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Parikh vectors
2FWT_1 3NZX_1 2NLI_1 Letter Amino acid
2 10 16 Y Tyrosine
8 17 23 E Glutamic acid
2 15 24 I Isoleucine
11 23 23 S Serine
7 19 16 T Threonine
20 18 40 A Alanine
3 6 14 N Asparagine
3 9 13 F Phenylalanine
5 16 24 V Valine
7 12 17 P Proline
7 14 25 D Aspartic acid
2 6 12 Q Glutamine
7 23 36 G Glycine
3 18 20 K Lycine
4 4 7 M Methionine
2 2 5 W Tryptophan
10 9 19 R Arginine
4 0 0 C Cysteine
4 3 9 H Histidine
14 26 25 L Leucine 

2FWT_1|Chain A|DHC, diheme cytochrome c|Rhodobacter sphaeroides (1063)
>3NZX_1|Chains A, O|Proteasome component Y7|Saccharomyces cerevisiae (4932)
>2NLI_1|Chains A, B|Lactate oxidase|Aerococcus viridans (1377)
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 \(w=f(c)\)
2FWT , Knot 62 125 0.79 40 98 120 
ALVVTDPLTRTECSACHMAYPAALLPARSWTALMADLPNHFGEDASLDEASRGQIESYLVANAADSSGAGRALRGLVQTDTPLRISELPWFKRKHADEVSPRMLEKARSMSNCAACHTGAERGLF
3NZX , Knot 111 250 0.81 38 159 239 
MTDRYSFSLTTFSPSGKLGQIDYALTAVKQGVTSLGIKATNGVVIATEKKSSSPLAMSETLSKVSLLTPDIGAVYSGMGPDYRVLVDKSRKVAHTSYKRIYGEYPPTKLLVSEVAKIMQEATQSGGVRPFGVSLLIAGHDEFNGFSLYQVDPSGSYFPWKATAIGKGSVAAKTFLEKRWNDELELEDAIHIALLTLKESVEGEFNGDTIELAIIGDENPDLLGYTGIPTDKGPRFRKLTSQEINDRLEAL
2NLI , Knot 158 368 0.84 38 208 348 
EYNAPSEIKYIDVVNTYDLEEEASKVVPHGGFNYIAGASGDEWTKRANDRAWKHKLLYPRLAQDVEAPDTSTEILGHKIKAPFIMAPIAAHGLAHTTKEAGTARAVSEFGTIMSISAYSGATFEEISEGLNGGPRWFQIYMAKDDQQNRDILDEAKSDGATAIILTADSTVSGNRDRDVKNKFVYPFGMPIVQRYLRGTAEGMSLNNIYGASKQKISPRDIEEIAGHSGLPVFVKGIQHPEDADMAIKRGASGIWVSNHGARQLYEAPGSFDTLPAIAERVNKRVPIVFDSGVRRGEHVAKALASGADVVALGRPVLFGLALGGWQGAYSVLDYFQKDLTRVMQLTGSQNVEDLKGLDLFDNPYGYEY

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 \(c\).

\(|P_{f(2FWT_1)}(2) \setminus P_{f(3NZX_1)}(2)|=48\), \(|P_{f(3NZX_1)}(2) \setminus P_{f(2FWT_1)}(2)|=109\). 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 1:11110011000000100110111111100101111011001100101001001010001110110001110110111000011010011110000100101011001001000110001100111
Pair \(Z_2\) Length of longest common subsequence
2FWT_1,3NZX_1 157 3
2FWT_1,2NLI_1 182 4
3NZX_1,2NLI_1 157 4

Newick tree

 
[
	2NLI_1:87.03,
	[
		2FWT_1:78.5,3NZX_1:78.5
	]:8.53
]

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 \((0.85)(0.8)(\frac{375 }{\log_{20} 375}-\frac{125}{\log_{20}125})=76.1\)
Status Protein1 Protein2 d d1/2
Query variables 2FWT_1 3NZX_1 95 70.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]