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Parikh vectors
3KCF_1 9HLC_1 8FLN_1 Letter Amino acid
11 11 14 M Methionine
6 20 5 C Cysteine
10 24 6 H Histidine
31 74 27 L Leucine
11 23 10 Q Glutamine
24 48 18 G Glycine
11 32 8 P Proline
23 49 22 S Serine
8 7 6 W Tryptophan
20 40 18 V Valine
24 34 11 A Alanine
25 35 13 I Isoleucine
9 36 12 F Phenylalanine
24 24 25 E Glutamic acid
20 21 19 K Lycine
19 32 10 T Threonine
11 28 15 Y Tyrosine
25 32 13 R Arginine
11 20 7 N Asparagine
19 38 14 D Aspartic acid 

3KCF_1|Chains A, B, C, D, E|TGF-beta receptor type-1|Homo sapiens (9606)
>9HLC_1|Chains A, B, C, D|Mucolipin-1|Homo sapiens (9606)
>8FLN_1|Chain A|Tyrosine-protein kinase BTK|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 \(w=f(c)\)
3KCF , Knot 146 342 0.83 40 213 326 
EDPSLDRPFISEGTTLKDLIYDMTTSGSGSGLPLLVQRTIARTIVLQESIGKGRFGEVWRGKWRGEEVAVKIFSSREERSWFREAEIYQTVMLRHENILGFIAADNKDNGTWTQLWLVSDYHEHGSLFDYLNRYTVTVEGMIKLALSTASGLAHLHMEIVGTQGKPAIAHRDLKSKNILVKKNGTCCIADLGLAVRHDSATDTIDIAPNHRVGTKRYMAPEVLDDSINMKHFESFKRADIYAMGLVFWEIARRCSIGGIHEDYQLPYYDLVPSDPSVEEMRKVVCEQKLRPNIPNRWQSCEALRVMAKIMRECWYANGAARLTALRIKKTLSQLSQQEGIKM
9HLC , Knot 240 628 0.82 40 277 555 
MHHHHHHHHGGSDYKDHDGDYKDHDIDYKDDDDKGGSGGSENLYFQGPGTAPAGPRGSETERLLTPNPGYGTQAGPSPAPPTPPEEEDLRRRLKYFFMSPCDKFRAKGRKPCKLMLQVVKILVVTVQLILFGLSNQLAVTFREENTIAFRHLFLLGYSDGADDTFAAYTREQLYQAIFHAVDQYLALPDVSLGRYAYVRGGGDPWTNGSGLALCQRYYHRGHVDPANDTFDIDPMVVTDCIQVDPPERPPPPPSDDLTLLESSSSYKNLTLKFHKLVNVTIHFRLKTINLQSLINNEIPDCYTFSVLITFDNKAHSGRIPISLETQAHIQECKHPSVFQHGDNSFRLLFDVVVILTCSLSFLLCARSLLRGFLLQNEFVGFMWRQRGRVISLWERLEFVNGWYILLVTSDVLTISGTIMKIGIEAKNLASYDVCSILLGTSTLLVWVGVIRYLTFFHNYNILIATLRVALPSVMRFCCCVAVIYLGYCFCGWIVLGPYHVKFRSLSMVSECLFSLINGDDMFVTFAAMQAQQGRSSLVWLFSQLYLYSFISLFIYMVLSLFIALITGAYDTIKHPGGAGAEESELQAYIAQCQDSPTSGKFRRGSGSACSLLCCCGRDPSEEHSLLVN
8FLN , Knot 124 273 0.85 40 189 263 
GMGLGYGSWEIDPKDLTFLKELGTGQFGVVKYGKWRGQYDVAIKMIKEGSMSEDEFIEEAKVMMNLSHEKLVQLYGVCTKQRPIFIITEYMANGSLLNYLREMRHRFQTQQLLEMCKDVCEAMEYLESKQFLHRDLAARNCLVNDQGVVKVSDFGLSRYVLDDEYTSSVGSKFPVRWSPPEVLMYSKFSSKSDIWAFGVLMWEIYSLGKMPYERFTNSETAEHIAQGLRLYRPHLASEKVYTIMYSCWHEKADERPTFKILLSNILDVMDEES

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 \(c\).

\(|P_{f(3KCF_1)}(2) \setminus P_{f(9HLC_1)}(2)|=48\), \(|P_{f(9HLC_1)}(2) \setminus P_{f(3KCF_1)}(2)|=112\). 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 1:001010011100100100110010001010111111000110011100011010110110101010011101100000001100101000111000011111110000010100111100000010110010000101011101110010111010101110010111100010000111000100011011111000010001011100011000011101100010100100100101011111110110000111100000110001110010100100110000101011001000011011101100010101110101101000100100001101
Pair \(Z_2\) Length of longest common subsequence
3KCF_1,9HLC_1 160 4
3KCF_1,8FLN_1 172 5
9HLC_1,8FLN_1 202 4

Newick tree

 
[
	8FLN_1:97.96,
	[
		3KCF_1:80,9HLC_1:80
	]:17.96
]

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 \((0.85)(0.8)(\frac{970 }{\log_{20} 970}-\frac{342}{\log_{20}342})=167.\)
Status Protein1 Protein2 d d1/2
Query variables 3KCF_1 9HLC_1 209 161.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]