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Parikh vectors
1KVS_1 3LDH_1 4RSD_1 Letter Amino acid
5 7 0 W Tryptophan
14 7 6 Y Tyrosine
24 25 4 D Aspartic acid
3 8 8 C Cysteine
18 20 3 I Isoleucine
18 25 15 S Serine
20 12 10 N Asparagine
30 24 3 G Glycine
9 11 4 M Methionine
21 10 4 P Proline
9 7 3 F Phenylalanine
16 12 10 T Threonine
26 19 13 A Alanine
16 9 4 R Arginine
12 11 7 Q Glutamine
27 35 2 L Leucine
14 13 5 E Glutamic acid
13 11 4 H Histidine
14 29 10 K Lycine
29 34 9 V Valine 

1KVS_1|Chain A|UDP-GALACTOSE 4-EPIMERASE|Escherichia coli (562)
>3LDH_1|Chain A|LACTATE DEHYDROGENASE|Squalus acanthias (7797)
>4RSD_1|Chain A|RIBONUCLEASE A|Bos taurus (9913)
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 \(w=f(c)\)
1KVS , Knot 146 338 0.83 40 215 330 
MRVLVTGGSGYIGSHTCVQLLQNGHDVIILDNLCNSKRSVLPVIERLGGKHPTFVEGDIRNEALMTEILHDHAIDTVIHFAGLKAVGESVQKPLEYYDNNVNGTLRLISAMRAANVKNFIFSSTATVYGDNPKIPYVESFPTGTPQSPYGKSKLMVEQILTDLQKAQPDWSIALLRYFNPVGAHPSGDMGEDPQGIPNNLMPYIAQVAVGRRDSLAIFGNDYPTEDGTGVRDYIHVMDLADGHVVAMEKLANKPGVHIYNLGAGVGNSVLDVVNAFSKACGKPVNYHFAPRREGDLPAYWADASKADRELNWRVTRTLDEMAQDTWHWQSRHPQGYPD
3LDH , Knot 139 330 0.81 42 198 314 
XTALKDKLIGHLATSQEPRSYNKITVVGCDAVGMADAISVLMKDLADEVALVDVMEDKLKGEMMDLEHGSLFLHTAKIVSGKDYSVSAGSKLVVITAGARQQEGESRLNLVQRNVNIFKFIIPNIVKHSPDCLKELHPELGTDKNKQDWKLSGLPMHRIIGSGCNLDSARFRYLMGERLGVHSCLVIGWVIGQHGDSVPSVWSGMWDAKLHKDVVDSAYEVIKLKGYTSWAIGLVVSNPVDVLTYVAWKGCSVADLAQTIMKDLCRVHPVSTMVKDFYGIKDNVFLSLPCVLNNGISHCNIVKMKLKPDEEQQLQKSATTLWDIQKDLKF
4RSD , Knot 63 124 0.81 38 106 120 
KETAAAKFERQHMDSSTSAASSSNYCNQMMKSRNLTKDRCKPVNTFVHESLADVQAVCSQKNVACKNGQTNCYQSYSTMSITDCRETGSSKYPNCAYKTTQANKHIIVACEGNPYVPVHFAASV

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 \(c\).

\(|P_{f(1KVS_1)}(2) \setminus P_{f(3LDH_1)}(2)|=98\), \(|P_{f(3LDH_1)}(2) \setminus P_{f(1KVS_1)}(2)|=81\). 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 1:10111011010110000101100100111100100000011111001110010110101000111001100011001101111011100100110000001010101101101101001110001010100101101001101010010100011100110010010101011110010111101010110010111001110110111100001111100010001011000101101101011110011001110100111111001101101100101011000111000101110110100100010101000100110001010000101010
Pair \(Z_2\) Length of longest common subsequence
1KVS_1,3LDH_1 179 4
1KVS_1,4RSD_1 203 4
3LDH_1,4RSD_1 178 3

Newick tree

 
[
	1KVS_1:97.81,
	[
		3LDH_1:89,4RSD_1:89
	]:8.81
]

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 \((0.85)(0.8)(\frac{668 }{\log_{20} 668}-\frac{330}{\log_{20}330})=93.2\)
Status Protein1 Protein2 d d1/2
Query variables 1KVS_1 3LDH_1 122 117
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]