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Parikh vectors
2KFE_1 3BLE_1 2YNJ_1 Letter Amino acid
2 17 18 N Asparagine
3 28 46 E Glutamic acid
3 26 17 S Serine
0 21 33 T Threonine
0 3 0 W Tryptophan
0 18 33 D Aspartic acid
0 0 3 C Cysteine
0 23 32 I Isoleucine
5 26 39 K Lycine
0 9 14 P Proline
0 19 70 A Alanine
3 26 50 G Glycine
2 33 41 L Leucine
1 14 7 F Phenylalanine
0 6 7 Y Tyrosine
1 26 58 V Valine
2 17 22 R Arginine
0 11 16 Q Glutamine
0 11 1 H Histidine
2 3 17 M Methionine 

2KFE_1|Chain A|meucin-24|null
>3BLE_1|Chain A|Citramalate synthase from Leptospira interrogans|Leptospira interrogans (173)
>2YNJ_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N|60 KDA CHAPERONIN|ESCHERICHIA COLI UTI89 (364106)
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 \(w=f(c)\)
2KFE , Knot 15 24 0.66 20 20 21 
GRGREFMSNLKEKLSGVKEKMKNS
3BLE , Knot 143 337 0.82 38 190 327 
GSHMGRSQKVSQMTKVETRLEILDVTLRDGEQTRGVSFSTSEKLNIAKFLLQKLNVDRVEIASARVSKGELETVQKIMEWAATEQLTERIEILGFVDGNKTVDWIKDSGAKVLNLLTKGSLHHLEKQLGKTPKEFFTDVSFVIEYAIKSGLKINVYLEDWSNGFRNSPDYVKSLVEHLSKEHIERIFLPDTLGVLSPEETFQGVDSLIQKYPDIHFEFHGHNDYDLSVANSLQAIRAGVKGLHASINGLGERAGNTPLEALVTTIHDKSNSKTNINEIAITEASRLVEVFSGKRISANRPIVGEDVFTQTAGVHADGDKKGNLYANPILPERFGRKR
2YNJ , Knot 201 524 0.80 38 201 465 
AAKDVKFGNDARVKMLRGVNVLADAVKVTLGPKGRNVVLDKSFGAPTITKDGVSVAREIELEDKFENMGAQMVKEVASKANDAAGDGTTTATVLAQAIITEGLKAVAAGMNPMDLKRGIDKAVTAAVEELKALSVPCSDSKAIAQVGTISANSDETVGKLIAEAMDKVGKEGVITVEDGTGLQDELDVVEGMQFDRGYLSPYFINKPETGAVELESPFILLADKKISNIREMLPVLEAVAKAGKPLLIIAEDVEGEALATLVVNTMRGIVKVAAVKAPGFGDRRKAMLQDIATLTGGTVISEEIGMELEKATLEDLGQAKRVVINKDTTTIIDGVGEEAAIQGRVAQIRQQIEEATSDYDREKLQERVAKLAGGVAVIKVGAATEVEMKEKKARVEDALHATRAAVEEGVVAGGGVALIRVASKLADLRGQNEDQNVGIKVALRAMEAPLRQIVLNCGEEPSVVANTVKGGDGNYGYNAATEEYGNMIDMGILDPTKVTRSALQYAASVAGLMITTECMVTDLP

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 \(c\).

\(|P_{f(2KFE_1)}(2) \setminus P_{f(3BLE_1)}(2)|=6\), \(|P_{f(3BLE_1)}(2) \setminus P_{f(2KFE_1)}(2)|=176\). 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 1:101001100100010110001000
Pair \(Z_2\) Length of longest common subsequence
2KFE_1,3BLE_1 182 3
2KFE_1,2YNJ_1 193 3
3BLE_1,2YNJ_1 141 4

Newick tree

 
[
	2KFE_1:10.35,
	[
		3BLE_1:70.5,2YNJ_1:70.5
	]:29.85
]

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 \((0.85)(0.8)(\frac{361 }{\log_{20} 361}-\frac{24}{\log_{20}24})=109.\)
Status Protein1 Protein2 d d1/2
Query variables 2KFE_1 3BLE_1 138 73.5
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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