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Parikh vectors
6VQM_1 1LSB_1 2OTY_1 Letter Amino acid
10 1 1 H Histidine
17 2 3 P Proline
6 6 3 W Tryptophan
19 6 9 V Valine
21 11 13 R Arginine
5 8 0 C Cysteine
11 3 5 Q Glutamine
20 7 10 D Aspartic acid
11 2 5 M Methionine
14 10 6 S Serine
13 7 12 T Threonine
19 12 17 A Alanine
6 14 11 N Asparagine
37 8 14 L Leucine
14 6 13 K Lycine
11 3 5 F Phenylalanine
6 3 6 Y Tyrosine
18 2 8 E Glutamic acid
22 12 11 G Glycine
9 6 10 I Isoleucine

6VQM_1|Chains A, B|Activated CDC42 kinase 1|Homo sapiens (9606)
>1LSB_1|Chain A|HEN EGG WHITE LYSOZYME|Gallus gallus (9031)
>2OTY_1|Chain A[auth X]|Lysozyme|Enterobacteria phage T4 (10665)
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 \(w=f(c)\)
6VQM , Knot 127 289 0.83 40 190 279
GSAGEGPLQSLTCLIGEKDLRLLEKLGDGSFGVVRRGEWDAPSGKTVSVAVKCLKPDVLSQPEAMDDFIREVNAMHSLDHRNLIRLYGVVLTPPMKMVTELAPLGSLLDRLRKHQGHFLLGTLSRYAVQVAEGMGYLESKRFIHRDLAARNLLLATRDLVKIGDFGLMRALPQNDDHYVMQEHRKVPFAWCAPESLKTRTFSHASDTWMFGVTLWEMFTYGQEPWIGLNGSQILHKIDKEGERLPRPEDCPQDIYNVMVQCWAHKPEDRPTFVALRDFLLEAQPTDMRA
1LSB , Knot 66 129 0.82 40 104 127
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
2OTY , Knot 76 162 0.79 38 115 155
MNIFEMLRIDEGLRLKIYKDTEGYYTIGIGHLLTKSPSLNAAKSELDKAIGRNTNGVITKDEAEKLFNQDVDAAVRGILRNAKLKPVYDSLDAVRRAAAINMVFQMGETGVAGFTNSLRMLQQKRWDEAAVNLAKSRWYNQTPNRAKRVITTFRTGTWDAYK

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 \(c\).

\(|P_{f(6VQM_1)}(2) \setminus P_{f(1LSB_1)}(2)|=137\), \(|P_{f(1LSB_1)}(2) \setminus P_{f(6VQM_1)}(2)|=51\). 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 1:1011011100100111000101100110101111001010110100101110010101100101100110010110010000110101111011101100111110110010000101111010001101101110100001100011100111100011011011110111000000110000011111011001000010010001111101101100100111110100110010001001101000100100111001100100010111100111010100101
Pair \(Z_2\) Length of longest common subsequence
6VQM_1,1LSB_1 188 3
6VQM_1,2OTY_1 183 3
1LSB_1,2OTY_1 139 3

Newick tree

 
[
	6VQM_1:99.30,
	[
		2OTY_1:69.5,1LSB_1:69.5
	]:29.80
]

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 \((0.85)(0.8)(\frac{418 }{\log_{20} 418}-\frac{129}{\log_{20}129})=87.0\)
Status Protein1 Protein2 d d1/2
Query variables 6VQM_1 1LSB_1 114 81.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]