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Parikh vectors
3THM_1 2QEE_1 5THA_1 Letter Amino acid
5 20 22 F Phenylalanine
20 18 48 T Threonine
5 13 23 W Tryptophan
9 10 19 Y Tyrosine
16 23 36 A Alanine
10 15 26 Q Glutamine
5 23 45 I Isoleucine
16 41 66 L Leucine
6 28 32 R Arginine
14 18 65 G Glycine
31 25 72 S Serine
7 22 25 N Asparagine
5 4 22 C Cysteine
0 17 9 M Methionine
16 17 45 P Proline
17 35 48 V Valine
9 29 40 D Aspartic acid
11 39 38 E Glutamic acid
2 19 37 H Histidine
12 21 40 K Lycine 

3THM_1|Chain A[auth L]|Fab EP6b_B01, light chain|Homo sapiens (9606)
>2QEE_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|BH0493 protein|Bacillus halodurans C-125 (272558)
>5THA_1|Chain A|Gem-associated protein 5|Homo sapiens (9606)
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 \(w=f(c)\)
3THM , Knot 97 216 0.80 38 143 208 
QSVLTQPPSVSEAPRQTVTISCSGNSSNIGRYPVNWYQQLPGKAPKLLIYSDNLRFSGVPDRFSGSKSGTTASLAIRDLLSEDEADYYCSTWDDTLEGWVFGGGTKVTVLGQPKAAPSVTLFPPSSEELQANKATLVCLISDFYPGAVTVAWKADSSPVKAGVETTTPSKQSNNKYAASSYLSLTPEQWKSHRSYSCQVTHEGSTVEKTVAPTECS
2QEE , Knot 183 437 0.84 40 239 420 
MSLSINSREVLAEKVKNAVNNQPVTDMHTHLFSPNFGEILLWDIDELLTYHYLVAEVMRWTDVSIEAFWAMSKREQADLIWEELFIKRSPVSEACRGVLTCLQGLGLDPATRDLQVYREYFAKKTSEEQVDTVLQLANVSDVVMTNDPFDDNERISWLEGKQPDSRFHAALRLDPLLNEYEQTKHRLRDWGYKVNDEWNEGSIQEVKRFLTDWIERMDPVYMAVSLPPTFSFPEESNRGRIIRDCLLPVAEKHNIPFAMMIGVKKRVHPALGDAGDFVGKASMDGVEHLLREYPNNKFLVTMLSRENQHELVVLARKFSNLMIFGCWWFMNNPEIINEMTRMRMEMLGTSFIPQHSDARVLEQLIYKWHHSKSIIAEVLIDKYDDILQAGWEVTEEEIKRDVADLFSRNFWRFVGRNDHVTSVKVEQQTEGHHHHHH
5THA , Knot 291 758 0.84 40 302 689 
MGSSHHHHHHSSGLVPRGSMGQEPRTLPPSPNWYCARCSDAVPGGLFGFAARTSVFLVRVGPGAGESPGTPPFRVIGELVGHTERVSGFTFSHHPGQYNLCATSSDDGTVKIWDVETKTVVTEHALHQHTISTLHWSPRVKDLIVSGDEKGVVFCYWFNRNDSQHLFIEPRTIFCLTCSPHHEDLVAIGYKDGIVVIIDISKKGEVIHRLRGHDDEIHSIAWCPLPGEDCLSINQEETSEEAEITNGNAVAQAPVTKGCYLATGSKDQTIRIWSCSRGRGVMILKLPFLKRRGGGIDPTVKERLWLTLHWPSNQPTQLVSSCFGGELLQWDLTQSWRRKYTLFSASSEGQNHSRIVFNLCPLQTEDDKQLLLSTSMDRDVKCWDIATLECSWTLPSLGGFAYSLAFSSVDIGSLAIGVGDGMIRVWNTLSIKNNYDVKNFWQGVKSKVTALCWHPTKEGCLAFGTDDGKVGLYDTYSNKPPQISSTYHKKTVYTLAWGPPVPPMSLGGEGDRPSLALYSCGGEGIVLQHNPWKLSGEAFDINKLIRDTNSIKYKLPVHTEISWKADGKIMALGNEDGSIEIFQIPNLKLICTIQQHHKLVNTISWHHEHGSQPELSYLMASGSNNAVIYVHNLKTVIESSPESPVTITEPYRTLSGHTAKITSVAWSPHHDGRLVSASYDGTAQVWDALREEPLCNFRGHQGRLLCVAWSPLDPDCIYSGADDFCVHKWLTSMQDHSRPPQGKKSIELEKKRLSQPKA

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 \(c\).

\(|P_{f(3THM_1)}(2) \setminus P_{f(2QEE_1)}(2)|=49\), \(|P_{f(2QEE_1)}(2) \setminus P_{f(3THM_1)}(2)|=145\). 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 1:001100110100110001010001000011001101000111011011100001010111001010001001011100110000100000010001011111110010111010111010111100001010010110110010111101110100011011100001000000001100010101001000000000100010010001110000
Pair \(Z_2\) Length of longest common subsequence
3THM_1,2QEE_1 194 4
3THM_1,5THA_1 203 4
2QEE_1,5THA_1 159 6

Newick tree

 
[
	3THM_1:10.04,
	[
		2QEE_1:79.5,5THA_1:79.5
	]:25.54
]

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 \((0.85)(0.8)(\frac{653 }{\log_{20} 653}-\frac{216}{\log_{20}216})=123.\)
Status Protein1 Protein2 d d1/2
Query variables 3THM_1 2QEE_1 158 116.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]