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Parikh vectors
8QHW_1 1RRO_1 6MTI_1 Letter Amino acid
15 2 3 R Arginine
4 6 4 I Isoleucine
11 11 5 S Serine
1 0 1 W Tryptophan
1 2 5 Y Tyrosine
0 1 0 C Cysteine
24 9 8 E Glutamic acid
4 8 10 F Phenylalanine
1 2 6 P Proline
9 2 11 V Valine
11 2 5 N Asparagine
10 15 10 D Aspartic acid
26 8 13 L Leucine
16 7 14 K Lycine
5 3 3 M Methionine
38 9 5 A Alanine
17 9 6 Q Glutamine
9 6 8 G Glycine
0 1 3 H Histidine
13 5 7 T Threonine 

8QHW_1|Chain A[auth E]|Membrane-associated protein Vipp1|Synechocystis sp. PCC 6803 (1148)
>1RRO_1|Chain A|RAT ONCOMODULIN|Rattus rattus (10117)
>6MTI_1|Chains A[auth 1], F[auth 6]|Synaptotagmin-1|Rattus norvegicus (10116)
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 \(w=f(c)\)
8QHW , Knot 91 215 0.75 36 121 198 
ALFDRLGRVVRANLNDLVSKAEDPEKVLEQAVIDMQEDLVQLRQAVARTIAEEKRTEQRLNQDTQEAKKWEDRAKLALTNGEENLAREALARKKSLTDTAAAYQTQLAQQRTMSENLRRNLAALEAKISEAKTKKNMLQARAKAAKANAELQQTLGGLGTSSATSAFERMENKVLDMEATSQAAGELAGFGIENQFAQLEASSGVEDELAALKAS
1RRO , Knot 53 108 0.76 38 84 103 
SITDILSAEDIAAALQECQDPDTFEPQKFFQTSGLSKMSASQVKDIFRFIDNDQSGYLDGDELKYFLQKFQSDARELTESETKSLMDAADNDGDGKIGADEFQEMVHS
6MTI , Knot 61 127 0.77 38 99 120 
KLGKLQYSLDYDFQNNQLLVGIIQAAELPALDMGGTSDPYVKVFLLPDKKKKFETKVHRKTLNPVFNEQFTFKVPYSELGGKTLVMAVYDFDRFSKHDIIGEFKVPMNTVDFGHVTEEWRDLQSAEK

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 \(c\).

\(|P_{f(8QHW_1)}(2) \setminus P_{f(1RRO_1)}(2)|=85\), \(|P_{f(1RRO_1)}(2) \setminus P_{f(8QHW_1)}(2)|=48\). 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 1:11100110110101001100100100110011101000110100111001100000000100000010010001011100100011001110000100011100001100001000100011110101001000001101010110101010001111100010011001000110101000111011111100011010100110001111010
Pair \(Z_2\) Length of longest common subsequence
8QHW_1,1RRO_1 133 3
8QHW_1,6MTI_1 138 3
1RRO_1,6MTI_1 123 3

Newick tree

 
[
	8QHW_1:69.72,
	[
		1RRO_1:61.5,6MTI_1:61.5
	]:8.22
]

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 \((0.85)(0.8)(\frac{323 }{\log_{20} 323}-\frac{108}{\log_{20}108})=66.8\)
Status Protein1 Protein2 d d1/2
Query variables 8QHW_1 1RRO_1 79 59.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]