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Parikh vectors
4CCN_1 7DKB_1 7USV_1 Letter Amino acid
5 4 4 C Cysteine
13 8 1 H Histidine
62 70 12 L Leucine
11 16 0 M Methionine
31 33 10 P Proline
7 10 0 W Tryptophan
30 49 17 V Valine
41 103 2 A Alanine
13 33 9 N Asparagine
34 32 16 E Glutamic acid
10 19 9 I Isoleucine
21 31 9 T Threonine
19 26 8 Y Tyrosine
38 32 1 R Arginine
25 34 1 Q Glutamine
21 29 5 F Phenylalanine
25 36 7 S Serine
27 46 6 D Aspartic acid
7 49 23 K Lycine
27 60 8 G Glycine 

4CCN_1|Chains A, B|BIFUNCTIONAL LYSINE-SPECIFIC DEMETHYLASE AND HISTIDYL-HYDROXYLASE NO66|HOMO SAPIENS (9606)
>7DKB_1|Chains A, B|Dipeptidyl-peptidase|Stenotrophomonas maltophilia (strain R551-3) (391008)
>7USV_1|Chains A, B|Gametocyte surface protein P230|Plasmodium falciparum (36329)
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 \(w=f(c)\)
4CCN , Knot 188 467 0.82 40 224 437 
MSPLRRVLAELNRIPSSRRRAARLFEWLIAPMPPDHFYRRLWEREAVLVRRQDHTYYQGLFSTADLDSMLRNEEVQFGQHLDAARYINGRRETLNPPGRALPAAAWSLYQAGCSLRLCSPQAFSTTVWQFLAVLQEQFGSMAGSNVYLTPPNSQGFAPHYDDIEAFVLQLEGRKLWRVYRPRAPTEELALTSSPNFSQDDLGEPVLQTVLEPGDLLYFPRGFIHQAECQDGVHSLHLTLSTYQRNTWGDFLEAILPLAVQAAMEENVEFRRGLPRDFMDYMGAQHSDSKDPRRTAFMEKVRVLVARLGHFAPVDAVADQRAKDFIHDSLPPVLTDRERALSVYGLPIRWEAGEPVNVGAQLTTETEVHMLQDGIARLVGEGGHLFLYYTVENSRVYHLEEPKCLEIYPQQADAMELLLGSYPEFVRVGDLPCDSVEDQLSLATTLYDKGLLLTKMPLALNAENLYFQ
7DKB , Knot 273 720 0.83 40 263 646 
MRSNLLAFSIVASLGLAQVAHAAEGMWVPQQLPEIAGPLQKAGLKLSPEQLANLTGDPMGAVVALGGCTASFVSPQGLVVTNHHCAYGAIQLNSTAQKNLIKDGFNAPTLKDELSAGPNARVFVLDQITDVTAQAKAAIAGAGNDPLARSRALDAFDKAQVAACEADAGFRCRLYSFSGGNTYRLFRNMEIKDVRLVYAPPGSVGKFGGDVDNWMWPRHTGDFSFYRAYVGKDGKPAAFAADNVPYQPKHFLKFADQPLGADDFVMVAGYPGRTNRYALAGEFNETASFTYPTIAKHYNAVLKMIADAGKADADVKVKYAATAASMNNVAKNYLGQLEGFKRIDAAGQKQAEEAAVLAWLKKQGAAGKPALAAHAQLLKHLDTSKSTRERDLFVGQFNNTSAVGAAITLYRLSIERSKPDAEREAGYQERDLTTIEGGLKQMDRRYVAKMDQQLQTYWLDQYVALPAAQRDNEVLNKWLAGSDAAAVKSLVNKLGGTELGSLDTRLKWFKADRAAFEASNDPAIQYAVAVMPALLKQEEQKKIREGESLTARPLYLQAVADYKKSQGEFVYPDANLSLRITFGNVMGYGKDGVKYTPFTTLEGVAAKETGEDPFDSPKALLDAVKAKRYGGLEDKRLGSVPVNFLSNLDITGGNSGSPVLDANGKLVGLAFDGNWESVSSNWVFDPVMTRMIAVDSRYMQWIMQEVAPAPQLLKELNLAK
7USV , Knot 70 148 0.78 36 98 142 
GASTNKEYVCDFTDQLKPTESGPKVKKCEVKVNEPLIKVKIICPLKGSVEKLYDNIEYVPKKSPYVVLTKEETKLKEKLLSKLIYGLLISPTVNEKENNFKEGVIEFTLPPVVHKATVFYFICDNSKTEDDNKKGNRGIVEVYVEPYG

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 \(c\).

\(|P_{f(4CCN_1)}(2) \setminus P_{f(7DKB_1)}(2)|=62\), \(|P_{f(7DKB_1)}(2) \setminus P_{f(4CCN_1)}(2)|=101\). 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 1:10110011101001100000110110111111110010001100011110000000001110010100110000101100101100101000010111011111110100110010100101100011011111000110111001010110001111000010111101010011010010110001110001010000110111001101101101101110010000110010101000000011011011111110111000101001110011001110000000100011100101111011011110111000100110001111100000110101111010110110111010000010110011101110110111000100001001001001010100101101111001011011011000100010110010001111001111101001010
Pair \(Z_2\) Length of longest common subsequence
4CCN_1,7DKB_1 163 5
4CCN_1,7USV_1 200 4
7DKB_1,7USV_1 217 3

Newick tree

 
[
	7USV_1:11.90,
	[
		4CCN_1:81.5,7DKB_1:81.5
	]:29.40
]

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 \((0.85)(0.8)(\frac{1187 }{\log_{20} 1187}-\frac{467}{\log_{20}467})=186.\)
Status Protein1 Protein2 d d1/2
Query variables 4CCN_1 7DKB_1 236 194
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]