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Parikh vectors
6CNM_1 1DUV_1 3VKR_1 Letter Amino acid
20 14 25 Q Glutamine
32 26 48 G Glycine
15 11 34 I Isoleucine
73 36 51 L Leucine
21 15 24 S Serine
22 22 25 T Threonine
10 13 24 N Asparagine
20 22 35 K Lycine
15 9 38 P Proline
8 10 11 Y Tyrosine
17 28 47 E Glutamic acid
15 12 15 M Methionine
13 13 19 F Phenylalanine
11 4 8 W Tryptophan
39 37 36 A Alanine
11 16 34 D Aspartic acid
9 4 11 C Cysteine
14 9 21 H Histidine
33 20 46 V Valine
29 12 39 R Arginine 

6CNM_1|Chains A, B, C, D|Intermediate conductance calcium-activated potassium channel protein 4|Homo sapiens (9606)
>1DUV_1|Chains A[auth G], B[auth H], C[auth I]|ORNITHINE TRANSCARBAMOYLASE|Escherichia coli (562)
>3VKR_1|Chain A|Nitrite reductase|Nicotiana tabacum (4097)
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
1DUV , Knot 145 333 0.84 40 208 323 
SGFYHKHFLKLLDFTPAELNSLLQLAAKLKADKKSGKEEAKLTGKNIALIFEKDSTRTRCSFEVAAYDQGARVTYLGPSGSQIGHKESIKDTARVLGRMYDGIQYRGYGQEIVETLAEYASVPVWNGLTNEFHPTQLLADLLTMQEHLPGKAFNEMTLVYAGDARNNMGNSMLEAAALTGLDLRLVAPQACWPEAALVTECRALAQQNGGNITLTEDVAKGVEGADFIYTDVWVSMGEAKEKWAERIALLREYQVNSKMMQLTGNPEVKFLHCLPAFHDDQTTLGKKMAEEFGLHGGMEVTDEVFESAASIVFDQAENRMHTIKAVMVATLSK
3VKR , Knot 234 591 0.84 40 263 550 
MGHHHHHHHHHHSSGHIEGRHMFSKNAVKLHATPPSVAAPPTGAPEVAAERLEPRVEEKDGYWILKEQFRKGINPQEKVKIEKEPMKLFMENGIEELAKIPIEEIDQSKLTKDDIDVRLKWLGLFHRRKNQYGRFMMRLKLPNGVTTSAQTRYLASVIRKYGKEGCADITTRQNWQIRGVVLPDVPEILKGLAEVGLTSLQSGMDNVRNPVGNPLAGIDPEEIVDTRPYTNLLSQFITGNSRGNPAVSNLPRKWNPCVVGSHDLYEHPHINDLAYMPATKDGRFGFNLLVGGFFSAKRCDEAIPLDAWVPADDVVPVCRAILEAFRDLGFRGNRQKCRMMWLIDELGVEGFRAEVEKRMPQQQLERASPEDLVQKQWERRDYLGVHPQKQEGYSFIGLHIPVGRVQADDMDELARLADEYGSGEIRLTVEQNIIIPNIETSKIEALLKEPVLSTFSPDPPILMKGLVACTGNQFCGQAIIETKARSLKITEEVQRQVSLTKPVRMHWTGCPNTCAQVQVADIGFMGCLTRDKNGKTVEGADVFLGGRIGSDSHLGEVYKKAVPCDDLVPLVVDLLVNNFGAVPREREETED

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 \(c\).

\(|P_{f(6CNM_1)}(2) \setminus P_{f(1DUV_1)}(2)|=91\), \(|P_{f(1DUV_1)}(2) \setminus P_{f(6CNM_1)}(2)|=75\). 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_1,1DUV_1 166 4
6CNM_1,3VKR_1 151 4
1DUV_1,3VKR_1 171 5

Newick tree

 
[
	1DUV_1:86.98,
	[
		6CNM_1:75.5,3VKR_1:75.5
	]:11.48
]

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 \((0.85)(0.8)(\frac{760 }{\log_{20} 760}-\frac{333}{\log_{20}333})=116.\)
Status Protein1 Protein2 d d1/2
Query variables 6CNM_1 1DUV_1 148 129.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]