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Parikh vectors
6XIE_1 1OZG_1 8JWS_1 Letter Amino acid
8 33 4 D Aspartic acid
8 43 11 G Glycine
1 17 1 M Methionine
8 55 5 V Valine
5 32 11 Q Glutamine
6 17 4 H Histidine
4 18 7 F Phenylalanine
5 29 9 P Proline
7 19 1 T Threonine
4 13 3 Y Tyrosine
11 71 4 A Alanine
8 30 8 R Arginine
0 23 5 N Asparagine
14 26 7 E Glutamic acid
5 19 11 K Lycine
6 33 8 S Serine
1 2 11 C Cysteine
2 28 5 I Isoleucine
18 52 12 L Leucine
1 6 0 W Tryptophan 

6XIE_1|Chain A|Proprotein convertase subtilisin/kexin type 9|Homo sapiens (9606)
>1OZG_1|Chains A, B|Acetolactate synthase, catabolic|Klebsiella pneumoniae (573)
>8JWS_1|Chains A, B, C|PHD finger protein 7|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 \(w=f(c)\)
6XIE , Knot 58 122 0.76 38 87 115 
QEDEDGDYEELVLALRSEEDGLAEAPEHGTTATFHRCAKDPWRLPGTYVVVLKEETHLSQSERTARRLQAQAARRGYLTKILHVFHGLLPGFLVKMSGDLLELALKLPHVDYIEEDSSVFAQ
1OZG , Knot 223 566 0.83 40 255 527 
MDKQYPVRQWAHGADLVVSQLEAQGVRQVFGIPGAKIDKVFDSLLDSSIRIIPVRHEANAAFMAAAVGRITGKAGVALVTSGPGCSNLITGMATANSEGDPVVALGGAVKRADKAKQVHQSMDTVAMFSPVTKYAIEVTAPDALAEVVSNAFRAAEQGRPGSAFVSLPQDVVDGPVSGKVLPASGAPQMGAAPDDAIDQVAKLIAQAKNPIFLLGLMASQPENSKALRRLLETSHIPVTSTYQAAGAVNQDNFSRFAGRVGLFNNQAGDRLLQLADLVICIGYSPVEYEPAMWNSGNATLVHIDVLPAYEERNYTPDVELVGDIAGTLNKLAQNIDHRLVLSPQAAEILRDRQHQRELLDRRGAQLNQFALHPLRIVRAMQDIVNSDVTLTVDMGSFHIWIARYLYTFRARQVMISNGQQTMGVALPWAIGAWLVNPERKVVSVSGDGGFLQSSMELETAVRLKANVLHLIWVDNGYNMVAIQEEKKYQRLSGVEFGPMDFKAYAESFGAKGFAVESAEALEPTLRAAMDVDGPAVVAIPVDYRDNPLLMGQLHLSQILEHHHHHH
8JWS , Knot 64 127 0.81 38 103 125 
GPGIQSSSPVCLLCLQEPGDPEKLGEFLQKDNLCVHYFCLILSSRLPQKGQPNRGLHGFMPEDIKREAVRASKKICFVCKKKGAAIRCQNDQCVQNFHLPCGQERGCLSQFFGEYKSYCRKHRPTQN

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 \(c\).

\(|P_{f(6XIE_1)}(2) \setminus P_{f(1OZG_1)}(2)|=17\), \(|P_{f(1OZG_1)}(2) \setminus P_{f(6XIE_1)}(2)|=185\). 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 1:00000100001111100000111011001001010001001101110011110000010000001001010110010100110110111111110101011011101101001000001110
Pair \(Z_2\) Length of longest common subsequence
6XIE_1,1OZG_1 202 4
6XIE_1,8JWS_1 128 3
1OZG_1,8JWS_1 216 3

Newick tree

 
[
	1OZG_1:11.94,
	[
		6XIE_1:64,8JWS_1:64
	]:50.94
]

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 \((0.85)(0.8)(\frac{688 }{\log_{20} 688}-\frac{122}{\log_{20}122})=162.\)
Status Protein1 Protein2 d d1/2
Query variables 6XIE_1 1OZG_1 205 122.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]