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Parikh vectors
2YGW_1 5KJT_1 5DKK_1 Letter Amino acid
29 23 6 R Arginine
16 15 10 N Asparagine
7 3 3 C Cysteine
21 36 5 P Proline
20 24 9 T Threonine
36 33 11 G Glycine
13 12 0 H Histidine
60 44 10 L Leucine
19 17 5 K Lycine
8 12 2 M Methionine
19 25 9 F Phenylalanine
33 30 6 S Serine
38 32 14 A Alanine
11 31 12 D Aspartic acid
17 23 6 I Isoleucine
8 6 1 W Tryptophan
32 27 11 V Valine
24 12 12 Q Glutamine
35 14 9 E Glutamic acid
14 14 4 Y Tyrosine 

2YGW_1|Chains A, B|MALONYL-COA DECARBOXYLASE, MITOCHONDRIAL|HOMO SAPIENS (9606)
>5KJT_1|Chain A|Shikimate O-hydroxycinnamoyltransferase|Arabidopsis thaliana (3702)
>5DKK_1|Chains A, B|LOV domain|Phaeodactylum tricornutum (2850)
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 \(w=f(c)\)
2YGW , Knot 187 460 0.83 40 229 438 
GTENLYFQSMDELLRRAVPPTPAYELRAATPAPAEGQCADFVSFYGGLAETAQRAELLGRLARGFGVDHGQVAEQSAGVLHLRQQQREAAVLLQAEDRLRYALVPRYRGLFHHISKLDGGVRFLVQLRADLLEAQALKLVEGPDVREMNGVLKGMLSEWFSSGFLNLERVTWHSPCEVLQKISEAEAVHPVKNWMDMKRRVGPYRRCYFFSHCSTPGEPLVVLHVALTGDISSNIQAIVKEHPPSETAAANKITAAIFYSISLTQQGLQGVELGTFLIKRVVKELQREFPHLGVFSSLSPIPGFTKWLLGLLNSQTKEHGRNELFTDSECKEISEITGGPINETLKLLLSSSEWVQSEKLVRALQTPLMRLCAWYLYGEKHRGYALNPVANFHLQNGAVLWRINWMADVSLRGITGSCGLMANYRYFLEETGPNSTSYLGSKIIKASEQVLSLVAQFQKN
5KJT , Knot 181 433 0.84 40 233 416 
MKINIRDSTMVRPATETPITNLWNSNVDLVIPRFHTPSVYFYRPTGASNFFDPQVMKEALSKALVPFYPMAGRLKRDDDGRIEIDCNGAGVLFVVADTPSVIDDFGDFAPTLNLRQLIPEVDHSAGIHSFPLLVLQVTFFKCGGASLGVGMQHHAADGFSGLHFINTWSDMARGLDLTIPPFIDRTLLRARDPPQPAFHHVEYQPAPSMKIPLDPSKSGPENTTVSIFKLTRDQLVALKAKSKEDGNTVSYSSYEMLAGHVWRSVGKARGLPNDQETKLYIATDGRSRLRPQLPPGYFGNVIFTATPLAVAGDLLSKPTWYAAGQIHDFLVRMDDNYLRSALDYLEMQPDLSALVRGAHTYKCPNLGITSWVRLPIYDADFGWGRPIFMGPGGIPYEGLSFVLPSPTNDGSLSVAIALQSEHMKLFEKFLFEI
5DKK , Knot 70 145 0.80 38 117 139 
GAMGDFSFIKALQTAQQNFVVTDPSLPDNPIVYASQGFLNLTGYSLDQILGRNCRFLQGPETDPKAVERIRKAIEQGNDMSVCLLNYRVDGTTFWNQFFIAALRDAGGNVTNFVGVQCKVSDQYAATVTKQQEEEEEAAANDDED

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 \(c\).

\(|P_{f(2YGW_1)}(2) \setminus P_{f(5KJT_1)}(2)|=78\), \(|P_{f(5KJT_1)}(2) \setminus P_{f(2YGW_1)}(2)|=82\). 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 1:1000101001001100111101100101101111010010110101111001001011101101111001011000111101000000111110100010011110001110010010111011101010110101101101101001011101110011001110100101001001100100101101100110100011100000110000011011111011101010001011100011000111001011110010100011011011011100110010001101111001011111001111110000000100011000000010010111100010111000011000011011001110101101010000101101110101001111101011101010110100111100001100011000001100110100011011101000
Pair \(Z_2\) Length of longest common subsequence
2YGW_1,5KJT_1 160 4
2YGW_1,5DKK_1 190 5
5KJT_1,5DKK_1 186 3

Newick tree

 
[
	5DKK_1:98.23,
	[
		2YGW_1:80,5KJT_1:80
	]:18.23
]

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 \((0.85)(0.8)(\frac{893 }{\log_{20} 893}-\frac{433}{\log_{20}433})=122.\)
Status Protein1 Protein2 d d1/2
Query variables 2YGW_1 5KJT_1 158 153.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]