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Parikh vectors
1FVX_1 8UWY_1 8XDN_1 Letter Amino acid
15 30 8 I Isoleucine
5 16 7 M Methionine
8 24 28 S Serine
3 16 9 Y Tyrosine
9 22 10 D Aspartic acid
3 6 6 C Cysteine
10 21 25 V Valine
2 18 9 R Arginine
5 26 26 T Threonine
2 11 16 Q Glutamine
13 28 42 G Glycine
8 27 40 L Leucine
10 19 15 K Lycine
6 29 31 A Alanine
9 12 16 N Asparagine
5 12 14 F Phenylalanine
3 19 31 P Proline
3 4 3 W Tryptophan
19 28 15 E Glutamic acid
0 9 10 H Histidine 

1FVX_1|Chain A|FLAVODOXIN|Clostridium beijerinckii (1520)
>8UWY_1|Chains A, B|Actin, alpha cardiac muscle 1|Sus scrofa (9823)
>8XDN_1|Chains A, B|Mitochondrial import receptor subunit TOM40 homolog|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 \(w=f(c)\)
1FVX , Knot 68 138 0.81 38 101 133 
MKIVYWSGTGNTEKMAELIAKGIIESGKDVNTINVSDVNIDELLNEDILILGCSAMNDEVLEESEFEPFIEEISTKISGKKVALFGSYGWGDGKWMRDFEERMNGYGCVVVETPLIVQNEPDEAEQDCIEFGKKIANI
8UWY , Knot 160 377 0.84 40 218 361 
MCDDEETTALVCDNGSGLVKAGFAGDDAPRAVFPSIVGRPRHQGVMVGMGQKDSYVGDEAQSKRGILTLKYPIEHGIITNWDDMEKIWHHTFYNELRVAPEEHPTLLTEAPLNPKANREKMTQIMFETFNVPAMYVAIQAVLSLYASGRTTGIVLDSGDGVTHNVPIYEGYALPHAIMRLDLAGRDLTDYLMKILTERGYSFVTTAEREIVRDIKEKLCYVALDFENEMATAASSSSLEKSYELPDGQVITIGNERFRCPETLFQPSFIGMESAGIHETTYNSIMKCDIDIRKDLYANNVLSGGTTMYPGIADRMQKEITALAPSTMKIKIIAPPERKYSVWIGGSILASLSTFQQMWISKQEYDEAGPSIVHRKCF
8XDN , Knot 148 361 0.80 40 193 336 
MGNVLAASSPPAGPPPPPAPALVGLPPPPPSPPGFTLPPLGGSLGAGTSTSRSSERTPGAATASASGAAEDGACGCLPNPGTFEECHRKCKELFPIQMEGVKLTVNKGLSNHFQVNHTVALSTIGESNYHFGVTYVGTKQLSPTEAFPVLVGDMDNSGSLNAQVIHQLGPGLRSKMAIQTQQSKFVNWQVDGEYRGSDFTAAVTLGNPDVLVGSGILVAHYLQSITPCLALGGELVYHRRPGEEGTVMSLAGKYTLNNWLATVTLGQAGMHATYYHKASDQLQVGVEFEASTRMQDTSVSFGYQLDLPKANLLFKGSVDSNWIVGATLEKKLPPLPLTLALGAFLNHRKNKFQCGFGLTIG

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 \(c\).

\(|P_{f(1FVX_1)}(2) \setminus P_{f(8UWY_1)}(2)|=44\), \(|P_{f(8UWY_1)}(2) \setminus P_{f(1FVX_1)}(2)|=161\). 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 1:101101010100001101110111001001001010010100110001111100110001100001011100100010100111110011101011001000101010111001111000100100001011001101
Pair \(Z_2\) Length of longest common subsequence
1FVX_1,8UWY_1 205 3
1FVX_1,8XDN_1 184 3
8UWY_1,8XDN_1 163 4

Newick tree

 
[
	1FVX_1:10.14,
	[
		8XDN_1:81.5,8UWY_1:81.5
	]:20.64
]

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 \((0.85)(0.8)(\frac{515 }{\log_{20} 515}-\frac{138}{\log_{20}138})=110.\)
Status Protein1 Protein2 d d1/2
Query variables 1FVX_1 8UWY_1 140 94.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]