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Parikh vectors
1CEN_1 5QOR_1 6LIV_1 Letter Amino acid
22 8 25 N Asparagine
9 2 13 Q Glutamine
16 12 36 G Glycine
29 13 23 K Lycine
10 8 33 T Threonine
11 12 24 P Proline
16 8 39 A Alanine
16 11 27 R Arginine
5 3 12 C Cysteine
28 13 49 L Leucine
8 3 13 M Methionine
22 10 28 F Phenylalanine
9 2 12 H Histidine
21 5 19 Y Tyrosine
15 5 45 V Valine
25 13 23 D Aspartic acid
32 11 37 E Glutamic acid
26 14 26 I Isoleucine
15 11 38 S Serine
8 3 9 W Tryptophan 

1CEN_1|Chain A|CELLULASE CELC|Clostridium thermocellum (1515)
>5QOR_1|Chain A|DCP2 (NUDT20)|Homo sapiens (9606)
>6LIV_1|Chains A, B, C, D, E, F|Tyrosine/DOPA decarboxylase 2|Papaver somniferum (3469)
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 \(w=f(c)\)
1CEN , Knot 152 343 0.86 40 215 336 
MVSFKAGINLGGWISQYQVFSKEHFDTFITEKDIETIAEAGFDHVRLPFDYPIIESDDNVGEYKEDGLSYIDRCLEWCKKYNLGLVLDMHHAPGYRFQDFKTSTLFEDPNQQKRFVDIWRFLAKRYINEREHIAFELLNQVVEPDSTRWNKLMLECIKAIREIDSTMWLYIGGNNYNSPDELKNLADIDDDYIVYNFHFYNPFFFTHQKAHWSESAMAYNRTVKYPGQYEGIEEFVKNNPKYSFMMELNNLKLNKELLRKDLKPAIEFREKKKCKLYCGEFGVIAIADLESRIKWHEDYISLLEEYDIGGAVWNYKKMDFEIYNEDRKPVSQELVNILARRKT
5QOR , Knot 81 167 0.82 40 127 164 
SMGVPTYGAIILDETLENVLLVQGYLAKSGWGFPKGKVNKEEAPHDCAAREVFEETGFDIKDYICKDDYIELRINDQLARLYIIPGIPKDTKFNPKTRREIRNIEWFSIEKLPCHRNDMTPKSKLGLAPNKFFMAIPFIRPLRDWLSRRFGDSSDSDNGFSSTGSTP
6LIV , Knot 216 531 0.85 40 254 498 
MGSLNTEDVLENSSAFGVTNPLDPEEFRRQGHMIIDFLADYYRDVEKYPVRSQVEPGYLRKRLPETAPYNPESIETILQDVTTEIIPGLTHWQSPNYYAYFPSSGSVAGFLGEMLSTGFNVVGFNWMSSPAATELESVVMDWFGKMLNLPESFLFSGSGGGVLQGTSCEAILCTLTAARDRKLNKIGREHIGRLVVYGSDQTHCALQKAAQVAGINPKNFRAIKTFKENSFGLSAATLREVILEDIEAGLIPLFVCPTVGTTSSTAVDPISPICEVAKEYEMWVHVDAAYAGSACICPEFRHFIDGVEEADSFSLNAHKWFFTTLDCCCLWVKDPSALVKALSTNPEYLRNKATESRQVVDYKDWQIALSRRFRSLKLWMVLRSYGVTNLRNFLRSHVKMAKTFEGLICMDGRFEITVPRTFAMVCFRLLPPKTIKVYDNGVHQNGNGVVPLRDENENLVLANKLNQVYLETVNATGSVYMTHAVVGGVYMIRFAVGSTLTEERHVIYAWKILQEHADLILGKFSEADFSS

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 \(c\).

\(|P_{f(1CEN_1)}(2) \setminus P_{f(5QOR_1)}(2)|=136\), \(|P_{f(5QOR_1)}(2) \setminus P_{f(1CEN_1)}(2)|=48\). 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 1:1101011101111100001100001001100001001101110010111001110000011000001100100010100000111110100111001001000011001000001101101110001000001110110011010000100111001011001000111011100000100100110100001100101001111000010100011100001001100011001100010001110100101000110001011101000000010010111111101000101000010110000111111000010101000000110001101110000
Pair \(Z_2\) Length of longest common subsequence
1CEN_1,5QOR_1 184 4
1CEN_1,6LIV_1 153 4
5QOR_1,6LIV_1 197 5

Newick tree

 
[
	5QOR_1:10.79,
	[
		1CEN_1:76.5,6LIV_1:76.5
	]:24.29
]

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 \((0.85)(0.8)(\frac{510 }{\log_{20} 510}-\frac{167}{\log_{20}167})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 1CEN_1 5QOR_1 128 95.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]