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Parikh vectors
2ECI_1 5JNF_1 3ZXK_1 Letter Amino acid
5 42 33 D Aspartic acid
9 39 18 E Glutamic acid
4 31 43 R Arginine
3 27 26 N Asparagine
7 42 22 I Isoleucine
2 15 9 M Methionine
4 22 23 F Phenylalanine
0 6 17 W Tryptophan
5 43 47 A Alanine
2 20 15 H Histidine
5 28 11 K Lycine
6 23 29 S Serine
1 30 47 T Threonine
2 26 29 Y Tyrosine
3 40 37 V Valine
3 19 14 Q Glutamine
6 43 56 G Glycine
7 57 33 L Leucine
6 27 31 P Proline
6 7 2 C Cysteine 

2ECI_1|Chain A|TNF receptor-associated factor 6|Homo sapiens (9606)
>5JNF_1|Chains A, B, C, D|Linear gramicidin synthase subunit A|Brevibacillus parabrevis (54914)
>3ZXK_1|Chains A, B|HIAXHD3|HUMICOLA INSOLENS (85995)
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 \(w=f(c)\)
2ECI , Knot 44 86 0.76 38 74 81 
GSSGSSGMEEIQGYDVEFDPPLESKYECPICLMALREAVQTPCGHRFCKACIIKSIRDAGHKCPVDNEILLENQLFPDNFAKREIL
5JNF , Knot 238 587 0.86 40 282 550 
GAMGRILFLTTFMSKGNKVVRYLESLHHEVVICQEKVHAQSANLQEIDWIVSYAYGYILDKEIVSRFRGRIINLHPSLLPWNKGRDPVFWSVWDETPKGVTIHLIDEHVDTGDILVQEEIAFADEDTLLDCYNKANQAIEELFIREWENIVHGRIAPYRQTAGGTLHFKADRDFYKNLNMTTVRELLALKRLCAEPKRGEKPIDKTFHQLFEQQVEMTPDHVAVVDRGQSLTYKQLNERANQLAHHLRGKGVKPDDQVAIMLDKSLDMIVSILAVMKAGGAYVPIDPDYPGERIAYMLADSSAAILLTNALHEEKANGACDIIDVHDPDSYSENTNNLPHVNRPDDLVYVMYTSGSTGLAKGVMIEHHNLVNFCEWYRPYFGVTPADKALVYSSFSFDGSALDIFTHLLAGAALHIVPSERKYDLDALNDYCNQEGITISYLPTGAAEQFMQMDNQSFRVVITGGDVLKKIERNGTYKLYNGYGPTECTIMVTMFEVDKPYANIPIGKPIDRTRILILDEALALQPIGVAGELFIVGEGLGRGYLNRPELTAEKFIVHPQTGERMYRTGDRARFLPDGNIEFLGRLD
3ZXK , Knot 221 542 0.85 40 253 503 
MPRQASTFTNPVLWEDHPALEVFRVGSVFYYSSSTFAYSPGAPVLKSYDLVHWTPVTHSVPRLNFGSNYDLPSGTPGAYVKGIWASTLRYRRSNDRFYWYGCVEGRTYLWTSPGGNALANNGEVPPSAWNWQHTATIDNCYYDAGLLIDDDDTMYIAYGNPTINVAQLSPDGTRQVRVQQRVYAHPQGQTVEGARMYKIRGNYYILVTRPADAEYVLRSTTGSPFGPYEARTLVSRIQGPLANAGFAHQGGIVDAPDGTWHYVAFMDAYPGGRIPVVAPLRWTADGWPEVVTDSQGRWGTSYPIPVRGAKNATEGLASTDLDEFRGTRFSEHWEWNHNPDTSKFTLLGGNEGGLILRTATVTGDLFAARNTLTRRIAGPKASGIFRLDVRGMRDGDRAGAVLFRDRAAYIGVWKQGNEARIVMVDDLRLNEDGWRTASTGRVAANGPVIDTNAQQDIWLRIDADITPAFGTNTERTTTFYYSIDGGRTYTRLGPAFAMTNSWRYFTGYRFGVFNFSTKSLGGEVKVKGFKMNMILEHHHHHH

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 \(c\).

\(|P_{f(2ECI_1)}(2) \setminus P_{f(5JNF_1)}(2)|=21\), \(|P_{f(5JNF_1)}(2) \setminus P_{f(2ECI_1)}(2)|=229\). 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 1:10010011001010010101110000001101111001100101001001011001001100011000111000111001100011
Pair \(Z_2\) Length of longest common subsequence
2ECI_1,5JNF_1 250 3
2ECI_1,3ZXK_1 239 3
5JNF_1,3ZXK_1 147 4

Newick tree

 
[
	2ECI_1:13.67,
	[
		3ZXK_1:73.5,5JNF_1:73.5
	]:61.17
]

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 \((0.85)(0.8)(\frac{673 }{\log_{20} 673}-\frac{86}{\log_{20}86})=171.\)
Status Protein1 Protein2 d d1/2
Query variables 2ECI_1 5JNF_1 221 124.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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