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Parikh vectors
8TAN_1 5RXI_1 1WHL_1 Letter Amino acid
17 7 0 M Methionine
30 24 7 F Phenylalanine
45 17 2 A Alanine
57 16 5 R Arginine
68 17 0 N Asparagine
45 21 5 D Aspartic acid
55 29 17 G Glycine
58 31 3 I Isoleucine
75 43 9 L Leucine
14 8 0 W Tryptophan
27 23 5 Q Glutamine
17 17 0 H Histidine
55 32 4 K Lycine
50 20 4 P Proline
48 16 1 Y Tyrosine
53 22 10 V Valine
39 6 3 C Cysteine
81 37 9 E Glutamic acid
63 38 9 S Serine
55 39 2 T Threonine 

8TAN_1|Chains A, B|Insulin-like growth factor 1 receptor|Homo sapiens (9606)
>5RXI_1|Chain A|Phosphatidylinositol 3,4,5-trisphosphate 5-phosphatase 1|Homo sapiens (9606)
>1WHL_1|Chain A|Cylindromatosis tumor suppressor CYLD|Homo sapiens (9606)
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 \(w=f(c)\)
8TAN , Knot 356 952 0.85 40 327 854 
EICGPGIDIRNDYQQLKRLENCTVIEGYLHILLISKAEDYRSYRFPKLTVITEYLLLFRVAGLESLGDLFPNLTVIRGWKLFYNYALVIFEMTNLKDIGLYNLRNITRGAIRIEKNADLCYLSTVDWSLILDAVSNNYIVGNKPPKECGDLCPGTMEEKPMCEKTTINNEYNYRCWTTNRCQKMCPSTCGKRACTENNECCHPECLGSCSAPDNDTACVACRHYYYAGVCVPACPPNTYRFEGWRCVDRDFCANILSAESSDSEGFVIHDGECMQECPSGFIRNGSQSMYCIPCEGPCPKVCEEEKKTKTIDSVTSAQMLQGCTIFKGNLLINIRRGNNIASELENFMGLIEVVTGYVKIRHSHALVSLSFLKNLRLILGEEQLEGNYSFYVLDNQNLQQLWDWDHRNLTIKAGKMYFAFNPKLCVSEIYRMEEVTGTKGRQSKGDINTRNNGERASCESDVLHFTSTTTSKNRIIITWHRYRPPDYRDLISFTVYYKEAPFKNVTEYDGQDACGSNSWNMVDVDLPPNKDVEPGILLHGLKPWTQYAVYVKAVTLTMVENDHIRGAKSEILYIRTNASVPSIPLDVLSASNSSSQLIVKWNPPSLPNGNLSYYIVRWQRQPQDGYLYRHNYCSKDKIPIRKYADGTIDIEEVTENPKTEVCGGEKGPCCACPKTEAEKQAEKEEAEYRKVFENFLHNSIFVPRPERKRRDVMQVANTTMSSRSRNTTAADTYNITDPEELETEYPFFESRVDNKERTVISNLRPFTLYRIDIHSCNHEAEKLGCSASNFVFARTMPAEGADDIPGPVTWEPRPENSIFLKWPEPENPNGLILMYEIKYGSQVEDQRECVSRQEYRKYGGAKLNRLNPGNYTARIQATSLSGNGSWTDPVFFYVQAKTGYENFIHRMKQLEDKVEELLSKNYHLENEVARLKKLVGERSSSEQKLISEEDLN
5RXI , Knot 195 463 0.86 40 252 447 
SMEQPEPDMITIFIGTWNMGNAPPPKKITSWFLSKGQGKTRDDSADYIPHDIYVIGTQEDPLSEKEWLEILKHSLQEITSVTFKTVAIHTLWNIRIVVLAKPEHENRISHICTDNVKTGIANTLGNKGAVGVSFMFNGTSLGFVNSHLTSGSEKKLRRNQNYMNILRFLALGDKKLSPFNITHRFTHLFWFGDLNYRVDLPTWEAETIIQKIKQQQYADLLSHDQLLTERREQKVFLHFEEEEITFAPTYRFERLTRDKYAYTKQKATGMKYNLPSWCDRVLWKSYPLVHVVCQSYGSTSDIMTSDHSPVFATFEAGVTSQFVSKNGPGTVDSQGQIEFLRCYATLKTKSQTKFYLEFHSSCLESFVKSQEGENEEGSEGELVVKFGETLPKLKPIISDPEYLLDQHILISIKSSDSDESYGEGCIALRLEATETQLPIYTPLTHHGELTGHFQGEIKLQTSQ
1WHL , Knot 46 95 0.73 32 74 89 
GSSGSSGIDVGCPVKVQLRSGEEKFPGVVRFRGPLLAERTVSGIFFGVELLEEGRGQGFTDGVYQGKQLFQCDEDCGVFVALDKLELIESGPSSG

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 \(c\).

\(|P_{f(8TAN_1)}(2) \setminus P_{f(5RXI_1)}(2)|=114\), \(|P_{f(5RXI_1)}(2) \setminus P_{f(8TAN_1)}(2)|=39\). 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 1:0101111010000001001000011010101111001000000011010110001111011110011011101011011011000111110100100111001001001110100010100100101011101100001110011000101011010001100000100000000100000001010001001000000000100110001100001011000000111011101100001011001000101011010000001111001001000101110010001001100110101000000000100100101101001101011101001001100100111110110101010000111010110010111100010100010110000100110100001010110101110101010010010010100100001010000010010000011010000000001110100001100001101010000111001000010010100010110101110001011111011011000110101101011000010110001101000101101110110100000011101011011010100011010001001010000000000111000101010100100010001011001100101000100010000100001100110001111010000001101100010000000011000010010010000111000100000011001011010010100000010011001001111001110110011111010101000111011010010111110010010010000001000000001110100101100010101001010101001111010100100011001001000100110000010001101001110000000011000010
Pair \(Z_2\) Length of longest common subsequence
8TAN_1,5RXI_1 153 5
8TAN_1,1WHL_1 269 4
5RXI_1,1WHL_1 216 3

Newick tree

 
[
	1WHL_1:13.73,
	[
		8TAN_1:76.5,5RXI_1:76.5
	]:57.23
]

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 \((0.85)(0.8)(\frac{1415 }{\log_{20} 1415}-\frac{463}{\log_{20}463})=243.\)
Status Protein1 Protein2 d d1/2
Query variables 8TAN_1 5RXI_1 310 230
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]