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Parikh vectors
6VQI_1 3HPF_1 7HBE_1 Letter Amino acid
16 12 9 Q Glutamine
8 7 7 M Methionine
6 3 1 W Tryptophan
34 29 11 V Valine
24 15 8 N Asparagine
26 33 16 D Aspartic acid
18 23 20 I Isoleucine
46 43 19 L Leucine
38 30 17 K Lycine
15 16 7 Y Tyrosine
14 17 8 R Arginine
11 26 15 G Glycine
3 3 0 C Cysteine
26 24 26 E Glutamic acid
6 14 10 H Histidine
15 18 10 F Phenylalanine
11 13 5 P Proline
24 26 16 S Serine
14 19 17 T Threonine
27 20 15 A Alanine 

6VQI_1|Chain A[auth G]|V-type proton ATPase subunit C 1|Rattus norvegicus (10116)
>3HPF_1|Chains A, B|Muconate cycloisomerase|Oceanobacillus iheyensis HTE831 (221109)
>7HBE_1|Chain A|Heat shock protein HSP 90-alpha|Homo sapiens (9606)
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 \(w=f(c)\)
6VQI , Knot 161 382 0.83 40 214 363 
MTEFWLISAPGEKTCQQTWEKLHAATTKNNNLAVSSKFNIPDLKVGTLDVLVGLSDELAKLDAFVEGVVKKVAQYMADVLEDSKDKVQENLLASGVDLVTYITRFQWDMAKYPIKQSLKNISEIIAKGVTQIDNDLKSRASAYNNLKGNLQNLERKNAGSLLTRSLAEIVKKDDFVLDSEYLVTLLVVVPKLNHNDWIKQYETLAEMVVPRSSNVLSEDQDSYLCNVTLFKKAVDDFRHKARENKFIVRDFQYNEEEMRADKEEMNRLSTDKKKQFGPLVRWLKVNFSEAFIAWIHIKALRVFVESVLRYGLPVNFQAMLLQPNKKSVKKLREVLHELYKHLDSSAAAIIDAPMDIPGLNLSQQEYYPYVYYKIDCNLLEFK
3HPF , Knot 165 391 0.84 40 213 368 
MKITDLELHAVGIPRHTGFVNKHVIVKIHTDEGLTGIGEMSDFSHLPLYSVDLHDLKQGLLSILLGQNPFDLMKINKELTDNFPETMYYFEKGSFIRNGIDNALHDLCAKYLDISVSDFLGGRVKEKIKVCYPIFRHRFSEEVESNLDVVRQKLEQGFDVFRLYVGKNLDADEEFLSRVKEEFGSRVRIKSYDFSHLLNWKDAHRAIKRLTKYDLGLEMIESPAPRNDFDGLYQLRLKTDYPISEHVWSFKQQQEMIKKDAIDIFNISPVFIGGLTSAKKAAYAAEVASKDVVLGTTQELSVGTAAMAHLGCSLTNINHTSDPTGPELYVGDVVKNRVTYKDGYLYAPDRSVKGLGIELDESLLAKYQVPDLSWDNVTVHQLQDRTADTKS
7HBE , Knot 105 237 0.80 38 157 225 
MDQPMEEEEVETFAFQAEIAQLMSLIINTFYSNKEIFLRELISNSSDALDKIRYESLTDPSKLDSGKELHINLIPNKQDRTLTIVDTGIGMTKADLINNLGTIAKSGTKAFMEALQAGADISMIGQFGVGFYSAYLVAEKVTVITKHNDDEQYAWESSAGGSFTVRTDTGEPMGRGTKVILHLKEDQTEYLEERRIKEIVKKHSQFIGYPITLFVEKERDKEVSDDEAELEHHHHHH

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 \(c\).

\(|P_{f(6VQI_1)}(2) \setminus P_{f(3HPF_1)}(2)|=80\), \(|P_{f(3HPF_1)}(2) \setminus P_{f(6VQI_1)}(2)|=79\). 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 1:1001111011100000001001011000000111000101101011010111110001101011101110011001101100000010001110110110010010101100110001001001110110010001000101000101010010000110110001101100001110000110111111010000110000011011110000110000000100101100110010001000011100100000010100001001000000011111011010100111111010110111001100111101011110100001001001100100010001111101110111101000000101000100011010
Pair \(Z_2\) Length of longest common subsequence
6VQI_1,3HPF_1 159 3
6VQI_1,7HBE_1 157 5
3HPF_1,7HBE_1 178 4

Newick tree

 
[
	3HPF_1:86.25,
	[
		6VQI_1:78.5,7HBE_1:78.5
	]:7.75
]

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 \((0.85)(0.8)(\frac{773 }{\log_{20} 773}-\frac{382}{\log_{20}382})=105.\)
Status Protein1 Protein2 d d1/2
Query variables 6VQI_1 3HPF_1 130 129
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]