Entropy & Information content By Thomas Nordahl Petersen

Entropy, Information contents &Logo plots

By Thomas Nordahl Petersen

Mutiple alignment of acceptor sites from 268 yeast DNA sequencesWhat is the biological signal around the site ?What are the important positionsHow can it be visualized ?

Biological information

Logo plot with Information Content

Exon Intron Exon

Entropy - Definition

Entropy of random variable is a measure of the uncertainty

In Thermodynamics G=H-TSThe entropy S of a system is the degree of disorder

Entropy - Definition

Entropy of a distribution of amino acidsThe Shannon entropy:

H(p) = - a pa log2(pa),where p is an amino

acid distribution.

H(p) is measured in bits: log2(2) = 1, log2(4)=2

Mutiple alignment of 3 sequences

Seq1: A L P K

Seq2: A V P R

Seq3: A I K R

High entropy - high disorder

Low entropy - low disorder

Entropy - example

H(p) = - a pa log2(pa)

Mutiple alignment of 3 sequences

Seq1: A L R

Seq2: A V R

Seq3: A I K

Pos1: H(p)= -[1*log2(1)] = 0

Pos2: H(p)= -[1/3*log2(1/3)+ 1/3*log2(1/3)+ 1/3*log2(1/3)]=

Pos3: H(p)= -[2/3*log2(2/3)+ 1/3*log2(1/3) =

Relative EntropyThe Kullback-Leiber distance D

How different is an amino acid distribution pa compared to a background distribution qa - i.e. distance D between them.

D(p||q) = a pa log2(pa/qa)

Normally a background distribution of the amino acids is

obtained as frequencies from a large database like UniProt.

Ala (A) 7.82 Gln (Q) 3.94 Leu (L) 9.62 Ser (S) 6.87

Arg (R) 5.32 Glu (E) 6.60 Lys (K) 5.93 Thr (T) 5.46

Asn (N) 4.20 Gly (G) 6.94 Met (M) 2.37 Trp (W) 1.16

Asp (D) 5.30 His (H) 2.27 Phe (F) 4.01 Tyr (Y) 3.07

Cys (C) 1.56 Ile (I) 5.90 Pro (P) 4.85 Val (V) 6.71

Information content

D(p||q) = a pa log2(pa/qa)

Often the Information content is used as a measure of the

degree of conservation.

I = a pa log2(pa/qa)

A special case is that where all amino acids have the same background distribution: qa = 1/20

Information content

I = a pa log2(pa/(1/20))

= a pa [log2pa - log2(1/20)]

= -H(p) - a palog2(1/20)

= -H(p) + a palog2(20)

= -H(p) + log2(20)

= -H(p) + 4.32

Information content

I = -H(p) + 4.32 = a palog2pa + 4.32

The Information content is at its maximum when then the entropy is zero - i.e. A fully conserved position in a multiple alignment.

Mutiple alignment of 3 sequences:

Seq1: A L R

Seq2: A V R

Seq3: A I K

Pos1: I = -[1*log2(1)]+ 4.32 = 4.32

Pos2: I = -[1/3*log2(1/3)+ 1/3*log2(1/3)+ 1/3*log2(1/3)] + 4.32 =

Pos3: I = -[2/3*log2(2/3)+ 1/3*log2(1/3) + 4.32=

Logo plots - HowTo

Logo plots - Information Content

Calculate Information Content

I = apalog2pa + log2(4), Maximal value is 2 bits

Total height at a position is the Information Content measured in bits. Height of letter is the proportional to the frequency of that letter. A Logo plot is a visualization of a mutiple alignment.

~0.5 each

Completely conserved

Programs to make a Logo plot

WebLogo Requires a mutiple alignment as input Protein or DNA sequences More output formats

Blast2Logo Requires a fasta file as input Only protein sequences Runs PSI-blast and makes a table of frequencies pdf logo plot

WebLogo - http://weblogo.berkeley.edu/

WebLogo - http://weblogo.berkeley.edu/

Find important positions

>sp|Q00017|RHA1_ASPAC Rhamnogalacturonan acetylesterase

MKTAALAPLFFLPSALATTVYLAGDSTMAKNGGGSGTNGWGEYLASYLSATVVNDAVAGR

SARSYTREGRFENIADVVTAGDYVIVEFGHNDGGSLSTDNGRTDCSGTGAEVCYSVYDGV

NETILTFPAYLENAAKLFTAKGAKVILSSQTPNNPWETGTFVNSPTRFVEYAELAAEVAG

VEYVDHWSYVDSIYETLGNATVNSYFPIDHTHTSPAGAEVVAEAFLKAVVCTGTSLKSVL

TTTSFEGTCL

What is the next step ?

Find homologous sequences - how ?

Blast or PsiBlast Download sequences Make a mutiple alignment ClustalW or others or use Blast2Logo program

Mutiple alignment programs

Blast2logo - http://www.cbs.dtu.dk/biotools/Blast2logo-1.0/

Important positions

Important positions in proteins are conserved

positions => high Information Content.

Conserved for a reason:

Functionally important positions Catalytic residues

Structurally important positions Manitain the correct fold of the protein

Blast2logo

Runs iterative blast i.e. Psi-Blast

Searching for homologues sequences by use

of Position Specific Scoring Matrices (PSSM).

Iteration - use Blosum62 scoring matrix Iteration - make profile of seq found in iteration 1 Iteration - make profile of seq found in iteration 2 Iteration - Calculate aa freq at each position in

query sequence. Correct for low counts and weight

seq such that very similar seq are down weighted

Important positions - counting

Example. Where is the active site?

Sequence profiles might show you where to look! The active site could be around S9, G42, N74, and H195

Exercise

Calculate nucleotide frequencies from a mutiple alignment of human donor sites

Calculate Entropy and Information content

Draw (by hand) a Logo plot

Use 2 Logo plot programs

Learn to interpret Logo & frequency plots

Active site residues & structural residues