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Parikh vectors
1LLP_1 8TVR_1 6FIZ_1 Letter Amino acid
23 48 11 V Valine
35 48 4 A Alanine
8 23 1 R Arginine
18 22 3 Q Glutamine
34 71 7 G Glycine
28 28 6 P Proline
19 51 11 I Isoleucine
10 34 4 K Lycine
22 50 5 S Serine
20 46 9 T Threonine
0 27 3 Y Tyrosine
11 41 2 N Asparagine
8 8 0 C Cysteine
16 22 3 E Glutamic acid
18 49 12 L Leucine
25 48 3 D Aspartic acid
10 10 1 H Histidine
8 9 0 M Methionine
27 25 9 F Phenylalanine
3 7 0 W Tryptophan 

1LLP_1|Chain A|LIGNIN PEROXIDASE|Phanerochaete chrysosporium (2822231)
>8TVR_1|Chains A, B, C, E[auth D], F[auth E], G[auth F], I[auth H], J[auth I], K[auth J], M[auth L], N[auth M], O[auth N], Q[auth P], R[auth Q], S[auth R], U[auth V], V[auth W], W[auth X]|Tail spike protein|Salmonella phage P22 (2908168)
>6FIZ_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P|Potassium channel protein,Cyclic nucleotide-gated olfactory channel,Potassium channel protein|Bacillus cereus (strain ATCC 14579 / DSM 31 / JCM 2152 / NBRC 15305 / NCIMB 9373 / NRRL B-3711) (226900)
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 \(w=f(c)\)
1LLP , Knot 145 343 0.82 38 198 332 
ATCANGKTVGDASCCAWFDVLDDIQANMFHGGQCGAEAHESIRLVFHDSIAISPAMEAKGKFGGGGADGSIMIFDTIETAFHPNIGLDEVVAMQKPFVQKHGVTPGDFIAFAGAVALSNCPGAPQMNFFTGRKPATQPAPDGLVPEPFHTVDQIIARVNDAGEFDELELVWMLSAHSVAAVNDVDPTVQGLPFDSTPGIFDSQFFVETQFRGTLFPGSGGNQGEVESGMAGEIRIQTDHTLARDSRTACEWQSFVGNQSKLVDDFQFIFLALTQLGQDPNAMTDCSDVIPLSKPIPGNGPFSFFPPGKSHSDIEQACAETPFPSLVTLPGPATSVARIPPHKA
8TVR , Knot 259 667 0.84 40 266 616 
MTDITANVVVSNPRPIFTESRSFKAVANGKIYIGQIDTDPVNPANQIPVYIENEDGSHVQITQPLIINAAGKIVYNGQLVKIVTVQGHSMAIYDANGSQVDYIANVLKYDPDQYSIEADKKFKYSVKLSDYPTLQDAASAAVDGLLIDRDYNFYGGETVDFGGKVLTIECKAKFIGDGNLIFTKLGKGSRIAGVFMESTTTPWVIKPWTDDNQWLTDAAAVVATLKQSKTDGYQPTVSDYVKFPGIETLLPPNAKGQNITSTLEIRECIGVEVHRASGLMAGFLFRGCHFCKMVDANNPSGGKDGIITFENLSGDWGKGNYVIGGRTSYGSVSSAQFLRNNGGFERDGGVIGFTSYRAGESGVKTWQGTVGSTTSRNYNLQFRDSVVIYPVWDGFDLGADTDMNPELDRPGDYPITQYPLHQLPLNHLIDNLLVRGALGVGFGMDGKGMYVSNITVEDCAGSGAYLLTHESVFTNIAIIDTNTKDFQANQIYISGACRVNGLRLIGIRSTDGQGLTIDAPNSTVSGITGMVDPSRINVANLAEEGLGNIRANSFGYDSAAIKLRIHKLSKTLDSGALYSHINGGAGSGSAYTQLTAISGSTPDAVSLKVNHKDCRGAEIPFVPDIASDDFIKDSSCFLPYWENNSTSLKALVKKPNGELVRLTLATL
6FIZ , Knot 50 94 0.80 34 74 91 
AKDKEFQVLFVLTILTLISGTIFYSTVEGLRPIDALYFSVVTLTTVGEAPPPQTDFGKIFTILYIFIGIGLVFGFIHKLAVNVQLPSILSNVVP

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 \(c\).

\(|P_{f(1LLP_1)}(2) \setminus P_{f(8TVR_1)}(2)|=48\), \(|P_{f(8TVR_1)}(2) \setminus P_{f(1LLP_1)}(2)|=116\). 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 1:1001010011010001110110010101101100110100010111000111011101010111111010111100100110101110011110011100011011011111111110001111010110100110011101111011001001110100110100101111101001111001010101111000111100011100010101111011001010011110101000001100000100100111000011001011111100110010110000011110011110111011111000001001010011101101111100110111001
Pair \(Z_2\) Length of longest common subsequence
1LLP_1,8TVR_1 164 4
1LLP_1,6FIZ_1 164 3
8TVR_1,6FIZ_1 206 4

Newick tree

 
[
	6FIZ_1:96.50,
	[
		1LLP_1:82,8TVR_1:82
	]:14.50
]

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 \((0.85)(0.8)(\frac{1010 }{\log_{20} 1010}-\frac{343}{\log_{20}343})=177.\)
Status Protein1 Protein2 d d1/2
Query variables 1LLP_1 8TVR_1 222 167
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]