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Parikh vectors
1GYC_1 6USW_1 5SKB_1 Letter Amino acid
5 16 12 C Cysteine
17 14 17 Q Glutamine
39 28 16 G Glycine
28 30 21 I Isoleucine
39 20 36 L Leucine
2 8 12 M Methionine
42 18 14 P Proline
31 35 14 N Asparagine
39 21 16 V Valine
30 11 15 F Phenylalanine
17 8 14 R Arginine
8 25 18 K Lycine
39 29 22 T Threonine
7 7 7 W Tryptophan
29 14 15 D Aspartic acid
8 21 24 E Glutamic acid
20 9 13 H Histidine
31 19 24 S Serine
15 8 12 Y Tyrosine
53 14 21 A Alanine 

1GYC_1|Chain A|LACCASE 2|TRAMETES VERSICOLOR (5325)
>6USW_1|Chain A|HIV-1 LM/HS clade A/E CRF01 gp120 core|Human immunodeficiency virus 1 (11676)
>5SKB_1|Chains A, B, C, D|cAMP and cAMP-inhibited cGMP 3',5'-cyclic phosphodiesterase 10A|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 \(w=f(c)\)
1GYC , Knot 198 499 0.82 40 224 466 
AIGPAASLVVANAPVSPDGFLRDAIVVNGVFPSPLITGKKGDRFQLNVVDTLTNHTMLKSTSIHWHGFFQAGTNWADGPAFVNQCPIASGHSFLYDFHVPDQAGTFWYHSHLSTQYCDGLRGPFVVYDPKDPHASRYDVDNESTVITLTDWYHTAARLGPRFPLGADATLINGLGRSASTPTAALAVINVQHGKRYRFRLVSISCDPNYTFSIDGHNLTVIEVDGINSQPLLVDSIQIFAAQRYSFVLNANQTVGNYWIRANPNFGTVGFAGGINSAILRYQGAPVAEPTTTQTTSVIPLIETNLHPLARMPVPGSPTPGGVDKALNLAFNFNGTNFFINNASFTPPTVPVLLQILSGAQTAQDLLPAGSVYPLPAHSTIEITLPATALAPGAPHPFHLHGHAFAVVRSAGSTTYNYNDPIFRDVVSTGTPAAGDNVTIRFQTDNPGPWFLHCHIDFHLEAGFAIVFAEDVADVKAANPVPKAWSDLCPIYDGLSEANQ
6USW , Knot 148 355 0.81 40 203 330 
VPVWKDADTTLFCASDAKAYETEVHNVWATHACVPTDPNPQEIHLENVTENFNMWKNNMVEQMHEDIISLWDQSLQPCVKLTGGSVIKQACPKISFDPIPIHYCTPAGYVILKCNDKNFNGTGPCKNVSSVQCTHGIKPVVSTQLLLNGSLAEEEIIIRSENLTNNAKTIIVHLNKSVEINCTRPSNGGSGSGGDIRKAYCEINGTKWNKVLKQVTEKLKEHFNNKTIIFQPPSGGDLEITMHSFNCRGEFFYCNTTQLFNNTCIGNETMKGCNGTITLPCKIKQIINMWQGTGQAMYAPPIDGKINCVSNITGILLTRDGGANNTSNETFRPGGGDMRDNWRSELYKYKVVQIE
5SKB , Knot 152 343 0.86 40 221 334 
GSSICTSEEWQGLMQFTLPVRLCKEIELFHFDIGPFENMWPGIFVYMVHRSCGTSCFELEKLCRFIMSVKKNYRRVPYHNWKHAVTVAHCMYAILQNNHTLFTDLERKGLLIACLCHDLDHRGFSNSYLQKFDHPLAALYSTSTMEQHHFSQTVSILQLEGHNIFSTLSSSEYEQVLEIIRKAIIATDLALYFGNRKQLEEMYQTGSLNLNNQSHRDRVIGLMMTACDLCSVTKLWPVTKLTANDIYAEFWAEGDEMKKLGIQPIPMMDRDKKDEVPQGQLGFYNAVAIPCYTTLTQILPPTEPLLKACRDNLSQWEKVIRGEETATWISSPSVAQKAAASED

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 \(c\).

\(|P_{f(1GYC_1)}(2) \setminus P_{f(6USW_1)}(2)|=105\), \(|P_{f(6USW_1)}(2) \setminus P_{f(1GYC_1)}(2)|=84\). 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 1:1111110111101110101110011110111101110100100101011001000011000010101110110011011111000111010011001011001101100001000000110111110010010100001000001101001000110111011111010110111001001011111101001000010110100010001010100101101011000111100101111000011101000110011010101101111111001110001111101000000011111000101110111110101111001101110101001110010101101111101101100100111110101111000101011101111111011010101111100110000000011100110010111100101010000111111000101010111111110011010110111011001011001100100
Pair \(Z_2\) Length of longest common subsequence
1GYC_1,6USW_1 189 5
1GYC_1,5SKB_1 181 4
6USW_1,5SKB_1 188 4

Newick tree

 
[
	6USW_1:95.46,
	[
		1GYC_1:90.5,5SKB_1:90.5
	]:4.96
]

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 \((0.85)(0.8)(\frac{854 }{\log_{20} 854}-\frac{355}{\log_{20}355})=134.\)
Status Protein1 Protein2 d d1/2
Query variables 1GYC_1 6USW_1 170 144.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]