What does mathematics contribute to bioinformatics?

1
What does mathematics contribute to
bioinformatics?

• Winfried Just
• Department of Mathematics
• Ohio University

2
A new microscope and a new physics

• In 2004 PLoS Biology published a paper by Joel E.
  Cohen
• Mathematics Is Biology's Next Microscope, Only
  Better
• Biology Is Mathematics' Next Physics, Only
  Better.
• Really?
• How does this new microscope differ from the
  traditional ones?
• How to use it?
• Why did mathematicians become seriously
  interested in
• biology?
• And how is all this related to bioinformatics?

3
More empirical observations

• NSF and NIH recently started to invest heavily in
  biomathematics.
• In 2002 the Mathematical Biosciences Institute
  (MBI, located at OSU) was founded this is the
  first and so far only NSF institute dedicated
  exclusively to applications of mathematics in one
  other area.
• Several other new research institutes in
  biomathematics are supported from public or
  private sources.
• A number of new journals specializing in
  biomathematics got started.
• The job market for biomathematicians is currently
  rather favorable, both in academia and industry,
  especially in the pharmaceutical industry.

4
What is behind this trend?

• And why do we observe this trend now, instead of
  30
• years ago or 30 years from now? There are two
  main
• reasons
• Contemporary biology generate a huge mountains of
  data. Drawing biologically meaningful inferences
  from these data requires analysis in the
  framework of good mathematical models. Hence
  mathematics has become a necessary tool for
  biology.
• Currently available computer power allows us to
  investigate sufficiently detailed mathematical
  models to draw biologically realistic inferences.
  Thus mathematics has become a useful tool for
  biology.

5
Biomathematics vs. bioinformatics

• Everything that has been said so far about
• biomathematics could also be said about
• bioinformatics.
• What is the difference between the two areas?
• Biomathematics Applications of mathematics to
  biology.
• Bioinformatics The design, implementation, and
  use of
• computer algorithms to draw inferences from
  massive sets of
• biomolecular data. It is an interdisciplinary
  field that draws on
• knowledge from biology, biochemistry, statistics,
  mathematics,
• and computer science.

6
Example of a huge data set Genbank

• The first viral genome was published in the
  1980s, the first
• bacterial genome, H. influenzae, 1.83 106 bp,
  in 1995,
• The first genome of a multicellular organism, C.
  elegans,
• 108 bp, w 1998. The sketch of our own genome,
• H. sapiens, p 109 bp, was announced in June
  2000.
• As of February 2008, Genbank contained 85 759 586
  764 bp
• of information.
• How to draw concrete inferences from such a huge
• mountains of information?

7
Where are the genes?

• Let us look, for example, at our own genome. The
  information
• about it is written in Genbank as a sequence p
  109 liter that
• would fill a million of tightly typed pages, the
  equivalent of
• several thousand novels
• ...actggtacctgtatatggacgctccatatttaatgcgcgatgcagga
  tctaaa...
• Less than 1.5 of this sequence codes proteins.
  How to find
• these genes?
• No human can read the whole sequence. A computer
  can read
• it easily, in a few seconds. So, maybe the
  computer will tell us
• where the genes are, where they start, and where
  they end.
• But what is the computer supposed to compute???

8
Honest Craigs Casino

• This is a casino in Nevada where one plays
  64-number
• roulette. In each round, a player bets chips on
  three
• among those 64 numbers. If one of these three
  chosen
• numbers comes up, honest Craig will pay a
  suitable
• premium. If not, the player loses the chips.
• QUESTION How long does it take, on average, for
  a
• winning number to come up?

9
Honest Craigs Casino

• This is a casino in Nevada where one plays
  64-number
• roulette. In each round, a player bets chips on
  three
• among those 64 numbers. If one of these three
  chosen
• numbers comes up, honest Craig will pay a
  suitable
• premium. If not, the player loses the chips.
• QUESTION How long does it take, on average, for
  a
• winning number to come up?
• ANSWER 64/3 21.33 rounds.

10
Probability of long waiting times

• Let us assume that Craig is as honest as he
  claims.
• Then the probability P(k) that our player will
  keep losing
• throughout the first k rounds is (61/64)k. In
  particular,
• starting from k 50 we obtain the following
  probabilities
• P(50) 0.0907 P(51) 0.0864 P(52) 0.0824
  P(53) 0.0785 P(54) 0.0748
• P(55) 0.0713 P(56) 0.0680 P(57) 0.0648
  P(58) 0.0618 P(59) 0.0589
• P(60) 0.0561 P(61) 0.0535 P(62) 0.0510
  P(63) 0.0486 P(64) 0.0463
• P(65) 0.0441 P(66) 0.0421 P(67) 0.0401
  P(68) 0.0382 P(69) 0.0364
• P(100) 0.0082 P(200) 0.000064 P(300)
  0.00000055

11
Some statistical terminology

• The assumption that Craig is as honest as he
  claims will
• be our null hypothesis. The suspicion that he is
  cheating
• after all is our alternative hypothesis. The
  number of
• losses that precede the first winning round will
  be our
• test statistics. The p-value is the probability
  that the
• test statistics takes the observed or a more
  extreme
• value under the assumption of the null
  hypothesis. If
• the p-value falls below our agreed upon
  significance
• level, we are justified in rejecting the null
  hypothesis. In
• science, the most commonly used significance
  level is
• 0.05. Falsely accusing honest Craiga about
  cheating
• would be a Type I error trusting him when he is
  in fact
• cheating would be a Type II error.

