The Information Processing

Mechanism of DNA and Efficient DNA Storage

Olgica Milenkovic University of Colorado, Boulder

Joint work with B. Vasic

Outline

PART I: HOW DOES DNA ENSURE ITS DATA INTEGRITY? Information Theory of Genetics: an emerging discipline Error-Correction and Proofreading in genetic processes

What type of codes “operate” at the level of bio-chemical processes of the Central Dogma?

Spin Glasses, Kaufmann’s “NK” Model, Regulatory Network of Gene Interactions and Low-Density Parity-Check (LDPC) Codes

Cancer, dysfunctional proofreading and chaos theory PART II: HOW DOES ONE STORE DNA? (DNA

COMPRESSION) Structure of DNA: Statistics and Modeling DNA Compression Genome Compression New Distance Measures and One-Way Communication PART III: NEW CODING PROBLEMS IN GENETICS

Information theory of genetics

2003: 50th Anniversary of discovery: DNA has a double-helix structure!

(Crick, Watson, Franklin, Wilkins 1953)

2003: Completion of the Human Genome Project (98% HDNA sequenced)

Every day an average of 15 new sequences added to SwissProt+GeneBank

Vast amount of genetic data just starting to be analyzed!

DNA is a CODE, but very little is known about its

• exact information content

• nature of redundancy

• statistical properties

• secondary structure

• influence on disease development and control

• underlying error-correcting mechanism

Information Theory of DNA

Helps in understanding the

EVOLUTION of DNA

FUNCTIONALITY of DNA

DISEASE DEVELOPMENT

IT community still not involved in this area!

Signal Processing Community is just getting involved:

Special Issue of Signal Processing Journal devoted to Genetics, 2003.

The League of Extraordinary Gentlemen…

I

How is information stored in a genetic sequence?

What are the atoms of information?

The DNA Polymer…

Sugar

Sugar

PO4

PO4

PO4

1’

S B

U A

G C

A K

R B

- O

P N

H E

O

S

P

H

A

T

E

Deoxiribose (Sugar)

O

OH H

OHCH2OH

H H

H H

2’3’

4’

5’

The Bases…

Purine Bases: Adenine (A); Guanine (G)

Pyramidine Bases:Thymine (T); Cytosine (C)

D

O

U

B

L

E

-

H

E

L

I

X

The Pairing Rule…

A and T paired through TWO hydrogen bonds

G and C paired through THREE hydrogen bonds

The Genetic Code…Second Letter

U C A G

First Letter

U

UUUleu

UCU

ser

UAUtyr

UGUcys U

CAG

Third Letter

UUC UCC UAC UGC

UUAleu

UCA UAA stop UGA stop

UUG UCG UAG stop UGG trp

C

CUA

leu

CCU

pro

CAUhis

CGU

arg

UCAG

CUC CCC CAC CGC

CUA CCA CAAgin

CGA

CUG CCG CAG CGG

A

AUU

ile

ACU

thr

AAUasn

AGUser U

CAG

AUC ACC AAC AGC

AUA ACA AAAlys

AGAarg

AUG met ACG AAG AGG

G

GUU

val

GCU

ala

GAUasp

GGU

gly

UCAG

GUC GCC GAC GGC

GUA GCA GAAglu

GGA

GUG GCG GAG GGG

Abbreviations

ala = alaninearg = arginineasn = asparagineasp = aspartic acidcys = cysteine

gln = glutamineglu = glutamic acidgly = glycinehis = histidineile = isoleucine

leu = leucinelys = lysinemet = methioninephy = phenylalaninepro = proline

ser = serinethr = threoninetrp = tryptophantyr = tyrosineval = valine

Genes, Exons, Introns (Junk DNA)…

Genes: Sequence of base pairs coding for chains of amino-acidsConsist of exons (coding) and introns (non-coding) regionsLength- anything between several tenths up to several millionsEXAMPLE: Among most complex identified genes isDYSTROPHINE (2 million bps, more than 60 exons, codes for 4000 amino acids)

Escherichia Coli: around 4000 genes; Humans: 35000-40000 genes

Junk DNA: “Disrespectful” name for introns Significant fraction of DNA

Shown (last year) to be “somewhat” responsible for RNA coding (Far from being “junk”, but function still not well understood…)

The Central Dogma…

DNA

Replication Transcription Translation

mRNA

Proteins

A Communication Theory Perspective:

Genetic ChannelDNA sequence

DNA sequence

mRNA

Proteins

What kind of errors are introduced by the Genetic Channel?

