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Parikh vectors
2ZRW_1 1VSR_1 5VKH_1 Letter Amino acid
0 5 4 C Cysteine
32 10 16 G Glycine
39 8 6 I Isoleucine
26 16 14 L Leucine
14 3 15 P Proline
34 10 18 A Alanine
14 1 5 N Asparagine
14 8 7 D Aspartic acid
8 2 9 Y Tyrosine
27 8 21 V Valine
16 6 5 F Phenylalanine
10 5 9 Q Glutamine
32 13 8 E Glutamic acid
12 0 2 M Methionine
17 13 5 R Arginine
3 6 4 H Histidine
4 5 6 W Tryptophan
29 5 13 K Lycine
25 4 31 S Serine
12 8 21 T Threonine 

2ZRW_1|Chains A, B, C, D|Isopentenyl-diphosphate delta-isomerase|Sulfolobus shibatae (2286)
>1VSR_1|Chain A|PROTEIN (VSR ENDONUCLEASE)|Escherichia coli (83333)
>5VKH_1|Chain A|Antibody Heavy Chain|Mus musculus (10090)
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 \(w=f(c)\)
2ZRW , Knot 157 368 0.84 38 197 345 
MPDIVNRKVEHVEIAAFENVDGLSSSTFLNDVILVHQGFPGISFSEINTKTKFFRKEISVPVMVTGMTGGRNELGRINKIIAEVAEKFGIPMGVGSQRVAIEKAEARESFAIVRKVAPTIPIIANLGMPQLVKGYGLKEFQDAIQMIEADAIAVHLNPAQEVFQPEGEPEYQIYALEKLRDISKELSVPIIVKESGNGISMETAKLLYSYGIKNFDTSGQGGTNWIAIEMIRDIRRGNWKAESAKNFLDWGVPTAASIMEVRYSVPDSFLVGSGGIRSGLDAAKAIALGADIAGMALPVLKSAIEGKESLEQFFRKIIFELKAAMMLTGSKDVDALKKTSIVILGKLKEWAEYRGINLSIYEKVRKRE
1VSR , Knot 69 136 0.83 38 110 132 
DTAIEKRLASLLTGQGLAFRVQDASLPGRPDFVVDEYRCVIFTHGCFWHHHHCYLFKVPATRTEFWLEKIGKNVERDRRDISRLQELGWRVLIVWECALRGREKLTDEALTERLEEWICGEGASAQIDTQGIHLLA
5VKH , Knot 100 219 0.82 40 147 205 
QVQLQQPGAELVKPGASVKLSCKASGYTFTSDWIHWVKQRPGHGLEWIGEIIPSYGRANYNEKIQKKATLTADKSSSTAFMQLSSLTSEDSAVYYCARERGDGYFAVWGAGTTVTVSSAKTTPPSVYPLAPGSAAQTNSMVTLGCLVKGYFPEPVTVTWNSGSLSSGVHTFPAVLQSDLYTLSSSVTVPSSSWPSETVTCNVAHPASSTKVDKKIVPRD

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 \(c\).

\(|P_{f(2ZRW_1)}(2) \setminus P_{f(1VSR_1)}(2)|=134\), \(|P_{f(1VSR_1)}(2) \setminus P_{f(2ZRW_1)}(2)|=47\). 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 1:11011000100101111001011000011001111001111101001000001100010111110110110001101001110110011111111000111001010001111001110111110111101101011001001101101011110101100110101010001011001001000101111100010110100101100011001000101100111101100100101010010011011110110110100011001111011100110110111111011111111100110100010011001110101111101000101100001111101001100011010100010000
Pair \(Z_2\) Length of longest common subsequence
2ZRW_1,1VSR_1 181 4
2ZRW_1,5VKH_1 172 5
1VSR_1,5VKH_1 171 3

Newick tree

 
[
	2ZRW_1:89.18,
	[
		5VKH_1:85.5,1VSR_1:85.5
	]:3.68
]

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 \((0.85)(0.8)(\frac{504 }{\log_{20} 504}-\frac{136}{\log_{20}136})=108.\)
Status Protein1 Protein2 d d1/2
Query variables 2ZRW_1 1VSR_1 133 91.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]