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Parikh vectors
2KHW_1 4WXG_1 4CPI_1 Letter Amino acid
7 34 16 G Glycine
4 24 3 I Isoleucine
7 32 7 L Leucine
10 26 4 K Lycine
12 20 19 T Threonine
4 12 3 Q Glutamine
11 29 5 E Glutamic acid
0 0 0 C Cysteine
8 18 2 H Histidine
5 15 2 F Phenylalanine
3 19 2 P Proline
4 15 6 Y Tyrosine
6 37 7 V Valine
8 51 18 A Alanine
3 21 7 N Asparagine
1 15 4 R Arginine
3 24 11 S Serine
2 2 6 W Tryptophan
9 23 5 D Aspartic acid
1 11 0 M Methionine 

2KHW_1|Chain A|Immunoglobulin G-binding protein G, DNA polymerase iota|Streptococcus sp. (1320)
>4WXG_1|Chains A, B[auth C]|Serine hydroxymethyltransferase|Streptococcus thermophilus (299768)
>4CPI_1|Chains A, B, C, D|STREPTAVIDIN|STREPTOMYCES AVIDINII (1895)
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 \(w=f(c)\)
2KHW , Knot 55 108 0.79 38 82 101 
MQYKLILNGKTLKGETTTEAVDAATAEKVFKQYANDNGVDGEWTYDDATKTFTVTEGSDEKITFPSDIDPQVFYELPEAVQKELLAEWKRTGSDFHIGHKLEHHHHHH
4WXG , Knot 176 428 0.83 38 213 408 
MRGSHHHHHHGSMIFDKEDYKAFDPELWNAIDAEAERQQNNIELIASENVVSKAVMAAQGTLLTNKYAEGYPGKRYYGGTAVIDVVETLAIERAKKLFGAKFANVQPHSGSQANAAVYMSLIQPGDTVMGMDLSAGGHLTHGAPVSFSGKTYNFVSYNVDKESELLDYDAILAQAKEVRPKLIVAGASAYSRIIDFAKFREIADAVGAYLMVDMAHIAGLVASGHHPSPVPYAHVTTTTTHKTLRGPRGGLILTDDEDIAKKLNSAVFPGLQGGPLEHVIAAKAVALKEALDPAFKEYGENVIKNAAAMADVFNQHPDFRVISGGTNNHLFLVDVTKVVENGKVAQNVLEEVNITLNKNSIPYEQLSPFKTSGIRVGSPAITSRGMGEAESRQIAEWMVEALENHDKPEVLERIRGDVKVLTDAFPLY
4CPI , Knot 60 127 0.76 36 89 120 
AEAGITGTWYNQLGSTFIVTAGADGALTGTYESAVGNAESRYVLTGRYDSAPATDGSGTALGWTVAWKNNYRNDHSATTWSGQYVGGAEARINTQWLLTSGTTEANAWKSTLVGHDTFTKVKPSAAS

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 \(c\).

\(|P_{f(2KHW_1)}(2) \setminus P_{f(4WXG_1)}(2)|=26\), \(|P_{f(4WXG_1)}(2) \setminus P_{f(2KHW_1)}(2)|=157\). 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 1:100011101001010000011011010011000100011010100001000101001000010110010101100110110001110100010010110010000000
Pair \(Z_2\) Length of longest common subsequence
2KHW_1,4WXG_1 183 6
2KHW_1,4CPI_1 111 4
4WXG_1,4CPI_1 192 4

Newick tree

 
[
	4WXG_1:10.43,
	[
		2KHW_1:55.5,4CPI_1:55.5
	]:47.93
]

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 \((0.85)(0.8)(\frac{536 }{\log_{20} 536}-\frac{108}{\log_{20}108})=126.\)
Status Protein1 Protein2 d d1/2
Query variables 2KHW_1 4WXG_1 159 98.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]