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Parikh vectors
1NBZ_1 2BEL_1 1AZF_1 Letter Amino acid
4 16 1 H Histidine
11 16 6 K Lycine
2 13 2 M Methionine
11 15 6 I Isoleucine
13 28 8 L Leucine
7 10 11 R Arginine
11 7 7 D Aspartic acid
10 8 3 Q Glutamine
10 17 2 E Glutamic acid
13 21 12 G Glycine
19 14 7 T Threonine
4 1 6 W Tryptophan
8 11 3 Y Tyrosine
10 5 2 P Proline
37 28 10 S Serine
10 25 6 V Valine
8 26 12 A Alanine
12 8 14 N Asparagine
5 4 8 C Cysteine
9 10 3 F Phenylalanine 

1NBZ_1|Chain A|antibody kappa light chain|Mus musculus (10090)
>2BEL_1|Chains A, B, C, D|CORTICOSTEROID 11-BETA-DEHYDROGENASE ISOZYME 1|HOMO SAPIENS (9606)
>1AZF_1|Chain A|LYSOZYME|Gallus gallus (9031)
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 \(w=f(c)\)
1NBZ , Knot 100 214 0.83 40 146 203 
DIVLTQSPATLSVTPGDSVSLSCRASQSISNNLHWYQQKSHESPRLLIKYASQSISGIPSRFSGSGSGTDFTLSINSVETEDFGMYFCQQSNSWPYTFGGGTKLEIKRADAAPTVSIFPPSSEQLTSGGASVVCFLNNFYPKDINVKWKIDGSERQNGVLNSWTDQDSKDSTYSMSSTLTLTKDEYERHNSYTCEATHKTSTSPIVKSFNRNEC
2BEL , Knot 125 283 0.83 40 176 271 
GSSHHHHHHSSGRENLYFQGHMEFRPEMLQGKKVIVTGASKGIGREMAYHLAKMGAHVVVTARSKETLQKVVSHCLELGAASAHYIAGTMEDMTFAEQFVAQAGKLMGGLDMLILNHITNTSLNLFHDDIHHVRKSMEVNFLSYVVLTVAALPMLKQSNGSIVVVSSLAGKVAYPMVAAYSASKFALDGFFSSIRKEYSVSRVNVSITLCVLGLIDTETAMKAVSGIVHMQAAPKEECALEIIKGGALRQEEVYYDSSLWTTLLIRNPCRKILEFLYSTSYGS
1AZF , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL

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 \(c\).

\(|P_{f(1NBZ_1)}(2) \setminus P_{f(2BEL_1)}(2)|=80\), \(|P_{f(2BEL_1)}(2) \setminus P_{f(1NBZ_1)}(2)|=110\). 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 1:0111000110101011001010001000100010100000000101110010001011100101010100101010010000111010000001100111100101001011101011110000100111011011001010010101010100000111001000000000001000101000000000000001000000011100100000
Pair \(Z_2\) Length of longest common subsequence
1NBZ_1,2BEL_1 190 3
1NBZ_1,1AZF_1 166 3
2BEL_1,1AZF_1 174 3

Newick tree

 
[
	2BEL_1:93.62,
	[
		1NBZ_1:83,1AZF_1:83
	]:10.62
]

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 \((0.85)(0.8)(\frac{497 }{\log_{20} 497}-\frac{214}{\log_{20}214})=81.8\)
Status Protein1 Protein2 d d1/2
Query variables 1NBZ_1 2BEL_1 105 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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]