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Parikh vectors
8BEC_1 1DSF_1 3QXZ_1 Letter Amino acid
14 4 21 R Arginine
13 6 12 D Aspartic acid
13 6 9 Q Glutamine
8 6 8 F Phenylalanine
9 6 18 P Proline
4 1 3 W Tryptophan
4 3 2 C Cysteine
9 3 14 E Glutamic acid
36 16 12 S Serine
15 2 50 A Alanine
38 10 23 G Glycine
0 3 7 H Histidine
10 4 11 I Isoleucine
11 3 2 Y Tyrosine
5 2 5 N Asparagine
21 13 26 L Leucine
7 6 2 K Lycine
3 1 6 M Methionine
15 8 16 T Threonine
16 9 18 V Valine 

8BEC_1|Chains A, C|pT1375 single-chain Fv|Homo sapiens (9606)
>1DSF_1|Chain A[auth L]|ANTICANCER ANTIBODY B1|Mus musculus (10090)
>3QXZ_1|Chains A, B, C|enoyl-CoA hydratase/isomerase|Mycobacterium abscessus (561007)
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 \(w=f(c)\)
8BEC , Knot 102 251 0.74 38 136 213 
EVQLVESGGGLIQPGGSLRLSCAASGLIVSRNYMNWVRQAPGKGLEWVSVLYSGGSTFYADSVKGRFTISRDNSKNTLYLQMNSLRAEDTAVYFCVRDLDVKGAFDIWGQGTMVSVSSGTGGSGGGGSGGGGSGGGASEIVLTQSPGTLSLSPGERATLSCRASQSVSSSYLAWYQQKPGQAPRLLIYGASSRATGIPDRFSGSVSGTDFTLTISRLEPEDFAVYYCQQYGSSPITFGQGTRLEIKDDDDK
1DSF , Knot 57 112 0.80 40 94 106 
DVVMTQTPLSLPVSLGDQASISCRSSQNLVHSDGKTYLHWFLQKPGQSPTLLIYKVSNRFSGVPDRFSGSGSGTDFILKISRVEAEDLGVYFCSQSTHVPLTFGCGTKLELK
3QXZ , Knot 114 265 0.80 40 150 248 
GPGSMVTELHEEIRDGVAVLTLHGPSTRNSFTVELGRQLGAAYQRLDDDPAVRVIVLTGAPPAFCSGAQISAAAETFAAPRNPDFSASPVQPAAFELRTPVIAAVNGHAIGIGMTLALHADIRILAEEGRYAIPQVRFGVAPDALAHWTLPRLVGTAVAAELLLTGASFSAQRAVETGLANRCLPAGKVLGAALRMAHDIATNVAPESAALTKRLLWDAQMTGMSAAEVAARETADHLRLMGSQDAAEGPRAFIDGRPPRWAGQR

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 \(c\).

\(|P_{f(8BEC_1)}(2) \setminus P_{f(1DSF_1)}(2)|=88\), \(|P_{f(1DSF_1)}(2) \setminus P_{f(8BEC_1)}(2)|=46\). 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 1:01011001111101110101001101111000010110011101101101100110010100101010100000000101010010100011010100101011101110101101001011011110111101111001110001101010110010100010001000011100001101101110110001011100101010100101010010100111000000100110110100101000000
Pair \(Z_2\) Length of longest common subsequence
8BEC_1,1DSF_1 134 7
8BEC_1,3QXZ_1 154 4
1DSF_1,3QXZ_1 162 3

Newick tree

 
[
	3QXZ_1:82.64,
	[
		8BEC_1:67,1DSF_1:67
	]:15.64
]

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 \((0.85)(0.8)(\frac{363 }{\log_{20} 363}-\frac{112}{\log_{20}112})=77.0\)
Status Protein1 Protein2 d d1/2
Query variables 8BEC_1 1DSF_1 82 59.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]