Finding Mathematics in Genes and Diseases

1. Finding Mathematics in Genes and Diseases Ming-Ying Leung Department of Mathematical Sciences University of Texas at El Paso (UTEP)

4. “1, 2, 3, … and Beyond” • A slideshow for HKU Open Day in 1980 • I did the narration and background music • The experience has a great impact on my journey Mathematics is beyond numbers… We find it in buildings, banks, and supermarkets… …in atoms, molecules, and genes …

6. Outline: • DNA and RNA • Genome, genes, and diseases • Palindromes and replication origins in viral genomes • Mathematics for prediction of replication origins Cytomegalovirus (CMV) Particle

7. A A C C G G A C T U T G DNA and RNA • DNA is deoxyribonucleic acid, made up of 4 nucleotide bases Adenine, Cytosine, Guanine, and Thymine. • RNA is ribonucleic acid, made up of 4 nucleotide bases Adenine, Cytosine, Guanine, and Uracil. • For uniformity of notation, all DNA and RNA data sequences deposited in GenBank are represented as sequences of A, C, G, and T. • The bases A and T form a complementary pair, so are C and G.

8. Genes and Genome

9. Genes and Diseases

10. Virus and Eye Diseases CMV Particle CMV Retinitis • inflammation of the retina • triggered by CMV particles • may lead to blindness Genome size ~ 230 kbp

11. Replication Origins and Palindromes • High concentration of palindromes exists around replication origins of other herpesviruses • Locating palindrome clusters on CMV genome sequence might reveal likely locations of its replication origins.

12. remove spaces and capitalize Palindromes in Letter Sequences Odd Palindrome: “A nut for a jar of tuna” ANUTFORA J AROFTUNA Even Palindrome: “Step on no pets” STEPON NOPETS

13. DNA Palindromes

14. Association of Palindrome Clusters with Replication Origins

15. Computational Prediction of Replication Origins in DNA Viruses • Palindrome distribution in a random sequence model • Criterion for identifying statistically significant palindrome clusters • Evaluate prediction accuracy • Try to improve…

16. A C G T Random Sequence Model • A mathematical model can be used to generate a DNA sequence • A DNA molecule is made up of 4 types of bases • It can be represented by a letter sequence with alphabet size = 4 • Adenosine • Cytosine • Guanine • Thymine Wheel of Bases (WOB)

17. A C G T Random Sequence Model Each type of the bases has its chance (or probability) of being used, depending on the base composition of the DNA molecule. • Adenosine • Cytosine • Guanine • Thymine Wheel of Bases (WOB)

18. A 1_3 C 1_3 1_6 1_6 G T Random Sequence Model Each type of the bases has its chance (or probability) of being used, depending on the base composition of the DNA molecule. • Adenosine • Cytosine • Guanine • Thymine Wheel of Bases (WOB)

19. Poisson Process Approximation of Palindrome Distribution

20. Use of the Scan Statistic to Identify Clusters of Palindromes

21. Measures of Prediction Accuracy Attempts to improve prediction accuracy by: • Adopting the best possible approximation to the scan statistic distribution • Taking the lengths of palindromes into consideration when counting palindromes • Using a better random sequence model

22. Markov Chain Sequence Models • More realistic random sequence model for DNA and RNA • It allows neighbor dependence of bases (i.e., the present base will affect the selection of bases for the next base) • A Markov chain of nucleotide bases can be generated using four WOBs in a “Sequence Generator (SG)”

23. Bases A C G T Sequence Generator (SG) Wheels of Bases (WOB)

24. Bases A C G T Sequence Generator (SG) Wheels of Bases (WOB)

25. Bases A C G T T Sequence Generator (SG) Wheels of Bases (WOB)

26. Bases A C G T T Sequence Generator (SG) Wheels of Bases (WOB)

27. Bases A C G T C T Sequence Generator (SG) Wheels of Bases (WOB)

28. Bases A C G T C T Sequence Generator (SG) Wheels of Bases (WOB)

29. Bases A C G T C T T Sequence Generator (SG) Wheels of Bases (WOB)

30. Bases A C G T C T T T Sequence Generator (SG) Wheels of Bases (WOB)

31. Bases A C G T C T T T T Sequence Generator (SG) Wheels of Bases (WOB)

32. Bases A C G T C T T T T A Sequence Generator (SG) Wheels of Bases (WOB)

33. Bases A C G T C T T T T A A Sequence Generator (SG) Wheels of Bases (WOB)

34. Bases A C G G G T C C C T T T T T T A A A A Sequence Generator (SG) Wheels of Bases (WOB)

35. Bases A C G G G T C C C T T T T T T A A A A Sequence Generator (SG) Wheels of Bases (WOB)

36. Results Obtained for Markov Sequence Models • Probabilities of occurrences of single palindromes • Probabilities of occurrences of overlapping palindromes • Mean and variance of palindrome counts

37. Related Work in Progress • Finding the palindrome distribution on Markov random sequences • Investigating other sequence patterns such as close repeats and inversions in relation to replication origins

38. Other Mathematical Topics in Genes and Diseases • Optimization Techniques – prediction of molecular structures • Differential Equations – molecular dynamics • Matrix Theory – analyzing gene expression data • Fourier Analysis – proteomic data

39. Acknowledgements Collaborators Louis H. Y. Chen (National University of Singapore) David Chew (National University of Singapore) Kwok Pui Choi (National University of Singapore) Aihua Xia (University of Melbourne, Australia) Funding Support NIH Grants S06GM08194-23, S06GM08194-24, and 2G12RR008124 NSF DUE9981104 W.M. Keck Center of Computational & Struct. Biol. at Rice University National Univ. of Singapore ARF Research Grant (R-146-000-013-112) Singapore BMRC Grants 01/21/19/140 and 01/1/21/19/217

40. St. Stephen’s Girls’ College

41. University of Hong Kong Department of Mathematics: A Beach Picnic

43. Continuing to Find Mathematics in Genes and Diseases Ming-Ying Leung Department of Mathematical Sciences University of Texas at El Paso (UTEP)