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Parikh vectors
5ZRV_1 4EYI_1 7TXN_1 Letter Amino acid
38 17 6 D Aspartic acid
9 11 3 H Histidine
71 30 6 K Lycine
31 12 5 F Phenylalanine
48 33 8 S Serine
64 40 6 V Valine
33 19 10 R Arginine
32 22 5 Q Glutamine
46 31 7 E Glutamic acid
68 22 7 G Glycine
46 20 6 I Isoleucine
19 12 2 M Methionine
63 22 9 A Alanine
13 11 3 C Cysteine
17 22 2 N Asparagine
75 52 17 L Leucine
33 18 5 P Proline
38 15 5 T Threonine
1 0 0 W Tryptophan
18 11 2 Y Tyrosine 

5ZRV_1|Chains A, C, E, G|Trifunctional enzyme subunit alpha, mitochondrial|Homo sapiens (9606)
>4EYI_1|Chain A[auth B]|DNA polymerase iota|Homo sapiens (9606)
>7TXN_1|Chains A, B|Transcriptional regulator|Nostoc sp. PCC 7120 = FACHB-418 (103690)
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 \(w=f(c)\)
5ZRV , Knot 284 763 0.82 40 266 687 
MVACRAIGILSRFSAFRILRSRGYICRNFTGSSALLTRTHINYGVKGDVAVVRINSPNSKVNTLSKELHSEFSEVMNEIWASDQIRSAVLISSKPGCFIAGADINMLAACKTLQEVTQLSQEAQRIVEKLEKSTKPIVAAINGSCLGGGLEVAISCQYRIATKDRKTVLGTPEVLLGALPGAGGTQRLPKMVGVPAALDMMLTGRSIRADRAKKMGLVDQLVEPLGPGLKPPEERTIEYLEEVAITFAKGLADKKISPKRDKGLVEKLTAYAMTIPFVRQQVYKKVEEKVRKQTKGLYPAPLKIIDVVKTGIEQGSDAGYLCESQKFGELVMTKESKALMGLYHGQVLCKKNKFGAPQKDVKHLAILGAGLMGAGIAQVSVDKGLKTILKDATLTALDRGQQQVFKGLNDKVKKKALTSFERDSIFSNLTGQLDYQGFEKADMVIEAVFEDLSLKHRVLKEVEAVIPDHCIFASNTSALPISEIAAVSKRPEKVIGMHYFSPVDKMQLLEIITTEKTSKDTSASAVAVGLKQGKVIIVVKDGPGFYTTRCLAPMMSEVIRILQEGVDPKKLDSLTTSFGFPVGAATLVDEVGVDVAKHVAEDLGKVFGERFGGGNPELLTQMVSKGFLGRKSGKGFYIYQEGVKRKDLNSDMDSILASLKLPPKSEVSSDEDIQFRLVTRFVNEAVMCLQEGILATPAEGDIGAVFGLGFPPCLGGPFRFVDLYGAQKIVDRLKKYEAAYGKQFTPCQLLADHANSPNKKFYQ
4EYI , Knot 175 420 0.84 38 228 401 
MELADVGAAASSQGVHDQVLPTPNASSRVIVHVDLDCFYAQVEMISNPELKDKPLGVQQKYLVVTCNYEARKLGVKKLMNVRDAKEKCPQLVLVNGEDLTRYREMSYKVTELLEEFSPVVERLGFDENFVDLTEMVEKRLQQLQSDELSAVTVSGHVYNNQSINLLDVLHIRLLVGSQIAAEMREAMYNQLGLTGCAGVASNKLLAKLVSGVFKPNQQTVLLPESCQHLIHSLNHIKEIPGIGYKTAKCLEALGINSVRDLQTFSPKILEKELGISVAQRIQKLSFGEDNSPVILSGPPQSFSEEDSFKKCSSEVEAKNKIEELLASLLNRVCQDGRKPHTVRLIIRRYSSEKHYGRESRQCPIPSHVIQKLGTGNYDVMTPMVDILMKLFRNMVNVKMPFHLTLLSVCFCNLKALNTAK
7TXN , Knot 58 114 0.80 38 95 112 
GSMRFLYHPDRKDISLPGVLYALGDPARLEIVRLLASKGEQCCAEFDFAIAKSTMSNHFKILRESGVVLTRKEGTQHINRLRREDLETLFPGLLDAVLRSAQPLLTCQQSAIVK

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 \(c\).

\(|P_{f(5ZRV_1)}(2) \setminus P_{f(4EYI_1)}(2)|=88\), \(|P_{f(4EYI_1)}(2) \setminus P_{f(5ZRV_1)}(2)|=50\). 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 1:1110011111001011011000101000101001110000100110101111010010001001000100010011001110001001111000110111110101111000100100100010011001000001111110100111110111000001100000011101011111111111000110111111110111010010100100111100110111111011000010010011101101110001010000111001010110111100010001000100000110111101101100110010011010000011011100000111110010110000011110001001111111111111101010011001100101011001000110110001000110010000110010101000110010111011100101000110010111100011100001111001111000100111100101100101101100000000001011111100101111100111100000111110011011001101001001000111111110110011101100110011011100111101011001100111100010110100011000010001001110101110001000001010110011001110100111101101011111111111011111011010110011001000011010010100111001001000100
Pair \(Z_2\) Length of longest common subsequence
5ZRV_1,4EYI_1 138 4
5ZRV_1,7TXN_1 213 3
4EYI_1,7TXN_1 191 4

Newick tree

 
[
	7TXN_1:10.79,
	[
		5ZRV_1:69,4EYI_1:69
	]:40.79
]

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 \((0.85)(0.8)(\frac{1183 }{\log_{20} 1183}-\frac{420}{\log_{20}420})=198.\)
Status Protein1 Protein2 d d1/2
Query variables 5ZRV_1 4EYI_1 252 194.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]