DNA within Chromosomes (tight packing):

• DNA wrapped around HISTONES (proteins)

• HISTONES are organized in NUCLEOSOMES

• NUCLEOSOMES CHROMATINE folded in CHROMOSOMES

Processing in the Genetic Channel: DNA REPLICATION

Untying the knots: Topoisomerases

Unwinding the helix: Helicases

Getting it all started: Primers

Doing the hard work: Polymerases

Sealing the segments: Ligases

Helping to keep two sides apart: SSB

Replication: more details

Rules: Replication always proceeds in 5’ to 3’ direction;

Replication is semi-conservative;

Replication is a parallel process for eukaryotes;

Facts: Polymerases can stitch together any combination

of bases (“Ps are a little bit sloppy’’)

Timing for replication:

E. Coli: 40 min

Humans (parallel): < 2 hours

Can be prolonged for proofreading purposes

Errors…Combination of substitution, deletion, insertion (replication fork), shift, reversal, etc

errors(Complete exon or intron deleted, or simple base pair deletions)1. Tautomeric shifts (transition/transvertion): *T-G, *G-T, *C-A, *A-C 2. Recombination between non-identical molecules (“HETERODUPLEX mismatches”)3. Spontaneous DEAMINATION (C to U, C to T, C-G to T-A), METHYLATION (CpG),

rare4. APURINIC/APYRAMIDINIC SITES (due to HYDROLISIS)5. CROSS-LINKS6. STRAND-BREAKAGE, OXIDATIVE DAMAGE ERRORS7. LOSS OF 5000-10000 PURINE and 200-500 PYRIMIDINE bases (20 hours) due to

radiation

Replication Errors: Polymerases miss-insertion probability between 10e-3/10e-5 Miscoding A-G-A-T-GC-T-G-C-T-A-C

Slippage

A-A-T-G

C-G-T-T A-C

T

Miscoding - Realignment

A-G-A-T-G C-T C-T-A-C

G

Slippage-Dislocation

G-A-A-T-G

C-G -T -T-T-A-C

Bio-chemical mechanism responsible for error correction?

Proofreading (Maroni, Molecular and Genetic Analysis of Human Traits): Replication polymerases error rate ; human DNA with bps, total of

106 errorsExample: C to U conversion causes presence of deoxyuridine, detected by uracil-DNA

GLYCOSYLASEGlycosylase process acts like erasure channel1. Proofreading based on semi-conservative nature of replication 2. Excision Repair Mechanisms: Arrays of Exonucleases Show large degree of pre-correction binding activity – correction performed by

EXCISION“Jumping’’ occurs between different genes !!! (Lin, Lloyd, Roberts, Nucleases)

Reduce error levels by an additional several orders of magnitudeMismatch-specific post-replication enzymesTotal number of errors per human DNA replication: on average JUST ONE

Replication and Repair have been optimized for balancing spontaneous mutational load:

Permitting evolution without threatening fitness or survival

310 9103

Characteristics of DNA ECC:

Error-correction performed on different levelsError correction performed in very short timeExtremely large number of very diverse errors

correctedError correction tied to global structure of DNA(not to consecutive base pairs) Error correction also depends on DNA topology

Identify ECCs of DNA… Error-Correcting Codes in DNA: Forsdyke (1981), Wolny (1983), Eigen

(1993), Liebovitch et al (1996), Battail (1997), Rosen and Moore (2003), McDonaill (2003)

Theories: Non-coding regions are in-series error detecting sequences! Ordering of coding/non-coding regions responsible for error-correction! Complementary base pairing corresponds to error-detecting code!

