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Parikh vectors
3TOT_1 3NWB_1 2JZL_1 Letter Amino acid
23 16 9 A Alanine
17 9 11 R Arginine
15 10 13 E Glutamic acid
14 18 10 D Aspartic acid
8 4 8 F Phenylalanine
6 13 1 Y Tyrosine
16 15 5 V Valine
10 11 5 T Threonine
4 4 1 C Cysteine
14 13 5 Q Glutamine
13 2 5 H Histidine
21 27 6 L Leucine
4 9 1 M Methionine
11 13 3 S Serine
6 2 2 W Tryptophan
6 6 9 N Asparagine
9 15 9 G Glycine
9 9 7 I Isoleucine
9 14 2 K Lycine
11 11 2 P Proline 

3TOT_1|Chains A, B|Glutathione s-transferase protein|Ralstonia solanacearum (305)
>3NWB_1|Chain A|Catechol O-methyltransferase|Rattus norvegicus (10116)
>2JZL_1|Chain A|Cyanovirin-N homolog|Neurospora crassa (5141)
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 \(w=f(c)\)
3TOT , Knot 100 226 0.80 40 157 212 
MVMKLIGSHASPYTRKVRVVLAEKKIDYQFVLEDVWNADTQIHQFNPLGKVPCLVMDDGGALFDSRVIAEYADTLSPVARLIPPSGRERVEVRCWEALADGLLDAAVALRVEQTQRTPEQRSESWITRQHHKIDEALKAMSRGLADRTWCNGNHLTLADIAVGCALAYLDFRQPQVDWREQHANLAAFYTRIEKRPSFLETQPQAENLYFQSHHHHHHWSHPQFEK
3NWB , Knot 102 221 0.83 40 149 214 
MGDTKEQRILRYVQQNAKPGDPQSVLEAIDTYCTQKEWAMNVGDAKGQIMDAVIREYSPSLVLELGAYCGYSAVRMARLLQPGARLLTMEMNPDYAAITQQMLNFAGLQDKVTILNGASQDLIPQLKKKYDVDTLDMVFLDHWKDRYLPDTLLLEKCGLLRKGTVLLADNVIVPGTPDFLAYVRGSSSFECTHYSSYLEYMKVVDGLEKAIYQGPSSPDKS
2JZL , Knot 58 114 0.80 40 88 106 
GSHMSFHVTAEDARIEVRDNRTILFARLRREDGEWNDASYELDQIIGNNDGHFQWGGQNFTETAEDIRFHPKEGAAEQPILRARLRDCNGEFHDRDVNLTEIVENVNGEFQAKF

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 \(c\).

\(|P_{f(3TOT_1)}(2) \setminus P_{f(3NWB_1)}(2)|=85\), \(|P_{f(3NWB_1)}(2) \setminus P_{f(3TOT_1)}(2)|=77\). 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 1:1110111001010000101111000100011100110100010010111011011100111110001110010010111011110100010100101110111011111010000001000000110000001001101100111000100100101101111011101010010101000010111100010001011000101001010000000010010100
Pair \(Z_2\) Length of longest common subsequence
3TOT_1,3NWB_1 162 3
3TOT_1,2JZL_1 161 3
3NWB_1,2JZL_1 163 3

Newick tree

 
[
	3NWB_1:81.49,
	[
		3TOT_1:80.5,2JZL_1:80.5
	]:0.99
]

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 \((0.85)(0.8)(\frac{447 }{\log_{20} 447}-\frac{221}{\log_{20}221})=65.8\)
Status Protein1 Protein2 d d1/2
Query variables 3TOT_1 3NWB_1 84 81.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]