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Parikh vectors
1ADL_1 1IKQ_1 3GPO_1 Letter Amino acid
1 38 8 P Proline
9 37 12 S Serine
2 11 2 W Tryptophan
11 56 11 G Glycine
6 66 8 L Leucine
4 21 8 N Asparagine
2 8 4 C Cysteine
0 15 8 H Histidine
8 26 6 I Isoleucine
14 15 13 K Lycine
4 6 5 M Methionine
7 69 16 A Alanine
7 44 9 R Arginine
2 18 7 Y Tyrosine
6 14 3 F Phenylalanine
11 36 9 D Aspartic acid
2 27 2 Q Glutamine
16 37 16 V Valine
8 43 11 E Glutamic acid
11 26 10 T Threonine 

1ADL_1|Chain A|ADIPOCYTE LIPID-BINDING PROTEIN|Mus musculus (10090)
>1IKQ_1|Chain A|EXOTOXIN A|Pseudomonas aeruginosa (287)
>3GPO_1|Chains A, B, C, D|Non-structural protein 3|Chikungunya virus (371094)
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 \(w=f(c)\)
1ADL , Knot 64 131 0.79 38 97 127 
CDAFVGTWKLVSSENFDDYMKEVGVGFATRKVAGMAKPNMIISVNGDLVTIRSESTFKNTEISFKLGVEFDEITADDRKVKSIITLDGGALVQVQKWDGKSTTIKRKRDGDKLVVECVMKGVTSTRVYERA
1IKQ , Knot 239 613 0.83 40 256 553 
AEEAFDLWNECAKACVLDLKDGVRSSRMSVDPAIADTNGQGVLHYSMVLEGGNDALKLAIDNALSITSDGLTIRLEGGVEPNKPVRYSYTRQARGSWSLNWLVPIGHEKPSNIKVFIHELNAGNQLSHMSPIYTIEMGDELLAKLARDATFFVRAHESNEMQPTLAISHAGVSVVMAQAQPRREKRWSEWASGKVLCLLDPLDGVYNYLAQQRCNLDDTWEGKIYRVLAGNPAKHDLDIKPTVISHRLHFPEGGSLAALTAHQACHLPLETFTRHRQPRGWEQLEQCGYPVQRLVALYLAARLSWNQVDQVIRNALASPGSGGDLGEAIREQPEQARLALTLAAAESERFVRQGTGNDEAGAANADVVSLTCPVAAGECAGPADSGDALLERNYPTGAEFLGDGGDVSFSTRGTQNWTVERLLQAHRQLEERGYVFVGYHGTFLEAAQSIVFGGVRARSQDLDAIWRGFYIAGDPALAYGYAQDQEPDARGRIRNGALLRVYVPRSSLPGFYRTSLTLAAPEAAGEVERLIGHPLPLRLDAITGPEEEGGRLETILGWPLAERTVVIPSAIPTDPRNVGGDLDPSSIPDKEQAISALPDYASQPGKPPREDLK
3GPO , Knot 82 168 0.83 40 129 163 
MKHHHHHHAPSYRVKRMDIAKNDEECVVNAANPRGLPGDGVCKAVYKKWPESFKNSATPVGTAKTVMCGTYPVIHAVGPNFSNYSESEGDRELAAAYREVAKEVTRLGVNSVAIPLLSTGVYSGGKDRLTQSLNHLFTAMDSTDADVVIYCRDKEWEKKISEAIQMRT

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 \(c\).

\(|P_{f(1ADL_1)}(2) \setminus P_{f(1IKQ_1)}(2)|=28\), \(|P_{f(1IKQ_1)}(2) \setminus P_{f(1ADL_1)}(2)|=187\). 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 1:00111101011000010001001111110001111101011101010110100000100001010111010010100001001101011111010010100001000001001110011011000010001
Pair \(Z_2\) Length of longest common subsequence
1ADL_1,1IKQ_1 215 3
1ADL_1,3GPO_1 140 3
1IKQ_1,3GPO_1 185 4

Newick tree

 
[
	1IKQ_1:10.51,
	[
		1ADL_1:70,3GPO_1:70
	]:38.51
]

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 \((0.85)(0.8)(\frac{744 }{\log_{20} 744}-\frac{131}{\log_{20}131})=174.\)
Status Protein1 Protein2 d d1/2
Query variables 1ADL_1 1IKQ_1 220 133
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]