Acceptor/Donor: hydrogen atom/lone electrons

1 represents donor, 0 acceptor

Additionally, add 0 or 1 for purine and pyramidine

Code: A 1010

G 0110

T 0101

C 1001

BEST ERROR CORRECTING MECHANISM: Deinococcus radiodurans • Microbe with extreme radiation resistance • Enabled to survive radiation doses thousands of times higher than would kill most organisms, including  humans. • Surpasses the cockroach by orders of magnitude!

Why? Because of its remarkable DNA-repair mechanism!!!  D. radiodurans flawlessly regenerates its radiation-shattered genome in about 24 hours.  

‘’Conan The Bacterium’’

(to conquer the Red Planet !)

Something seemingly unrelated…

nm nnnnmnmiji xhxxJhJxE

,,, 2

1)}{,}{;(

Spin Glasses, the Ising Model, Hopfield Networks or “Boltzmann Machines”:

State x of a spin glass with N spins that may take values in {-1,+1}

Energy of the state x: E, external field h

nm nnnm xhxxJhJxE

,2

1),;(

The Hamiltonian

Hamiltonian for Ising model

+

++

+

-+

Example:

Water exists as a gas, liquid or solid, but

all microscopic elements are H2O molecules

This is due to intermolecular interactions depending on temperature, pressure etc.

“frustration”

Something seemingly unrelated…Codes on graphs: the most powerful class of error correcting codes in information theory, including Turbo, Low-Density Parity-Check (LDPC), Repeat-Accumulate (RA) Codes

1 1 1 0 1 0 0

0 1 1 1 0 1 0

1 0 1 1 0 0 1

H

Most important consequence of graphical

description: efficient iterative decoding

Variable nodes communicate to check nodes their reliabilityCheck nodes decide which variables are unreliable and “suppress” their inputsNumber of edges in graph = density of H Sparse = small complexity

Detrimental for convergence of decoder: presence of short cycle in code graphApplications of LDPC codes: for cryptography, compression, distributed source coding for sensor networks, error control coding in optical, wireless comm and magnetic and optical storage…

Variables Checks

Gallager’s Decoding Algorithm A

Works for (Binary Symmetric Channel) BSC:

Each variable sends its channel reliability unless all incoming messages from checks say “change”

Each check sends estimate of the bit based on modulo two sum of other bits participating in the check

Alternative view: Variables=Atoms; Binary Values=Spins;

Variables “align” or “misalign” according to interaction patterns

)(...2121 ... iiiiiiii ShSSSJH

rr

LDPC equivalent to diluted spin glasses

Ground state search for above Hamiltonian = maximum aposteriori decoding of codeword

Average magnetization at a site = MAP decision for individual variable

Something seemingly unrelated…

The regulatory Network of Gene Interactions (RNGI)

Kaufmann (1960’s): “NK” Evolution through Changing Interactions between Genes

Life exists at the Edge of Chaos! BASED ON SPIN GLASSES!

RANDOM BOOLEAN FUNCTION MODEL:

Evolution carried by genes, not base pairs, and the way genes interact!

T T+1

G1 G2 G3 G1 G2 G3

0 0 0 0 0 1

0 0 1 0 0 1

0 1 0 1 0 1

0 1 1 0 0 0

1 0 0 1 0 1

1 0 1 0 1 0

1 1 0 0 0 1

1 1 1 0 1 1

G1

G2

G3

G1G2 G1

0 0 0

0 1 1

1 0 0

1 1 0

G1G3 G2

0 0 0

0 1 0

1 0 0

1 1 1

G1G2 G3

0 0 1

0 1 1

1 0 0

1 1 1

Chaos, Attractors, ConnectivityBoolean networks: dynamical systems Attractors: point and periodic

