KKK-S  KKK+S KKK-S  KKK * +S 2 nd Stage (1 st Phosphate group) KK+KKK *  KK-KKK * KK-KKK *  KK+KKK * KK-KKK *  KKK * +KK * 2 nd Stage (2 nd Phosphate group) KKK * +KK *  KK * -KKK * KK * -KKK *  KKK * +KK * KK * -KKK *  KKK * +KK ** 3 rd Stage (1 st Phosphate group) K+KK **  K-KK ** K-KK **  K+KK ** K-KK **  KK ** +K * 3 rd Stage (2 nd Phosphate group) KK ** +K *  K * -KK ** K * -KK **  KK ** +K * K * -KK **  KK ** +K ** Phosphatase Reactions 1 st Stage KKK * +Ph 1  KKK * -Ph 1 KKK * -Ph 1  KKK+ Ph 1 KKK * - Ph 1  KKK * + Ph 1 2 nd Stage (1 st Phosphate group) KK * +Ph 2  KK * -Ph 2 KK * - Ph 2  KK+ Ph 2 KK * - Ph 2  KK * + Ph 2 2 nd Stage (2 nd Phosphate group) KK ** +Ph 2  KK ** -Ph 2 KK ** - Ph 2  KK * + Ph 2 KK ** - Ph 2  KK ** + Ph 2 3rd Stage (1 st Phosphate group) K * +Ph 3  K * -Ph 3 K * - Ph 3  K+ Ph 3 K * - Ph 3  K * + Ph 3 3 rd Stage (2 nd Phosphate group) K ** +Ph 3  K ** -Ph 3 K ** - Ph 3  K * + Ph 3 K ** - Ph 3  K ** + Ph 3

29. 29 MAPK Cascade on Scaffold Scaffold binding significantly increases the rate of phosphorylation Scaffold has 3 slots: one for each kinase Each slot can be in different states –Slot 1: empty, KKK, or KKK* bound –Slot 2: empty, KK, KK*, or KK** bound –Slot 3: empty, K, K*, K** bound Enter/leave scaffold in any order KKK* and either KK or KK* must be bound at same time produce KK**, etc. Number of reactions increases exponentially with number of slots

30. 30 Effect of Scaffold on Simulations

31. 31 Reactions in MAP Kinase Cascade Phosphorylation in Solution Binding to Scaffold Phosphorylation in Scaffold

32. 32 Effect of Scaffold on MAPK

33. 33 Stochastic Comments When the number of molecules is small the continuous approach is unrealistic –Differential equations describe probabilities and not concentrations At intermediate concentrations the continuous approach has some validity but there will still be noise due to stochastic effects. –Langevin Approach:

34. 34 Direct Stochastic Algorithm Gillespie Algorithm (1/3): At any given time, determine which reaction is going to occur next, and modify numbers of molecules accordingly Gillespie DT (1977) J. Phys. Chem. 81: 2340-2361.

35. 35 Gillespie Algorithm (2/3)

36. 36 Gillespie Algorithm (3/3) Let t=0 While t<t max { Calculate all the ai=h i k i and a 0 =  a j Generate two random numbers r 1, r 2 on (0, 1) The time until the next reaction is  =(1/a 0 )ln(1/r 1 ) Set t = t +  Reaction R j occurs at t, where j satisfies a 1 +a 2 +…+a j-1 < r 2 a 0 ≤ a j +a j+1 +…+a n Update the X 1,X 2,…,X n to reflect the occurance of reaction R j }

37. 37 Stochastic MAPK Simulation (1/3)

38. 38 Stochastic MAPK Simulation (2/3)

39. 39 Stochastic MAPK Simulation (3/3)

40. 40 Analysis of Multi-step reactions Steady State Solution Adding steps increases sensitivity Simplify to:

41. 41 Analysis of multi-stage reactions Consider two stages of a cascade with m and n steps Steady State:

42. 42 Analysis of multi-stage reactions If [X]<<K x Hill exponent is product of m and n –E.g., a three-step stage followed by a four-step stage behaves like a 12-step stage By incorporating negative feedback can produce high-gain amplification (see refs).

