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Parikh vectors
6OFV_1 8DSI_1 9CFO_1 Letter Amino acid
12 29 10 I Isoleucine
38 46 18 L Leucine
20 21 9 P Proline
12 7 0 W Tryptophan
9 26 6 N Asparagine
4 5 2 C Cysteine
5 15 4 Q Glutamine
24 36 19 G Glycine
40 25 24 A Alanine
6 3 5 M Methionine
33 14 7 R Arginine
26 19 4 H Histidine
7 47 16 K Lycine
23 38 19 V Valine
20 30 8 T Threonine
14 27 3 Y Tyrosine
24 25 9 D Aspartic acid
29 39 15 E Glutamic acid
14 18 3 F Phenylalanine
21 29 10 S Serine 

6OFV_1|Chain A|Epi-isozizaene synthase|Streptomyces coelicolor (strain ATCC BAA-471 / A3(2) / M145) (100226)
>8DSI_1|Chains A, B|Nicotinamide phosphoribosyltransferase|Homo sapiens (9606)
>9CFO_1|Chain A|Parkinson disease protein 7|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 \(w=f(c)\)
6OFV , Knot 157 381 0.81 40 204 353 
MGSSHHHHHHSSGLVPRGSHMHAFPHGTTATPTAIAVPPSLRLPVIEAAFPRQLHPYWPKLQETTRTWLLEKRLMPADKVEEYADGLCYTDLMAGYYLGAPDEVLQAIADYSAWQFVWDDRHDRDIVHGRAGAWRRLRGLLHTALDSPGDHLHHEDTLVAGFADSVRRLYAFLPATWNARFARHFHTVIEAYDREFHNRTRGIVPGVEEYLELRRLTFAHWIWTDLLEPSSGCELPDAVRKHPAYRRAALLSQEFAAWYNDLCSLPKEIAGDEVHNLGISLITHHSLTLEEAIGEVRRRVEECITEFLAVERDALRFADELADGTVRGKELSGAVRANVGNMRNWFSSVYWFHHESGRYMVDSWDDRSTPPYVNNEAAGEK
8DSI , Knot 202 499 0.83 40 239 470 
MNPAAEAEFNILLATDSYKVTHYKQYPPNTSKVYSYFECREKKTENSKLRKVKYEETVFYGLQYILNKYLKGKVVTKEKIQEAKDVYKEHFQDDVFNEKGWNYILEKYDGHLPIEIKAVPEGFVIPRGNVLFTVENTDPECYWLTNWIETILVQSWYPITVATNSREQKKILAKYLLETSGNLDGLEYKLHDFGYRGVSSQETAGIGASAHLVNFKGTDTVAGLALIKKYYGTKDPVPGYSVPAAEHSTITAWGKDHEKDAFEHIVTQFSSVPVSVVSDSYDIYNACEKIWGEDLRHLIVSRSTQAPLIIRPDSGNPLDTVLKVLEILGKKFPVTENSKGYKLLPPYLRVIQGDGVDINTLQEIVEGMKQKMWSIENIAFGSGGGLLQKLTRDLLNCSFKCSYVVTNGLGINVFKDPVADPNKRSKKGRLSLHRTPAGNFVTLEEGKGDLEEYGQDLLHTVFKNGKVTKSYSFDEIRKNAQLNIELEAAHHLEHHHHHH
9CFO , Knot 89 192 0.81 40 127 185 
GSHMASKRALVILAKGAEEMETVIPVDVMRRAGIKVTVAGLAGKDPVQCSRDVVICPDASLEDAKKEGPYDVVVLPGGNLGAQNLSESAAVKEILKEQENRKGLIAAIXAGPTALLAHEIGFGSKVTTHPLAKDKMMNGGHYTYSENRVEKDGLILTSRGPGTSFEFALAIVEALNGKEVAAQVKAPLVLKD

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 \(c\).

\(|P_{f(6OFV_1)}(2) \setminus P_{f(8DSI_1)}(2)|=65\), \(|P_{f(8DSI_1)}(2) \setminus P_{f(6OFV_1)}(2)|=100\). 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 1:110000000000111101001011101001010111111010111101111001010110100000011100011110010001011000011110011110011011100011011100000001101011110010111001100110010000011111100100101111101010110010011010000100000111111000101001011011100110100100110110001100011110001111000100110011100100111011000010100111010001000100111100011011001101010100101110101101001100101100001001100100000110100011100
Pair \(Z_2\) Length of longest common subsequence
6OFV_1,8DSI_1 165 6
6OFV_1,9CFO_1 175 4
8DSI_1,9CFO_1 172 6

Newick tree

 
[
	9CFO_1:88.12,
	[
		6OFV_1:82.5,8DSI_1:82.5
	]:5.62
]

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 \((0.85)(0.8)(\frac{880 }{\log_{20} 880}-\frac{381}{\log_{20}381})=133.\)
Status Protein1 Protein2 d d1/2
Query variables 6OFV_1 8DSI_1 170 147.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]