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Parikh vectors
2PZE_1 1XHJ_1 6MTF_1 Letter Amino acid
12 9 9 V Valine
4 3 10 C Cysteine
6 3 11 Q Glutamine
20 10 12 E Glutamic acid
3 8 5 H Histidine
5 5 7 P Proline
13 6 14 T Threonine
8 4 13 R Arginine
6 1 23 N Asparagine
10 5 16 D Aspartic acid
11 3 10 F Phenylalanine
20 7 12 G Glycine
24 9 13 L Leucine
8 2 3 M Methionine
2 0 6 W Tryptophan
8 0 11 Y Tyrosine
12 4 13 A Alanine
18 5 12 I Isoleucine
18 2 17 K Lycine
21 2 14 S Serine 

2PZE_1|Chains A, B|Cystic fibrosis transmembrane conductance regulator|Homo sapiens (9606)
>1XHJ_1|Chain A|Nitrogen Fixation Protein NifU|Staphylococcus epidermidis (176280)
>6MTF_1|Chains A, B|26.7 kDa salivary protein|Phlebotomus duboscqi (37738)
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 \(w=f(c)\)
2PZE , Knot 107 229 0.84 40 156 224 
SLTTTEVVMENVTAFWEEGGTPVLKDINFKIERGQLLAVAGSTGAGKTSLLMMIMGELEPSEGKIKHSGRISFCSQFSWIMPGTIKENIIFGVSYDEYRYRSVIKACQLEEDISKFAEKDNIVLGEGGITLSGGQRARISLARAVYKDADLYLLDSPFGYLDVLTEKEIFESCVCKLMANKTRILVTSKMEHLKKADKILILHEGSSYFYGTFSELQNLQPDFSSKLMG
1XHJ , Knot 46 88 0.78 36 68 81 
MPTENPTMFDQVAEVIERLRPFLLRDGGDCTLVDVEDGIVKLQLHGACGTCPSSTITLKAGIERALHEEVPGVIEVEQVFLEHHHHHH
6MTF , Knot 110 231 0.86 40 168 225 
MWRYPRNADQTFWAFRTCQRQSEGAKSLREWYRWNLPNDEDTHCYVKCVWLHLGLYNEQNKSLRVDRIMEQFNSRSVAIPGGINTISGPTDGTCKDIYDKTINFFNNNVNDLRTAFYGIKKLSDEWFTQNSNTKPKGTKISDFCNAENREKGGADCQHACSAYYYRLVDEDNEPIHFRNLNILGITDEQFASCVKASKNKQGCKVADTMYNCVEKHNSQALKILDNQSPTY

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 \(c\).

\(|P_{f(2PZE_1)}(2) \setminus P_{f(1XHJ_1)}(2)|=116\), \(|P_{f(1XHJ_1)}(2) \setminus P_{f(2PZE_1)}(2)|=28\). 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 1:0100001110010111001101110010101001011111100111000111111101010010100010101000101111101000111110000000001101001000100110000111101110101100101011011000101011001110101100001100010011100001110001001001001111001000101010010010101000111
Pair \(Z_2\) Length of longest common subsequence
2PZE_1,1XHJ_1 144 3
2PZE_1,6MTF_1 172 3
1XHJ_1,6MTF_1 176 3

Newick tree

 
[
	6MTF_1:91.46,
	[
		2PZE_1:72,1XHJ_1:72
	]:19.46
]

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 \((0.85)(0.8)(\frac{317 }{\log_{20} 317}-\frac{88}{\log_{20}88})=72.0\)
Status Protein1 Protein2 d d1/2
Query variables 2PZE_1 1XHJ_1 92 62
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]