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Parikh vectors
8DPZ_1 7RUJ_1 4KNS_1 Letter Amino acid
15 34 21 G Glycine
1 18 11 H Histidine
13 45 13 L Leucine
11 22 15 T Threonine
2 5 4 W Tryptophan
10 19 11 R Arginine
8 4 1 C Cysteine
6 25 15 E Glutamic acid
15 15 12 Y Tyrosine
0 13 7 M Methionine
16 21 16 S Serine
10 10 15 K Lycine
12 44 27 A Alanine
18 20 17 N Asparagine
6 14 7 Q Glutamine
12 18 16 P Proline
17 32 26 V Valine
9 15 23 D Aspartic acid
9 21 17 I Isoleucine
15 11 11 F Phenylalanine 

8DPZ_1|Chain A[auth B]|Spike protein S1|Severe acute respiratory syndrome coronavirus 2 (2697049)
>7RUJ_1|Chain A|Cysteine desulfurase|Escherichia coli (562)
>4KNS_1|Chains A, B, C, D, E, F|Multicopper oxidase type 1|Nitrosomonas europaea (228410)
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 \(w=f(c)\)
8DPZ , Knot 95 205 0.82 38 144 198 
IVRFPNITNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKKS
7RUJ , Knot 172 406 0.84 40 216 389 
MIFSVDKVRADFPVLSREVNGLPLAYLDSAASAQKPSQVIDAEAEFYRHGYAAVHRGIHTLSAQATEKMENVRKRASLFINARSAEELVFVRGTTEGIALVANSWGNSNVRAGDNIIISQMEHHANIVPWQMLCARVGAELRVIPLNPDGTLQLETLPTLFDEKTRLLAITHVSNVLGTENPLAEMITLAHQHGAKVLVDGAQAVMHHPVDVQALDCDFYVFSGHKLYGPTGIGILYVKEALLQEMPPWEGGGSMIATVSLSEGTTWTKAPWRFEAGTPNTGGIIGLGAALEYVSALGLNNIAEYEQNLMHYALSQLESVPDLTLYGPQNRLGVIAFNLGKHHAYDVGSFLDNYGIAVRTGHHCAMPLMAYYNVPAMCRASLAMYNTHEEVDRLVTGLQRIHRLLG
4KNS , Knot 128 285 0.84 40 188 276 
KTVQVTLHAVETDVAYDNKGSTYRAWTFDGKVPGPVVRVTEGDTVEFTLINDKNSKNSHSMDFHAARLDVVEDFESIKPGETKKYTFTADNPGVFFYHCGSDPMIQHIARGMYGVIIVDPKDANALPKADREYVLIQAEHYENPDDKTAMMQNKWSNVVFNGGVFKYDPVHDSEATSWLQAKPGERVRIYFVNAGPNELSSLHPIAGIWDRVYPSGNPKNVQYALQSYLIGAGDAATLDLISPVEGANAIVDHSMRHAHSGAIAVIMFTNDADPEAGRGENILIR

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 \(c\).

\(|P_{f(8DPZ_1)}(2) \setminus P_{f(7RUJ_1)}(2)|=58\), \(|P_{f(7RUJ_1)}(2) \setminus P_{f(8DPZ_1)}(2)|=130\). 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 1:1101101001011101101001101011000010001100011000101001000110100100101001010011101001001111001011000001100101011110000100011100001001100001011000100010011001001101100011100011010011100100111101011011101011000
Pair \(Z_2\) Length of longest common subsequence
8DPZ_1,7RUJ_1 188 3
8DPZ_1,4KNS_1 176 4
7RUJ_1,4KNS_1 170 3

Newick tree

 
[
	8DPZ_1:92.97,
	[
		4KNS_1:85,7RUJ_1:85
	]:7.97
]

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 \((0.85)(0.8)(\frac{611 }{\log_{20} 611}-\frac{205}{\log_{20}205})=115.\)
Status Protein1 Protein2 d d1/2
Query variables 8DPZ_1 7RUJ_1 144 108.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]