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Parikh vectors
2FHS_1 1SBI_1 7WYW_1 Letter Amino acid
36 38 83 A Alanine
5 6 14 H Histidine
9 13 64 T Threonine
7 10 25 Y Tyrosine
14 11 61 D Aspartic acid
4 1 23 C Cysteine
5 9 31 Q Glutamine
20 38 67 S Serine
20 30 70 V Valine
10 16 41 N Asparagine
15 5 65 E Glutamic acid
12 11 57 K Lycine
7 14 44 P Proline
12 4 46 F Phenylalanine
1 3 12 W Tryptophan
12 2 48 R Arginine
25 32 76 G Glycine
16 13 86 I Isoleucine
22 15 91 L Leucine
10 4 24 M Methionine 

2FHS_1|Chains A, B|enoyl-[acyl-carrier-protein] reductase, NADH-dependent|Escherichia coli (316407)
>1SBI_1|Chain A|SUBTILISIN 8397|Bacillus amyloliquefaciens (1390)
>7WYW_1|Chains A, D[auth C]|Sodium/potassium-transporting ATPase subunit alpha|Squalus acanthias (7797)
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 \(w=f(c)\)
2FHS , Knot 118 262 0.83 40 164 251 
MGFLSGKRILVTGVASKLSIAYGIAQAMHREGAELAFTYQNDKLKGRVEEFAAQLGSDIVLQCDVAEDASIDTMFAELGKVWPKFDGFVHSIGFAPGDQLDGDYVNAVTREGFKIAHDISSYSFVAMAKACRSMLNPGSALLTLSYLGAERAIPNYNVMGLAKASLEANVRYMANAMGPEGVRVNAISAGPIRTLAASGIKDFRKMLAHCEAVTPIRRTVTIEDVGNSAAFLCSDLSAGISGEVVHVDGGFSIAAMNELELK
1SBI , Knot 115 275 0.78 40 148 254 
AQSVPYGVSQIKAPALHSQGYTGSNVKVAVIDSGIDSSHPDLKVAGGASFVPSETNPFQDNNSHGTHVAGTVAALDNSIGVLGVAPSASLYAVKVLGADGSGQYSWIINGIEWAIANNMDVINMSLGGPSGSAALKAAVDKAVASGVVVVAAAGNEGTSGSSSTVGYPAKYPSVIAVGAVDSSNQRASFSSVGPELDVMAPGVSICSTLPGNKYGAYSGTSMASPHVAGAAALILSKHPNWTNTQVRSSLENTTTKLGDSFYYGKGLINVQAAAQ
7WYW , Knot 379 1028 0.85 40 311 914 
MGKGTASDKYEPAATSENATKSKKKGKKDKIDKKRDLDELKKEVSMDDHKLSLDELHNKYGTDLTRGLTNARAKEILARDGPNSLTPPPTTPEWIKFCRQLFGGFSILLWIGAILCFLAYGIQAATEDEPANDNLYLGVVLSTVVIVTGCFSYYQEAKSSRIMDSFKNMVPQQALVIRDGEKSTINAEFVVAGDLVEVKGGDRIPADLRIISAHGCKVDNSSLTGESEPQTRSPEFSSENPLETRNIAFFSTNCVEGTARGVVVYTGDRTVMGRIATLASGLEVGRTPIAIEIEHFIHIITGVAVFLGVSFFILSLILGYSWLEAVIFLIGIIVANVPEGLLATVTVCLTLTAKRMARKNCLVKNLEAVETLGSTSTICSDKTGTLTQNRMTVAHMWFDNQIHEADTTENQSGAAFDKTSATWSALSRIAALCNRAVFQAGQDNVPILKRSVAGDASESALLKCIELCCGSVQGMRDRNPKIVEIPFNSTNKYQLSIHENEKSSESRYLLVMKGAPERILDRCSTILLNGAEEPLKEDMKEAFQNAYLELGGLGERVLGFCHFALPEDKYNEGYPFDADEPNFPTTDLCFVGLMAMIDPPRAAVPDAVGKCRSAGIKVIMVTGDHPITAKAIAKGVGIISEGNETIEDIAARLNIPIGQVNPRDAKACVVHGSDLKDLSTEVLDDILHYHTEIVFARTSPQQKLIIVEGCQRQGAIVAVTGDGVNDSPALKKADIGVAMGISGSDVSKQAADMILLDDNFASIVTGVEEGRLIFDNLKKSIAYTLTSNIPEITPFLVFIIGNVPLPLGTVTILCIDLGTDMVPAISLAYEQAESDIMKRQPRNPKTDKLVNERLISMAYGQIGMIQALGGFFSYFVILAENGFLPMDLIGKRVRWDDRWISDVEDSFGQQWTYEQRKIVEFTCHTSFFISIVVVQWADLIICKTRRNSIFQQGMKNKILIFGLFEETALAAFLSYCPGTDVALRMYPLKPSWWFCAFPYSLIIFLYDEMRRFIIRRSPGGWVEQETYY

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 \(c\).

\(|P_{f(2FHS_1)}(2) \setminus P_{f(1SBI_1)}(2)|=86\), \(|P_{f(1SBI_1)}(2) \setminus P_{f(2FHS_1)}(2)|=70\). 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 1:1111010011101110010110111011000110111000000101010011101100111000110010100111011011101011100111111001010010110001101100100001111101000110110111010011100111000111110101010100110111101101011011110011101100100111000110110001010011001111000101110101101011101111001010
Pair \(Z_2\) Length of longest common subsequence
2FHS_1,1SBI_1 156 3
2FHS_1,7WYW_1 199 5
1SBI_1,7WYW_1 193 4

Newick tree

 
[
	7WYW_1:10.82,
	[
		2FHS_1:78,1SBI_1:78
	]:25.82
]

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 \((0.85)(0.8)(\frac{537 }{\log_{20} 537}-\frac{262}{\log_{20}262})=78.1\)
Status Protein1 Protein2 d d1/2
Query variables 2FHS_1 1SBI_1 97 96
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]