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Parikh vectors
3ZRZ_1 1DSJ_1 1SWU_1 Letter Amino acid
2 1 2 H Histidine
2 3 7 L Leucine
2 0 0 M Methionine
1 2 3 F Phenylalanine
3 0 11 S Serine
5 1 5 Y Tyrosine
3 0 7 N Asparagine
8 0 0 C Cysteine
13 3 16 G Glycine
3 0 2 P Proline
9 1 19 T Threonine
3 1 6 W Tryptophan
2 1 7 V Valine
3 2 19 A Alanine
7 2 4 R Arginine
5 1 4 D Aspartic acid
1 2 3 Q Glutamine
7 1 5 E Glutamic acid
6 5 3 I Isoleucine
5 0 4 K Lycine 

3ZRZ_1|Chains A, B|FIBRONECTIN|HOMO SAPIENS (9606)
>1DSJ_1|Chain A|VPR PROTEIN|Human immunodeficiency virus 1 (11676)
>1SWU_1|Chains A, B, C, D|STREPTAVIDIN|Streptomyces avidinii (1895)
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 \(w=f(c)\)
3ZRZ , Knot 49 90 0.81 40 75 87 
AEETCFDKYTGNTYRVGDTYERPKDSMIWDCTCIGAGRGRISCTIANRCHEGGQSYKIGDTWRRPHETGGYMLECVCLGNGKGEWTCKPI
1DSJ , Knot 19 28 0.75 30 26 26 
XYGDTWAGVEAIIRILQQLLFIHFRIGX
1SWU , Knot 59 127 0.75 36 87 120 
AEAGITGTWYNQLGSTFIVTAGADGALTGTFESAVGNAESRYVLTGRYDSAPATDGSGTALGWTVAWKNNYRNAHSATTWSGQYVGGAEARINTQWLLTSGTTEANAWKSTLVGHDTFTKVKPSAAS

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 \(c\).

\(|P_{f(3ZRZ_1)}(2) \setminus P_{f(1DSJ_1)}(2)|=69\), \(|P_{f(1DSJ_1)}(2) \setminus P_{f(3ZRZ_1)}(2)|=20\). 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 1:100001000010000110000010001110000111101010001100000110000110010010001101100101101010100011
Pair \(Z_2\) Length of longest common subsequence
3ZRZ_1,1DSJ_1 89 4
3ZRZ_1,1SWU_1 124 2
1DSJ_1,1SWU_1 97 2

Newick tree

 
[
	1SWU_1:58.91,
	[
		3ZRZ_1:44.5,1DSJ_1:44.5
	]:14.41
]

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 \((0.85)(0.8)(\frac{118 }{\log_{20} 118}-\frac{28}{\log_{20}28})=33.2\)
Status Protein1 Protein2 d d1/2
Query variables 3ZRZ_1 1DSJ_1 42 27.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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