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Parikh vectors
6VDT_1 5XSE_1 1YIC_1 Letter Amino acid
10 8 4 F Phenylalanine
11 10 4 D Aspartic acid
7 4 2 Q Glutamine
2 1 4 H Histidine
12 13 16 K Lycine
6 7 2 M Methionine
6 5 7 N Asparagine
2 15 4 P Proline
7 8 8 T Threonine
1 1 1 W Tryptophan
4 1 5 Y Tyrosine
12 8 5 S Serine
8 22 3 V Valine
12 34 7 A Alanine
6 7 3 R Arginine
6 10 7 E Glutamic acid
9 18 12 G Glycine
13 22 8 L Leucine
1 4 2 C Cysteine
2 10 4 I Isoleucine 

6VDT_1|Chains A, B|Dehaloperoxidase B|Amphitrite ornata (129555)
>5XSE_1|Chains A, B, C|KHG/KDPG aldolase|Zymomonas mobilis subsp. mobilis (strain ATCC 31821 / ZM4 / CP4) (264203)
>1YIC_1|Chain A|CYTOCHROME C, ISO-1|Saccharomyces cerevisiae (4932)
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 \(w=f(c)\)
6VDT , Knot 68 137 0.81 40 105 134 
GFKQDIATLRGDLRTYAQDIFLAFLNKYPDEKRNFKNYVGKSDQELKSMAKFGDHTEKVFNLMMEVADRATDCVPLASDASTLVQMKQHSGLTTGNFEKLFVALVEYMRASGQSFDSQSWDRFGKNLVSALSSAGMK
5XSE , Knot 96 208 0.82 40 128 201 
MRDIDSVMRLAPVMPVLVIEDIADAKPIAEALVAGGLNVLEVTLRTPCALEAIKIMKEVPGAVVGAGTVLNAKMLDQAQEAGCEFFVSPGLTADLGKHAVAQKAALLPGVANAADVMLGLDLGLDRFKFFPAENIGGLPALKSMASVFRQVRFCPTGGITPTSAPKYLENPSILCVGGSWVVPAGKPDVAKITALAKEASAFKRAAVA
1YIC , Knot 59 108 0.85 40 93 105 
TEFKAGSAKKGATLFKTRCLQCHTVEKGGPHKVGPNLHGIFGRHSGQAEGYSYTDANIKKNVLWDENNMSEYLTNPKKYIPGTKMAFGGLKKEKDRNDLITYLKKASE

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 \(c\).

\(|P_{f(6VDT_1)}(2) \setminus P_{f(5XSE_1)}(2)|=57\), \(|P_{f(5XSE_1)}(2) \setminus P_{f(6VDT_1)}(2)|=80\). 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 1:11000110101010001001111110001000001000110000010011011000001101110110010001111001001101000011001010011111100101010010000100110011011001110
Pair \(Z_2\) Length of longest common subsequence
6VDT_1,5XSE_1 137 5
6VDT_1,1YIC_1 126 3
5XSE_1,1YIC_1 143 3

Newick tree

 
[
	5XSE_1:72.20,
	[
		6VDT_1:63,1YIC_1:63
	]:9.20
]

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 \((0.85)(0.8)(\frac{345 }{\log_{20} 345}-\frac{137}{\log_{20}137})=63.5\)
Status Protein1 Protein2 d d1/2
Query variables 6VDT_1 5XSE_1 78 64.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]