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Parikh vectors
4DNL_1 7TUO_1 5YXY_1 Letter Amino acid
7 26 32 V Valine
8 20 21 R Arginine
0 6 4 C Cysteine
11 39 42 E Glutamic acid
8 21 28 G Glycine
4 10 13 M Methionine
8 10 14 T Threonine
5 19 22 Y Tyrosine
10 16 22 D Aspartic acid
2 8 15 H Histidine
6 10 26 I Isoleucine
13 38 45 L Leucine
15 15 30 K Lycine
4 17 39 A Alanine
8 26 15 N Asparagine
3 25 5 Q Glutamine
10 19 19 P Proline
4 10 2 W Tryptophan
4 20 15 F Phenylalanine
10 30 19 S Serine 

4DNL_1|Chain A|Protein kinase C alpha type|Homo sapiens (9606)
>7TUO_1|Chain A|Ubiquitin carboxyl-terminal hydrolase 28|Homo sapiens (9606)
>5YXY_1|Chains A, B, C|Cytosolic NiFe-hydrogenase, alpha subunit|Thermococcus kodakarensis (strain ATCC BAA-918 / JCM 12380 / KOD1) (69014)
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 \(w=f(c)\)
4DNL , Knot 71 140 0.83 38 115 135 
GHTEKRGRIYLKAEVADEKLHVTVRDAKNLIPMDPNGLSDPYVKLKLIPDPKNESKQKTKTIRSTLNPQWNESFTFKLKPSDKDRRLSVEIWDWDRTTRNDFMGSLSFGVSELMKMPASGWYKLLNQEEGEYYNVPIPEG
7TUO , Knot 166 385 0.85 40 220 371 
SPNPNDWRRVDGWPVGLKNVGNTCWFSAVIQSLFQLPEFRRLVLSYSLPQNVLENCRSHTEKRNIMFMQELQYLFALMMGSNRKFVDPSAALDLLKGAFRSSEEQQQDVSEFTHKLLDWLEDAFQLAVNVNSPRNKSENPMVQLFYGTFLTEGVREGKPFCNNETFGQYPLQVNGYRNLDECLEGAMVEGDVELLPSDHSVKYGQERWFTKLPPVLTFELSRFEFNQSLGQPEKIHNKLEFPQIIYMDRYMYGSGSGSRQVPYRLHAVLVHEGQANAGHYWAYIYNQPRQSWLKYNDISVTESSWEEVERDSYGGLRNVSAYCLMYINDKLPYFNAEAAPTESDQMSEVEALSVELKHYIQEDNWRFEQEVEEWEEEQSCKIQLN
5YXY , Knot 178 428 0.84 40 214 407 
MKNVYLPITVDHIARVEGKGGVEIVVGDDGVKEVKLNIIEGPRFFEAITLGKKLDEALAIYPRICSFCSAAHKLTAVEAAEKAIGFTPREEIQALREVLYIGDMIESHALHLYLLVLPDYLGYSGPLHMIDEYKKEMSIALDLKNLGSWMMDELGSRAIHQENAVLGGFGKLPDKSVLENMKRRLKEALPKAEYTFELFTKLEQYEEVEGPITHIAVKPRNGVYGIYGDYLKASDGNEFPSEEYREHIKEFVVEHSFAKHSHYHGKPFMVGAISRLVNNADTLYGRAKELYESYKDLLRSTNPFANNLAQALELVYFTERAIDLIDEALAKWPIRPRDEVALKDGFGVSTTEAPRGVLVYALKVENGRVSYADIITPTAFNLAMMEQHVRMMAEKHYNDDPEKLKLLAEMVVRAYDPCISCSVHVARL

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 \(c\).

\(|P_{f(4DNL_1)}(2) \setminus P_{f(7TUO_1)}(2)|=45\), \(|P_{f(7TUO_1)}(2) \setminus P_{f(4DNL_1)}(2)|=150\). 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 1:10000010101010110001010100100111101011001010101110100000000001000101010001010101000000101011010000000111010111001101110110011000010000111101
Pair \(Z_2\) Length of longest common subsequence
4DNL_1,7TUO_1 195 5
4DNL_1,5YXY_1 177 4
7TUO_1,5YXY_1 164 4

Newick tree

 
[
	4DNL_1:96.52,
	[
		5YXY_1:82,7TUO_1:82
	]:14.52
]

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 \((0.85)(0.8)(\frac{525 }{\log_{20} 525}-\frac{140}{\log_{20}140})=113.\)
Status Protein1 Protein2 d d1/2
Query variables 4DNL_1 7TUO_1 143 97
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]