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Parikh vectors
5NMY_1 6SWD_1 3NAN_1 Letter Amino acid
16 0 53 K Lycine
12 0 45 P Proline
21 0 62 T Threonine
15 0 82 V Valine
6 0 48 R Arginine
16 24 36 N Asparagine
11 0 93 L Leucine
16 0 22 Y Tyrosine
0 276 24 C Cysteine
36 393 68 G Glycine
20 0 57 S Serine
10 0 23 Q Glutamine
7 0 37 F Phenylalanine
8 0 13 W Tryptophan
9 0 12 H Histidine
13 0 72 I Isoleucine
7 0 33 M Methionine
10 199 86 A Alanine
7 0 50 D Aspartic acid
5 0 78 E Glutamic acid 

5NMY_1|Chain A|Lysostaphin|Staphylococcus simulans (1286)
>6SWD_1|Chain A[auth 2]|16S ribosomal RNA|Pyrococcus abyssi GE5 (272844)
>3NAN_1|Chain A|SERCA1a|Oryctolagus cuniculus (9986)
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 \(w=f(c)\)
5NMY , Knot 112 245 0.83 38 157 232 
GSHEHSAQWLNNYKKGYGYGPYPLGINGGMHYGVDFFMNIGTPVKAISSGKIVEAGWSNYGGGNQIGLIENDGVHRQWYMHLSKYNVKVGDYVKAGQIIGWSGSTGYSTAPHLHFQRMVNSFSNSTAQDPMPFLKSAGYGKAGGTVTPTPNTGWKTNKYGTLYKSESASFTPNTDIITRTTGPFRSMPQSGVLKAGQTIHYDEVMKQDGHVWVGYTGNSGQRIYLPVRTWNKSTNTLGVLWGTIK
6SWD , Knot 199 1044 0.44 12 20 76 
AUUCNGGUUGAUCCUGCCGGAGGCCACUGCUAUGGGGGUCNGACUAAGCCAUGCGAGUCAAGGGGGCGUCCCUUCUGGGACGCCACCGGCGGACGGCUCAGUAACACGUCGGUAACCUACCCUCGGGAGGGGGAUAACCCCGGGAAACUGGGGCUAAUCCCCCAUAGGCCUGGGGUACUGGAAGGUCCCCAGGCCGAAAGGGAGCCGUAAGGCUCCGCCCGAGGAUGGGCCGGCGGCCGAUUAGGUAGUUGGUGGGGUAACGGCCCACCAAGCNGAAGAUCGGUACGGGCNGUGAGAGCGGGAGCCNGGAGAUGGACACUGAGACACGGGUCCAGGCCCUACGGGGCGCAGCAGGCGCGAAACCUCNGCAAUGCGGGAAACNGCGACGGGGGGACCCCCAGUGCCGUGCCUCUGGCACGGCUUUUCCGGAGUGUAAAAAGCUCCGGGAAUAAGGGCUGGGCAAGGCNGGUGGCAGCCGCCGCGGUAAUACCGGCGGCCNGAGUGGUGGCCACUAUUAUUGGGCCUAAAGCGGCNGUAGCCGGGCCCGUAAGUCCCUGGCGAAAUCCCACGGCUCAACNGUGGGGCUCGCUGGGGAUACUGCGGGCCUUGGGACNGGGAGAGGCNGGGGGUACCCCNGGGGUAGGGGUGAAAUCCUAUAAUCCCGGGGGGACCGCCAGUGGCGAAGGCGCCNGGCUGGAACGGGUCNGACGGUGAGGGCNGAAGGCCAGGGGAGCGAACNGGAUUAGAUACCCGGGUAGUCCUGGCUGUAAAGGAUGCGGGCUAGGUGUCGGGCGAGCUUCGAGCUCGCCCGGUGCNGUAGGGAAGCNGUUAAGCCNGCNGCCUGGGGAGUACGGCNGCAAGGCUGAAACUUAAAGGAAUUGGCGGGGGAGCCUGCUCCUUGCACACACCGCCXGUCACUCCACCCGAGCGGGGCCUAGGUGAGGCCCGAUCUCCUUCGGGAGGUCGGGUCGAGCCUAGGCUCCGUGAGGGGGGAGAAGUCGUAACAAGGUAGCNGUAGGGGAACCUACGGCUCGAUCACCUCCU
