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Parikh vectors
5BWF_1 2EEA_1 2CWG_1 Letter Amino acid
25 7 2 I Isoleucine
29 13 13 S Serine
34 7 10 A Alanine
22 5 10 N Asparagine
37 7 4 L Leucine
29 3 3 F Phenylalanine
26 10 5 T Threonine
12 0 3 W Tryptophan
26 7 8 Y Tyrosine
5 3 32 C Cysteine
14 1 0 H Histidine
24 6 7 K Lycine
7 0 2 M Methionine
27 2 4 R Arginine
41 14 41 G Glycine
28 5 4 E Glutamic acid
28 9 5 P Proline
23 7 1 V Valine
37 8 5 D Aspartic acid
11 1 12 Q Glutamine 

5BWF_1|Chains A, B|Beta-1,4-glucosidase|Trichoderma harzianum (5544)
>2EEA_1|Chain A|Filamin-B|Homo sapiens (9606)
>2CWG_1|Chains A, C[auth B]|AGGLUTININ ISOLECTIN 1 (WGA1)|Triticum aestivum (4565)
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 \(w=f(c)\)
5BWF , Knot 200 485 0.85 40 247 457 
MGSSHHHHHHSSGLVPRGSHMLPKDFQWGFATAAYQIEGAIDKDGRGPSIWDTFCAIPGKIADGTSGVTACDSYNRTAEDIALLKSLGAKSYRFSISWSRIIPKGGRDDPVNQLGIDHYAQFVDDLLEAGITPFITLFHWDLPEELHQRYGGLLNRTEFPLDFENYARVMFKALPKVRNWITFNEPLCSAIPGYGSGTFAPGRQSTTEPWIVGHNLLVAHGRAVKVYRDEFKDLNDGQIGIVLNGDFTYPWDSSDPLDREAAERRLEFFTAWYADPIYLGDYPASMRKQLGDRLPEFTPEEKAFVLGSNDFYGMNHYTSNYIRHRTSPATADDTVGNVDVLFYNKEGQCIGPETESSWLRPCPAGFRDFLVWISKRYNYPKIYVTENGTSLKGENDLPKEKILEDDFRVNYYNEYIRAMFTAATLDGVNVKGYFAWSLMDNFEWADGYVTRFGVTYVDYENGQQRFPKKSAKSLKPLFDELIAKE
2EEA , Knot 59 115 0.81 36 94 110 
GSSGSSGPESPLQFYVNYPNSGSVSAYGPGLVYGVANKTATFTIVTEDAGEGGLDLAIEGPSKAEISCIDNKDGTCTVTYLPTLPGDYSILVKYNDKHIPGSPFTAKITDDSRRC
2CWG , Knot 64 171 0.64 38 88 129 
QRCGEQGSNMECPNNLCCSQYGYCGMGGDYCGKGCQNGACWTSKRCGSQAGGATCTNNQCCSQYGYCGFGAEYCGAGCQGGPCRADIKCGSQAGGKLCPNNLCCSQWGFCGLGSEFCGGGCQSGACSTDKPCGKDAGGRVCTNNYCCSKWGSCGIGPGYCGAGCQSGGCDG

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 \(c\).

\(|P_{f(5BWF_1)}(2) \setminus P_{f(2EEA_1)}(2)|=174\), \(|P_{f(2EEA_1)}(2) \setminus P_{f(5BWF_1)}(2)|=21\). 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 1:11000000000011110100111001011110110010111000101101100101111011010011010000000100111100111000010101001110110001100111000101100110111011101101011001000011110000111010001011101110100110100110011110101011110000001111100111101011010000100100101111101010011000011000110001011011010110110011010001100110101000111110001011000000010000011010001101011100001001110000011010111100111110000001010100010010100011000110001010000001011101101011010101110110010110101001110010000100011000100101110011100
Pair \(Z_2\) Length of longest common subsequence
5BWF_1,2EEA_1 195 3
5BWF_1,2CWG_1 223 4
2EEA_1,2CWG_1 132 3

Newick tree

 
[
	5BWF_1:11.77,
	[
		2EEA_1:66,2CWG_1:66
	]:48.77
]

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 \((0.85)(0.8)(\frac{600 }{\log_{20} 600}-\frac{115}{\log_{20}115})=141.\)
Status Protein1 Protein2 d d1/2
Query variables 5BWF_1 2EEA_1 178 108.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]