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Parikh vectors
9FUT_1 6BGP_1 4ISO_1 Letter Amino acid
25 13 15 Q Glutamine
26 10 3 M Methionine
21 17 5 N Asparagine
30 27 15 D Aspartic acid
28 30 10 E Glutamic acid
38 25 28 G Glycine
13 17 14 P Proline
50 20 17 S Serine
13 13 6 Y Tyrosine
38 20 13 R Arginine
2 8 7 C Cysteine
47 23 14 I Isoleucine
23 22 6 K Lycine
36 19 12 T Threonine
5 11 8 W Tryptophan
47 20 16 V Valine
48 19 16 A Alanine
9 15 7 H Histidine
60 30 19 L Leucine
23 23 10 F Phenylalanine 

9FUT_1|Chains A, B|ATP-dependent lipid A-core flippase|Escherichia coli (562)
>6BGP_1|Chains A, B, C, D|Calpain-3|Homo sapiens (9606)
>4ISO_1|Chain A|Suppressor of tumorigenicity 14 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 \(w=f(c)\)
9FUT , Knot 231 582 0.84 40 254 532 
MHNDKDLSTWQTFRRLWPTIAPFKAGLIVAGVALILNAASDTFMLSLLKPLLDDGFGKTDRSVLVWMPLVVIGLMILRGITSYVSSYCISWVSGKVVMTMRRRLFGHMMGMPVSFFDKQSTGTLLSRITYDSEQVASSSSGALITVVREGASIIGLFIMMFYYSWQLSIILIVLAPIVSIAIRVVSKRFRNISKNMQNTMGQVTTSAEQMLKGHKEVLIFGGQEVETKRFDKVSNRMRLQGMKMVSASSISDPIIQLIASLALAFVLYAASFPSVMDSLTAGTITVVFSSMIALMRPLKSLTNVNAQFQRGMAACQTLFTILDSEQEKDEGKRVIERATGDVEFRNVTFTYPGRDVPALRNINLKIPAGKTVALVGRSGSGKSTIASLITRFYDIDEGEILMDGHDLREYTLASLRNQVALVSQNVHLFNDTVANNIAYARTEQYSREQIEEAARMAYAMDFINKMDNGLDTVIGENGVLLSGGQRQRIAIARALLRDSPILILDEATSALDTESERAIQAALDELQKNRTSLVIAHRLSTIEKADEIVVVEDGVIVERGTHNDLLEHRGVYAQLHKMQFGQ
6BGP , Knot 163 382 0.84 40 236 365 
IISRNFPIIGVKEKTFEQLHKKCLEKKVLYVDPEFPPDETSLFYSQKFPIQFVWKRPPEICENPRFIIDGANRTDICQGELGDAWFLAAIACLTLNQHLLFRVIPHDQSFIENYAGIFHFQFWRYGEWVDVVIDDCLPTYNNQLVFTKSNHRNEFWSALLEKAYAKLHGSYEALKGGNTTEAMEDFTGGVAEFFEIRDAPSDMYKIMKKAIERGSLMGCSIDDGTNMTYGTSPSGLNMGELIARMVRNMDNSLLQDSDLDPRGSDERPTRTIIPVQYETRMACGLVRGHAYSVTGLDEVPFKGEKVKLVRLRNPWGQVEWNGSWSDRWKDWSFVDKDEKARLQHQVTEDGEFWMSYEDFIYHFTKLEICNLTADLEHHHHHH
4ISO , Knot 113 241 0.85 40 183 236 
VVGGTDADEGEWPWQVSLHALGQGHICGASLISPNWLVSAAHCYIDDRGFRYSDPTQWTAFLGLHDQSQRSAPGVQERRLKRIISHPFFNDFTFDYDIALLELEKPAEYSSMVRPICLPDASHVFPAGKAIWVTGWGHTQYGGTGALILQKGEIRVIQQTTCENLLPQQITPRMMCVGFLSGGVDSCQGDSGGPLSSVEADGRIFQAGVVSWGDGCAQRNKPGVYTRLPLFRDWIKENTGV

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 \(c\).

\(|P_{f(9FUT_1)}(2) \setminus P_{f(6BGP_1)}(2)|=83\), \(|P_{f(6BGP_1)}(2) \setminus P_{f(9FUT_1)}(2)|=65\). 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 1:100000100100100111011110111111111111011000111011011100111000001111111111111110110001000010110101110100011101111110110000010110010000001100001111011001101111111110001010111111111101110110001001000100011010001001101000111111001000010010001010110110100100111011101111111011011011001011010111001111101100100101010011110001101100000000100110010101010010100110011110010101111001111100101000110110010010010111010010000110100011110001011000110011010000000001001101101101100100110011100111101100001111011100011111001001100000011011100100000011110010010010011110011110010000110001101010010110
Pair \(Z_2\) Length of longest common subsequence
9FUT_1,6BGP_1 148 4
9FUT_1,4ISO_1 169 4
6BGP_1,4ISO_1 185 4

Newick tree

 
[
	4ISO_1:92.94,
	[
		9FUT_1:74,6BGP_1:74
	]:18.94
]

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 \((0.85)(0.8)(\frac{964 }{\log_{20} 964}-\frac{382}{\log_{20}382})=154.\)
Status Protein1 Protein2 d d1/2
Query variables 9FUT_1 6BGP_1 195 161
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]