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Parikh vectors
4EAH_1 8JSX_1 5YRK_1 Letter Amino acid
30 45 10 I Isoleucine
23 37 10 S Serine
16 15 7 Y Tyrosine
19 12 12 K Lycine
12 27 3 F Phenylalanine
27 23 11 T Threonine
29 43 9 A Alanine
28 39 16 G Glycine
26 54 8 L Leucine
11 12 6 Q Glutamine
28 10 6 E Glutamic acid
9 6 2 H Histidine
17 22 1 M Methionine
19 26 3 P Proline
18 11 8 R Arginine
22 15 7 D Aspartic acid
6 9 5 C Cysteine
12 14 4 N Asparagine
4 4 4 W Tryptophan
21 33 10 V Valine 

4EAH_1|Chains A[auth D], F, G, H|Actin, alpha skeletal muscle|Oryctolagus cuniculus (9986)
>8JSX_1|Chain A|Synaptic vesicular amine transporter|Homo sapiens (9606)
>5YRK_1|Chains A, B|PPL3-b|Pteria penguin (113549)
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 \(w=f(c)\)
4EAH , Knot 160 377 0.84 40 220 362 
MCDEDETTALVCDNGSGLVKAGFAGDDAPRAVFPSIVGRPRHQGVMVGMGQKDSYVGDEAQSKRGILTLKYPIEHGIITNWDDMEKIWHHTFYNELRVAPEEHPTLLTEAPLNPKANREKMTQIMFETFNVPAMYVAIQAVLSLYASGRTTGIVLDSGDGVTHNVPIYEGYALPHAIMRLDLAGRDLTDYLMKILTERGYSFVTTAEREIVRDIKEKLCYVALDFENEMATAASSSSLEKSYELPDGQVITIGNERFRCPETLFQPSFIGMESAGIHETTYNSIMKCDIDIRKDLYANNVMSGGTTMYPGIADRMQKEITALAPSTMKIKIIAPPERKYSVWIGGSILASLSTFQQMWITKQEYDEAGPSIVHRKCF
8JSX , Knot 185 457 0.82 40 217 425 
SRKLILFIVFLALLLDNMLLTVVVPIIPSYLYSIKHEKNATEIQTARPVHTASISDSFQSIFSYYDNSTMVTGNATRDLTLHQTATQHMVTNASAVPSDCPSEDKDLLNENVQVGLLFASKATVQLITNPFIGLLTNRIGYPIPIFAGFCIMFVSTIMFAFSSSYAFLLIARSLQGIGSSCSSVAGMGMLASVYTDDEERGNVMGIALGGLAMGVLVGPPFGSVLYEFVGKTAPFLVLAALVLLDGAIQLFVLQPSRVQPESQKGTPLTTLLKDPYILIAAGSICFANMGIAMLEPALPIWMMETMCSRKWQLGVAFLPASISYLIGTNIFGILAHKMGRWLCALLGMIIVGVSILCIPFAKNIYGLIAPNFGVGFAIGMVDSSMMPIMGYLVDLRHVSVYGSVYAIADVAFCMGYAIGPSAGGAIAKAIGFPWLMTIIGIIDILFAPLCFFLRSPP
5YRK , Knot 71 142 0.82 40 116 137 
QVASEYLGGPGGDAFDDKALAQNGDITRIEMQCTDVATYIKLRYGKVDSRQWGWANENCIQWSKKGVKVVHELSSGEYITSAIVTYGKYVQSITFKTNKRTLPRCGTSATEKSVTVLIPGGLKYISGRWGCRIDGLRFHAKC

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 \(c\).

\(|P_{f(4EAH_1)}(2) \setminus P_{f(8JSX_1)}(2)|=81\), \(|P_{f(8JSX_1)}(2) \setminus P_{f(4EAH_1)}(2)|=78\). 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 1:10000000111000101110111110011011110111010001111111000001100100001110100110011100100100110001000101110001011001110101000010011100101111011101110101010001111001011000111001011101110101110010001101100010011001000110010001001110100011011000010000011010110110001001001101011110011100000001100010100010100110110010111100100010111100101011111000001111101110100100111000000011101100001
Pair \(Z_2\) Length of longest common subsequence
4EAH_1,8JSX_1 159 4
4EAH_1,5YRK_1 194 3
8JSX_1,5YRK_1 195 3

Newick tree

 
[
	5YRK_1:10.48,
	[
		4EAH_1:79.5,8JSX_1:79.5
	]:22.98
]

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 \((0.85)(0.8)(\frac{834 }{\log_{20} 834}-\frac{377}{\log_{20}377})=123.\)
Status Protein1 Protein2 d d1/2
Query variables 4EAH_1 8JSX_1 152 142
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]