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Parikh vectors
8CYE_1 3LJW_1 3GNB_1 Letter Amino acid
79 12 7 A Alanine
11 10 7 E Glutamic acid
44 4 10 G Glycine
36 10 13 L Leucine
53 9 7 S Serine
65 4 2 T Threonine
45 5 1 N Asparagine
48 5 3 D Aspartic acid
0 0 1 C Cysteine
30 6 6 Q Glutamine
29 14 5 K Lycine
7 3 3 M Methionine
29 13 2 I Isoleucine
10 3 2 F Phenylalanine
12 7 0 Y Tyrosine
31 4 4 V Valine
12 2 10 R Arginine
0 3 10 H Histidine
7 6 3 P Proline
0 0 0 W Tryptophan 

8CYE_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V|Flagellin|Escherichia coli O127:H6 (168807)
>3LJW_1|Chains A, B|Protein polybromo-1|Homo sapiens (9606)
>3GNB_1|Chain A|V(D)J recombination-activating protein 1|Mus musculus (10090)
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 \(w=f(c)\)
8CYE , Knot 205 548 0.78 34 189 477 
MAQVINTNSLSLITQNNINKNQSALSSSIERLSSGLRINSAKDDAAGQAIANRFTSNIKGLTQAARNANDGISVAQTTEGALSEINNNLQRIRELTVQASTGTNSDSDLDSIQDEIKSRLDEIDRVSGQTQFNGVNVLAKDGSMKIQVGANDGQTITIDLKKIDSDTLGLNGFNVNGKGETANTAATLKDMSGFTAAAAPGGTVGVTQYTDKSAVASSVDILNAVAGADGNKVTTSADVGFGTPAAAVTYTYNKDTNSYSAASDDISSANLAAFLNPQARDTTKATVTIGGKDQDVNIDKSGNLTAADDGAVLYMDATGNLTKNNAGGDTQATLAKVATATGAKAATIQTDKGTFTSDGTAFDGASMSIDANTFANAVKNDTYTATVGAKTYSVTTGSAAADTAYMSNGVLSDTPPTYYAQADGSITTTEDAAAGKLVYKGSDGKLTTDTTSKAESTSDPLAALDDAISQIDKFRSSLGAVQNRLDSAVTNLNNTTTNLSEAQSRIQDADYATEVSNMSKAQIIQQAGNSVLAKANQVPQQVLSLLQG
3LJW , Knot 60 120 0.79 36 99 117 
TEGSSPAYLKEILEQLLEAIVVATNPSGRLISELFQKLPSKVQYPDYYAIIKEPIDLKTIAQRIQNGSYKSIHAMAKDIDLLAKNAKTYNEPGSQVFKDANSIKKIFYMKKAEIEHHEMA
3GNB , Knot 47 96 0.74 36 72 90 
MGSSHHHHHHSSGLVPGSHMGGRPRQHLLSLTRRAQKHRLRELKIQVKEFADKEEGGDVKAVCLTLFLLALRARNEHRQADELEAIMQGRGSGLQP

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 \(c\).

\(|P_{f(8CYE_1)}(2) \setminus P_{f(3LJW_1)}(2)|=120\), \(|P_{f(3LJW_1)}(2) \setminus P_{f(8CYE_1)}(2)|=30\). 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 1:11011000010110000100000110001001001101001000111011100100010110011001001101100001110010001001001010100100000010010001000100100101000101101110010101011100100101010010000111011010101001001101001011011111110111000000011100101101111101001000101111011111000000000000110001001011111010100000101011100001010001010110011110101010100001110001011011010110110100001010001011011010101001101100000010111000010010111001010011100011000101010100000111101100100101000000010000011111001100100100011110001001100100000010010001001001001001001011001100111010011001101101
Pair \(Z_2\) Length of longest common subsequence
8CYE_1,3LJW_1 150 4
8CYE_1,3GNB_1 177 4
3LJW_1,3GNB_1 121 4

Newick tree

 
[
	8CYE_1:88.04,
	[
		3LJW_1:60.5,3GNB_1:60.5
	]:27.54
]

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 \((0.85)(0.8)(\frac{668 }{\log_{20} 668}-\frac{120}{\log_{20}120})=158.\)
Status Protein1 Protein2 d d1/2
Query variables 8CYE_1 3LJW_1 186 114
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]