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Parikh vectors
3BLC_1 1UMN_1 9CST_1 Letter Amino acid
10 5 2 H Histidine
12 9 4 I Isoleucine
29 16 8 L Leucine
12 8 5 K Lycine
11 8 2 F Phenylalanine
25 8 8 Q Glutamine
22 5 10 N Asparagine
18 10 7 D Aspartic acid
5 6 3 M Methionine
18 14 12 S Serine
18 12 9 V Valine
35 15 24 A Alanine
28 9 21 G Glycine
5 3 6 W Tryptophan
0 0 0 C Cysteine
11 16 4 E Glutamic acid
23 6 3 P Proline
28 4 20 T Threonine
14 5 6 Y Tyrosine
6 6 5 R Arginine 

3BLC_1|Chains A, B|Inner membrane protein oxaA|Escherichia coli K12 (83333)
>1UMN_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|DPS-LIKE PEROXIDE RESISTANCE PROTEIN|STREPTOCOCCUS SUIS (1307)
>9CST_1|Chain A|Streptavidin|Streptomyces avidinii (1895)
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 \(w=f(c)\)
3BLC , Knot 140 330 0.82 38 192 309 
MDKNPQPQAQQTTQTTTTAAGSAADQGVPASGQGKLISVKTDVLDLTINTRGGDVEQALLPAYPKELNSTQPFQLLETSPQFIYQAQSGLTGRDGPDNPANGPRPLYNVEKDAYVLAEGQNELQVPMTYTDAAGNTFTKTFVLKRGDYAVNVNYNVQNAGEKPLEISSFGQLKQSITLPPHLDTGSSNFALHTFRGAAYSTPDAAYAAYAFDTIADNENLNISSKGGWVAMLQQYFATAWIPHNDGTNNFYTANLGNGIAAIGYKSQPVLVQPGQTGAMNSTLWVGPEIQDKMAAVAPHLDLTVDYGWLWFISQPLLVPRGSLEHHHHHH
1UMN , Knot 79 165 0.81 38 122 162 
QSPAEIASFSPRPSLADSKAVLNQAVADLSVAHSILHQVHWYMRGRGFMIWHPKMDEYMEEIDGYLDEMSERLITLGGAPFSTLKEFSENSQLKEVLGDYNVTIEEQLARVVEVFRYLAALFQKGFDVSDEEGDSVTNDIFNVAKASIEKHIWMLQAELGQAPKL
9CST , Knot 74 159 0.78 38 111 154 
MASMTGGQQMGRDEAGITGTWYNQLGSTFIVTAGADGALTGTYESAVGNAESRYVLTGRYDSAPATDGSGTALGWTVAWKNNYRNAHSATTWSGQYVGGAQARINTQWLLTSGTTEANAWASTLVGHDTFTKVKPSAASIDAAKKAGVNNGNPLDAVQQ

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 \(c\).

\(|P_{f(3BLC_1)}(2) \setminus P_{f(1UMN_1)}(2)|=122\), \(|P_{f(1UMN_1)}(2) \setminus P_{f(3BLC_1)}(2)|=52\). 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 1:100010101000000000111011001111010101101000110101000110100111110100100001101100010110010011010011001101101100100010111010001011100001110010001110010011010001001100110100110100010111010010001110010111000101101101100110000101000111111100011011110001000100101101111110000111101100111000111110100011111101010100111111001111101010000000
Pair \(Z_2\) Length of longest common subsequence
3BLC_1,1UMN_1 174 3
3BLC_1,9CST_1 153 4
1UMN_1,9CST_1 159 3

Newick tree

 
[
	1UMN_1:85.49,
	[
		3BLC_1:76.5,9CST_1:76.5
	]:8.99
]

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 \((0.85)(0.8)(\frac{495 }{\log_{20} 495}-\frac{165}{\log_{20}165})=96.6\)
Status Protein1 Protein2 d d1/2
Query variables 3BLC_1 1UMN_1 123 91.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]