Finding Mathematics in
Genes and Diseases
Ming-Ying Leung
Department of Mathematical Sciences
University of Texas at El Paso (UTEP)
“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 …
Outline:
• DNA and RNA
• Genome, genes, and diseases
• Palindromes and replication
origins in viral genomes
• Mathematics for prediction
of replication origins
Cytomegalovirus
(CMV) Particle
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.
A
C
G
T
A
C
G
U
A
C
T
G
Genes and Genome
Genes and Diseases
Virus and Eye Diseases
CMV Particle
CMV Retinitis
Genome size
~ 230 kbp
• inflammation of the retina
• triggered by CMV particles
• may lead to blindness
Replication Origins and Palindromes
• High concentration of palindromes
exists around replication origins of
other herpesviruses
• Locating clusters of palindromes (above
a minimal length) on CMV genome
sequence might reveal likely locations of
its replication origins.
Palindromes in Letter Sequences
Odd Palindrome:
“A nut for a jar of tuna”
remove spaces and capitalize
ANUTFORA JAROFTUNA
Even Palindrome:
“Step on no pets”
STEPON NOPETS
DNA Palindromes
Association of Palindrome Clusters
with Replication Origins
Computational Prediction of
Replication Origins
• Palindrome distribution in a random
sequence model
• Criterion for identifying statistically
significant palindrome clusters
• Evaluate prediction accuracy
• Try to improve…
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
A
C
G
T
Wheel of Bases
(WOB)
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
A
C
G
T
Wheel of Bases
(WOB)
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
A
C
G
T
Wheel of Bases
(WOB)
Poisson Process Approximation of
Palindrome Distribution
Use of the Scan Statistic to Identify
Clusters of Palindromes
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
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)”
Sequence Generator (SG)
Bases
A
C
G
T
Wheels of Bases (WOB)
Sequence Generator (SG)
Bases
A
C
G
T
Wheels of Bases (WOB)
Sequence Generator (SG)
Bases
A
C
G
T
T
Wheels of Bases (WOB)
Sequence Generator (SG)
Bases
A
C
G
T
T
Wheels of Bases (WOB)
Sequence Generator (SG)
Bases
A
C
G
T
T C
Wheels of Bases (WOB)
Sequence Generator (SG)
Bases
A
C
G
T
T C
Wheels of Bases (WOB)
Sequence Generator (SG)
Bases
A
C
G
T
T C T
Wheels of Bases (WOB)
Sequence Generator (SG)
Bases
A
C
G
T
T C T T
Wheels of Bases (WOB)
Sequence Generator (SG)
Bases
A
C
G
T
Wheels of Bases (WOB)
T C T T T
Sequence Generator (SG)
Bases
A
C
G
T
Wheels of Bases (WOB)
T C T T T A
Sequence Generator (SG)
Bases
A
C
G
T
Wheels of Bases (WOB)
T C T T T A A
Sequence Generator (SG)
Bases
A
C
G
T
Wheels of Bases (WOB)
T C T T T A A C A A G C T T G
Sequence Generator (SG)
Bases
A
C
G
T
Wheels of Bases (WOB)
T C T T T A A C A A G C T T G
Results Obtained for Markov
Sequence Models
• Probabilities of occurrences of single
palindromes
• Probabilities of occurrences of
overlapping palindromes
• Mean and variance of palindrome counts
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
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 – proteomics data
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
St. Stephen’s
Girls’ College
University of Hong Kong
Department of Mathematics: A Beach Picnic
Continuing to Find Mathematics
in Genes and Diseases
Ming-Ying Leung
Department of Mathematical Sciences
University of Texas at El Paso (UTEP)