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Parikh vectors
5NNT_1 7LYX_1 3LVV_1 Letter Amino acid
0 21 44 A Alanine
1 20 18 Q Glutamine
3 10 36 I Isoleucine
1 8 41 N Asparagine
0 3 9 C Cysteine
6 30 50 K Lycine
2 27 36 V Valine
1 26 30 T Threonine
0 11 11 W Tryptophan
5 30 34 R Arginine
2 27 54 D Aspartic acid
3 26 43 E Glutamic acid
4 63 63 L Leucine
4 38 39 F Phenylalanine
2 32 48 S Serine
0 15 30 Y Tyrosine
2 32 30 G Glycine
0 26 25 H Histidine
0 19 15 M Methionine
1 25 36 P Proline 

5NNT_1|Chains A, B|Cathelicidin antimicrobial peptide|Homo sapiens (9606)
>7LYX_1|Chain A|7-alpha-hydroxycholest-4-en-3-one 12-alpha-hydroxylase|Homo sapiens (9606)
>3LVV_1|Chain A|Glutamate--cysteine ligase|Saccharomyces cerevisiae (4932)
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 \(w=f(c)\)
5NNT , Knot 21 37 0.68 28 33 35 
LLGDFFRKSKEKIGKEFKRIVQRIKDFLRNLVPRTES
7LYX , Knot 199 489 0.84 40 248 459 
MAKKTSSRQRRPWEPPLDKGTVPWLGHAMAFRKNMFEFLKRMRTKHGDVFTVQLGGQYFTFVMDPLSFGSILKDTQRKLDFGQYAKKLVLKVFGYRSVQGDHEMIHSASTKHLRGDGLKDLNETMLDSLSFVMLTSKGWSLDASCWHEDSLFRFCYYILFTAGYLSLFGYTKDKEQDLLQAGELFMEFRKFDLLFPRFVYSLLWPREWLEVGRLQRLFHKMLSVSHSQEKEGISNWLGNMLQFLREQGVPSAMQDKFNFMMLWASQGNTGPTSFWALLYLLKHPEAIRAVREEATQVLGEARLETKQSFAFKLGALQHTPVLDSVVEETLRLRAAPTLLRLVHEDYTLKMSSGQEYLFRHGDILALFPYLSVHMDPDIHPEPTVFKYDRFLNPNGSRKVDFFKTGKKIHHYTMPWGSGVSICPGRFFALSEVKLFILLMVTHFDLELVDPDTPLPHVDPQRWGFGTMQPSHDVRFRYRLHPTEHHHHHH
3LVV , Knot 272 692 0.85 40 293 647 
MGLLALGTPLQWFESRTYNEHIRDEGIEQLLYIFQAAGKRDNDPLFWGDELEYMVVDFDDKERNSMLDVCHDKILTELNMEDSSLCEANDVSFHPEYGRYMLEATPASPYLNYVGSYVEVNMQKRRAIAEYKLSEYARQDSKNNLHVGSRSVPLTLTVFPRMGCPDFINIKDPWNHKNAASRSLFLPDEVINRHVRFPNLTASIRTRRGEKVCMNVPMYKDIATPETDDSIYDRDWFLPEDKEAKLASKPGFIYMDSMGFGMGCSCLQVTFQAPNINKARYLYDALVNFAPIMLAFSAAAPAFKGWLADQDVRWNVISGAVDDRTPKERGVAPLLPKYNKNGFGGIAKDVQDKVLEIPKSRYSSVDLFLGGSKFFNRTYNDTNVPINEKVLGRLLENDKAPLDYDLAKHFAHLYIRDPVSTFEELLNQDNKTSSNHFENIQSTNWQTLRFKPPTQQATPDKKDSPGWRVEFRPFEVQLLDFENAAYSVLIYLIVDSILTFSDNINAYIHMSKVWENMKIAHHRDAILFEKFHWKKSFRNDTDVETEDYSISEIFHNPENGIFPQFVTPILCQKGFVTKDWKELKHSSKHERLYYYLKLISDRASGELPTTAKFFRNFVLQHPDYKHDSKISKSINYDLLSTCDRLTHLDDSKGELTSFLGAEIAEYVKKNKPSIESKCDKLAAALGHHHHHH

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 \(c\).

\(|P_{f(5NNT_1)}(2) \setminus P_{f(7LYX_1)}(2)|=8\), \(|P_{f(7LYX_1)}(2) \setminus P_{f(5NNT_1)}(2)|=223\). 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 1:1110110000001100100110010011001110000
Pair \(Z_2\) Length of longest common subsequence
5NNT_1,7LYX_1 231 3
5NNT_1,3LVV_1 268 3
7LYX_1,3LVV_1 151 6

Newick tree

 
[
	5NNT_1:13.71,
	[
		7LYX_1:75.5,3LVV_1:75.5
	]:62.21
]

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 \((0.85)(0.8)(\frac{526 }{\log_{20} 526}-\frac{37}{\log_{20}37})=150.\)
Status Protein1 Protein2 d d1/2
Query variables 5NNT_1 7LYX_1 191 102.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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