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Parikh vectors
6EBB_1 7VVB_1 6AFW_1 Letter Amino acid
2 14 22 Y Tyrosine
6 26 15 R Arginine
0 8 10 C Cysteine
5 23 79 G Glycine
5 20 25 P Proline
5 27 46 T Threonine
12 31 64 I Isoleucine
7 56 58 S Serine
8 31 67 V Valine
4 31 9 Q Glutamine
12 34 29 E Glutamic acid
0 15 6 H Histidine
13 45 75 L Leucine
11 43 35 K Lycine
0 5 6 W Tryptophan
8 28 99 A Alanine
3 21 23 N Asparagine
10 35 32 D Aspartic acid
4 9 20 M Methionine
5 20 46 F Phenylalanine 

6EBB_1|Chain A|DNA-binding response regulator|Streptococcus pneumoniae serotype 4 (strain ATCC BAA-334 / TIGR4) (170187)
>7VVB_1|Chain A[auth B]|Target of rapamycin complex 2 subunit MAPKAP1|Homo sapiens (9606)
>6AFW_1|Chains A, B|Pyrophosphate-energized vacuolar membrane proton pump|Vigna radiata var. radiata (3916)
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 \(w=f(c)\)
6EBB , Knot 60 120 0.79 34 95 115 
MKKILIVDDEKPISDIIKFNMTKEGYEVVTAFNGREALEQFEAEQPDIIILDLMLPEIDGLEVAKTIRKTSSVPILMLSAKDSEFDKVIGLELGADDYVTKPFSNRELQARVKALLRRSQ
7VVB , Knot 217 522 0.86 40 264 492 
MAFLDNPTIILAHIRQSHVTSDDTGMCEMVLIDHDVDLEKIHPPSMPGDSGSEIQGSNGETQGYVYAQSVDITSSWDFGIRRRSNTAQRLERLRKERQNQIKCKNIQWKERNSKQSAQELKSLFEKKSLKEKPPISGKQSILSVRLEQCPLQLNNPFNEYSKFDGKGHVGTTATKKIDVYLPLHSSQDRLLPMTVVTMASARVQDLIGLICWQYTSEGREPKLNDNVSAYCLHIAEDDGEVDTDFPPLDSNEPIHKFGFSTLALVEKYSSPGLTSKESLFVRINAAHGFSLIQVDNTKVTMKEILLKAVKRRKGSQKVSGPQYRLEKQSEPNVAVDLDSTLESQSAWEFCLVRENSSRADGVFEEDSQIDIATVQDMLSSHHYKSFKVSMIHRLRFTTDVQLGISGDKVEIDPVTNQKASTKFWIKQKPISIDSDLLCACDLAEEKSPSHAIFKLTYLSNHDYKHLYFESDAATVNEIVLKVNYILESRASTARADYFAQKQRKLNRRTSFSFQKEKKSGQQ
6AFW , Knot 285 766 0.82 40 264 665 
MGAAILPDLGTEILIPVCAVIGIAFALFQWLLVSKVKLSAVRDASPNAAAKNGYNDYLIEEEEGINDHNVVVKCAEIQNAISEGATSFLFTEYKYVGIFMVAFAILIFLFLGSVEGFSTSPQACSYDKTKTCKPALATAIFSTVSFLLGGVTSLVSGFLGMKIATYANARTTLEARKGVGKAFITAFRSGAVMGFLLAANGLLVLYIAINLFKIYYGDDWGGLFEAIDGYGLGGSSMALFGRVGGGIYTKAADVGADLVGKVERNIPEDDPRNPAVIADNVGDNVGDIAGMGSDLFGSYAESSCAALVVASISSFGLNHELTAMLYPLIVSSVGILVCLLTTLFATDFFEIKAVKEIEPALKKQLVISTVLMTIGVAVVSFVALPTSFTIFNFGVQKDVKSWQLFLCVAVGLWAGLIIGFVTEYYTSNAYSPVQDVADSCRTGAATNVIFGLALGYKSVIIPIFAIAISIFVSFTFAAMYGIAVAALGMLSTIATGLAIDAYGPISDNAGGIAEMAGMSHRIRERTDALDAAGNTTAAIGKGFAIGSAALVSLALFGAFVSRASITTVDVLTPKVFIGLIVGAMLPYWFSAMTMKSVGSAALKMVEEVRRQFNTIPGLMEGTAKPDYATCVKISTDASIKEMIPPGALVMLTPLVVGILFGVETLSGVLAGSLVSGVQIAISASNTGGAWDNAKKYIEAGASEHARSLGPKGSDCHKAAVIGDTIGDPLKDTSGPSLNILIKLMAVESLVFAPFFATHGGLLFKIF

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 \(c\).

\(|P_{f(6EBB_1)}(2) \setminus P_{f(7VVB_1)}(2)|=16\), \(|P_{f(7VVB_1)}(2) \setminus P_{f(6EBB_1)}(2)|=185\). 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 1:100111100001100110101000100110110100110010100101111011110101101100100000111111010000100111101110001001100001010101110000
Pair \(Z_2\) Length of longest common subsequence
6EBB_1,7VVB_1 201 4
6EBB_1,6AFW_1 197 4
7VVB_1,6AFW_1 128 4

Newick tree

 
[
	6EBB_1:10.79,
	[
		6AFW_1:64,7VVB_1:64
	]:44.79
]

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 \((0.85)(0.8)(\frac{642 }{\log_{20} 642}-\frac{120}{\log_{20}120})=151.\)
Status Protein1 Protein2 d d1/2
Query variables 6EBB_1 7VVB_1 192 115
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]