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Parikh vectors
6WHS_1 4KDQ_1 4WKH_1 Letter Amino acid
66 18 20 V Valine
81 26 36 L Leucine
30 5 7 M Methionine
42 10 20 F Phenylalanine
77 26 24 S Serine
15 6 9 W Tryptophan
72 13 33 A Alanine
44 14 22 D Aspartic acid
56 18 16 E Glutamic acid
57 21 23 K Lycine
41 11 25 Q Glutamine
23 10 10 H Histidine
54 24 8 I Isoleucine
55 20 26 T Threonine
28 12 16 Y Tyrosine
63 11 15 R Arginine
42 32 16 N Asparagine
12 9 4 C Cysteine
61 18 32 G Glycine
40 17 15 P Proline 

6WHS_1|Chains A, C|Glutamate receptor ionotropic, NMDA 1|Rattus norvegicus (10116)
>4KDQ_1|Chains A, B[auth C], C[auth E]|Hemagglutinin|Influenza A virus (577543)
>4WKH_1|Chain A|Chitotriosidase-1|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 \(w=f(c)\)
6WHS , Knot 362 959 0.86 40 329 882 
MSTMHLLTFALLFSCSFARAASDPKIVNIGAVLSTRKHEQMFREAVNQANKRHGSWKIQLQATSVTHKPNAIQMALSVCEDLISSQVYAILVSHPPTPNDHFTPTPVSYTAGFYRIPVLGLTTRMSIYSDKSIHLSFLRTVPPYSHQSSVWFEMMRVYNWNHIILLVSDDHEGRAAQKRLETLLEERESKSKKRNYENLDQLSYDNKRGPKAEKVLQFDPGTKNVTALLMEARELEARVIILSASEDDAATVYRAAAMLDMTGSGYVWLVGEREISGNALRYAPDGIIGLQLINGKNESAHISDAVGVVAQAVHELLEKENITDPPRGCVGNTNIWKTGPLFKRVLMSSKYADGVTGRVEFNEDGDRKFAQYSIMNLQNRKLVQVGIYNGTHVIPNDRKIIWPGGETEKPRGYQMSTRLKIVTIHQEPFVYVKPTMSDGTCKEEFTVNGDPVKKVICTGPNDTSPGSPRHTVPQCCYGFCIDLLIKLARTMQFTYEVHLVADGKFGTQERVQNSNKKEWNGMMGELLSGQADMIVAPLTINNERAQYIEFSKPFKYQGLTILVKKEIPRSTLDSFMQPFQSTLWLLVGLSVHVVAVMLYLLDRFSPFGRFKVNSQSESTDALTLSSAMWFSWGVLLNSGIGEGAPRSFSARILGMVWAGFAMIIVASYTANLAAFLVLDRPEERITGINDPRLRNPSDKFIYATVKQSSVDIYFRRQVELSTMYRHMEKHNYESAAEAIQAVRDNKLHAFIWDSAVLEFEASQKCDLVTTGELFFRSGFGIGMRKDSPWKQQVSLSILKSHENGFMEDLDKTWVRYQECDSRSNAPATLTCENMAGVFMLVAGGIVAGIFLIFIEIAYKRHKDARRKQMQLAFAAVNVWRKNLQDRKSGRAEPDPKKKATFRAITSTLASSFKRRRSSKDTSTGGGRGALQNQKDTVLPRRAIEREEGQLQLCSRHRES
4KDQ , Knot 143 321 0.85 40 206 309 
SDQICIGYHANNSTEQVDTIMEKNVTVTHAQDILEKTHNGKLCDLDGVKPLILRDCSVAGWLLGNPMCDEFLNVPEWSYIVEKINPVNDLCYPGNFNDYEELKHLLSRINHFEKIQIIPKSSWSDHEASSGVSSACPYQGRSSFFRNVVWLIKKNNTYPTIKRSYNNTNQEDLLVLWGIHHPNDAAEQTRLYQNPTTYISVGTSTLNQRLVPKLATRSKVNGQSGRMEFFWTILKPNDAINFESNGNFIAPENAYKIVKKGDSTIMKSELEYGNCNTKCQTPIGAINSSMPFHNIHPLTIGECPKYVKSNRLVLATGLRNS
4WKH , Knot 160 377 0.84 40 221 358 
AKLVCYFTNWAQYRQGEARFLPKDLDPSLCTHLIYAFAGMTNHQLSTTEWNDETLYQEFNGLKKMNPKLKTLLAIGGWNFGTQKFTDMVATANNRQTFVNSAIRFLRKYSFDGLDLDWEYPGSQGSPAVDKERFTTLVQDLANAFQQEAQTSGKERLLLSAAVPAGQTYVDAGYEVDKIAQNLDFVNLMAYDFHGSWEKVTGHNSPLYKRQEESGAAASLNVDAAVQQWLQKGTPASKLILGMPTYGRSFTLASSSDTRVGAPATGSGTPGPFTKEGGMLAYYEVCSWKGATKQRIQDQKVPYIFRDNQWVGFDDVESFKTKVSYLKQKGLGGAMVWALDLDDFAGFSCNQGRYPLIQTLRQELSLVPRGSHHHHHH

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 \(c\).

\(|P_{f(6WHS_1)}(2) \setminus P_{f(4KDQ_1)}(2)|=151\), \(|P_{f(4KDQ_1)}(2) \setminus P_{f(6WHS_1)}(2)|=28\). 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 1:10010110111110001101100101101111100000001100110010000101010101001000101101110100011000101111001101000101011000111001111110001010000010101100111000000111011010010011111000001011000100110000000000000010010000001101001101011000101111010010101111010000110100111110101010111110001010110011011111011010000101001111110110011000010011010110001100111100111000010110101010001000110001101000011011100100111000011111100001010010001011010001110101010010000010101011001100110000110100011000011010111011001010001011101011000010000000101111011010101111110100001001010011000110111000110001001101100011111110101111110110010111010100000000110100111101111100111011100101011111111111111100010111111100100010110010100100011010100001010100010100100010000000110110110000101111001110101000001100101110011111100001100010101100000111001000110000000000111010000111111111111111111111011000000100001011111101100010000010101010001010110001100100000000000111011100000011100110000101010000000
Pair \(Z_2\) Length of longest common subsequence
6WHS_1,4KDQ_1 179 4
6WHS_1,4WKH_1 152 5
4KDQ_1,4WKH_1 165 3

Newick tree

 
[
	4KDQ_1:89.17,
	[
		6WHS_1:76,4WKH_1:76
	]:13.17
]

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 \((0.85)(0.8)(\frac{1280 }{\log_{20} 1280}-\frac{321}{\log_{20}321})=251.\)
Status Protein1 Protein2 d d1/2
Query variables 6WHS_1 4KDQ_1 322 213.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]