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Parikh vectors
5RUS_1 2VDK_1 6KSC_1 Letter Amino acid
7 22 9 Y Tyrosine
13 54 22 G Glycine
2 2 2 M Methionine
5 22 10 F Phenylalanine
6 18 17 T Threonine
11 39 11 S Serine
0 9 5 W Tryptophan
3 6 2 C Cysteine
3 21 10 Q Glutamine
19 46 34 L Leucine
13 6 4 K Lycine
5 27 18 P Proline
2 27 34 R Arginine
9 27 23 D Aspartic acid
6 8 18 H Histidine
6 10 9 I Isoleucine
15 38 38 A Alanine
16 12 5 N Asparagine
6 21 31 E Glutamic acid
22 37 25 V Valine 

5RUS_1|Chains A, B|Non-structural protein 3|Severe acute respiratory syndrome coronavirus 2 (2697049)
>2VDK_1|Chain A|INTEGRIN ALPHA-IIB|HOMO SAPIENS (9606)
>6KSC_1|Chain A|DNA ligase A|Mycobacterium tuberculosis H37Rv (83332)
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 \(w=f(c)\)
5RUS , Knot 80 169 0.81 38 119 162 
SMVNSFSGYLKLTDNVYIKNADIVEEAKKVKPTVVVNAANVYLKHGGGVAGALNKATNNAMQVESDDYIATNGPLKVGGSCVLSGHNLAKHCLHVVGPNVNKGEDIQLLKSAYENFNQHEVLLAPLLSAGIFGADPIHSLRVCVDTVRTNVYLAVFDKNLYDKLVSSFL
2VDK , Knot 179 452 0.80 40 218 406 
LNLDPVQLTFYAGPNGSQFGFSLDFHKDSHGRVAIVVGAPRTLGPSQEETGGVFLCPWRAEGGQCPSLLFDLRDETRNVGSQTLQTFKARQGLGASVVSWSDVIVACAPWQHWNVLEKTEEAEKTPVGSCFLAQPESGRRAEYSPCRGNTLSRIYVENDFSWDKRYCEAGFSSVVTQAGELVLGAPGGYYFLGLLAQAPVADIFSSYRPGILLWHVSSQSLSFDSSNPEYFDGYWGYSVAVGEFDGDLNTTEYVVGAPTWSWTLGAVEILDSYYQRLHRLRGEQMASYFGHSVAVTDVNGDGRHDLLVGAPLYMESRADRKLAEVGRVYLFLQPRGPHALGAPSLLLTGTQLYGRFGSAIAPLGDLDRDGYNDIAVAAPYGGPSGRGQVLVFLGQSEGLRSRPSQVLDSPFPTGSAFGFSLRGAVDIDDNGYPDLIVGAYGANQVAVYRAQP
6KSC , Knot 136 327 0.80 40 173 301 
QTAPEVLRQWQALAEEVREHQFRYYVRDAPIISDAEFDELLRRLEALEEQHPELRTPDSPTQLVGGAGFATDFEPVDHLERMLSLDNAFTADELAAWAGRIHAEVGDAAHYLCELRIDGVALSLVYREGRLTRASTRGDGRTGEDVTLNARTIADVPERLTPGDDYPVPEVLEVRGEVFFRLDDFQALNASLVEEGKAPFANPRNSAAGSLRQKDPAVTARRRLRMICHGLGHVEGFRPATLHQAYLALRAWGLPVSEHTTLATDLAGVRERIDYWGEHRHEVDHEIDGVVVKVDEVALQRRLGSTSRAPRWAIAYKYPPEHHHHHH

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 \(c\).

\(|P_{f(5RUS_1)}(2) \setminus P_{f(2VDK_1)}(2)|=40\), \(|P_{f(2VDK_1)}(2) \setminus P_{f(5RUS_1)}(2)|=139\). 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 1:0110010101010001010010110010010101110110101001111111100100011010000011001110111001101001100010111101001001011001000100001111111011111101100101010010001011110001000110011
Pair \(Z_2\) Length of longest common subsequence
5RUS_1,2VDK_1 179 3
5RUS_1,6KSC_1 168 3
2VDK_1,6KSC_1 151 4

Newick tree

 
[
	5RUS_1:90.24,
	[
		6KSC_1:75.5,2VDK_1:75.5
	]:14.74
]

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 \((0.85)(0.8)(\frac{621 }{\log_{20} 621}-\frac{169}{\log_{20}169})=129.\)
Status Protein1 Protein2 d d1/2
Query variables 5RUS_1 2VDK_1 161 111.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]