1. A short tutorial on DNA structure and functions
The DNA double helix
A short tutorial on DNA structure
and functions

2. Nucleotides: the bricks of a ladder structure
The DNA helix is made assemblying four structural
units called nucleotides. Each nucleotide has 3 building
blocks:
Adenine (A)
Guanine (G)
Cytosine (C)
Thymine (T)
Base:
purine
pyrimidine
Phosphate
group
Sugar ring

3. The double helix A T C G A T ... G T A C
Four-letter alphabet {A,C,G,T}
Hidrogen bonds

5. The central dogma of molecular biology
The DNA molecule
contains the information
for coding all the proteins
(more than 100,000 in
humans)
The DNA expression
(# and type of proteins
produced) is different
in different cells

6. The DNA molecule is divided in regions that code
for proteins (GENES) and non-coding regions
chromatine

7. The “words” in genes are triplets called codons
Each codon on the gene corresponds to an amynoacid
on the corresponding protein
Serine
The message encoding
is not as simple as that!
Strong degeneracy
43=64 different triplets
20 different amynoacids

8. The C paradox (“complete genome size” paradox)
The size of complete genomes (=total length of DNA
on all chromosomes) does not seem to be correlated
with the Phenotypic complexity of the species
Homo Sapiens …. C ~ 3.4 x 109 bp
Pinus resinosa ….. C ~ 6.8 x 1010 bp
Amoeba dubia …. C ~ 6.7 x 1011 bp

9. Coding to non-coding ratio
The coding to non-coding ratio generally decreases
from simpler to higher organisms

10. Is it all junk? The higher the organism is on the evolutionary scale,
the greater the amount of “junk” DNA
Is it all junk?
Regulatory sequences
instructions concerning
transcription and expression
Specific protein binding
sites
Interplay between DNA
sequence and structure

11. An example: promoters Transcription factors bind to specific sequences
upstream with respect to the gene (promoters)
Then the RNA-polymerase
“recognises” where to start
transcription
Regulation of gene expression through promoters and transcription factors. Adjacent to genes are promoters, DNA regions that control how much RNA is transcribed from that gene. (1) The enzyme RNA polymerase II initiates trancription at a specific site in the promoter. (2) Certain 'basal' transcription factors control the binding of RNA polymerase to this site. Other proteins that bind to specific short DNA stretches in the promoter and basal factors work together to activate or inhibit transcription. Drugs such as alcohol may modify the activity of those factors. (3) Here, alcohol is arbitrarily assumed to increase the activity of an activating transcription factor (ATF), resulting in (4) increased RNA synthesis from the alcohol-responsive gene. (5) Genes lacking this particular ATF binding site would not respond to alcohol.

12. The role of non-coding DNA
… is still an open problem
tools from Computational Linguistics
techniques of information theory
Analysis of DNA sequences with
Statistical physics

13. Statistical analysis of DNA sequences
A short account of how physicists abuse of biology

14. Restatement of the problem
A long sequence is given of symbols drawn from
a four-letter alphabet {A,T,C,G}. How is it organised?
Is there a language?
Space of investigation at least two-dimensional:
Functional bipartition of the sequence - coding
and non-coding regions (x axis)
Position of the organism on the evolutionary
scale (y axis)

15. Methodological pre-requisite
When searching for rules that hold in the biological world, physicists must switch to
the biological definition of law
A law in biology is a rule that holds for
describing a given phenomenon with the
least number of exceptions

16. Long-range correlations
Long-range correlations in a sequence are the first signature
of some non-trivial auto-organisation (ex: languages); as opposed to a sequence generated by a process characterised by local order.
Scale-free “fractal” process => power-law decay of correlations
Counter-example: Markov process of order n0
There exists the characteristic length n0 in the organisation of
the sequence produced by such a process. Correlations decay
exponentially as exp(-n/ n0)

