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Parikh vectors
2MQU_1 8FFN_1 5ENL_1 Letter Amino acid
6 15 1 C Cysteine
0 32 15 P Proline
0 42 20 T Threonine
2 26 19 N Asparagine
0 38 9 Q Glutamine
2 46 25 E Glutamic acid
3 39 22 I Isoleucine
5 23 36 K Lycine
1 35 16 F Phenylalanine
0 31 31 D Aspartic acid
3 41 32 S Serine
2 13 5 W Tryptophan
0 24 5 M Methionine
1 43 14 R Arginine
3 45 37 G Glycine
0 20 11 H Histidine
2 99 40 L Leucine
1 24 9 Y Tyrosine
1 47 34 V Valine
3 56 55 A Alanine 

2MQU_1|Chain A|Neurotoxin Hm-3|Heriaeus melloteei (2337432)
>8FFN_1|Chains A, B, C, D|Transient receptor potential cation channel subfamily V member 5|Oryctolagus cuniculus (9986)
>5ENL_1|Chain A|ENOLASE|Saccharomyces cerevisiae (4932)
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 \(w=f(c)\)
2MQU , Knot 22 35 0.74 28 34 33 
GCIAKNKECAWFSGEWCCGALSCKYSIKRNLKICV
8FFN , Knot 281 739 0.83 40 305 676 
MGACPPKAKGPWAQLQKLLISWPVGEQDWEQYRDRVNMLQQERIRDSPLLQAAKENDLRLLKILLLNQSCDFQQRGAVGETALHVAALYDNLEAATLLMEAAPELAKEPALCEPFVGQTALHIAVMNQNLNLVRALLARGASVSARATGAAFRRSPHNLIYYGEHPLSFAACVGSEEIVRLLIEHGADIRAQDSLGNTVLHILILQPNKTFACQMYNLLLSYDEHSDHLQSLELVPNHQGLTPFKLAGVEGNTVMFQHLMQKRKHVQWTCGPLTSTLYDLTEIDSWGEELSFLELVVSSKKREARQILEQTPVKELVSFKWKKYGRPYFCVLASLYILYMICFTTCCIYRPLKLRDDNRTDPRDITILQQKLLQEAYVTHQDNIRLVGELVTVTGAVIILLLEIPDIFRVGASRYFGQTILGGPFHVIIITYASLVLLTMVMRLTNMNGEVVPLSFALVLGWCSVMYFARGFQMLGPFTIMIQKMIFGDLMRFCWLMAVVILGFASAFHITFQTEDPNNLGEFSDYPTALFSTFELFLTIIDGPANYSVDLPFMYCITYAAFAIIATLLMLNLFIAMMGDTHWRVAQERDELWRAQVVATTVMLERKMPRFLWPRSGICGYEYGLGDRWFLRVENHHDQNPLRVLRYVEAFKCSDKEDGQEQLSEKRPSTVESGMLSRASVAFQTPSLSRTTSQSSNSHRGWEILRRNTLGHLNLGLDLGEGDGEEVYHFTETSQVAPA
5ENL , Knot 174 436 0.80 40 212 406 
AVSKVYARSVYDSRGNPTVEVELTTEKGVFRSIVPSGASTGVHEALEMRDGDKSKWMGKGVLHAVKNVNDVIAPAFVKANIDVSDQKAVDDFLISLDGTANKSKLGANAILGVSLAASRAAAAEKNVPLYKHLADLSKSKTSPYVLPVPFLNVLNGGSHAGGALALQEFMIAPTGAKTFAEALRIGSEVYHNLKSLTKKRYGASAGNVGDEGGVAPNIQTAEEALDLIVDAIKAAGHDGKVKIGLDCASSEFFKDGKYDLDFKNPNSDKSKWLTGPQLADLYHSLMKRYPIVSIEDPFAEDDWEAWSHFFKTAGIQIVADDLTVTNPKRIATAIEKKAADALLLKVNQIGTLSESIKAAQDSFAAGWGVMVSHRSGETEDTFIADLVVGLRTGQIKTGAPARSERLAKLNQLLRIEEELGDNAVFAGENFHHGDKL

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 \(c\).

\(|P_{f(2MQU_1)}(2) \setminus P_{f(8FFN_1)}(2)|=9\), \(|P_{f(8FFN_1)}(2) \setminus P_{f(2MQU_1)}(2)|=280\). 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 1:10110000011101010011100000100010101
Pair \(Z_2\) Length of longest common subsequence
2MQU_1,8FFN_1 289 3
2MQU_1,5ENL_1 210 3
8FFN_1,5ENL_1 155 4

Newick tree

 
[
	2MQU_1:13.80,
	[
		5ENL_1:77.5,8FFN_1:77.5
	]:61.30
]

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 \((0.85)(0.8)(\frac{774 }{\log_{20} 774}-\frac{35}{\log_{20}35})=216.\)
Status Protein1 Protein2 d d1/2
Query variables 2MQU_1 8FFN_1 270 141.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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