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Parikh vectors
2VNK_1 7OVJ_1 5VHO_1 Letter Amino acid
20 25 17 G Glycine
26 26 27 L Leucine
13 12 1 F Phenylalanine
18 15 6 P Proline
3 3 2 W Tryptophan
6 8 12 N Asparagine
13 20 11 D Aspartic acid
26 18 18 E Glutamic acid
15 18 11 I Isoleucine
22 20 13 V Valine
20 16 28 A Alanine
17 19 7 R Arginine
2 8 5 C Cysteine
6 11 7 Q Glutamine
5 13 9 H Histidine
12 30 16 K Lycine
9 12 5 M Methionine
12 16 12 S Serine
20 18 10 T Threonine
7 10 6 Y Tyrosine 

2VNK_1|Chains A, B, C, D|NADPH\:FERREDOXIN REDUCTASE|RHODOBACTER CAPSULATUS (1061)
>7OVJ_1|Chain A|Dual specificity mitogen-activated protein kinase kinase 7|Homo sapiens (9606)
>5VHO_1|Chain A[auth G]|26S proteasome non-ATPase regulatory subunit 10|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 \(w=f(c)\)
2VNK , Knot 119 272 0.81 40 174 260 
TTVNETTPIAPAKVLPDAQTVTSVRHWTDTLFSFRVTRPQTLRFRSGEFVMIGLLDDNGKPIMRAYSIASPAWDEELEFYSIKVPDGPLTSRLQHIKVGEQIILRPKPVGTLVIDALLPGKRLWFLATGTGIAPFASLMREPEAYEKFDEVIMMHACRTVAELEYGRQLVEALQEDPLIGELVEGKLKYYPTTTREEFHHMGRITDNLASGKVFEDLGIAPMNPETDRAMVCGSLAFNVDVMKVLESYGLREGANSEPREFVVEKAFVGEGI
7OVJ , Knot 137 318 0.82 40 204 304 
MGHHHHHHSAKQTGYLTIGGQRYQAEINDLENLGEMGSGTCGQVWKMRFRKTGHVIAVKQMRRSGNKEENKRILMDLDVVLKSHDCPYIVQCFGTFITNTDVFIAMELMGTCAEKLKKRMQGPIPERILGKMTVAIVKALYYLKEKHGVIHRDVKPSNILLDERGQIKLCDFGISGRLVDSKAKTRSAGCAAYMAPERIDPPDPTKPDYDIRADVWSLGISLVELATGQFPYKNCKTDFEVLTKVLQEEPPLLPGHMGFSGDFQSFVKDCLTKDHRKRPKYNKLLEHSFIKRYETLEVDVASWFKDVMAKTRSPRTSG
5VHO , Knot 97 223 0.78 40 142 213 
CVSNLMVCNLAYSGKLEELKESILADKSLATRTDQDSRTALHWACSAGHTEIVEFLLQLGVPVNDKDDAGWSPLHIAASAGRDEIVKALLGKGAQVNAVNQNGCTPLHYAASKNRHEIAVMLLEGGANPDAKDHYEATAMHRAAAKGNLKMIHILLYYKASTNIQDTEGNTPLHLACDEERVEEAKLLVSQGASIYIENKEEKTPLQVAKGGLGLILKRMVEG

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 \(c\).

\(|P_{f(2VNK_1)}(2) \setminus P_{f(7OVJ_1)}(2)|=77\), \(|P_{f(7OVJ_1)}(2) \setminus P_{f(2VNK_1)}(2)|=107\). 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 1:00100001111101110100100100100011010100100101001011111110001011101001101110001010010110111000100101100111010111011101111100111110101111110110010100010011110100011010010011011000111101101010001000000100110100011010110011111101000011101011101011011000110011000100111001111011
Pair \(Z_2\) Length of longest common subsequence
2VNK_1,7OVJ_1 184 4
2VNK_1,5VHO_1 172 3
7OVJ_1,5VHO_1 184 5

Newick tree

 
[
	7OVJ_1:93.91,
	[
		2VNK_1:86,5VHO_1:86
	]:7.91
]

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 \((0.85)(0.8)(\frac{590 }{\log_{20} 590}-\frac{272}{\log_{20}272})=89.5\)
Status Protein1 Protein2 d d1/2
Query variables 2VNK_1 7OVJ_1 117 106.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]