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Parikh vectors
8IVW_1 4CAP_1 7RLZ_1 Letter Amino acid
21 14 0 H Histidine
10 11 0 M Methionine
12 11 0 C Cysteine
46 25 6 G Glycine
46 36 0 L Leucine
24 19 0 F Phenylalanine
36 23 0 I Isoleucine
26 24 0 V Valine
31 20 6 A Alanine
29 22 1 R Arginine
25 18 1 N Asparagine
33 25 0 E Glutamic acid
45 26 0 S Serine
32 25 0 T Threonine
13 13 0 W Tryptophan
26 16 0 Y Tyrosine
36 26 1 D Aspartic acid
27 18 2 Q Glutamine
27 27 0 K Lycine
38 23 1 P Proline 

8IVW_1|Chains A, D, G, J|Neuropilin-2|Homo sapiens (9606)
>4CAP_1|Chains A, B|NITRIC OXIDE SYNTHASE, BRAIN|RATTUS NORVEGICUS (10116)
>7RLZ_1|Chains A[auth P], F[auth R]|peptide from Circumsporozoite protein variant VK210|Plasmodium vivax (5855)
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 \(w=f(c)\)
8IVW , Knot 230 583 0.83 40 279 548 
HHHHHHENLYFQDPPCGGRLNSKDAGYITSPGYPQDYPSHQNCEWIVYAPEPNQKIVLNFNPHFEIEKHDCKYDFIEIRDGDSESADLLGKHCGNIAPPTIISSGSMLYIKFTSDYARQGAGFSLRYEIFKTGSEDCSKNFTSPNGTIESPGFPEKYPHNLDCTFTILAKPKMEIILQFLIFDLEHDPLQVGEGDCKYDWLDIWDGIPHVGPLIGKYCGTKTPSELRSSTGILSLTFHTDMAVAKDGFSARYYLVHQEPLENFQCNVPLGMESGRIANEQISASSTYSDGRWTPQQSRLHGDDNGWTPNLDSNKEYLQVDLRFLTMLTAIATQGAISRETQNGYYVKSYKLEVSTNGEDWMVYRHGKNHKVFQANNDATEVVLNKLHAPLLTRFVRIRPQTWHSGIALRLELFGCRVTDAPCSNMLGMLSGLIADSQISASSTQEYLWSPSAARLVSSRSGWFPRIPQAQPGEEWLQVDLGTPKTVKGVIIQGARGGDSITAVEARAFVRKFKVSYSLNGKDWEYIQDPRTQQPKLFEGNMHYDTPDIRRFDPIPAQYVRVYPERWSPAGIGMRLEVLGCDWT
4CAP , Knot 186 422 0.88 40 262 411 
CPRFLKVKNWETDVVLTDTLHLKSTLETGCTEHICMGSIMLPSQHTRKPEDVRTKDQLFPLAKEFLDQYYSSIKRFGSKAHMDRLEEVNKEIESTSTYQLKDTELIYGAKHAWRNASRCVGRIQWSKLQVFDARDCTTAHGMFNYICNHVKYATNKGNLRSAITIFPQRTDGKHDFRVWNSQLIRYAGYKQPDGSTLGDPANVQFTEICIQQGWKAPRGRFDVLPLLLQANGNDPELFQIPPELVLEVPIRHPKFDWFKDLGLKWYGLPAVSNMLLEIGGLEFSACPFSGWYMGTEIGVRDYCDNSRYNILEEVAKKMDLDMRKTSSLWKDQALVEINIAVLYSFQSDKVTIVDHHSATESFIKHMENEYRCRGGCPADWVWIVPPMSGSITPVFHQEMLNYRLTPSFEYQPDPWNTHVWKG
7RLZ , Knot 12 18 0.64 14 13 16 
GDRAAGQPAGNGAGGQAA

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 \(c\).

\(|P_{f(8IVW_1)}(2) \setminus P_{f(4CAP_1)}(2)|=81\), \(|P_{f(4CAP_1)}(2) \setminus P_{f(8IVW_1)}(2)|=64\). 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 1:0000000010100110110100001101001101000100000011101101000111010101010000000011010010000101110001011110110010110101000010011110100011001000000010010101001111000100100010111010101110111101000110110100000110110111011111100010001001000011101010001111001101000110001100100011111001011000101000000101010000101000110101000000101010110110111001110000001001000010100010011100010000110100010011100101111001101010010011110101110010011000111110111100010100000011010110110000111101101011001101011010010111101101100101101011100101000101001001001000010110101000010100101111001010100101111110101110010
Pair \(Z_2\) Length of longest common subsequence
8IVW_1,4CAP_1 145 4
8IVW_1,7RLZ_1 270 3
4CAP_1,7RLZ_1 255 2

Newick tree

 
[
	7RLZ_1:14.72,
	[
		8IVW_1:72.5,4CAP_1:72.5
	]:73.22
]

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 \((0.85)(0.8)(\frac{1005 }{\log_{20} 1005}-\frac{422}{\log_{20}422})=153.\)
Status Protein1 Protein2 d d1/2
Query variables 8IVW_1 4CAP_1 191 166.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]