43. 43 Oscillators in Nature Where they occur (to name a few): –Circadian rhythms –Mitotic oscillations –Calcium oscillations –Glycolysis –cAMP –Hormone levels How they occur: feedback –Both negative & positive feedback systems –Some have feed-forward loops also

44. 44 Negative feedback: canonical model* Equivalent second order system:0 Characteristic equation: *Hoffmann et al (2002) Science 298:1241 a=0.25,b=10,  =1,S=5

45. 45 2-species ring oscillator

46. 46 3-species ring oscillator

47. 47 3-species Ring Oscillator v=10K M Robust oscillations v=K M Damped oscillations

48. 48 Repressilator Constructed in E. coli Elowitz & Leibler, Nature 403:335 (2000)

49. 49 Repressilator Model Simulations

50. 50 Cell Division - Canonical Model Goldbeter (1991) PNAS USA, 88:9107 “Minimal” Model of Cell Division

51. 51 Cell Division - Canonical Model

52. 52 Multi-cellular networks Intracellular Network e.g., of mass action, etc. Transport, ligand/receptor interactions, etc Diffusion Tensor Connection matrix Set of neighbors of cell j Species x i in cell j

53. 53 Example - coupled oscillators Two uncoupled Oscillators Two Coupled Oscillators,  =.1

54. 54 Example - 105 coupled oscillators

55. 55 Example - 105 coupled oscillators

56. 56 Coupled nonlinear oscillators Arbitrarily let species X in CMX model diffuse to adjacent cells

57. 57 Coupled CMX Oscillators All oscillating at same frequency But different phases What happens if you have 105 coupled oscillators with random phase shifts?

58. 58 105 CMX Oscillators: uncoupled

59. 59 105 CMX Oscillators: uncoupled

60. 60 105 CMX Oscillators: low coupling

61. 61 105 CMX Oscillators: higher coupling

62. 62 105 CMX Oscillators: higher coupling

63. 63 105 CMX Oscillators: Random Period Uncoupled motion

64. 64 105 CMX Oscillators: Random Period

65. 65 105 CMX Oscillators: Random Period Uncoupled Coupled Oscillators

66. 66 Pattern Formation Activator-Inhibitor Models Single Diffusing Species Self-activating (locally) Self-inhibitory (externally) Two Diffusing Species X: Activator Y: Inhibitor

67. 67 Two species pattern formation model Continuous model*: Discrete implementation: *See Murray Chapter 14 for detailed analysis

68. 68 Single species pattern formation model Continuous (logistic) model: Discrete implementation: Steady State Equation (v=K=v M =K M =1, x=[A])

69. 69 Single species pattern formation model is a steady state only if all neighbors are at x=0 Suppose that there are n x neighbors in state x and all other neighbors are in state x=0, 0≤n x 6 Example: case when n x =1 (exactly one neighbor at x, all others at 0): Question: what other combinations are possible?

70. 70 Single Species Model - 105 Cells

71. 71 Single Species Model - 105 Cells

72. 72 Provide: most of our food and fiber all of our paper, cellulose, rayon pharmaceuticals feed stock waxes perfumes Shoot Apical Meristem growing tip of a plant Image courtesy of E. M. Meyerowitz, Caltech Division of Biology Computable Plant Project

73. 73 Computable Plant Project NSF (USA) Frontiers in Integrative Biological Research (FIBR) Program S/W Architecture: Production-scale model inference –Models formulated as cellerator reactions or SBML –C++ simulation code autogenerated from models –Mathematical framework combining transcriptional regulation, signal transduction, and dynamical mechanical models –Simulation engine including standard numerical solvers and plot capability –Nonlinear optimization and parameter estimation –ad hoc image processing and data mining tools Image Acquisition –Dedicated Zeiss LSM 510 meta upright laser scanning confocal microscope. http://www.computableplant.org

74. 74 Computable Plant Project

75. 75 Model Organism Arabidopsis Thaliana

76. 76 Cell Identification in image z-stack

77. 77 Identification of Cell Birth

78. 78 Image courtesy of E. M. Meyerowitz, Caltech Division of Biology Shoot Apical Meristem

79. 79 Meristem Pattern Maintenance Model

80. 80 Simulation of Meristem Growth

81. 81 Systems Biology Markup Language http://sbml.org libsbml (C++) MathSBML (Mathematica)

82. 82 The Standard Paradigm of Biology RNA

83. 83 Microarrays Produce a lot of data! Affymetrix GeneChip® microarray. Images courtesy of Affymetrix.

