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Parikh vectors
8HDN_1 8SLB_1 6BHN_1 Letter Amino acid
44 4 12 A Alanine
12 9 7 N Asparagine
2 1 2 C Cysteine
11 17 7 K Lycine
8 5 1 M Methionine
14 14 8 S Serine
11 32 16 E Glutamic acid
10 15 6 F Phenylalanine
2 2 5 W Tryptophan
10 19 7 D Aspartic acid
8 8 17 Q Glutamine
28 11 14 G Glycine
5 13 4 H Histidine
10 12 5 P Proline
12 16 12 T Threonine
9 21 9 R Arginine
16 20 9 I Isoleucine
21 29 14 L Leucine
1 13 6 Y Tyrosine
18 27 19 V Valine 

8HDN_1|Chains A, B, C, D|7-alpha-hydroxysteroid dehydrogenase|Brucella melitensis biotype 1 (strain 16M / ATCC 23456 / NCTC 10094) (224914)
>8SLB_1|Chain A|Cobalt/magnesium transport protein CorA|Thermotoga maritima MSB8 (243274)
>6BHN_1|Chain A|Methyl-accepting chemotaxis sensory transducer with phytochrome sensor|Nostoc punctiforme (strain ATCC 29133 / PCC 73102) (63737)
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 \(w=f(c)\)
8HDN , Knot 111 252 0.81 40 154 233 
MSPFHLNDAVAIVTGAAAGIGRAIAGTFAKAGASVVVTDLKSEGAEAVAAAIRQAGGKAIGLECNVTDEQHREAVIKAALDQFGKITVLVNNAGGGGPKPFDMPMSDFEWAFKLNLFSLFRLSQLAAPHMQKAGGGAILNISSMAGENTNVRMASYGSSKAAVNHLTRNIAFDVGPMGIRVNAIAPGAIKTDALATVLTPEIERAMLKHTPLGRLGEAQDIANAALFLCSPAAAWISGQVLTVSGGGVQELD
8SLB , Knot 124 288 0.81 40 171 269 
MGSSHHHHHHSSGRENLYFQGHMEEKRLSAKKGLPPGTLVYTGKYREDFEIEVMNYSIEEFREFKTTDVESVLPFRDSSTPTWINITGIHRTDVVQRVGEFFGIHPLVLEDILNVHQRPKVEFFENYVFIVLKMFTYDKNLHELESEQVSLILTKNCVLMFQEKIGDVFDPVRERIRYNRGIIRKKRADYLLYSLIDALVDDYFVLLEKIDDEIDVLEEEVLERPEKETVQRTHQLKRNLVELRKTIWPLREVLSSLYRDVPPLIEKETVPYFRDVYDHTIQIADTVE
6BHN , Knot 90 180 0.86 40 140 176 
MGEKAVTKISNRIRQSSDVEEIFKTTTQEVRQLLRCDRVAVYRFNPNWTGEFVAESVAHTWVKLVGPDIKTVWEDTHLQETQGGRYAQGENFVVNDIYQVGHSPCHIEILEQFEVKAYVIVPVFAGEQLWGLLAAYQNSGTRDWDESEVTLLARIGNQLGLALQQTEYLQQVQGQSAKPG

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 \(c\).

\(|P_{f(8HDN_1)}(2) \setminus P_{f(8SLB_1)}(2)|=79\), \(|P_{f(8SLB_1)}(2) \setminus P_{f(8HDN_1)}(2)|=96\). 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 1:101101001111101111111011110110111011100100011011111100111011110001000000011101110011010111001111110110111001011101011011010011110100111111101001110000101100100011100100011101111110101111111000111011010100111000111011010011011111001111110101101011110010
Pair \(Z_2\) Length of longest common subsequence
8HDN_1,8SLB_1 175 3
8HDN_1,6BHN_1 160 3
8SLB_1,6BHN_1 167 4

Newick tree

 
[
	8SLB_1:87.28,
	[
		8HDN_1:80,6BHN_1:80
	]:7.28
]

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 \((0.85)(0.8)(\frac{540 }{\log_{20} 540}-\frac{252}{\log_{20}252})=82.0\)
Status Protein1 Protein2 d d1/2
Query variables 8HDN_1 8SLB_1 104 98.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]