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Parikh vectors
6BOH_1 6BPO_1 7ZAF_1 Letter Amino acid
13 62 7 A Alanine
10 32 4 R Arginine
5 54 5 N Asparagine
9 74 9 G Glycine
0 15 4 H Histidine
3 10 6 M Methionine
4 55 3 S Serine
9 37 5 I Isoleucine
3 22 6 F Phenylalanine
5 81 2 T Threonine
18 44 8 V Valine
6 52 5 D Aspartic acid
3 29 3 Q Glutamine
6 37 12 L Leucine
0 9 0 W Tryptophan
1 2 1 C Cysteine
8 23 10 E Glutamic acid
11 27 15 K Lycine
6 29 7 P Proline
2 33 4 Y Tyrosine 

6BOH_12|Chains L, QB|50S ribosomal protein L14|Thermus thermophilus (strain HB27 / ATCC BAA-163 / DSM 7039) (262724)
>6BPO_1|Chains A, B, C, D|Catecholate siderophore receptor Fiu|Escherichia coli (strain K12) (83333)
>7ZAF_1|Chain A[auth AAA]|KH domain-containing, RNA-binding, signal transduction-associated protein 1|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 \(w=f(c)\)
6BOH , Knot 62 122 0.81 36 89 117 
MIQPQTYLEVADNTGARKIMCIRVLKGSNAKYATVGDVIVASVKEAIPRGAVKEGDVVKAVVVRTKKEVKRPDGSAIRFDDNAAVIINNQLEPRGTRVFGPVARELREKGFMKIVSLAPEVL
6BPO , Knot 274 727 0.82 40 264 645 
AEGQTNADDTLVVEASTPSLYAPQQSADPKFSRPVADTTRTMTVISEQVIKDQGATNLTDALKNVPGVGAFFAGENGNSTTGDAIYMRGADTSNSIYIDGIRDIGSVSRDTFNTEQVEVIKGPSGTDYGRSAPTGSINMISKQPRNDSGIDASASIGSAWFRRGTLDVNQVIGDTTAVRLNVMGEKTHDAGRDKVKNERYGVAPSVAFGLGTANRLYLNYLHVTQHNTPDGGIPTIGLPGYSAPSAGTAALNHSGKVDTHNFYGTDSDYDDSTTDTATMRFEHDINDNTTIRNTTRWSRVKQDYLMTAIMGGASNITQPTSDVNSWTWSRTANTKDVSNKILTNQTNLTSTFYTGSIGHDVSTGVEFTRETQTNYGVNPVTLPAVNIYHPDSSIHPGGLTRNGANANGQTDTFAIYAFDTLQITRDFELNGGIRLDNYHTEYDSATACGGSGRGAITCPTGVAKGSPVTTVDTAKSGNLMNWKAGALYHLTENGNVYINYAVSQQPPGGNNFALAQSGSGNSANRTDFKPQKANTSEIGTKWQVLDKRLLLTAALFRTDIENEVEQNDDGTYSQYGKKRVEGYEISVAGNITPAWQVIGGYTQQKATIKNGKDVAQDGSSSLPYTPEHAFTLWSQYQATDDISVGAGARYIGSMHKGSDGAVGTPAFTEGYWVADAKLGYRVNRNLDFQLNVYNLFDTDYVASINKSGYRYHPGEPRTFLLTANMHF
7ZAF , Knot 61 116 0.83 38 95 114 
NYLDLFSHKNMKLKERVLIPVKQYPKFNFVGKILGPQGNTIKRLQEETGAKISVLGKGSMRDKAKEEELRKGGDPKYAHLNMDLHVFIEVFGPPCEAYALMAHAMEEVKKFLVPDM

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 \(c\).

\(|P_{f(6BOH_1)}(2) \setminus P_{f(6BPO_1)}(2)|=14\), \(|P_{f(6BPO_1)}(2) \setminus P_{f(6BOH_1)}(2)|=189\). 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 1:11010001011000110011010110100100101101111010011101110010110111100000100101011010001111100010101001111110010001110110111011
Pair \(Z_2\) Length of longest common subsequence
6BOH_1,6BPO_1 203 3
6BOH_1,7ZAF_1 116 4
6BPO_1,7ZAF_1 215 3

Newick tree

 
[
	6BPO_1:11.97,
	[
		6BOH_1:58,7ZAF_1:58
	]:57.97
]

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 \((0.85)(0.8)(\frac{849 }{\log_{20} 849}-\frac{122}{\log_{20}122})=204.\)
Status Protein1 Protein2 d d1/2
Query variables 6BOH_1 6BPO_1 252 146.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]