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Parikh vectors
7LRT_1 1NHK_1 2FLQ_1 Letter Amino acid
7 0 4 C Cysteine
18 10 24 G Glycine
5 14 27 I Isoleucine
14 11 17 K Lycine
14 15 24 A Alanine
6 7 24 R Arginine
12 10 27 L Leucine
4 3 10 M Methionine
7 8 16 F Phenylalanine
21 6 22 T Threonine
8 4 17 Y Tyrosine
8 5 13 N Asparagine
2 4 13 H Histidine
12 6 19 P Proline
5 0 12 W Tryptophan
11 6 18 Q Glutamine
7 12 34 E Glutamic acid
26 9 21 V Valine
7 6 15 D Aspartic acid
34 8 18 S Serine 

7LRT_1|Chains A[auth D], C[auth F], E[auth H]|antibody A23-58.1 heavy chain|Homo sapiens (9606)
>1NHK_1|Chains A[auth L], B[auth R]|NUCLEOSIDE DIPHOSPHATE KINASE|Myxococcus xanthus (34)
>2FLQ_1|Chains A, B|Nitric Oxide Synthase|Geobacillus stearothermophilus (1422)
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 \(w=f(c)\)
7LRT , Knot 102 228 0.81 40 151 217 
QMQLVQSGPEVKKPGTSVKVSCKASGFTFTSSAVQWVRQARGQRLEWIGWIVVGSGNTNYAQKFQERVTITRDMSTSTAYMELSSLRSEDTAVYYCAAPNCSNVVCYDGFDIWGQGTMVTVSSASTKGPSVFPLAPSSKSTSGGTAALGCLVKDYFPEPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKKVEPKSCDK
1NHK , Knot 71 144 0.81 36 112 138 
AIERTLSIIKPDGLEKGVIGKIISRFEEKGLKPVAIRLQHLSQAQAEGFYAVHKARPFFKDLVQFMISGPVVLMVLEGENAVLANRDIMGATNPAQAAEGTIRKDFATSIDKNTVHGSDSLENAKIEIAYFFRETEIHSYPYQK
2FLQ , Knot 168 375 0.88 40 238 367 
MVTIESDKLRQHDEQLMTKAEQFIIASYRELGKSEQEIKRRVNEIRWEVEQTGTYRHTYEELSYGAKMAWRHSNRCIGRLFWQSLHVIDAREAVTEEEVFSYLFHHIEVATNGGKIRPTITIFRPNGEVRIWNHQLIRYAGYETEEGIIGDSSSLTFTRACEQLGWKGEKTPFDVLPLVIQVGGQKPVWTPIPKELVLEVPIEHPEFPWFRDLQLKWYAVPIISDMCLEIGGIRYMAAPFNGWYMGTEIGARNFADDYRYNMLPKVASCMGLDTNSNASLWKDKALVELNIAVLYSYKKAGVSIVDHHTAARQFQLFEQQEKAAGRHVTGDWTWLIPPLSPATTHIFHRSYDNTMMLPNFFYQDRPYEPQRGEEQ

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 \(c\).

\(|P_{f(7LRT_1)}(2) \setminus P_{f(1NHK_1)}(2)|=104\), \(|P_{f(1NHK_1)}(2) \setminus P_{f(7LRT_1)}(2)|=65\). 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 1:010110011010011001010001011010001101100101001011111111010000100100010100010000101010010000011000111000011000110111010110100100011011111100000011011110110001101101010011100110011111000110010011011000110000100100010000100010100000
Pair \(Z_2\) Length of longest common subsequence
7LRT_1,1NHK_1 169 3
7LRT_1,2FLQ_1 199 4
1NHK_1,2FLQ_1 204 3

Newick tree

 
[
	2FLQ_1:10.62,
	[
		7LRT_1:84.5,1NHK_1:84.5
	]:21.12
]

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 \((0.85)(0.8)(\frac{372 }{\log_{20} 372}-\frac{144}{\log_{20}144})=69.0\)
Status Protein1 Protein2 d d1/2
Query variables 7LRT_1 1NHK_1 89 71.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]