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Parikh vectors
1DGQ_1 8KDA_1 5CJZ_1 Letter Amino acid
6 8 41 Y Tyrosine
4 3 16 H Histidine
17 21 48 L Leucine
21 11 26 K Lycine
4 6 30 P Proline
13 7 43 T Threonine
1 2 18 W Tryptophan
3 14 30 R Arginine
5 8 39 N Asparagine
9 10 43 G Glycine
15 10 33 F Phenylalanine
12 9 46 S Serine
0 0 6 C Cysteine
14 12 19 I Isoleucine
11 16 28 V Valine
13 12 55 A Alanine
16 12 32 D Aspartic acid
7 2 17 Q Glutamine
12 21 44 E Glutamic acid
5 5 12 M Methionine 

1DGQ_1|Chain A|LEUKOCYTE FUNCTION ASSOCIATED ANTIGEN-1|Homo sapiens (9606)
>8KDA_1|Chains A, G[auth B], H[auth C], I[auth D], J[auth E], K[auth F], L[auth G], M[auth H], N[auth I], O[auth J], P[auth K], Q[auth L]|RNA-free ribonuclease P|Hydrogenobacter thermophilus DSM 653 (608538)
>5CJZ_1|Chain A|SusD homolog|Bacteroides thetaiotaomicron VPI-5482 (226186)
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 \(w=f(c)\)
1DGQ , Knot 86 188 0.79 38 129 181 
MASKGNVDLVFLFDGSMSLQPDEFQKILDFMKDVMKKLSNTSYQFAAVQFSTSYKTEFDFSDYVKWKDPDALLKHVKHMLLLTNTFGAINYVATEVFREELGARPDATKVLIIITDGEATDSGNIDAAKDIIRYIIGIGKHFQTKESQETLHKFASKPASEFVKILDTFEKLKDLFTELQKKIYVIEG
8KDA , Knot 88 189 0.81 38 123 181 
MDTFVLDTSVFTNPDVYHQFEEDQLGAIENFISLASHTNANFFMPTSVYYEFTKMVSLGDLAPKFELVVRIRSPRKWGLMVPAEFLYEFIEEVRYRINKGLRIAEEHTKEAGKLAEEEVGRVVNRLREKYREALRAGIIDSKEDVDVLLLSYELDAILVSGDEGLRKWADRVGIKLIDPKNLRYIMENL
5CJZ , Knot 253 626 0.86 40 283 588 
MGSSHHHHHHSSENLYFQGHMASAMSSPTRALGFLYSCYGGVSTDLPSAYLGEINSTTDEYVLPYSWNTDGYWGAYAFNTASSTNQDWLWGTTYQYIGQCYLFLQKLENAGSDIASDAEKEQWRAECQFLVAYYHFATLRRYGPIPITDSYIPMDTPTSEYNGRFHFDYCVDWIANQLDEAAKVLPANRTATNEWGRATSTIAKAVKARLLLYAASPLWNGSFPYPNWQNENFETPGYGKALVSNTYDKSKWERALAACQEALTLATTSGDRELYDDDEYYSRQSLNLPFVPGVADVEDNKEFLKNVMKMRYAVSTRESEGNKEIIWGLSNQFDFYSRYPLRILKKSDGTWHAGYSGVSPTLYTFEHFYTANGKLPEKDLDFTPSSEWFESAGISSREDIIKLNVGREPRFYAWMAFDGGDYGTKFAAGSPLKLEMRNSEMHGYNPSLFNRDHSVTGFLTQKFVDPVTEFYTAGGSTSGTSAPTILFRLAELYLNVAECHAALGNTQEAIDALNPVRERAGIPKLTLADITNNMTIKDWVHNERFVELWNEGHRFFDVRRWAEGAKYFGANKREGLNAEVQSPTFEEFNKRTTVDAPYVWENRMYLNPVFYNEVYKNPQMVQAPGY

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 \(c\).

\(|P_{f(1DGQ_1)}(2) \setminus P_{f(8KDA_1)}(2)|=70\), \(|P_{f(8KDA_1)}(2) \setminus P_{f(1DGQ_1)}(2)|=64\). 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 1:11001010111110101010100100110110011001000000111101000000010100010100101110010011110001111001100110001110101001111100101000101011001100111110010000000010011001100110110010010011001000101101
Pair \(Z_2\) Length of longest common subsequence
1DGQ_1,8KDA_1 134 3
1DGQ_1,5CJZ_1 204 3
8KDA_1,5CJZ_1 202 4

Newick tree

 
[
	5CJZ_1:11.63,
	[
		1DGQ_1:67,8KDA_1:67
	]:43.63
]

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 \((0.85)(0.8)(\frac{377 }{\log_{20} 377}-\frac{188}{\log_{20}188})=56.3\)
Status Protein1 Protein2 d d1/2
Query variables 1DGQ_1 8KDA_1 70 69.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]