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Parikh vectors
6DGE_1 1NGN_1 3LHH_1 Letter Amino acid
14 9 8 R Arginine
8 6 10 N Asparagine
33 19 21 L Leucine
11 5 7 T Threonine
11 1 9 M Methionine
7 7 8 F Phenylalanine
6 7 1 Y Tyrosine
28 6 17 V Valine
9 3 8 Q Glutamine
23 11 13 E Glutamic acid
22 5 11 G Glycine
16 6 5 H Histidine
11 9 7 I Isoleucine
19 15 4 K Lycine
28 8 7 A Alanine
18 8 15 D Aspartic acid
6 1 2 C Cysteine
14 11 5 P Proline
24 10 14 S Serine
0 8 0 W Tryptophan 

6DGE_1|Chain A|N(G),N(G)-dimethylarginine dimethylaminohydrolase 1|Homo sapiens (9606)
>1NGN_1|Chain A|methyl-CpG binding protein MBD4|Mus musculus (10090)
>3LHH_1|Chain A|CBS domain protein|Shewanella oneidensis (211586)
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 \(w=f(c)\)
6DGE , Knot 132 308 0.81 38 187 287 
MGSSHHHHHHSSGLVPRGSHMASMAGLGHPAAFGRATHAVVRALPESLGQHALRSAKGEEVDVARAERQHQLYVGVLGSKLGLQVVELPADESLPDCVFVEDVAVVCEETALITRPGAPSRRKEVDMMKEALEKLQLNIVEMKDENATLDGGDVLFTGREFFVGLSKRTNQRGAEILADTFKDYAVSTVPVADGLHLKSFCSMAGPNLIAIGSSESAQKALKIMQQMSDHRYDKLTVPDDIAANCIYLNIPNKGHVLLHRTPEEYPESAKVYEKLKDHMLIPVSMSELEKVDGLLTCCSVLINKKVDS
1NGN , Knot 77 155 0.83 40 128 151 
ALSPPRRKSFKKWTPPRSPFNLVQEILFHDPWKLLIATIFLNRTSGKMAIPVLWEFLEKYPSAEVARAADWRDVSELLKPLGLYDLRAKTIIKFSDEYLTKQWRYPIELHGIGKYGNDSYRIFCVNEWKQVHPEDHKLNKYHDWLWENHEKLSLS
3LHH , Knot 81 172 0.80 38 119 168 
GHLDDNVTQEDIQAMLQEGSSAGVIEHNEHAMVKNVFRLDERTISSLMVPRSDIVFLDLNLPLDANLRTVMQSPHSRFPVCRNNVDDMVGIISAKQLLSESIAGERLELVDLVKNCNFVPNSLSGMELLEHFRTTGSQMVFVVDEYGDLKGLVTLQDMMDALTGEFFQEDGS

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 \(c\).

\(|P_{f(6DGE_1)}(2) \setminus P_{f(1NGN_1)}(2)|=115\), \(|P_{f(1NGN_1)}(2) \setminus P_{f(6DGE_1)}(2)|=56\). 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 1:11000000000011110100110111110111110100111011100110011001010010110100000101111100111011011100011001110011110000111001111000001011001100101011010000101011011101001111100000001101110010001100111101101001001111011111000010011011001000000010110011100101011001011100010001001010001000111110100100101110000111000100
Pair \(Z_2\) Length of longest common subsequence
6DGE_1,1NGN_1 171 4
6DGE_1,3LHH_1 170 3
1NGN_1,3LHH_1 169 3

Newick tree

 
[
	6DGE_1:85.49,
	[
		3LHH_1:84.5,1NGN_1:84.5
	]:0.99
]

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 \((0.85)(0.8)(\frac{463 }{\log_{20} 463}-\frac{155}{\log_{20}155})=91.0\)
Status Protein1 Protein2 d d1/2
Query variables 6DGE_1 1NGN_1 114 87
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]