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Parikh vectors
7DXI_1 8RJX_1 6PNE_1 Letter Amino acid
31 1 24 P Proline
43 18 25 S Serine
31 14 25 T Threonine
9 0 13 W Tryptophan
37 24 24 V Valine
43 17 23 D Aspartic acid
39 10 27 E Glutamic acid
23 20 26 K Lycine
36 19 21 A Alanine
22 1 17 Y Tyrosine
34 2 19 F Phenylalanine
53 3 22 R Arginine
19 14 24 G Glycine
51 7 34 L Leucine
19 9 14 H Histidine
22 6 24 I Isoleucine
13 2 11 M Methionine
26 7 19 N Asparagine
10 0 11 C Cysteine
20 8 18 Q Glutamine 

7DXI_1|Chains A, B|FI02838p|Drosophila melanogaster (7227)
>8RJX_1|Chain A|Osmotically-inducible protein Y|Escherichia coli (562)
>6PNE_1|Chains A, B|Nitric oxide synthase, brain|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 \(w=f(c)\)
7DXI , Knot 231 581 0.84 40 265 551 
EEGQTGGFSSACTAIDAVYTWVNGSDPNFIEDIRRFDDKYDPSRFDDKNELRYSLRSLEKHAAWIRHVYIVTNGQIPSWLDLSYERVTVVPHEVLAPDPDQLPTFSSSAIETFLHRIPKLSKRFLYLNDDIFLGAPLYPEDLYTEAEGVRVYQAWMVPDCALDCPWTYIGDGACDRHCNIDACQFDGGDCSETGPASDAHVIPPSKEVLEVQPAAVPQSRVHRFPQMGLQKLFRRSSANFKDVMRHRNVSTLKELRRIVERFNKAKLMSLNPELETSSSEPQTTQRHGLRKEDFKSSTDIYSHSLIATNMLLNRAYGFKARHVLAHVGFLIDKDIVEAMQRRFHQQILDTAHQRFRAPTDLQYAFAYYSFLMSETKVMSVEEIFDEFDTDGSATWSDREVRTFLTRIYQPPLDWSAMRYFEEVVQNCTRNLGMHLKVDTVEHSTLVYERYEDSNLPTITRDLVVRCPLLAEALAANFAVRPKYNFHVSPKRTSHSNFMMLTSNLTEVVESLDRLRRNPRKFNCINDNLDANRGEDNEMVRHLLEDFYLSFFPRRSKFELPPQYRNRFESWRDFQRWKRRKR
8RJX , Knot 79 182 0.75 36 112 168 
ENNAQTTNESAGQKVDSSMNKVGNFMDDSAITAKVKAALVDHDNIKSTDISVKTDQKVVTLSGFVESQAQAEEAVKVAKGVEGVTSVSDKLHVRDAKEGSVKGYAGDTATTSEIKAKLLADDIVPSRHVKVETTDGVVQLSGTVDSQAQSDRAESIAKAVDGVKSVKNDLKTKGGGHHHHHH
6PNE , Knot 184 421 0.88 40 261 408 
CPRFLKVKNWETEVVLTDTLHLKSTLETGCTEYICMGSIMHPSQHARRPEDVATKDQLFPLAKEFIDQYYSSIKRFGSKAHMERLEEVNKEIDTTSTYQLKDTELIYGAKHAWRNASRCVGRIQWSKLQVFDARDCTTAHGMFNYICNHVKYATNKGNLRSAITIFPQRTDGKHDFRVWNSQLIRYAGYKQPDGSTLGDPANVQFTEICIQQGWKPPRGRFDVLPLLLQANGNDPELFQIPPELVLEVPIRHPKFEWFKDLGLKWYGLPAVSNMLLEIGGLEFSACPFSGWYMGTEIGVRDYCDNSRYNILEEVAKKMNLDMRKTSSLWKDQALVEINIAVLYSFQSDKVTIVDHHSATESFIKHMENEYRCRGGCPADWVWIVPPMSGSITPVFHQEMLNYRLTPSFEYQPDPWNTHVWK

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 \(c\).

\(|P_{f(7DXI_1)}(2) \setminus P_{f(8RJX_1)}(2)|=179\), \(|P_{f(8RJX_1)}(2) \setminus P_{f(7DXI_1)}(2)|=26\). 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 1:00100111001001101100110100101100100100000100100000100010010001111001011001011011010000101110011110100110100011001100110100011010001111111010010001011010011111001100110011011000000101001011000001110010111100011010111110001001101110011000010100110000100100100110010010110101010000001000000110000100000100001110011100101101001110111110001101100010001100100010110010011100011100001101001100100010101000010011001001110101100100110000001110101001000011000000001101000111001111011110111010001010100000001111000100110010010001001001000101001000011001100101011100001011100000100100100100000
Pair \(Z_2\) Length of longest common subsequence
7DXI_1,8RJX_1 205 4
7DXI_1,6PNE_1 144 4
8RJX_1,6PNE_1 201 4

Newick tree

 
[
	8RJX_1:10.58,
	[
		7DXI_1:72,6PNE_1:72
	]:37.58
]

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 \((0.85)(0.8)(\frac{763 }{\log_{20} 763}-\frac{182}{\log_{20}182})=162.\)
Status Protein1 Protein2 d d1/2
Query variables 7DXI_1 8RJX_1 210 135.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]