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Parikh vectors
2BQN_1 5YZH_1 7XHG_1 Letter Amino acid
11 33 10 G Glycine
1 8 7 H Histidine
5 6 0 W Tryptophan
8 24 15 V Valine
8 27 7 D Aspartic acid
6 7 1 C Cysteine
3 28 5 E Glutamic acid
5 30 3 I Isoleucine
5 25 5 T Threonine
6 18 7 Y Tyrosine
10 15 11 N Asparagine
17 31 8 A Alanine
14 16 8 R Arginine
6 18 8 Q Glutamine
8 27 11 L Leucine
5 38 7 K Lycine
2 15 6 M Methionine
2 16 8 F Phenylalanine
2 12 5 P Proline
6 20 7 S Serine 

2BQN_1|Chain A|LYSOZYME|Homo sapiens (9606)
>5YZH_1|Chains A, B|Isocitrate dehydrogenase [NADP] cytoplasmic|Mus musculus (10090)
>7XHG_1|Chains A, B, C, D|Ras GTPase-activating protein-binding protein 1|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 \(w=f(c)\)
2BQN , Knot 68 130 0.84 40 108 126 
KVFERCELARTLKRLGMDGYRGISLANWMCLAKWESGYNTRATNYNAGDRSTDYGIFQINSRYWCNDGKTPGAVNAAHLSCSALLQDNIADAAAAAKRVVRDPQGIRAWVAWRNRCQNRDVRQYVQGCGV
5YZH , Knot 176 414 0.85 40 228 399 
MSRKIQGGSVVEMQGDEMTRIIWELIKEKLILPYVELDLHSYDLGIENRDATNDQVTKDAAEAIKKYNVGVKCATITPDEKRVEEFKLKQMWKSPNGTIRNILGGTVFREAIICKNIPRLVTGWVKPIIIGRHAYGDQYRATDFVVPGPGKVEITYTPKDGTQKVTYMVHDFEEGGGVAMGMYNQDKSIEDFAHSSFQMALSKGWPLYLSTKNTILKKYDGRFKDIFQEIYDKKYKSQFEAQKICYEHRLIDDMVAQAMKSEGGFIWACKNYDGDVQSDSVAQGYGSLGMMTSVLICPDGKTVEAEAAHGTVTRHYRMYQKGQETSTNPIASIFAWSRGLAHRAKLDNNTELSFFAKALEDVCIETIEAGFMTKDLAACIKGLPNVQRSDYLNTFEFMDKLGENLKAKLAQAKL
7XHG , Knot 69 139 0.81 38 114 135 
MVMEKPSPLLVGREFVRQYYTLLNQAPDMLHRFYGKNSSYVHGGLDSNGKPADAVYGQKEIHRKVMSQNFTNCHTKIRHVDAHATLNDGVVVQVMGLLSNNNQALRRFMQTFVLAPEGSVANKFYVHNDIFRYQDEVFG

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 \(c\).

\(|P_{f(2BQN_1)}(2) \setminus P_{f(5YZH_1)}(2)|=43\), \(|P_{f(5YZH_1)}(2) \setminus P_{f(2BQN_1)}(2)|=163\). 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 1:0110000110010011101001101101101101001000010000110000001110100001000100111101101000111000110111110011001011011111000000001000101011
Pair \(Z_2\) Length of longest common subsequence
2BQN_1,5YZH_1 206 4
2BQN_1,7XHG_1 158 4
5YZH_1,7XHG_1 188 3

Newick tree

 
[
	5YZH_1:10.32,
	[
		2BQN_1:79,7XHG_1:79
	]:25.32
]

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 \((0.85)(0.8)(\frac{544 }{\log_{20} 544}-\frac{130}{\log_{20}130})=121.\)
Status Protein1 Protein2 d d1/2
Query variables 2BQN_1 5YZH_1 158 103
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]