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Parikh vectors
3DPL_1 5DAN_1 5FDI_1 Letter Amino acid
20 15 13 A Alanine
14 16 19 D Aspartic acid
18 5 11 Q Glutamine
13 6 2 M Methionine
15 5 12 F Phenylalanine
6 4 7 W Tryptophan
27 19 9 I Isoleucine
39 18 24 K Lycine
12 11 17 P Proline
16 17 12 T Threonine
22 19 7 R Arginine
41 31 13 E Glutamic acid
21 13 17 V Valine
22 8 10 N Asparagine
2 2 1 C Cysteine
10 20 22 G Glycine
8 6 12 H Histidine
46 37 26 L Leucine
21 11 18 S Serine
9 11 8 Y Tyrosine 

3DPL_1|Chain A[auth C]|Cullin-5|Homo sapiens (9606)
>5DAN_1|Chain A|2,5-diketo-D-gluconic acid reductase|Thermotoga maritima (strain ATCC 43589 / MSB8 / DSM 3109 / JCM 10099) (243274)
>5FDI_1|Chain A|Carbonic anhydrase 2|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 \(w=f(c)\)
3DPL , Knot 159 382 0.82 40 208 363 
GSESKCPEELANYCDMLLRKTPLSKKLTSEEIEAKLKEVLKKLKYVQNKDVFMRYHKAHLTRRLILDISADSEIEENMVEWLREVGMPADYVNKLARMFQDIKVSEDLNQAFKEMHKNNKLALPADSVNIKILNAGAWSRSSEKVFVSLPTELEDLIPEVEEFYKKNHSGRKLHWHHLMSNGIITFKNEVGQYDLEVTTFQLAVLFAWNQRPREKISFENLKLATELPDAELRRTLWSLVAFPKLKRQVLLYEPQVNSPKDFTEGTLFSVNQEFSLIKNAKVQKRGKINLIGRLQLTTERMREEENEGIVQLRILRTQEAIIQIMKMRKKISNAQLQTELVEILKNMFLPQKKMIKEQIEWLIEHKYIRRDESDINTFIYMA
5DAN , Knot 121 274 0.82 40 171 261 
MLYKELGRTGEEIPALGLGTWGIGGFETPDYSRDEEMVELLKTAIKMGYTHIDTAEYYGGGHTEELIGKAIKDFRREDLFIVSKVWPTHLRRDDLLRSLENTLKRLDTDYVDLYLIHWPNPEIPLEETLSAMAEGVRQGLIRYIGVSNFDRRLLEEAISKSQEPIVCDQVKYNIEDRDPERDGLLEFCQKNGVTLVAYSPLRRTLLSEKTKRTLEEIAKNHGATIYQIMLAWLLAKPNVVAIPKAGRVEHLRENLKATEIKLSEEEMKLLDSLG
5FDI , Knot 112 260 0.79 40 176 249 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK

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 \(c\).

\(|P_{f(3DPL_1)}(2) \setminus P_{f(5DAN_1)}(2)|=98\), \(|P_{f(5DAN_1)}(2) \setminus P_{f(3DPL_1)}(2)|=61\). 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 1:1000001001100001110001100010000101010011001001000011100001010001110101000100011011001111100100110110010100010011001000001111100101011011110000001110110010011101001000000100101001100111010001100010100101111111000100010100101100110101000110111110100011100101001001001011010001011001010001010111010100001000000111010110000111011010001001010001101100111100011000101110000100000010011011
Pair \(Z_2\) Length of longest common subsequence
3DPL_1,5DAN_1 159 5
3DPL_1,5FDI_1 178 4
5DAN_1,5FDI_1 169 3

Newick tree

 
[
	5FDI_1:89.07,
	[
		3DPL_1:79.5,5DAN_1:79.5
	]:9.57
]

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 \((0.85)(0.8)(\frac{656 }{\log_{20} 656}-\frac{274}{\log_{20}274})=106.\)
Status Protein1 Protein2 d d1/2
Query variables 3DPL_1 5DAN_1 134 114
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]