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Parikh vectors
8ZHG_1 8RUO_1 4NAE_1 Letter Amino acid
42 3 12 R Arginine
51 12 18 E Glutamic acid
71 10 14 I Isoleucine
105 14 5 S Serine
12 2 3 W Tryptophan
93 16 23 V Valine
78 14 25 A Alanine
66 10 7 Q Glutamine
26 1 5 H Histidine
103 15 20 L Leucine
76 7 6 F Phenylalanine
61 10 19 D Aspartic acid
94 21 14 G Glycine
58 23 9 K Lycine
53 5 11 Y Tyrosine
88 11 6 N Asparagine
30 0 4 C Cysteine
10 6 3 M Methionine
66 11 9 P Proline
95 14 12 T Threonine 

8ZHG_1|Chain A[auth C]|Spike glycoprotein,Fibritin,Expression Tag|Severe acute respiratory syndrome coronavirus 2 (2697049)
>8RUO_1|Chain A|Outer membrane protein MIP|Legionella pneumophila (446)
>4NAE_1|Chain A|Heptaprenylglyceryl phosphate synthase|Geobacillus kaustophilus (235909)
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 \(w=f(c)\)
8ZHG , Knot 451 1278 0.84 40 328 1094 
VSSQCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHVSGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLGVYYHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPINLVRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFLPFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQDVNCTEVPVAIHADQLTPTWRVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSPGSASSVASQSIIAYTMSLGAENSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGIAVEQDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSPIEDLLFNKVTLADAGFIKQYGDCLGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGPALQIPFPMQMAYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLSSTPSALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDILSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGRSLEVLFQGPGHHHHHHHHSAWSHPQFEKGGGSGGGGSGGSAWSHPQFEK
8RUO , Knot 91 205 0.78 38 139 196 
TDKDKLSYSIGADLGKNFKNQGIDVNPEAMAKGMQEAMSGAQLALTEQQMKDVLNKFQKDLMAKRTAEFNKKADENKVKGEAFLTENKNKPGVVVLPSGLQYKVINSGNGVKPGKSDTVTVEYTGRLIDGTVFDSTEKTGKPATFQVSQVIPGWTEALQLMPAGSTWEIYVPSGLAYGPRSVGGPIGPNETLIFKIHLISVKKSS
4NAE , Knot 102 225 0.81 40 146 213 
EIRAWRHVFKLDPNKPIDDERLERLCESGTDAVIVGGTDGVTIDNVLDLLARIRRFSVPCALEVTDVEALTPGFDVYLVPIVLNSRQAEWIIGRHHEAVKQYGDMMNWDEIAAEGYCILNPECKAAKLTRADTELDVDDIVAYARLAEHLYKLPIFYLEYSGVYGDPSVVEKVKQALDQTQLFYGGGITTPEQAEHMARYADTVVVGNAIYDAFEQALATVAAVK

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 \(c\).

\(|P_{f(8ZHG_1)}(2) \setminus P_{f(8RUO_1)}(2)|=196\), \(|P_{f(8RUO_1)}(2) \setminus P_{f(8ZHG_1)}(2)|=7\). 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 1:100001010000011110000100110010011000110000011111100101101101010010001001111100110110000001101111100100000011110010011101001010001111100000000110001010001000010010011110101001010010011100101010100000110110011011011011101111101001001111000010110000110111110011010100111000001010011001101100000010010100110000010101000110110100101110110100110101100001000110001100010100100011010010010100101001110100100111100101100000110010101111000010001110000100110000101100010001001100100110110001110001101001110010011110101101110101100000110000101010110101110000001111001100110000110010010110101001111011011000000111100010000111110100101010100010011000110111100100000001111111010000000011010011000111001011100011000001111001010100011110100001000101010000000111001010001001101111000000001110100100011100111101001110100100001100111001011011110001001101110011010010110111111000111000011111010011011111110111110110010111100011000001110010011101000100010111010011000101100110010001111001100110010110101010011010100100010001101101010101110010001110000101010100110110011011111010011100001001111000101011001111001001110000100101100000110100011111100010011010100100010001000001010110101101011010001001001100100011010011000010101101100101010001011110011100101110111000000000110010100111011110110110010100
Pair \(Z_2\) Length of longest common subsequence
8ZHG_1,8RUO_1 203 4
8ZHG_1,4NAE_1 210 5
8RUO_1,4NAE_1 151 3

Newick tree

 
[
	8ZHG_1:11.98,
	[
		8RUO_1:75.5,4NAE_1:75.5
	]:35.48
]

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 \((0.85)(0.8)(\frac{1483 }{\log_{20} 1483}-\frac{205}{\log_{20}205})=335.\)
Status Protein1 Protein2 d d1/2
Query variables 8ZHG_1 8RUO_1 418 240.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]