Characterized by network topology+ Number and period lengths

choices of Boolean node functions

100 111

000 011

001 101

110 010

Kimatograph of the network

MOST IMPORTANT topological factor:

CONNECTIVITY

KEY: Sparse connectivity allows enough variability for evolutionary processes, produces self-organizing structures, but doesn’t allow the system to “get trapped in” chaotic behavior

MOST IMPORTANT Boolean function factors:

BIAS (number of 1 outputs)

CANNALIZATION (depends on number of inputs determining output)

The NK model and RNGI

N= number of genes; K=number of genes co-interacting with one given gene

K=2 critical value (mainly frozen states with islands of changing interaction)

Interaction between genes in regulatory network: very limited in scope

K ranges everywhere between 2-3 to 10-15: If we check carefully, logarithmic in N, i.e. number of genes

Between 2 and 3 for Escherichia Coli (around a thousand genes)

4 and 8 for higher metazoea (several thousand genes) Can explain the process of cell differentiation: genetic material of each cell the same, yet cells functionally and morphologically very differentEach cell type CORRESPONDS TO ONE GIVEN ATTRACTOR of the RNGI

Counting attractors for networks with N=40000 genes, K=2 gives Cell types (correct number 258).

260

KEY IDEA: LDPC Code with Given Decoding Algorithm is a BOOLEAN

NETWORK, SPIN GLASS,…

Example: LDPC Code under Gallager’s A Algorithm

LDPC Code:

Variables and Checks

LDPC Code:

The Control Graph

In the Control Graph, edge (i,j) exists if i-th bit controls j-th bit (i.e. if i and j are at distance exactly two)Boolean function determined by decoding algorithm: For Gallager’s A algorithm, takes form of truncated/periodically repeated MORSE-THUE sequence

Morse-Thue: 0 1 2 3 4 5 6 7 …

0 1 10 11 100 101 110 111 …

0 1 1 0 1 0 0 1 …

Properties:

Self-Similar (fractal)

Results in unbiased Boolean functions

G1

G2

G3

G4

Use Boolean Network Analysis for LDPC Codes No cycles of length four, code regular: uniform choice for Boolean

function Cycles of length four: Boolean functions vary, many more

attractors In no case are the functions canalizing1 2 1

( 1) ... (1 (1 )...(1 )) mod2s si i i i i i i iF G N N N G N N

1 2, ,...,

si i iN N N1 2, ,...,

si i iC C Cmodulo two sums of variable nodes connected to controls

( ,0) ( ,1) ( , )( ) 1 1 1, ,..., , , ,...,j j j lj j j N

j

fff x f x x x x l x x

x

Can use mean-field theorems to see when initial perturbations in the codewords disappear in the limit: use the Boolean derivative, sensitivity analysis, iterative Jacobian and Lyapunov exponent (as in Schmulevich et.al):

N N ( )ijfJacobian F is a

matrix with

in entry (i,j).

1 1( 1) ( ) ( ) ( ( ) ( )), (( ,..., )) ( ( ),..., ( )),

0, 0( )

1, 0

N N

ii

i

d t F t d t H F t d t H g g H g H g

gH g

g

( ) (0) (1) ... ( )IJ t F F F t

1( ) log ( )

tT t

T 2

,,

( ) | ( 1)|/ | ( )|, | | (1/ )N

i ji j

t IJ t IJ t M N M

, ix 1 2, ,..., Nx x x

( ) [ / ] { / 1}i i iI f E f x P f x

The influence of variable on the Boolean function f is defined as the expectation of the partial derivative with respect to the distribution of the variables

.

Influence carries important information about frozen states, error susceptibility etc.

