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Parikh vectors
1QHH_1 1ELV_1 4FKL_1 Letter Amino acid
8 15 8 N Asparagine
6 6 9 Q Glutamine
11 21 21 K Lycine
12 18 17 T Threonine
16 21 19 A Alanine
6 20 16 D Aspartic acid
1 10 3 C Cysteine
12 25 17 E Glutamic acid
4 4 10 H Histidine
14 13 16 I Isoleucine
18 21 39 L Leucine
6 10 16 F Phenylalanine
7 27 23 V Valine
9 17 14 S Serine
13 15 16 R Arginine
8 35 16 G Glycine
6 9 5 M Methionine
7 25 19 P Proline
2 8 4 W Tryptophan
1 13 10 Y Tyrosine 

1QHH_1|Chain A|PROTEIN (PCRA (SUBUNIT))|Geobacillus stearothermophilus (1422)
>1ELV_1|Chain A|COMPLEMENT C1S COMPONENT|Homo sapiens (9606)
>4FKL_1|Chain A|Cyclin-dependent kinase 2|Homo sapiens (9606)
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 \(w=f(c)\)
1QHH , Knot 83 167 0.84 40 126 164 
MNFLSEQLLAHLNKEQQEAVRTTEGPLLIMAGAGSGKTRVLTHRIAYLMAEKHVAPWNILAITFTNKAAREMRERVQSLLGGAAEDVWISTFHSMCVRILRRDIDRIGINRNFSILDPTDQLSVMKTILKEKNIDPKKFEPRTILGTISAAKNELLPPEQFAKRAST
1ELV , Knot 142 333 0.82 40 208 317 
DLDCGIPESIENGKVEDPESTLFGSVIRYTCEEPYYYMENGGGGEYHCAGNGSWVNEVLGPELPKCVPVCGVPREPFEEKQRIIGGSDADIKNFPWQVFFDNPWAGGALINEYWVLTAAHVVEGNREPTMYVGSTSVQTSRLAKSKMLTPEHVFIHPGWKLLAVPEGRTNFDNDIALVRLKDPVKMGPTVSPICLPGTSSDYNLMDGDLGLISGWGRTEKRDRAVRLKAARLPVAPLRKCKEVKVEKPTADAEAYVFTPNMICAGGEKGMDSCKGDSGGAFAVQDPNDKTKFYAAGLVSWGPQCGTYGLYTRVKNYVDWIMKTMQENSTPRED
4FKL , Knot 135 299 0.85 42 192 285 
XMENFQKVEKIGEGTYGVVYKARNKLTGEVVALKKIRLDTETEGVPSTAIREISLLKELNHPNIVKLLDVIHTENKLYLVFEFLHQDLKKFMDASALTGIPLPLIKSYLFQLLQGLAFCHSHRVLHRDLKPQNLLINTEGAIKLADFGLARAFGVPVRTYTHEVVTLWYRAPEILLGCKYYSTAVDIWSLGCIFAEMVTRRALFPGDSEIDQLFRIFRTLGTPDEVVWPGVTSMPDYKPSFPKWARQDFSKVVPPLDEDGRSLLSQMLHYDPNKRISAKAALAHPFFQDVTKPVPHLRL

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 \(c\).

\(|P_{f(1QHH_1)}(2) \setminus P_{f(1ELV_1)}(2)|=46\), \(|P_{f(1ELV_1)}(2) \setminus P_{f(1QHH_1)}(2)|=128\). 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 1:10110001110100000011000011111111110100011000110111000111101111010001100100010011111100111001001010110001001110001011010001011001100001010010100111010110001111001100100
Pair \(Z_2\) Length of longest common subsequence
1QHH_1,1ELV_1 174 4
1QHH_1,4FKL_1 162 3
1ELV_1,4FKL_1 180 4

Newick tree

 
[
	1ELV_1:90.87,
	[
		1QHH_1:81,4FKL_1:81
	]:9.87
]

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 \((0.85)(0.8)(\frac{500 }{\log_{20} 500}-\frac{167}{\log_{20}167})=97.4\)
Status Protein1 Protein2 d d1/2
Query variables 1QHH_1 1ELV_1 126 93
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]