1. Entropy, Information contents & Logo plots By Thomas Nordahl Petersen

2. Biological information
GTTCTTCGTGTTTATTTTTAGGAAATTGATGA
TTGTTTCTCCTTTTAAAATAGTACTGCTGTTT
TTTACTAACGACACATTGAAGAAATCACTTTG
GATACGCTTACCGTTATCCAGAGCTACAGCGC
TACTAATATGTAATACTTCAGCTCCCCTTAAT
ATTGAGATCTTTTTTAACTAGTTAGGTCTACC
TTCTCCCCTTCTTCATTTTAGCCTGTTTGGAC
TAACATAACTTATTTACATAGTGCCATTGAAC
GATATTTCCCGTTGTGTTAAGGCTGAGAAGAA
TTTTCCCGACCATCAAGACAGGTGATTTATCA
TGCAAAAACTTTTTTTCACAGGGCTAACTTGC
GTTTATTGTGTTTCCACTCAGTTAAAAAACGA
AACGTACTTTAATATTTATAGTACTTCATTCG
AACATGCTATTTTTCATACAGCAACCTCACAT
CTGCACTCATCATTAGATTAGAGGAACATGGA
TACTTTTCTTTATCTAAGCAGCTAACTCAACT
ATCAACATGCTATTGAACTAGAGATCCACCTA
TAACTAACATGACTTTAACAGGGCTAATTTAC
AGTACTAACTAATTAACTTAGAACATTAACAT
GATCACCGTCACATTTATTAGAATTTCAAACG
CAGTGGAATTTTTTTTTCTAGAAATGGTATCG
CTCTATGACCAATAAAAACAGACTGTACTTTC
AAATGGTATTATTTATAACAGTTGAACATTTC
ATAAATATGCGATCAATATAGACCGTTGATAT
ATTTTACTTTTTTTTTTTTAGGAGCTCCAAGA
ATTTATTTCCTTATAATACAGACACGGTTACA
TCGCAATTAATTTTCTAATAGTTTTTCATTTT
GACCATCTTTCTTTTCCCCAGTGCTAAACACG
AACCTTCTTTCTCATTCGTAGATTACTGTTGC
AATTACTAACAGCTGTAATAGCCGACAAATTT
CTCTCTGCGCGTCCAATTTAGCTATACTGTTG
TTGTTTTGTTTTGTCGTACAGTGTTTGGAGAA
AAACTTCCATTTCTTACATAGATCATCGCCAT
TCCTTTCCATAATTTATTCAGCGCTTTGGTAT
CGATTTACTATTTCCATTTAGACGTTGTTCAA
AATTTACTAACAATACTTCAGTTTATAATGGA
TCCTATACTAACAATTTGTAGTTCATAAATAA
Exon Intron Exon
Mutiple alignment of acceptor sites from 268 yeast DNA sequences
What is the biological signal around the site ?
What are the important positions
How can it be visualized ?
Logo plot with Information Content
Sequence-logo

3. Entropy - Definition
Entropy of random variable is a measure of the uncertainty
In Thermodynamics G=H-TS
The entropy S of a system is the degree of disorder

4. Entropy - Definition Entropy of a distribution of amino acids
The 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

5. 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)]=1.58
Pos3: H(p)= -[2/3*log2(2/3)+ 1/3*log2(1/3) =0.92

6. Relative Entropy The 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) Gln (Q) Leu (L) Ser (S) 6.87
Arg (R) Glu (E) Lys (K) Thr (T) 5.46
Asn (N) Gly (G) Met (M) Trp (W) 1.16
Asp (D) His (H) Phe (F) Tyr (Y) 3.07
Cys (C) Ile (I) Pro (P) Val (V) 6.71

