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Parikh vectors
6SGD_1 6IRF_1 1UOW_1 Letter Amino acid
14 64 16 V Valine
10 63 8 A Alanine
21 49 5 R Arginine
15 37 8 D Aspartic acid
5 11 1 C Cysteine
8 38 5 F Phenylalanine
10 36 6 P Proline
15 64 7 S Serine
16 26 6 Y Tyrosine
11 41 8 N Asparagine
17 53 14 G Glycine
18 46 19 K Lycine
10 28 4 M Methionine
7 28 6 Q Glutamine
12 21 3 H Histidine
32 75 15 L Leucine
24 54 9 E Glutamic acid
16 51 9 I Isoleucine
16 48 8 T Threonine
2 14 2 W Tryptophan 

6SGD_1|Chain A|Serine/threonine-protein kinase Nek2|Homo sapiens (9606)
>6IRF_1|Chains A, C|Glutamate receptor ionotropic, NMDA 1|Homo sapiens (9606)
>1UOW_1|Chain A|SYNAPTOTAGMIN I|RATTUS NORVEGICUS (10116)
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 \(w=f(c)\)
6SGD , Knot 126 279 0.84 40 183 267 
MPSRAEDYEVLYTIGTGSYGRCQKIRRKSDGKILVWKELDYGSMTEAEKQMLVSEVNLLRELKHPNIVRYYDRIIDRTNTTLYIVMEYCEGGDLASVITKGTKERQYLDEEFVLRVMTQLTLALKECHRRSDGGHTVLHRDLKPANVFLDGKQNVKLGDFGLARILNHDTSFAKTFVGTPYYMSPEQMNRMSYNEKSDIWSLGCLLYELCALMPPFTAFSQKELAGKIREGKFRRIPYRYSDELNEIITRMLNLKDYHRPSVEEILENPLILEHHHHHH
6IRF , Knot 323 847 0.85 40 322 785 
MSTMRLLTLALLFSCSVARAACDPKIVNIGAVLSTRKHEQMFREAVNQANKRHGSWKIQLNATSVTHKPNAIQMALSVCEDLISSQVYAILVSHPPTPNDHFTPTPVSYTAGFYRIPVLGLTTRMSIYSDKSIHLSFLRTVPPYSHQSSVWFEMMRVYSWNHIILLVSDDHEGRAAQKRLETLLEERESKAEKVLQFDPGTKNVTALLMEAKELEARVIILSASEDDAATVYRAAAMLNMTGSGYVWLVGEREISGNALRYAPDGILGLQLINGKNESAHISDAVGVVAQAVHELLEKENITDPPRGCVGNTNIWKTGPLFKRVLMSSKYADGVTGRVEFNEDGDRKFANYSIMNLQNRKLVQVGIYNGTHVIPNDRKIIWPGGETEKPRGYQMSTRLKIVTIHQEPFVYVKPTLSDGTCKEEFTVNGDPVKKVICTGPNDTSPGSPRHTVPQCCYGFCIDLLIKLARTMNFTYEVHLVADGKFGTQERVNNSNKKEWNGMMGELLSGQADMIVAPLTINNERAQYIEFSKPFKYQGLTILVKKEIPRSTLDSFMQPFQSTLWLLVGLSVHVVAVMLYLLDRFSPFGRFKVNSEEEEEDALTLSSAMWFSWRVLLNSGIGEGAPRSFSARILGMVWAGFAMIIVASYTANLAAFLVLDRPEERITGINDPRLRNPSDKFIYATVKQSSVDIYFRRQVELSTMYRHMEKHNYESAAEAIQAVRDNKLHAFIWDSAVLEFEASQKCDLVTTGELFFRSGFGIGMRKDSPWKQNVSLSILKSHENGFMEDLDKTWVRYQECDSRSNAPATLTFENMAGVFMLVAGGIVAGIFLIFIEIAYKRHKDARRKQ
1UOW , Knot 77 159 0.81 40 115 150 
SGGGGGILEKLGDICFSLRYVPTAGKLTVVILEAKNLKKMDVGGLSDPYVKIHLMQNGKRLKKKKTTIKKNTLNPYYNESFSFEVPFEQIQKVQVVVTVLDYDKIGKNDAIGKVFVGYNSTGAELRHWSDMLANPRRPIAQWHTLQVEEEVDAMLAVKK

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 \(c\).

\(|P_{f(6SGD_1)}(2) \setminus P_{f(6IRF_1)}(2)|=25\), \(|P_{f(6IRF_1)}(2) \setminus P_{f(6SGD_1)}(2)|=164\). 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 1:110010000110011010010000100000101111001001010010001110010110010010110000011000000101110000110110110010000001000111011001011100000000110011000101101110100010110111101100000110011101001010010010000000110110110010111111011000011101001010011000000100110011010000010100110011110000000
Pair \(Z_2\) Length of longest common subsequence
6SGD_1,6IRF_1 189 5
6SGD_1,1UOW_1 170 4
6IRF_1,1UOW_1 233 3

Newick tree

 
[
	6IRF_1:11.22,
	[
		6SGD_1:85,1UOW_1:85
	]:27.22
]

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 \((0.85)(0.8)(\frac{1126 }{\log_{20} 1126}-\frac{279}{\log_{20}279})=225.\)
Status Protein1 Protein2 d d1/2
Query variables 6SGD_1 6IRF_1 285 187
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]