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Parikh vectors
5IDJ_1 8BSW_1 6JXX_1 Letter Amino acid
5 9 2 H Histidine
13 15 5 I Isoleucine
9 20 8 K Lycine
12 15 6 T Threonine
12 18 9 Q Glutamine
14 18 8 E Glutamic acid
28 25 5 L Leucine
8 20 3 S Serine
1 5 0 W Tryptophan
20 12 5 R Arginine
6 19 4 N Asparagine
3 4 4 M Methionine
10 16 4 F Phenylalanine
4 20 2 Y Tyrosine
0 2 1 C Cysteine
23 19 7 G Glycine
11 20 4 P Proline
21 22 6 V Valine
24 28 4 A Alanine
18 24 8 D Aspartic acid 

5IDJ_1|Chain A|Histidine kinase|Caulobacter vibrioides (155892)
>8BSW_1|Chain A|Isopenicillin N synthase|Aspergillus nidulans FGSC A4 (227321)
>6JXX_1|Chain A|Small ubiquitin-related modifier 2|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 \(w=f(c)\)
5IDJ , Knot 105 242 0.79 38 150 230 
LSQAQKMQAIGQLAGGVAHDFNNLLTAIQLRLDELLHRHPVGDPSYEGLNEIRQTGVRAADLVRKLLAFSRKQTVQREVLDLGELISEFEVLLRRLLREDVKLITDYGRDLPQVRADKSQLETAVMNLAVNARDAVRAAKGGGVVRIRTARLTRDEAIQLGFPAADGDTAFIEVSDDGPGIPPDVMGKIFDPFFTTKPVGEGTGLGLATVYGIVKQSDGWIHVHSRPNEGAAFRIFLPVYEA
8BSW , Knot 149 331 0.87 40 204 319 
MGSVSKANVPKIDVSPLFGDDQAAKMRVAQQIDAASRDTGFFYAVNHGINVQRLSQKTKEFHMSITPEEKWDLAIRAYNKEHQDQVRAGYYLSIPGKKAVESFCYLNPNFTPDHPRIQAKTPTHEVNVWPDETKHPGFQDFAEQYYWDVFGLSSALLKGYALALGKEENFFARHFKPDDTLASVVLIRYPYLDPYPEAAIKTAADGTKLSFEWHEDVSLITVLYQSNVQNLQVETAAGYQDIEADDTGYLINCGSYMAHLTNNYYKAPIQRVKWVNAERQSLPFFVNLGYDSVIDPFDPREPNGKSDREPLSYGDYLQNGLVSLINKNGQT
6JXX , Knot 51 95 0.81 38 82 93 
MADEKPKEGVKTENNDHINLKVAGQDGSVVQFKIKRHTPLSKLMKAYCERQGLSMRQIRFRFDGQPINETDTPAQLEMEDEDTIDVFQQQTGGVY

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 \(c\).

\(|P_{f(5IDJ_1)}(2) \setminus P_{f(8BSW_1)}(2)|=55\), \(|P_{f(8BSW_1)}(2) \setminus P_{f(5IDJ_1)}(2)|=109\). 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 1:10010010111011111100100110110101001100011101000110010001101101100111100000100011011011001011100110001011000100110101000010011101110100110110111110100101000011011111101001110100011111101110110111000111010111110101110000111010001001111011111001
Pair \(Z_2\) Length of longest common subsequence
5IDJ_1,8BSW_1 164 4
5IDJ_1,6JXX_1 158 3
8BSW_1,6JXX_1 178 3

Newick tree

 
[
	8BSW_1:87.65,
	[
		5IDJ_1:79,6JXX_1:79
	]:8.65
]

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 \((0.85)(0.8)(\frac{573 }{\log_{20} 573}-\frac{242}{\log_{20}242})=93.9\)
Status Protein1 Protein2 d d1/2
Query variables 5IDJ_1 8BSW_1 119 100.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]