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Parikh vectors
4JAT_1 4HMM_1 3WGB_1 Letter Amino acid
4 0 0 W Tryptophan
6 11 8 Y Tyrosine
16 7 17 E Glutamic acid
33 10 33 G Glycine
9 10 14 H Histidine
22 4 13 P Proline
2 2 4 C Cysteine
48 4 47 L Leucine
9 3 10 F Phenylalanine
26 9 18 S Serine
18 5 11 I Isoleucine
7 3 5 K Lycine
6 7 11 M Methionine
4 10 16 Q Glutamine
16 9 12 T Threonine
30 11 29 V Valine
39 6 43 A Alanine
26 8 23 R Arginine
6 8 6 N Asparagine
26 5 21 D Aspartic acid 

4JAT_1|Chains A, B, C, D|Alpha-pyrone synthesis polyketide synthase-like Pks11|Mycobacterium tuberculosis (1773)
>4HMM_1|Chains A, B|Major prion protein|Oryctolagus cuniculus (9986)
>3WGB_1|Chains A, B, C, D|L-allo-threonine aldolase|Aeromonas jandaei (650)
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 \(w=f(c)\)
4JAT , Knot 148 353 0.82 40 178 324 
MSVIAGVFGALPPHRYSQSEITDSFVEFPGLKEHEEIIRRLHAAAKVNGRHLVLPLQQYPSLTDFGDANEIFIEKAVDLGVEALLGALDDANLRPSDIDMIATATVTGVAVPSLDARIAGRLGLRPDVRRMPLFGLGSVAGAAGVARLRDYLRGAPDDVAVLVSVELCSLTYPAVKPTVSSLVGTALFGDGAAAVVAVGDRRAEQVRAGGPDILDSRSSLYPDSLHIMGWDVGSHGLRLRLSPDLTNLIERYLANDVTTFLDAHRLTKDDIGAWVSHPGGPKVIDAVATSLALPPEALELTWRSLGEIGNLSSASILHILRDTIEKRPPSGSAGLMLAMGPGFCTELVLLRWR
4HMM , Knot 64 132 0.79 38 104 125 
MGSSHHHHHHSSGLVPRGSHMAVVGGLGGYMLGSAMSRPLIHFGNDYEDRYYRENMYRYPNQVYYRPVDQYSNQNNFVHDCVNITVKQHTVTTTTKGENFTETDIKIMERVVEQMCITQYQQESQAAYQRAA
3WGB , Knot 143 341 0.81 38 181 316 
GSHMRYIDLRSDTVTQPTDAMRQCMLHAEVGDDVYGEDPGVNALEAYGADLLGKEAALFVPSGTMSNLLAVMSHCQRGEGAVLGSAAHIYRYEAQGSAVLGSVALQPVPMQADGSLALADVRAAIAPDDVHFTPTRLVCLENTHNGKVLPLPYLREMRELVDEHGLQLHLDGARLFNAVVASGHTVRELVAPFDSVSICLSKGLGAPVGSLLVGSHAFIARARRLRKMVGGGMRQAGILAQAGLFALQQHVVRLADDHRRARQLAEGLAALPGIRLDLAQVQTNMVFLQLTSGESAPLLAFMKARGILFSGYGELRLVTHLQIHDDDIEEVIDAFTEYLGA

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 \(c\).

\(|P_{f(4JAT_1)}(2) \setminus P_{f(4HMM_1)}(2)|=125\), \(|P_{f(4HMM_1)}(2) \setminus P_{f(4JAT_1)}(2)|=51\). 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 1:10111111111110000000100011011110000011001011101010011111000101001101001110011011101111110010101001011101010111110101011101110101001111111011111111010001011100111110101001001110101001110111101111111110001001011110110000010100101111011001101010101001100011001001101001000011111001111011011100111110110101001101101001011011000100011010111111111110001111010
Pair \(Z_2\) Length of longest common subsequence
4JAT_1,4HMM_1 176 3
4JAT_1,3WGB_1 157 5
4HMM_1,3WGB_1 175 4

Newick tree

 
[
	4HMM_1:90.62,
	[
		4JAT_1:78.5,3WGB_1:78.5
	]:12.12
]

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 \((0.85)(0.8)(\frac{485 }{\log_{20} 485}-\frac{132}{\log_{20}132})=104.\)
Status Protein1 Protein2 d d1/2
Query variables 4JAT_1 4HMM_1 129 88.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]