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Parikh vectors
3NAY_1 8DIY_1 8JUQ_1 Letter Amino acid
8 5 20 N Asparagine
13 9 34 D Aspartic acid
12 8 15 Q Glutamine
27 12 20 E Glutamic acid
14 4 1 H Histidine
15 17 13 T Threonine
15 9 25 Y Tyrosine
19 10 37 A Alanine
4 0 0 C Cysteine
17 11 46 G Glycine
18 31 12 I Isoleucine
15 11 13 R Arginine
34 24 20 L Leucine
27 7 13 K Lycine
4 13 9 M Methionine
20 27 16 F Phenylalanine
18 9 8 P Proline
15 19 21 S Serine
2 4 10 W Tryptophan
14 27 17 V Valine 

3NAY_1|Chains A, B|3-phosphoinositide-dependent protein kinase 1|Homo sapiens (9606)
>8DIY_1|Chain A|Ion transport protein|Aliarcobacter butzleri RM4018 (367737)
>8JUQ_1|Chains A, B, C, D[auth G], E[auth H], F[auth I]|Chitoporin|Vibrio harveyi (669)
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 \(w=f(c)\)
3NAY , Knot 131 311 0.80 40 185 289 
MHHHHHHENLYFQGQHAQPPPQPRKKRPEDFKFGKILGEGSFSTVVLARELATSREYAIKILEKRHIIKENKVPYVTRERDVMSRLDHPFFVKLYFTFQDDEKLYFGLSYAKNGELLKYIRKIGSFDETCTRFYTAEIVSALEYLHGKGIIHRDLKPENILLNEDMHIQITDFGTAKVLSPESKQARANSFVGTAQYVSPELLTEKSACKSSDLWALGCIIYQLVAGLPPFRAGNEYLIFQKIIKLEYDFPEKFFPKARDLVEKLLVLDATKRLGCEEMEGYGPLKAHPFFESVTWENLHQQTPPKLTAYL
8DIY , Knot 115 257 0.82 38 158 242 
MDYKDDDDKGSLVPRGSHMYLRITNIVESSFFTKFIIYLIVLNGITMGLETSKTFMQSFGVYTTLFNQIVITIFTIEIILRIYVHRISFFKDPWSLFDFFVVAISLVPTSSGFEILRVSRVLRLFRLVTAVPQMRKIVSALISVIPGMLSVIALMTLFFYIFAIMATQLFGERFPEWFGTLGESFYTLFQVMTLESWSMGIVRPLMEVYPYAWVFFIPFIFVVTFVMINLVVAIIVDAMAILNQKEEQHIIDEVQSH
8JUQ , Knot 150 350 0.83 38 191 322 
DGANSDAAKEYLTKDSFSYEVYGIIAMQAAYRDYDSGDAKQDDNLGGMQLNNESRIGFRGKKQFANFEPTFIWQIEGGYVDPSFGGEGAGLGERDTFVGFESASWGQVRLGRVLTPMYELVDWPASNPGLGDVYDWGGAIGGAKYQDRQSNTIRWDSPMYADKFSIDAAVGAGDKAGLGAGDDYWGGIAAHYKLGPLQLDAAYEGNRNIEAEGQTWENNTYLVGVQGWFENGISFFAQYKYMEADASNGVNEKQDAMSAGLMYTTGDWQYKLGYAANFDLERDGKTLSNTSDDVVSAQIMYFVDPSAVLYARARMNDFNEGLDGLDDAARWTSGTNGDYNEYSVGVEYYF

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 \(c\).

\(|P_{f(3NAY_1)}(2) \setminus P_{f(8DIY_1)}(2)|=98\), \(|P_{f(8DIY_1)}(2) \setminus P_{f(3NAY_1)}(2)|=71\). 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 1:10000000010101001011101000010010110111010100111100110000011011000011000011010000011001001111010101000001011100100101100100110100000010010110110010101110001010011100010101001101011010000101001110100101011000010000011111011001111111101100011100110100011001110100110011110100011000101011101011100101001000011010101
Pair \(Z_2\) Length of longest common subsequence
3NAY_1,8DIY_1 169 5
3NAY_1,8JUQ_1 178 4
8DIY_1,8JUQ_1 173 4

Newick tree

 
[
	8JUQ_1:88.81,
	[
		3NAY_1:84.5,8DIY_1:84.5
	]:4.31
]

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 \((0.85)(0.8)(\frac{568 }{\log_{20} 568}-\frac{257}{\log_{20}257})=88.0\)
Status Protein1 Protein2 d d1/2
Query variables 3NAY_1 8DIY_1 107 98.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]