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Parikh vectors
5JVM_1 8PAZ_1 3TUE_1 Letter Amino acid
11 6 17 L Leucine
1 5 7 M Methionine
3 8 13 P Proline
1 1 5 C Cysteine
4 3 6 Q Glutamine
4 8 18 G Glycine
0 3 10 H Histidine
1 5 22 S Serine
0 0 3 W Tryptophan
4 12 17 V Valine
5 1 8 R Arginine
4 7 9 N Asparagine
13 10 15 E Glutamic acid
3 13 13 A Alanine
8 5 8 D Aspartic acid
2 5 6 T Threonine
2 4 6 Y Tyrosine
6 11 10 I Isoleucine
6 13 14 K Lycine
4 3 12 F Phenylalanine 

5JVM_1|Chains A, B|Chimera protein of Kinesin-like protein KIF3C and Microtubule-associated protein RP/EB family member 1|Mus musculus (10090)
>8PAZ_1|Chain A|PSEUDOAZURIN|Alcaligenes faecalis (511)
>3TUE_1|Chains A, B, C, D, E|Tryparedoxin peroxidase|Leishmania major (5664)
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 \(w=f(c)\)
5JVM , Knot 45 82 0.80 36 72 79 
LSNEDPKDTLLREFQEEIARLKAQLEKKGMLVEDLEKERDFYFGKLRNIELICQENEGENDPVLQRIVDILYATDEGFVIPD
8PAZ , Knot 61 123 0.79 38 97 117 
ENIEVHMLNKGAEGAMVFEPAYIKANPGDTVTFIPVDKGHNVESIKDMIPEGAEKFKSKINENYVLTVTQPGAYLVKCTPHYAMGMIALIAVGDSPANLDQIVSAKKPKIVQERLEKVIASAK
3TUE , Knot 103 219 0.84 40 163 209 
MGSSHHHHHHSSGLVPRGSHMSCGNAKINSPAPSFEEVALMPNGSFKKISLSSYKGKWVVLFFYPLDFTFVCPTEVIAFSDSVSRFNELNCEVLACSIDSEYAHLQWTLQDRKKGGLGTMAIPILADKTKNIARSYGVLEESQGVAYRGLFIIDPHGMLRQITVNDMPVGRSVEEVLRLLEAFQFVEKHGEVCPANWKKGDPGMKPEPNASVEGYFSKQ

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 \(c\).

\(|P_{f(5JVM_1)}(2) \setminus P_{f(8PAZ_1)}(2)|=43\), \(|P_{f(8PAZ_1)}(2) \setminus P_{f(5JVM_1)}(2)|=68\). 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 1:1000010001100100011010101000111100100000101101001011000001000111001101101000111110
Pair \(Z_2\) Length of longest common subsequence
5JVM_1,8PAZ_1 111 3
5JVM_1,3TUE_1 163 3
8PAZ_1,3TUE_1 156 3

Newick tree

 
[
	3TUE_1:86.35,
	[
		5JVM_1:55.5,8PAZ_1:55.5
	]:30.85
]

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 \((0.85)(0.8)(\frac{205 }{\log_{20} 205}-\frac{82}{\log_{20}82})=40.5\)
Status Protein1 Protein2 d d1/2
Query variables 5JVM_1 8PAZ_1 50 41
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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