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Parikh vectors
5SID_1 8JRC_1 3VAF_1 Letter Amino acid
21 16 9 I Isoleucine
18 3 9 K Lycine
12 2 5 Y Tyrosine
14 20 7 R Arginine
15 16 11 D Aspartic acid
17 6 12 Q Glutamine
16 11 18 G Glycine
36 20 20 L Leucine
7 1 0 W Tryptophan
21 21 15 A Alanine
14 8 10 N Asparagine
14 11 9 P Proline
24 14 7 S Serine
22 11 7 T Threonine
16 22 9 V Valine
13 5 2 H Histidine
15 6 12 F Phenylalanine
12 8 5 M Methionine
12 0 1 C Cysteine
24 8 6 E Glutamic acid 

5SID_1|Chains A, B, C, D|cAMP and cAMP-inhibited cGMP 3',5'-cyclic phosphodiesterase 10A|Homo sapiens (9606)
>8JRC_1|Chains A, B|Nif-specific regulatory protein|Rhodopseudomonas palustris (1076)
>3VAF_1|Chains A, B|Splicing factor U2AF 65 kDa subunit|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 \(w=f(c)\)
5SID , Knot 152 343 0.86 40 221 334 
GSSICTSEEWQGLMQFTLPVRLCKEIELFHFDIGPFENMWPGIFVYMVHRSCGTSCFELEKLCRFIMSVKKNYRRVPYHNWKHAVTVAHCMYAILQNNHTLFTDLERKGLLIACLCHDLDHRGFSNSYLQKFDHPLAALYSTSTMEQHHFSQTVSILQLEGHNIFSTLSSSEYEQVLEIIRKAIIATDLALYFGNRKQLEEMYQTGSLNLNNQSHRDRVIGLMMTACDLCSVTKLWPVTKLTANDIYAEFWAEGDEMKKLGIQPIPMMDRDKKDEVPQGQLGFYNAVAIPCYTTLTQILPPTEPLLKACRDNLSQWEKVIRGEETATWISSPSVAQKAAASED
8JRC , Knot 96 209 0.81 38 134 202 
AMAQREIRLVDNEYPSPSMTHPPIPLSDIALTGIFEISKILTSPARLEITLANVVNLLQSFLQMRNGVVSLLADDGVPDITVGVGWNEGSDNRYRARLPQKAIDQIVATAVPLVADNVSAHPMFTAADAMALGATDEIRVSFIGVPIRIDSRVVGTLSIDRVRDGRSHFRMDADVRFLTMVANLIGQTVKLHRVVARDRERLMAESHRL
3VAF , Knot 85 174 0.84 38 135 169 
GPLGSARRLYVGNIPFGITEEAMMDFFNAQMRLGGLTQAPGNPVLAVQINQDKNFAFLEFRSVDETTQAMAFDGIIFQGQSLKIRRPHDYQPLPGAHKLFIGGLPNYLNDDQVKELLTSFGPLKAFNLVKDSATGLSKGYAFCEYVDINVTDQAIAGLNGMQLGDKKLLVQRAS

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 \(c\).

\(|P_{f(5SID_1)}(2) \setminus P_{f(8JRC_1)}(2)|=138\), \(|P_{f(8JRC_1)}(2) \setminus P_{f(5SID_1)}(2)|=51\). 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 1:1001000001011101011101000101101011110011111110110000100010100100111010000001100010011011001011100000110010001111101000100011000010010011111000001000010001011010100110010000000110110011110011101100001001000101010000000011111101001001001111001010010101110100100111011111000000011010111001111100001001111001110100001001001101000101100101100111000
Pair \(Z_2\) Length of longest common subsequence
5SID_1,8JRC_1 189 3
5SID_1,3VAF_1 186 3
8JRC_1,3VAF_1 145 3

Newick tree

 
[
	5SID_1:99.83,
	[
		3VAF_1:72.5,8JRC_1:72.5
	]:27.33
]

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 \((0.85)(0.8)(\frac{552 }{\log_{20} 552}-\frac{209}{\log_{20}209})=98.4\)
Status Protein1 Protein2 d d1/2
Query variables 5SID_1 8JRC_1 127 100
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]