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Parikh vectors
3BNW_1 6KJF_1 2NXO_1 Letter Amino acid
10 8 9 H Histidine
11 22 17 P Proline
14 18 12 S Serine
17 46 19 G Glycine
16 17 14 F Phenylalanine
2 7 5 W Tryptophan
7 19 21 R Arginine
1 3 4 C Cysteine
11 16 8 Q Glutamine
11 17 20 E Glutamic acid
6 35 29 A Alanine
9 30 23 D Aspartic acid
14 50 37 L Leucine
8 29 17 T Threonine
6 12 6 Y Tyrosine
13 21 25 V Valine
3 10 5 N Asparagine
7 14 7 I Isoleucine
9 16 9 K Lycine
6 10 4 M Methionine 

3BNW_1|Chains A, B|Riboflavin kinase, putative|Trypanosoma brucei (5691)
>6KJF_1|Chains A, B|Pc15g00720 protein|Penicillium rubens Wisconsin 54-1255 (500485)
>2NXO_1|Chains A, B, C, D|Hypothetical protein SCO4506|Streptomyces coelicolor (1902)
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 \(w=f(c)\)
3BNW , Knot 89 181 0.85 40 138 173 
MAHHHHHHMRQTGSFQPFFLRGKVVHGKGRGGSQLGFPTANIGLDKDVMECLQPYKNLVVYGWGTVSQVPGKERESFGPYPFAASIGFNMQFDEKTLTVEPYFLHEFGWDFYGAVVKIIVLGEIRSMGSFHSLQALVDTIKSDVQFTRDMLQKPQLQEFSRHSLFESPSSTIPYFEDLPDE
6KJF , Knot 166 400 0.83 40 216 372 
HMDTTFQAAIDTGKINGAVVCATDAQGHFVYNKATGERTLLSGEKQPQQLDDVLYLASATKLITTIAALQCVEDGLLSLDGDLSSIAPELAAKYVLTGFTDDESPLDDPPARPITLKMLLTHSSGTSYHFLDPSIAKWRAQYANPENEKPRLVEEMFTYPLSFQPGTGWMYGPGLDWAGRVVERVTGGTLMEFMQKRIFDPLGITDSQFYPVTREDLRARLVDLNPSDPGALGSAVIGGGGEMNLRGRGAFGGHGLFLTGLDFVKILRSLLANDGMLLKPAAVDNMFQQHLGPEAAASHRAALASPLGPFFRVGTDPETKVGYGLGGLLTLEDVDGWYGERTLTWGGGLTLTWFIDRKNNLCGVGAIQAVLPVDGDLMADLKQTFRHDIYRKYSAWKGQQ
2NXO , Knot 125 291 0.81 40 173 274 
SLDNSRTRPRVGHIQFLNCLPLYWGLARTGTLLDFELTKDTPEKLSEQLVRGDLDIGPVTLVEFLKNADDLVAFPDIAVGCDGPVMSCVIVSQVPLDRLDGARVALGSTSRTSVRLAQLLLSERFGVQPDYYTCPPDLSLMMQEADAAVLIGDAALRANMIDGPRYGLDVHDLGALWKEWTGLPFVFAVWAARRDYAEREPVITRKVHEAFLASRNLSLEEVEKVAEQAARWEAFDEDTLAKYFTTLDFRFGAPQLEAVTEFARRVGPTTGFPADVKVELLKPEGHHHHHH

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 \(c\).

\(|P_{f(3BNW_1)}(2) \setminus P_{f(6KJF_1)}(2)|=48\), \(|P_{f(6KJF_1)}(2) \setminus P_{f(3BNW_1)}(2)|=126\). 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 1:1100000010001010111101011010101100111101011100011001010001110111010011100000111011110111010100001010101100111010111101111101001101001011100100010100011001010010000110010001101001100
Pair \(Z_2\) Length of longest common subsequence
3BNW_1,6KJF_1 174 4
3BNW_1,2NXO_1 155 6
6KJF_1,2NXO_1 163 4

Newick tree

 
[
	6KJF_1:86.44,
	[
		3BNW_1:77.5,2NXO_1:77.5
	]:8.94
]

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 \((0.85)(0.8)(\frac{581 }{\log_{20} 581}-\frac{181}{\log_{20}181})=115.\)
Status Protein1 Protein2 d d1/2
Query variables 3BNW_1 6KJF_1 146 105.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]