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Parikh vectors
4YGR_1 7YVF_1 4RJK_1 Letter Amino acid
4 5 25 Q Glutamine
16 2 35 K Lycine
4 3 5 W Tryptophan
14 6 56 V Valine
9 6 28 P Proline
11 7 28 T Threonine
13 3 13 Y Tyrosine
5 4 23 N Asparagine
8 2 12 H Histidine
14 4 38 I Isoleucine
9 4 19 F Phenylalanine
10 7 62 A Alanine
15 5 23 R Arginine
20 6 41 D Aspartic acid
31 9 50 L Leucine
8 21 26 S Serine
0 2 2 C Cysteine
17 2 33 E Glutamic acid
16 12 41 G Glycine
5 0 11 M Methionine 

4YGR_1|Chain A|Hydrolase|Thermococcus onnurineus (strain NA1) (523850)
>7YVF_1|Chain A|TH027 Fab light chain|Homo sapiens (9606)
>4RJK_1|Chains A, B, C, D, E, F, G, H|Acetolactate synthase|Bacillus subtilis (1415167)
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 \(w=f(c)\)
4YGR , Knot 102 229 0.80 38 150 217 
MKAVLFDIDGTILTEEPLIMLFLPQVYDKLSRKLGISKDEARERFLSEILGRRDSYDWHDWNFFFKLFDLDLKYEELLERYPHKLQVYPDTIPTLEWLRDTGYKLGIVTSGPKYQRLKLKLTGLLDYFDVVITRDDVNAIKPEPKIFLYTIERLGVEPGEAVMVGDSLSQDVYGAKSVGMTAVWINRNGDRGYNMADYEIRTLYELRKILGGERVDKLAAALEHHHHHH
7YVF , Knot 52 110 0.74 38 83 104 
QSVLTQSPSASGTPGQRVTISCSGSRSNIGSNFVYWFHQLPGTAPKLLIHSNDQRPSGVPDRFSGSNSGTSASLAISGLRSEDEADYYCAAWDDSLSSWVFGGGTKLTVL
4RJK , Knot 225 571 0.83 40 249 534 
MLTKATKEQKSLVKNRGAELVVDCLVEQGVTHVFGIPGAKIDAVFDALQDKGPEIIVARHEQNAAFMAQAVGRLTGKPGVVLVTSGPGASNLATGLLTANTEGDPVVALAGNVIRADRLKRTHQSLDNAALFQPITKYSVEVQDVKNIPEAVTNAFRIASAGQAGAAFVSFPQDVVNEVTNTKNVRAVAAPKLGPAADDAISAAIAKIQTAKLPVVLVGMKGGRPEAIKAVRKLLKKVQLPFVETYQAAGTLSRDLEDQYFGRIGLFRNQPGDLLLEQADVVLTIGYDPIEYDPKFWNINGDRTIIHLDEIIADIDHAYQPDLELIGDIPSTINHIEHDAVKVEFAEREQKILSDLKQYMHEGEQVPADWKSDRAHPLEIVKELRNAVDDHVTVTCDIGSHAIWMSRYFRSYEPLTLMISNGMQTLGVALPWAIGASLVKPGEKVVSVSGDGGFLFSAMELETAVRLKAPIVHIVWNDSTYDMVAFQQLKKYNRTSAVDFGNIDIVKYAESFGATGLRVESPDQLADVLRQGMNAEGPVIIDVPVDYSDNINLASDKLPKEFGELMKTKAL

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 \(c\).

\(|P_{f(4YGR_1)}(2) \setminus P_{f(7YVF_1)}(2)|=110\), \(|P_{f(7YVF_1)}(2) \setminus P_{f(4YGR_1)}(2)|=43\). 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 1:1011110101011000111111110100010001110000100011001110000001001011101101010000110001001010100110101100010011110011000010101011100101110000101101010111001001110110111110010001011001110111100010010011000100100100111100100111110000000
Pair \(Z_2\) Length of longest common subsequence
4YGR_1,7YVF_1 153 4
4YGR_1,4RJK_1 163 5
7YVF_1,4RJK_1 210 4

Newick tree

 
[
	4RJK_1:99.13,
	[
		4YGR_1:76.5,7YVF_1:76.5
	]:22.63
]

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 \((0.85)(0.8)(\frac{339 }{\log_{20} 339}-\frac{110}{\log_{20}110})=70.8\)
Status Protein1 Protein2 d d1/2
Query variables 4YGR_1 7YVF_1 89 65.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]