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Parikh vectors
4YZK_1 5WIN_1 9HLB_1 Letter Amino acid
50 15 34 A Alanine
25 21 24 E Glutamic acid
10 19 35 I Isoleucine
28 33 74 L Leucine
10 4 11 M Methionine
41 14 48 G Glycine
12 6 28 Y Tyrosine
3 19 20 N Asparagine
13 14 36 F Phenylalanine
20 11 32 T Threonine
6 4 7 W Tryptophan
35 12 32 R Arginine
31 16 38 D Aspartic acid
4 7 20 C Cysteine
6 8 23 Q Glutamine
14 9 24 H Histidine
7 22 21 K Lycine
35 14 32 P Proline
22 15 49 S Serine
27 14 40 V Valine 

4YZK_1|Chain A|Tryptophan dimethylallyltransferase|Streptomyces blastmyceticus (68180)
>5WIN_1|Chain A|Tyrosine-protein kinase JAK2|Homo sapiens (9606)
>9HLB_1|Chains A, B, C, D|Mucolipin-1|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 \(w=f(c)\)
4YZK , Knot 162 399 0.81 40 194 371 
MESAGPGTGPQPPRTSGDFTPDTGVIAEMTGRPMRFDSDRYRPTDTYAEVACDKVCRAYEGLGADGGDRESLLAFLRDLTDPWGELPVGTPPEDACWVSIDGMPLETSVAWAGRKAGVRLSLESPRGPAKRRMEDGMALTRRLAGRPGVSVDPCLRVEDLFTDDDPQGYFTIAHAVAWTPGGHPRYKIFLNPAVRGREQAAARTEEAMIRLGLEQPWRALTEHLGGAYGPEHEPAALAMDLVPGDDFRVQVYLAHSGVSAEAIDAKSAVAADHVPGSFARALRGINGADDTPEWKRKPPVTAFSFGPGRAVPGATLYVPMIPVHGSDAAARDRVAAFLRSEGMDAVGYEAVLDAISDRSLPESHTQNFISYRGGDSPRFSVYLAPGVYREALEHHHHHH
5WIN , Knot 124 277 0.84 40 185 271 
VFHKIRNEDLIFNESLGQGTFTKIFKGVRREVGDYGQLHETEVLLKVLDKAHRNYSESFFEAASMMSKLSHKHLVLNYGVCVCGDENILVQEFVKFGSLDTYLKKNKNCINILWKLEVAKQLAAAMHFLEENTLIHGNVCAKNILLIREEDRKTGNPPFIKLSDPGISITVLPKDILQERIPWVPPECIENPKNLNLATDKWSFGTTLWEICSGGDKPLSALDSQRKLQFYEDRHQLPAPKAAELANLINNCMDYEPDHRPSFRAIIRDLNSLFTPD
9HLB , Knot 240 628 0.82 40 277 555 
MHHHHHHHHGGSDYKDHDGDYKDHDIDYKDDDDKGGSGGSENLYFQGPGTAPAGPRGSETERLLTPNPGYGTQAGPSPAPPTPPEEEDLRRRLKYFFMSPCDKFRAKGRKPCKLMLQVVKILVVTVQLILFGLSNQLAVTFREENTIAFRHLFLLGYSDGADDTFAAYTREQLYQAIFHAVDQYLALPDVSLGRYAYVRGGGDPWTNGSGLALCQRYYHRGHVDPANDTFDIDPMVVTDCIQVDPPERPPPPPSDDLTLLESSSSYKNLTLKFHKLVNVTIHFRLKTINLQSLINNEIPDCYTFSVLITFDNKAHSGRIPISLETQAHIQECKHPSVFQHGDNSFRLLFDVVVILTCSLSFLLCARSLLRGFLLQNEFVGFMWRQRGRVISLWERLEFVNGWYILLVTSDVLTISGTIMKIGIEAKNLASYDVCSILLGTSTLLVWVGVIRYLTFFHNYNILIATLRVALPSVMRFCCCVAVIYLGYCFCGWIVLGPYHVKFRSLSMVSECLFSLINGDDMFVTFAAMQAQQGRSSLVWLFSQLYLYSFISLFIYMVLSLFIALITGAYDTIKHPGGAGAEESELQAYIAQCQDSPTSGKFRRGSGSACSLLCCCGRDPSEEHSLLVN

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 \(c\).

\(|P_{f(4YZK_1)}(2) \setminus P_{f(5WIN_1)}(2)|=95\), \(|P_{f(5WIN_1)}(2) \setminus P_{f(4YZK_1)}(2)|=86\). 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 1:100111101101100010101001111010101101000000100001011000100100111101100001111100100111011110110010110101111000111110011101010010111000100111100011101110101010100110000101010110111101110100011101110100011100001110111001101100011110110001111110111100101010110011010110100111100111011011011011000101000111011011110111110101111110100111000111110001101110011101100001100000011000110010101011111000110000000
Pair \(Z_2\) Length of longest common subsequence
4YZK_1,5WIN_1 181 4
4YZK_1,9HLB_1 173 6
5WIN_1,9HLB_1 172 4

Newick tree

 
[
	4YZK_1:89.34,
	[
		9HLB_1:86,5WIN_1:86
	]:3.34
]

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 \((0.85)(0.8)(\frac{676 }{\log_{20} 676}-\frac{277}{\log_{20}277})=110.\)
Status Protein1 Protein2 d d1/2
Query variables 4YZK_1 5WIN_1 136 118.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]