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Parikh vectors
4XSQ_1 2BBI_1 8QGE_1 Letter Amino acid
18 2 22 L Leucine
8 5 17 K Lycine
7 6 10 P Proline
4 2 12 Y Tyrosine
4 3 6 Q Glutamine
15 0 20 G Glycine
11 2 19 I Isoleucine
8 1 19 V Valine
12 8 18 D Aspartic acid
8 14 1 C Cysteine
8 2 15 F Phenylalanine
17 9 20 S Serine
15 4 15 A Alanine
10 2 13 R Arginine
8 4 19 E Glutamic acid
9 2 12 T Threonine
2 0 2 W Tryptophan
9 3 7 N Asparagine
5 1 17 H Histidine
5 1 8 M Methionine 

4XSQ_1|Chains A, B|variable lymphocyte receptor-like protein Bf66946|Branchiostoma floridae (7739)
>2BBI_1|Chain A|TRYPSIN/CHYMOTRYPSIN BOWMAN-BIRK INHIBITOR|Glycine max (3847)
>8QGE_1|Chain A|NAD kinase 1|Listeria monocytogenes EGD-e (169963)
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 \(w=f(c)\)
4XSQ , Knot 89 183 0.84 40 133 173 
MCPSACKCTVSLYGEMVVACGGMGLTEIPEDIPHRAVYLVLKDNNITKITSYSFKGLRNLQGIDLSNNKINHISSAALRHLGHLDDIDLSRNELTSVSEKLFDFPISSAKAQGRRFFVYLANNPWGCDCRMAWLAQELAGGSKTFGDRHMECATPAALAGRGLSEIPQTSFVCTGRDISFDSD
2BBI , Knot 35 71 0.70 36 59 69 
DDESSKPCCDQCACTKSNPPQCRCSDMRLNSCHSACKSCICALSYPAQCFCVDITDFCYEPCKPSEDDKEN
8QGE , Knot 123 272 0.84 40 185 261 
MKYMITSKGDEKSDLLRLNMIAGFGEYDMEYDDVEPEIVISIGGDGTFLSAFHQYEERLDEIAFIGIHTGHLGFYADWRPAEADKLVKLLAKGEYQKVSYPLLKTTVKYGIGKKEATYLALNESTVKSSGGPFVVDVVINDIHFERFRGDGLCMSTPSGTTAYNKSLGGALMHPSIEAMQLTEMASINNRVYRTIGSPLVFPKHHVVSLQPVNDKDFQISVDHLSILHRDVQEIRYEVSAKKIHFARFRSFPFWRRVHDSFIEDLEHHHHHH

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 \(c\).

\(|P_{f(4XSQ_1)}(2) \setminus P_{f(2BBI_1)}(2)|=107\), \(|P_{f(2BBI_1)}(2) \setminus P_{f(4XSQ_1)}(2)|=33\). 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 1:101010000101010111101111100110011001101110000100100001011001011010000100100111001101001010000100100011011100101010011101100111000011111001111000110001001011111101100110001100100101000
Pair \(Z_2\) Length of longest common subsequence
4XSQ_1,2BBI_1 140 4
4XSQ_1,8QGE_1 170 3
2BBI_1,8QGE_1 192 3

Newick tree

 
[
	8QGE_1:96.57,
	[
		4XSQ_1:70,2BBI_1:70
	]:26.57
]

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 \((0.85)(0.8)(\frac{254 }{\log_{20} 254}-\frac{71}{\log_{20}71})=59.5\)
Status Protein1 Protein2 d d1/2
Query variables 4XSQ_1 2BBI_1 77 52
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]