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Parikh vectors
1ODM_1 4YET_1 4DIQ_1 Letter Amino acid
4 4 11 M Methionine
16 11 21 F Phenylalanine
5 9 8 W Tryptophan
12 4 41 R Arginine
18 12 35 E Glutamic acid
10 8 13 H Histidine
15 11 10 I Isoleucine
20 11 33 P Proline
28 13 43 A Alanine
24 9 29 D Aspartic acid
19 20 30 G Glycine
20 13 7 K Lycine
20 14 25 S Serine
15 9 23 T Threonine
20 8 20 Y Tyrosine
22 6 31 V Valine
19 15 14 N Asparagine
17 3 26 Q Glutamine
25 19 64 L Leucine
2 4 5 C Cysteine 

1ODM_1|Chain A|ISOPENICILLIN N SYNTHASE|Emericella nidulans (strain FGSC A4 / ATCC 38163 / CBS 112.46 / NRRL 194 / M139) (227321)
>4YET_1|Chains A, B|Superoxide dismutase|Babesia bovis (5865)
>4DIQ_1|Chains A, B|Lysine-specific demethylase NO66|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 \(w=f(c)\)
1ODM , Knot 149 331 0.87 40 206 319 
MGSVSKANVPKIDVSPLFGDDQAAKMRVAQQIDAASRDTGFFYAVNHGINVQRLSQKTKEFHMSITPEEKWDLAIRAYNKEHQDQVRAGYYLSIPGKKAVESFCYLNPNFTPDHPRIQAKTPTHEVNVWPDETKHPGFQDFAEQYYWDVFGLSSALLKGYALALGKEENFFARHFKPDDTLASVVLIRYPYLDPYPEAAIKTAADGTKLSFEWHEDVSLITVLYQSNVQNLQVETAAGYQDIEADDTGYLINCGSYMAHLTNNYYKAPIHRVKWVNAERQSLPFFVNLGYDSVIDPFDPREPNGKSDREPLSYGDYLQNGLVSLINKNGQT
4YET , Knot 96 203 0.83 40 156 197 
GPGSMAFKLPALPYGMRELIPHISEETLSFHYGKHHAGYVNKLNSLIKGTPMESCTIEELILGQTGAVFNNAAQIWNHTFYWNSMGPNCGGEPTGPIRKKIEEKFGSFSAFKTDFSNLLAGHFGSGWGWLVLKDDGTADIVQTHDAGSPLKENLGRPLLCCDVWEHAYYIDYKNDRLSYINSWWNLVNWDFANKNLEAPFKWS
4DIQ , Knot 199 489 0.84 40 232 456 
LRRRYTMASGPQVDNTGGEPAWDSPLRRVLAELNRIPSSRRRAARLFEWLIAPMPPDHFYRRLWEREAVLVRRQDHTYYQGLFSTADLDSMLRNEEVQFGQHLDAARYINGRRETLNPPGRALPAAAWSLYQAGCSLRLLCPQAFSTTVWQFLAVLQEQFGSMAGSNVYLTPPNSQGFAPHYDDIEAFVLQLEGRKLWRVYRPRAPTEELALTSSPNFSQDDLGEPVLQTVLEPGDLLYFPRGFIHQAECQDGVHSLHLTLSTYQRNTWGDFLEAILPLAVQAAMEENVEFRRGLPRDFMDYMGAQHSDSKDPRRTAFMEKVRVLVARLGHFAPVDAVADQRAKDFIHDSLPPVLTDRERALSVYGLPIRWEAGEPVNVGAQLTTETEVHMLQDGIARLVGEGGHLFLYYTVENSRVYHLEEPKCLEIYPQQADAMELLLGSYPEFVRVGDLPCDSVEDQLSLATTLYDKGLLLTKMPLALNAENLYFQ

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 \(c\).

\(|P_{f(1ODM_1)}(2) \setminus P_{f(4YET_1)}(2)|=104\), \(|P_{f(4YET_1)}(2) \setminus P_{f(1ODM_1)}(2)|=54\). 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 1:1101001011010101111000110101100101100001110110011010010000001010101000101110100000000101100101110011001001010101001010100100010111000001110011000010111100111010111110000111001010001101111001010101011100110100101010001011011000010010100111000101000101100100110100000011100101101000011111011000110110100101000001100100100111011000100
Pair \(Z_2\) Length of longest common subsequence
1ODM_1,4YET_1 158 3
1ODM_1,4DIQ_1 168 4
4YET_1,4DIQ_1 190 4

Newick tree

 
[
	4DIQ_1:92.95,
	[
		1ODM_1:79,4YET_1:79
	]:13.95
]

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 \((0.85)(0.8)(\frac{534 }{\log_{20} 534}-\frac{203}{\log_{20}203})=95.3\)
Status Protein1 Protein2 d d1/2
Query variables 1ODM_1 4YET_1 120 95.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]