Use Boolean Network Analysis for LDPC Codes

Iterated Jacobian:

Lyapunov exponent:

( ) ( ) (1 ( ))K i iKs t c s t s t

i

Control of the chaotic phase in the a Boolean network by means of periodic pulses (with period T) that “freeze” a fraction of nodes

iterative change of size of “stable core”

LDPC Codes and Gallager’s A Decoding Algorithm

1 2 3 1 2 3 1 2 3( , , ) (1 ) (1 )(1 )f z z z z z z z z z 1 2 3 1 2 3 1 2 3 2 1 3

1 2 3 3 1 2

( , , )/ 1 , ( , , )/ ,

( , , )/

f z z z z z z f z z z z z z

f z z z z z z

A (B)=C1 (C)=C2 (D)=C3 F3(A) A (B C) D C1 C2 F1(A) A (B C D) C1 F2(A)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 1 1

0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 1 1

0 0 1 1 0 0 0 1 1 1 1 1 0 0 1 1 0 0

0 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 1

0 1 0 1 0 0 1 0 1 1 1 1 0 1 0 1 0 0

0 1 1 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0

0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 1 1 1

1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0

1 0 0 1 1 1 0 0 1 0 1 1 1 0 0 1 1 1

1 0 1 0 1 1 0 1 0 1 0 1 1 0 1 0 1 1

1 0 1 1 1 1 0 1 1 1 1 1 1 0 1 1 0 0

1 1 0 0 1 1 1 0 0 1 0 1 1 1 0 0 1 1

1 1 0 1 1 1 1 0 1 1 1 1 1 1 0 1 0 0

1 1 1 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0

1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1

New decoding methods for LDPC and other Block Codes…

Work in Progress:

• Decoders that don’t operate on the frozen core• Decoders that periodically freeze some variables to avoid chaotic behavior• Iterative decoders that work for asymmetric channels and channels with insertion/deletion errors

Bold Conjecture:

The ECC of DNA Replication operates on multiple levels

Carrier of information is gene, not base pair

The Global level involves Genes; Local levels may involve exons or base pairs in general;

The Global Code is an LDPC Code!

Wigner observed that the same mathematical concepts turn up in entirely unexpected connections in whole of

science…

(no explanation as of yet)LDPC related to statistical physics (spin glasses) to neural

networks to self-organizing systems to …

R. Sole and B. Goodwin, Signs of Life: How Complexity Pervades Biology

Table 1: Example of 15-node regulatory network in terms of gene controls

Gene Controls Control (after addition)

1 2,3,13 2,313

2 1,3 1,3

3 4,5,6 4,5,6,1,2

4 5,6 5,6,3

5 4,9,6 4,9,6

6 5,9,3,4,7,8 3,4,5,7,8,9

7 15,8,6 15,8,6

8 7,9 7,9

9 4,5,6 4,5,6

10 9,13,15 9,13,15

11 8,12,13 8,12,13

12 11,13,15 8,11,13,15

13 8,14,15 8,14,15,11,12

14 X X

15 11,12 11,12,13,14,10,7

The Corresponding LDPC Code

1 2 3 4 5 6 7 8 9 101

11

21

31

41

5

1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0

2 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0

3 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0

4 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0

5 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0

6 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1

7 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0

8 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0

9 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1

100 0 0 0 0 0 0 1 0 0 1 1 1 0 0

110 0 0 0 0 0 0 0 0 0 0 0 1 1 1

120 0 0 0 0 0 0 0 0 0 1 1 0 0 1

15-gene interaction example by Hashimoto (Shmulevich, Anderson

Cancer Center)

Need q-ary LDPC code corresponding to different levels of interaction

Cancer: genetic disorder of somatic cellsHuman cancer: INDUCED and SPONTANEOUS

• Accumulation of mutant (erroneous) genes that control cell cycle, maintain genomic stability, and mediate apoptosis

• Causes of mutation: depurination and depyrimidation of DNA; proofreading and mismatch errors during DNA replication

•Deamination of 5-methylcytosine to produce C to T base pair substitutions; and damage to DNA and its replication imposed by products of metabolism (notably oxidative damage caused by oxygen free radicals)

