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Parikh vectors
9BLO_1 7NYM_1 1OIE_1 Letter Amino acid
1 0 4 C Cysteine
5 2 27 Q Glutamine
14 1 40 K Lycine
2 2 15 M Methionine
6 0 19 S Serine
14 3 25 T Threonine
1 1 7 W Tryptophan
17 4 39 V Valine
9 7 43 A Alanine
1 5 19 R Arginine
15 9 35 E Glutamic acid
3 3 25 P Proline
14 1 27 N Asparagine
2 3 16 H Histidine
4 2 28 I Isoleucine
12 5 37 L Leucine
4 5 21 Y Tyrosine
14 4 38 D Aspartic acid
14 3 49 G Glycine
4 3 18 F Phenylalanine 

9BLO_1|Chains A, B|Surface anchored protein|Clostridium perfringens B str. ATCC 3626 (451754)
>7NYM_1|Chains A[auth AAA], B[auth BBB], C[auth CCC], D[auth DDD]|SH3 domain of JNK-interacting Protein 1 (JIP1)|Homo sapiens (9606)
>1OIE_1|Chain A|PROTEIN USHA|ESCHERICHIA COLI (562)
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 \(w=f(c)\)
9BLO , Knot 73 156 0.78 40 108 148 
GAMDPEFNLPEVKDGTLRCTVIADGVNGSSEKEALVSFENSKDGVDVKDTINYEGLVANQNYTLTGTLMHVKADGSLEEIATKTTNVTAGENGNGTWGLDFGNQKLQVGEKYVVFENAESVENLIDTDKDYNLDTKQVVKHEDKNDKAQTLVVEKP
7NYM , Knot 36 63 0.79 36 54 60 
GHMEQTHRAIFRFVPRHEDELELEVDDPLLAELQAEDYWYEAYNMRTGARGVFPAYYAIEVTK
1OIE , Knot 217 532 0.85 40 266 495 
YEQDKTYKITVLHTNDHHGHFWRNEYGEYGLAAQKTLVDGIRKEVAAEGGSVLLLSGGDINTGVPESDLQDAEPDFRGMNLVGYDAMAIGNHEFDNPLTVLRQQEKWAKFPLLSANIYQKSTGERLFKPWALFKRQDLKIAVIGLTTDDTAKIGNPEYFTDIEFRKPADEAKLVIQELQQTEKPDIIIAATHMGHYDNGEHGCNAPGDVEMARALPAGSLAMIVGGHSQDPVCMAAENKKQVDYVPGTPCKPDQQNGIWIVQAHEWGKYVGRADFEFRNGEMKMVNYQLIPVNLKKKVTWEDGKSERVLYTPEIAENQQMISLLSPFQNKGKAQLEVKIGETNGRLEGDRDKVRFVQTNMGRLILAAQMDRTGADFAVMSGGGIRDSIEAGDISYKNVLKVQPFGNVVVYADMTGKEVIDYLTAVAQMKPDSGAYPQFANVSFVAKDGKLNDLKIKGEPVDPAKTYRMATLNFNATGGDGYPRLDNKCGYVNTGFIDAEVLKAYIQKSSPLDVSVYEPKGEVSWQEHHHHHH

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 \(c\).

\(|P_{f(9BLO_1)}(2) \setminus P_{f(7NYM_1)}(2)|=89\), \(|P_{f(7NYM_1)}(2) \setminus P_{f(9BLO_1)}(2)|=35\). 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 1:111010101101001010001110110100000111010000011010001000111100000101011010101010011000001011001010111011000101100011100100100110000000100001100000000100111001
Pair \(Z_2\) Length of longest common subsequence
9BLO_1,7NYM_1 124 3
9BLO_1,1OIE_1 184 3
7NYM_1,1OIE_1 234 4

Newick tree

 
[
	1OIE_1:11.13,
	[
		9BLO_1:62,7NYM_1:62
	]:54.13
]

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 \((0.85)(0.8)(\frac{219 }{\log_{20} 219}-\frac{63}{\log_{20}63})=51.8\)
Status Protein1 Protein2 d d1/2
Query variables 9BLO_1 7NYM_1 62 43.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]