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Parikh vectors
8QRH_1 6VXR_1 8BRI_1 Letter Amino acid
19 20 20 I Isoleucine
16 10 15 F Phenylalanine
19 13 6 P Proline
24 19 11 E Glutamic acid
23 22 20 D Aspartic acid
20 14 2 H Histidine
45 47 29 L Leucine
42 12 12 T Threonine
10 5 0 W Tryptophan
45 18 20 V Valine
18 12 8 N Asparagine
32 27 13 K Lycine
13 12 17 M Methionine
12 8 0 C Cysteine
18 13 10 R Arginine
11 14 6 Q Glutamine
52 17 26 G Glycine
28 20 7 S Serine
10 19 3 Y Tyrosine
35 17 28 A Alanine 

8QRH_1|Chains A, B, C|Genome polyprotein|Orthoflavivirus encephalitidis (3052465)
>6VXR_1|Chain A|Maternal embryonic leucine zipper kinase|Homo sapiens (9606)
>8BRI_1|Chains A, B, C, D, E|Chemotaxis protein PomA|Vibrio alginolyticus (663)
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 \(w=f(c)\)
8QRH , Knot 201 492 0.84 40 250 460 
SRCTHLENRDFVTGTQGTTRVTLVLELGGCVTITAEGKPSMDVWLDSIYQENPAKTREYCLHAKLSDTKVAARCPTMGPATLAEEHQSGTVCKRDQSDRGWGNHCGLFGKGSIVTCVKASCEAKKKATGHVYDANKIVYTVKVEPHTGDYVAANETHSGRKTASFTVSSEKTILTMGDYGDVSLLCRVASGVDLAQTVILELDKTSEHLPTAWQVHRDWFNDLALPWKHEGAQNWNNAERLVEFGAPHAVKMDVYNLGDHTGVLLKSLAGVPVAHIDGTKYHLKSGHVTCEVGLEKLKMKGLTYTMCDKTKFTWKRIPTDSGHDTVVMEVAFSGTKPCRIPVRAVAHGSPDVNVAMLITPNPIIENNGGGFIEMQLPPGDNIIYVGELSHQWFQKGSSIGRVFQKTRKGIERLTVIGEHAWDFGSAGGFLTSVGKALHTVLGGAFNSLFGGVGFLPKILVGVVLAWLGLNMRNPTMSMSFLLAGGLVLAMTL
6VXR , Knot 148 339 0.84 40 214 323 
SMKDYDELLKYYELHETIGTGGFAKVKLACHILTGEMVAIKIMDKNTLGSDLPRIKTEIEALKNLRHQHICQLYHVLETANKIFMVLEYCPGGELFDYIISQDRLSEEETRVVFRQIVSAVAYVHSQGYAHRDLKPENLLFDEYHKLKLIDFGLCAKPKGNKDYHLQTCCGSLAYAAPELIQGKSYLGSEADVWSMGILLYVLMCGFLPFDDDNVMALYKKIMRGKYDVPKWLSPSSILLLQQMLQVDPKKRISMKNLLNHPWIMQDYNYPVEWQSKNPFIHLDDDCVTELSVHHRNNRQTMEDLISLWQYDHLTATYLLLLAKKARGKPVRLRLSSFS
8BRI , Knot 110 253 0.80 36 148 244 
MDLATLLGLIGGFAFVIMAMVLGGSIGMFVDVTSILIVVGGSIFVVLMKFTMGQFFGATKIAGKAFMFKADEPEDLIAKIVEMADAARKGGFLALEEMEINNTFMQKGIDLLVDGHDADVVRAALKKDIALTDERHTQGTGVFRAFGDVAPAMGMIGTLVGLVAMLSNMDDPKAIGPAMAVALLTTLYGAILSNMVFFPIADKLSLRRDQETLNRRLIMDGVLAIQDGQNPRVIDSYLKNYLNEGKRALEIDE

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 \(c\).

\(|P_{f(8QRH_1)}(2) \setminus P_{f(6VXR_1)}(2)|=102\), \(|P_{f(6VXR_1)}(2) \setminus P_{f(8QRH_1)}(2)|=66\). 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 1:000001000011010010001011101110101010101010111001000011000000101010000111001011110110000010100000000111000111101011001010001000101010010011001010100100111000001000101010000011011001010110011011011001110100000011011010001100111110001100100100110111101101010011000111100111111101010000100101000111001010110001000001010011000100011101110100100111011101010101111101011100011111010111100110110100011001001101100000110010111001101101111100110110011111100111111111011111111111101001010101111111111101
Pair \(Z_2\) Length of longest common subsequence
8QRH_1,6VXR_1 168 4
8QRH_1,8BRI_1 186 4
6VXR_1,8BRI_1 170 4

Newick tree

 
[
	8BRI_1:90.72,
	[
		8QRH_1:84,6VXR_1:84
	]:6.72
]

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 \((0.85)(0.8)(\frac{831 }{\log_{20} 831}-\frac{339}{\log_{20}339})=133.\)
Status Protein1 Protein2 d d1/2
Query variables 8QRH_1 6VXR_1 170 144.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]