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Parikh vectors
4DMH_1 4LNH_1 5EQZ_1 Letter Amino acid
28 22 16 L Leucine
11 11 22 K Lycine
11 11 5 T Threonine
19 25 8 V Valine
30 21 11 A Alanine
24 22 8 G Glycine
17 11 6 H Histidine
9 16 9 I Isoleucine
18 7 3 F Phenylalanine
19 5 1 P Proline
5 1 1 W Tryptophan
8 14 12 N Asparagine
12 12 12 D Aspartic acid
2 6 0 C Cysteine
12 13 4 Q Glutamine
23 20 13 E Glutamic acid
9 8 3 M Methionine
17 10 1 R Arginine
14 18 9 S Serine
13 13 2 Y Tyrosine 

4DMH_1|Chains A, B, C, D|Putative hydrolase|Pseudomonas aeruginosa (208963)
>4LNH_1|Chain A|Uridine phosphorylase|Vibrio fischeri (312309)
>5EQZ_1|Chain A|Rev protein|Borrelia burgdorferi (strain ATCC 35210 / B31 / CIP 102532 / DSM 4680) (224326)
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 \(w=f(c)\)
4DMH , Knot 135 301 0.85 40 200 292 
AEEFPVPNGFESAYREVDGVKLHYVKGGQGPLVMLVHGFGQTWYEWHQLMPELAKRFTVIAPDLPGLGQSEPPKTGYSGEQVAVYLHKLARQFSPDRPFDLVAHDIGIWNTYPMVVKNQADIARLVYMEAPIPDARIYRFPAFTAQGESLVWHFSFFAADDRLAETLIAGKERFFLEHFIKSAASNTEVFSERLLDLYARSYAKPHSLNASFEYYRALNESVRQNAELAKTRLQMPTMTLAGGGHGGMGTFQLEQMKAYAEDVEGHVLPGCGHWLPEECAAPMNRLVIDFLSRGRHHHHHH
4LNH , Knot 118 266 0.82 40 171 259 
MHHHHHHSSGVDLGTENLYFQSMSKQPHICVDENQVSPYVIVCGEPDRVNRIVELMDNVELLAENREYRVFNGVYKGTTITICSTGIGAPSMIIAVEELKLCGATHVIRVGSAGAMQDHIQLGELIVAEGAVRDEGGSKAYISSAYPAYASFALLKEISHFLENQSVKYYFGVVRSHDSFYTDEEDQLCQYWNKKGILGADMETSALFTVGRLRGLQVASILNNVVLYQQDVKEGVNQYVNDNKAMMNGERLAAITALETLCKQGA
5EQZ , Knot 66 146 0.75 38 99 135 
MAHHHHHHYVEEKKEIDSLMEDVLALVNDSSGGKFKDYKDKINELKENLKDIGNAELKEKLLNLQNSFQDKLAAKLAALKAAKNTIENITDKDQDISKRKIWSEAKLVGVTVPLLGSNTSGNGDKMSKNAVEQIDKVIKFLEEGTN

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 \(c\).

\(|P_{f(4DMH_1)}(2) \setminus P_{f(4LNH_1)}(2)|=104\), \(|P_{f(4LNH_1)}(2) \setminus P_{f(4DMH_1)}(2)|=75\). 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 1:1001111011001000101101001011011111110111001001001110110010111101111100011001001001110100110010100110111001111000111100010110110101111010100111101010011101011110001100111100011100110011000011000110101000101001010100001100010001011000101101011111011110101001010100101011110101110001111001110110010000000
Pair \(Z_2\) Length of longest common subsequence
4DMH_1,4LNH_1 179 6
4DMH_1,5EQZ_1 185 6
4LNH_1,5EQZ_1 158 6

Newick tree

 
[
	4DMH_1:94.67,
	[
		4LNH_1:79,5EQZ_1:79
	]:15.67
]

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 \((0.85)(0.8)(\frac{567 }{\log_{20} 567}-\frac{266}{\log_{20}266})=85.1\)
Status Protein1 Protein2 d d1/2
Query variables 4DMH_1 4LNH_1 110 101
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]