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Parikh vectors
6XUL_1 6ZHD_1 6HVT_1 Letter Amino acid
34 105 23 S Serine
17 95 19 T Threonine
5 12 2 W Tryptophan
7 61 14 D Aspartic acid
5 30 0 C Cysteine
21 94 23 G Glycine
9 79 9 F Phenylalanine
12 63 12 P Proline
8 66 6 Q Glutamine
4 26 3 H Histidine
2 11 4 M Methionine
20 95 16 V Valine
11 58 18 K Lycine
9 53 10 Y Tyrosine
15 81 18 A Alanine
9 88 6 N Asparagine
9 51 17 E Glutamic acid
5 71 15 I Isoleucine
17 107 26 L Leucine
11 42 9 R Arginine 

6XUL_1|Chains A, C, E, G, I[auth H], K[auth J]|Heavy chain|Homo sapiens (9606)
>6ZHD_1|Chains A, B, C|Spike glycoprotein,Fibritin|Severe acute respiratory syndrome coronavirus 2 (2697049)
>6HVT_1|Chains A, O|Proteasome subunit alpha type-2|Saccharomyces cerevisiae (strain ATCC 204508 / S288c) (559292)
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 \(w=f(c)\)
6XUL , Knot 102 230 0.80 40 150 215 
QVQLVESGGGSVQPGRSLRLSCEASGFTFEAYAMHWVRQPPGKGLEWVSSINWNSGRIAYADSVKGRFTISRDNARNSLYLQMNSLRLEDTAFYYCAKDIRRFSTGGAEFEYWGQGTLVTVSSASTKGPSVFPLAPSSKSTSGGTAALGCLVKDYFPEPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKRVEPKSCDKTH
6ZHD , Knot 451 1288 0.83 40 329 1098 
MFVFLVLLPLVSSQCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHVSGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLGVYYHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPINLVRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFLPFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQDVNCTEVPVAIHADQLTPTWRVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSPGSASSVASQSIIAYTMSLGAENSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGIAVEQDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSFIEDLLFNKVTLADAGFIKQYGDCLGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGAALQIPFAMQMAYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLSSTASALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDILSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGRSLEVLFQGPGHHHHHHHHGSAWSHPQFEKGGGSGGGSGGSAWSHPQFEK
6HVT , Knot 111 250 0.81 38 159 239 
MTDRYSFSLTTFSPSGKLGQIDYALTAVKQGVTSLGIKATNGVVIATEKKSSSPLAMSETLSKVSLLTPDIGAVYSGMGPDYRVLVDKSRKVAHTSYKRIYGEYPPTKLLVSEVAKIMQEATQSGGVRPFGVSLLIAGHDEFNGFSLYQVDPSGSYFPWKATAIGKGSVAAKTFLEKRWNDELELEDAIHIALLTLKESVEGEFNGDTIELAIIGDENPDLLGYTGIPTDKGPRFRKLTSQEINDRLEAL

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 \(c\).

\(|P_{f(6XUL_1)}(2) \setminus P_{f(6ZHD_1)}(2)|=10\), \(|P_{f(6ZHD_1)}(2) \setminus P_{f(6XUL_1)}(2)|=189\). 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 1:01011001110101100101000101101010110110011101101100101001011010010101010000100010101001010001100010010010011101001101011010010001101111110000001101111011000110110101001110011001111100011001001101100011000010010001000010001010000000
Pair \(Z_2\) Length of longest common subsequence
6XUL_1,6ZHD_1 199 5
6XUL_1,6HVT_1 159 3
6ZHD_1,6HVT_1 188 4

Newick tree

 
[
	6ZHD_1:10.90,
	[
		6XUL_1:79.5,6HVT_1:79.5
	]:22.40
]

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 \((0.85)(0.8)(\frac{1518 }{\log_{20} 1518}-\frac{230}{\log_{20}230})=335.\)
Status Protein1 Protein2 d d1/2
Query variables 6XUL_1 6ZHD_1 427 249
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]