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Parikh vectors
5GVH_1 2MRN_1 1VFH_1 Letter Amino acid
15 4 13 S Serine
2 1 8 W Tryptophan
36 6 32 V Valine
18 3 30 R Arginine
31 3 39 G Glycine
20 6 5 K Lycine
9 0 4 F Phenylalanine
6 4 6 N Asparagine
14 8 19 D Aspartic acid
1 0 4 C Cysteine
21 5 32 L Leucine
3 0 9 Y Tyrosine
6 1 10 Q Glutamine
4 8 16 H Histidine
12 2 7 M Methionine
14 3 22 P Proline
33 3 66 A Alanine
29 3 27 E Glutamic acid
24 6 15 I Isoleucine
16 1 22 T Threonine 

5GVH_1|Chain A|Enoyl-[acyl-carrier-protein] reductase [FMN]|Thermotoga maritima (243274)
>2MRN_1|Chains A, B|Antitoxin MazE|Escherichia coli K-12 (83333)
>1VFH_1|Chain A|alanine racemase|Streptomyces lavendulae (1914)
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 \(w=f(c)\)
5GVH , Knot 136 314 0.83 40 168 300 
MTVRTRVTDLLEIEHPILMGGMAWAGTPTLAAAVSEAGGLGIIGSGAMKPDDLRKAISELRQKTDKPFGVNIILVSPWADDLVKVCIEEKVPVVTFGAGNPTKYIRELKENGTKVIPVVASDSLARMVERAGADAVIAEGMESGGHIGEVTTFVLVNKVSRSVNIPVIAAGGIADGRGMAAAFALGAEAVQMGTRFVASVESDVHPVYKEKIVKASIRDTVVTGAKLGHPARVLRTPFARKIQEMEFENPMQAEEMLVGSLRRAVVEGDLERGSFMVGQSAGLIDEIKPVKQIIEDILKEFKETVEKLRGYIEE
2MRN , Knot 36 67 0.75 34 54 61 
NHKVHHHHHHMSDDDDKGIHSSVKRWGNSPAVRIPATLMQALNLNIDDEVKIDLVDGKLIIEPVRKE
1VFH , Knot 154 386 0.79 40 185 348 
MNETPTRVYAEIDLDAVRANVRALRARAPRSALMAVVKSNAYGHGAVPCARAAQEAGAAWLGTATPEEALELRAAGIQGRIMCWLWTPGGPWREAIETDIDVSVSGMWALDEVRAAARAAGRTARIQLKADTGLGRNGCQPADWAELVGAAVAAQAEGTVQVTGVWSHFACADEPGHPSIRLQLDAFRDMLAYAEKEGVDPEVRHIANSPATLTLPETHFDLVRTGLAVYGVSPSPELGTPAQLGLRPAMTLRASLALVKTVPAGHGVSYGHHYVTESETHLALVPAGYADGIPRNASGRGPVLVAGKIRRAAGRIAMDQFVVDLGEDLAEAGDEAVILGDAERGEPTAEDWAQAAHTIAYEIVTRIGGRVPRVYLGGLEHHHHHH

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 \(c\).

\(|P_{f(5GVH_1)}(2) \setminus P_{f(2MRN_1)}(2)|=134\), \(|P_{f(2MRN_1)}(2) \setminus P_{f(5GVH_1)}(2)|=20\). 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 1:10100010011010011111111111010111110011111111011101001001100100000011110111101110011010100011110111101000100100010011111100011011001110111101100110110100111100100010111111111101011111111110110110011101000101100001101010001101101101101100111001001010011010011110100111010100101111001111001011001100110010001001010100
Pair \(Z_2\) Length of longest common subsequence
5GVH_1,2MRN_1 154 3
5GVH_1,1VFH_1 131 3
2MRN_1,1VFH_1 181 6

Newick tree

 
[
	2MRN_1:89.34,
	[
		5GVH_1:65.5,1VFH_1:65.5
	]:23.84
]

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 \((0.85)(0.8)(\frac{381 }{\log_{20} 381}-\frac{67}{\log_{20}67})=98.1\)
Status Protein1 Protein2 d d1/2
Query variables 5GVH_1 2MRN_1 123 73
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]