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Parikh vectors
1LMK_1 1ORH_1 2CHD_1 Letter Amino acid
7 14 9 R Arginine
6 13 12 N Asparagine
10 27 9 D Aspartic acid
10 23 7 I Isoleucine
9 17 5 F Phenylalanine
34 19 10 S Serine
19 22 10 T Threonine
8 23 7 A Alanine
9 25 9 E Glutamic acid
2 12 2 M Methionine
11 7 6 Q Glutamine
9 10 4 P Proline
6 5 1 W Tryptophan
4 11 3 C Cysteine
27 16 8 G Glycine
4 10 3 H Histidine
21 24 17 L Leucine
15 27 12 K Lycine
14 18 4 Y Tyrosine
13 30 4 V Valine 

1LMK_1|Chains A, B[auth C], C[auth E], D[auth G]|ANTI-PHOSPHATIDYLINOSITOL SPECIFIC PHOSPHOLIPASE C DIABODY|Mus musculus (10090)
>1ORH_1|Chain A|Protein arginine N-methyltransferase 1|Rattus norvegicus (10116)
>2CHD_1|Chain A|RABPHILIN-3A|RATTUS NORVEGICUS (10116)
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 \(w=f(c)\)
1LMK , Knot 107 238 0.82 40 151 220 
VQLQQSGTELMKPGRSLKISCKTTGYIFSNYWIEWVKQRPGHGLEWIGKILPGGGSNTYNDKFKGKATFTADTSSNIAYMQLSSLTSEDSAVYYCARGEDYYAYWYVLDYWGQGTTVTVSSGGGGSDIELTQSPLSLPVSLGDQASISCRSSQSLVHSNGNTSLHWYLKKPGQSPKLLIYKVSTRFSGVPDRFSGSGSGTDFTLKISRVEAEDLGVYFCSQSTHVPFTFGSGTKLELK
1ORH , Knot 153 353 0.84 40 213 342 
MAAAEAANCIMEVSCGQAESSEKPNAEDMTSKDYYFDSYAHFGIHEEMLKDEVRTLTYRNSMFHNRHLFKDKVVLDVGSGTGILCMFAAKAGARKVIGIECSSISDYAVKIVKANKLDHVVTIIKGKVEEVELPVEKVDIIISEWMGYCLFYQSMLNTVLHARDKWLAPDGLIFPDRATLYVTAIEDRQYKDYKIHWWENVYGFDMSCIKDVAIKEPLVDVVDPKQLVTNACLIKEVDIYTVKVEDLTFTSPFCLQVKRNDYVHALVAYFNIEFTRCHKRTGFSTSPESPYTHWKQTVFYMEDYLTVKTGEEIFGTIGMRPNAKNNRDLDFTIDLDFKGQLCELSCSTDYRMR
2CHD , Knot 67 142 0.78 40 109 136 
GSEANSYDSDQATTLGALEFSLLYDQDNSNLQCTIIRAKGLKPMDSNGLADPYVKLHLLPGASKSNKLRTKTLRNTRNPVWNETLQYHGITEEDMQRKTLRISVCDEDKFGHNEFIGETRFSLKKLKANQRKNFNICLERVI

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 \(c\).

\(|P_{f(1LMK_1)}(2) \setminus P_{f(1ORH_1)}(2)|=52\), \(|P_{f(1ORH_1)}(2) \setminus P_{f(1LMK_1)}(2)|=114\). 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 1:1010001001101100101000001011000110110001101101110111111000000010101010100000110101001000001100010100001010110011010010100111100101000110111011001010000000110001000101010011001011100100010111001010101001010100101001110100000011101101001010
Pair \(Z_2\) Length of longest common subsequence
1LMK_1,1ORH_1 166 4
1LMK_1,2CHD_1 156 4
1ORH_1,2CHD_1 200 3

Newick tree

 
[
	1ORH_1:96.07,
	[
		1LMK_1:78,2CHD_1:78
	]:18.07
]

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 \((0.85)(0.8)(\frac{591 }{\log_{20} 591}-\frac{238}{\log_{20}238})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 1LMK_1 1ORH_1 127 104
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]