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Parikh vectors
2ZIQ_1 3MAJ_1 2HLP_1 Letter Amino acid
0 24 30 V Valine
0 5 9 N Asparagine
0 7 12 Q Glutamine
2 6 8 F Phenylalanine
5 41 20 L Leucine
3 4 8 K Lycine
0 1 2 W Tryptophan
3 20 35 D Aspartic acid
1 0 0 C Cysteine
4 43 31 G Glycine
1 25 11 P Proline
1 3 8 Y Tyrosine
1 53 28 A Alanine
6 27 26 E Glutamic acid
0 9 7 H Histidine
3 26 17 S Serine
2 16 15 T Threonine
3 39 16 R Arginine
1 28 16 I Isoleucine
0 5 4 M Methionine 

2ZIQ_1|Chain A[auth L]|Thrombin light chain|Homo sapiens (9606)
>3MAJ_1|Chain A|DNA processing chain A|Rhodopseudomonas palustris (1076)
>2HLP_1|Chains A, B|MALATE DEHYDROGENASE|Haloarcula marismortui (2238)
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 \(w=f(c)\)
2ZIQ , Knot 24 36 0.79 28 34 34 
TFGSGEADCGLRPLFEKKSLEDKTERELLESYIDGR
3MAJ , Knot 153 382 0.79 38 172 345 
GHMDVGERSSDQGTTVLTEAQRIDWMRLIRAENVGPRTFRSLINHFGSARAALERLPELARRGGAARAGRIPSEDEARREIEAGRRIGVELVAPGETGYPTRLATIDDAPPLLGVHALPEALAVMARPMIAIVGSRNASGAGLKFAGQLAADLGAAGFVVISGLARGIDQAAHRASLSSGTVAVLAGGHDKIYPAEHEDLLLDIIQTRGAAISEMPLGHVPRGKDFPRRNRLISGASVGVAVIEAAYRSGSLITARRAADQGREVFAVPGSPLDPRAAGTNDLIKQGATLITSASDIVEAVASILERPIELPGREPEHAPPEGEPDTGDRTRILALLGPSPVGIDDLIRLSGISPAVVRTILLELELAGRLERHGGSLVSLS
2HLP , Knot 133 303 0.83 38 182 291 
TKVSVVGAAGTVGAAAGYNIALRDIADEVVFVDIPDKEDDTVGQAADTNHGIAYDSNTRVRQGGYEDTAGSDVVVITAGIPRQPGQTRIDLAGDNAPIMEDIQSSLDEHNDDYISLTTSNPVDLLNRHLYEAGDRSREQVIGFGGRLDSARFRYVLSEEFDAPVQNVEGTILGEHGDAQVPVFSKVRVDGTDPEFSGDEKEQLLGDLQESAMDVIERKGATEWGPARGVAHMVEAILHDTGRVLPASVKLEGEFGHEDTAFGVPVRLGSNGVEEIVEWDLDDYEQDLMADAAEKLSDQYDKIS

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 \(c\).

\(|P_{f(2ZIQ_1)}(2) \setminus P_{f(3MAJ_1)}(2)|=13\), \(|P_{f(3MAJ_1)}(2) \setminus P_{f(2ZIQ_1)}(2)|=151\). 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 1:011010100110111000010000000110001010
Pair \(Z_2\) Length of longest common subsequence
2ZIQ_1,3MAJ_1 164 3
2ZIQ_1,2HLP_1 174 2
3MAJ_1,2HLP_1 110 4

Newick tree

 
[
	2ZIQ_1:92.30,
	[
		3MAJ_1:55,2HLP_1:55
	]:37.30
]

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 \((0.85)(0.8)(\frac{418 }{\log_{20} 418}-\frac{36}{\log_{20}36})=120.\)
Status Protein1 Protein2 d d1/2
Query variables 2ZIQ_1 3MAJ_1 144 79
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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