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Parikh vectors
8BDO_1 1WNR_1 4CZT_1 Letter Amino acid
1 0 6 H Histidine
3 1 12 M Methionine
0 0 4 W Tryptophan
4 3 10 Q Glutamine
1 2 13 Y Tyrosine
7 4 34 A Alanine
7 5 20 R Arginine
10 5 24 D Aspartic acid
2 0 4 C Cysteine
8 5 15 P Proline
0 1 25 N Asparagine
5 11 37 G Glycine
5 8 25 I Isoleucine
10 8 39 L Leucine
9 4 30 T Threonine
5 13 32 V Valine
8 12 35 E Glutamic acid
7 10 49 K Lycine
7 1 24 F Phenylalanine
5 1 26 S Serine 

8BDO_1|Chains A, D|Elongin-B|Homo sapiens (9606)
>1WNR_1|Chains A, B, C, D, E, F, G|10 kDa chaperonin|Thermus thermophilus (274)
>4CZT_1|Chains A, B, C, D|CBL-INTERACTING SERINE/THREONINE-PROTEIN KINASE 23|ARABIDOPSIS THALIANA (3702)
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 \(w=f(c)\)
8BDO , Knot 55 104 0.81 36 88 101 
MDVFLMIRRHKTTIFTDAKESSTVFELKRIVEGILKRPPDEQRLYKDDQLLDDGKTLGECGFTSQTARPQAPATVGLAFRADDTFEALCIEPFSSPPELPDVMK
1WNR , Knot 46 94 0.74 34 70 92 
MIKPLGDRVVVKRIEEEPKTKGGIVLPDTAKEKPQKGKVIAVGTGRVLENGQRVPLEVKEGDIVVFAKYGGTEIEIDGEEYVILSERDLLAVLQ
4CZT , Knot 191 464 0.84 40 226 439 
PGIHSGRTRVGKYELGRTLGEGTFAKVKFARNVENGDNVAIKVIDKEKVLKNKMIAQIKREISTMKLIKHPNVIRMFEVMASKTKIYFVLEFVTGGELFDKISSNGRLKEDEARKYFQQLINAVDYCHSRGVYHRDLKPENLLLDANGALKVSDFGLSALPQQVREDGLLHTTCGTPNYVAPEVINNKGYDGAKADLWSCGVILFVLMAGYLPFEDSNLTSLYKKIFKAEFTCPPWFSASAKKLIKRILDPNPATRITFAEVIENEWFKKGYKAPKFENADVSLDDVDAIFDDSGESKNLVVERREEGLKTPVTMNAFELISTSQGLNLGSLFEKQMGLVKRKTRFTSKSSANEIVTKIEAAAAPMGFDVKTNNYKMKLTGEKSGRKGQLAVATEVFQVAPSLYMVEMRKSGGDTLEFHKFYKNLTTGLKDIVWKTIDEEKEEGTDGGGTNGAMANRTIAKQST

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 \(c\).

\(|P_{f(8BDO_1)}(2) \setminus P_{f(1WNR_1)}(2)|=62\), \(|P_{f(1WNR_1)}(2) \setminus P_{f(8BDO_1)}(2)|=44\). 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 1:10111110000001100100000110100110111001100001000001100100110011000010101110111110100010110101100110110110
Pair \(Z_2\) Length of longest common subsequence
8BDO_1,1WNR_1 106 3
8BDO_1,4CZT_1 176 4
1WNR_1,4CZT_1 180 3

Newick tree

 
[
	4CZT_1:98.11,
	[
		8BDO_1:53,1WNR_1:53
	]:45.11
]

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 \((0.85)(0.8)(\frac{198 }{\log_{20} 198}-\frac{94}{\log_{20}94})=34.1\)
Status Protein1 Protein2 d d1/2
Query variables 8BDO_1 1WNR_1 41 38.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]