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XIE Huimin ( 谢惠民 ) Department of Mathematics, Suzhou University and HAO Bailin ( 郝柏林 )

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### Visualization of K-Tuple Distribution in Prokaryote Complete Genomes and Their Randomized Counterparts

### Poisson Approximation forthe Expectation Curve

XIE Huimin (谢惠民)

Department of Mathematics, Suzhou University

and

HAO Bailin (郝柏林)

Ｔ－Life Research Center, Fudan University

Beijing Genomics Institute, Academia Sinica

Institute of Theoretical Physics, Academia Sinica

Prokaryote Complete Genomes ( PCG )

K

Biology-inspired

mathematics

Avioded and Rare

K-strings

K=6-9,15,18

Phylogeny

Based on PCG

Compositional

Distance

( Success)

Species-specificity

of avoidance

Combinatorics

Goulden-Jackson

cluster method

Decomposition

and

Reconstruction of

AA sequences

Factorizable

Language

Phylogeny

( Failure )

Graph theory:

Euler paths

The Algorithm (Hao Histogram) Implemented at:

- National Institute for Standard and Technology (NIST)

http://math.nist.gov/~FHunt/GenPatterns/

- European Bioinformatics Institute (EBI)

http://industry.ebi.ac.uk/openBSA/bsa_viewers

However, 2D only, no 1D histograms.

Two Mathematical Problems

- Dimensions of the complementary sets of portraits of tagged strings.
- Number of true and redundant missing strings.
- The two problems turn out to be one and the same, the first being graphic representation of the second.

Two Methods to Solve the Problem

- Combinatorial solution: Goulden-Jackson cluster method (1979); number of dirty and clean words.
- Language theory solution: factorizable language, minimal deterministic finite-state automaton.

2. 1D Histogram of K-Tuples

- Collect those K-tuples whose count fall in a bin from to ,
- Plot the number of such K-tuples versus the counts,
- This is a 1D histogram or
- An expectation curve.

The effect of c+g content in 2D histograms of

original genome and randomized sequence:

3. Three Artificial Models Generating Sequences

- Eiid: equal-probability independently and identically distributed model.
- Niid: nonequal-probability independently and identically distributed model.
- MMn: Markov model of order n

Monte Carlo Methodestimation of expectation (ex) and standard deviation (sd) for an niid model

(the compositions of a,c,g,t are 15:35:35:15, the length of

sequence is , the value of K=8.)

Validation about the Robustness of K-Histograms: a comparison of absolute error from ex in an experiment with sd as reference

Compare the population of shuffling a given sequence and the population of sequence generated from a stochastic model.

F-test

t-test

4. A Theory for the Expectation Curve (1)

Definition. For each , define a random variable

(1)

Where random variable takes value 1 if the i-th K-tuple occurs

exactly n times in the sequence, or takes value 0 if it does not

occur.

A Theory for the Expectation Curve (2)

Theorem. For each , the mathematical expectation

of random variable is given by

(1)

Where the random variable is the occurrence number of

K-tuples of I-th type.

The Exact Computation of Expectation Curve

In order to compute the expectation curve we need to know the probability for each and .

The Goulden-Jackson cluster method can be used successfully for the model of eiid.

It is still difficult to do the computation for other models.

Two Experiments (for the model of eiid):

compare with a K-histogram compare with Monte Carlo method

the red curves are the standard deviation estimation

obtained by Monte Carlo method.

For each K-tuple calculate its expected number of appearing in sequence of length N, then use the formula of probability function of Poisson distribution and sum them for all K-tuples:

Remark. This follows from a theorem in Percus and Whitlock, ACM

Transaction on Modeling and Computer Simulation, 5 (1995) 87—100

(the model, however, can only be eiid, and the tuples must be overlapless).

Comparison of Poissonapproximation with 7-histogram for Haemophilus influenzae

Comparison of Poissonapproximation with 8-histogram for Haemophilus influenzae

A comparison of Poisson approximation withMonte Carlo method

In this computation the model is an niid, in which the parameters

are taken from the randomized sequence of H. influenzae.

5. Analysis of the Mechanism of Multi-Modal K-histograms

An example for H. influenzae. The length of its genome is

1830023. Under the simplified conditions of

for , there are only 9 types of different of as shown

in the following list.

The following map shows the nine individual probability functions and their sum

Notice that the effect from the ratio of successive modes:

6. Analysis of Short-Range Correlation by K-Histograms

Two 8-histograms for E. coli,

the left one is from its genome,

and the right one is from its

Markov model of order 1.

Compare the 8-histograms of Markov Models of order from 2—7 for E. coli

Using Markov model of order 5 and Monte Carlo method

to compare the 8-histogram of E. coli’s complete genome

sequence with the ex and sd of MM5.

this is the ratio curve

for

the red curve is the expectation

curve estimated by doing 50

times of simulation.

Reference:

Huimin Xie, Bailin Hao, “Visualization of K-tuple distribution in prokaryote complete genomes and their randomized counterparts”, CSB2002: IEEE Computer Systems Bioinformatics Conference Proceedings, IEEE Computer Society, Los Alamitos, 2002, 31-42.

7. Discussion

Most of the results shown above are of experimental nature, many problems are left for future study.

- How to select reasonably the value of K.
- How to use 1D visualization to protein?
- What are the properties of random variables ?
- How to compute exactly the expectation curve for the model of niid and MMn?
- Why the Poisson approximation is effective without considering the overlap of K-tuples?

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