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Parikh vectors
2TPI_1 4OLA_1 2ZRI_1 Letter Amino acid
8 58 13 P Proline
10 40 21 D Aspartic acid
12 22 0 C Cysteine
10 57 13 Q Glutamine
3 39 11 F Phenylalanine
25 53 37 G Glycine
3 32 4 H Histidine
14 61 30 L Leucine
2 21 9 M Methionine
4 6 1 W Tryptophan
10 28 5 Y Tyrosine
18 70 28 V Valine
2 57 18 R Arginine
4 41 27 E Glutamic acid
15 48 25 I Isoleucine
34 43 20 S Serine
14 54 38 A Alanine
16 22 10 N Asparagine
15 53 22 K Lycine
10 54 17 T Threonine 

2TPI_1|Chain A[auth Z]|TRYPSINOGEN|null (9913)
>4OLA_1|Chain A|Protein argonaute-2|Homo sapiens (9606)
>2ZRI_1|Chain A|Protein recA|Mycobacterium smegmatis str. MC2 155 (246196)
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 \(w=f(c)\)
2TPI , Knot 102 229 0.80 40 144 221 
VDDDDKIVGGYTCGANTVPYQVSLNSGYHFCGGSLINSQWVVSAAHCYKSGIQVRLGEDNINVVEGNEQFISASKSIVHPSYNSNTLNNDIMLIKLKSAASLNSRVASISLPTSCASAGTQCLISGWGNTKSSGTSYPDVLKCLKAPILSDSSCKSAYPGQITSNMFCAGYLEGGKDSCQGDSGGPVVCSGKLQGIVSWGSGCAQKNKPGVYTKVCNYVSWIKQTIASN
4OLA , Knot 326 859 0.85 40 325 795 
MYSGAGPALAPPAPPPPIQGYAFKPPPRPDFGTSGRTIKLQANFFEMDIPKIDIYHYELDIKPEKCPRRVNREIVEHMVQHFKTQIFGDRKPVFDGRKNLYTAMPLPIGRDKVELEVTLPGEGKDRIFKVSIKWVSCVSLQALHDALSGRLPSVPFETIQALDVVMRHLPSMRYTPVGRSFFTASEGCSNPLGGGREVWFGFHQSVRPSLWKMMLNIDVSATAFYKAQPVIEFVCEVLDFKSIEEQQKPLTDSQRVKFTKEIKGLKVEITHCGQMKRKYRVCNVTRRPASHQTFPLQQESGQTVECTVAQYFKDRHKLVLRYPHLPCLQVGQEQKHTYLPLEVCNIVAGQRCIKKLTDNQTSTMIRATARSAPDRQEEISKLMRSADFNTDPYVREFGIMVKDEMTDVTGRVLQPPSILYGGRNKAIATPVQGVWDMRNKQFHTGIEIKVWAIACFAPQRQCTEVHLKSFTEQLRKISRDAGMPIQGQPCFCKYAQGADSVEPMFRHLKNTYAGLQLVVVILPGKTPVYAEVKRVGDTVLGMATQCVQMKNVQRTTPQTLSNLCLKINVKLGGVNNILLPQGRPPVFQQPVIFLGADVTHPPAGDGKKPSIAAVVGSMDAHPNRYCATVRVQQHRQEIIQDLAAMVRELLIQFYKSTRFKPTRIIFYRDGVSEGQFQQVLHHELLAIREACIKLEKDYQPGITFIVVQKRHHTRLFCTDKNERVGKSGNIPAGTTVDTKITHPTEFDFYLCSHAGIQGTSRPSHYHVLWDDNRFSSDELQILTYQLCHTYVRCTRSVSIPAPAYYAHLVAFRARYHLVDKEHDSAEGSHTSGQSNGRDHQALAKAVQVHQDTLRTMYFA
2ZRI , Knot 150 349 0.84 38 186 328 
MAQQAPDREKALELAMAQIDKNFGKGSVMRLGEEVRQPISVIPTGSISLDVALGIGGLPRGRVIEIYGPESSGKTTVALHAVANAQAAGGIAAFIDAEHALDPEYAKKLGVDTDSLLVSQPDTGEQALEIADMLVRSGALDIIVIDSVAALVPRAEIEGEMGDSHVGLQARLMSQALRKMTGALNNSGTTAIFINALREKIGVMFGSPETTTGGKALKFYASVRLDVRRIETLKDGTDAVGNRTRVKVVKNKVSPPFKQAEFDILYGQGISREGSLIDMGVEHGFIRKSGSWFTYEGEQLGQGKENARKFLLENTDVANEIEKKIKEKLGIGAVVTAEADDVLPAPVDF

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 \(c\).

\(|P_{f(2TPI_1)}(2) \setminus P_{f(4OLA_1)}(2)|=15\), \(|P_{f(4OLA_1)}(2) \setminus P_{f(2TPI_1)}(2)|=196\). 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 1:1000001111000110011001010010010110110001110110000011010110001011010001101000110100000010001111010011010001101011000101100011011100000100010110010111100000001011010001101101011000001001111100101011101101010000111000100010110001100
Pair \(Z_2\) Length of longest common subsequence
2TPI_1,4OLA_1 211 3
2TPI_1,2ZRI_1 170 4
4OLA_1,2ZRI_1 159 4

Newick tree

 
[
	2TPI_1:10.64,
	[
		2ZRI_1:79.5,4OLA_1:79.5
	]:21.14
]

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 \((0.85)(0.8)(\frac{1088 }{\log_{20} 1088}-\frac{229}{\log_{20}229})=231.\)
Status Protein1 Protein2 d d1/2
Query variables 2TPI_1 4OLA_1 300 187.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]