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Parikh vectors
3HNQ_1 1HTM_1 4NXW_1 Letter Amino acid
9 3 15 T Threonine
6 0 23 R Arginine
7 2 21 D Aspartic acid
7 1 14 Q Glutamine
18 0 15 I Isoleucine
6 1 14 K Lycine
8 2 32 A Alanine
6 2 21 P Proline
0 0 5 W Tryptophan
11 0 20 E Glutamic acid
4 2 5 H Histidine
10 4 54 L Leucine
4 0 5 M Methionine
4 0 12 Y Tyrosine
10 2 22 V Valine
5 3 9 N Asparagine
1 1 7 C Cysteine
8 3 17 G Glycine
8 0 8 F Phenylalanine
12 1 23 S Serine 

3HNQ_1|Chains A, B, C, D|Virulence protein STM3117|Salmonella enterica subsp. enterica serovar Typhimurium (90371)
>1HTM_1|Chains A, C, E|HEMAGGLUTININ HA1 CHAIN|uncultured beta proteobacterium UMTRA-608 (140147)
>4NXW_1|Chain A|Mitochondrial dynamic protein MID51|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 \(w=f(c)\)
3HNQ , Knot 73 144 0.84 38 122 142 
MLFFNVASLKYKHHESIQMIIDRIDHLVLTVSDISTTIRFYEEVLGFSAVTFKQNRKALIFGAQKINLHQQEMEFEPKASRPTPGSADLCFITSTPINDVVSEILQAGISIVEGPVERTGATGEIMSIYIRDPDGNLIEISQYV
1HTM , Knot 20 27 0.81 26 25 25 
QDLPGNDNSTATLCLGHHAVPNGTLVK
4NXW , Knot 147 342 0.83 40 191 329 
GSARKGQVDLKKSRLRMSLQEKLLTYYRNRAAIPAGEQARAKQAAVDICAELRSFLRAKLPDMPLRDMYLSGSLYDDLQVVTADHIQLIVPLVLEQNLWSCIPGEDTIMNVPGFFLVRRENRGSSYWDRCVVGGYLSPKTVADTFEKVVAGSINWPAIGSLLDYVIRPAPPPEALTLEVQYERDKHLFIDFLPSVTLGDTVLVAKPHRLAQYDNLWRLSLRPAETARLRALDQADSGCRSLCLKILKAICKSTPALGHLTASQLTNVILHLAQEEADWSPDMLADRFLQALRGLISYLEAGVLPSALNPKVNLFAELTPEEIDELGYTLYCSLSEPEVLLQT

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 \(c\).

\(|P_{f(3HNQ_1)}(2) \setminus P_{f(1HTM_1)}(2)|=111\), \(|P_{f(1HTM_1)}(2) \setminus P_{f(3HNQ_1)}(2)|=14\). 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 1:111101101000000010111001001110100100010100011110110100000111111001010000101010100101101010110001100110011011101101110001101011010100101011010001
Pair \(Z_2\) Length of longest common subsequence
3HNQ_1,1HTM_1 125 2
3HNQ_1,4NXW_1 161 3
1HTM_1,4NXW_1 188 3

Newick tree

 
[
	4NXW_1:94.38,
	[
		3HNQ_1:62.5,1HTM_1:62.5
	]:31.88
]

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 \((0.85)(0.8)(\frac{171 }{\log_{20} 171}-\frac{27}{\log_{20}27})=51.0\)
Status Protein1 Protein2 d d1/2
Query variables 3HNQ_1 1HTM_1 65 39
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]