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Parikh vectors
2QTQ_1 7FJL_1 3ZUE_1 Letter Amino acid
10 27 13 E Glutamic acid
5 10 5 H Histidine
10 12 26 I Isoleucine
20 29 21 D Aspartic acid
12 30 68 G Glycine
6 18 11 M Methionine
14 20 47 S Serine
6 13 24 Q Glutamine
29 27 32 L Leucine
9 15 27 F Phenylalanine
10 14 17 Y Tyrosine
18 46 56 A Alanine
8 18 36 N Asparagine
2 6 1 C Cysteine
11 16 5 K Lycine
6 31 50 P Proline
8 30 58 T Threonine
0 13 10 W Tryptophan
12 34 53 V Valine
17 30 19 R Arginine 

2QTQ_1|Chains A, B, C, D|Transcriptional regulator, TetR family|Novosphingobium aromaticivorans (279238)
>7FJL_1|Chains A, B, C, D, E, F|Rieske (2Fe-2S) domain protein|Comamonas testosteroni (strain DSM 14576 / KF-1) (399795)
>3ZUE_1|Chains A, B, C|CAPSID STRUCTURAL PROTEIN VP60|RABBIT HEMORRHAGIC DISEASE VIRUS (11976)
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 \(w=f(c)\)
2QTQ , Knot 99 213 0.83 38 149 206 
GMSSDVQKGDNLETPGARDLLLQTASNIMREGDVVDISLSELSLRSGLNSALVKYYFGNKAGLLKALLDRDMENIVKSVDALLAKDDMSPEAKLRRHISKCIDTYYDYPYLNRLLMRLVRDSDEAEAKRIADQYLLPLHRAYNRFIGEGVKAGVFRPINPQLFYFTVTGAADRFFSARLVLKHCFDQDTLTEQLRDSYREHTVDFIMAGILAH
7FJL , Knot 187 439 0.86 40 249 424 
MMTHEENELLCRVEGDAPMGRLMRRHWTPICLVEEVGEPDGTPVKARAFGEDLVVFRDSEGRVGVMDEYCPHRRASLVYGRNEEGGLRCLYHGWKMDVDGNVLEMASEPAASGMVDKVKHTAYPTQEWAGMVWAYMGPKETMPEFLPPAWAPTADTRVSIAKVLLPCNWAQILEGAIDSAHSSSLHSSDMVPARVDGAKATDKTWLRPSTDKAPRMQVQRTGYGFRYAALRRPLSNAAENDYVRSTVFVAPATALIPPNNLYNVANINVPMDDTNTAFYFIAWGHPSQTPETETWRKFLRQTVGVDLDQNYRPLRNEANKFWQDRNAMKAGNFTGITGFPNQDVAMWLTMGPIADRTHDRLGASDLAIVEFRKQMLDAVKAFEQGAPAIGTGVEAATPTVCSFQAIVPKTTDWRTYDAHYVWLDGQDRPEFEPSYSVKS
3ZUE , Knot 227 579 0.83 40 237 516 
MEGKARTAPQGEAAGTATTASVPGTTTDGLDPGVVATTSVVTAENSSASIATAGIGGPPQQVDQQETWRTNFYYNDVFTWSVADAPGSILYTVQHSPQNNPFTAVLSQMYAGWAGGMQFRFIVAGSGVFGGRLVAAVIPPGIEIGPGLEVRQFPHVVIDARSLEPVTNTMPDLRPNMYHPTGDPGLVPTLVLSVYNNLINPFGGSTSAIQVTVETRPSEDFEFVMIRAPSSKTVDSISPAGLLTTPVLTGVGNDNRWNGQIVGLQPVPGGFSTCNRHWNLNGSTYGWSSPRFADIDHRRGSASYPGNNATNVLQFWYANAGSAVDNPISQVAPDGFPDMSFVPFNGPGIPAAGWVGFGAIWNSNSGAPNVTTVQAYELGFATGAPGNLQPTTNTSGAQTVAKSIYAVVTGTAQNPAGLFVMASGVISTPNANAITYTPQPDRIVTTPGTPAAAPVGKNTPIMFASVVRRTGDVNATAGSASGTQYGTGSQPLPVTIGLSLNNYSSSLMPGQFFVWQLTFASGFMEIGLSVDGYFYAGTGALTTLIDLTELIDVRPVGPRPSKSTLVFNLGGTANGFSYV

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 \(c\).

\(|P_{f(2QTQ_1)}(2) \setminus P_{f(7FJL_1)}(2)|=43\), \(|P_{f(7FJL_1)}(2) \setminus P_{f(2QTQ_1)}(2)|=143\). 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 1:110001001001001110011100100110010110101001010011001110001100111101110001001100101111000101010100010001000000101001110110000010100110001111001000111011011110110101101010111001101011100010000100010000000010111111110
Pair \(Z_2\) Length of longest common subsequence
2QTQ_1,7FJL_1 186 3
2QTQ_1,3ZUE_1 162 4
7FJL_1,3ZUE_1 142 4

Newick tree

 
[
	2QTQ_1:91.97,
	[
		3ZUE_1:71,7FJL_1:71
	]:20.97
]

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 \((0.85)(0.8)(\frac{652 }{\log_{20} 652}-\frac{213}{\log_{20}213})=124.\)
Status Protein1 Protein2 d d1/2
Query variables 2QTQ_1 7FJL_1 162 117.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]