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Parikh vectors
5LFZ_1 4PEG_1 4PNJ_1 Letter Amino acid
6 9 4 R Arginine
9 14 5 Q Glutamine
17 31 19 K Lycine
17 16 8 V Valine
3 18 1 N Asparagine
2 7 0 C Cysteine
18 33 18 L Leucine
6 27 6 S Serine
7 28 3 Y Tyrosine
17 15 4 P Proline
18 14 5 T Threonine
19 10 17 A Alanine
13 20 14 E Glutamic acid
13 24 11 G Glycine
18 17 12 H Histidine
2 9 3 M Methionine
7 15 6 F Phenylalanine
18 17 7 D Aspartic acid
10 31 9 I Isoleucine
3 5 2 W Tryptophan 

5LFZ_1|Chain A|ArCE4A|Arthrobacter sp. AW19M34-1 (1914985)
>4PEG_1|Chains A, B, C, D, E|RNA lariat debranching enzyme, putative|Entamoeba histolytica (5759)
>4PNJ_1|Chain A|Myoglobin|Physeter catodon (9755)
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 \(w=f(c)\)
5LFZ , Knot 102 223 0.82 40 152 217 
HHHHHHAGQPEPVATPPAVDCATTKCVALTFDDGPGEYTNRLLDELSEQHTPATFFVLGKNVKKYPKTLKRMVDEGHQIGSHTFDHKDITKLTAEGIEHEVQWTDEAIEQAAGVKPQILRPPYGAHGAVYDRLIPYPLVLWDVDTLDWKHHDPQKTVRIALEEAKPGSIILMHDIHESSVKAVPQLVSKLHDAGYTLVTVDQLFAGTDFKPAKAYDHRFKTNP
4PEG , Knot 154 360 0.84 40 221 348 
MATEQIQHIAIVGCVHGKYREMYRQLSEYEKSTGKEISFVICTGDMQTLRYEADLVYLKVPPKYKQMGDFHLYYEGKEKAPYLTLFIGGNHESSNVLLHLYNGGFVCFNMYYLGVCSCININGLRIVGVSGIYKSFDEKKPYTYPPSPNDVVSLFHTRNYVIQMLSNLSQSSQIDISLSHDWPQGIVMKGNYKQLYRFQPGFKKDGASLGSPINKVILNTLKPKYWISGHMHCEYHAEEGPTHFIALGKIGYKNAISYLDLPLKQKTDLEYDKDWVCNLIMTWPAFSNKAQFPDLSYSISELLSKRTKELDKKIIELWEKYIGLKIIYDSDTFDIQFTSRRFYIEKIYNELNINHHHHHH
4PNJ , Knot 74 154 0.80 38 112 149 
MVLSEGEWQLVLHVWAKVEADVAGHGQDILIRLFKSHPETLEKFDRFKHLKTEAEMKASEDLKKHGVTVLTALGAILKKKGHHEAELKPLAQSHATKHKIPIKYLEFISEAIIHVLHSRHPGDFGADAQGAMNKALELFRKDIAAKYKELGYQG

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 \(c\).

\(|P_{f(5LFZ_1)}(2) \setminus P_{f(4PEG_1)}(2)|=58\), \(|P_{f(4PEG_1)}(2) \setminus P_{f(5LFZ_1)}(2)|=127\). 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 1:0000001101011101111001000011101001110000011001000001101111100100010010011001001100010000100101011000101000110011110101101101101110001110111110100101000010001011100101101111001000010111011001001100110100111100101101000010001
Pair \(Z_2\) Length of longest common subsequence
5LFZ_1,4PEG_1 185 6
5LFZ_1,4PNJ_1 134 3
4PEG_1,4PNJ_1 191 3

Newick tree

 
[
	4PEG_1:10.42,
	[
		5LFZ_1:67,4PNJ_1:67
	]:34.42
]

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 \((0.85)(0.8)(\frac{583 }{\log_{20} 583}-\frac{223}{\log_{20}223})=102.\)
Status Protein1 Protein2 d d1/2
Query variables 5LFZ_1 4PEG_1 135 106.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]