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Parikh vectors
5MCH_1 3UAM_1 3NHJ_1 Letter Amino acid
14 10 15 E Glutamic acid
21 7 14 K Lycine
9 7 6 W Tryptophan
28 11 16 V Valine
13 15 8 R Arginine
32 7 6 N Asparagine
24 16 11 P Proline
18 0 4 C Cysteine
48 16 18 G Glycine
13 8 10 I Isoleucine
22 14 20 L Leucine
19 7 14 F Phenylalanine
45 15 15 T Threonine
18 9 8 Y Tyrosine
29 21 18 A Alanine
12 10 12 Q Glutamine
11 3 4 M Methionine
26 17 14 S Serine
37 16 11 D Aspartic acid
6 7 6 H Histidine 

5MCH_1|Chains A, B|Cellobiohydrolase CHBI|Daphnia pulex (6669)
>3UAM_1|Chains A, B, C, D, E, F|Chitin binding domain|Burkholderia pseudomallei (320372)
>3NHJ_1|Chains A, B|Ribosyldihydronicotinamide dehydrogenase [quinone]|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 \(w=f(c)\)
5MCH , Knot 184 445 0.84 40 236 417 
QNVGTQAAEEPLNLPISVCTAPGNCQTEADAVVLDSNWRWAHTTTGYTNCYTGNLWDTTLCPTPETCTTNCAIDGVPLADWSGTYGGSVTGNKFNLKFVTVGPYSTNIGARTFLLDSTKTRYRMFQLLNREFTYDVDVSSLDCGLNGALYFVSMDADGGAAKYPTNKGGAKYGTGYCDAQCPHDVKWINGLANSKDWTPIPGDANSGKGYYGNCCAELDIWEANKQSQAFTTHPCTPNDQTRCEGVVCGDNDSGDRYNGMCDKDGCDFASYRMNDHTFYGPGSTFKLDSTKPFTVVSQFITTDGTDNGDFKEFRRFYVQNGVRIENSKVNFPGITAYDSITDEMCAATKGLFGDLDDHKNKGGMKQMGEAMRKGMALVMSIWDDHDVNMLWLDSNYPPTGNPSTPGVARGPCPTTSGVPSEVEVTQANAVVSFGNIKFGPIGSTV
3UAM , Knot 100 216 0.83 38 155 209 
GSMSTTDSRSATKSSSISPRHGRVITPESRAVYLYEAGRLDFGQVNELEGGKFFPATQSGLRDPDAPDDVANGMPPRDGEIASGGRTADARAQLNEPDSVAHWQKHAVRSGQSLQISWSYSMPHKTRRWTYWITKPGWDTQARLARAHFEPDPLKVYLNTYQPYWGPDADKELIPQGETIHEFNLPTRTGYHVLLAVWDVADTANAFYQVIDLNFA
3NHJ , Knot 108 230 0.85 40 169 224 
AGKKVLIVYAHQEPKSFNGSLKNVAVDELSRQGCTVTVSDLYAMNFEPRATDKDITGTLSNPEVFNYGVETHEAYKQRSLASDITDEQKKVREADLVIFQFPLYWFSVPAILKGWMDRVLCQGFAFDIPGFYDSGLLQGKLALLSVTTGGTAEMYTKTGVNGDSRYFLWPLQHGTLHFCGFKVLAPQISFAPEIASEEERKGMVAAWSQRLQTIWKEEPIPCTAHWHFGQ

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 \(c\).

\(|P_{f(5MCH_1)}(2) \setminus P_{f(3UAM_1)}(2)|=137\), \(|P_{f(3UAM_1)}(2) \setminus P_{f(5MCH_1)}(2)|=56\). 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 1:0011001100110111010011100000101111000101100001000000101100010101000000011011111010100110101001010110111000011100111000000001101100010001010010011011101101010111100100011100101000100100101101110000101111010010100100010101101000001100010010000000111010000100001100001001100010000101110010100001101100110001000101001001010011010000101111010001000101100111101000000111001101100111111011000010111100001101010011110110100011100101001011101101011111001
Pair \(Z_2\) Length of longest common subsequence
5MCH_1,3UAM_1 193 3
5MCH_1,3NHJ_1 183 4
3UAM_1,3NHJ_1 164 4

Newick tree

 
[
	5MCH_1:97.71,
	[
		3NHJ_1:82,3UAM_1:82
	]:15.71
]

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 \((0.85)(0.8)(\frac{661 }{\log_{20} 661}-\frac{216}{\log_{20}216})=125.\)
Status Protein1 Protein2 d d1/2
Query variables 5MCH_1 3UAM_1 161 119
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]