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Parikh vectors
5PHD_1 6HVF_1 7TRP_1 Letter Amino acid
27 21 13 G Glycine
20 15 27 T Threonine
14 19 35 V Valine
7 7 10 C Cysteine
24 30 34 L Leucine
8 6 8 W Tryptophan
19 13 16 Y Tyrosine
14 11 11 D Aspartic acid
18 4 13 H Histidine
20 10 23 I Isoleucine
23 16 15 K Lycine
11 13 10 M Methionine
18 9 22 S Serine
18 17 17 R Arginine
20 7 19 N Asparagine
15 14 9 Q Glutamine
26 27 5 E Glutamic acid
19 10 19 F Phenylalanine
18 17 13 P Proline
25 20 30 A Alanine 

5PHD_1|Chain A|Lysine-specific demethylase 4D|Homo sapiens (9606)
>6HVF_1|Chains A, B|Proto-oncogene tyrosine-protein kinase Src|Gallus gallus (9031)
>7TRP_1|Chain A[auth R]|Muscarinic acetylcholine receptor M4|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 \(w=f(c)\)
5PHD , Knot 160 364 0.86 40 229 347 
MHHHHHHSSGVDLGTENLYFQSMETMKSKANCAQNPNCNIMIFHPTKEEFNDFDKYIAYMESQGAHRAGLAKIIPPKEWKARETYDNISEILIATPLQQVASGRAGVFTQYHKKKKAMTVGEYRHLANSKKYQTPPHQNFEDLERKYWKNRIYNSPIYGADISGSLFDENTKQWNLGHLGTIQDLLEKECGVVIEGVNTPYLYFGMWKTTFAWHTEDMDLYSINYLHLGEPKTWYVVPPEHGQRLERLARELFPGSSRGCGAFLRHKVALISPTVLKENGIPFNRITQEAGEFMVTFPYGYHAGFNHGFNCAEAINFATPRWIDYGKMASQCSCGEARVTFSMDAFVRILQPERYDLWKRGQDR
6HVF , Knot 134 286 0.88 40 196 276 
GHMQTQGLAKDAWEIPRESLRLEVKLGQGCFGEVWMGTWNGTTRVAIKTLKPGTMSPEAFLQEAQVMKKLRHEKLVQLYAVVSEEPIYIVMEYMSKGCLLDFLKGEMGKYLRLPQLVDMAAQIASGMAYVERMNYVHRDLRAANILVGENLVCKVADFGLARLIEDNEYTARQGAKFPIKWTAPEAALYGRFTIKSDVWSFGILLTELTTKGRVPYPGMVNREVLDQVERGYRMPCPPECPESLHDLMCQCWRKDPEERPTFEYLQAFLEDYFTSTEPQYQPGENL
7TRP , Knot 148 349 0.82 40 205 331 
DYKDDDDAMANFTPVNGSSGNQSVRLVTSSSHNRYETVEMVFIATVTGSLSLVTVVGNILVMLSIKVNRQLQTVNNYFLFSLACADLIIGAFSMNLYTVYIIKGYWPLGAVVCDLWLALDYVVSNASVMNLLIISFDRYFCVTKPLTYPARRTTKMAGLMIAAAWVLSFVLWAPAILFWQFVVGKRTVPDNQCFIQFLSNPAVTFGTAIAAFYLPVVIMTVLYIHISLASRSRVHKHRPEGPKEKKAKTKRQMAARERKVTRTIFAILLAFILTWTPYNVMVLVNTFCQSCIPDTVWSIGYWLCYVNSTINPACYALCNATFKKTFRHLLLCQYRNIGTARHHHHHHHH

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 \(c\).

\(|P_{f(5PHD_1)}(2) \setminus P_{f(6HVF_1)}(2)|=103\), \(|P_{f(6HVF_1)}(2) \setminus P_{f(5PHD_1)}(2)|=70\). 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 1:1000000001101100010100100100010010010001111010000100100011010001100111101111001010000001001111011001101011110000000011011000011000000011000100100001000100011011010101100000010110110100110000111101100101011110001110000101001001011010010111100100100110011110001011110001111010110001111001000110111011010011100110010110110101100101100000101010101011101101000011001000
Pair \(Z_2\) Length of longest common subsequence
5PHD_1,6HVF_1 173 4
5PHD_1,7TRP_1 190 6
6HVF_1,7TRP_1 191 4

Newick tree

 
[
	7TRP_1:97.99,
	[
		5PHD_1:86.5,6HVF_1:86.5
	]:11.49
]

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 \((0.85)(0.8)(\frac{650 }{\log_{20} 650}-\frac{286}{\log_{20}286})=101.\)
Status Protein1 Protein2 d d1/2
Query variables 5PHD_1 6HVF_1 131 116
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]