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Parikh vectors
9CAS_1 2WGJ_1 5WSV_1 Letter Amino acid
46 15 21 E Glutamic acid
70 16 8 I Isoleucine
63 15 3 P Proline
108 19 5 S Serine
89 14 12 T Threonine
46 13 6 R Arginine
70 16 17 D Aspartic acid
11 4 0 W Tryptophan
58 8 6 Q Glutamine
83 15 8 F Phenylalanine
93 26 7 V Valine
102 34 9 L Leucine
56 20 7 K Lycine
15 10 10 M Methionine
83 14 10 A Alanine
86 11 6 N Asparagine
30 6 0 C Cysteine
89 20 13 G Glycine
23 20 1 H Histidine
50 10 2 Y Tyrosine 

9CAS_1|Chains A, B, C|Spike glycoprotein|Bat SARS-like coronavirus RsSHC014 (1415851)
>2WGJ_1|Chain A|HEPATOCYTE GROWTH FACTOR RECEPTOR|HOMO SAPIENS (9606)
>5WSV_1|Chains A, C|Calmodulin|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 \(w=f(c)\)
9CAS , Knot 448 1271 0.84 40 330 1104 
MKLLVLVFATLVSSYTIEKCLDFDDRTPPANTQFLSSHRGVYYPDDIFRSNVLHLVQDHFLPFDSNVTRFITFGLNFDNPIIPFRDGIYFAATEKSNVIRGWVFGSTMNNKSQSVIIMNNSTNLVIRACNFELCDNPFFVVLKSNNTQIPSYIFNNAFNCTFEYVSKDFNLDLGEKPGNFKDLREFVFRNKDGFLHVYSGYQPISAASGLPTGFNALKPIFKLPLGINITNFRTLLTAFPPRPDYWGTSAAAYFVGYLKPTTFMLKYDENGTITDAVDCSQNPLAELKCSVKSFEIDKGIYQTSNFRVAPSKEVVRFPNITNLCPFGEVFNATTFPSVYAWERKRISNCVADYSVLYNSTSFSTFKCYGVSATKLNDLCFSNVYADSFVVKGDDVRQIAPGQTGVIADYNYKLPDDFLGCVLAWNTNSKDSSTSGNYNYLYRWVRRSKLNPYERDLSNDIYSPGGQSCSAVGPNCYNPLRPYGFFTTAGVGHQPYRVVVLSFELLNAPATVCGPKLSTDLIKNQCVNFNFNGLTGTGVLTPSSKRFQPFQQFGRDVSDFTDSVRDPKTSEILDISPCSFGGVSVITPGTNTSSEVAVLYQDVNCTDVPVAIHADQLTPSWRVYSTGNNVFQTQAGCLIGAEHVDTSYECDIPIGAGICASYHTVSSLRSTSQKSIVAYTMSLGADSSIAYSNNTIAIPTNFSISITTEVMPVSMAKTSVDCNMYICGDSTECANLLLQYGSFCTQLNRALSGIAVEQDRNTREVFAQVKQMYKTPTLKDFGGFNFSQILPDPLKPTKRSFIEDLLFNKVTLADAGFMKQYGECLGDINARDLICAQKFNGLTVLPPLLTDDMIAAYTAALVSGTATAGWTFGAGAALQIPFAMQMAYRFNGIGVTQNVLYENQKQIANQFNKAISQIQESLTTTSTALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDILSRLDKVEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQAAPHGVVFLHVTYVPSQERNFTTAPAICHEGKAYFPREGVFVFNGTSWFITQRNFFSPQIITTDNTFVSGSCDVVIGIINNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGRSLEVLFQGPGHHHHHHHHSAWSHPQFEKGGGSGGGGSGGSAWSHPQFEK
2WGJ , Knot 134 306 0.83 40 201 292 
MVHIDLSALNPELVQAVQHVVIGPSSLIVHFNEVIGRGHFGCVYHGTLLDNDGKKIHCAVKSLNRITDIGEVSQFLTEGIIMKDFSHPNVLSLLGICLRSEGSPLVVLPYMKHGDLRNFIRNETHNPTVKDLIGFGLQVAKGMKYLASKKFVHRDLAARNCMLDEKFTVKVADFGLARDMYDKEYYSVHNKTGAKLPVKWMALESLQTQKFTTKSDVWSFGVLLWELMTRGAPPYPDVNTFDITVYLLQGRRLLQPEYCPDPLYEVMLKCWHPKAEMRPSFSELVSRISAIFSTFIGEHHHHHHHH
5WSV , Knot 69 151 0.76 36 99 136 
GPGSMADQLTEEQIAEFKEAFSLFDKDGDGTITTKELGTVMRSLGQNPTEAELQDMINEVDADGNGTIDFPEFLTMMARKMKDTDSEEEIREAFRVFDKDGNGYISAAELRHVMTNLGEKLTDEEVDEMIREADIDGDGQVNYEEFVQMMT

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 \(c\).

\(|P_{f(9CAS_1)}(2) \setminus P_{f(2WGJ_1)}(2)|=153\), \(|P_{f(2WGJ_1)}(2) \setminus P_{f(9CAS_1)}(2)|=24\). 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 1:10111111101100001000101000011100011000011001001100011011000111100010011011101001111100110111000001101111100100000011110000011101001010001111110000001100110011000100100010101100110100100111000011101001001101101110110110111011111010010011011110100110011101110101001110000010100110000011101000100101001100000101110001101101001011101101001101011000010001100011000001001000110100100101001010011101001001111001111000001100111011110000000000100001001100001010000100010011100001111000011010111001111001001111010110111010110100011000010101011010111010000101100110010010001001000011010100111101101100000011110001000011111010010101010001001100011011110010000000111111101000010010000000111001011100011000001111001010100011110110001000101010000010111001010001001101111000000001110100100010100111101001110110100001100111001011011110001001101010011010010110111111000111100111101010111011111110111110110010111100011000000110010011001000100000111010011000101100110010001111001100110010010101010011010100100010001101101010101110010001110000101010100110110111011111010011000001001111000101011001111101001110000110101100000110100011111100010011010100100010001000001010110101101011010001001001100100011010011000010101101100101010001011110011100101110111000000000110010100111011110110110010100
Pair \(Z_2\) Length of longest common subsequence
9CAS_1,2WGJ_1 177 8
9CAS_1,5WSV_1 241 5
2WGJ_1,5WSV_1 186 3

Newick tree

 
[
	5WSV_1:11.29,
	[
		9CAS_1:88.5,2WGJ_1:88.5
	]:24.79
]

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 \((0.85)(0.8)(\frac{1577 }{\log_{20} 1577}-\frac{306}{\log_{20}306})=327.\)
Status Protein1 Protein2 d d1/2
Query variables 9CAS_1 2WGJ_1 412 253.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]