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Parikh vectors
5RHB_1 4QDD_1 5UWW_1 Letter Amino acid
17 31 17 A Alanine
12 6 3 C Cysteine
9 30 13 E Glutamic acid
7 10 13 H Histidine
29 20 19 L Leucine
16 21 9 S Serine
24 23 12 T Threonine
27 24 22 V Valine
11 21 10 R Arginine
21 16 10 N Asparagine
11 23 10 I Isoleucine
11 17 18 K Lycine
17 17 11 F Phenylalanine
13 20 13 P Proline
3 12 3 W Tryptophan
17 31 15 D Aspartic acid
14 16 9 Q Glutamine
26 29 17 G Glycine
10 7 5 M Methionine
11 16 8 Y Tyrosine 

5RHB_1|Chain A|3C-like proteinase|Severe acute respiratory syndrome coronavirus 2 (2697049)
>4QDD_1|Chain A|3-ketosteroid 9alpha-hydroxylase oxygenase|Rhodococcus rhodochrous (1829)
>5UWW_1|Chain A|GTP-binding nuclear protein Ran|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 \(w=f(c)\)
5RHB , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ
4QDD , Knot 169 390 0.86 40 237 375 
MSIDTARSGSDDDVEIREIQAAAAPTRFARGWHCLGLLRDFQDGKPHSIEAFGTKLVVFADSKGQLNVLDAYCRHMGGDLSRGEVKGDSIACPFHDWRWNGKGKCTDIPYARRVPPIAKTRAWTTLERNGQLYVWNDPQGNPPPEDVTIPEIAGYGTDEWTDWSWKSLRIKGSHCREIVDNVVDMAHFFYIHYSFPRYFKNVFEGHTATQYMHSTGREDVISGTNYDDPNAELRSEATYFGPSYMIDWLESDANGQTIETILINCHYPVSNNEFVLQYGAIVKKLPGVSDEIAAGMAEQFAEGVQLGFEQDVEIWKNKAPIDNPLLSEEDGPVYQLRRWYQQFYVDVEDITEDMTKRFEFEIDTTRAVASWQKEVAENLAKQAEGSTATP
5UWW , Knot 109 237 0.83 40 166 226 
METGSSHHHHHHSSGLPRGSHMAAQGEPQVQFKLVLVGDGGTGKTTFVKRHLTGEFEKKYVATLGVEVHPLVFHTNRGPIKFNVWDTAGQEKFGGLRDGYYIQAQCAIIMFDVTSRVTYKNVPNWHRDLVRVCENIPIVLCGNKVDIKDRKVKAKSIVFHRKKNLQYYDISAKSNYNFEKPFLWLARKLIGDPNLEFVAMPALAPPEVVMDPALAAQYEHDLEVAQTTALPDEDDDL

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 \(c\).

\(|P_{f(5RHB_1)}(2) \setminus P_{f(4QDD_1)}(2)|=74\), \(|P_{f(4QDD_1)}(2) \setminus P_{f(5RHB_1)}(2)|=115\). 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 1:011001111010101011010010001011110011001001100000110100001110000001110110101011100100011010100101001000110101100101110001010110001101010101011010010111010000101001001011011011001010101111000010111000010101111101111010011100100010010111100000110000101111101001111101010100110011010011101110001011011000011010
Pair \(Z_2\) Length of longest common subsequence
5RHB_1,4QDD_1 189 4
5RHB_1,5UWW_1 172 3
4QDD_1,5UWW_1 181 3

Newick tree

 
[
	4QDD_1:94.59,
	[
		5RHB_1:86,5UWW_1:86
	]:8.59
]

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 \((0.85)(0.8)(\frac{696 }{\log_{20} 696}-\frac{306}{\log_{20}306})=107.\)
Status Protein1 Protein2 d d1/2
Query variables 5RHB_1 4QDD_1 143 124.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]