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Parikh vectors
8BMS_1 3FCQ_1 3IYM_1 Letter Amino acid
19 28 49 A Alanine
10 10 25 R Arginine
10 19 10 N Asparagine
14 13 19 Q Glutamine
20 26 52 S Serine
3 3 7 W Tryptophan
10 25 25 T Threonine
29 25 22 D Aspartic acid
17 8 17 E Glutamic acid
21 36 32 G Glycine
3 8 10 H Histidine
5 2 6 M Methionine
14 8 24 P Proline
18 11 14 K Lycine
8 10 30 F Phenylalanine
26 22 35 V Valine
0 0 3 C Cysteine
21 18 8 I Isoleucine
31 16 40 L Leucine
3 28 6 Y Tyrosine 

8BMS_1|Chain A|Energy-coupling factor transporter ATP-binding protein EcfA1|Lactobacillus delbrueckii subsp. bulgaricus ATCC 11842 = JCM 1002 (390333)
>3FCQ_1|Chain A|Thermolysin|Bacillus thermoproteolyticus (1427)
>3IYM_1|Chains A, B|Capsid protein|Penicillium stoloniferum virus S (216371)
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 \(w=f(c)\)
8BMS , Knot 124 282 0.82 38 167 269 
GSDNIISFDHVTFTYPDSPRPALSDLSFAIERGSWTALIGHNGSGKSTVSKLINGLLAPDDLDKSSITVDGVKLGADTVWEVREKVGIVFQNPDNQFVGATVSDDVAFGLENRAVPRPEMLKIVAQAVADVGMADYADSEPSNLSGGQKQRVAIAGILAVKPQVIILDQSTSMLDPEGKEQILDLVRKIKEDNNLTVISITHDLEEAAGADQVLVLDDGQLLDQGKPEEIFPKVEMLKRIGLDIPFVYRLKQLLKERGIVLPDEIDDDEKLVQSLWQLNSKM
3FCQ , Knot 131 316 0.79 38 180 301 
ITGTSTVGVGRGVLGDQKNINTTYSTYYYLQDNTRGDGIFTYDAKYRTTLPGSLWADADNQFFASYDAPAVDAHYYAGVTYDYYKNVHNRLSYDGNNAAIRSSVHYSQGYNNAFWNGSEMVYGDGDGQTFIPLSGGIDVVAHELTHAVTDYTAGLIYQNESGAINEAISDIFGTLVEFYANKNPDWEIGEDVYTPGISGDSLRSMSDPAKYGDPDHYSKRYTGTQDNGGVHINSGIINKAAYLISQGGTHYGVSVVGIGRDKLGKIFYRALTQYLTPTSNFSQLRAAAVQSATDLYGSTSQEVASVKQAFDAVGVK
3IYM , Knot 175 434 0.81 40 210 403 
MSSIAPTDSVSSSGKRSKPGKRERQQARSAVGSAGGKPASASKAAAFAQGGSSDPVPMPGKYPVVFSTGAGEPTRDQEFALPVHKAFPLFGSVSDKYRRNPRYAEFRAHSEFTDGVFGTHLAVSSLLRLAQQLVHAHVNMGLPLGDFAPLASSDVRIPSALASVVNQFGEFSSPSIGTRFLLRDFEHAVSRVVFLADQLWTNGNSHHIFARSWLPMSNNDGNFKTIVASRLLEFISAGDLSILPTVLEDAVLSGEVPEAWEQVKDLLGDAPGVGQVDRRDRFDFLFKSYADVGQFTTAFTTQAASDVLTELGLPWNSPSAGHLNWQYSTKQRFTFLADTWAKLSAAYSQFFELSSGLATRQSATGSHAQMVDLTSVEGVTVLKAALALSAPEFSLAACFPPSCIFVGGLTRRVVVTTSLSVSQRATEFCQMDWR

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 \(c\).

\(|P_{f(8BMS_1)}(2) \setminus P_{f(3FCQ_1)}(2)|=63\), \(|P_{f(3FCQ_1)}(2) \setminus P_{f(8BMS_1)}(2)|=76\). 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 1:100011010010100100101110010111001010111100101000100110111110010000101011011100110100011111001000111101000111110001110101101110111011110010001001011000011111111101011110000011010100011011001000001011010001001111001111001011001010011101011001110111100100110001111100100000110011010001
Pair \(Z_2\) Length of longest common subsequence
8BMS_1,3FCQ_1 139 4
8BMS_1,3IYM_1 145 4
3FCQ_1,3IYM_1 142 4

Newick tree

 
[
	3IYM_1:72.48,
	[
		8BMS_1:69.5,3FCQ_1:69.5
	]:2.98
]

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 \((0.85)(0.8)(\frac{598 }{\log_{20} 598}-\frac{282}{\log_{20}282})=88.7\)
Status Protein1 Protein2 d d1/2
Query variables 8BMS_1 3FCQ_1 111 104.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]