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Systems Biology and Numerical Analysis

Systems Biology and Numerical Analysis. Stephanie Taylor Ph.D. Candidate Department of Computer Science, UCSB March 1, 2006. We are Living In a Bacterial World. Staphylococcal Enterotoxin B (SEB). (CDC MMWR 1983). We are Living In a Bacterial World. Staphylococcal Enterotoxin B (SEB).

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Systems Biology and Numerical Analysis

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  1. Systems Biology and Numerical Analysis Stephanie Taylor Ph.D. Candidate Department of Computer Science, UCSB March 1, 2006

  2. We are LivingIn a Bacterial World Staphylococcal Enterotoxin B (SEB) (CDC MMWR 1983)

  3. We are LivingIn a Bacterial World Staphylococcal Enterotoxin B (SEB) (CDC MMWR 1983)

  4. SEB SEB is cleared from the system by the kidneys But when there is too much SEB, the kidney cells die (apoptosis).

  5. SEB-Induced Apoptosis • How does a kidney cell react to SEB? • Can we stop it from killing the cell? Truth in Advertising : These questions will not be answered during this talk. This is an active area of research!

  6. Capture interactions between components Quantify dynamics Elucidate more complicated mechanisms Determine components of system Perform experiments Reveal certain mechanisms Understanding Biological Systems Systems Biology

  7. Process • Experiment • Explain with Cartoon • Translate Cartoon Interactions into Mathematical Expressions • Combine Expressions into an ODE Model • Analyze the Model (Fall, Marland, Wagner, & Tyson, Computational Cell Biology, 2002)

  8. SEB Experimentation cluster analysis gene arrays (Zhang, Bio-SPICE Technical Report, 2005), (TJU, Bio-SPICE Use Case Reports, 2005)

  9. Process • Experiment • Explain with Cartoon • Translate Cartoon Interactions into Mathematical Expressions • Combine Expressions into an ODE Model • Analyze the Model (Fall, Marland, Wagner, & Tyson, Computational Cell Biology, 2002)

  10. SEB-Induced Apoptosis Cartoon (Jason Shoemaker,2005)

  11. Process • Experiment • Explain with Cartoon • Translate Cartoon Interactions into Mathematical Expressions • Combine Expressions into an ODE Model • Analyze the Model (Fall, Marland, Wagner, & Tyson, Computational Cell Biology, 2002)

  12. SEB-Induced Apoptosis Model • Each node represents an entity (such as a protein) • Each edge represents a reactions • We construct one ODE for each node. r14_k r21_k (Jason Shoemaker,2005)

  13. SEB-Induced Apoptosis Model parameter r14_k r21_k state (Jason Shoemaker,2005)

  14. Process • Experiment • Explain with Cartoon • Translate Cartoon Interactions into Mathematical Expressions • Combine Expressions into an ODE Model • Analyze the Model (Fall, Marland, Wagner, & Tyson, Computational Cell Biology, 2002)

  15. SEB-Induced Apoptosis Model … …

  16. SEB-Induced Apoptosis Model …

  17. SEB-Induced Apoptosis Model …

  18. Process • Experiment • Explain with Cartoon • Translate Cartoon Interactions into Mathematical Expressions • Combine Expressions into an ODE Model • Analyze the Model (Fall, Marland, Wagner, & Tyson, Computational Cell Biology, 2002)

  19. Simulating the Model • How do we integrate the ODE’s over time to see the dynamics? • If we know what the protein concentrations are at time 0, we can use the rate information to determine the concentrations at later times. • An analytical solution is too difficult (or impossible) to find. • So, we solve it numerically

  20. Forward (Explicit) Euler • First step – Discretize Time y time t1 tn h

  21. Forward (Explicit) Euler • Taylor Series Expansion X If the timestep h is small, then this term will be close to 0.

  22. Forward (Explicit) Euler • Taylor Series Expansion • Forward Euler Method X

  23. Forward (Explicit) Euler y time t1 tn h

  24. Forward (Explicit) Euler y time t1 tn h

  25. Forward (Explicit) Euler y time t1 tn h

  26. Forward (Explicit) Euler • If the true solution behaves well, and if our timestep h is small enough, then Forward Euler will give us a reasonable result. • However, it is inefficient and has only first order precision in h. • Taylor Expansion • Forward Euler

  27. How to get Fancier • Variable timestep sizes • Higher Order Methods • Multi-Step Methods • Runge-Kutta Methods • Adams-Moulton Methods • Backward Differentiation Formula (BDF) • Software for solving ODE’s • XPP • DASPK

  28. Simulation Results BAD state indicating apoptosis state indicating apoptosis

  29. Simulation Results • Didn’t have expected dynamics • Cell died whether or not SEB was present • What is wrong with the model? • Either our kinetics are incorrect • Or we aren’t capturing enough of the players • We are pretty sure the kinetics are correct. So how do we choose where to expand the model? • Use human intuition • Is there a way the computer can help? With 77 states, we have lots of information.

  30. Refining the SEB Model • What if we knew what parts of the model were having the strongest effect on the behavior? Where are the hotspots? • We can find the hotspots using sensitivity analysis

  31. Sensitivity Analysis • What effect does a small perturbation in a parameter have upon the state of the system at time t? p=0.2+Dp Dp = 0.05 p=0.2

  32. Sensitivity Analysis • Solve the sensitivity equations along with the original system

  33. Sensitivity Analysis NT NY NP

  34. Fisher Information Matrix NP Weighted Norms of the Sensitivities NP S1*T S4*T S2*T S3*T X V3 V2 V1 V4 X S3* S1* S4* S2* F3 F1 F4 F2

  35. Fisher Information Matrix NP NP Overall Parameter Sensitivities NP

  36. Fisher Information Matrix Overall Parameter Sensitivities NP 323 0.001 0.023 3E5 3E6 3.245 748 0.547 23E3 276 10 The system is highly sensitive to parameters P4, P5, and P9!

  37. Hotspots • So, we did the sensitivity analysis, identifying some parts of the pathway that had a large effect upon the dynamics of the system • Did a literature and database search for genes/proteins related to the ones in the hotspots. • Expanded the model

  38. Model Refinement

  39. Model Refinement

  40. Model Refinement

  41. Simulation PI3K_A PI3K_A

  42. Conclusion • Process • Experiment • Explain with Cartoon • Translate Cartoon Interactions into Mathematical Expressions • Combine Expressions into an ODE Model • Analyze the Model (Simulation, Sensitivity Analysis) • REPEAT • And now, for a peek into my software …

  43. BioSens SBML FIM Model dx/dt = f(x,p,t) BioSens Simulation & Sensitivity Computation Sensitivity Ranking x(t,p) Si,j Measurement Selection

  44. Software Dependencies BioSens 2.0 XPP Matlab DASPK 3.0 libSBML Xerces Tapenade g77, gcc make cygpath Cygwin Java VM

  45. Acknowledgements • Dr. Rudi Gunawan • Dr. Tingting Zhang • Jason Shoemaker • Dr. Francis J. Doyle, III • Dr. Linda Petzold www.cse.ucsb.edu

  46. Thank You! Questions?

  47. Fisher Information Matrix • FIM represents the amount of information contained in data. • Rankings: Diagonal entries represent the effect each parameter has on the overall system. (# states) (# parameters) (# timesteps) (# parameters)

  48. Sensitivity Analysis NT NT

  49. Sensitivity Analysis SiT Si X X

  50. Sensitivity Analysis NP NP

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