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Parikh vectors
1PPO_1 6FDP_1 4HDR_1 Letter Amino acid
20 2 32 V Valine
11 11 16 E Glutamic acid
31 10 40 G Glycine
22 14 15 K Lycine
0 2 13 M Methionine
13 3 12 P Proline
14 4 18 S Serine
14 16 44 A Alanine
10 5 15 N Asparagine
10 6 21 I Isoleucine
4 0 1 W Tryptophan
3 5 15 D Aspartic acid
4 1 7 H Histidine
9 6 20 T Threonine
12 5 7 Y Tyrosine
11 7 15 R Arginine
7 2 6 C Cysteine
7 5 11 Q Glutamine
11 11 33 L Leucine
3 5 7 F Phenylalanine 

1PPO_1|Chain A|PROTEASE OMEGA|Carica papaya (3649)
>6FDP_1|Chain A|RNA polymerase II-associated protein 3|Homo sapiens (9606)
>4HDR_1|Chains A, C|ArsA|Sporomusa ovata (2378)
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 \(w=f(c)\)
1PPO , Knot 98 216 0.81 38 145 206 
LPENVDWRKKGAVTPVRHQGSCGSCWAFSAVATVEGINKIRTGKLVELSEQELVDCERRSHGCKGGYPPYALEYVAKNGIHLRSKYPYKAKQGTCRAKQVGGPIVKTSGVGRVQPNNEGNLLNAIAKQPVSVVVESKGRPFQLYKGGIFEGPCGTKVDHAVTAVGYGKSGGKGYILIKNSWGTAWGEKGYIRIKRAPGNSPGVCGLYKSSYYPTKN
6FDP , Knot 60 120 0.79 38 95 115 
GPHMQAISEKDRGNGFFKEGKYERAIECYTRGIAADGANALLPANRAMAYLKIQKYEEAEKDCTQAILLDGSYSKAFARRGTARTFLGKLNEAKQDFETVLLLEPGNKQAVTELSKIKKK
4HDR , Knot 150 348 0.84 40 192 325 
GGMSLLQATVAKIMRPDTVIKDQVKTKLAGVLQSAGSLGRLEDMVEQYAGITGELNPALPKPCMVVASADHGVARRVVSAYPIETTIHMTANYLISQGASANAFANFCGADMVVVDMGVAGDLSYVPGLWHRKIAYGTQDFTEGPAMTREQAIQAVETGIDIVNDRVKHGNRCFCLGEMGIGNTTSSATIVGAFTGLAPEKVTGRGTGISDSRLKTKMEIVGRALAVNKPNPQDGLDVLAKVGGFELGALAGVILGSAANRCAVVIDGLNTTAAALIANVIHPLSKEYMFASHLSGEPAHSIALRQLQLEACLELGVRLGEGIGASMVVDMLYVAIKLLNNRGGKANA

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 \(c\).

\(|P_{f(1PPO_1)}(2) \setminus P_{f(6FDP_1)}(2)|=99\), \(|P_{f(6FDP_1)}(2) \setminus P_{f(1PPO_1)}(2)|=49\). 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 1:110010100011101100010010011101110101100100101101000011000000010011011011001100110100001001001000100111111000111010100010110111001101110001011010011110110100100110111010011010111000110111001010100111001110110000001000
Pair \(Z_2\) Length of longest common subsequence
1PPO_1,6FDP_1 148 3
1PPO_1,4HDR_1 161 3
6FDP_1,4HDR_1 147 3

Newick tree

 
[
	1PPO_1:78.54,
	[
		6FDP_1:73.5,4HDR_1:73.5
	]:5.04
]

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 \((0.85)(0.8)(\frac{336 }{\log_{20} 336}-\frac{120}{\log_{20}120})=66.6\)
Status Protein1 Protein2 d d1/2
Query variables 1PPO_1 6FDP_1 81 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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]