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Parikh vectors
4HKX_1 1GGN_1 5VFC_1 Letter Amino acid
10 63 3 R Arginine
7 22 8 H Histidine
10 51 19 I Isoleucine
21 78 27 L Leucine
10 48 33 K Lycine
0 21 0 M Methionine
13 36 9 Y Tyrosine
11 62 21 V Valine
5 49 22 D Aspartic acid
4 9 10 C Cysteine
8 38 11 F Phenylalanine
15 36 11 P Proline
22 29 37 S Serine
6 12 10 W Tryptophan
13 63 18 A Alanine
16 48 20 G Glycine
9 36 21 T Threonine
16 46 17 N Asparagine
4 30 6 Q Glutamine
20 65 8 E Glutamic acid 

4HKX_1|Chain A[auth E]|Hemagglutinin HA1|Influenza A virus (464623)
>1GGN_1|Chain A|PROTEIN (GLYCOGEN PHOSPHORYLASE)|Oryctolagus cuniculus (9986)
>5VFC_1|Chain A|WD repeat-containing protein 5|Homo sapiens (9606)
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 \(w=f(c)\)
4HKX , Knot 101 220 0.82 38 147 212 
ADPGYLLEAPLQLGNCSVAGWILGNPECELLISRESWSYIVEKPNPENGTCYPGHFADYEELREQLSSVSSFERFEIFPKESSWPNHTTTGVSASCSHNGESSFYKNLLWLTGKNGLYPNLSKSYANNKEKEVLVLWGVHHPPNIGDQRALYHTENAYVSVVSSHYSRKFTPEIAKRPKVRDQEGRINYYWTLLEPGDTIIFEANGNLIAPRYAFALSRG
1GGN , Knot 321 842 0.85 40 316 771 
SRPLSDQEKRKQISVRGLAGVENVTELKKNFNRHLHFTLVKDRNVATPRDYYFALAHTVRDHLVGRWIRTQQHYYEKDPKRIYYLSLEFYMGRTLQNTMVNLALENACDEATYQLGLDMEELEEIEEDAGLGNGGLGRLAACFLDSMATLGLAAYGYGIRYEFGIFNQKICGGWQMEEADDWLRYGNPWEKARPEFTLPVHFYGRVEHTSQGAKWVDTQVVLAMPYDTPVPGYRNNVVNTMRLWSAKAPNDFNLKDFNVGGYIQAVLDRNLAENISRVLYPNDNFFEGKELRLKQEYFVVAATLQDIIRRFKSSKFGCRDPVRTNFDAFPDKVAIQLNDTHPSLAIPELMRVLVDLERLDWDKAWEVTVKTCAYTNHTVIPEALERWPVHLLETLLPRHLQIIYEINQRFLNRVAAAFPGDVDRLRRMSLVEEGAVKRINMAHLCIAGSHAVNGVARIHSEILKKTIFKDFYELEPHKFQNKTNGITPRRWLVLCNPGLAEIIAERIGEEYISDLDQLRKLLSYVDDEAFIRDVAKVKQENKLKFAAYLEREYKVHINPNSLFDVQVKRIHEYKRQLLNCLHVITLYNRIKKEPNKFVVPRTVMIGGKAAPGYHMAKMIIKLITAIGDVVNHDPVVGDRLRVIFLENYRVSLAEKVIPAADLSEQISTAGTEASGTGNMKFMLNGALTIGTMDGANVEMAEEAGEENFFIFGMRVEDVDRLDQRGYNAQEYYDRIPELRQIIEQLSSGFFSPKQPDLFKDIVNMLMHHDRFKVFADYEEYVKCQERVSALYKNPREWTRMVIRNIATSGKFSSDRTIAQYAREIWGVEPSRQRLPAPDEKIP
5VFC , Knot 122 311 0.75 38 162 267 
TQSKPTPVKPNYALKFTLAGHTKAVSSVKFSPNGEWLASSSADKLIKIWGAYDGKFEKTISGHKLGISDVAWSSDSNLLVSASDDKTLKIWDVSSGKCLKTLKGHSNYVFCCNFNPQSNLIVSGSFDESVRIWDVKTGKCLKTLPAHSDPVSAVHFNRDGSLIVSSSYDGLCRIWDTASGQCLKTLIDDDNPPVSFVKFSPNGKYILAATLDNTLKLWDYSKGKCLKTYTGHKNEKYCIFANFSVTGGKWIVSGSEDNLVYIWNLQTKEIVQKLQGHTDVVISTACHPTENIIASAALENDKTIKLWKSDC

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 \(c\).

\(|P_{f(4HKX_1)}(2) \setminus P_{f(1GGN_1)}(2)|=16\), \(|P_{f(1GGN_1)}(2) \setminus P_{f(4HKX_1)}(2)|=185\). 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 1:1011011011101100011111110100011100001001100101001000110110000100010010010010111000011000001101000001000100011110100110101000010000001111111001101100011000001010110000000101011001010000101000101101100111010101111001111001
Pair \(Z_2\) Length of longest common subsequence
4HKX_1,1GGN_1 201 6
4HKX_1,5VFC_1 161 4
1GGN_1,5VFC_1 208 4

Newick tree

 
[
	1GGN_1:10.55,
	[
		4HKX_1:80.5,5VFC_1:80.5
	]:28.05
]

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 \((0.85)(0.8)(\frac{1062 }{\log_{20} 1062}-\frac{220}{\log_{20}220})=227.\)
Status Protein1 Protein2 d d1/2
Query variables 4HKX_1 1GGN_1 283 176
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]