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Parikh vectors
3PHB_1 2ENN_1 1WFL_1 Letter Amino acid
15 6 1 Q Glutamine
14 5 2 H Histidine
3 1 0 W Tryptophan
18 2 4 E Glutamic acid
10 1 3 Y Tyrosine
23 7 3 A Alanine
11 2 3 N Asparagine
4 7 6 C Cysteine
13 8 8 K Lycine
16 2 6 P Proline
16 6 12 S Serine
19 1 3 D Aspartic acid
32 7 9 G Glycine
28 1 4 L Leucine
15 5 3 F Phenylalanine
17 4 1 T Threonine
23 5 1 V Valine
19 3 4 R Arginine
12 4 0 I Isoleucine
16 0 1 M Methionine 

3PHB_1|Chains A[auth E], B[auth Q], C[auth S], D[auth T], E[auth U], F[auth Y]|Purine nucleoside phosphorylase|Homo sapiens (9606)
>2ENN_1|Chain A|Protein kinase C theta type|Homo sapiens (9606)
>1WFL_1|Chain A|Zinc finger protein 216|Mus musculus (10090)
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 \(w=f(c)\)
3PHB , Knot 141 324 0.83 40 207 307 
MRGSHHHHHHGMASMTGGQQMGRDLYDDDDKDPTLMENGYTYEDYKNTAEWLLSHTKHRPQVAIICGSGLGGLTDKLTQAQIFDYSEIPNFPRSTVPGHAGRLVFGFLNGRACVMMQGRFHMYEGYPLWKVTFPVRVFHLLGVDTLVVTNAAGGLNPKFEVGDIMLIRDHINLPGFSGQNPLRGPNDERFGDRFPAMSDAYDRTMRQRALSTWKQMGEQRELQEGTYVMVAGPSFETVAECRVLQKLGADAVGMSTVPEVIVARHCGLRVFGFSLITNKVIMDYESLEKANHEEVLAAGKQAAQKLEQFVSILMASIPLPDKAS
2ENN , Knot 41 77 0.77 38 61 71 
GSSGSSGQRRGAIKQAKVHHVKCHEFTATFFPQPTFCSVCHEFVWGLNKQGYQCRQCNAAIHKKCIDKVIAKCTGSA
1WFL , Knot 38 74 0.73 36 56 66 
GSSGSSGPSSSQSEEKAPELPKPKKNRCFMCRKKVGLTGFDCRCGNLFCGLHRYSDKHNCPYDYKAEASGPSSG

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 \(c\).

\(|P_{f(3PHB_1)}(2) \setminus P_{f(2ENN_1)}(2)|=177\), \(|P_{f(2ENN_1)}(2) \setminus P_{f(3PHB_1)}(2)|=31\). 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 1:101000000011101011001100100000001011001000000000101110000001011110101111100010010110000110110001110110111111010101110101010010111010111011011110011100111110101011011110001011110100110110000110011110010000100011001001100001001001111110100110001100111011110011011110001101111011000111000010010000111110011001001101111011110010
Pair \(Z_2\) Length of longest common subsequence
3PHB_1,2ENN_1 208 3
3PHB_1,1WFL_1 197 3
2ENN_1,1WFL_1 97 7

Newick tree

 
[
	3PHB_1:11.55,
	[
		1WFL_1:48.5,2ENN_1:48.5
	]:65.05
]

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 \((0.85)(0.8)(\frac{401 }{\log_{20} 401}-\frac{77}{\log_{20}77})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 3PHB_1 2ENN_1 132 79
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]