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Parikh vectors
2DMA_1 5GZF_1 6ZJD_1 Letter Amino acid
18 6 15 A Alanine
0 6 4 H Histidine
11 13 15 V Valine
20 10 14 R Arginine
37 9 13 E Glutamic acid
9 14 15 G Glycine
5 4 4 M Methionine
8 10 18 T Threonine
1 1 3 W Tryptophan
9 15 11 S Serine
5 10 13 D Aspartic acid
0 2 0 C Cysteine
6 8 11 Q Glutamine
23 17 14 I Isoleucine
4 10 7 F Phenylalanine
0 12 14 P Proline
8 11 7 N Asparagine
15 16 25 L Leucine
17 11 16 K Lycine
2 4 8 Y Tyrosine 

2DMA_1|Chain A|V-type ATP synthase subunit E|Pyrococcus horikoshii (70601)
>5GZF_1|Chain A|Galectin-8|Homo sapiens (9606)
>6ZJD_1|Chain A|GTP:AMP phosphotransferase AK3, mitochondrial|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 \(w=f(c)\)
2DMA , Knot 84 198 0.74 34 111 184 
MNGAELIIQEINKEAERKIEYILNEARQQAEKIKEEARRNAEAKAEWIIRRAKTQAELEKQRIIANARLEVRRKRLAIQEEIISSVLEEVKRRLETMSEDEYFESVKALLKEAIKELNEKKVRVMSNEKTLGLIASRIEEIKSELGDVSIELGETVDTMGGVIVETEDGRIRIDNTFEARMERFEGEIRSTIAKVLFG
5GZF , Knot 88 189 0.81 40 136 181 
GSHMMLSLNNLQNIIYNPVIPFVGTIPDQLDPGTLIVIRGHVPSDADRFQVDLQNGSSMKPRADVAFHFNPRFKRAGCIVCNTLINEKWGREEITYDTPFKREKSFEIVIMVLKDKFQVAVNGKHTLLYGHRIGPEKIDTLGIYGKVNIHSIGFSFSSDLQSTQASSLELTEISRENVPKSGTPQLRLP
6ZJD , Knot 102 227 0.81 38 159 216 
MGASARLLRAVIMGAPGSGKGTVSSRITTHFELKHLSSGDLLRDNMLRGTEIGVLAKAFIDQGKLIPDDVMTRLALHELKNLTQYSWLLDGFPRTLPQAEALDRAYQIDTVINLNVPFEVIKQRLTARWIHPASGRVYNIEFNPPKTVGIDDLTGEPLIQREDDKPETVIKRLKAYEDQTKPVLEYYQKKGVLETFSGTETNKIWPYVYAFLQTKVPQRSQKASVTP

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 \(c\).

\(|P_{f(2DMA_1)}(2) \setminus P_{f(5GZF_1)}(2)|=67\), \(|P_{f(5GZF_1)}(2) \setminus P_{f(2DMA_1)}(2)|=92\). 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 1:101101110010001000100110010001001000100010101011100100010100001110101010000111000110011001000100100000100101110011001000010110000011111001001000110101011001001111110000101010001010100101010001101111
Pair \(Z_2\) Length of longest common subsequence
2DMA_1,5GZF_1 159 3
2DMA_1,6ZJD_1 146 3
5GZF_1,6ZJD_1 145 3

Newick tree

 
[
	2DMA_1:77.55,
	[
		6ZJD_1:72.5,5GZF_1:72.5
	]:5.05
]

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 \((0.85)(0.8)(\frac{387 }{\log_{20} 387}-\frac{189}{\log_{20}189})=58.8\)
Status Protein1 Protein2 d d1/2
Query variables 2DMA_1 5GZF_1 74 74
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]