3NAN , Knot 367 994 0.85 40 320 881 
MEAAHSKSTEECLAYFGVSETTGLTPDQVKRHLEKYGHNELPAEEGKSLWELVIEQFEDLLVRILLLAACISFVLAWFEEGEETITAFVEPFVILLILIANAIVGVWQERNAENAIEALKEYEPEMGKVYRADRKSVQRIKARDIVPGDIVEVAVGDKVPADIRILSIKSTTLRVDQSILTGESVSVIKHTEPVPDPRAVNQDKKNMLFSGTNIAAGKALGIVATTGVSTEIGKIRDQMAATEQDKTPLQQKLDEFGEQLSKVISLICVAVWLINIGHFNDPVHGGSWIRGAIYYFKIAVALAVAAIPEGLPAVITTCLALGTRRMAKKNAIVRSLPSVETLGCTSVICSDKTGTLTTNQMSVCKMFIIDKVDGDFCSLNEFSITGSTYAPEGEVLKNDKPIRSGQFDGLVELATICALCNDSSLDFNETKGVYEKVGEATETALTTLVEKMNVFNTEVRNLSKVERANACNSVIRQLMKKEFTLEFSRDRKSMSVYCSPAKSSRAAVGNKMFVKGAPEGVIDRCNYVRVGTTRVPMTGPVKEKILSVIKEWGTGRDTLRCLALATRDTPPKREEMVLDDSSRFMEYETDLTFVGVVGMLDPPRKEVMGSIQLCRDAGIRVIMITGDNKGTAIAICRRIGIFGENEEVADRAYTGREFDDLPLAEQREACRRACCFARVEPSHKSKIVEYLQSYDEITAMTGDGVNDAPALKKAEIGIAMGSGTAVAKTASEMVLADDNFSTIVAAVEEGRAIYNNMKQFIRYLISSNVGEVVCIFLTAALGLPEALIPVQLLWVNLVTDGLPATALGFNPPDLDIMDRPPRSPKEPLISGWLFFRYMAIGGYVGAATVGAAAWWFMYAEDGPGVTYHQLTHFMQCTEDHPHFEGLDCEIFEAPEPMTMALSVLVTIEMCNALNSLSENQSLMRMPPWVNIWLLGSICLSMSLHFLILYVDPLPMIFKLKALDLTQWLMVLKISLPVIGLDEILKFIARNYLEG

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 \(c\).

\(|P_{f(5NMY_1)}(2) \setminus P_{f(6SWD_1)}(2)|=154\), \(|P_{f(6SWD_1)}(2) \setminus P_{f(5NMY_1)}(2)|=17\). 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 1:10000010110000010101101111011100110111011011011001011011100011100111100011000101010000101100101101111010010001101010011001000010011111001101011101010100110000010100000101010001100001110011001110110010000110001011110010010010111001000000111111010
Pair \(Z_2\) Length of longest common subsequence
5NMY_1,6SWD_1 171 3
5NMY_1,3NAN_1 217 4
6SWD_1,3NAN_1 322 4

Newick tree

 
[
	3NAN_1:15.63,
	[
		5NMY_1:85.5,6SWD_1:85.5
	]:65.13
]

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 \((0.85)(0.8)(\frac{1289 }{\log_{20} 1289}-\frac{245}{\log_{20}245})=275.\)
Status Protein1 Protein2 d d1/2
Query variables 5NMY_1 6SWD_1 195 151.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]