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Parikh vectors
3VXT_1 2MCF_1 1ZSO_1 Letter Amino acid
9 4 9 Y Tyrosine
11 5 4 A Alanine
4 6 5 C Cysteine
8 17 16 E Glutamic acid
0 4 7 H Histidine
11 4 10 F Phenylalanine
3 0 4 W Tryptophan
12 11 9 L Leucine
10 13 9 V Valine
9 4 9 T Threonine
6 8 8 R Arginine
13 7 19 N Asparagine
17 10 11 D Aspartic acid
13 0 2 Q Glutamine
9 13 12 I Isoleucine
12 13 9 K Lycine
11 10 4 G Glycine
4 4 4 M Methionine
10 8 3 P Proline
33 7 10 S Serine 

3VXT_1|Chains A, C|T36-5 TCR alpha chain|Homo sapiens (9606)
>2MCF_1|Chain A|TGAM_1934|Thermococcus gammatolerans (593117)
>1ZSO_1|Chains A, B|hypothetical protein|Plasmodium falciparum (5833)
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 \(w=f(c)\)
3VXT , Knot 96 205 0.83 38 140 198 
MQKEVEQNSGPLSVPEGAIASLNCTYSDRGSQSFFWYRQYSGKSPELIMSIYSNGDKEDGRFTAQLNKASQYVSLLIRDSQPSDSATYLWGTYNQGGKLIFGQGTELSVKPNIQNPDPAVYQLRDSKSSDKSVCLFTDFDSQTNVSQSKDSDVYITDKCVLDMRSMDFKSNSAVAWSNKSDFACANAFNNSIIPEDTFFPSPESS
2MCF , Knot 74 148 0.83 36 113 143 
MKYDVVIIPESFHRFDKHNMEHICPPMVIGDRSYDIAMEIVNGVDRVIKASFNASVEELEGEDCDVLYRKYTLEKEGKKGIVHVKLRKITENCPPVDGNRCSVLEFERDIECIVKAIEECLAKGELNSKLEGKPIPNPLLGLDSTRTG
1ZSO , Knot 82 164 0.85 40 128 158 
MAHHHHHHMKNTVVRIKAELENVKRLFCDDEYLWIFNIRDSTSSLTRDNIQFRKTDILEIPNSRGTANFMIKWTEYPKYSTINFVNTKNSCSYEEVNNNEWRDFASFECRGIELIDFFPSNNFIVEDTKGKLYYDVNLSDQNWCDYNEEHEMCVGIYNLEYEVN

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 \(c\).

\(|P_{f(3VXT_1)}(2) \setminus P_{f(2MCF_1)}(2)|=93\), \(|P_{f(2MCF_1)}(2) \setminus P_{f(3VXT_1)}(2)|=66\). 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 1:1000100001110110111101000000010001110000010010111010001000010101010010001011100001000100111000011011110100101010100101110010000000001011001000001000000010100001101001010000111100000110101100011100011101000
Pair \(Z_2\) Length of longest common subsequence
3VXT_1,2MCF_1 159 4
3VXT_1,1ZSO_1 170 4
2MCF_1,1ZSO_1 157 3

Newick tree

 
[
	3VXT_1:83.52,
	[
		2MCF_1:78.5,1ZSO_1:78.5
	]:5.02
]

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 \((0.85)(0.8)(\frac{353 }{\log_{20} 353}-\frac{148}{\log_{20}148})=62.2\)
Status Protein1 Protein2 d d1/2
Query variables 3VXT_1 2MCF_1 79 68
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]