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Parikh vectors
2MAF_1 1THJ_1 2HXC_1 Letter Amino acid
19 16 15 G Glycine
8 5 4 F Phenylalanine
18 11 12 T Threonine
14 14 9 D Aspartic acid
16 16 4 I Isoleucine
8 6 7 L Leucine
3 1 3 W Tryptophan
17 8 4 Y Tyrosine
21 5 4 R Arginine
10 15 7 P Proline
17 28 5 V Valine
17 19 7 A Alanine
0 1 14 C Cysteine
5 6 2 Q Glutamine
10 20 5 E Glutamic acid
8 5 7 H Histidine
13 5 3 K Lycine
0 6 1 M Methionine
15 14 12 S Serine
19 13 10 N Asparagine 

2MAF_1|Chain A|Opacity protein opA60|Neisseria gonorrhoeae (485)
>1THJ_1|Chains A, B, C|CARBONIC ANHYDRASE|Methanosarcina thermophila (2210)
>2HXC_1|Chains A[auth D], B[auth H]|Aromatic amine dehydrogenase|Alcaligenes faecalis (511)
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 \(w=f(c)\)
2MAF , Knot 109 238 0.83 36 160 229 
ASEDGGRGPYVQADLAYAYEHITHDYPEPTAPNKNKISTVSDYFRNIRTRSVHPRVSVGYDFGGWRIAADYARYRKWNNNKYSVNIENVRIRKENGIRIDRKTENQENGTFHAVSSLGLSAIYDFQINDKFKPYIGARVAYGHVRHSIDSTKKTIEVTTVPSNAPNGAVTTYNTDPKTQNDYQSNSIRRVGLGVIAGVGFDITPKLTLDAGYRYHNWGRLENTRFKTHEASLGVRYRF
1THJ , Knot 98 214 0.82 40 136 207 
MQEITVDEFSNIRENPVTPWNPEPSAPVIDPTAYIDPEASVIGEVTIGANVMVSPMASIRSDEGMPIFVGDRSNVQDGVVLHALETINEEGEPIEDNIVEVDGKEYAVYIGNNVSLAHQSQVHGPAAVGDDTFIGMQAFVFKSKVGNNCVLEPRSAAIGVTIPDGRYIPAGMVVTSQAEADKLPEVTDDYAYSHTNEAVVYVNVHLAEGYKETS
2HXC , Knot 62 135 0.75 40 103 129 
AGGGGSSSGADHISLNPDLANEDEVNSCDYWRHCAVDGFLCSCCGGTTTTCPPGSTPSPISWIGTCHNPHDGKDYLISYHDCCGKTACGRCQCNTQTRERPGYEFFLHNDVNWCMANENSTFHCTTSVLVGLAKN

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 \(c\).

\(|P_{f(2MAF_1)}(2) \setminus P_{f(1THJ_1)}(2)|=89\), \(|P_{f(1THJ_1)}(2) \setminus P_{f(2MAF_1)}(2)|=65\). 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 1:1000110110101011010001000010101100001001000100100001010101100111101110010000100000010100101000011010000000001010110011101100101000101011101101010001000000101001100110111000000100000000001001111111111101010101011000001101000010000101110001
Pair \(Z_2\) Length of longest common subsequence
2MAF_1,1THJ_1 154 3
2MAF_1,2HXC_1 171 3
1THJ_1,2HXC_1 167 3

Newick tree

 
[
	2HXC_1:86.86,
	[
		2MAF_1:77,1THJ_1:77
	]:9.86
]

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 \((0.85)(0.8)(\frac{452 }{\log_{20} 452}-\frac{214}{\log_{20}214})=69.3\)
Status Protein1 Protein2 d d1/2
Query variables 2MAF_1 1THJ_1 91 84.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]