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Parikh vectors
3DSB_1 3VQM_1 4ZZE_1 Letter Amino acid
12 3 29 D Aspartic acid
11 9 25 I Isoleucine
7 8 14 R Arginine
13 4 25 N Asparagine
2 3 23 Q Glutamine
12 10 25 L Leucine
6 2 10 M Methionine
8 7 40 A Alanine
2 0 4 H Histidine
14 10 26 K Lycine
6 3 25 T Threonine
15 10 12 Y Tyrosine
17 14 20 E Glutamic acid
6 9 25 G Glycine
5 2 27 F Phenylalanine
0 6 19 P Proline
5 4 23 S Serine
3 0 5 W Tryptophan
11 11 18 V Valine
2 0 1 C Cysteine 

3DSB_1|Chains A, B|Putative acetyltransferase|Clostridium difficile (272563)
>3VQM_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N|Small heat shock protein StHsp14.0|Sulfolobus tokodaii (273063)
>4ZZE_1|Chains A, B|Sugar binding protein of ABC transporter system|Bifidobacterium animalis subsp. lactis (580050)
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 \(w=f(c)\)
3DSB , Knot 73 157 0.78 38 116 148 
SNAEELIEIREARMDDLDTIAKFNYNLAKETEGKELDMDVLTKGVKALLLDERKGKYHVYTVFDKVVAQIMYTYEWSDWRNGNFLWIQSVYVDKEYRRKGIFNYLFNYIKNICDKDENIVGMRLYVEKENINAKATYESLNMYECDYNMYEYEVIHS
3VQM , Knot 59 115 0.81 34 91 113 
MYYLGKELQKRSEELSRGFYELVYPPVDMYEEGGYLVVVADLAGFNKEKIKARVSGQNELIIEAEREITEPGVKYLTQRPKYVRKVIRLPYNVAKDAEISGKYENGVLTIRIPIA
4ZZE , Knot 167 396 0.84 40 217 379 
GSHMGSGQVTLDFFQFKAEAADWFKQAAQEFEKENPDIRININNSANAQTDLRTRFVKDRVPDVITFNGDYSFGTFAASGVFHDFTDDPLVSELNEGMVNIAKNLVQTSDPAKKRLYGLPFAGNASGYIYNKDLFRKVGLDPDNPPQTWDEFIAMLKKFRDAGINPVQATLADAWTTQAPLASLAGTLVPESEYAALKSGDTTFKQIWTEPIEKEIELFKYADSEKGVTYQQGTQNFAKGTAAIIPLGTYAIPQITMVNKDIDLGFAQMPATNDASKQILTAGDDVILTMGANSRHKEQSMRFIRFLMSKKQLENYADAQSAITPLKETYFGNKALEPVRPFFESNRVADFCDHYIPSSINIGGYLQSAIMSGNVNQFIDSMQNEWNKVQARDFRK

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 \(c\).

\(|P_{f(3DSB_1)}(2) \setminus P_{f(3VQM_1)}(2)|=70\), \(|P_{f(3VQM_1)}(2) \setminus P_{f(3DSB_1)}(2)|=45\). 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 1:0010011010010100100110100011000010010101100110111100001000100110011101100001001001011110010100000001110011001001000000111101010000101010000101000000100001100
Pair \(Z_2\) Length of longest common subsequence
3DSB_1,3VQM_1 115 4
3DSB_1,4ZZE_1 187 4
3VQM_1,4ZZE_1 182 4

Newick tree

 
[
	4ZZE_1:10.22,
	[
		3DSB_1:57.5,3VQM_1:57.5
	]:43.72
]

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 \((0.85)(0.8)(\frac{272 }{\log_{20} 272}-\frac{115}{\log_{20}115})=49.4\)
Status Protein1 Protein2 d d1/2
Query variables 3DSB_1 3VQM_1 58 50.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]