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Parikh vectors
4ESM_1 4ONO_1 1PSJ_1 Letter Amino acid
21 41 12 G Glycine
6 4 3 M Methionine
10 40 6 S Serine
14 16 4 T Threonine
16 15 4 R Arginine
11 11 5 N Asparagine
20 18 5 P Proline
7 12 3 W Tryptophan
11 15 7 Y Tyrosine
10 26 4 Q Glutamine
11 15 10 K Lycine
7 17 1 H Histidine
35 38 4 L Leucine
18 25 5 V Valine
30 16 7 A Alanine
3 8 14 C Cysteine
12 14 5 I Isoleucine
17 20 6 F Phenylalanine
30 19 11 D Aspartic acid
13 25 8 E Glutamic acid 

4ESM_1|Chain A|Phenylalanine-4-hydroxylase|Chromobacterium violaceum (243365)
>4ONO_1|Chain A|Beta-2-microglobulin/T-cell surface glycoprotein CD1c/T-cell surface glycoprotein CD1b chimeric protein|Homo sapiens (9606)
>1PSJ_1|Chain A|PHOSPHOLIPASE A2|Gloydius halys (8714)
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 \(w=f(c)\)
4ESM , Knot 134 302 0.84 40 188 287 
GLPGSMNDRADFVVPDITTRKNVGLSHDANDFTLPQPLDRYSAEDHATWATLYQRQCKLLPGRACDEFLEGLERLEVDADRVPDFNKLNEKLMAATGWKIVAVPGLIPDDVFFEHLANRRFPVTWWLREPHQLDYLQEPDVFHDLFGHVPLLINPVFADALEAYGKGGVKAKALGALPMLARLYWYTVEFGLINTPAGMRIYGAGILSSKSESIYCLDSASPNRVGFDLMRIMNTRYRIDTFQKTYFVIDSFKQLFDATAPDFAPLYLQLADAQPWGAGDIAPDDLVLNAGDHQGWADTEDV
4ONO , Knot 164 395 0.82 40 224 368 
PIQRTPKIQVYSRHPAENGKSNFLNCYVSGFHPSDIEVDLLKNGERIEKVEHSDLSFSKDWSFYLLYYTEFTPTEKDEYACRVNHVTLSQPKIVKWDRDMGGGGSGGSGSGGGSSADASQEHVSFHVIQIFSFVNQSWARGQGSGWLDELQTHGWDSESGTIIFLHQWSKGQFSNEELSDLELLFRFYLFGLTREIQDHASQDYSKYPFEVQVKAGCELHSGGSPEGFFQVAFNGLDLLSFQQTTWVPSPGCGSLAQSVCHLLNHQYEGVTETVYNLIRSTCPRFLLGLLDAGKMYVHRQVKPEAWLSSGPSPGPGRLQLVCHVSGFYPKPVWVMWMRGEQEQQGTQLGDILPNAQGTWYLRATLDVADGEAAGLSCRVKHSSLEGQDIILYWHH
1PSJ , Knot 65 124 0.84 40 99 122 
SLIQFETLIMKVAKKSGMFWYSNYGCYCGWGGQGRPQDATDRCCFVHDCCYGKVTGCDPKMDVYSFSEENGDIVCGGDDPCKKEICECDRAAAICFRDNLTLYNDKKYWAFGAKNCPQEESEPC

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 \(c\).

\(|P_{f(4ESM_1)}(2) \setminus P_{f(4ONO_1)}(2)|=73\), \(|P_{f(4ONO_1)}(2) \setminus P_{f(4ESM_1)}(2)|=109\). 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 1:11110100010111101000001110001001011011000010001011010000001111010001101100101010011010010001111011011111111100111001100011101110010010010010110011101111101111011010101110101111111110101001011110011110101111100000010010010100111011011000001001000011100100110101101111010110101111101110011101100011100001
Pair \(Z_2\) Length of longest common subsequence
4ESM_1,4ONO_1 182 3
4ESM_1,1PSJ_1 193 3
4ONO_1,1PSJ_1 195 3

Newick tree

 
[
	1PSJ_1:98.92,
	[
		4ESM_1:91,4ONO_1:91
	]:7.92
]

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 \((0.85)(0.8)(\frac{697 }{\log_{20} 697}-\frac{302}{\log_{20}302})=109.\)
Status Protein1 Protein2 d d1/2
Query variables 4ESM_1 4ONO_1 138 122
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]