# Visualization of K-Tuple Distribution in Prokaryote Complete Genomes and Their Randomized Counterparts XIE Huimin ( 谢惠民 ) Department of Mathematics, Suzhou.

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Visualization of K-Tuple Distribution in Prokaryote Complete Genomes and Their Randomized Counterparts 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 Combinatorics Goulden-Jackson cluster method Factorizable Language Avioded and Rare K-strings K=6-9,15,18 Species-specificity of avoidance Phylogeny ( Failure ) Phylogeny Based on PCG Compositional Distance ( Success ) Decomposition and Reconstruction of AA sequences Graph theory: Euler paths

1. 2D Histogram of K-Tuples

gc a t

Thermoanaerobacter tengchongenic (K = 8) g c at

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:

Escherichia coli original genome

Escherichia coli randomized sequence

Haemophilus influenzae randomized sequence

Mycobacterium leprae original genome

Mycobacterium laprae randomized sequence

Mycobacterium tuberculosis original genome

Mycobacterium tuberculosis randomized sequence

G+C Content of Some Bacteria SpeciesG+C Content H. influenzae38.15% E. coli50.79% M. laprae57.80% M. tuberculosis65.61%

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 Method estimation 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-testt-test

Definition. For each, define a random variable 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. (1) 4. A Theory for the Expectation Curve (1)

Theorem. For each, the mathematical expectation of random variable is given by Where the random variable is the occurrence number of K-tuples of I-th type. (1) A Theory for the Expectation Curve (2)

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.

Poisson Approximation for the Expectation Curve 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 Poisson approximation with K- histogram for U. urealyticum

Comparison of Poisson approximation with 7- histogram for Haemophilus influenzae

Comparison of Poisson approximation with 8- histogram for Haemophilus influenzae

A comparison of Poisson approximation with Monte 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 0 1 2 3 4 5 6 7 8 153.1 94.4 58.2 35.9 22.213.7 8.4 5.2 3.2 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:

For E. coli the ratio is 0.968931, hence the result is quite different

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. the red curve is the expectation curve estimated by doing 50 times of simulation. this is the ratio curve for

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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