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Parikh vectors
2ODR_1 4AJY_1 3SWE_1 Letter Amino acid
49 10 38 L Leucine
15 3 10 M Methionine
23 9 14 P Proline
21 9 33 T Threonine
19 1 5 Y Tyrosine
15 1 10 H Histidine
43 5 38 I Isoleucine
60 7 18 K Lycine
18 7 22 R Arginine
23 1 14 N Asparagine
6 2 6 C Cysteine
9 7 10 Q Glutamine
34 6 38 G Glycine
34 11 22 D Aspartic acid
51 9 30 E Glutamic acid
38 8 22 S Serine
41 6 36 V Valine
25 9 46 A Alanine
29 7 11 F Phenylalanine
3 0 1 W Tryptophan 

2ODR_1|Chain A|phosphoseryl-tRNA synthetase|Methanococcus maripaludis (267377)
>4AJY_1|Chain A[auth B]|TRANSCRIPTION ELONGATION FACTOR B POLYPEPTIDE 2|HOMO SAPIENS (9606)
>3SWE_1|Chain A|UDP-N-acetylglucosamine 1-carboxyvinyltransferase|Haemophilus influenzae (71421)
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 \(w=f(c)\)
2ODR , Knot 214 665 0.69 42 241 513 
MGSHHHHHHSSGLVPRGSHMFKREEIIEMANKDFEKAWIETKDLIKAKKINESYPRIKPVFGKTHPVNDTIENLRQAYLRMGFEEYINPVIVDERDIYKQFGPEAMAVLDRCFYLAGLPRPDVGLSDEKISQIEKLGIKVSEHKESLQKILHGYKKGTLDGDDLVLEISNALEISSEMGLKILEDVFPEFKDLTAVSSKLTLRSHMTSGWFLTVSDLMNKKPLPFKLFSIDRCFRREQKEDKSHLMTYHSASCAIAGEGVDINDGKAIAEGLLSQFGFTNFKFIPDEKKSKYYTPETQTEVYAYHPKLKEWLEVATFGVYSPVALSKYGIDVPVMNLGLGVERLAMISGNFADVREMVYPQFYEHKLNDRNVASMVKLDKVPVMDEIYDLTKELIESCVKNKDLKSPCELAIEKTFSFGKTKKNVKINIFEKEEGKNLLGPSILNEIYVYDGNVIGIPESFDGVKEEFKDFLEKGKSEGVATGIRYIDALCFKITSKLEEAFVSNTTEFKVKVPIVRSLSDINLKIDDIALKQIMSKNKVIDVRGPVFLNVEVKIEXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
4AJY , Knot 61 118 0.82 38 98 115 
MDVFLMIRRHKTTIFTDAKESSTVFELKRIVEGILKRPPDEQRLYKDDQLLDDGKTLGECGFTSQTARPQAPATVGLAFRADDTFEALCIEPFSSPPELPDVMKPQDSGSSANEQAVQ
3SWE , Knot 171 424 0.81 40 201 388 
MDKFRVYGQSRLSGSVNISGAKNAALPILFAAILATEPVKLTNVPELKDIETTLKILRQLGVVVDRDATGAVLLDASNINHFTAPYELVKTMRASIWALAPLVARFHQGQVSLPGGCSIGARPVDLHISGLEKLGADIVLEEGYVKAQVSDRLVGTRIVIEKVSVGATLSIMMAATLAKGTTVIENAAREPEIVDTADFLNKMGAKITGAGSAHITIEGVERLTGCEHSVVPDRIETGTFLIAAAISGGCVVCQNTKADTLDAVIDKLREAGAQVDVTENSITLDMLGNRPKAVNIRTAPHPGFPTDMQAQFTLLNMVAEGTSIITETIFENRFMHIPELIRMGGKAEIEGNTAVCHGVEQLSGTEVIATDLRASISLVLAGCIATGETIVDRIYHIDRGYEHIEDKLRGLGAKIERFSGSDEA

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 \(c\).

\(|P_{f(2ODR_1)}(2) \setminus P_{f(4AJY_1)}(2)|=163\), \(|P_{f(4AJY_1)}(2) \setminus P_{f(2ODR_1)}(2)|=20\). 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 1:11000000000111101001100001101100010011100001101001000010101111000110001001001010111000101111000010001110111110001011111010111000010010011101000000100110100010101001110100110100011101100111010010110001010001001111010011000111101101000100000000001100001001111011010010111011100111001011100000000010000010100101001101101110011110001101111011111001111010110100110101000010000110110100111100100100011000100001001001110001011000001010110000100111101100101001011111001011000100110010001110110010110101000100111000001010111100100101010011100110000110101111101010100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
Pair \(Z_2\) Length of longest common subsequence
2ODR_1,4AJY_1 183 3
2ODR_1,3SWE_1 146 5
4AJY_1,3SWE_1 177 3

Newick tree

 
[
	4AJY_1:95.00,
	[
		2ODR_1:73,3SWE_1:73
	]:22.00
]

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 \((0.85)(0.8)(\frac{783 }{\log_{20} 783}-\frac{118}{\log_{20}118})=188.\)
Status Protein1 Protein2 d d1/2
Query variables 2ODR_1 4AJY_1 195 117
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]