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Parikh vectors
3OTP_1 1FCD_1 4ZKU_1 Letter Amino acid
34 11 13 N Asparagine
17 24 11 E Glutamic acid
52 42 16 G Glycine
14 9 3 F Phenylalanine
38 33 18 V Valine
2 5 0 C Cysteine
7 9 3 H Histidine
0 6 1 W Tryptophan
5 20 6 Y Tyrosine
36 26 17 S Serine
22 18 24 T Threonine
43 46 35 A Alanine
13 11 12 R Arginine
22 24 17 D Aspartic acid
31 11 12 Q Glutamine
30 29 14 I Isoleucine
26 25 10 K Lycine
35 20 18 L Leucine
14 6 2 M Methionine
18 26 5 P Proline 

3OTP_1|Chains A, B, C, D, E, F|Protease do|Escherichia coli (562)
>1FCD_1|Chains A, C[auth B]|FLAVOCYTOCHROME C SULFIDE DEHYDROGENASE (FLAVIN-BINDING SUBUNIT)|Allochromatium vinosum (1049)
>4ZKU_1|Chains A, B|Tail needle protein gp26|Enterobacteria phage P22 (10754)
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 \(w=f(c)\)
3OTP , Knot 184 459 0.82 38 205 423 
AETSSATTAQQMPSLAPMLEKVMPSVVSINVEGSTTVNTPRMPRNFQQFFGDDSPFCQEGSPFQSSPFCQGGQGGNGGGQQQKFMALGSGVIIDADKGYVVTNNHVVDNATVIKVQLSDGRKFDAKMVGKDPRSDIALIQIQNPKNLTAIKMADSDALRVGDYTVAIGNPFGLGETVTSGIVSALGRSGLNAENYENFIQTDAAINRGNAGGALVNLNGELIGINTAILAPDGGNIGIGFAIPSNMVKNLTSQMVEYGQVKRGELGIMGTELNSELAKAMKVDAQRGAFVSQVLPNSSAAKAGIKAGDVITSLNGKPISSFAALRAQVGTMPVGSKLTLGLLRDGKQVNVNLELQQSSQNQVDSSSIFNGIEGAEMSNKGKDQGVVVNNVKTGTPAAQIGLKKGDVIIGANQQAVKNIAELRKVLDSKPSVLALNIQRGDSTIYLLMQAAALEHHHHHH
1FCD , Knot 167 401 0.83 40 206 378 
AGRKVVVVGGGTGGATAAKYIKLADPSIEVTLIEPNTDYYTCYLSNEVIGGDRKLESIKHGYDGLRAHGIQVVHDSATGIDPDKKLVKTAGGAEFGYDRCVVAPGIELIYDKIEGYSEEAAAKLPHAWKAGEQTAILRKQLEDMADGGTVVIAPPAAPFRCPPGPYERASQVAYYLKAHKPMSKVIILDSSQTFSKQSQFSKGWERLYGFGTENAMIEWHPGPDSAVVKVDGGEMMVETAFGDEFKADVINLIPPQRAGKIAQIAGLTNDAGWCPVDIKTFESSIHKGIHVIGDASIANPMPKSGYSANSQGKVAAAAVVVLLKGEEPGTPSYLNTCYSILAPAYGISVAAIYRPNADGSAIESVPDSGGVTPVDAPDWVLEREVQYAYSWYNNIVHDTFG
4ZKU , Knot 105 237 0.80 38 149 221 
GPGSMADPSLNNPVVIQATRLDASILPRNVFSKSYLLYVIAQGTDVGAIAGKANEAGQGAYDAQVKNDEQDVELADHEARIKQLRIDVDDHESRITANTKAITALNVRVTTAEGEIASLQTNVSALDGRVTTAENNISALQADYVSKTATTSQSLASPLNVTTSYSVGGKKVLGARQTGWTAATGTANKGVFDADLTFAVSDTYTQSEIQAIANALITERRRTKAMEDALRAHGLID

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 \(c\).

\(|P_{f(3OTP_1)}(2) \setminus P_{f(1FCD_1)}(2)|=71\), \(|P_{f(1FCD_1)}(2) \setminus P_{f(3OTP_1)}(2)|=72\). 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 1:100001001001101111100111011010101000100101100100111000110001011000110011011011100001111101111010010110000110010110101001001010111001000111101001001011011000110110001111011111001001110111001101000001100011100101111110101011110011111011011111111001100100011001010010111110010001101101010011110011100011011101101100101011001111010110111100101111001001010101000000010000110110110100010001111001001011101110010111110001100110100110001011110100100010111011110000000
Pair \(Z_2\) Length of longest common subsequence
3OTP_1,1FCD_1 143 4
3OTP_1,4ZKU_1 146 4
1FCD_1,4ZKU_1 143 5

Newick tree

 
[
	4ZKU_1:72.50,
	[
		3OTP_1:71.5,1FCD_1:71.5
	]:1.00
]

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 \((0.85)(0.8)(\frac{860 }{\log_{20} 860}-\frac{401}{\log_{20}401})=122.\)
Status Protein1 Protein2 d d1/2
Query variables 3OTP_1 1FCD_1 152 143.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]