17. Long-range correlations: methods
Spectral analysis of DNA sequences
Walk representations of DNA sequences

18. Mapping rules
Map a nucleotide sequence {ni} (n {A,C,T,G}, i=1,2,…,L) onto a numerical sequence {ui}
Purine-pyrimidine
Weak-strong

19. Spectral analysis Divide the sequence of L nucleotides into K=[ L/N ]
non-overlapping sub-sequences of size N, and compute
the power spectrum of each one
where qf is the Discrete Fourier Transform of set ui

20. Finally, average S( f ) over the K subsequences
If a sequence has long-range power-law correlations
Hence, a log-log plot of S( f ) vs f is a straight line
with slope -b

21. Low-frequency portion: spurious
contributions for f < order 1/N
Medium-frequency portion:
power-law behaviour
High-frequency portion: peaks
at f =1/3 and f =1/9 bp-1

22. The measured distributions are found to be peaked
on non-zero values of b in non-coding regions only
Purine-pyrimidine mapping rule, N=512
Practical use? This
result may be useful
in gene-finding
algorithms based on
the different correlation
properties of coding and
non-coding regions
S.V. Buldyrev et al., PRE,
51, 5084 (1995)
S.M.Ossadnik et al, Biophys. J.,
67, 64 (1994).

23. Walk representation Mapping rule => displacement Displacement y(l)
Nucleotide distance
Displacement y(l)

24. One then studies the root mean square fluctuations
F(l) about the average of the displacement
where
and the bars indicate averages over l0

25. The mean square fluctuation is related to the autocorrelation
function
in the following way

26. There are three possible behaviours
Random base sequence => C(l )=0 on average (except C(0)=1). Hence (normal random walk)
Local correlations up to a distance L =>
However, asymptotically ( l >> L)
If there is no characteristic length (“infinite-length” correlations) one gets power laws of the form with (anomalous random walk)

27. Techniques from statistical Linguistic
Zipf plots
Shannon entropy and redundancy
Mutual information

28. Zipf plots Construct the Rank vector R, by ordering the words from
Conventional Zipf analysis:
take all words in a text written in a given language
measure their frequency of occurrence ( f )
Construct the Rank vector R, by ordering the words from
the most frequent to the least, assigning ranks R=1, 2, ...
The Zipf plot is f (R)
It is found that in several natural languages
with z close to 1

29. n-tuple Zipf analysis What if the words of the language are not known?
One can fix a word length n and perform Zipf analysis
of a sequence of symbols of length N by taking into
account all N-n+1 words
Application of n -tuple Zipf analysis to natural and
formal languages gives encouraging results. Yet, the
success of such analysis provides just a necessary
condition for non-Markovian structure in a language

30. n-tuple Zipf analysis of English R. N
n-tuple Zipf analysis of English R.N. Mantegna et al, PRE 52, 2939 (1995)
English text of about
106 words from an
Encyclopedia. Alphabet
of 32 character: 26
english letters and 6
punctuation symbols,
including the “blank”
character.
Fit gives
To a good extent
independent of n

31. n-tuple Zipf analysis of Unix binary code R. N
n-tuple Zipf analysis of Unix binary code R.N. Mantegna et al, PRE 52, 2939 (1995)
Compiled version of the
Unix operating system
(9x106 characters). The
alphabet is binary.
Fit gives
To a good extent
independent of n

32. n-tuple Zipf analysis still shows power-law behaviour,
but the exponent z is different from conventional Zipf
analysis
The difference observed in the exponent z may be
due to the vocabulary enlargement of the n -tuple analysis.
Conventional Zipf scheme: only REAL WORDS
n-tuple scheme: all n-long combinations

33. n-tuple Zipf analysis of human DNA sequence R. N
n-tuple Zipf analysis of human DNA sequence R.N. Mantegna et al, PRE 52, 2939 (1995)
Fit gives
Primarily non-coding
human DNA sequence
(2x105 bases) from
GenBank (coding=1.5 %)
To a good extent
independent of n

34. Coding => logarithmic fit
Extensive n-tuple Zipf analysis of coding and non-coding sequences R.N. Mantegna et al, PRE 52, 2939 (1995)
A statistically significant functional difference is found
in the functional form of the n-tuple Zipf plots of coding
and non-coding sequences in most organisms.
Non-coding =>
power-law fit
Coding => logarithmic fit

35. Entropy and Redundancy
Shannon, 1948: the foundations of information theory.
The concept of entropy of an information source
In order to introduce Shannon entropy, let us
give a mathematical definition of
surprise!

