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Parikh vectors
5SGX_1 6QMA_1 6ZDI_1 Letter Amino acid
24 50 14 S Serine
7 17 2 W Tryptophan
16 56 25 V Valine
21 38 13 I Isoleucine
18 39 8 K Lycine
15 48 6 F Phenylalanine
14 33 17 P Proline
24 61 16 E Glutamic acid
16 44 31 G Glycine
12 22 6 Y Tyrosine
36 70 26 L Leucine
22 33 17 T Threonine
21 68 31 A Alanine
14 35 16 R Arginine
14 23 9 N Asparagine
17 40 10 Q Glutamine
15 21 10 D Aspartic acid
12 7 6 C Cysteine
13 16 2 H Histidine
12 14 5 M Methionine 

5SGX_1|Chains A, C|cAMP and cAMP-inhibited cGMP 3',5'-cyclic phosphodiesterase 10A|Homo sapiens (9606)
>6QMA_1|Chains A, B|Predicted protein|Nectria haematococca (660122)
>6ZDI_1|Chain A|17-beta-hydroxysteroid dehydrogenase 14|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 \(w=f(c)\)
5SGX , Knot 152 343 0.86 40 221 334 
GSSICTSEEWQGLMQFTLPVRLCKEIELFHFDIGPFENMWPGIFVYMVHRSCGTSCFELEKLCRFIMSVKKNYRRVPYHNWKHAVTVAHCMYAILQNNHTLFTDLERKGLLIACLCHDLDHRGFSNSYLQKFDHPLAALYSTSTMEQHHFSQTVSILQLEGHNIFSTLSSSEYEQVLEIIRKAIIATDLALYFGNRKQLEEMYQTGSLNLNNQSHRDRVIGLMMTACDLCSVTKLWPVTKLTANDIYAEFWAEGDEMKKLGIQPIPMMDRDKKDEVPQGQLGFYNAVAIPCYTTLTQILPPTEPLLKACRDNLSQWEKVIRGEETATWISSPSVAQKAAASED
6QMA , Knot 286 735 0.85 40 298 684 
MSNLKDFSQPGSGQESNFGVDFVIHYKVPAAERDEAEAGFVQLIRALTTVGLATEVRHGENESLLVFVKVASPDLFAKQVYRARLGDWLHGVRVSAPHNDIAQALQDEPVVEAERLRLIYLMITKPHNEGGAGVTPTNAKWKHVESIFPLHSHSFNKEWIKKWSSKYTLEQTDIDNIRDKFGESVAFYFAFLRSYFRFLVIPSAFGFGAWLLLGQFSYLYALLCGLWSVVFFEYWKKQEVDLAVQWGVRGVSSIQQSRPEFEWEHEAEDPITGEPVKVYPPMKRVKTQLLQIPFALACVVALGALIVTCNSLEVFINEVYSGPGKQYLGFLPTIFLVIGTPTISGVLMGAAEKLNAMENYATVDAHDAALIQKQFVLNFMTSYMALFFTAFVYIPFGHILHPFLNFWRATAQTLTFSEKELPTREFQINPARISNQMFYFTVTAQIVNFATEVVVPYIKQQAFQKAKQLKSGSKVQEDHEEEAEFLQRVREECTLEEYDVSGDYREMVMQFGYVAMFSVAWPLAACCFLVNNWVELRSDALKIAISSRRPIPWRTDSIGPWLTALSFLSWLGSITSSAIVYLCSNSKNGTQGEASPLKAWGLLLSILFAEHFYLVVQLAVRFVLSKLDSPGLQKERKERFQTKKRLLQENLGQDAAEEAAAPGIEHSEKITREALEEEARQASIRGHGTPEEMFWQRQRGMQETIEIGRRMIEQQLAAGKNGKKSAPAVPSEKAS
6ZDI , Knot 118 270 0.81 40 170 259 
MATGTRYAGKVVVVTGGGRGIGAGIVRAFVNSGARVVICDKDESGGRALEQELPGAVFILCDVTQEDDVKTLVSETIRRFGRLDCVVNNAGHHPPPQRPEETSAQGFRQLLELNLLGTYTLTKLALPYLRKSQGNVINISSLVGAIGQAQAVPYVATKGAVTAMTKALALDESPYGVRVNCISPGNIWTPLWEELAALMPDPRASIREGMLAQPLGRMGQPAEVGAAAVFLASEANFCTGIELLVTGGAELGYGCKASRSTPVDAPDIPS

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 \(c\).

\(|P_{f(5SGX_1)}(2) \setminus P_{f(6QMA_1)}(2)|=42\), \(|P_{f(6QMA_1)}(2) \setminus P_{f(5SGX_1)}(2)|=119\). 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
5SGX_1,6QMA_1 161 4
5SGX_1,6ZDI_1 185 4
6QMA_1,6ZDI_1 176 4

Newick tree

 
[
	6ZDI_1:93.30,
	[
		5SGX_1:80.5,6QMA_1:80.5
	]:12.80
]

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 \((0.85)(0.8)(\frac{1078 }{\log_{20} 1078}-\frac{343}{\log_{20}343})=194.\)
Status Protein1 Protein2 d d1/2
Query variables 5SGX_1 6QMA_1 251 183
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]