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Parikh vectors
2ZRA_1 5DXT_1 5HWX_1 Letter Amino acid
37 45 31 G Glycine
17 35 6 T Threonine
5 37 10 Y Tyrosine
28 50 29 V Valine
37 42 28 A Alanine
21 55 9 D Aspartic acid
13 44 3 Q Glutamine
13 42 6 P Proline
0 29 2 C Cysteine
22 68 10 K Lycine
11 38 15 F Phenylalanine
25 64 39 I Isoleucine
30 110 46 L Leucine
9 36 9 M Methionine
4 27 3 H Histidine
20 53 22 S Serine
1 18 4 W Tryptophan
18 51 6 R Arginine
10 55 15 N Asparagine
28 63 8 E Glutamic acid 

2ZRA_1|Chain A|Protein recA|Mycobacterium smegmatis str. MC2 155 (246196)
>5DXT_1|Chain A|Phosphatidylinositol 4,5-bisphosphate 3-kinase catalytic subunit alpha isoform|Homo sapiens (9606)
>5HWX_1|Chain A|sodium,calcium exchanger|Methanocaldococcus jannaschii (strain ATCC 43067 / DSM 2661 / JAL-1 / JCM 10045 / NBRC 100440) (243232)
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 \(w=f(c)\)
2ZRA , Knot 150 349 0.84 38 186 329 
MAQQAPDREKALELAMAQIDKNFGKGSVMRLGEEVRQPISVIPTGSISLDVALGIGGLPRGRVIEIYGPESSGKTTVALHAVANAQAAGGIAAFIDAEHALDPEYAKKLGVDTDSLLVSQPDTGEQALEIADMLVRSGALDIIVIDSVAALVPRAEIEGEMGDSHVGLQARLMSQALRKMTGALNNSGTTAIFINELREKIGVMFGSPETTTGGKALKFYASVRLDVRRIETLKDGTDAVGNRTRVKVVKNKVSPPFKQAEFDILYGQGISREGSLIDMGVEHGFIRKSGSWFTYEGEQLGQGKENARKFLLENTDVANEIEKKIKEKLGIGAVVTAEADDVLPAPVDF
5DXT , Knot 365 962 0.86 40 338 878 
NREEKILNREIGFAIGMPVCEFDMVKDPEVQDFRRNILNVCKEAVDLRDLNSPHSRAMYVYPPNVESSPELPKHIYNKLDKGQIIVVIWVIVSPNNDKQKYTLKINHDCVPEQVIAEAIRKKTRSMLLSSEQLKLCVLEYQGKYILKVCGCDEYFLEKYPLSQYKYIRSCIMLGRMPNLMLMAKESLYSQLPMDCFTMPSYSRRISTATPYMNGETSTKSLWVINSALRIKILCATYVNVNIRDIDKIYVRTGIYHGGEPLCDNVNTQRVPCSNPRWNEWLNYDIYIPDLPRAARLCLSICSVKGRKGAKEEHCPLAWGNINLFDYTDTLVSGKMALNLWPVPHGLEDLLNPIGVTGSNPNKETPCLELEFDWFSSVVKFPDMSVIEEHANWSVSREAGFSYSHAGLSNRLARDNELRENDKEQLKAISTRDPLSEITEQEKDFLWSHRHYCVTIPEILPKLLLSVKWNSRDEVAQMYCLVKDWPPIKPEQAMELLDCNYPDPMVRGFAVRCLEKYLTDDKLSQYLIQLVQVLKYEQYLDNLLVRFLLKKALTNQRIGHFFFWHLKSEMHNKTVSQRFGLLLESYCRACGMYLKHLNRQVEAMEKLINLTDILKQEKKDETQKVQMKFLVEQMRRPDFMDALQGFLSPLNPAHQLGNLRLEECRIMSSAKRPLWLNWENPDIMSELLFQNNEIIFKNGDDLRQDMLTLQIIRIMENIWQNQGLDLRMLPYGCLSIGDCVGLIEVVRNSHTIMQIQCKGGLKGALQFNSHTLHQWLKDKNKGEIYDAAIDLFTRSCAGYCVATFILGIGDRHNSNIMVKDDGQLFHIDFGHFLDHKKKKFGYKRERVPFVLTQDFLIVISKGAQECTKTREFERFQEMCYKAYLAIRQHANLFINLFSMMLGSGMPELQSFDDIAYIRKTLALDKTEQEALEYFMKQMNDAHHGGWTTKMDWIFHTIKQHALN
5HWX , Knot 124 301 0.78 40 145 272 
MVILGVGYFLLGLILLYYGSDWFVLGSERIARHFNVSNFVIGATVMAIGTSLPEILTSAYASYMHAPGISIGNAIGSCICNIGLVLGLSAIISPIIVDKNLQKNILVYLLFVIFAAVIGIDGFSWIDGVVLLILFIIYLRWTVKNGSAEIEENNDKNNPSVVFSLVLLIIGLIGVLVGAELFVDGAKKIALALDISDKVIGFTLVAFGTSLPELMVSLAAAKRNLGGMVLGNVIGSNIADIGGALAVGSLFMHLPAENVQMAVLVIMSLLLYLFAKYSKIGRWQGILFLALYIIAIASLRM

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 \(c\).

\(|P_{f(2ZRA_1)}(2) \setminus P_{f(5DXT_1)}(2)|=22\), \(|P_{f(5DXT_1)}(2) \setminus P_{f(2ZRA_1)}(2)|=174\). 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 1:1100110000110111101000110101101100100110111010101011111111101011010110001000111011101011111111101001101001001110000111001001001101101110011101111001111110101010110001110101100110010111000100111100100011111101000011011010101010100100100100111000010110001011100101011010110001011011100111000101100010011010001001110000110010001000111111101010011111101
Pair \(Z_2\) Length of longest common subsequence
2ZRA_1,5DXT_1 196 5
2ZRA_1,5HWX_1 157 4
5DXT_1,5HWX_1 221 4

Newick tree

 
[
	5DXT_1:11.75,
	[
		2ZRA_1:78.5,5HWX_1:78.5
	]:33.25
]

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 \((0.85)(0.8)(\frac{1311 }{\log_{20} 1311}-\frac{349}{\log_{20}349})=250.\)
Status Protein1 Protein2 d d1/2
Query variables 2ZRA_1 5DXT_1 327 217
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]