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Parikh vectors
1ICF_1 1GNL_1 4WAM_1 Letter Amino acid
6 11 4 C Cysteine
15 27 15 E Glutamic acid
8 20 6 F Phenylalanine
15 26 13 S Serine
12 61 15 A Alanine
7 22 16 N Asparagine
8 13 8 Q Glutamine
9 56 25 L Leucine
9 41 15 K Lycine
4 10 5 M Methionine
8 18 4 P Proline
7 35 10 T Threonine
2 5 5 W Tryptophan
10 18 7 Y Tyrosine
13 39 16 V Valine
4 14 10 R Arginine
11 35 14 D Aspartic acid
2 13 12 H Histidine
20 50 12 G Glycine
5 30 17 I Isoleucine 

1ICF_1|Chains A, C|PROTEIN (CATHEPSIN L: HEAVY CHAIN)|Homo sapiens (9606)
>1GNL_1|Chains A, B|HYBRID CLUSTER PROTEIN|DESULFOVIBRIO DESULFURICANS (876)
>4WAM_1|Chains A, B|beta-carbonic anhydrase|Haemophilus influenzae (71421)
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 \(w=f(c)\)
1ICF , Knot 82 175 0.80 40 132 168 
APRSVDWREKGYVTPVKNQGQCGSCWAFSATGALEGQMFRKTGRLISLSEQNLVDCSGPQGNEGCNGGLMDYAFQYVQDNGGLDSEESYPYEATEESCKYNPKYSVANDTGFVDIPKQEKALMKAVATVGPISVAIDAGHESFLFYKEGIYFEPDCSSEDMDHGVLVVGYGFEST
1GNL , Knot 218 544 0.84 40 251 510 
SNAMFCYQCQETVGNKGCTQVGVCGKKPETAALQDALIYVTKGLGQIATRLRAEGKAVDHRIDRLVTGNLFATITNANFDDDILAERVRMTCAAKKELAASLTDKSGLSDAALWEASEKSAMLAKAGTVGVMATTDDDVRSLRWLITFGLKGMAAYAKHADVLGKHENSLDAFMQEALAKTLDDSLSVADLVALTLETGKFGVSAMALLDAANTGTYGHPEITKVNIGVGSNPGILISGHDLRDLEMLLKQTEGTGVDVYTHSEMLPAHYYPAFKKYAHFKGNYGNAWWKQKEEFESFNGPVLLTTNCLVPPKDSYKDRVYTTGIVGFTGCKHIPGEIGEHKDFSAIIAHAKTCPAPTEIESGEIIGGFAHNQVLALADKVIDAVKSGAIKKFVVMAGCDGRAKSRSYYTDFAEGLPKDTVILTAGCAKYRYNKLNLGDIGGIPRVLDAGQCNDSYSLAVIALKLKEVFGLEDVNDLPIVYNIAWYEQKAVIVLLALLSLGVKNIHLGPTLPAFLSPNVAKVLVEQFNIGGITSPQDDLKAFFG
4WAM , Knot 106 229 0.83 40 169 222 
MDKIKQLFANNYSWAQRMKEENSTYFKELADHQTPHYLVIACSDSRVSPEKLTNLEPGELFVHRNVANQVIHTDFNCLSVVQYAVDVLKIEHIIICGHTNCGGIHAAMADKDLGLINNWLLHIRDIWFKHGHLLGKLSPEKRADMLTKINVAEQVYNLGRTSIVKSAWERGQKLSLHGWVYDVNDGFLVDQGVMATSRETLEISYRNAIARLSILDEENILKKDHLENT

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 \(c\).

\(|P_{f(1ICF_1)}(2) \setminus P_{f(1GNL_1)}(2)|=33\), \(|P_{f(1GNL_1)}(2) \setminus P_{f(1ICF_1)}(2)|=152\). 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 1:1100101000101011000100100111010111010110001011010000110001101001001111001100100011100000010010000000010001100011101100001110111011110111011000111000110101000000100111111011000
Pair \(Z_2\) Length of longest common subsequence
1ICF_1,1GNL_1 185 3
1ICF_1,4WAM_1 181 3
1GNL_1,4WAM_1 170 5

Newick tree

 
[
	1ICF_1:93.57,
	[
		4WAM_1:85,1GNL_1:85
	]:8.57
]

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 \((0.85)(0.8)(\frac{719 }{\log_{20} 719}-\frac{175}{\log_{20}175})=153.\)
Status Protein1 Protein2 d d1/2
Query variables 1ICF_1 1GNL_1 193 125.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]