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Parikh vectors
6CNM_1 3ZHE_1 9LIU_1 Letter Amino acid
15 32 7 I Isoleucine
15 19 2 M Methionine
11 6 0 W Tryptophan
33 18 6 V Valine
32 16 6 G Glycine
10 22 1 N Asparagine
9 7 1 C Cysteine
17 31 7 E Glutamic acid
73 45 12 L Leucine
20 33 13 K Lycine
13 21 4 F Phenylalanine
22 17 10 T Threonine
39 20 19 A Alanine
8 16 3 Y Tyrosine
21 30 6 S Serine
20 22 8 Q Glutamine
11 23 4 D Aspartic acid
14 8 2 H Histidine
15 15 6 P Proline
29 19 18 R Arginine 

6CNM_1|Chains A, B, C, D|Intermediate conductance calcium-activated potassium channel protein 4|Homo sapiens (9606)
>3ZHE_1|Chains A, C|NONSENSE-MEDIATED MRNA DECAY PROTEIN|CAENORHABDITIS ELEGANS (6239)
>9LIU_1|Chains A, E|Histone H3|Xenopus laevis (8355)
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 \(w=f(c)\)
6CNM , Knot 173 427 0.81 40 224 397 
MGGDLVLGLGALRRRKRLLEQEKSLAGWALVLAGTGIGLMVLHAEMLWFGGCSWALYLFLVKCTISISTFLLLCLIVAFHAKEVQLFMTDNGLRDWRVALTGRQAAQIVLELVVCGLHPAPVRGPPCVQDLGAPLTSPQPWPGFLGQGEALLSLAMLLRLYLVPRAVLLRSGVLLNASYRSIGALNQVRFRHWFVAKLYMNTHPGRLLLGLTLGLWLTTAWVLSVAERQAVNATGHLSDTLWLIPITFLTIGYGDVVPGTMWGKIVCLCTGVMGVCCTALLVAVVARKLEFNKAEKHVHNFMMDIQYTKEMKESAARVLQEAWMFYKHTRRKESHAARRHQRKLLAAINAFRQVRLKHRKLREQVNSMVDISKMHMILYDLQQNLSSSHRALEKQIDTLAGKLDALTELLSTALGPRQLPEPSQQSK
3ZHE , Knot 181 420 0.86 40 243 402 
MQKSDEVTEKFKRYCNQLEKYGQTENVHSPVMAMLRRKGRKQLIEIMKRDGDCTSSINKLWIVGYYHPFQFFIRDKEKNMAIAVLLTMFCGELQEMLSLPDDKYPALWNMYIGDFHRYMPDEEIQKCLAVGYYSRAIDLDPNQGRAFHVLAGLRADLNVAQKLRLMILGQLADAPYKKGTELLEYLKFPQKESTDKLMVDFVIWALNEKSKRMDYQMTGIKIVNEFKAEIEQKLEFDWSLIMSTCRLASKLAMKKFGFQQFYNCFDTISTLYITIYSRTISSKCLLAEAISWISDSAEILGHLDEQKNEPHFQKLSVFAKTKWNELNDLVMNHINSVFTSMSLTINPSISMTSFLLNGPISEPNVEFLSQLINYLVSVEFPPMEIIHDREESGPLLRRINQSEQKRLDIQIKTQNDEVNR
9LIU , Knot 65 135 0.78 38 102 126 
ARTKQTARKSTGGKAPRKQLATKAARKSAPATGGVKKPHRYRPGTVALREIRRYQKSTELLIRKLPFQRLVREIAQDFKTDLRFQSSAVMALQEASEAYLVALFEDTNLCAIHAKRVTIMPKDIQLARRIRGERA

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 \(c\).

\(|P_{f(6CNM_1)}(2) \setminus P_{f(3ZHE_1)}(2)|=79\), \(|P_{f(3ZHE_1)}(2) \setminus P_{f(6CNM_1)}(2)|=98\). 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 1:1110111111110000011000001111111111011111110101111110011101111000101001111011111010010111000110010111010011011101110110111101110100111110010111111101011101111101011101111001111010000111100101001111010100011011111011111001111011000110101010001111110110110101111011101101001111100011111111001010010001001110100000100011011001111000000000011000000111110110010100001000100110100101110010001000001100010011101011001100111100110100000
Pair \(Z_2\) Length of longest common subsequence
6CNM_1,3ZHE_1 177 4
6CNM_1,9LIU_1 186 4
3ZHE_1,9LIU_1 205 4

Newick tree

 
[
	9LIU_1:10.79,
	[
		6CNM_1:88.5,3ZHE_1:88.5
	]:12.29
]

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 \((0.85)(0.8)(\frac{847 }{\log_{20} 847}-\frac{420}{\log_{20}420})=114.\)
Status Protein1 Protein2 d d1/2
Query variables 6CNM_1 3ZHE_1 142 140.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]