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Parikh vectors
6HPA_1 4GBD_1 5JIG_1 Letter Amino acid
13 32 11 R Arginine
18 22 5 I Isoleucine
27 8 7 K Lycine
12 13 2 M Methionine
14 23 3 P Proline
8 7 5 Y Tyrosine
0 5 0 C Cysteine
20 26 9 E Glutamic acid
8 13 4 F Phenylalanine
18 32 5 V Valine
18 32 13 D Aspartic acid
11 19 7 S Serine
16 15 10 T Threonine
6 7 2 W Tryptophan
16 55 6 A Alanine
14 12 4 N Asparagine
13 21 2 Q Glutamine
12 37 6 G Glycine
9 23 10 H Histidine
20 56 16 L Leucine 

6HPA_1|Chain A|Putative polysaccharide deacetylase|Bacillus anthracis (1392)
>4GBD_1|Chains A, B|Putative uncharacterized protein|Pseudomonas aeruginosa (208964)
>5JIG_1|Chain A|Ubiquitin and WLM domain-containing metalloprotease SPCC1442.07c|Schizosaccharomyces pombe (strain 972 / ATCC 24843) (284812)
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 \(w=f(c)\)
6HPA , Knot 120 273 0.82 38 182 262 
QDNLYEEIQKHAKQYEIAPQNAMIDKIWKATPGYNGRQVDMEASYNNMKKLKKFDQKHLEFKEVSPSVHLEDLSPAPIYRGHPNKKMVGLTIDVAWGNEYLPRILEILKKHDVKATFFLEGRWVKENLRFAKMIVDANQEVGNHSYTHPNMKTLSSDEIRDQLQKTNRMIEAATNQKVRWFAPPSGSFRDEVVKIADDFQMGTIMWTVDTIDWKRPEPDVLLQRVMRKIHPGAIVLMHPTSSTTEALDTMITKLKEQGYKVGNITELLDEKRV
4GBD , Knot 187 458 0.83 40 224 426 
MHHHHHHENLYFQGMPNVRNPFDLLLLPTWIVPVEPAGVVLRDHALGIRDGQIALVAPREQAMRHGATEIRELPGMLLAPGLVNAHGHSAMSLFRGLADDLPLMTWLQDHIWPAEGQWVSEDFIRDGTELAIAEQVKGGITCFSDMYFYPQAICGVVHDSGVRAQVAIPVLDFPIPGARDSAEAIRQGMALFDDLKHHPRIRIAFGPHAPYTVSDDKLEQILVLTEELDASIQMHVHETAFEVEQAMERNGERPLARLHRLGLLGPRFQAVHMTQVDNDDLAMLVETNSSVIHCPESNLKLASGFCPVEKLWQAGVNVAIGTDGAASNNDLDLLGETRTAALLAKAVYGQATALDAHRALRMATLNGARALGLERLIGSLEAGKAADLVAFDLSGLAQQPVYDPVSQLIYASGRDCVRHVWVGGRQLLDDGRLLRHDEQRLIARAREWGAKIAASDRS
5JIG , Knot 65 127 0.82 38 103 123 
GSIYTFNELVVLDYPHKDRALRYLERLRDDTGIKKIMDSHRWTVPLLSEMDPAEHTRHDSKTLGLNHNQGAHIELRLRTDRYDGFRDYKTVKSTLIHELTHNVHGEHDSSFWELFRQLTKEADAADL

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 \(c\).

\(|P_{f(6HPA_1)}(2) \setminus P_{f(4GBD_1)}(2)|=69\), \(|P_{f(4GBD_1)}(2) \setminus P_{f(6HPA_1)}(2)|=111\). 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 1:000100010001000011100111001101011001001010100001001001000010100101010100101111001010001111010111100011011011000010101110101100010110111010001100000010100100001000100000110110000101111101010001101100101101110100101001010111001100101111111010000001100110010001001101001100001
Pair \(Z_2\) Length of longest common subsequence
6HPA_1,4GBD_1 180 3
6HPA_1,5JIG_1 179 3
4GBD_1,5JIG_1 191 4

Newick tree

 
[
	4GBD_1:93.86,
	[
		6HPA_1:89.5,5JIG_1:89.5
	]:4.36
]

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 \((0.85)(0.8)(\frac{731 }{\log_{20} 731}-\frac{273}{\log_{20}273})=126.\)
Status Protein1 Protein2 d d1/2
Query variables 6HPA_1 4GBD_1 160 129
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]