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Parikh vectors
8DTE_1 7DYA_1 8HWS_1 Letter Amino acid
16 12 10 K Lycine
13 4 9 T Threonine
10 15 12 L Leucine
7 23 5 E Glutamic acid
5 6 0 M Methionine
6 9 15 F Phenylalanine
1 3 2 W Tryptophan
0 0 8 C Cysteine
8 7 10 R Arginine
16 7 14 G Glycine
11 9 7 I Isoleucine
7 9 12 S Serine
22 13 14 A Alanine
7 8 9 D Aspartic acid
10 9 5 Q Glutamine
6 4 2 H Histidine
7 2 12 P Proline
7 9 15 Y Tyrosine
14 9 17 V Valine
10 6 18 N Asparagine 

8DTE_1|Chain A|Periplasmic transport protein|Neisseria gonorrhoeae NCCP11945 (521006)
>7DYA_1|Chains A, B, C, D, E, F, G, H|Ferritin|Thermotoga maritima (strain ATCC 43589 / MSB8 / DSM 3109 / JCM 10099) (243274)
>8HWS_1|Chain A|Spike protein S2'|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
8DTE , Knot 82 183 0.77 38 129 171 
MAHHHHHHMSPADAVGQIRQNATQVLTILKSGDAASARPKAEAYAVPYFDFQRMTALAVGNPWRTASDAQKQALAKEFQTLLIRTYSGTMLKFKNATVNVKDNPIVNKGGKEIVVRAEVGIPGQKPVNMDFTTYQSGGKYRTYNVAIEGTSLVTVYRNQFGEIIKAKGIDGLIAELKAKNGGK
7DYA , Knot 78 164 0.80 38 121 161 
MMVISEKVRKALNDQLNREIYSSYLYLSMATYFDAEGFKGFAHWMKKQAQEELTHAMKFYEYIYERGGRVELEAIEKPPSNWNGIKDAFEAALKHEEFVTQSIYNILELASEEKDHATVSFLKWFVDEQVEEEDQVREILDLLEKANGQMSVIFQLDRYLGQRE
8HWS , Knot 91 196 0.81 38 135 189 
TNLCPFDEVFNATRFASVYAWNRKRISNCVADYSVLYNFAPFFAFKCYGVSPTKLNDLCFTNVYADSFVIRGNEVSQIAPGQTGNIADYNYKLPDDFTGCVIAWNSNKLDSKVGGNYNYRYRLFRKSNLKPFERDISTEIYQAGNKPCNGVAGVNCYFPLQSYGFRPTYGVGHQPYRVVVLSFELLHAPATVCGPK

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 \(c\).

\(|P_{f(8DTE_1)}(2) \setminus P_{f(7DYA_1)}(2)|=81\), \(|P_{f(7DYA_1)}(2) \setminus P_{f(8DTE_1)}(2)|=73\). 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 1:110000001011011101000100110110010110101010101110101001011111011001001000111001001110000101101001010100011100110011101011111001101010000011000000111010011010000110110101101111010100110
Pair \(Z_2\) Length of longest common subsequence
8DTE_1,7DYA_1 154 3
8DTE_1,8HWS_1 170 3
7DYA_1,8HWS_1 158 3

Newick tree

 
[
	8HWS_1:83.67,
	[
		8DTE_1:77,7DYA_1:77
	]:6.67
]

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 \((0.85)(0.8)(\frac{347 }{\log_{20} 347}-\frac{164}{\log_{20}164})=55.3\)
Status Protein1 Protein2 d d1/2
Query variables 8DTE_1 7DYA_1 70 66.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]