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Parikh vectors
8CYC_1 8PJZ_1 4JJR_1 Letter Amino acid
79 21 13 A Alanine
48 18 16 E Glutamic acid
14 10 10 M Methionine
42 18 22 R Arginine
62 24 16 D Aspartic acid
97 15 11 T Threonine
99 19 19 S Serine
88 15 13 N Asparagine
82 40 25 G Glycine
76 6 19 I Isoleucine
108 15 32 L Leucine
77 0 16 F Phenylalanine
60 15 9 P Proline
12 6 3 W Tryptophan
54 18 18 Y Tyrosine
40 0 5 C Cysteine
62 0 9 Q Glutamine
17 22 8 H Histidine
60 0 23 K Lycine
96 36 12 V Valine 

8CYC_1|Chains A, B, C|Spike glycoprotein|Severe acute respiratory syndrome coronavirus 2 (2697049)
>8PJZ_1|Chains A, B|SAKe6EEtal|synthetic construct (32630)
>4JJR_1|Chains A, B|Casein kinase I isoform delta|Mus musculus (10090)
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 \(w=f(c)\)
8CYC , Knot 453 1273 0.84 40 338 1104 
MFVFLVLLPLVSSQCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHAIHVSGTNGTKRFDNPVLPFNDGVYFASTEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLGVYYHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPINLVRDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLTGTGVLTESNKKFLPFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQDVNCTEVPVAIHADQLTPTWRVYSTGSNVFQTRAGCLIGAEHVNNSYECDIPIGAGICASYQTQTNSPRRARSVASQSIIAYTMSLGAENSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLNRALTGIAVEQDKNTQEVFAQVKQIYKTPPIKDFGGFNFSQILPDPSKPSKRSFIEDLLFNKVTLADAGFIKQYGDCLGDIAARDLICAQKFNGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGAALQIPFAMQMAYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLSSTASALGKLQDVVNQNAQALNTLVKQLSSNFGAISSVLNDILSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQYIKWPWYIWLGFIAGLIAIVMVTIMLCCMTSCCSCLKGCCSCGSCCKFDEDDSEPVLKGVKLHYT
8PJZ , Knot 38 298 0.24 32 50 63 
GSHMNGHIYAVGGYDGSPDGHTHLNSVEAYDPERDEWSLVAPMNTRRSGVGVAVLDGHIYAVGGYDGHTHLNSVEAYDPERDEWHMVTPMNTRRSGVGVAVLNGHIYAVGGYDGSPDGHTHLNSVEAYDPERDEWSLVAPMNTRRSGVGVAVLDGHIYAVGGYDGHTHLNSVEAYDPERDEWHMVTPMNTRRSGVGVAVLNGHIYAVGGYDGSPDGHTHLNSVEAYDPERDEWSLVAPMNTRRSGVGVAVLDGHIYAVGGYDGHTHLNSVEAYDPERDEWHMVTPMNTRRSGVGVAVL
4JJR , Knot 135 299 0.85 40 191 294 
MELRVGNRYRLGRKIGSGSFGDIYLGTDIAAGEEVAIKLECVKTKHPQLHIESKIYKMMQGGVGIPTIRWCGAEGDYNVMVMELLGPSLEDLFNFCSRKFSLKTVLLLADQMISRIEYIHSKNFIHRDVKPDNFLMGLGKKGNLVYIIDFGLAKKYRDARTHQHIPYRENKNLTGTARYASINTHLGIEQSRRDDLESLGYVLMYFNLGSLPWQGLKAATKRQKYERISEKKMSTPIEVLCKGYPSEFATYLNFCRSLRFDDKPDYSYLRQLFRNLFHRQGFSYDYVFDWNMLKFGASR

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 \(c\).

\(|P_{f(8CYC_1)}(2) \setminus P_{f(8PJZ_1)}(2)|=297\), \(|P_{f(8PJZ_1)}(2) \setminus P_{f(8CYC_1)}(2)|=9\). 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 1:1111111111100001010000011110000100110010011000110000011111100101101101010010001001111100110110000001101111100100000011110010011101001010001111100000000110001010001000010010011110101001010010011100101010100000110110011011011011101111101001001111000010110000110111110011010100111000001010011001101100000010010100110000010101000110110100101110110100110101100001000110001100010100100011010010010100101001110100100111100101100000110010101111000010001110000100110000101100010001001100100110110001110001101001110010011110101101110101100000110000101010110101110000001111001100110000110010010110101001111011011000000111100010000111110100101010100010011000110111100100000001111111010000000010010011000111001011100011000001111001010100011110100001000101010000000111001010001001101111000000001110100100011100111101001110100100001100111001011011110001001101110011010010110111111000111000011111010011011111110111110110010111100011000001110010011101000100010111010011000101100110010001111001100110010110101010011010100100010001101101010101110010001110000101010100110110011011111010011100001001111000101011001111001001110000100101100000110100011111100010011010100100010001000001010110101101011010001001001100100011010011000001011101111111111111110111001000000101000010000100000011101101000
Pair \(Z_2\) Length of longest common subsequence
8CYC_1,8PJZ_1 306 4
8CYC_1,4JJR_1 173 4
8PJZ_1,4JJR_1 191 3

Newick tree

 
[
	8PJZ_1:13.53,
	[
		8CYC_1:86.5,4JJR_1:86.5
	]:52.03
]

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 \((0.85)(0.8)(\frac{1571 }{\log_{20} 1571}-\frac{298}{\log_{20}298})=328.\)
Status Protein1 Protein2 d d1/2
Query variables 8CYC_1 8PJZ_1 442 233
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]