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Parikh vectors
5OSW_1 3FFC_1 1QGA_1 Letter Amino acid
13 7 12 N Asparagine
57 24 26 E Glutamic acid
33 23 16 T Threonine
20 13 10 Y Tyrosine
48 22 16 A Alanine
43 21 18 D Aspartic acid
35 4 5 C Cysteine
19 16 8 Q Glutamine
18 9 4 H Histidine
62 18 23 L Leucine
27 7 14 F Phenylalanine
60 7 34 K Lycine
34 12 25 V Valine
22 26 9 R Arginine
18 19 25 G Glycine
14 7 15 I Isoleucine
4 3 11 M Methionine
28 16 14 P Proline
26 14 16 S Serine
2 9 7 W Tryptophan 

5OSW_1|Chain A|Albumin|Capra hircus (9925)
>3FFC_1|Chains A, F|HLA class I histocompatibility antigen, B-8 alpha chain|Homo sapiens (9606)
>1QGA_1|Chains A, B|PROTEIN (FERREDOXIN:NADP+ REDUCTASE)|Pisum sativum (3888)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
5OSW , Knot 234 583 0.85 40 257 530 
DTHKSEIAHRFNDLGEENFQGLVLIAFSQYLQQCPFDEHVKLVKELTEFAKTCVADESHAGCDKSLHTLFGDELCKVATLRETYGDMADCCEKQEPERNECFLKHKDDSPDLPKLKPEPDTLCAEFKADEKKFWGKYLYEVARRHPYFYAPELLYYANKYNGVFQECCQAEDKGACLLPKIETMREKVLASSARQRLRCASIQKFGERALKAWSVARLSQKFPKADFTDVTKIVTDLTKVHKECCHGDLLECADDRADLAKYICDHQDTLSSKLKECCDKPVLEKSHCIAEIDKDAVPENLPPLTADFAEDKEVCKNYQEAKDVFLGSFLYEYSRRHPEYAVSVLLRLAKEYEATLEDCCAKEDPHACYATVFDKLKHLVDEPQNLIKKNCELFEKHGEYGFQNALIVRYTRKAPQVSTPTLVEISRSLGKVGTKCCAKPESERMPCTEDYLSLILNRLCVLHEKTPVSEKVTKCCTESLVNRRPCFSDLTLDETYVPKPFDGESFTFHADICTLPDTEKQIKKQTALVELLKHKPKATDEQLKTVMENFVAFVDKCCAADDKEGCFLLEGPKLVASTQAALA
3FFC , Knot 120 277 0.81 40 180 266 
GSHSMRYFDTAMSRPGRGEPRFISVGYVDDTQFVRFDSDAASPREEPRAPWIEQEGPEYWDRNTQIFKTNTQTDRESLRNLRGYYNQSEAGSHTLQSMYGCDVGPDGRLLRGHNQYAYDGKDYIALNEDLRSWTAADTAAQITQRKWEAARVAEQDRAYLEGTCVEWLRRYLENGKDTLERADPPKTHVTHHPISDHEATLRCWALGFYPAEITLTWQRDGEDQTQDTELVETRPAGDRTFQKWAAVVVPSGEEQRYTCHVQHEGLPKPLTLRWEPS
1QGA , Knot 136 308 0.84 40 196 294 
QVTTEAPAKVVKHSKKQDENIVVNKFKPKEPYVGRCLLNTKITGDDAPGETWHMVFSTEGEVPYREGQSIGIVPDGIDKNGKPHKLRLYSIASSAIGDFGDSKTVSLCVKRLVYTNDAGEVVKGVCSNFLCDLKPGSEVKITGPVGKEMLMPKDPNATVIMLGTGTGIAPFRSFLWKMFFEKHEDYQFNGLAWLFLGVPTSSSLLYKEEFEKMKEKAPENFRLDFAVSREQVNDKGEKMYIQTRMAQYAEELWELLKKDNTFVYMCGLKGMEKGIDDIMVSLAAKDGIDWIEYKRTLKKAEQWNVEVW

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(5OSW_1)}(2) \setminus P_{f(3FFC_1)}(2)|=128\), \(|P_{f(3FFC_1)}(2) \setminus P_{f(5OSW_1)}(2)|=51\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:0000001100100110001011111110001000110001011001001100011000011000010011100100110100001011000000010000011000000101101010100101010100001110010011000101011011001000011100000100011011101001000111001000100101001100110110110100011010100100110010010000001011001000101100100000010001000000111000001101000111001111010110000100000010011110110000000100110111011000010100001000101001011001001100100110000011000100110011110000011010010110100011011000010100001100000101110010110000110001000000011000101001010000110110100101010100110000010000111011000101000010011001111100001100001011101101110001111
Pair \(Z_2\) Length of longest common subsequence
5OSW_1,3FFC_1 179 4
5OSW_1,1QGA_1 175 4
3FFC_1,1QGA_1 198 3

Newick tree

 
[
	3FFC_1:96.55,
	[
		5OSW_1:87.5,1QGA_1:87.5
	]:9.05
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{860 }{\log_{20} 860}-\frac{277}{\log_{20}277})=158.\)
Status Protein1 Protein2 d d1/2
Query variables 5OSW_1 3FFC_1 202 147
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]