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Parikh vectors
7TJA_1 1GCS_1 1TLH_1 Letter Amino acid
3 9 6 F Phenylalanine
3 5 6 T Threonine
0 4 0 W Tryptophan
2 8 0 P Proline
6 6 6 V Valine
6 13 4 D Aspartic acid
1 5 1 H Histidine
4 6 10 I Isoleucine
12 2 7 K Lycine
6 14 4 G Glycine
9 13 4 S Serine
8 2 3 A Alanine
0 6 9 N Asparagine
2 10 2 Q Glutamine
5 9 11 E Glutamic acid
0 15 3 Y Tyrosine
3 20 6 R Arginine
1 7 0 C Cysteine
3 13 6 L Leucine
2 7 2 M Methionine 

7TJA_1|Chains A, E, G[auth I], K[auth M], O[auth R]|Phycoerythrin alpha-subunit 1|Proteomonas sulcata (77928)
>1GCS_1|Chain A|GAMMA-B CRYSTALLIN|Bos taurus (9913)
>1TLH_1|Chain A|10 kDa anti-sigma factor|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 \(w=f(c)\)
7TJA , Knot 43 76 0.81 34 65 73 
GMDKSAKAPAITIFDHRGCSRAPKESSAKSGSQDDEMLVKVASTKVTVSEDVAAKKLQEFIGFKEKGLDGSVIRKK
1GCS , Knot 84 174 0.83 40 129 166 
GKITFYEDRGFQGHCYECSSDCPNLQPYFSRCNSIRVDSGCWMLYERPNYQGHQYFLRRGDYPDYQQWMGFNDSIRSCRLIPQHTGTFRMRIYERDDFRGQMSEITDDCPSLQDRFHLTEVHSLNVLEGSWVLYEMPSYRGRQYLLRPGEYRRYLDWGAMNAKVGSLRRVMDFY
1TLH , Knot 48 90 0.80 34 75 88 
MNKNIDTVREIITVASILIKFSREDIVENRANFIAFLNEIGVTHEGRKLNQNSFRKIVSELTQEDKKTLIDEFNEGFEGVYRYLEMYTNK

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 \(c\).

\(|P_{f(7TJA_1)}(2) \setminus P_{f(1GCS_1)}(2)|=44\), \(|P_{f(1GCS_1)}(2) \setminus P_{f(7TJA_1)}(2)|=108\). 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 1:1100010111101100010001100001001000001110110001010001110010011110001101011000
Pair \(Z_2\) Length of longest common subsequence
7TJA_1,1GCS_1 152 2
7TJA_1,1TLH_1 98 3
1GCS_1,1TLH_1 156 3

Newick tree

 
[
	1GCS_1:84.29,
	[
		7TJA_1:49,1TLH_1:49
	]:35.29
]

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 \((0.85)(0.8)(\frac{250 }{\log_{20} 250}-\frac{76}{\log_{20}76})=56.4\)
Status Protein1 Protein2 d d1/2
Query variables 7TJA_1 1GCS_1 72 52
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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