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Parikh vectors
2BHU_1 3NIJ_1 8DBE_1 Letter Amino acid
66 7 17 G Glycine
66 3 22 L Leucine
8 5 20 K Lycine
18 3 24 S Serine
47 5 13 T Threonine
2 12 9 C Cysteine
43 7 16 E Glutamic acid
13 1 2 W Tryptophan
39 4 28 V Valine
44 3 16 R Arginine
38 6 22 D Aspartic acid
26 8 9 H Histidine
9 0 9 M Methionine
35 3 10 P Proline
26 1 5 Y Tyrosine
52 2 31 A Alanine
14 4 15 N Asparagine
20 1 9 Q Glutamine
9 4 32 I Isoleucine
27 3 9 F Phenylalanine 

2BHU_1|Chain A|MALTOOLIGOSYLTREHALOSE TREHALOHYDROLASE|DEINOCOCCUS RADIODURANS (243230)
>3NIJ_1|Chain A|E3 ubiquitin-protein ligase UBR1|Saccharomyces cerevisiae (4932)
>8DBE_1|Chains A, B, C, D, E, F|Ribose-phosphate pyrophosphokinase 1|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 \(w=f(c)\)
2BHU , Knot 233 602 0.82 40 244 547 
MTQTQPVTPTPPASFQTQHDPRTRLGATPLPGGAGTRFRLWTSTARTVAVRVNGTEHVMTSLGGGIYELELPVGPGARYLFVLDGVPTPDPYARFLPDGVHGEAEVVDFGTFDWTDADWHGIKLADCVFYEVHVGTFTPEGTYRAAAEKLPYLKELGVTAIQVMPLAAFDGQRGWGYDGAAFYAPYAPYGRPEDLMALVDAAHRLGLGVFLDVVYNHFGPSGNYLSSYAPSYFTDRFSSAWGMGLDYAEPHMRRYVTGNARMWLRDYHFDGLRLDATPYMTDDSETHILTELAQEIHELGGTHLLLAEDHRNLPDLVTVNHLDGIWTDDFHHETRVTLTGEQEGYYAGYRGGAEALAYTIRRGWRYEGQFWAVKGEEHERGHPSDALEAPNFVYCIQNHDQIGNRPLGERLHQSDGVTLHEYRGAAALLLTLPMTPLLFQGQEWAASTPFQFFSDHAGELGQAVSEGRKKEFGGFSGFSGEDVPDPQAEQTFLNSKLNWAEREGGEHARTLRLYRDLLRLRREDPVLHNRQRENLTTGHDGDVLWVRTVTGAGERVLLWNLGQDTRAVAEVKLPFTVPRRLLLHTEGREDLTLGAGEAVLVG
3NIJ , Knot 45 82 0.80 38 71 80 
GSVHKHTGRNCGRKFKIGEPLYRCHECGCDDTCVLCIHCFNPKDHVNHHVCTDICTEFTSGICDCGDEEAWNSPLHCKAEEQ
8DBE , Knot 137 318 0.82 40 204 302 
MPNIKIFSGSSHQDLSQKIADRLGLELGKVVTKKFSNQETCVEIGESVRGEDVYIVQSGCGEINDNLMELLIMINACKIASASRVTAVIPCFPYARQDKKDKSRAPISAKLVANMLSVAGADHIITMDLHASQIQGFFDIPVDNLYAEPAVLKWIRENISEWRNCTIVSPDAGGAKRVTSIADRLNVDFALIHKERKKANEVDRMVLVGDVKDRVAILVDDMADTCGTICHAADKLLSAGATRVYAILTHGIFSGPAISRINNACFEAVVVTNTIPQEDKMKHCSKIQVIDISMILAEAIRRTHNGESVSYLFSHVPL

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 \(c\).

\(|P_{f(2BHU_1)}(2) \setminus P_{f(3NIJ_1)}(2)|=203\), \(|P_{f(3NIJ_1)}(2) \setminus P_{f(2BHU_1)}(2)|=30\). 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 1:10000110101110100000100011101111111001011000100111010100011001111100101111111001111011101010101110110101011011010100101011011001100101101010100011100110100111011011111110100111001111011011010100111110110011111110110001110100100011001000100111111001010100010101011100001011010101010000000110011001001110011110000011011010010111000100000101010001001100111011100100110001011110100000101001101101100100000110011100100001101000011111110111011110100111001101100011011011001000011110110100110101000110001011000110010010100011010000111000000010010010111100101110011110110000111010111011001110001000101111011111
Pair \(Z_2\) Length of longest common subsequence
2BHU_1,3NIJ_1 233 3
2BHU_1,8DBE_1 200 4
3NIJ_1,8DBE_1 193 3

Newick tree

 
[
	2BHU_1:11.29,
	[
		8DBE_1:96.5,3NIJ_1:96.5
	]:15.79
]

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 \((0.85)(0.8)(\frac{684 }{\log_{20} 684}-\frac{82}{\log_{20}82})=175.\)
Status Protein1 Protein2 d d1/2
Query variables 2BHU_1 3NIJ_1 221 125.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]