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Parikh vectors
2JNR_1 4CFP_1 9JWS_1 Letter Amino acid
1 9 8 E Glutamic acid
4 12 7 P Proline
0 26 21 T Threonine
0 23 11 R Arginine
0 14 7 Q Glutamine
1 23 30 L Leucine
1 18 4 K Lycine
4 11 5 F Phenylalanine
1 26 13 S Serine
0 19 11 Y Tyrosine
2 25 28 A Alanine
2 24 31 I Isoleucine
1 22 15 V Valine
1 21 3 N Asparagine
2 0 0 C Cysteine
0 25 25 G Glycine
0 7 8 H Histidine
0 9 9 M Methionine
0 3 8 W Tryptophan
0 24 8 D Aspartic acid 

2JNR_1|Chain A|VIR165|synthetic construct (32630)
>4CFP_1|Chains A, B|MEMBRANE-BOUND LYTIC MUREIN TRANSGLYCOSYLASE C|ESCHERICHIA COLI (562)
>9JWS_1|Chains A, B|Bacteriorhodopsin-II-like protein|Haloquadratum walsbyi DSM 16790 (362976)
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 \(w=f(c)\)
2JNR , Knot 15 20 0.75 22 17 18 
LEAIPCSIPPCFAFNKPFVF
4CFP , Knot 146 341 0.83 38 199 326 
MTKKGDTYNEAWVKDTNGFDILMGQFAHNIENIWGFKEVVIAGPKDYVKYTDQYQTRSHINFDDGTITIETIAGTEPAAHLRRAIIKTLLMGDDPSSVDLYSDVDDITISKEPFLYGQVVDNTGQPIRWEGRASNFADYLLKNRLKSRSNGLRIIYSVTINMVPNHLDKRAHKYLGMVRQASRKYGVDESLILAIMQTQSSFNPYAVSRSDALGLMQVVQHTAGKDVFRSQGKSGTPSRSFLFDPASNIDTGTAYLAMLNNVYLGGIDNPTSRRYAVITAYNGGAGSVLRVFSNDKIQAANIINTMTPGDVYQTLTTRHPSAESRRYLYKVNTAQKSYRRR
9JWS , Knot 108 252 0.79 38 147 236 
MATPGSEATWLWIGTIGMVLGTVYFAVRGRGSTDPEQQTYYIITTLIPAIAAAAYLAMATGLGVISMPIRGTEVIDIYWARYADWLLTTPLLIIDLALVAGARKQTLYKLIIIDAIMILGGLAGSMMQQGAVIRIVWWAVSTAAFIILLYYLLGELSERARSRSAETGIVFNRLRNITLGLWALYPIVWILGTGGGFGIIAVTTEIMLYVMLDIGTKIGFGAVLLESQDVLQAASHPSSTNDIKSHHHHHHH

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 \(c\).

\(|P_{f(2JNR_1)}(2) \setminus P_{f(4CFP_1)}(2)|=11\), \(|P_{f(4CFP_1)}(2) \setminus P_{f(2JNR_1)}(2)|=193\). 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 1:10111001110111001111
Pair \(Z_2\) Length of longest common subsequence
2JNR_1,4CFP_1 204 2
2JNR_1,9JWS_1 148 2
4CFP_1,9JWS_1 150 5

Newick tree

 
[
	4CFP_1:94.13,
	[
		2JNR_1:74,9JWS_1:74
	]:20.13
]

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 \((0.85)(0.8)(\frac{361 }{\log_{20} 361}-\frac{20}{\log_{20}20})=111.\)
Status Protein1 Protein2 d d1/2
Query variables 2JNR_1 4CFP_1 139 74.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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