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Parikh vectors
4ZKL_1 2KJZ_1 5NHD_1 Letter Amino acid
9 10 20 V Valine
5 5 13 R Arginine
8 10 11 H Histidine
9 5 40 K Lycine
2 9 23 T Threonine
1 2 8 W Tryptophan
2 5 20 Y Tyrosine
5 5 18 Q Glutamine
7 8 11 P Proline
2 0 3 C Cysteine
9 11 27 L Leucine
4 5 17 M Methionine
6 12 16 S Serine
4 10 21 F Phenylalanine
10 14 41 A Alanine
2 4 19 N Asparagine
10 7 30 D Aspartic acid
6 7 37 E Glutamic acid
13 14 38 G Glycine
12 1 24 I Isoleucine 

4ZKL_1|Chains A, B, C, D|Histidine triad nucleotide-binding protein 1|Homo sapiens (9606)
>2KJZ_1|Chains A, B|ATC0852|Agrobacterium tumefaciens (176299)
>5NHD_1|Chains A, B, C, D|Xylose isomerase|Piromyces sp. E2 (73868)
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 \(w=f(c)\)
4ZKL , Knot 65 126 0.83 40 101 122 
MADEIAKAQVARPGGDTIFGKIIRKEIPAKIIFEDDRCLAFHDISPQAPTHFLVIPKKHISQISVAEDDDESLLGHLMIVGKKCAADLGLNKGYRMVVNEGSDGGQSVYHVHLHVLGGRQMHWPPG
2KJZ , Knot 69 144 0.79 38 110 136 
MGSSHHHHHHSSGRENLYFQGHMTHPDFTILYVDNPPASTQFYKALLGVDPVESSPTFSLFVLANGMKLGLWSRHTVEPKASVTGGGGELAFRVENDAQVDETFAGWKASGVAMLQQPAKMEFGYTFTAADPDSHRLRVYAFAG
5NHD , Knot 185 437 0.85 40 226 412 
MAKEYFPQIQKIKFEGKDSKNPLAFHYYDAEKEVMGKKMKDWLRFAMAWWHTLCAEGADQFGGGTKSFPWNEGTDAIEIAKQKVDAGFEIMQKLGIPYYCFHDVDLVSEGNSIEEYESNLKAVVAYLKEKQKETGIKLLWSTANVFGHKRYMNGASTNPDFDVVARAIVQIKNAIDAGIELGAENYVFWGGREGYMSLLNTDQKREKEHMATMLTMARDYARSKGFKGTFLIEPKPMEPTKHQYDVDTETAIGFLKAHNLDKDFKVNIEVNHATLAGHTFEHELACAVDAGMLGSIDANRGDYQNGWDTDQFPIDQYELVQAWMEIIRGGGFVTGGTNFDAKTRRNSTDLEDIIIAHVSGMDAMARALENAAKLLQESPYTKMKKERYASFDSGIGKDFEDGKLTLEQVYEYGKKNGEPKQTSGKQELYEAIVAMYQ

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 \(c\).

\(|P_{f(4ZKL_1)}(2) \setminus P_{f(2KJZ_1)}(2)|=70\), \(|P_{f(2KJZ_1)}(2) \setminus P_{f(4ZKL_1)}(2)|=79\). 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 1:110011010110111001110110001110111000001110010101100111110001001011000000111011111000110111001001110010011001001010111100101111
Pair \(Z_2\) Length of longest common subsequence
4ZKL_1,2KJZ_1 149 3
4ZKL_1,5NHD_1 211 4
2KJZ_1,5NHD_1 198 4

Newick tree

 
[
	5NHD_1:11.01,
	[
		4ZKL_1:74.5,2KJZ_1:74.5
	]:35.51
]

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 \((0.85)(0.8)(\frac{270 }{\log_{20} 270}-\frac{126}{\log_{20}126})=45.1\)
Status Protein1 Protein2 d d1/2
Query variables 4ZKL_1 2KJZ_1 57 53
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]