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Parikh vectors
8HLF_1 1KAF_1 2MIM_1 Letter Amino acid
8 12 6 I Isoleucine
11 9 12 L Leucine
1 7 5 M Methionine
5 0 5 P Proline
11 5 14 S Serine
10 5 12 A Alanine
2 1 7 Q Glutamine
10 10 9 E Glutamic acid
6 4 11 T Threonine
3 0 2 W Tryptophan
5 4 6 Y Tyrosine
6 6 19 D Aspartic acid
9 2 6 H Histidine
8 5 11 V Valine
13 6 15 G Glycine
12 11 19 K Lycine
2 5 4 R Arginine
6 11 9 N Asparagine
0 1 4 C Cysteine
8 4 2 F Phenylalanine 

8HLF_1|Chains A, B|Novel protein with potential Cupin domain|Candidatus Pelagibacter ubique HTCC1062 (335992)
>1KAF_1|Chains A, B, C, D, E, F|Transcription regulatory protein MOTA|Enterobacteria phage T4 (10665)
>2MIM_1|Chain A|CD3 epsilon protein,CD3 glycoprotein|Gallus gallus (9031)
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 \(w=f(c)\)
8HLF , Knot 68 136 0.81 38 107 127 
MIFVKNLASVLSQEWSSTEKYPGVRWKFLIDADFDGSSGLSLGFAEIAPGGDLTLHYHSPAEIAVVTNGKGILNKSGKLETIKKGDVVYIAGNAEHALKNNGKETLEFYWIFPTDRFSEVEYFPAKQKSGHHHHHH
1KAF , Knot 56 108 0.81 36 89 104 
MEITSDMEEDKDLMLKLLDKNGFVLKKVEIYRSNYLAILEKRTNGIRNFEINNNGNMRIFGYKMMEHHIQKFTDIGMSCKIAKNGNVYLDIKRSAENIEAVITVASEL
2MIM , Knot 82 178 0.79 40 129 165 
GSHMGQEEFAVEISGTTVTITCPSSGDDIKWKPDPALGDNNKYIIQNHDSSPLTVSCTAGDQEHTMYLNAKVGSADDAKKDAAKKDDAKKDDAKKDGSVAKLGVHGLSMSVKEVSGKVFLQCQESKDLNTNYLWKKGKEELGNMRQLDLGAIYDDPRGTYTCQRDENVNSTLHVHYRM

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 \(c\).

\(|P_{f(8HLF_1)}(2) \setminus P_{f(1KAF_1)}(2)|=74\), \(|P_{f(1KAF_1)}(2) \setminus P_{f(8HLF_1)}(2)|=56\). 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 1:1111001101100010000001110101110101010011011110111110101000011011110010111000101001001011011101001100010001010111100010010011100001000000
Pair \(Z_2\) Length of longest common subsequence
8HLF_1,1KAF_1 130 3
8HLF_1,2MIM_1 136 3
1KAF_1,2MIM_1 150 4

Newick tree

 
[
	2MIM_1:73.65,
	[
		8HLF_1:65,1KAF_1:65
	]:8.65
]

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 \((0.85)(0.8)(\frac{244 }{\log_{20} 244}-\frac{108}{\log_{20}108})=43.4\)
Status Protein1 Protein2 d d1/2
Query variables 8HLF_1 1KAF_1 54 48.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]