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Parikh vectors
1QUF_1 3WTP_1 1JSR_1 Letter Amino acid
13 21 19 R Arginine
3 1 0 C Cysteine
27 11 32 G Glycine
18 6 25 T Threonine
22 19 28 L Leucine
7 1 9 M Methionine
18 11 12 P Proline
15 1 13 Y Tyrosine
13 7 8 Q Glutamine
24 8 14 E Glutamic acid
8 6 6 H Histidine
22 4 20 I Isoleucine
12 11 18 S Serine
18 5 35 V Valine
21 12 31 A Alanine
11 1 14 N Asparagine
16 3 21 D Aspartic acid
24 7 16 K Lycine
7 6 6 F Phenylalanine
5 2 0 W Tryptophan 

1QUF_1|Chain A|FERREDOXIN-NADP+ REDUCTASE|Nostoc sp. (1168)
>3WTP_1|Chain A|Histone H3-like centromeric protein A|Homo sapiens (9606)
>1JSR_1|Chains A, B, C, D|L-asparaginase|Erwinia chrysanthemi (556)
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 \(w=f(c)\)
1QUF , Knot 138 304 0.86 40 200 297 
MTQAKAKHADVPVNLYRPNAPFIGKVISNEPLVKEGGIGIVQHIKFDLTGGNLKYIEGQSIGIIPPGVDKNGKPEKLRLYSIASTRHGDDVDDKTISLCVRQLEYKHPESGETVYGVCSTYLTHIEPGSEVKITGPVGKEMLLPDDPEANVIMLATGTGIAPMRTYLWRMFKDAERAANPEYQFKGFSWLVFGVPTTPNILYKEELEEIQQKYPDNFRLTYAISREQKNPQGGRMYIQDRVAEHADQLWQLIKNEKTHTYICGLRGMEEGIDAALSAAAAKEGVTWSDYQKDLKKAGRWHVETY
3WTP , Knot 64 143 0.74 40 99 131 
GSHMGPRRRSRKPEAPRRRSPSPTPTPGPSRRGPSLGASSHQHSRRRQGWLKEIRKLQKSTHLLIRKLPFSRLAREICVKFTRGVDFNWQAQALLALQEAAEAFLVHLFEDAYLLTLHAGRVTLFPKDVQLARRIRGLEEGLG
1JSR , Knot 138 327 0.81 36 184 310 
ADKLPNIVILATGGTIAGSAATGTQTTGYKAGALGVDTLINAVPEVKKLANVKGEQFSNMASENMTGDVVLKLSQRVNELLARDDVDGVVITHGTDTVEESAYFLHLTVKSDKPVVFVAAMRPATAISADGPMNLLEAVRVAGDKQSRGRGVMVVINDRIGSARYITKTNASTLDTFRANEEGYLGVIIGNRIYYQNRIDKLHTTRSVFDVRGLTSLPKVDILYGYQDDPEYLYDAAIQHGVKGIVYAGMGAGSVSVRGIAGMRKALEKGVVVMRSTRTGNGIVPPDEELPGLVSDSLNPAHARILLMLALTRTSDPKVIQEYFHTY

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 \(c\).

\(|P_{f(1QUF_1)}(2) \setminus P_{f(3WTP_1)}(2)|=139\), \(|P_{f(3WTP_1)}(2) \setminus P_{f(1QUF_1)}(2)|=38\). 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 1:1001010010111010010111110110001110011111100101010110100101001111111100010100101001100001001000010101001000010010010110000100101100101011110011110010101111101011111000110110010011010001011011111110010110000100100001001010011000000101101010001100100110110000000010110110011011101111001101000000100110101000
Pair \(Z_2\) Length of longest common subsequence
1QUF_1,3WTP_1 177 3
1QUF_1,1JSR_1 136 4
3WTP_1,1JSR_1 175 3

Newick tree

 
[
	3WTP_1:93.72,
	[
		1QUF_1:68,1JSR_1:68
	]:25.72
]

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 \((0.85)(0.8)(\frac{447 }{\log_{20} 447}-\frac{143}{\log_{20}143})=90.5\)
Status Protein1 Protein2 d d1/2
Query variables 1QUF_1 3WTP_1 117 83.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]