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Parikh vectors
1HON_1 3WUY_1 3VEC_1 Letter Amino acid
10 18 7 Q Glutamine
12 11 16 F Phenylalanine
32 37 36 A Alanine
13 13 7 N Asparagine
29 23 20 E Glutamic acid
28 12 16 R Arginine
31 12 35 D Aspartic acid
4 5 3 C Cysteine
44 33 35 L Leucine
22 16 9 K Lycine
17 20 24 P Proline
16 15 29 S Serine
33 22 16 T Threonine
46 27 34 G Glycine
6 13 9 H Histidine
20 20 11 I Isoleucine
9 13 5 M Methionine
4 4 2 W Tryptophan
16 11 8 Y Tyrosine
39 24 31 V Valine 

1HON_1|Chains A, B|ADENYLOSUCCINATE SYNTHETASE|Escherichia coli (562)
>3WUY_1|Chains A, B|Nitrilase|Synechocystis (1111708)
>3VEC_1|Chains A, B, C|DypB|Rhodococcus jostii (101510)
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 \(w=f(c)\)
1HON , Knot 177 431 0.83 40 218 408 
GNNVVVLGTQWGDEGKGKIVDLLTERAKYVVRYQGGHNAGHTLVINGEKTVLHLIPSGILRENVTSIIGNGVVLSPAALMKEMKELEDRGIPVRERLLLSEACPLILDYHVALDNAREKARGAKAIGTTGRGIGPAYEDKVARRGLRVGDLFDKETFAEKLKEVMEYHNFQLVNYYKAEAVDYQKVLDDTMAVADILTSMVVDVSDLLDQARQRGDFVMFEGAQGTLLDIDHGTYPYVTSSNTTAGGVATGSGLGPRYVDYVLGILKAYSTRVGAGPFPTELFDETGEFLCKQGNEFGATTGRRRRTGWLDTVAVRRAVQLNSLSGFCLTKLDVLDGLKEVKLCVAYRMPDGREVTTTPLAADDWKGVEPIYETMPGWSESTFGVKDRSGLPQAALNYIKRIEELTGVPIDIISTGPDRTETMILRDPFDA
3WUY , Knot 149 349 0.83 40 208 338 
GSHMLGKIMLNYTKNIRAAAAQISPVLFSQQGTMEKVLDAIANAAKKGVELIVFPETFVPYYPYFSFVEPPVLMGKSHLKLYQEAVTVPGKVTQAIAQAAKTHGMVVVLGVNEREEGSLYNTQLIFDADGALVLKRRKITPTYHERMVWGQGDGAGLRTVDTTVGRLGALACWEHYNPLARYALMAQHEQIHCGQFPGSMVGQIFADQMEVTMRHHALESGCFVINATGWLTAEQKLQITTDEKMHQALSGGCYTAIISPEGKHLCEPIAEGEGLAIADLDFSLIAKRKRMMDSVGHYARPDLLQLTLNNQPWSALEANPVTPNAIPAVSDPELTETIEALPNNPIFSH
3VEC , Knot 147 353 0.81 40 190 328 
GSHMPGPVARLAPQAVLTPPSAASLFLVLVAGDSDDDRATVCDVISGIDGPLKAVGFRELAGSLSCVVGVGAQFWDRVSASSKPAHLHPFVPLSGPVHSAPSTPGDLLFHIKAARKDLCFELGRQIVSALGSAATVVDEVHGFRYFDSRDLLGFVAGTENPTDDDAADSALIGDEDPDFRGGSYVIVQKYLHDMSAWNTLSTEEQERVIGRTKLENVELDDDAQPSNSHVTLNTIVDDDGVEHDILRDNMAFGSLGEAEYGTYFIGYAKDPAVTELMLRRMFLGEPPGNYDRVLDFSTAATGTLFFVPSRDVLESLGDEPAGAESAPEDPVEPAAAGPYDLSLKIGGLKGVSQ

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 \(c\).

\(|P_{f(1HON_1)}(2) \setminus P_{f(3WUY_1)}(2)|=81\), \(|P_{f(3WUY_1)}(2) \setminus P_{f(1HON_1)}(2)|=71\). 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 1:10011111001100101011011000100110001100110011101000110111011100010011101111011111001001000111100011100101111000111001000101101110010111110000110011011011000011001001100001011000010110000110001111011001110100110010001011110110101101001001010000001111101011110010011111010000111111100110001011000100111001000001110011100110100101101001011011001010110011010010001111001011011000111100001110000111011100100100101111011001100000111001101
Pair \(Z_2\) Length of longest common subsequence
1HON_1,3WUY_1 152 5
1HON_1,3VEC_1 134 4
3WUY_1,3VEC_1 164 4

Newick tree

 
[
	3WUY_1:82.68,
	[
		1HON_1:67,3VEC_1:67
	]:15.68
]

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 \((0.85)(0.8)(\frac{780 }{\log_{20} 780}-\frac{349}{\log_{20}349})=117.\)
Status Protein1 Protein2 d d1/2
Query variables 1HON_1 3WUY_1 147 134.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]