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Parikh vectors
7WJZ_1 8YLM_1 3UZY_1 Letter Amino acid
93 14 20 G Glycine
64 2 23 K Lycine
11 3 7 M Methionine
43 21 19 R Arginine
30 0 6 C Cysteine
63 6 10 Q Glutamine
81 0 12 F Phenylalanine
68 11 21 P Proline
12 2 5 W Tryptophan
53 2 14 Y Tyrosine
83 4 18 N Asparagine
61 9 16 D Aspartic acid
105 33 32 L Leucine
52 14 24 E Glutamic acid
73 6 23 I Isoleucine
92 12 13 T Threonine
94 5 23 V Valine
77 24 19 A Alanine
27 4 18 H Histidine
103 14 23 S Serine 

7WJZ_1|Chains A[auth B], D[auth I], G[auth J], J[auth K], M[auth L], P[auth M]|Spike glycoprotein|Severe acute respiratory syndrome coronavirus 2 (2697049)
>8YLM_1|Chains A, B, C, D|Transcriptional regulator, MarR family|Deinococcus radiodurans (1299)
>3UZY_1|Chains A, B|3-oxo-5-beta-steroid 4-dehydrogenase|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 \(w=f(c)\)
7WJZ , Knot 452 1285 0.84 40 332 1112 
MFVFLVLLPLVSSQCVNLTTRTQLPPAYTNSFTRGVYYPDKVFRSSVLHSTQDLFLPFFSNVTWFHVISGTNGTKRFDNPVLPFNDGVYFASIEKSNIIRGWIFGTTLDSKTQSLLIVNNATNVVIKVCEFQFCNDPFLDHKNNKSWMESEFRVYSSANNCTFEYVSQPFLMDLEGKQGNFKNLREFVFKNIDGYFKIYSKHTPIIVREPEDLPQGFSALEPLVDLPIGINITRFQTLLALHRSYLTPGDSSSGWTAGAAAYYVGYLQPRTFLLKYNENGTITDAVDCALDPLSETKCTLKSFTVEKGIYQTSNFRVQPTESIVRFPNITNLCPFDEVFNATRFASVYAWNRKRISNCVADYSVLYNLAPFFTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGNIADYNYKLPDDFTGCVIAWNSNKLDSKVSGNYNYLYRLFRKSNLKPFERDISTEIYQAGNKPCNGVAGFNCYFPLRSYSFRPTYGVGHQPYRVVVLSFELLHAPATVCGPKKSTNLVKNKCVNFNFNGLKGTGVLTESNKKFLPFQQFGRDIADTTDAVRDPQTLEILDITPCSFGGVSVITPGTNTSNQVAVLYQGVNCTEVPVAIHADQLTPTWRVYSTGSNVFQTRAGCLIGAEYVNNSYECDIPIGAGICASYQTQTKSHGSASSVASQSIIAYTMSLGAENSVAYSNNSIAIPTNFTISVTTEILPVSMTKTSVDCTMYICGDSTECSNLLLQYGSFCTQLKRALTGIAVEQDKNTQEVFAQVKQIYKTPPIKYFGGFNFSQILPDPSKPSKRSPIEDLLFNKVTLADAGFIKQYGDCLGDIAARDLICAQKFKGLTVLPPLLTDEMIAQYTSALLAGTITSGWTFGAGPALQIPFPMQMAYRFNGIGVTQNVLYENQKLIANQFNSAIGKIQDSLSSTPSALGKLQDVVNHNAQALNTLVKQLSSKFGAISSVLNDIFSRLDPPEAEVQIDRLITGRLQSLQTYVTQQLIRAAEIRASANLAATKMSECVLGQSKRVDFCGKGYHLMSFPQSAPHGVVFLHVTYVPAQEKNFTTAPAICHDGKAHFPREGVFVSNGTHWFVTQRNFYEPQIITTDNTFVSGNCDVVIGIVNNTVYDPLQPELDSFKEELDKYFKNHTSPDVDLGDISGINASVVNIQKEIDRLNEVAKNLNESLIDLQELGKYEQGSGYIPEAPRDGQAYVRKDGEWVFLSTFLSGLEVLFQGPGGWSHPQFEKGGGSGGGSGGSAWSHPQFEKGGSHHHHHHHH
8YLM , Knot 81 186 0.75 36 109 172 
SAKDPVSARMDNDTAALLERIRSDWARLNHGQGPDSDGLTPSAGPMLTLLLLERLHAALGREIERTYAASGLNAAGWDLLLTLYRSAPPEGLRPTELSALAAISGPSTSNRIVRLLEKGLIERREDERDRRSASIRLTPQGRALVTHLLPAHLATTQRVLAPLSAQEQRTLEELAGRMLAGLEQGV
3UZY , Knot 153 346 0.86 40 220 329 
MGSSHHHHHHSSGLVPRGSHMDLSAASHRIPLSDGNSIPIIGLGTYSEPKSTPKGACATSVKVAIDTGYRHIDGAYIYQNEHEVGEAIREKIAEGKVRREDIFYCGKLWATNHVPEMVRPTLERTLRVLQLDYVDLYIIHVPMAFKPGDEIYPRDENGKWLYHKSNLCATWEAMEACKDAGLVKSLGVSNFNRRQLELILNKPGLKHKPVSNQVECHPYFTQPKLLKFCQQHDIVITAYSPLGTSRNPIWVNVSSPPLLKDALLNSLGKRYNKTAAQIVLRFNIQRGVVVIPKSFNLERIKENFQIFDFSLTEEEMKDIEALNKNVRFVELLMWRDHPEYPFHDEY

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 \(c\).

\(|P_{f(7WJZ_1)}(2) \setminus P_{f(8YLM_1)}(2)|=230\), \(|P_{f(8YLM_1)}(2) \setminus P_{f(7WJZ_1)}(2)|=7\). 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 1:1111111111100001010000011110000100110010011000110000011111100101101101001000100111110011011010000110111110010000001111001001110100101000111000000011000101000100001001001111010100101001001110010101010000011110010011011011011101111101001001111000010110000110111110011010100111000001010011001101100000010010100110000010101000110110100101100110100110101100001000110001100111110100011010010010100101001110100100111100101100000110010101111000010001010000100110000101100010001001100100111110001110000101001110010011110101101110101100000110000101010110101110000001111001100110000110010010110101001111011011000000111100110000111110100101010100010011000110111100100000001111111010000000001010011000111001011100011000001111001010100011110100001000101010000000111001010001001101111000000001110100100011100111101001110100100001100111001011011110001001101110011010010110111111000111000011111010011011111110111110110010111100011000001110010011101000100010111010011000101100110010001111001100110010110101010011010100100010001101101010101110010001110000101010100110110011011111010011100001001111000101011001111001001110000100101100000110100011111100010011010100100010001000001010110101101011010001001001100100011010011000010101101100101010001011110011011011101111100101001110111011011001010011000000000
Pair \(Z_2\) Length of longest common subsequence
7WJZ_1,8YLM_1 237 4
7WJZ_1,3UZY_1 162 7
8YLM_1,3UZY_1 189 4

Newick tree

 
[
	8YLM_1:11.57,
	[
		7WJZ_1:81,3UZY_1:81
	]:33.57
]

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 \((0.85)(0.8)(\frac{1471 }{\log_{20} 1471}-\frac{186}{\log_{20}186})=338.\)
Status Protein1 Protein2 d d1/2
Query variables 7WJZ_1 8YLM_1 429 243.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]