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Parikh vectors
8ULH_1 7QOD_1 7LNQ_1 Letter Amino acid
14 9 5 Q Glutamine
31 10 20 G Glycine
14 31 17 L Leucine
14 5 4 P Proline
5 1 5 C Cysteine
4 14 3 H Histidine
13 10 43 K Lycine
12 7 16 Y Tyrosine
7 5 6 W Tryptophan
16 17 11 A Alanine
7 9 7 R Arginine
8 10 21 D Aspartic acid
36 13 20 S Serine
9 3 11 F Phenylalanine
19 6 12 T Threonine
26 11 20 V Valine
7 8 31 N Asparagine
9 16 23 E Glutamic acid
3 7 30 I Isoleucine
3 4 7 M Methionine 

8ULH_1|Chain A[auth H]|1G12 Fab heavy chain|Homo sapiens (9606)
>7QOD_1|Chains A, B|Guanosine polyphosphate pyrophosphohydrolases/synthetases|Corynebacterium glutamicum ATCC 13032 (196627)
>7LNQ_1|Chain A|Beta-lactamase|Clostridioides difficile (1496)
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 \(w=f(c)\)
8ULH , Knot 106 257 0.76 40 152 227 
QVQLVQSGAEVKKSGASVQVSCKASGYPFGNYGITWVRQAPGQGLEWMGWISAYNGQTYYAEKFQGRVTMTTDTSTSTGYMELRSLRSDDTAVYFCARDVPVVAAVLRDYWGQGVTVSSSSASTKGPSVFPLAPGTAALGCLVKDYFPEPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKRVEPKSCDKGSENLYFQGSWSHPQFEKGGGSGGGSGGGSWSHPQFEK
7QOD , Knot 89 196 0.80 40 136 185 
MNTLSPRLRKAMNTAAWAHRHHVRKGGGIPYVSHLYSVMYLLASVTNDEDVLIAGLLHDTLEDVPEEYNSAQLEADFGPRVRELVEELTKQPLKSWKARADAYLLHLSAGASLEAVLISTADKLHNLMSILDDLEIHGEDLWQRFNAGKEQQIWWYSEVYQISLQRLGFNELNKQLGLCVEKLLKQSALEHHHHHH
7LNQ , Knot 132 312 0.81 40 180 289 
MKRKKNFIWIAILLVGVVMVMYYAVKKHNDINQKTDKNYIKSELNKSKIKNNDKNKKESVNIVDYSDCFEGISGGAIFCNTKNKEYNIYNKELIETRRSPCSTFKIVSTLIGLEKGVINSKESVMGYDGTDYPNKNWNKNLSLEEAFKESCVWYYKKLINKVDAKSVQNILDDLKYGNCDISEWEGDLKNGKGHLNGFWLESSLQISPKEQVQTMAKIFEGDTNFKKEHINILRDIMAIDVNDANINVYGKTGTGFDEKNKRVDAWFVGMLEREGDTYYFAIKSDDSNKEITGPKVKEIAINIIKKYYSVRE

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 \(c\).

\(|P_{f(8ULH_1)}(2) \setminus P_{f(7QOD_1)}(2)|=94\), \(|P_{f(7QOD_1)}(2) \setminus P_{f(8ULH_1)}(2)|=78\). 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 1:01011001101000110101000101011100110110011101101111101001000010010101010000000010101001000001101010011111111000110110100001000110111111101111011000110110101001110011001111100011001001101100011000010010001000010001010000010001010101001010011101110111010010100
Pair \(Z_2\) Length of longest common subsequence
8ULH_1,7QOD_1 172 4
8ULH_1,7LNQ_1 172 3
7QOD_1,7LNQ_1 176 5

Newick tree

 
[
	7LNQ_1:87.33,
	[
		8ULH_1:86,7QOD_1:86
	]:1.33
]

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 \((0.85)(0.8)(\frac{453 }{\log_{20} 453}-\frac{196}{\log_{20}196})=75.2\)
Status Protein1 Protein2 d d1/2
Query variables 8ULH_1 7QOD_1 92 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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]