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Parikh vectors
6MTZ_1 2CWJ_1 5MXU_1 Letter Amino acid
22 7 33 S Serine
17 2 18 Q Glutamine
19 6 22 F Phenylalanine
16 0 15 H Histidine
22 6 35 I Isoleucine
23 6 30 K Lycine
18 7 20 Y Tyrosine
17 13 42 V Valine
8 2 4 C Cysteine
19 10 47 G Glycine
25 12 27 E Glutamic acid
10 5 23 T Threonine
19 13 31 A Alanine
12 7 27 R Arginine
37 7 51 L Leucine
9 3 21 M Methionine
5 9 36 P Proline
3 0 14 W Tryptophan
30 4 28 N Asparagine
24 4 36 D Aspartic acid 

6MTZ_1|Chains A, B|HepC.19480.a.B1|Helicobacter pylori (strain G27) (563041)
>2CWJ_1|Chain A|putative endonuclease|Aeropyrum pernix (272557)
>5MXU_1|Chains A, B|Vanillyl-alcohol oxidase|Penicillium simplicissimum (69488)
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 \(w=f(c)\)
6MTZ , Knot 151 355 0.83 40 208 337 
MAHHHHHHMQVYHLSHIDLDGYACQLVSKQFFKNIQCYNANYGREVSARIYEILNAIAQSKESEFLILVSDLNLNLNEAEYLQDKIQEHRLQNKNIQIQLLDHHISGKEVAESFHWYFLDTNRCATKIVYEFLKKHYAILEPKNTTWLEPLVEMVNSVDIWDTQGYGFELGKVCMRMITQSSELNRFMFDDENRDYKLKLLEEVKNYLFLENAPVAYDNDLFRLKKIALGGDPDTETMDNISSNAQTHLLSLKKHDCSVYYQDKKGFLSYSMGGISVLANLFLTQNPDFDFYIDVNAKGNVSLRANGNCDVCELSQMCFNGGGHRNASGGKIDGFRESFNYRDIKEQIEEIFNNA
2CWJ , Knot 62 123 0.80 36 98 121 
METAPKPVGPYSQAVESGCFMFVSGQIPINPETGALEEGGFKESAKRALDNLKAIVEGAGYSMDDIVKVTVYITDISRFSEFNEVYREYFNRPYPARAVVGVAALPLGAPLEVEAVLYTCRRK
5MXU , Knot 228 560 0.86 40 266 523 
MSKTQEFRPLTLPPKLSLSDFNEFIQDIIRIVGSENVEVISSKDQIVDGSYMKPTHTHDPHHVMDQDYFLASAIVAPRNVADVQSIVGLANKFSFPLWPISIGRNSGYGGAAPRVSGSVVLDMGKNMNRVLEVNVEGAYCVVEPGVTYHDLHNYLEANNLRDKLWLDVPDLGGGSVLGNAVERGVGYTPYGDHWMMHSGMEVVLANGELLRTGMGALPDPKRPETMGLKPEDQPWSKIAHLFPYGFGPYIDGLFSQSNMGIVTKIGIWLMPNPRGYQSYLITLPKDGDLKQAVDIIRPLRLGMALQNVPTIRHILLDAAVLGDKRSYSSRTEPLSDEELDKIAKQLNLGRWNFYGALYGPEPIRRVLWETIKDAFSAIPGVKFYFPEDTPENSVLRVRDKTMQGIPTYDELKWIDWLPNGAHLFFSPIAKVSGEDAMMQYAVTKKRCQEAGLDFIGTFTVGMREMHHIVCIVFNKKDLIQKRKVQWLMRTLIDDCAANGWGEFRTHLAFMDQIMETYNWNNSSFLRFNEVLKNAVDPNGIIAPGKSGVWPSQYSHVTWKL

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 \(c\).

\(|P_{f(6MTZ_1)}(2) \setminus P_{f(2CWJ_1)}(2)|=149\), \(|P_{f(2CWJ_1)}(2) \setminus P_{f(6MTZ_1)}(2)|=39\). 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 1:1100000010100100101010100110001100100001001001010100110111000000111110010101001001000100001000010101100010100110010101100000100110011000011101000011011101100101100010110110101011000001001110000000010110010001110011110000110100111110100001001000100011010000001000000111000111101110111000101010101010101010101000100100101011100010110101100010000100010011001
Pair \(Z_2\) Length of longest common subsequence
6MTZ_1,2CWJ_1 188 3
6MTZ_1,5MXU_1 174 4
2CWJ_1,5MXU_1 216 4

Newick tree

 
[
	2CWJ_1:10.56,
	[
		6MTZ_1:87,5MXU_1:87
	]:18.56
]

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 \((0.85)(0.8)(\frac{478 }{\log_{20} 478}-\frac{123}{\log_{20}123})=105.\)
Status Protein1 Protein2 d d1/2
Query variables 6MTZ_1 2CWJ_1 136 91.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]