36. A mathematical formulation of surprise
Suppose that the surprise one feels upon learning that an event
E has occurred depends only on the probability p(E), 0 < p < 1.
It is natural to agree on four basic axioms for the function S(p)
S(1) = 0
S(p) is a strictly decreasing function of p
S(p) is a continuous function of p
S(pq) = S(p) + S(q), 0 < p,q < 1
Theorem:

37. , then surprise is measured in bits
If we take
, then surprise is measured in bits
Suppose an observable X=(x1,x2,…,xN) is associated with
a probability distribution P=(p1,p2,…,pN). Then, the average
amount of surprise felt upon learning the actual value of X
will be
H(X) can be also regarded as the average amount of information
received when X is observed, and hence “carried” by X

38. Entropy of an information source
Suppose a source of information emits symbols drawn from an
alphabet with l characters. The source is characterised by a
probability distribution P=(p1,p2,…,pl). It is natural to define
the entropy of the source (in bits) as
H(source) measures the information content of the source
associated with the probability distribution P .

39. Maximum entropy of an information source
The maximum entropy of P (maximum degree of uncertainty) is
attained when
In this case one has

40. Block (n-gram) entropy
Let us define the probability to observe a given string of length n
The n-gram entropy is then defined as

41. The largest-disorder (maximum information) case corresponds to
The maximum-disorder case corresponds to linear n-gram entropy
Deviations from randomness (presence of constraints) in a given
sequence can be revealed in the form of non-linear n-gram entropy
Important result

42. Estimating n-gram entropies from data
* Start with a sequence of length L
* Slide a window of length n by steps of 1 symbol (base)
* Store all the L - n + 1 encountered words
* Calculate where ni is the number
of occurrencies of the i-th word
Of course, in order to achieve statistical significance, one must
make sure that

43. n-gram entropy measures in human chromosome 1
Two symbols alphabet (purines = {A,G}, pyrimidines = {C,T}).
Maximum n-gram entropy (in bits) Smax = n log2 2
Coding
Non-coding intergenic

44. Redundancy
The flexibility of a language can be quantified in terms of its
redundancy, i.e. the limit relative deviation from the random case
The greater the redundancy, the farther is the symbolic structure
under study from the random case, i.e. the more constraints
(grammar) are present

45. Redundancy of DNA sequences R.N. Mantegna et al, PRE 52, 2939 (1995)
Non-coding regions
Coding regions

46. Correlations part II: mutual information
The traditional methods for studying correlations are not invariant
under change of the mapping rule. Moreover, correlation functions
measure linear correlations by definition
The mutual information generalises the measure of correlations
between two symbols (bases) separated by a distance k in the
sequence

47. The thermodinamical analogyX Y
Joint system [ X, Y ]
Statistical dependence
Statistical independence

48. The mutual information can be defined as the distance in “entropy
space” between statistical independence and dependence
If kB is replaced by 1/ln 2 then I(X,Y) quantifies the amount of
mutual information between X and Y in bits

49. Mutual information in DNA sequencesk
ni, nj [A,C,T,G]
Y (ni)
X (nj)
Pij(k)
Joint probability of finding nucleotide ni and nj
spaced by k in the sequence

50. Mutual information in human DNA sequences I
Mutual information in human DNA sequences I. Grosse et al, PRE 61, 5624 (2000)
Mutual information of human coding and non-coding sequences from Genbank.
Mutual information calculated by averaging over fragments of length 500 bp
The mutual information
clearly distinguishes
between coding
and non-coding regions
Period 3 bp