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Parikh vectors
1NPJ_1 5ARM_1 7BCL_1 Letter Amino acid
26 8 21 T Threonine
34 6 25 V Valine
8 11 29 R Arginine
9 5 15 Q Glutamine
22 1 20 P Proline
19 3 13 K Lycine
13 2 16 N Asparagine
1 18 14 C Cysteine
15 6 12 I Isoleucine
37 6 30 G Glycine
14 1 11 F Phenylalanine
4 0 13 W Tryptophan
13 0 13 Y Tyrosine
38 19 20 A Alanine
19 7 18 D Aspartic acid
21 13 22 E Glutamic acid
5 6 30 S Serine
14 7 17 H Histidine
21 8 40 L Leucine
10 6 4 M Methionine 

1NPJ_1|Chains A, B, C|Copper-containing nitrite reductase|Alcaligenes faecalis (511)
>5ARM_1|Chain A|CSP3|METHYLOSINUS TRICHOSPORIUM (595536)
>7BCL_1|Chain A|Palmitoleoyl-protein carboxylesterase NOTUM|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 \(w=f(c)\)
1NPJ , Knot 142 343 0.80 40 189 323 
QGAVRKATAAEIAALPRQKVELVDPPFVHAHSQVAEGGPKVVEFTMVIEEKKIVIDDAGTEVHAMAFNGTVPGPLMVVHQDDYLELTLINPETNTLMHNIDFHAATGALGGGGLTEINPGEKTILRFKATKPGVFVYHCAPPGMVPWAVVSGMNGAIMVLPREGLHDGKGKALTYDKIYYVGEQDFYVPRDENGKYKKYEAPGDAYEDTVKVMRTLTPTHVVFNGAVGALTGDKAMTAAVGEKVLIVHSQANRDTRPHLIGGHGDYVWATGKFNTPPDVDQETWFIPGGAAGAAFYTFQQPGIYAYVNHNLIEAFELGAAAHFKVTGEWNDDLMTSVLAPSGT
5ARM , Knot 64 133 0.78 36 95 124 
MHVEAMISKHPQARGQTDRSLVQCVEMCFDCAQTCAACADACLGEDKVADLRHCIRLNLDCAEICVAAGSIASRAAGTEESILRTMLQTCAEMCRMCEEECRRHAGNHEHCRICADVCKECETACRSATGLTH
7BCL , Knot 162 383 0.83 40 219 364 
ETGSAQQLNEDLRLHLLLNTSVTCNDGSPAGYYLKESRGSRRWLLFLEGGWYCFNRENCDSRYDTMRRLMSSRDWPRTRTGTGILSSQPEENPYWWNANMVFIPYCSSDVWSGASSKSEKNEYAFMGALIIQEVVRELLGRGLSGAKVLLLAGSSAGGTGVLLNVDRVAEQLEKLGYPAIQVRGLADSGWFLDNKQYRHTDCVDTITCAPTEAIRRGIRYWNGVVPERCRRQFQEGEEWNCFFGYKVYPTLRSPVFVVQWLFDEAQLTVDNVHLTGQPVQEGLRLYIQNLGRELRHTLKDVPASFAPACLSHEIIIRSHWTDVQVKGTSLPRALHCWDRSLHDSHKASKTPLKGCPVHLVDSCPWPHCNPSCPTGTKHHHHHH

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 \(c\).

\(|P_{f(1NPJ_1)}(2) \setminus P_{f(5ARM_1)}(2)|=149\), \(|P_{f(5ARM_1)}(2) \setminus P_{f(1NPJ_1)}(2)|=55\). 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 1:0111001011011111000101101111010001101110110101110000111001100101111010111111110000010101101000011001010110111111110010110001101010011111000111111111110110111111100110010101100001001100010110000100000011101000010110010100111011111101001101111001111000100000101111010011101010011010000111111111111001001110101000110110111110101010100011001111010
Pair \(Z_2\) Length of longest common subsequence
1NPJ_1,5ARM_1 204 4
1NPJ_1,7BCL_1 188 3
5ARM_1,7BCL_1 196 4

Newick tree

 
[
	5ARM_1:10.94,
	[
		1NPJ_1:94,7BCL_1:94
	]:7.94
]

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 \((0.85)(0.8)(\frac{476 }{\log_{20} 476}-\frac{133}{\log_{20}133})=101.\)
Status Protein1 Protein2 d d1/2
Query variables 1NPJ_1 5ARM_1 130 90
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]