12
Craiga Venters Lab

• In 1995 Craig Venters team sequenced the genome
  of the
• bacterium H. influenzae. If we want to detect
  the positions of
• its 1740 genes that code proteins in its sequence
  of 1 830 140
• base pairs, we can reason as follows In bacteria
  almost all the
• genome codes proteins. Let us start from
  position n and read
• triplets (n, n1, n2), (n3, n4, n5),

13
Craiga Venters Lab

• In 1995 Craig Venters team sequenced the genome
  of the
• bacterium H. influenzae. If we want to detect
  the positions of
• its 1740 genes that code proteins in its sequence
  of 1 830 140
• base pairs, we can reason as follows In bacteria
  almost all the
• genome codes proteins. Let us start from
  position n and read
• triplets (n, n1, n2), (n3, n4, n5), If
  we read in the
• correct reading frame, we will read a sequence of
  codons that
• ends with a STOP codon, that is, TAA, TGA, TAG.

14
Craiga Venters Lab

• In 1995 Craig Venters team sequenced the genome
  of the
• bacterium H. influenzae. If we want to detect
  the positions of
• its 1740 genes that code proteins in its sequence
  of 1 830 140
• base pairs, we can reason as follows In bacteria
  almost all the
• genome codes proteins. Let us start from
  position n and read
• triplets (n, n1, n2), (n3, n4, n5), If
  we read in the
• correct reading frame, we will read a sequence of
  codons that
• ends with a STOP codon, that is, TAA, TGA, TAG.
  Such a
• STOP codon will appear on average once in about
  300 triplets.
• If we read in one of the other five reading
  frames, we will read
• garbage, that is, a more or less random sequence
  of triplets
• and one of the triplets TAA, TGA, TAG will be
  encountered on
• average once every 64/3 21.33 positions.
• Rings a bell?

15
This is the same problem!

• With minor modifications Now our null hypothesis
  will be that
• we read in the wrong reading frame, the
  alternative hypothesis
• will be that we read a coding sequence in the
  correct reading
• frame. If we dont encounter a STOP codon while
  reading 63
• successive triplets, we can reject the null
  hypothesis at
• significance level 0.05 and conclude that we
  found a sequence
• that codes a protein whose end is easy to find.
• So we can design an easy gene-finding algorithm
  based on
• finding these so-called ORFs (open reading
  frames).

16
Some caveats

• The beginning of the gene is somewhat more
  difficult to determine, since ATG is both the
  START codon and the codon for methionine, and the
  promoter is also part of the gene.
• The garbage in the other five reading frames is
  not completely random.
• This approach will miss all genes that code
  proteins shorter than 63 amino acids (type ?
  error) and will sometimes discover spurious genes
  (type ? error).
• This approach is unsuitable for discovering
  RNA-coding genes.
• However, the above problems can be solved, and
  there
• exist good gene-finding algorithms based on this
  idea.

17
Craiga Venters lab in 2000

• But now let us look at the genome of H. sapiens
• Protein-coding regions constitute only a small
  fraction of our genome.
• All by itself, this would lead to a lot more Type
  I errors
• than in prokaryotes.

18
Craiga Venters lab in 2000

• But now let us look at the genome of H. sapiens
• Protein-coding regions constitute only a small
  fraction of our genome.
• The coding sequences, exons, are interspersed
  with introns.
• A given codon may be split by an intron.
• Consecutive exons dont have to sit in the same
  reading frame.
• Introns look similar to random sequences.
• So we are faced with a much more difficult
  problem.
• Nowadays there exist pretty good algorithms for
  finding genes
• in eukaryotes. But
• No algorithm for finding genes in prokaryotes
  will work here.

19
Mathematics and mathematicians

• Mathematics is a great language for elucidating
  the common structure in apparently unrelated
  problems.
• Mathematicians have a tendency to talk about
  complicated theories in their jargon instead of
  giving simple and concrete answers.
• Mathematical microscopes often dont come with
  a simple users manual. In order to successfully
  use them, one needs to understand to some extent
  how they work. The choice of the most
  appropriate mathematical microscope for a given
  biological problem often requires active
  cooperation between mathematicians and
  biologists.
• The key to success in this type of cooperation is
  finding a common language and mutual
  understanding of and respect for the two
  different intellectual approaches.
• Mathematical models form the basis for
  formulating hypotheses, often in the form of
  probabilities.
• The final interpretation of these hypotheses and
  their experimental verification belongs to the
  biologists. Thus mathematical microscopes will
  not make the more traditional ones redundant.
• In points 3-6, feel free to substitute
  bioinformatics for mathematics.

20
Biomathematics vs. bioinformatics

• Biomathematics Applications of mathematics to
• biology.
• Bioinformatics The design, implementation, and
  use of
• computer algorithms to draw inferences from
  massive
• sets of biomolecular data. It is an
  interdisciplinary field
• that draws on knowledge from biology,
  biochemistry,
• statistics, mathematics, and computer science.
• The design of all bioinformatics tools is based
  on
• mathematical models. In order to choose the most
• appropropriate among the available tools and draw
  
• proper inferences, one needs to understand these
  models.