A Bayesian Statistical Approach to Modeling Gene Regulatory Pathways in Human Placental Data

1. A Bayesian Statistical Approach to Modeling Gene Regulatory Pathways in Human Placental Data Elinor Velasquez Dept. of Biology San Francisco State University

2. Outline of talk • Introduction • The experimental approach: Obtaining placenta data • The experimental approach: Modeling gene regulatory networks • Results from experiments • Conclusions and future work • Acknowledgements

3. Introduction

4. Overall goal To use a bioinformatics model for which to better understand the human placenta http://www.biotechnologycenter.org/hio/assets/hisimages/placenta/placenta44.jpg

5. The human placenta http://www.uchsc.edu/winnlab/index.html

6. The basal plate in the placenta Site of known anatomical abnormalities in preeclampsia http://www.uchsc.edu/winnlab/projects.html

7. EGFR pathway • EGFR, cell surface receptor for epidermal growth factors • Potentially important gene for the placenta British Journal of Cancer (2006) 94, 184 – 188

8. EGFR regulates gene expression EGFR CSPG2 DCN ANGPT2

9. Causal relationships EGFR CSPG2 DCN ANGPT2

10. Example of a gene regulatory network Gene 1 Gene 3 Gene 2 Gene 5 Gene 6 Gene 4

11. Definition of a Bayesian network • There exist nodes (disks) • There are edges (arrows) between some of the nodes • Causality is implied by the edges • Acyclic Gene 1 Gene 3 Gene 2 Gene 5 Gene 6 Gene 4

12. The experimental approach: Obtaining placenta data

13. Data collected from microarrays • Data comes from 36 experiments conducted by Virginia Winn et al.at the SJ Fisher lab, UCSF • Gene expression profiling experiments cRNA hybridization 45000 dots (25-mer oligo probe sets) representing the human genome

14. Traditional “spotted” arrays

15. What is a probe set? • Several oligonucleotides designed to hybridize to various parts of the mRNA generated from a single gene Probe set mRNA gene

16. Affymetrix GeneChips

17. Microarray data The normalized log 2 intensity values were centered to the median value of each probe set, by Virginia Winn et al. 5 time segments: 1 2 3 4 5 A probe setx1 ... x6 y1 ... y9 z1 ...z6 w1...w6 s1 ... s9 36 data points per probe set

18. Microarray data • Red denotes the up regulated expression and green denotes the down regulated expression relative to the median value • Genes differentially expressed in the basal plate of placentas: Rows contain data from a single basal plate cRNA sample and columns correspond to a single probe set. http://www.uchsc.edu/winnlab/index.html

19. Summary of data used in bioinformatics experiments • 36 placentas • 45, 000 probe sets • Time-series data from 14-16 weeks to term Gene egfr

20. The experimental approach: Modeling gene regulatory networks

21. Step 1. Create a naïve Bayesian network using the probe set data Step 2. Score the naïve Bayesian network Step 3. Randomly add/delete an edge and rescore the Bayesian network Step 4. Continue until best score reached Step 5. Combine probe sets to create the gene regulatory network Outline of bioinformatics experimental design PS 1 PS 2 PS 4 PS 3

22. Four probe sets (Three genes)

23. Define naïve Bayesian network • Choose a root node • All other nodes branch off of the root node • PS1 is the parent node PS 1 PS 2 PS 4 PS 3

24. Step 1: Create a naïve Bayesian network using probe set data PS1 • Use data from one time segment • Choose Weeks 23-24 data (6 placentas) • Choose 4 probe sets PS3 PS4 PS2

25. Placenta data for Weeks 23-24 PS1 corresponds to 201984 which corresponds to EGFR PS2 corresponds to 236034, PS3 corresponds to 211148: PS2 and PS3 both correspond to ANGPT2 PS4 corresponds to 204620 which corresponds to CSPG2

26. Step 2: Score the naïve Bayesian network • We want to score this network: PS1 PS4 PS3 PS2

27. The network score is a function of conditional probabilities • Conditional probability, Prob(N | Pa(N)), is the probability of child node N given parent of N • Example: Given a parent PS1’s node has an associated expression value 10, what is the probability that its child node, PS4, has an expression value of 8? PS1 PS4

