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Parikh vectors
7ZWM_1 4KYW_1 6LQP_1 Letter Amino acid
0 3 0 W Tryptophan
23 8 0 Y Tyrosine
33 19 0 E Glutamic acid
5 4 0 M Methionine
38 16 0 S Serine
6 12 0 Q Glutamine
40 24 0 K Lycine
26 18 0 F Phenylalanine
26 12 0 V Valine
11 8 87 A Alanine
33 23 0 N Asparagine
24 11 0 D Aspartic acid
42 17 0 I Isoleucine
9 15 0 R Arginine
16 6 65 C Cysteine
15 13 68 G Glycine
22 11 0 T Threonine
10 5 0 H Histidine
31 21 0 L Leucine
18 11 0 P Proline 

7ZWM_1|Chains A, F|Gametocyte surface protein P45/48|Plasmodium falciparum (5833)
>4KYW_1|Chain A|Type-2 restriction enzyme DpnI|Streptococcus pneumoniae (170187)
>6LQP_1|Chain A[auth 3A]|U3 snoRNA|Saccharomyces cerevisiae S288C (559292)
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 \(w=f(c)\)
7ZWM , Knot 173 428 0.81 38 211 400 
MMLYISAKKAQVAFILYIVLVLRIISGNNDFCKPSSLNSEISGFIGYKCNFSNEGVHNLKPDMRERRSIFCTIHSYFIYDKIRLIIPKKSSSPEFKILPEKCFQKVYTDYENRVETDISELGLIEYEIEENDTNPNYNERTITISPFSPKDIEFFCFCDNTEKVISSIEGRSAMVHVRVLKYPHNILFTNLTNDLFTYLPKTYNESNFVSNVLEVELNDGELFVLACELINKKCFQEGKEKALYKSNKIIYHKNLTIFKAPFYVTSKDVNTECTCKFKNNNYKIVLKPKYEKKVIHGCNFSSNVSSKHTFTDSLDISLVDDSAHISCNVHLSEPKYNHLVGLNCPGDIIPDCFFQVYQPESEELEPSNIVYLDSQINIGDIEYYEDAEGDDKIKLFGIVGSIPKTTSFTCICKKDKKSAYMTVTIDSA
4KYW , Knot 120 257 0.86 40 168 251 
GPHMELHFNLELVETYKSNSQKARILTEDWVYRQSYCPNCGNNPLNHFENNRPVADFYCNHCSEEFELKSKKGNFSSTINDGAYATMMKRVQADNNPNFFFLTYTKNFEVNNFLVLPKQFVTPKSIIQRKPLAPTANRAGWIGCNIDLSQVPSKGRIFLVQDGQVRDPEKVTKEFKQGLFLRKSSLSSRGWTIEILNCIDKIEGSEFTLEDMYRFESDLKNIFVKNNHIKEKIRQQLQILRDKEIIEFKGRGKYRKL
6LQP , Knot 80 333 0.46 8 16 64 
GUCGACGUACUUCAUAGGAUCAUUUCUAUAGGAAUCGUCACUCUUUGACUCUUCAAAAGAGCCACUGAAUCCAACUUGGUUGAUGAGUCCCAUAACCUUUGUACCCCAGAGUGAGAAACCGAAAUUGAAUCUAAAUUAGCUUGGUCCGCAAUCCUUAGCGGUUCGGCCAUCUAUAAUUUUGAAUAAAAAUUUUGCUUUGCCGUUGCAUUUGUAGUUUUUUCCUUUGGAAGUAAUUACAAUAUUUUAUGGCGCGAUGAUCUUGACCCAUCCUAUGUACUUCUUUUUUGAAGGGAUAGGGCUCUAUGGGUGGGUACAAAUGGCAGUCUGACAAGU

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 \(c\).

\(|P_{f(7ZWM_1)}(2) \setminus P_{f(4KYW_1)}(2)|=106\), \(|P_{f(4KYW_1)}(2) \setminus P_{f(7ZWM_1)}(2)|=63\). 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 1:11101010010111110111110110100010010010001011110000100011001010100000110010001100010111100000101011100010010000000100010011110001000000100000010101101001011010000001100101001110101100100111001000110011000000011001101010010111110011000010010001100000110000101101110100001000000010000001110100000110100100010000010001010110001010001010010000111100110111001101001000010100110100010110100000101000101111110110000100100000001010101001
Pair \(Z_2\) Length of longest common subsequence
7ZWM_1,4KYW_1 169 4
7ZWM_1,6LQP_1 223 2
4KYW_1,6LQP_1 176 2

Newick tree

 
[
	6LQP_1:10.21,
	[
		7ZWM_1:84.5,4KYW_1:84.5
	]:20.71
]

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 \((0.85)(0.8)(\frac{685 }{\log_{20} 685}-\frac{257}{\log_{20}257})=119.\)
Status Protein1 Protein2 d d1/2
Query variables 7ZWM_1 4KYW_1 144 116.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]