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Parikh vectors
3DKP_1 6WVN_1 1VGC_1 Letter Amino acid
18 24 1 S Serine
13 11 0 F Phenylalanine
2 7 0 W Tryptophan
2 5 1 C Cysteine
14 12 1 Q Glutamine
26 32 2 L Leucine
3 12 0 Y Tyrosine
11 19 2 V Valine
12 9 0 R Arginine
7 5 0 H Histidine
7 12 0 M Methionine
9 8 0 E Glutamic acid
13 20 2 G Glycine
22 18 1 I Isoleucine
19 18 0 K Lycine
13 10 2 P Proline
20 21 1 A Alanine
9 22 0 N Asparagine
13 18 0 D Aspartic acid
12 18 0 T Threonine 

3DKP_1|Chain A|Probable ATP-dependent RNA helicase DDX52|Homo sapiens (9606)
>6WVN_1|Chain A|2'-O-methyltransferase|Severe acute respiratory syndrome coronavirus 2 (2697049)
>1VGC_1|Chain A|GAMMA CHYMOTRYPSIN|Bos taurus (9913)
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 \(w=f(c)\)
3DKP , Knot 112 245 0.83 40 168 230 
SMKINFLRNKHKIHVQGTDLPDPIATFQQLDQEYKINSRLLQNILDAGFQMPTPIQMQAIPVMLHGRELLASAPTGSGKTLAFSIPILMQLKQPANKGFRALIISPTRELASQIHRELIKISEGTGFRIHMIHKAAVAAKKFGPKSSKKFDILVTTPNRLIYLLKQDPPGIDLASVEWLVVDESDKLFEDGKTGFRDQLASIFLACTSHKVRRAMFSATFAYDVEQWCKLNLDNVISVSIGARNS
6WVN , Knot 137 301 0.86 40 200 290 
SNASSQAWQPGVAMPNLYKMQRMLLEKCDLQNYGDSATLPKGIMMNVAKYTQLCQYLNTLTLAVPYNMRVIHFGAGSDKGVAPGTAVLRQWLPTGTLLVDSDLNDFVSDADSTLIGDCATVHTANKWDLIISDMYDPKTKNVTKENDSKEGFFTYICGFIQQKLALGGSVAIKITEHSWNADLYKLMGHFAWWTAFVTNVNASSSEAFLIGCNYLGKPREQIDGYVMHANYIFWRNTNPIQLSSYSLFDMSKFPLKLRGTAVMSLKEGQINDMILSLLSKGRLIIRENNRVVISSDVLVNN
1VGC , Knot 11 13 0.72 18 12 11 
CGVPAIQPVLSGL

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 \(c\).

\(|P_{f(3DKP_1)}(2) \setminus P_{f(6WVN_1)}(2)|=69\), \(|P_{f(6WVN_1)}(2) \setminus P_{f(3DKP_1)}(2)|=101\). 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 1:01010110000010101001101110100100000100011001101110110110101111110100111011010100111011111010011001101111010001100100011010010110101100111110011100000101110010011011000111101101011110000011001001100011011110000010011101011001001001010011010111000
Pair \(Z_2\) Length of longest common subsequence
3DKP_1,6WVN_1 170 4
3DKP_1,1VGC_1 168 2
6WVN_1,1VGC_1 196 2

Newick tree

 
[
	6WVN_1:94.16,
	[
		3DKP_1:84,1VGC_1:84
	]:10.16
]

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 \((0.85)(0.8)(\frac{546 }{\log_{20} 546}-\frac{245}{\log_{20}245})=85.7\)
Status Protein1 Protein2 d d1/2
Query variables 3DKP_1 6WVN_1 109 98
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]