• Defective DNA excision-repair; low levels of antioxidants, antioxidant enzymes, and nucleophiles that trap DNA-reactive electrophiles; and enzymes that conjugate nucleophiles with DNA-damaging electrophiles

Cancer Research

To summarize: Various forms of cancer tightly linked to malfunctioning of proofreading (ECC) mechanism

Cancer cells: correspond to a special type of attractor of the RNGI

(A cancer cell is “just another configuration” of RNGI)

(Schmulevich et.al., Anderson Cancer Research Center)

This attractor has genes interacting in a way that results in uncontrolled cell division

Key observation: C-Change in RNGI results in further weakening of the proofreading system, and VV

Example 1: Cancer cells cheat the proofreading mechanism regulating reduction in length of

telomeres

Aging: during each cell division, telomeres get shorter and shorter…

When they become too short, errors in replication happen, leading to cancer

(a time bomb in our body)

Cancer cells “cheat” proofreading mechanism and allow telomeres to maintain constant length

Finding the error-control mechanism: classifying diseases accurately, curing diseases (including cancer) by gene therapy, making telomer lengths constant over long time…

Example 2: Breast Cancer Oncogene BRCA1 tightly linked to error-control of DNA and cell division regulation

How to obtain results practically? DNA Microarrays!

Figure taken from Schmulevich et.al.

II

How can one efficiently store DNA sequences?

DNA Storage: Compression

GenBank/Swiss-Prot: storage of large number of DNA and protein sequences (17471 million sequences in GenBank, 2002)

Every day, an average of 15 new sequences added to database

DNA compression absolutely necessary to maintain banks Fractal DNA structure to be exploited Possible use of Tsallis entropy Need novel compression algorithms DNA sequences of related species differ in very small

percent of base pairs: need cross-reference compression Need meaningful definition of DNA distance

-- major paradigm shift from base-pair distance to chromosomal distance --

Statistical properties of DNA sequencesBases within the human mitochondrion (length approximately 17000) appear with the following frequencies:

A T G C

0.31 0.13 0.25 0.31

while within different regions of human fetal globin gene:

Introns A T G C

0.27 0.29 0.27 0.17

Exons A T G C

0.24 0.22 0.28 0.25

Parts of genetic sequences can be modeled by Markov chains of given order and transition probabilities; order 2-7

BPs, like CG, have very small probability: most notorious triplet repeats, related to Huntington’s disease and Fragile-X mental retardation, consist of these very unlikely “CG” pairs: (CGG)m ,(CCG)m, m = number of repetitions;

Regions of uniform distribution: isochors; can stretch in length up to hundreds KbpsRepetitive patterns: tandem repeats (TR), random repeats (RR), short interspersed repeat sequences (SINE’s, 9% of DNA), long interspersed repeat sequences (LINE’s).

Junk-DNA seems to have long-range (fractal) characteristics.

A fractal patterns arises from the so-called DNA walk: a graphical representation of the DNA sequence in which one moves up for C or

T and down for A or G.          

Can have two, three-dimensional random walk: further differentiation A,G,C,T

         

C A T GFractal dimension of the DNA molecule:0.85 for higher species, 1 for lower

Use lingual analysis of human languages for exploring DNA "language" (Zipf method)

http://library.thinkquest.org/26242/full/ap/ap13.htmlDNAWalker http://athena.bioc.uvic.ca/pbr/walk/

DNA and Cantor Sets

Provata and Almirantis, 2003: Fractal Cantor pattern in DNA

Exons - filled regions

Introns - empty regions

Random, fractal, Cantor-like set

Implication: atom (carrier of information) exon/intron pairs

History-based random walk and DNA description in terms of urn models

Only introns in higher species have higher complexity than in lower species

Both coding and non-coding regions exhibit long range correlation, with spectral density of introns

bf/1

Known algorithms

FILE COMPRESSION

RATE (ACHIEVABLE

)

