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Parikh vectors
9UXE_1 3KAS_1 4HKZ_1 Letter Amino acid
77 50 14 A Alanine
71 27 6 I Isoleucine
53 24 10 Y Tyrosine
61 37 10 D Aspartic acid
30 4 4 C Cysteine
107 64 14 L Leucine
76 39 9 F Phenylalanine
56 44 13 K Lycine
65 25 12 P Proline
100 49 31 S Serine
95 38 20 T Threonine
88 36 6 N Asparagine
64 21 15 Q Glutamine
49 34 9 E Glutamic acid
24 12 3 H Histidine
95 48 16 V Valine
42 26 7 R Arginine
85 41 11 G Glycine
11 10 1 M Methionine
10 11 2 W Tryptophan 

9UXE_1|Chains A, B, C|Spike glycoprotein|Severe acute respiratory syndrome coronavirus 2 (2697049)
>3KAS_1|Chain A|Transferrin receptor protein 1|Homo sapiens (9606)
>4HKZ_1|Chain A|Trastuzumab light chain|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 \(w=f(c)\)
9UXE , Knot 446 1259 0.84 40 326 1088 
MFVFLVLLPLVSSQCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHVSGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLGVYYHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPINLVRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFLPFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQDVNCTEVPVAIHADQLTPTWRVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSPGSASSVASQSIIAYTMSLGAENSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGIAVEQDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSPIEDLLFNKVTLADAGFIKQYGDCLGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGPALQIPFPMQMAYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLSSTPSALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDILSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVLLSTFLGRSLEVLFQGPGHHHHHHHHSA
3KAS , Knot 256 640 0.86 40 285 592 
RLYWDDLKRKLSEKLDSTDFTSTIKLLNENSYVPREAGSQKDENLALYVENQFREFKLSKVWRDQHFVKIQVKDSAQNSVIIVDKNGRLVYLVENPGGYVAYSKAATVTGKLVHANFGTKKDFEDLYTPVNGSIVIVRAGKITFAEKVANAESLNAIGVLIYMDQTKFPIVNAELSFFGHAHLGTGDPYTPGFPSFNHTQFPPSRSSGLPNIPVQTISRAAAEKLFGNMEGDCPSDWKTDSTCRMVTSESKNVKLTVSNVLKEIKILNIFGVIKGFVEPDHYVVVGAQRDAWGPGAAKSGVGTALLLKLAQMFSDMVLKDGFQPSRSIIFASWSAGDFGSVGATEWLEGYLSSLHLKAFTYINLDKAVLGTSNFKVSASPLLYTLIEKTMQNVKHPVTGQFLYQDSNWASKVEKLTLDNAAFPFLAYSGIPAVSFCFCEDTDYPYLGTTMDTYKELIERIPELNKVARAAAEVAGQFVIKLTHDVELNLDYERYNSQLLSFVRDLNQYRADIKEMGLSLQWLYSARGDFFRATSRLTTDFGNAEKTDRFVMKKLNDRVMRVEYHFLSPYVSPKESPFRHVFWGSGSHTLPALLENLKLRKQNNGAFNETLFRNQLALATWTIQGAANALSGDVWDIDNEF
4HKZ , Knot 101 213 0.84 40 153 204 
DIQMTQSPSSLSASVGDRVTITCRASQDVNTAVAWYQQKPGKAPKLLIYSASFLYSGVPSRFSGSRSGTDFTLTISSLQPEDFATYYCQQHYTTPPTFGQGTKVEIKRTVAAPSVFIFPPSDEQLKSGTASVVCLLNNFYPREAKVQWKVDNALQSGNSQESVTEQDSKDSTYSLSSTLTLSKADYEKHKVYACEVTHQGLSSPVTKSFNRGE

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 \(c\).

\(|P_{f(9UXE_1)}(2) \setminus P_{f(3KAS_1)}(2)|=74\), \(|P_{f(3KAS_1)}(2) \setminus P_{f(9UXE_1)}(2)|=33\). 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 1:11111111111000010100000111100001001100100110001100000111111001011011010100100010011111001101100000011011111001000000111100100111010010100011111000000001100010100010000100100111101010010100100111001010101000001101100110110110111011111010010011110000101100001101111100110101001110000010100110011011000000100101001100000101010001101101001011101101001101011000010001100011000101001000110100100101001010011101001001111001011000001100101011110000100011100001001100001011000100010011001001101100011100011010011100100111101011011101011000001100001010101101011100000011110011001100001100100101101010011110110110000001111000100001111101001010101000100110001101111001000000011111110100000000110100110001110010111000110000011110010101000111101000010001010100000001110010100010011011110000000011101001000111001111010011101001000011001110010110111100010011011100110100101101111110001110000111110100110111111101111101100101111000110000011100100111010001000101110100110001011001100100011110011001100101101010100110101001000100011011010101011100100011100001010101001101100110111110100111000010011110001010110011110010011100001001011000001101000111111000100110101001000100010000010101101011010110100010010011001000110100110000101011011001010100010111100111001011101110000000001
Pair \(Z_2\) Length of longest common subsequence
9UXE_1,3KAS_1 107 4
9UXE_1,4HKZ_1 197 5
3KAS_1,4HKZ_1 180 4

Newick tree

 
[
	4HKZ_1:10.47,
	[
		9UXE_1:53.5,3KAS_1:53.5
	]:50.97
]

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 \((0.85)(0.8)(\frac{1899 }{\log_{20} 1899}-\frac{640}{\log_{20}640})=310.\)
Status Protein1 Protein2 d d1/2
Query variables 9UXE_1 3KAS_1 396 296.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]