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Parikh vectors
4PXK_1 3OZZ_1 7EOQ_1 Letter Amino acid
10 3 30 R Arginine
6 2 31 M Methionine
14 3 48 F Phenylalanine
23 6 57 A Alanine
0 0 12 C Cysteine
21 5 51 T Threonine
14 4 62 S Serine
10 1 19 W Tryptophan
9 4 27 Y Tyrosine
20 4 80 V Valine
2 2 24 Q Glutamine
9 10 47 E Glutamic acid
39 6 74 L Leucine
3 5 47 K Lycine
16 6 48 I Isoleucine
7 3 38 P Proline
5 2 36 N Asparagine
9 6 44 D Aspartic acid
33 5 58 G Glycine
8 5 20 H Histidine 

4PXK_1|Chain A|Bacteriorhodopsin|Haloarcula marismortui (272569)
>3OZZ_1|Chain A[auth B]|Cytochrome b5|Bos taurus (9913)
>7EOQ_1|Chains A, C|Glutamate receptor ionotropic, NMDA 2A|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 \(w=f(c)\)
4PXK , Knot 111 258 0.79 38 143 238 
MPAPGSEGIWLWLGTAGMFLGMLYFIARGWGETDGRRQKFYIATILITAIAFVNYLAMALGFGLTFIEFGGEQHPIYWARYTDWLFTTPLLLYNLGLLAGADRNTIYSLVSLDVLMIGTGVVATLSAGSGVLSAGAERLVWWGISTAFLLVLLYFLFSSLSGRVANLPSDTRSTFKTLRNLVTVVWLVYPVWWLVGSEGLGLVGIGIETAGFMVIDLVAKVGFGIILLRSHGVLDGAAETTGTGATPADDLEHHHHHH
3OZZ , Knot 46 82 0.82 38 74 80 
AVKYYTLEEIQKHNNAKSTWMIIHYKVYDITRFLSEHPGGEEVLLEQAGADATESFEDVGHSPDAREMLKTFIIGELHPDDR
7EOQ , Knot 325 853 0.85 40 323 789 
MGRVGYWTLLVLPALLVWRGPAPSAAAEKGPPALNIAVMLGHSHDVTERELRTLWGPEQAAGLPLDVNVVALLMNRTDPKSLITHVCDLMSGARIHGLVFGDDTDQEAVAQMLDFISSHTFVPILGIHGGASMIMADKDPTSTFFQFGASIQQQATVMLKIMQDYDWHVFSLVTTIFPGYREFISFVKTTVDNSFVGWDMQNVITLDTSFEDAKTQVQLKKIHSSVILLYCSKDEAVLILSEARSLGLTGYDFFWIVPSLVSGNTELIPKEFPSGLISVSYDDWDYSLEARVRDGIGILTTAASSMLEKFSYIPEAKASCYGQMERPEVPMHTLHPFMVNVTWDGKDLSFTEEGYQVHPRLVVIVLNKDREWEKVGKWENHTLSLRHAVWPRYKSFSDCEPDDNHLSIVTLEEAPFVIVEDIDPLTETCVRNTVPCRKFVKINNSTNEGMNVKKCCKGFCIDILKKLSRTVKFTYDLYLVTNGKHGKKVNNVWNGMIGEVVYQRAVMAVGSLTINEERSEVVDFSVPFVETGISVMVSRSNGTVSPSAFLEPFSASVWVMMFVMLLIVSAIAVFVFEYFSPVGYNRNLAKGKAPHGPSFTIGKAIWLLWGLVFNNSVPVQNPKGTTSKIMVSVWAFFAVIFLASYTANLAAFMIQEEFVDQVTGLSDKKFQRPHDYSPPFRFGTVPNGSTERNIRNNYPYMHQYMTKFNQKGVEDALVSLKTGKLDAFIYDAAVLNYKAGRDEGCKLVTIGSGYIFATTGYGIALQKGSPWKRQIDLALLQFVGDGEMEELETCWLTGICHNEKNEVMSSQLDIDNMAGVFYMLAAAMALSLITFIWEHLFYKSRAEAKRMKG

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 \(c\).

\(|P_{f(4PXK_1)}(2) \setminus P_{f(3OZZ_1)}(2)|=106\), \(|P_{f(3OZZ_1)}(2) \setminus P_{f(4PXK_1)}(2)|=37\). 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 1:111110011111110111111110111011100010000101101110111110011111111101101110001101100001110011110011111110000100110101111101111010110111011100111111001111111011100101011011000000100100110111110111111100111111111001111110111011111111000111011100010110110010000000
Pair \(Z_2\) Length of longest common subsequence
4PXK_1,3OZZ_1 143 4
4PXK_1,7EOQ_1 200 4
3OZZ_1,7EOQ_1 267 4

Newick tree

 
[
	7EOQ_1:12.78,
	[
		4PXK_1:71.5,3OZZ_1:71.5
	]:58.28
]

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 \((0.85)(0.8)(\frac{340 }{\log_{20} 340}-\frac{82}{\log_{20}82})=80.9\)
Status Protein1 Protein2 d d1/2
Query variables 4PXK_1 3OZZ_1 97 64
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]