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Parikh vectors
2MHK_1 7EEJ_1 5MUD_1 Letter Amino acid
29 15 11 Q Glutamine
10 8 9 E Glutamic acid
13 24 20 G Glycine
14 6 14 K Lycine
29 11 15 A Alanine
11 6 4 R Arginine
14 17 9 N Asparagine
0 2 5 C Cysteine
14 22 32 S Serine
6 11 4 W Tryptophan
4 15 9 Y Tyrosine
8 15 18 V Valine
11 11 6 I Isoleucine
26 10 12 L Leucine
13 14 20 T Threonine
11 8 15 P Proline
16 11 6 D Aspartic acid
8 6 2 H Histidine
8 6 1 M Methionine
5 9 4 F Phenylalanine 

2MHK_1|Chain A|Penicillin-binding protein activator LpoA|Escherichia coli (83333)
>7EEJ_1|Chain A|glycoside hydrolase family 12 beta-1,3-1,4-glucanase|Chaetomium sp. (1769349)
>5MUD_1|Chains A, B|light chain dimer,IGL@ protein|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 \(w=f(c)\)
2MHK , Knot 110 250 0.81 38 155 237 
MGSSHHHHHHSSGLVPRGSHMGTHTPDQSTAYMQGTAQADSAFYLQQMQQSSDDTRINWQLLAIRALVKEGKTGQAVELFNQLPQELNDAQRREKTLLAVEIKLAQKDFAGAQNLLAKITPADLEQNQQARYWQAKIDASQGRPSIDLLRALIAQEPLLGAKEKQQNIDATWQALSSMTQEQANTLVINADENILQGWLDLQRVWFDNRNDPDMMKAGIADWQKRYPNNPGAKMLPTQLVNVKAFKPAST
7EEJ , Knot 109 227 0.86 40 169 223 
LDKRQQVTLCEQYGYWSGNGYEINNNLWGRDSATSGWQCSYLDGSSDSGIQWHTTWEWQGGQHDVKSYVYSGKQFPRGQRITSINSMQTSVSWYYDTTNVRANVAYDIFTAADPNHVNSSGDYELMIWLAKYGDVQPIGSPVGTVHVNGRNWELWIGMNGNMKVFSFIAPSPLNSWSGEVKEFFNYLQYNQGYPAGDQHLIVFQMGTEAFTGGPATMTVSHFSANIY
5MUD , Knot 95 216 0.78 40 140 205 
QSVLTQPPSVSAAPGQKVTISCSGNNSNIGKNYVSWYQQLPGRTPKVIMYENNKRSSGIPDRFSGSKSGTSATLGITGLQTGDEADYYCGVWDSSLSGGVFGGGTKVTVLGQPKAAPSVTLFPPSSEELQANKATLVCLISDFYPGAVTVAWKADSSPVKAGVETTTPSKQSNNKYAASSYLSLTPEQWKSHRSYSCQVTHEGSTVEKTVAPTECS

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 \(c\).

\(|P_{f(2MHK_1)}(2) \setminus P_{f(7EEJ_1)}(2)|=80\), \(|P_{f(7EEJ_1)}(2) \setminus P_{f(2MHK_1)}(2)|=94\). 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 1:1100000000001111010011000100001010101010011010010000000010101111011100100101101100110010010000001111010110001111001110101101000001001010101001010101101111001111100000010101011001000010011101000110111010011100000101101111010000100111011100110101101100
Pair \(Z_2\) Length of longest common subsequence
2MHK_1,7EEJ_1 174 3
2MHK_1,5MUD_1 169 3
7EEJ_1,5MUD_1 169 4

Newick tree

 
[
	7EEJ_1:86.17,
	[
		2MHK_1:84.5,5MUD_1:84.5
	]:1.67
]

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 \((0.85)(0.8)(\frac{477 }{\log_{20} 477}-\frac{227}{\log_{20}227})=72.3\)
Status Protein1 Protein2 d d1/2
Query variables 2MHK_1 7EEJ_1 89 87.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]