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Parikh vectors
3PHM_1 3QVW_1 7RYQ_1 Letter Amino acid
11 14 20 N Asparagine
18 19 29 D Aspartic acid
24 31 21 G Glycine
22 15 20 T Threonine
15 17 27 Y Tyrosine
26 27 34 V Valine
15 24 29 I Isoleucine
16 12 15 M Methionine
22 15 27 P Proline
17 25 27 S Serine
3 5 4 W Tryptophan
21 37 57 A Alanine
13 36 77 E Glutamic acid
11 8 23 H Histidine
11 33 42 K Lycine
15 22 23 F Phenylalanine
15 12 35 R Arginine
10 1 12 C Cysteine
7 2 26 Q Glutamine
18 37 73 L Leucine 

3PHM_1|Chain A|PROTEIN (PEPTIDYLGLYCINE ALPHA-HYDROXYLATING MONOOXYGENASE)|Rattus norvegicus (10116)
>3QVW_1|Chain A|Myo-inositol-1-phosphate synthase (Ino1)|Archaeoglobus fulgidus (2234)
>7RYQ_1|Chain A[auth B]|KIF-binding protein|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 \(w=f(c)\)
3PHM , Knot 138 310 0.85 40 205 297 
NECLGTIGPVTPLDASDFALDIRMPGVTPKESDTYFCMSMRLPVDEEAFVIDFKPRASMDTVHHMLLFGCNMPSSTGSYWFCDEGTCTDKANILYAWARNAPPTRLPKGVGFRVGGETGSKYFVLQVHYGDISAFRDNHKDCSGVSVHLTRVPQPLIAGMYLMMSVDTVIPPGEKVVNADISCQYKMYPMHVFAYRVHTHHLGKVVSGYRVRNGQWTLIGRQNPQLPQAFYPVEHPVDVTFGDILAARCVFTGEGRTEATHIGGTSSDEMCNLYIMYYMEAKYALSFMTCTKNVAPDMFRTIPAEANIPI
3QVW , Knot 162 392 0.82 40 209 367 
MKVWLVGAYGIVSTTAMVGARAIERGIAPKIGLVSELPHFEGIEKYAPFSFEFGGHEIRLLSNAYEAAKEHWELNRHFDREILEAVKSDLEGIVARKGTALNCGSGIKELGDIKTLEGEGLSLAEMVSRIEEDIKSFADDETVVINVASTEPLPNYSEEYHGSLEGFERMIDEDRKEYASASMLYAYAALKLGLPYANFTPSPGSAIPALKELAEKKGVPHAGNDGKTGETLVKTTLAPMFAYRNMEVVGWMSYNILGDYDGKVLSARDNKESKVLSADKVLEKMLGYSPYSITEIQYFPSLVDNKTAFDFVHFKGFLGKLMKFYFIWDAIDAIVAAPLILDIARFLLFAKKKGVKGVVKEMAFFFKSPMDTNVINTHEQFVVLKEWYSNLK
7RYQ , Knot 243 621 0.84 40 263 575 
MANVPWAEVCEKFQAALALSRVELHKNPEKEPYKSKYSARALLEEVKALLGPAPEDEDERPEAEDGPGAGDHALGLPAEVVEPEGPVAQRAVRLAVIEFHLGVNHIDTEELSAGEEHLVKCLRLLRRYRLSHDCISLCIQAQNNLGILWSEREEIETAQAYLESSEALYNQYMKEVGSPPLDPTERFLPEEEKLTEQERSKRFEKVYTHNLYYLAQVYQHLEMFEKAAHYCHSTLKRQLEHNAYHPIEWAINAATLSQFYINKLCFMEARHCLSAANVIFGQTGKISATEDTPEAEGEVPELYHQRKGEIARCWIKYCLTLMQNAQLSMQDNIGELDLDKQSELRALRKKELDEEESIRKKAVQFGTGELCDAISAVEEKVSYLRPLDFEEARELFLLGQHYVFEAKEFFQIDGYVTDHIEVVQDHSALFKVLAFFETDMERRCKMHKRRIAMLEPLTVDLNPQYYLLVNRQIQFEIAHAYYDMMDLKVAIADRLRDPDSHIVKKINNLNKSALKYYQLFLDSLRDPNKVFPEHIGEDVLRPAMLAKFRVARLYGKIITADPKKELENLATSLEHYKFIVDYCEKHPEAAQEIEVELELSKEMVSLLPTKMERFRTKMALT

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 \(c\).

\(|P_{f(3PHM_1)}(2) \setminus P_{f(3QVW_1)}(2)|=93\), \(|P_{f(3QVW_1)}(2) \setminus P_{f(3PHM_1)}(2)|=97\). 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 1:0001101111011010011101011110100000010101011100011110101010100100111110011000100110001000001011011100111001101111011100100011101001010110000000011010100110111111011101001111100110101000001011011100100001101101001001010111000101101101100110101101111001101010001001110000010010110010100110110000011101100111010111
Pair \(Z_2\) Length of longest common subsequence
3PHM_1,3QVW_1 190 4
3PHM_1,7RYQ_1 182 3
3QVW_1,7RYQ_1 162 4

Newick tree

 
[
	3PHM_1:96.69,
	[
		7RYQ_1:81,3QVW_1:81
	]:15.69
]

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 \((0.85)(0.8)(\frac{702 }{\log_{20} 702}-\frac{310}{\log_{20}310})=108.\)
Status Protein1 Protein2 d d1/2
Query variables 3PHM_1 3QVW_1 136 123.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]