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Parikh vectors
5YXL_1 3UEE_1 8XBC_1 Letter Amino acid
10 5 11 H Histidine
12 5 21 S Serine
13 7 14 T Threonine
9 6 7 R Arginine
11 7 25 D Aspartic acid
14 4 25 Q Glutamine
6 7 44 G Glycine
31 11 33 L Leucine
12 11 29 F Phenylalanine
10 10 14 P Proline
11 14 54 A Alanine
24 20 29 E Glutamic acid
15 5 19 I Isoleucine
8 3 13 M Methionine
9 2 23 V Valine
9 5 17 N Asparagine
3 6 0 C Cysteine
15 15 31 K Lycine
2 3 10 W Tryptophan
5 0 10 Y Tyrosine 

5YXL_1|Chains A, C|Bile acid receptor|Homo sapiens (9606)
>3UEE_1|Chains A, B[auth C]|Baculoviral IAP repeat-containing protein 5|Homo sapiens (9606)
>8XBC_1|Chain A|XylA|Vibrio sp. EA2 (3079860)
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 \(w=f(c)\)
5YXL , Knot 104 229 0.82 40 154 219 
ELTPDQQTLLHFIMDSYNKQRMPQEITNKILKEEFSAEENFLILTEMATNHVQVLVEFTKKLPGFQTLDHEDQIALLKGSAVEAMFLRSAEIFNKKLPSGHSDLLEERIRNSGISDEYITPMFSFYKSIGELKMTQEEYALLTAIVILSPDRQYIKDREAVEKLQEPLLDVLQKLCKIHQPENPQHFACLLGRLTELRTFNHHHAEMLMSWRVNDHKFTPLLCEIWDVQ
3UEE , Knot 71 146 0.80 38 110 138 
GSHEMGAPTLPPAWQPFLKDHRISTFKNWPFLEGCACTPERMAEAGFIHCPTENEPDLAQCFFCFAELEGWEPDDDPIEEHKKHSSGCAFLSVKKQFEELTLGEFLKLDRERAKNKIAKETNNKKKEFEETAKKVRRAIEQLAAMD
8XBC , Knot 175 429 0.82 38 213 389 
MLRNHKLSKVFSATAISLFLSSTAFADTTVRMMHIETDPNVLGVWEEIAKDFEAKNPDIKVNLEFLENEAFKAKLPTLLQSQQKPDLFYSWGGGNFQVRAESGLLEDMEGYSATLNQELSAAGMNAFKIDGKQYGAPYMVSQVGFWYNKKLFKQAGIDGESIQTWDEFLTAIEKLKAAGITPIAVGGADKWPMHFYWSYLAMRAGGQEAFAAAMQDQGDGFAGEAFVRAGEELKRLAALEPFQPGFMAAGYGESAGLFGDYKAAIHLMGDWDYNFQAQQAVDKKGVVDSDLGFMNFPVLKGGAGAGSDTLGGINGFAFAKGAKPEAAKWLEFFLNENSQTKLAEIDQIIPVAKGADKGLKNPFKQKISQTISSAQWHQVFFDQALGADVGGVVNDISVGIVNGDVTPKEAAEQVQEAWEMRLEHHHHHH

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 \(c\).

\(|P_{f(5YXL_1)}(2) \setminus P_{f(3UEE_1)}(2)|=98\), \(|P_{f(3UEE_1)}(2) \setminus P_{f(5YXL_1)}(2)|=54\). 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 1:0101000011011100000001100100011000101000111100110001011101000111100100000111101011011110010110001101000110001000110000101110100011010100000111011111010000100001100100111011001001001001001101110100100100001011101010000101110011010
Pair \(Z_2\) Length of longest common subsequence
5YXL_1,3UEE_1 152 3
5YXL_1,8XBC_1 175 4
3UEE_1,8XBC_1 175 3

Newick tree

 
[
	8XBC_1:91.01,
	[
		5YXL_1:76,3UEE_1:76
	]:15.01
]

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 \((0.85)(0.8)(\frac{375 }{\log_{20} 375}-\frac{146}{\log_{20}146})=69.2\)
Status Protein1 Protein2 d d1/2
Query variables 5YXL_1 3UEE_1 88 71.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]