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Parikh vectors
7NLP_1 1KZM_1 3WHI_1 Letter Amino acid
35 23 45 A Alanine
12 8 12 E Glutamic acid
31 35 18 L Leucine
3 5 16 Y Tyrosine
8 6 24 K Lycine
8 4 6 M Methionine
8 26 48 S Serine
2 1 4 W Tryptophan
39 17 32 V Valine
3 8 0 C Cysteine
7 13 14 Q Glutamine
32 17 41 G Glycine
8 2 8 H Histidine
15 13 19 I Isoleucine
7 20 5 F Phenylalanine
22 25 23 T Threonine
17 20 4 R Arginine
7 27 17 N Asparagine
21 21 13 D Aspartic acid
12 17 16 P Proline 

7NLP_1|Chain A|Acetylglutamate kinase|Mycobacterium tuberculosis H37Rv (83332)
>1KZM_1|Chain A|Peroxidase C1A|Armoracia rusticana (3704)
>3WHI_1|Chains A, B|Subtilisin E|Bacillus subtilis subsp. subtilis (224308)
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 \(w=f(c)\)
7NLP , Knot 124 297 0.79 40 164 272 
GSMVSRIEALPTHIKAQVLAEALPWLKQLHGKVVVVKYGGNAMTDDTLRRAFAADMAFLRNCGIHPVVVHGGGPQITAMLRRLGIEGDFKGGFRVTTPEVLDVARMVLFGQVGRELVNLINAHGPYAVGITGEDAQLFTAVRRSVTVDGVATDIGLVGDVDQVNTAAMLDLVAAGRIPVVSTLAPDADGVVHNINADTAAAAVAEALGAEKLLMLTDIDGLYTRWPDRDSLVSEIDTGTLAQLLPTLESGMVPKVEACLRAVIGGVPSAHIIDGRVTHCVLVELFTDAGTGTKVVRG
1KZM , Knot 135 308 0.83 40 185 295 
QLTPTFYDNSCPNVSNIVRDTIVNELRSDPRIAASILSLHFEDCFVNGCDASILLDNTTSFRTEKDAFGNANSARGFPVIDRMKAAVESACPRTVSCADLLTIAAQQSVTLAGGPSWRVPLGRRDSLQAFLDLANANLPAPFFTLPQLKDSFRNVGLNRSSDLVALSGGHTFGKNQCRFIMDRLYNFSNTGLPDPTLNTTYLQTLRGLCPLNGNLSALVDFDLRTPTIFDNKYYVNLEEQKGLIQSDQELFSSPNATDTIPLVRSFANSTQTFFNAFVEAMDRMGNITPLTGTQGQIRLNCRVVNSNS
3WHI , Knot 150 365 0.80 38 180 334 
AGKSSTEKKYIVGFKQTMSAMSSAKKKDVISEKGGKVQKQFKYVNAAAATLDEKAVKELKKDPSVAYVEEDHIAHEYGKPSWLGGGSTQPAQSVPYGISQIKAPALHSQGYTGSNVKVAVIDSGIDSSHPDLNVRGGASFVPSETNPYQDGSSHGTHVAGTIAALNNSIGVLGVSPSASLYAVKVLDSTGSGQYSWIINGIEWAISNNMDVINMSLGGPTGSTALKTVVDKAVSSGIVVAAAAGNEGSSGSTSTVGYPAKYPSTIAVGAVNSSNQRASFSSAGSELDVMAPGVSIQSTLPGGTYGAYNGTAMATPHVAGAAALILSKHPTWTNAQVRDRLESTATYLGNSFYYGKGLINVQAAAQ

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 \(c\).

\(|P_{f(7NLP_1)}(2) \setminus P_{f(1KZM_1)}(2)|=63\), \(|P_{f(1KZM_1)}(2) \setminus P_{f(7NLP_1)}(2)|=84\). 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 1:101100101110010101110111110010101111001101100001001111011110001101111011110101110011101010111010010110110111110110011011010110111101001011011000101011100111110100100111101111101111001110101110010100111111011110011110010110001100001100100101101110100111101010101111111010110101000111011001101001101
Pair \(Z_2\) Length of longest common subsequence
7NLP_1,1KZM_1 147 4
7NLP_1,3WHI_1 156 4
1KZM_1,3WHI_1 161 4

Newick tree

 
[
	3WHI_1:81.08,
	[
		7NLP_1:73.5,1KZM_1:73.5
	]:7.58
]

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 \((0.85)(0.8)(\frac{605 }{\log_{20} 605}-\frac{297}{\log_{20}297})=86.1\)
Status Protein1 Protein2 d d1/2
Query variables 7NLP_1 1KZM_1 109 104.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]