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Parikh vectors
6WTV_1 4HGI_1 8XEF_1 Letter Amino acid
21 12 15 Y Tyrosine
12 24 15 V Valine
8 13 3 H Histidine
14 20 8 R Arginine
26 19 17 N Asparagine
12 23 13 G Glycine
29 22 8 I Isoleucine
16 24 14 F Phenylalanine
0 5 2 W Tryptophan
9 35 14 A Alanine
4 3 8 C Cysteine
21 36 5 E Glutamic acid
25 50 12 L Leucine
37 36 11 K Lycine
6 14 0 M Methionine
17 17 15 S Serine
19 33 8 D Aspartic acid
5 26 13 P Proline
13 21 9 T Threonine
13 22 5 Q Glutamine 

6WTV_1|Chains A, D, G, L[auth J]|reticulocyte binding protein 2b|Plasmodium vivax (strain Salvador I) (126793)
>4HGI_1|Chains A, B|Bifunctional P-450/NADPH-P450 reductase|Bacillus megaterium (1404)
>8XEF_1|Chain A|Spike protein S2'|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
6WTV , Knot 133 307 0.82 38 182 296 
GAMGSTNTTDNIDYFDISDESNYYLISQLRPHFSNIYFFDEFKRYASYHTEIKRYEDIHKTKVNSLLNEASRAIGICNRAKNTVKGLINILENPQKFKTQRESYDVKLRQYEEKKEAFRGCLLNKNRKNLDQIKKINNEIRDLLEKLKCSQDCQTNVYFDMIKIYLVDFKKMPYENYDTFIKQYKNSYLSGVDMIRKIEKQIDNPVTINAIKFTQKEMGYIIDRFEYHLQKVKHSIDQVTALSDGVKPKQVTKNRLKEYYFNIGNYYSIFKFGKDSLNMLNKALIHKEKIVHNLLGELFGHLEERIS
4HGI , Knot 189 455 0.84 40 239 437 
TIKEMPQPKTFGELKNLPLLNTDKPVQALMKIADELGEIFKFEAPGRVTRYLSSQRLIKEACDESRFDKNLSQALKFVRDFAGDGLATSWTHEKNWKKAHNILLPSFSQQAMKGYHAMMVDIAVQLVQKWERLNADEHIEVPEDMTRLTLDTIGLCGFNYRFNSFYRDQPHPFITSMVRALDEAMNKLQRANPDDPAYDENKRQFQEDIKVMNDLVDKIIADRKASGEQSDDLLAHMLNGKDPETGEPLDDENIRYQIITFLIAGHETTSGLLSFALYFLVKNPHVLQKAAEEAARVLVDPVPSYKQVKQLKYVGMVLNEALRLWPTAPAFSLYAKEDTVLGGEYPLEKGDELMVLIPQLHRDKTIWGDDVEEFRPERFENPSAIPQHAFKPFGNGQRACIGQQFALHEATLVLGMMLKHFDFEDHTNYELDIKETLTLKPEGFVVKAKSKKIPL
8XEF , Knot 90 195 0.81 38 133 187 
NLCPFHEVFNATTFASVYAWNRKRISNCVADYSVIYNFAPFFAFKCYGVSPTKLNDLCFTNVYADSFVIRGNEVSQIAPGQTGNIADYNYKLPDDFTGCVIAWNSNKLDSKPSGNYNYLYRLFRKSKLKPFERDISTEIYQAGNKPCNGVAGSNCYSPLQSYGFRPTYGVGHQPYRVVVLSFELLHAPATVCGPK

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 \(c\).

\(|P_{f(6WTV_1)}(2) \setminus P_{f(4HGI_1)}(2)|=53\), \(|P_{f(4HGI_1)}(2) \setminus P_{f(6WTV_1)}(2)|=110\). 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 1:1111000000010010100000001100101010010110010001000001000001000010011001001111000100010111011001001000000001010000000011010110000001001001000100110010000000001010110101101001100000011000000010110110010001001101011010000110110010001001000100101100110100100001000010110000110110001011001110000110011101110100010
Pair \(Z_2\) Length of longest common subsequence
6WTV_1,4HGI_1 163 4
6WTV_1,8XEF_1 173 4
4HGI_1,8XEF_1 198 3

Newick tree

 
[
	8XEF_1:96.47,
	[
		6WTV_1:81.5,4HGI_1:81.5
	]:14.97
]

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 \((0.85)(0.8)(\frac{762 }{\log_{20} 762}-\frac{307}{\log_{20}307})=124.\)
Status Protein1 Protein2 d d1/2
Query variables 6WTV_1 4HGI_1 159 132.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]