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Parikh vectors
1REQ_1 1ASC_1 4NQI_1 Letter Amino acid
11 5 0 W Tryptophan
29 16 22 Q Glutamine
36 18 28 K Lycine
58 38 28 L Leucine
22 11 4 Y Tyrosine
43 22 12 R Arginine
12 6 3 H Histidine
22 8 6 M Methionine
24 20 9 F Phenylalanine
38 21 15 S Serine
51 27 19 E Glutamic acid
54 30 10 G Glycine
47 19 17 D Aspartic acid
2 5 0 C Cysteine
40 17 17 I Isoleucine
38 15 8 P Proline
42 22 18 T Threonine
53 26 10 V Valine
83 47 20 A Alanine
22 23 18 N Asparagine 

1REQ_1|Chains A, C|METHYLMALONYL-COA MUTASE|Propionibacterium freudenreichii subsp. shermanii (1752)
>1ASC_1|Chain A|ASPARTATE AMINOTRANSFERASE|Escherichia coli (562)
>4NQI_1|Chains A, B, C, D|SH3 domain-containing protein|Dictyostelium discoideum (44689)
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 \(w=f(c)\)
1REQ , Knot 282 727 0.85 40 281 664 
STLPRFDSVDLGNAPVPADAARRFEELAAKAGTGEAWETAEQIPVGTLFNEDVYKDMDWLDTYAGIPPFVHGPYATMYAFRPWTIRQYAGFSTAKESNAFYRRNLAAGQKGLSVAFDLPTHRGYDSDNPRVAGDVGMAGVAIDSIYDMRELFAGIPLDQMSVSMTMNGAVLPILALYVVTAEEQGVKPEQLAGTIQNDILKEFMVRNTYIYPPQPSMRIISEIFAYTSANMPKWNSISISGYHMQEAGATADIEMAYTLADGVDYIRAGESVGLNVDQFAPRLSFFWGIGMNFFMEVAKLRAARMLWAKLVHQFGPKNPKSMSLRTHSQTSGWSLTAQDVYNNVVRTCIEAMAATQGHTQSLHTNSLDEAIALPTDFSARIARNTQLFLQQESGTTRVIDPWSGSAYVEELTWDLARKAWGHIQEVEKVGGMAKAIEKGIPKMRIEEAAARTQARIDSGRQPLIGVNKYRLEHEPPLDVLKVDNSTVLAEQKAKLVKLRAERDPEKVKAALDKITWAAGNPDDKDPDRNLLKLCIDAGRAMATVGEMSDALEKVFGRYTAQIRTISGVYSKEVKNTPEVEEARELVEEFEQAEGRRPRILLAKMGQDGHDRGQKVIATAYADLGFDVDVGPLFQTPEETARQAVEADVHVVGVSSLAGGHLTLVPALRKELDKLGRPDILITVGGVIPEQDFDELRKDGAVEIYTPGTVIPESAISLVKKLRASLDA
1ASC , Knot 168 396 0.84 40 219 370 
MFENITAAPADPILGLADLFRADERPGKINLGIGVYKDETGKTPVLTSVKKAEQYLLENETTKNYLGIDGIPEFGRCTQELLFGKGSALINDKRARTAQTPGGTGALRVAADFLAKNTSVKRVWVSNPSWPNHKSVFNSAGLEVREYAYYDAENHTLDFDALINSLNEAQAGDVVLFHGCCHNPTGIDPTLEQWQTLAQLSVEKGWLPLFAFAYQGFARGLEEDAEGLRAFAAMHKELIVASSYSKNFGLYNERVGACTLVAADSETVDRAFSQMKAAIRANYSNPPAHGASVVATILSNDALRAIWEQELTDMRQRIQRMRQLFVNTLQEKGANRDFSFIIKQNGMFSFSGLTKEQVLRLREEFGVYAVASGRVNVAGMTPDNMAPLCEAIVAVL
4NQI , Knot 116 264 0.81 36 162 256 
GPLGSMSNAKKQQNPVIEITLKTINNLKVNSPPLFTEVIKAANKYQQQAQALSQAGLVLADTLTRLTIHNGGDFGEGFKKLADAIKDLENRRDDVAKVLLNEFITPNKQAIEDDQKAIATFEKNYKKDRDQMRQDILKLEAKTRKAGKKTTPEVLKQQITELNDKIKESEQLNANKLRDVVLMERRKHATFLSQFNQFLEKEIELSADTMSKFSTNLNTHRDLINSQSQLPLEMESMISKQERTLVQIQPQGDTGSDAYRISYA

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 \(c\).

\(|P_{f(1REQ_1)}(2) \setminus P_{f(1ASC_1)}(2)|=105\), \(|P_{f(1ASC_1)}(2) \setminus P_{f(1REQ_1)}(2)|=43\). 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 1:0011010010110111110110010011101101011001001111011000100010110001111111011010101101101000111001000011000011110011011101100010000010111011111111001001001111111001010101011111111101101000110100111010001100111000010110101011001110001011010010101001001110101011001101100101100111010011101011111110111011010110111101100111001001010000000110101001000110001011110010000100001001111100101011000011100001000110110101010010101100111010010011111011001110101001110001010010011111000010001110110100001110001011010100010010111001011110100001000110101011011101101001100111000101001011000010001010010011001001010010111101100100010011101010111010111110010001001101010111100111101011111000100110101110111111000100100011101001101110011011001010101
Pair \(Z_2\) Length of longest common subsequence
1REQ_1,1ASC_1 148 4
1REQ_1,4NQI_1 171 4
1ASC_1,4NQI_1 165 4

Newick tree

 
[
	4NQI_1:87.09,
	[
		1REQ_1:74,1ASC_1:74
	]:13.09
]

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 \((0.85)(0.8)(\frac{1123 }{\log_{20} 1123}-\frac{396}{\log_{20}396})=190.\)
Status Protein1 Protein2 d d1/2
Query variables 1REQ_1 1ASC_1 241 186
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]