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Parikh vectors
7UMG_1 8CRY_1 1CMJ_1 Letter Amino acid
16 0 21 Q Glutamine
15 0 21 S Serine
7 0 8 M Methionine
9 0 2 W Tryptophan
15 0 8 Y Tyrosine
16 0 31 V Valine
14 0 16 R Arginine
14 0 23 D Aspartic acid
12 0 18 I Isoleucine
25 0 40 L Leucine
11 0 26 K Lycine
10 0 19 F Phenylalanine
16 1 22 T Threonine
13 1 42 A Alanine
9 0 19 N Asparagine
4 1 5 C Cysteine
21 0 25 E Glutamic acid
17 4 19 G Glycine
13 0 7 H Histidine
14 0 30 P Proline 

7UMG_1|Chain A|Major histocompatibility complex class I-related gene protein|Homo sapiens (9606)
>8CRY_1|Chains A, B|DNA (5'-D(*CP*TP*(JSP)P*(1W5)P*(1W5)P*(1WA)P*(IGU)P*(JSP)P*(IGU)P*(JSP)P*(1W5)P*(1WA)P*(1WA)P*(IGU)P*AP*G)-3')|synthetic construct (32630)
>1CMJ_1|Chain A|CYTOCHROME P450|Fusarium oxysporum (5507)
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 \(w=f(c)\)
7UMG , Knot 127 271 0.87 40 195 261 
MRTHSLRYFRLGVSDPIHGVPEFISVGYVDSHPITTYDSVTRQKEPRAPWMAENLAPDHWERYTQLLRGWQQMFKVELKRLQRHYNHSGSHTYQRMIGCELLEDGSTTGFLQYAYDGQDFLIFNKDTLSWLAVDNVAHTIKQAWEANQHELLYQKNWLEEECIAWLKRFLEYGKDTLQRTEPPLVRVNRKETFPGVTALFCKAHGFYPPEIYMTWMKNGEEIVQEIDYGDILPSGDGTYQAWASIELDPQSSNLYSCHVEHSGVHMVLQVP
8CRY , Knot 7 16 0.40 10 7 9 
CTXXXXGXGXXXXGAG
1CMJ , Knot 169 402 0.84 40 222 382 
ASGAPSFPFSRASGPEPPAEFAKLRATNPVSQVKLFDGSLAWLVTKHKDVCFVATSEKLSKVRTRQGFPELSASGKQAAKAKPTFVDMDPPEHMHQRSMVEPTFTPEAVKNLQPYIQRTVDDLLEQMKQKGCANGPVDLVKEFALPVPSYIIYTLLGVPFNDLEYLTQQNAIRTNGSSTAREASAANQELLDYLAILVEQRLVEPKDDIISKLCTEQVKPGNIDKSDAVQIAFLLLVAGNATMVNMIALGVATLAQHPDQLAQLKANPSLAPQFVEELCRYHTATALAIKRTAKEDVMIGDKLVRANEGIIASNQSANRDEEVFENPDEFNMNRKWPPQDPLGFGFGDHRCIAEHLAKAELTTVFSTLYQKFPDLKVAVPLGKINYTPLNRDVGIVDLPVIF

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 \(c\).

\(|P_{f(7UMG_1)}(2) \setminus P_{f(8CRY_1)}(2)|=195\), \(|P_{f(8CRY_1)}(2) \setminus P_{f(7UMG_1)}(2)|=7\). 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 1:1000010010111001101110110110100011000001000001011111001110010000011011001101010010000000100000011100110010001110010010011110000101111001100100110100001100001100001111001100100010000111101000001111011100101101101010110010011001001011101010001110101010000100001000110111011
Pair \(Z_2\) Length of longest common subsequence
7UMG_1,8CRY_1 202 1
7UMG_1,1CMJ_1 177 4
8CRY_1,1CMJ_1 223 2

Newick tree

 
[
	8CRY_1:11.70,
	[
		7UMG_1:88.5,1CMJ_1:88.5
	]:23.20
]

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 \((0.85)(0.8)(\frac{287 }{\log_{20} 287}-\frac{16}{\log_{20}16})=91.5\)
Status Protein1 Protein2 d d1/2
Query variables 7UMG_1 8CRY_1 124 64.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]