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Parikh vectors
2LAR_1 1VFH_1 8BPU_1 Letter Amino acid
1 66 12 A Alanine
0 30 11 R Arginine
0 6 14 N Asparagine
0 10 3 Q Glutamine
0 4 3 F Phenylalanine
0 22 2 P Proline
0 8 6 W Tryptophan
0 19 7 D Aspartic acid
2 4 8 C Cysteine
0 16 1 H Histidine
0 13 10 S Serine
3 22 7 T Threonine
2 39 12 G Glycine
0 5 6 K Lycine
0 9 3 Y Tyrosine
0 27 2 E Glutamic acid
0 15 6 I Isoleucine
0 32 8 L Leucine
0 7 2 M Methionine
0 32 6 V Valine 

2LAR_1|Chain A|DNA_(5'-D(*AP*TP*GP*GP*TP*BGR*CP*TP*C)-3')_|
>1VFH_1|Chain A|alanine racemase|Streptomyces lavendulae (1914)
>8BPU_1|Chain A[auth AAA]|Lysozyme|Gallus gallus (9031)
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 \(w=f(c)\)
2LAR , Knot 7 9 0.57 10 8 7 
ATGGTXCTC
1VFH , Knot 154 386 0.79 40 185 348 
MNETPTRVYAEIDLDAVRANVRALRARAPRSALMAVVKSNAYGHGAVPCARAAQEAGAAWLGTATPEEALELRAAGIQGRIMCWLWTPGGPWREAIETDIDVSVSGMWALDEVRAAARAAGRTARIQLKADTGLGRNGCQPADWAELVGAAVAAQAEGTVQVTGVWSHFACADEPGHPSIRLQLDAFRDMLAYAEKEGVDPEVRHIANSPATLTLPETHFDLVRTGLAVYGVSPSPELGTPAQLGLRPAMTLRASLALVKTVPAGHGVSYGHHYVTESETHLALVPAGYADGIPRNASGRGPVLVAGKIRRAAGRIAMDQFVVDLGEDLAEAGDEAVILGDAERGEPTAEDWAQAAHTIAYEIVTRIGGRVPRVYLGGLEHHHHHH
8BPU , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL

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 \(c\).

\(|P_{f(2LAR_1)}(2) \setminus P_{f(1VFH_1)}(2)|=4\), \(|P_{f(1VFH_1)}(2) \setminus P_{f(2LAR_1)}(2)|=181\). 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 1:101100000
Pair \(Z_2\) Length of longest common subsequence
2LAR_1,1VFH_1 185 2
2LAR_1,8BPU_1 108 2
1VFH_1,8BPU_1 183 3

Newick tree

 
[
	1VFH_1:10.55,
	[
		2LAR_1:54,8BPU_1:54
	]:47.55
]

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 \((0.85)(0.8)(\frac{395 }{\log_{20} 395}-\frac{9}{\log_{20}9})=126.\)
Status Protein1 Protein2 d d1/2
Query variables 2LAR_1 1VFH_1 152 78
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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