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Parikh vectors
3WJL_1 6ASU_1 3OXC_1 Letter Amino acid
9 16 1 Y Tyrosine
23 59 6 V Valine
6 88 5 A Alanine
11 29 3 N Asparagine
6 3 0 C Cysteine
6 14 1 H Histidine
18 65 10 L Leucine
22 34 1 S Serine
9 60 13 G Glycine
16 48 8 T Threonine
7 46 4 R Arginine
9 26 5 Q Glutamine
4 41 15 I Isoleucine
2 19 2 M Methionine
21 36 6 P Proline
4 10 2 W Tryptophan
15 62 4 D Aspartic acid
15 36 4 E Glutamic acid
20 26 7 K Lycine
7 23 2 F Phenylalanine 

3WJL_1|Chains A, B|Ig gamma-1 chain C region|Homo sapiens (9606)
>6ASU_1|Chain A|Malate synthase G|Mycobacterium tuberculosis (1773)
>3OXC_1|Chains A, B|Protease|Human immunodeficiency virus 1 (11676)
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 \(w=f(c)\)
3WJL , Knot 104 230 0.82 40 156 224 
EPKSSDKTHTCPPCPAPDLLGDDSVFLFPPKPKDTLMISRTPEVTCVVVDVSDEDGEVKFNWYVDGVEVHNAKTKPREEQYNSTYRVVSVLTVLHQDWLNGKEYKCKVSNKALPRPIEKTISKAKGQPREPQVYTLPPSRDELTKNQVSLTCLVKGFYPSDIAVEWESNGQPENNYKTTPPVLDSDGSFFLYSKLTVDKSRWQQGNVFSCSVMHEALHNHYTQKSLSLSP
6ASU , Knot 277 741 0.82 40 268 652 
MTDRVSVGNLRIARVLYDFVNNEALPGTDIDPDSFWAGVDKVVADLTPQNQALLNARDELQAQIDKWHRRRVIEPIDMDAYRQFLTEIGYLLPEPDDFTITTSGVDAEITTTAGPQLVVPVLNARFALNAANARWGSLYDALYGTDVIPETDGAEKGPTYNKVRGDKVIAYARKFLDDSVPLSSGSFGDATGFTVQDGQLVVALPDKSTGLANPGQFAGYTGAAESPTSVLLINHGLHIEILIDPESQVGTTDRAGVKDVILESAITTIMDFEDSVAAVDAADKVLGYRNWLGLNKGDLAAAVDKDGTAFLRVLNRDRNYTAPGGGQFTLPGRSLMFVRNVGHLMTNDAIVDTDGSEVFEGIMDALFTGLIAIHGLKASDVNGPLINSRTGSIYIVKPKMHGPAEVAFTCELFSRVEDVLGLPQNTMKIGIMDEERRTTVNLKACIKAAADRVVFINTGFLDRTGDEIHTSMEAGPMVRKGTMKSQPWILAYEDHNVDAGLAAGFSGRAQVGKGMWTMTELMADMVETKIAQPRAGASTAWVPSPTAATLHALHYHQVDVAAVQQGLAGKRRATIEQLLTIPLAKELAWAPDEIREEVDNNCQSILGYVVRWVDQGVGASKVPDIHDVALMEDRATLRISSQLLANWLRHGVITSADVRASLERMAPLVDRQNAGDVAYRPMAPNFDDSIAFLAAQELILSGAQQPNGYTEPILHRRRREFKARAAEKPAPSDRAGDDAAR
3OXC , Knot 53 99 0.82 38 77 95 
PQITLWKRPLVTIKIGGQLKEALLDTGADDTVIEEMSLPGRWKPKMIGGIGGFIKVRQYDQIIIEIAGHKAIGTVLVGPTPVNIIGRNLLTQIGATLNF

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 \(c\).

\(|P_{f(3WJL_1)}(2) \setminus P_{f(6ASU_1)}(2)|=38\), \(|P_{f(6ASU_1)}(2) \setminus P_{f(3WJL_1)}(2)|=150\). 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 1:01000000000110111011100011111101000111000101001110100001010101010110100100010000000000110110110001101000000100011101100010010101001010011100001000010100110110100111010001010000000111100010111000101000010010110001100110000000010101
Pair \(Z_2\) Length of longest common subsequence
3WJL_1,6ASU_1 188 4
3WJL_1,3OXC_1 165 3
6ASU_1,3OXC_1 213 4

Newick tree

 
[
	6ASU_1:10.75,
	[
		3WJL_1:82.5,3OXC_1:82.5
	]:23.25
]

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 \((0.85)(0.8)(\frac{971 }{\log_{20} 971}-\frac{230}{\log_{20}230})=201.\)
Status Protein1 Protein2 d d1/2
Query variables 3WJL_1 6ASU_1 256 167.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]