GZIP ARITHM. VPS2A UNIXCOMPRESS

BIO-COMPRESS

BWT GTAC

Human GrowthHormone

(HUMGHCSA)

2.00 2.065 2.052 1.607 2.19 1.31 1.608 1.1

GenCompress (Chen, ’97)Biocompress (Grumbach/Tachi, ‘94)Fact (Rivals, ’00)GenomeSequenceCompress (Sato et.al 00’)

Use characteristics of DNA like repeats, reverse complements…Compression rate is about 1.74 bits per base (78% in compression ratio) 

Two classes: statistical and grammar based compression algorithmsHuffman, Lempel-Ziv, Arithmetic Coding, Burrows-Wheeler,

Kieffer’s Grammar Based Schemes(with DNA specific modifications)

No known algorithm specially suited for fractal nature of DNA, although 90% fractal!

Different Entropy Measures:

Shannon Entropy: i

iiS pppH log)(

Renyi Entropy:

)1/(1)( qppHi

qiT

Tsallis Entropy:

i

qiR p

qpH log

1

1)(

Hausdorff Dimension:

TE non-additive in the way that for two independent PS A,B

)()()1()()()( BHAHqBHAHBAH TTTTT

n

zzpHpH

n

nT

derivedS

1limlog)()(

)/1(log

loglim

0 r

Nr

Approach: Use “Fractal Grammars”

Barthel, Brandau, Hermesmeier, Heising: Fractal Prediction, 1997Zerotree wavelet coding using fractal prediction

Inference of context-free grammars from fractal data setsSyntactic generation of fractalsTheory of formal languages can be used to state the problem of "syntactic fractal pattern recognition" Explore Connections with Wavelets(ideas by Jacques Blanc-Talon)

Example: Heighway dragon and Koch curve

haghhbcabbabhaa

hfdbxxx

aggegeeccaca

ahgfedcbaG

)(,...,)(,)(

},,,{,)(

,)(,)(,)(,)(

}}{},,{},,,,,,,,{{

222

1

1111

21

How does one compress sets of related DNA Sequences?

Distributed Source Coding Problem: Peculiar Correlation Patterns

Could explore Wavelet Based CompressionDistributed Source Coding with LDPC Codes…

Genomic Distance and One-Way Communication

Major paradigm shift in genetic distance measure: From base-pair distance (involving deletion, insertion and substitution):

Sankoff, Kruskal,Time Warps, String Edits, and Macromolecules) to Chromosomal Distance based on global arrangements of genes

Inversions are primary mechanism of genome rearrangement!

REVERSAL DISTANCEThe smallest number of inversions necessary to transform one genome

into anotherFinding the minimum number of reversals needed to “sort” a

permutation Permutations are signed, indicating direction of transcriptionExample: (+1 +3 +2) (+1 -2 -3) (+1 +2 -3) (+1

+2 +3)

How does one perform one-way communication (SENDING INFORMATION TO A RECEIVER WHO POSESESS CORRELATED INFORMATION) under the reversal distance measure?

Uses molecular sequence reduction (MSR) algorithms similar to those used to match patterns in the study of DNA.

The algorithms identify and eliminate repetitions previously undetected in network traffic in wide area networks (Wans) to give compression ratios of between 1.2:1 for voice and video and 5:1 for SQL traffic.

The other way around:DNA compression methods increase network efficiency by up to 10

timesPeribit's SR-50 compressor

III

Additional Coding Problems in Genetics

DNA Computing

Codes with Constant GC Content and invariant under Watson-Crick Inversion

Microarray Error Control Coding

Using design theory to reduce error rate of DNA array data

Use novel clustering algorithms for DNA Array Data

Conclusion

Genetics is the most exciting source of new ideas for coding theory

The atom of information is a gene, not a base pair or a triple of base pairs

The error control code of the genome is to be found operating on the level of genes

Compression, phylogenic tree construction: comparison of species has to be performed on the level of genes first

Once the genes are compared, can move to local base pair comparisons