Türkçe bilmiyorum 😟

I am

• Mathematical Engineer, U. of Chile
• PhD Informatics, U Rennes 1, France
• PhD Mathematical Modeling, U. of Chile
• not a Biologist
• but an Applied Mathematician who can speak “biologist language”

• Big and small computers
• Telecommunication Networks
• Between 2003 and 2014 I was the chief research engineer
  • on the main bioinformatic group in my country (MATHomics)
  • in the top research center (Center for Mathematical Modeling)
  • in the top university of my country (University of Chile)

• Peach:
  • response to cold stress
• Grapefruit:
  • development related to seed and grape size (Sultaniye)
• Wine:
  • quality control on exported wine,
  • avoid secondary fermentation

• Salmon:
  • effect of diet on metabolism,
  • selection of stress tolerant families.
  • Whole genome sequencing
    • 10M dollars project,
    • harder than human genome
    • Chile, Canada and Norway
• Mining:
  • copper extraction helped by bacteria

We had a research contract with the main mining company

State owned, big enough to pay for long term research

Few papers, many patents

In Biology every rule has an exception

A biological “law” is verified ~80% of time

Theory is in constant change

And yet it moves

• Classic Biology deals with organisms that we can see
  • Ecology
  • Predator - Prey
  • Epidemiology
• Cell Biology focus on single cells and their pieces
  • Microscopy
  • Image processing
  • Immunology
• Molecular Biology

Focus on the pieces that form these pieces. Things that can not be observed on the microscope

• Metabolites
  • small molecules
  • ~100 atoms
• Proteins
  • large molecules with some structure
  • ~10.000 to 100.000 atoms
• Nucleic acids (DNA, RNA)

The tree of life has two main branches

• Prokaryotes (e.g. bacteria)
  • Single cell
  • Single chromosome
  • Single compartment
• Eukaryotes (e.g. humans)
  • Single- or multi-cellular
  • Several chromosomes
  • Several compartments such as the Nucleus

• The membrane creates an environment with limited interaction with the environment.
• In a first approach we can assume that there is no interaction
• The “soup” inside the cell contains molecules in different concentrations
• The state of the cell is a vector of length m+p+n
  • m metabolites, p proteins, n nucleic acids
    [c₁, …, c_m, …, c_m+p, …, c_m+p+n]

This is the whole picture

In some cases we focus only on the relation between DNA and proteins

• Some events can trigger production of RNA and proteins.
• It is usually assumed that protein concentration is proportional to RNA concentration.

We can abstract the chemical nature of these molecules and look them as sequences of symbols

• Each protein is a chain of amino-acids (LEGO pieces)
• There are 20 types of amino-acids
• Each protein corresponds to a word in an alphabet of 20 symbols
• Length between 20 and 1000
• In the cell the protein will fold and adopt a specific shape

• Each chromosome is a double-chain of nucleotides
• There are 4 types of nucleotides: A, C, G, T
• The length is between 10⁵ and 10⁸
• There is an A on one strand iff there is a T in the other. Same with C and G.
• Some sub-words of DNA encode the “recipe” to make proteins
  • these sub-words are called genes

• When the cell needs to produce a protein, the “recipe” is copied from DNA to a messenger RNA.
  • This is done by a specific protein
• There is a mapping 1-to-1 between DNA and RNA.
• There is a mapping of 3-tuples of DNA to amino-acids
  4³ → 20
• This mapping is called genetic code.
• There are proteins that read the RNA and build the encoded protein

• The genome is the set of all chromosomes in a cell
• A chromosome is a single DNA molecule
• DNA contains many sub-words called genes

When the cell needs a protein

• a gene is copied to a RNA molecule
• the RNA molecule is translated to a protein

Each object involves a mathematical problem

Current technology allows us to read DNA in runs of ~100-600 letters. Imagine a book of 1000 pages:

• several copies of the book are cut randomly in one million pieces
• different pieces may overlap. No “disjoint” condition
• half of the pieces are lost
• the remaining half is splashed with ink in the middle

The problem is to reconstruct the original book

The classical approach is to see each “piece” as a vertex of a graph. There is an edge when the two “pieces” overlap.

Lander & Waterman (1988) proved that the expected number of connected components is

\[E( C) = N e^{-\frac{NL}{G}}\] where N is the number of “pieces”, L the average length of the “pieces”, and G is the length of the chromosome.

But experimental results did not match the theory. There was a wrong hypothesis.

It was assumed that sequence overlap corresponded to physical overlap.

But some genes have multiples copies in the genome. A kind of backup.

The question is: How to traverse the graph and reconstruct the original sequence?

Once the complete genome has been assembled we need to find the “words” in the text. There are no “spaces”.

The usual approach is to see genes as a realization of a Markov Chain, the intergenic region as another chain, and the transition between both controlled by a hidden Markov chain.

This is the Hidden Markov Model. In practice the problem is how to find the good parameters. Good in Prokaryotes, not so good in Eukaryotes.

If we have found all “words”, what is their meaning?

It is observed that most genes are homolog to genes on other species. This homology is determined by an edit distance. We can “transform” a gene into another by

• substitution: ACGT → ACTT
• insertion: ACGT → ACTGT
• deletion: ACGT → ACT

Each edition has a cost. The distance is the minimal cost (Method of Smith & Waterman).

What is a “reasonable distance”?

To evaluate significance we need a “null hypothesis”.

Karlin & Altschul described a model for the expected number of sequences within a given distance using substitution and arbitrary scores.

The general problem including insertions and deletion has not been formally solved, although there are some “rule of thumb” approaches.

Now we are in condition to evaluate (partially) the state of the cell by measuring the concentration of RNA.

The expression of a gene is the concentration of the RNA transcribed from the gene.

There are several techniques to do that. Some are based on the chemistry of nucleic acids:

• DNA is stable in double strand config
• RNA exists in single strand config
• If DNA is on single strand then RNA can hybridize

• RNA hybridized to DNA can be observed by fluorescence
• Fluorescence is increasing (non-linearly) with concentration
• There are many sources of noise
• Careful statistical analysis is required
• Moreover, the experiments should be planned considering the analysis step
  • Technical replicas
  • Biological replicas

Gene expression experiments result in a matrix.

Each row is a gene, each column an experiment.

The problem is to find structures in this matrix.

Classical case: clustering of genes by linear correlation

But correlation may be non-linear: entropy based mutual information

\[\int_{Y}\int_{X} p(x,y) \log \frac{p(x,y)}{p(x)p(y)} dx\,dy\]

Determining true binding sites is hard. Current methods produce too many false positives.

Im my research I built a putative regulatory network for the well studied bacteria E.coli. We expected ~4K regulations. We got 25K regulations.

I integrated this model with microarray data to find the “most probable” regulatory network using a parsimony criterium.

Once we get a map of the regulatory interactions, we can use it how the cell will evolve. Considering only the vector of RNA concentrations, we have the status \[C_t = (c_1,\ldots,c_n)_{t}\] Then the regulatory network defines \(F\) such that \[C_{t+1} = F(C_{t})\] Finding \(F\) is an open problem.

Broadly speaking, there are three kinds of proteins

• Structural pieces:
  • keratine (hair), tubuline (tubes)
• Information handlers:
  • cell replication (mitosis),
  • transcription, translation
• Enzymes:
  • catalyzers that trigger chemical reactions
  • transform one metabolite into another

• Molecular biology inspire many mathematical problems
• Some come from the experimental procedures
  • So they will evolve with technology
• Some come from the very basic nature of Nature
• Without proper mathematical models it is hard to understand the meaning of experimental results
• Math come to Molecular Biology to stay