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Parikh vectors
7ALH_1 5NXX_1 4MNP_1 Letter Amino acid
11 3 5 R Arginine
12 0 0 C Cysteine
26 18 16 G Glycine
11 32 39 K Lycine
10 10 14 M Methionine
13 9 10 P Proline
24 19 20 T Threonine
21 10 18 N Asparagine
14 9 12 Q Glutamine
16 19 11 S Serine
9 17 26 E Glutamic acid
29 16 27 L Leucine
27 15 17 V Valine
17 27 33 A Alanine
17 18 13 D Aspartic acid
7 4 10 H Histidine
11 19 10 I Isoleucine
17 3 17 F Phenylalanine
3 8 0 W Tryptophan
11 12 14 Y Tyrosine 

7ALH_1|Chain A|3C-like proteinase|Severe acute respiratory syndrome coronavirus 2 (2697049)
>5NXX_1|Chains A[auth C], B[auth D]|Glycine betaine ABC transport system glycine betaine-binding protein OpuAC|Bacillus subtilis (1423)
>4MNP_1|Chain A|N-acetylneuraminate-binding protein|Fusobacterium nucleatum subsp. nucleatum (190304)
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 \(w=f(c)\)
7ALH , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ
5NXX , Knot 119 268 0.82 38 170 259 
DENASAAEQVNKTIIGIDPGSGIMSLTDKAMKDYDLNDWTLISASSAAMTATLKKSYDRKKPIIITGWTPHWMFSRYKLKYLDDPKQSYGSAEEIHTITRKGFSKEQPNAAKLLSQFKWTQDEMGEIMIKVEEGEKPAKVAAEYVNKHKDQIAEWTKGVQKVKGDKINLAYVAWDSEIASTNVIGKVLEDLGYEVTLTQVEAGPMWTAIATGSADASLSAWLPNTHKAYAAKYKGKYDDIGTSMTGVKMGLVVPQYMKNVNSIEDLKK
4MNP , Knot 131 312 0.80 36 176 290 
KYNLKMGMTAGTSQNEYKAAEVFAKELKKRSNGEIELKLYPNAQLGKDDLAMMQQLEGGALDFTFAETGRFSTFFPEAEVFTLPYMIKDFNHMKKAVNTKFGKDLFKKVHDKKGMTVLAQAYNGTRQTTSNKAIKSLADMKGMKLRVPGAAANLAYAKYTEAAPTPMAFSEVYLALQTNAVDGQENPLSTIKAQKFYEVQKYLAMTNHILNDQLYLVSNITMEELPENLQKVVKESAEVAAEYHTKLFMDEEKSLKDFFKSKGVTITEPNLVDFKKAMKPFYDEYIKKNGKVGENAIKAIEAVRLEHHHHHH

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 \(c\).

\(|P_{f(7ALH_1)}(2) \setminus P_{f(5NXX_1)}(2)|=108\), \(|P_{f(5NXX_1)}(2) \setminus P_{f(7ALH_1)}(2)|=82\). 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 1:011001111010101011010010001011110011001001100000110100001110000001110110101011100100011010100101001000110101100101110001010110001101010101011010010111010000101001001011011011001010101111000010111000010101111101111010011100100010010111100000110000101111101001111101010100110011010011101110001011011000011010
Pair \(Z_2\) Length of longest common subsequence
7ALH_1,5NXX_1 190 3
7ALH_1,4MNP_1 162 3
5NXX_1,4MNP_1 158 5

Newick tree

 
[
	7ALH_1:91.16,
	[
		4MNP_1:79,5NXX_1:79
	]:12.16
]

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 \((0.85)(0.8)(\frac{574 }{\log_{20} 574}-\frac{268}{\log_{20}268})=86.4\)
Status Protein1 Protein2 d d1/2
Query variables 7ALH_1 5NXX_1 112 103
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]