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Parikh vectors
2QUY_1 3LFT_1 4XEH_1 Letter Amino acid
27 19 27 L Leucine
12 6 14 F Phenylalanine
16 15 11 P Proline
19 30 24 A Alanine
11 0 17 R Arginine
21 14 7 N Asparagine
16 14 30 E Glutamic acid
23 25 19 S Serine
24 34 28 V Valine
20 17 18 D Aspartic acid
25 23 31 G Glycine
4 3 9 H Histidine
24 22 30 I Isoleucine
18 5 11 Y Tyrosine
3 0 1 C Cysteine
19 25 28 K Lycine
11 6 13 M Methionine
3 1 4 W Tryptophan
9 15 8 Q Glutamine
30 21 13 T Threonine 

2QUY_1|Chains A, B, C, D, E, F, G, H|Penicillin acylase|Bacillus sphaericus
>3LFT_1|Chains A, B|uncharacterized protein|Streptococcus pneumoniae (170187)
>4XEH_1|Chain A|Ketol-acid reductoisomerase|Ignisphaera aggregans (583356)
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 \(w=f(c)\)
2QUY , Knot 143 335 0.82 40 201 325 
CSSLSIRTTDDKSLFARTMDFTMEPDSKVIIVPRNYGIRLLEKENVVINNSYAFVGMGSTDITSPVLYDGVNEKGLMGAMLYYATFATYADEPKKGTRGINPVYVISQVLGNCVTVDDVIEKLTSYTLLNEANIILGFAPPLHYTFTDASGESIVIEPDKTGITIHRKTIGVMTASPGYEWHQTNLRAYIGVTPNPPQDIMMGDLDLTPFGQGAGGLGLPGDFTPSARFLRVAYWKKYTEKAKNETEGVTNLFHILSSVNIPKGVVLTNEGKTDYTIYTSAMCAQSKNYYFKLYDNSRISAVSLMAENLNSQDLITFEWDRKQDIKQLNQVNVMS
3LFT , Knot 128 295 0.82 36 166 280 
SNAKIGVLQFVSHPSLDLIYKGIQDGLAEEGYKDDQVKIDFMNSEGDQSKVATMSKQLVANGNDLVVGIATPAAQGLASATKDLPVIMAAITDPIGANLVKDLKKPGGNVTGVSDHNPAQQQVELIKALTPNVKTIGALYSSSEDNSKTQVEEFKAYAEKAGLTVETFAVPSTNEIASTVTVMTSKVDAIWVPIDNTIASGFPTVVSSNQSSKKPIYPSATAMVEVGGLASVVIDQHDLGVATGKMIVQVLKGAKPADTPVNVFSTGKSVINKKIAQELGITIPESVLKEAGQVI
4XEH , Knot 147 343 0.83 40 201 329 
MAKIYKDEDISLEPIKNKTIAILGYGSQGRAWALNLRDSGLNVVVGLERQGDSWRRAIDDGFKPMYTKDAVAIADIIVFLVPDMVQKSLWLNSVKDFMKKGADLVFAHGFNIHFKIIEPPKDSDVYMIAPKSPGPIVRRSYEMGGGVPALVAVYQNVSGEALQKALAIAKGIGCARAGVIESTFKEETETDLFGEQVILVGGIMELIKASFETLVEEGYQPEVAYFETVNELKLIVDLIYEKGLTGMLRAVSDTAKYGGITVGKFIIDKSVRDKMKIVLERIRSGEFAREWIKEYERGMPTVFKELSELEGSTIETVGRKLREMMFRGMKQISSHLEHHHHHH

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 \(c\).

\(|P_{f(2QUY_1)}(2) \setminus P_{f(3LFT_1)}(2)|=96\), \(|P_{f(3LFT_1)}(2) \setminus P_{f(2QUY_1)}(2)|=61\). 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 1:00010100000001110010101010001111100011011000011100001111110001001110011000111111100101100100100100110110110011100101001100100001100101111111110001001010011101000110100001111010110010000101011101011001111010101110111111111010101011011010000001000001100110110010110111100010000010001101000000101000001011011100100001101010000010010010110
Pair \(Z_2\) Length of longest common subsequence
2QUY_1,3LFT_1 157 4
2QUY_1,4XEH_1 156 4
3LFT_1,4XEH_1 155 5

Newick tree

 
[
	2QUY_1:78.49,
	[
		4XEH_1:77.5,3LFT_1:77.5
	]:0.99
]

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 \((0.85)(0.8)(\frac{630 }{\log_{20} 630}-\frac{295}{\log_{20}295})=93.4\)
Status Protein1 Protein2 d d1/2
Query variables 2QUY_1 3LFT_1 120 111
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]