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Parikh vectors
5AWB_1 7JUM_1 3AAC_1 Letter Amino acid
26 4 7 A Alanine
43 12 3 D Aspartic acid
49 5 4 F Phenylalanine
28 21 12 V Valine
14 4 0 C Cysteine
30 11 10 G Glycine
22 1 0 H Histidine
132 16 10 L Leucine
40 26 11 K Lycine
6 0 2 M Methionine
35 11 6 P Proline
76 14 5 S Serine
6 0 0 W Tryptophan
38 1 8 R Arginine
80 12 4 N Asparagine
29 3 3 Q Glutamine
43 20 15 E Glutamic acid
51 11 9 I Isoleucine
42 12 4 T Threonine
21 11 10 Y Tyrosine 

5AWB_1|Chain A|Toll-like receptor 8|Homo sapiens (9606)
>7JUM_1|Chains A, B, C|Gametocyte surface protein P230|Plasmodium falciparum (isolate 3D7) (36329)
>3AAC_1|Chains A, B|Putative uncharacterized protein ST1653|Sulfolobus tokodaii (273063)
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 \(w=f(c)\)
5AWB , Knot 287 811 0.79 40 270 664 
RSPWEENFSRSYPCDEKKQNDSVIAECSNRRLQEVPQTVGKYVTELDLSDNFITHITNESFQGLQNLTKINLNHNPNVQHQNGNPGIQSNGLNITDGAFLNLKNLRELLLEDNQLPQIPSGLPESLTELSLIQNNIYNITKEGISRLINLKNLYLAWNCYFNKVCEKTNIEDGVFETLTNLELLSLSFNSLSHVPPKLPSSLRKLFLSNTQIKYISEEDFKGLINLTLLDLSGNCPRCFNAPFPCVPCDGGASINIDRFAFQNLTQLRYLNLSSTSLRKINAAWFKNMPHLKVLDLEFNYLVGEIASGAFLTMLPRLEILDLSFNYIKGSYPQHINISRNFSKLLSLRALHLRGYVFQELREDDFQPLMQLPNLSTINLGINFIKQIDFKLFQNFSNLEIIYLSENRISPLVKDTRQSYANSSSFQRHIRKRRSTDFEFDPHSNFYHFTRPLIKPQCAAYGKALDLSLNSIFFIGPNQFENLPDIACLNLSANSNAQVLSGTEFSAIPHVKYLDLTNNRLDFDNASALTELSDLEVLDLSYNSHYFRIAGVTHHLEFIQNFTNLKVLNLSHNNIYTLTDKYNLESKSLVELVFSGNRLDILWNDDDNRYISIFKGLKNLTRLDLSLNRLKHIPNEAFLNLPASLTELHINDNMLKFFNWTLLQQFPRLELLDLRGNKLLFLTDSLSDFTSSLRTLLLSHNRISHLPSGFLSEVSSLKHLDLSSNLLKTINKSALETKTTTKLSMLELHGNPFECTCDIGDFRRWMDEHLNVKIPRLVDVICASPGDQRGKSIVSLELTTCVSDVTEFLVPR
7JUM , Knot 88 195 0.79 36 126 187 
SVLQSGALPSVGVDELDKIDLSYETTESGDTAVSEDSYDKYASQNTNKEYVCDFTDQLKPTESGPKVKKCEVKVNEPLIKVKIICPLKGSVEKLYDNIEYVPKKSPYVVLTKEETKLKEKLLSKLIYGLLISPTVNEKENNFKEGVIEFTLPPVVHKATVFYFICDNSKTEDDNKKGNRGIVEVYVEPYGNKING
3AAC , Knot 62 123 0.80 34 98 121 
MYYLGKELQKRSEELSRGFYELVYPPVDMYEEGGYLVVVADLAGFNKEKIKARVSGQNELIIEAEREITEPGVKYLTQRPKYVRKVIRLPYNVAKDAEISGKYENGVLTIRIPIAGTSVFKFE

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 \(c\).

\(|P_{f(5AWB_1)}(2) \setminus P_{f(7JUM_1)}(2)|=161\), \(|P_{f(7JUM_1)}(2) \setminus P_{f(5AWB_1)}(2)|=17\). 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 1:0011000100001000000000111000000100110011001001010001100100001011001001010001010000101110001101001111010010011100001101101110010010110001001000110011010010111000100100000100111001001011010100100111011001001110000100100001011101011010100100101111011001110101001110010010010100001001011110011010110101001110110111101110101101010010100100101000100110101101010110010000101110110100101110110010101100100101101000010111000000010000100010000000101010001001001110100110101101010011111100100110110101010001011010010111010010100001010010110010010110100000010111100010110010010110100001001000001000011011101001011100000001011011001001010100100110011101110100101000110110101100110101101010011110001001000100111000010011011100100100101000110010001100000001011010101100000110100110001010110110110101100010011010100010010011110
Pair \(Z_2\) Length of longest common subsequence
5AWB_1,7JUM_1 178 4
5AWB_1,3AAC_1 212 4
7JUM_1,3AAC_1 130 3

Newick tree

 
[
	5AWB_1:10.59,
	[
		7JUM_1:65,3AAC_1:65
	]:41.59
]

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 \((0.85)(0.8)(\frac{1006 }{\log_{20} 1006}-\frac{195}{\log_{20}195})=221.\)
Status Protein1 Protein2 d d1/2
Query variables 5AWB_1 7JUM_1 268 166
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]