7. 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

8. Information contents amino acids
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) = a palog2pa

9. Information content I = -H(p) + 4.32 = a palog2pa + 4.32
General formula:
a palog2pa + log2(N), where N is number of letters
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
Pos2: I = [1/3*log2(1/3)+ 1/3*log2(1/3)+ 1/3*log2(1/3)] =2.74
Pos3: I = [2/3*log2(2/3)+ 1/3*log2(1/3) = 3.40

10. Logo plots - HowTo Count nucleotides at each position:
GTTCTTCGTGTTTATTTTTAGGAAATTGATGA
TTGTTTCTCCTTTTAAAATAGTACTGCTGTTT
TTTACTAACGACACATTGAAGAAATCACTTTG
GATACGCTTACCGTTATCCAGAGCTACAGCGC
TACTAATATGTAATACTTCAGCTCCCCTTAAT
ATTGAGATCTTTTTTAACTAGTTAGGTCTACC
TTCTCCCCTTCTTCATTTTAGCCTGTTTGGAC
TAACATAACTTATTTACATAGTGCCATTGAAC
GATATTTCCCGTTGTGTTAAGGCTGAGAAGAA
TTTTCCCGACCATCAAGACAGGTGATTTATCA
TGCAAAAACTTTTTTTCACAGGGCTAACTTGC
GTTTATTGTGTTTCCACTCAGTTAAAAAACGA
AACGTACTTTAATATTTATAGTACTTCATTCG
AACATGCTATTTTTCATACAGCAACCTCACAT
CTGCACTCATCATTAGATTAGAGGAACATGGA
TACTTTTCTTTATCTAAGCAGCTAACTCAACT
ATCAACATGCTATTGAACTAGAGATCCACCTA
TAACTAACATGACTTTAACAGGGCTAATTTAC
AGTACTAACTAATTAACTTAGAACATTAACAT
GATCACCGTCACATTTATTAGAATTTCAAACG
CAGTGGAATTTTTTTTTCTAGAAATGGTATCG
CTCTATGACCAATAAAAACAGACTGTACTTTC
AAATGGTATTATTTATAACAGTTGAACATTTC
ATAAATATGCGATCAATATAGACCGTTGATAT
ATTTTACTTTTTTTTTTTTAGGAGCTCCAAGA
ATTTATTTCCTTATAATACAGACACGGTTACA
TCGCAATTAATTTTCTAATAGTTTTTCATTTT
GACCATCTTTCTTTTCCCCAGTGCTAAACACG
AACCTTCTTTCTCATTCGTAGATTACTGTTGC
AATTACTAACAGCTGTAATAGCCGACAAATTT
CTCTCTGCGCGTCCAATTTAGCTATACTGTTG
TTGTTTTGTTTTGTCGTACAGTGTTTGGAGAA
AAACTTCCATTTCTTACATAGATCATCGCCAT
TCCTTTCCATAATTTATTCAGCGCTTTGGTAT
CGATTTACTATTTCCATTTAGACGTTGTTCAA
AATTTACTAACAATACTTCAGTTTATAATGGA
TCCTATACTAACAATTTGTAGTTCATAAATAA
Count nucleotides at each position:
Convert to frequencies:
Frequency-logo:

11. Logo plots - Information Content
Calculate Information Content
I = apalog2pa + log2(4), Maximal value is 2 bits
Sequence-logo
Completely conserved
~0.5 each
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.

12. 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

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

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

15. 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, Mafft or others
or use Blast2Logo program

16. Mutiple alignment programs

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

18. 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

19. 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

20. Psi-Blast Iterative Blast
An iterative process to search for remote homologs
Capture and use evolutionary conserved information
Scoring matrix is refined by use of gap-free multiple alignment
Input sequence
Sequence
database
Blast
E < threshold
4 iterations
PSSM
Multiple alignment
PSSM: Position Specific Scoring Matrix 20

21. Important positions - counting

22. Blast2logo Important amino acids: G24, D25 & S26
G89, N91 & D92
Important amino acids:
D209 & H212 22

23. 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