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Parikh vectors
4NRV_1 8EEL_1 1NDD_1 Letter Amino acid
25 41 7 G Glycine
36 29 8 L Leucine
20 22 2 S Serine
7 13 2 Y Tyrosine
9 41 4 D Aspartic acid
17 16 4 Q Glutamine
23 30 9 E Glutamic acid
4 31 4 T Threonine
21 41 3 A Alanine
27 20 4 R Arginine
6 18 1 N Asparagine
12 9 1 H Histidine
9 26 8 I Isoleucine
10 21 9 K Lycine
26 13 3 P Proline
4 17 0 W Tryptophan
7 3 0 C Cysteine
3 11 2 M Methionine
17 15 0 F Phenylalanine
14 32 5 V Valine 

4NRV_1|Chain A|Endonuclease 8-like 1|Homo sapiens (9606)
>8EEL_1|Chains A, B|Amine oxidase|Corynebacterium ammoniagenes (1697)
>1NDD_1|Chains A, B, C, D|PROTEIN (UBIQUITIN-LIKE PROTEIN NEDD8)|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 \(w=f(c)\)
4NRV , Knot 127 297 0.81 40 173 279 
PEGPELHLASQFVNEACRALVFGGCVEKSSVSRNPEVPFESSAYRISASARGKELRLILSPLPGAQPQQEPLALVFRFGMSGSFQLVPREELPRHAHLRFYTAPPGPRLALCFVDIRRFGRWDLGGKWQPGRGPCVLQEYQQFRESVLRNLADKAFDRPICEALLDQRFFNGIGNYLRAEILYRLKIPPFEKARSVLEALQQHRPSPELTLSQKIRTKLQNPDLLELCHSVPKEVVQLGGKGYGSESGEEDFAAFRAWLRCYGMPGMSSLQDRHGRTIWFQGDPGPLAPLEHHHHHH
8EEL , Knot 183 449 0.83 40 239 423 
MSKNKVVIIGAGFAGLVAARELQTAGIEYEILEAKDRIGGRAWTEERMGRPLELGATWVHWFQAHTWTEIMRYGQRTEITASPSGNDAHWVTDGKVVKGTEDDLDEKLTAAMGVTYEGSEEYFPNPHDPLWVLSDDFDGPAEVRERFLSDDQTNAIDLVKEAGFDQETIDLVDAFWCAGYIGDPYTGSALMAKQWGALSDNRYRVMEDITLKWKLNNGMRSLYDGIAGDLNTDIRLNTPVAKVEHHDNGATVTTESGEVIEASAVICTVPVGALSNIEFSPALPDAVQSVIDDKWNSQGAKIWIKIKGHHRFLGYAPKPAKMSVVRSEYFMDDDTTILVGFGYDNTNIDLNSIEDAQAVINQWRDDLEVVDTTGHNWVADKWAGQAWGTLRKGQFTQGWSLFDDIDSQLFFAGSDYAYGWRGVCVDGALEKGMTTARQVINSMRETKEQ
1NDD , Knot 38 76 0.72 34 60 74 
MLIKVKTLTGKEIEIDIEPTDKVERIKERVEEKEGIPPQQQRLIYSGKQMNDEKTAADYKILGGSVLHLVLALRGG

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 \(c\).

\(|P_{f(4NRV_1)}(2) \setminus P_{f(8EEL_1)}(2)|=60\), \(|P_{f(8EEL_1)}(2) \setminus P_{f(4NRV_1)}(2)|=126\). 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 1:101101011001100100111111010000100010111000100101010100101110111110100011111101110101011100011001010100111110111011010011010111010110110110000010001100110011001100111000110111001010110010111100100110110000101010100010001001011010001100110111010100010001111011100011111001000010011101011111110000000
Pair \(Z_2\) Length of longest common subsequence
4NRV_1,8EEL_1 186 4
4NRV_1,1NDD_1 155 3
8EEL_1,1NDD_1 213 3

Newick tree

 
[
	8EEL_1:10.42,
	[
		4NRV_1:77.5,1NDD_1:77.5
	]:28.92
]

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 \((0.85)(0.8)(\frac{746 }{\log_{20} 746}-\frac{297}{\log_{20}297})=123.\)
Status Protein1 Protein2 d d1/2
Query variables 4NRV_1 8EEL_1 159 130.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]