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Parikh vectors
6AVO_1 3VNI_1 2ZOL_1 Letter Amino acid
20 25 22 G Glycine
5 6 5 M Methionine
20 22 26 L Leucine
14 19 11 K Lycine
8 9 7 F Phenylalanine
23 24 20 A Alanine
7 14 9 N Asparagine
7 20 12 D Aspartic acid
11 6 14 Q Glutamine
12 20 4 I Isoleucine
10 9 14 P Proline
11 13 22 R Arginine
18 23 29 E Glutamic acid
4 10 8 H Histidine
14 15 13 S Serine
2 6 12 W Tryptophan
2 5 4 C Cysteine
13 10 17 T Threonine
14 16 16 Y Tyrosine
19 22 13 V Valine 

6AVO_10|Chains BA[auth P], P[auth b]|Proteasome subunit alpha type-2|Homo sapiens (9606)
>3VNI_1|Chains A, B, C, D|Xylose isomerase domain protein TIM barrel|Clostridium cellulolyticum (394503)
>2ZOL_1|Chains A, B[auth C]|H-2 class I histocompatibility antigen, D-B alpha chain|Mus musculus (10090)
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 \(w=f(c)\)
6AVO , Knot 106 234 0.82 40 160 226 
MAERGYSFSLTTFSPSGKLVQIEYALAAVAGGAPSVGIKAANGVVLATEKKQKSILYDERSVHKVEPITKHIGLVYSGMGPDYRVLVHRARKLAQQYYLVYQEPIPTAQLVQRVASVMQEYTQSGGVRPFGVSLLICGWNEGRPYLFQSDPSGAYFAWKATAMGKNYVNGKTFLEKRYNEDLELEDAIHTAILTLKESFEGQMTEDNIEVGICNEAGFRRLTPTEVKDYLAAIA
3VNI , Knot 133 294 0.85 40 194 284 
AMKHGIYYAYWEQEWEADYKYYIEKVAKLGFDILEIAASPLPFYSDIQINELKACAHGNGITLTVGHGPSAEQNLSSPDPDIRKNAKAFYTDLLKRLYKLDVHLIGGALYSYWPIDYTKTIDKKGDWERSVESVREVAKVAEACGVDFCLEVLNRFENYLINTAQEGVDFVKQVDHNNVKVMLDTFHMNIEEDSIGGAIRTAGSYLGHLHTGECNRKVPGRGRIPWVEIGEALADIGYNGSVVMEPFVRMGGTVGSNIKVWRDISNGADEKMLDREAQAALDFSRYVLECHKHS
2ZOL , Knot 127 278 0.85 40 186 270 
GPHSMRYFETAVSRPGLEEPRYISVGYVDNKEFVRFDSDAENPRYEPRAPWMEQEGPEYWERETQKAKGQEQWFRVSLRNLLGYYNQSAGGSHTLQQMSGCDLGSDWRLLRGYLQFAYEGRDYIALNEDLKTWTAADMAAQITRRKWEQSGAAEHYKAYLEGECVEWLHRYLKNGNATLLRTDSPKAHVTHHPRSKGEVTLRCWALGFYPADITLTWQLNGEELTQDMELVETRPAGDGTFQKWASVVVPLGKEQNYTCRVYHEGLPEPLTLRWERWE

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 \(c\).

\(|P_{f(6AVO_1)}(2) \setminus P_{f(3VNI_1)}(2)|=61\), \(|P_{f(3VNI_1)}(2) \setminus P_{f(6AVO_1)}(2)|=95\). 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 1:110010010100101010110100111111111101110110111110000000110000010010110001111001111000111001001100001100011101011001101100000011101111011101100101011000101101110101110001010011000000010100110011101000101010000101110001110010100100011111
Pair \(Z_2\) Length of longest common subsequence
6AVO_1,3VNI_1 156 4
6AVO_1,2ZOL_1 170 4
3VNI_1,2ZOL_1 172 4

Newick tree

 
[
	2ZOL_1:87.85,
	[
		6AVO_1:78,3VNI_1:78
	]:9.85
]

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 \((0.85)(0.8)(\frac{528 }{\log_{20} 528}-\frac{234}{\log_{20}234})=84.1\)
Status Protein1 Protein2 d d1/2
Query variables 6AVO_1 3VNI_1 110 97.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]