84. 84 RNA Fragments are Selectively Sticky

85. 85 Affymetrix GeneChip® Scanner 3000 with workstation Data from an experiment showing the expression of thousands of genes on a single GeneChip® probe array. Images courtesy of Affymetrix.

86. 86 Model Inference: Fitting A Model to Data Cluster to reduce data size Use simplest possible mathematical possible to determine connectivity –Fit parameters with some optimization process: simulated annealing, least squares, steepest descent, etc. –Refine model with biological knowledge –Refine with better accurate math model –… and repeat until done …

87. 87 Clustering Time Gene

88. 88 Data clusters in two dimensions x y Concentration at time t 1 Concentration at time t 2 Plot ([X[t 3 1],X[t 2 ],X[t 3 ],…,X[t n ]) for every species

89. 89 Data clusters in two dimensions x y Concentration at time t 1 Concentration at time t 2

90. 90 Clusters (may) correspond to functional modules 4: Input 0: Output Signal Transduction Network 1 2 3

91. 91 Approximation Models Linear S-Systems (Savageau) Generalized Mass Action

92. 92 Approximation Models Generalized Continuous Sigma-Pi Networks

93. 93 Approximation Models Recurrent Artificial Neural Networks Recurrent Artificial Neural Networks with controlled degradation

94. 94 Approximation Models Recurrent Artificial Neural Networks with biochemical knowledge about some species Known or hypothesized interactions due to mass action, Michaelis-Menten, or other reactions (A priori knowledge or assumptions)

95. 95 Approximation Models Multicellular Artificial Neural Networks with biochemical knowledge about some species: Resources Diffusion Lower index: species; Upper Index: Cell Geometric Connections

96. 96 Stripe Formation in Drosophila J Exp Zoology 271:47-56 Dashes- Observations Solid - Model Patterson, JT Studies in the genetics of drosophila, University of Texas Press (1943); http://flybase.bio.indiana.edu:82/anatomy/Drosophila Observed Reinitz, Sharp, Mjolsness Exper. Zoo. 271:47-56 (1995)

97. 97 Some Important Meetings ISMB-2004, Scotland, ≈ 30 July 04 Intelligent Systems in Molecular Biology 2003: Australia; 2005: US; 2006:Brazil ICSB-2004, Heidelberg, Oct 04 International Conference on Systems Biology SBML Forum held as satellite meeting 2003:US; 2002:Sweden; 2001:US; 2000: Japan PSB-2005, Hawaii, Jan 05 Pacific Symposium on Biocomputing RECOMB, Spring 05 Research in Computational Molecular Biology BGRS-04, July 04, Semiannually in Novosibirsk Bioinformatics of Genome Regulation and Structure Satellite meetings of many major biology and computer science meetings: SIAM, ACB, IEEE, ASCB (US), Neuroscience, IBRO,..

98. 98 Collaborators CelleratorCellerator –Eric Mjolsness, U. California, Irvine (Computer ) –Andre Levchenko, Johns Hopkins (Bioengineering) Computable Plant - Eric Mjolsness, PIComputable Plant - Eric Mjolsness, PI –Elliot Meyerowitz, Caltech (Biology) –Venu Reddy, Caltech (Biology) –Marcus Heisler, Caltech (Biology) –Henrik Jonsson, Lund, Sweden (Physics) –Victoria Gor, JPL (Machine Learning) SBML (John Doyle, PI, Caltech; H. Kitano, Japan)SBML (John Doyle, PI, Caltech; H. Kitano, Japan) –Mike Hucka, Caltech (Control & Dynamical Systems) –Andrew Finney, University of Hertfordshire, UK