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Parikh vectors
5DJE_1 8JKV_1 7GNW_1 Letter Amino acid
10 33 10 F Phenylalanine
8 34 7 P Proline
12 31 10 R Arginine
14 26 12 D Aspartic acid
1 19 5 Q Glutamine
2 15 6 H Histidine
19 30 7 K Lycine
3 8 6 M Methionine
4 24 7 Y Tyrosine
10 57 12 A Alanine
16 27 6 E Glutamic acid
4 27 17 T Threonine
4 20 0 W Tryptophan
8 48 16 V Valine
4 72 18 G Glycine
3 25 10 N Asparagine
1 14 3 C Cysteine
3 44 13 I Isoleucine
4 93 13 L Leucine
10 59 4 S Serine 

5DJE_1|Chains A, B|Zuotin|Saccharomyces cerevisiae (strain ATCC 204508 / S288c) (559292)
>8JKV_1|Chains A, B|Heparan-alpha-glucosaminide N-acetyltransferase|Homo sapiens (9606)
>7GNW_1|Chains A, B|Protease 3C|enterovirus D68 (42789)
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 \(w=f(c)\)
5DJE , Knot 67 140 0.78 40 104 133 
GSSCDFVADVPPPKKGTDYDFYEAWGPVFEAEARFSKKTPIPSLGNMDSSKKEVEQFYAFWHRFDSWRTFEFLDEDVPDDSSNRDHKRYIERKNKAARDKKKTADMARLVKLVERAVSEDPRIKMFKEEEKKEKERRKWE
8JKV , Knot 262 706 0.81 40 286 614 
MTGARASAAEQRRAGRSGQARAAERAAGMSGAGRALAALLLAASVLSAALLAPGGSSGRDAQAAPPRDLDKKRHAELKMDQALLLIHNELLWTNLTVYWKSECCYHCLFQVLVNVPQSPKAGKPSAAAASVSTQHGSILQLNDTLEEKEVCRLEYRFGEFGNYSLLVKNIHNGVSEIACDLAVNEDPVDSNLPVSIAFLIGLAVIIVISFLRLLLSLDDFNNWISKAISSRETDRLINSELGSPSRTDPLDGDVQPATWRLSALPPRLRSVDTFRGIALILMVFVNYGGGKYWYFKHASWNGLTVADLVFPWFVFIMGSSIFLSMTSILQRGCSKFRLLGKIAWRSFLLICIGIIIVNPNYCLGPLSWDKVRIPGVLQRLGVTYFVVAVLELLFAKPVPEHCASERSCLSLRDITSSWPQWLLILVLEGLWLGLTFLLPVPGCPTGYLGPGGIGDFGKYPNCTGGAAGYIDRLLLGDDHLYQHPSSAVLYHTEVAYDPEGILGTINSIVMAFLGVQAGKILLYYKARTKDILIRFTAWCCILGLISVALTKVSENEGFIPVNKNLWSLSYVTTLSSFAFFILLVLYPVVDVKGLWTGTPFFYPGMNSILVYVGHEVFENYFPFQWKLKDNQSHKEHLTQNIVATALWVLIAYILYRKKIFWKIGSGGGGSGGGGSGGGWSHPQFEKGGGSGGGSGGSAWSHPQFEK
7GNW , Knot 86 182 0.82 38 139 179 
MGPGFDFAQAIMKKNTVIARTEKGEFTMLGVYDRVAVIPTHASVGEIIYINDVETRVLDACALRDLTDTNLEITIVKLDRNQKFRDIRHFLPRCEDDYNDAVLSVHTSKFPNMYIPVGQVTNYGFLNLGGTPTHRILMYNFPTRAGQCGGVVTTTGKVIGIHVGGNGAQGFAAMLLHSYFTD

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 \(c\).

\(|P_{f(5DJE_1)}(2) \setminus P_{f(8JKV_1)}(2)|=23\), \(|P_{f(8JKV_1)}(2) \setminus P_{f(5DJE_1)}(2)|=205\). 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 1:10000111011110010000100111111010101000011101101000000100101110010010010110001100000000000100000110000001011011011001100010101100000000000010
Pair \(Z_2\) Length of longest common subsequence
5DJE_1,8JKV_1 228 3
5DJE_1,7GNW_1 167 3
8JKV_1,7GNW_1 205 3

Newick tree

 
[
	8JKV_1:11.51,
	[
		5DJE_1:83.5,7GNW_1:83.5
	]:32.01
]

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 \((0.85)(0.8)(\frac{846 }{\log_{20} 846}-\frac{140}{\log_{20}140})=197.\)
Status Protein1 Protein2 d d1/2
Query variables 5DJE_1 8JKV_1 244 146.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]