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Parikh vectors
4IRD_1 6SSZ_1 3BPR_1 Letter Amino acid
0 16 9 N Asparagine
6 8 6 C Cysteine
0 18 18 K Lycine
0 11 9 F Phenylalanine
2 12 12 A Alanine
0 7 9 Q Glutamine
0 5 16 M Methionine
0 13 13 Y Tyrosine
0 12 19 V Valine
0 4 4 W Tryptophan
0 18 22 E Glutamic acid
0 4 13 H Histidine
0 19 16 I Isoleucine
0 9 14 P Proline
0 15 23 S Serine
5 6 12 T Threonine
0 7 17 R Arginine
0 16 20 D Aspartic acid
5 25 21 G Glycine
0 17 40 L Leucine 

4IRD_1|Chains A[auth B], C[auth G]|DNA (5'-D(*TP*CP*TP*TP*GP*GP*GP*TP*CP*CP*TP*AP*GP*GP*AP*CP*CP*C)-3')|
>6SSZ_1|Chains A, B|Cysteine proteinase falcipain 2a|Plasmodium falciparum (5833)
>3BPR_1|Chains A, B, C, D|Proto-oncogene tyrosine-protein kinase MER|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 \(w=f(c)\)
4IRD , Knot 9 18 0.48 8 11 16 
TCTTGGGTCCTAGGACCC
6SSZ , Knot 116 242 0.87 40 174 236 
GPQMNYEEVIKKYRGEENFDHAAYDWRLHSGVTPVKDQKNCGSAWAFSSIGSVESQYAIRKNKLITLSEQELVDCSFKNYGCNGGLINNAFEDMIELGGICPDGDYPYVSDAPNLCNIDRCTEKYGIKNYLSVPDNKLKEALRFLGPISISVAVSDDFAFYKEGIFDGECGDQLNHAVMLVGFGMKEIVNPLTKKGEKHYYYIIKNSWGQQWGERGFINIETDESGLMRKCGLGTDAFIPLI
3BPR , Knot 136 313 0.83 40 196 303 
MGSSHHHHHHSSGLVPRGSEELQNKLEDVVIDRNLLILGKILGEGEFGSVMEGNLKQEDGTSLKVAVKTMKLDNSSQREIEEFLSEAACMKDFSHPNVIRLLGVCIEMSSQGIPKPMVILPFMKYGDLHTYLLYSRLETGPKHIPLQTLLKFMVDIALGMEYLSNRNFLHRDLAARNCMLRDDMTVCVADFGLSKKIYSGDYYRQGRIAKMPVKWIAIESLADRVYTSKSDVWAFGVTMWEIATRGMTPYPGVQNHEMYDYLLHGHRLKQPEDCLDELYEIMYSCWRTDPLDRPTFSVLRLQLEKLLESLPDV

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 \(c\).

\(|P_{f(4IRD_1)}(2) \setminus P_{f(6SSZ_1)}(2)|=8\), \(|P_{f(6SSZ_1)}(2) \setminus P_{f(4IRD_1)}(2)|=171\). 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 1:000011100001111000
Pair \(Z_2\) Length of longest common subsequence
4IRD_1,6SSZ_1 179 2
4IRD_1,3BPR_1 201 2
6SSZ_1,3BPR_1 182 3

Newick tree

 
[
	3BPR_1:97.89,
	[
		4IRD_1:89.5,6SSZ_1:89.5
	]:8.39
]

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 \((0.85)(0.8)(\frac{260 }{\log_{20} 260}-\frac{18}{\log_{20}18})=82.5\)
Status Protein1 Protein2 d d1/2
Query variables 4IRD_1 6SSZ_1 114 60
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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