28. Conditional probability • EGFR (PS1) is the parent node and has value 10. • CSPG2 (PS4) is the child node and has value 8 two times • Conditional probability = 2/6 PS1 PS4

29. Score for a Bayesian network The score of the naive network equals the product of all the nonzero conditional probabilities associated with the network: P(N1, N2, N3, N4) = Π P(Ni | pa(Ni)) 4 i=1

30. Score for the naïve Bayesian network P(N1, N2, N3, N4) = 1/3966 = 2.54 x 10-5 PS1 PS2 PS4 PS3

31. Step 3: Randomly add/delete an edge and rescore the Bayesian network PS1 The score becomes 1/78732 = 1.27 x 10-5. PS2 PS3 PS4

32. Step 4. Continue until best score reached • Since the score is a probability, we want the score to be high. • The naïve network is the better choice between the two networks, so we pick it as our final network. PS1 PS2 PS4 PS3

33. Step 5. Combine probe sets to create the gene regulatory network EGFR ANGPT2 CSPG2

34. 40 probe sets (26 genes)

35. Gene regulatory pathwayfor 26 genes Step 1. Create a naïve Bayesian network using 40 probe sets for each time segment Step 2. Score the naïve Bayesian network Step 3. Randomly add/delete an edge and rescore the Bayesian network Step 4. Continue until best score reached Step 5. Combine probe sets to create the gene regulatory network for the placenta

36. Step 1. Create a naïve Bayesian network using 40 probe sets for each time segment

37. Create a naïve Bayesian network PS 7 PS 8 PS 6 PS 9 PS 1 PS 5 PS 2 PS 4 PS 3

38. Step 2. Score the naïve Bayesian network

39. Score for a Bayesian network The score of the naive network equals the product of all the nonzero conditional probabilities associated with the network: P(N1, N2, N3, N4) = Π P(Ni | pa(Ni)) 40 i=1

40. Step 3. Randomly add/delete an edge and rescore the Bayesian network Step 4. Continue until best score reached

41. With four probe sets, at least two Bayesian networks were constructed: PS1 PS1 PS2 PS3 PS2 PS4 PS3 PS4

42. Exhaustive search • To be certain that we have the best scoring network, we need to construct all possible networks from our naïve networks • With four probe sets, we only constructed one other network than the naïve network • How to construct all possible networks?

43. How do we construct all possible networks? • 1 probe set 1 Bayesian network • 2 probe sets 2 possible Bayesian networks • 3 probe sets 12 possible Bayesian networks • 4 probe sets 144 possible Bayesian networks • 5 probe sets > 4800 possible Bayesian networks! • 6 probe sets … ?? • And so on…

44. Welcome to “Modern Heuristics” • Step 1. Representation of a model • Step 2. The scoring function • Step 3. Defining the search problem • Step 4. Consider local optima score local change

45. Step 1: Representation of the model • The model is a gene regulatory pathway. • We are going to assume a Bayesian model for our probe set: • The number of possible pathways is so large as to forbid an exhaustive search for the best Bayesian network. PS 1 PS 2 PS 4 PS 3

46. Step 2: The scoring function • The fair coin, p(X = heads) = ½ • What happens if the coin is unfairly weighted? • We need to re-think probability: p(X) = ∫p(x) r(x) dx • r(x) is a weight function.

47. Step 2. The scoring function • The scoring function is a probability • Assume the network has a Dirichlet distribution which is the weight function used to weight the conditional probabilities. www.wikipedia.com

48. Step 2. The scoring function Probability of a fixed network equals product of conditional probabilities times the Dirichlet distribution: 40 P(N) = Π P(Ni | pa(Ni)) D(Ni) i = 1 such that D(Ni) = ∏ Θiάi-1(N i)

49. Step 3: Defining the search problem What it means to search: a. Construct a first network (Use a naïve Bayesian network) b. Score the first network using the scoring function c. Perform the Hill-climbing algorithm.

50. Step 3. Defining the search problem The Hill-climbing Algorithm: • Randomly choose a node • “Search” in the neighborhood of that node for the best scoring network