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Parikh vectors
6NGP_1 2AYD_1 2WMN_1 Letter Amino acid
16 6 20 Y Tyrosine
20 0 28 A Alanine
18 2 21 Q Glutamine
25 2 43 E Glutamic acid
25 6 19 G Glycine
14 5 12 H Histidine
36 3 39 L Leucine
13 1 2 W Tryptophan
24 6 29 V Valine
11 1 16 M Methionine
26 8 20 S Serine
25 5 19 T Threonine
26 5 24 D Aspartic acid
11 2 7 C Cysteine
23 6 13 P Proline
22 6 28 R Arginine
18 1 12 N Asparagine
23 3 22 I Isoleucine
27 7 34 K Lycine
19 1 20 F Phenylalanine 

6NGP_1|Chains A, B|Nitric oxide synthase, brain|Rattus norvegicus (10116)
>2AYD_1|Chain A|WRKY transcription factor 1|Arabidopsis thaliana (3702)
>2WMN_1|Chain A|DEDICATOR OF CYTOKINESIS PROTEIN 9|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 \(w=f(c)\)
6NGP , Knot 186 422 0.88 40 262 411 
CPRFLKVKNWETDVVLTDTLHLKSTLETGCTEHICMGSIMLPSQHTRKPEDVRTKDQLFPLAKEFLDQYYSSIKRFGSKAHMDRLEEVNKEIESTSTYQLKDTELIYGAKHAWRNASRCVGRIQWSKLQVFDARDCTTAHGMFNYICNHVKYATNKGNLRSAITIFPQRTDGKHDFRVWNSQLIRYAGYKQPDGSTLGDPANVQFTEICIQQGWKAPRGRFDVLPLLLQANGNDPELFQIPPELVLEVPIRHPKFDWFKDLGLKWYGLPAVSNMLLEIGGLEFSACPFSGWYMGTEIGVRDYCDNSRYNILEEVAKKMDLDMRKTSSLWKDQALVEINIAVLYSFQSDKVTIVDHHSATESFIKHMENEYRCRGGCPADWVWIVPPMSGSITPVFHQEMLNYRLTPSFEYQPDPWNTHVWKG
2AYD , Knot 41 76 0.77 38 65 74 
SRIVVHTQTLFDIVNDGYRWRKYGQKSVKGSPYPRSYYRCSSPGCPVKKHVERSSHDTKLLITTYEGKHDHDMPPG
2WMN , Knot 179 428 0.84 40 238 403 
KSYASTPELRKTWLDSMARIHVKNGDLSEAAMCYVHVTALVAEYLTRKGVFRQGCTAFRVITPNIDEEASMMEDVGMQDVHFNEDVLMELLEQCADGLWKAERYELIADIYKLIIPIYEKRRDFERLAHLYDTLHRAYSKVTEVMHSGRRLLGTYFRVAFFGQGFFEDEDGKEYIYKEPKLTPLSEISQRLLKLYSDKFGSENVKMIQDSGKVNPKDLDSKYAYIQVTHVIPFFDEKELQERKTEFERSHNIRRFMFEMPFTQTGKRQGGVEEQCKRRTILTAIHCFPYVKKRIPVMYQHHTDLNPIEVAIDEMSKKVAELRQLCSSAEVDMIKLQLKLQGSVSVQVNAGPLAYARAFLDDTNTKRYPDNKVKLLKEVFRQFVEACGQALAVNERLIKEDQLEYQEEMKANYREMAKELSEIMHEQLG

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 \(c\).

\(|P_{f(6NGP_1)}(2) \setminus P_{f(2AYD_1)}(2)|=216\), \(|P_{f(2AYD_1)}(2) \setminus P_{f(6NGP_1)}(2)|=19\). 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 1:01011010010001110001010001001000010110111100000010010000011111001100000010011001010010010001000000010000110110011001000110101001011010000010111001000100100010100110111000010001011000110011000101001101101010010100110110101011111101010010110111011101110010101100111010111110011101111010101101101100111000000000011001100101010000011000111010111100100001011000010001100100000001101101111111101010111000110001010100010110001101
Pair \(Z_2\) Length of longest common subsequence
6NGP_1,2AYD_1 235 4
6NGP_1,2WMN_1 172 4
2AYD_1,2WMN_1 217 3

Newick tree

 
[
	2AYD_1:12.77,
	[
		6NGP_1:86,2WMN_1:86
	]:34.77
]

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 \((0.85)(0.8)(\frac{498 }{\log_{20} 498}-\frac{76}{\log_{20}76})=127.\)
Status Protein1 Protein2 d d1/2
Query variables 6NGP_1 2AYD_1 170 98
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]