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Parikh vectors
1TNG_1 1RAG_1 1OEX_1 Letter Amino acid
16 15 6 N Asparagine
12 1 2 C Cysteine
8 12 13 P Proline
10 18 47 T Threonine
2 15 1 R Arginine
4 14 5 E Glutamic acid
15 15 18 I Isoleucine
14 38 19 L Leucine
15 15 11 K Lycine
2 8 0 M Methionine
4 2 5 W Tryptophan
14 34 29 A Alanine
25 15 38 G Glycine
3 12 20 F Phenylalanine
10 8 13 Y Tyrosine
10 21 21 D Aspartic acid
3 11 1 H Histidine
34 20 49 S Serine
18 22 23 V Valine
10 14 9 Q Glutamine 

1TNG_1|Chain A|TRYPSIN|Bos taurus (9913)
>1RAG_1|Chains A, C|Aspartate carbamoyltransferase catalytic chain|Escherichia coli (83333)
>1OEX_1|Chain A|ENDOTHIAPEPSIN|CRYPHONECTRIA PARASITICA (5116)
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 \(w=f(c)\)
1TNG , Knot 102 229 0.80 40 144 221 
VDDDDKIVGGYTCGANTVPYQVSLNSGYHFCGGSLINSQWVVSAAHCYKSGIQVRLGEDNINVVEGNEQFISASKSIVHPSYNSNTLNNDIMLIKLKSAASLNSRVASISLPTSCASAGTQCLISGWGNTKSSGTSYPDVLKCLKAPILSDSSCKSAYPGQITSNMFCAGYLEGGKDSCQGDSGGPVVCSGKLQGIVSWGSGCAQKNKPGVYTKVCNYVSWIKQTIASN
1RAG , Knot 139 310 0.85 40 190 299 
ANPLYQKHIISINDLSRDDLNLVLATAAKLKANPQPELLKHKVIASCFFEASTRTRLSFETSMHRLGASVVGFSDSANTSLGKKGETLADTISVISTYVDAIVMRHPQEGAARLATEFSGNVPVLNAGDGSNQHPTQTLLDLFTIQETQGRLDNLHVAMVGDLKYGRTVHSLTQALAKFDGNRFYFIAPDALAMPQYILDMLDEKGIAWSLHSSIEEVMAEVDILYMTRVQKERLDPSEYANVKAQFVLRASDLHNAKANMKVLHPLPRVDEIATDVDKTPHAWYFQQAGNGIFARQALLALVLNRDLVL
1OEX , Knot 132 330 0.77 38 160 297 
STGSATTTPIDSLDDAYITPVQIGTPAQTLNLDFDTGSSDLWVFSSETTASEVDGQTIYTPSKSTTAKLLSGATWSISYGDGSSSSGDVYTDTVSVGGLTVTGQAVESAKKVSSSFTEDSTIDGLLGLAFSTLNTVSPTQQKTFFDNAKASLDSPVFTADLGYHAPGTYNFGFIDTTAYTGSITYTAVSTKQGFWEWTSTGYAVGSGTFKSTSIDGIADTGTTLLYLPATVVSAYWAQVSGAKSSSSVGGYVFPCSATLPSFTFGVGSARIVIPGDYIDFGPISTGSSSCFGGIQSSAGIGINIFGDVALKAAFVVFNGATTPTLGFASK

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 \(c\).

\(|P_{f(1TNG_1)}(2) \setminus P_{f(1RAG_1)}(2)|=70\), \(|P_{f(1RAG_1)}(2) \setminus P_{f(1TNG_1)}(2)|=116\). 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
1TNG_1,1RAG_1 186 3
1TNG_1,1OEX_1 146 4
1RAG_1,1OEX_1 152 4

Newick tree

 
[
	1RAG_1:88.54,
	[
		1TNG_1:73,1OEX_1:73
	]:15.54
]

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 \((0.85)(0.8)(\frac{539 }{\log_{20} 539}-\frac{229}{\log_{20}229})=88.7\)
Status Protein1 Protein2 d d1/2
Query variables 1TNG_1 1RAG_1 118 101
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]