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Parikh vectors
5IDR_1 7CCD_1 7UCU_1 Letter Amino acid
7 19 12 R Arginine
9 2 8 N Asparagine
26 23 25 L Leucine
4 0 9 M Methionine
14 5 26 S Serine
12 11 24 T Threonine
16 11 17 D Aspartic acid
14 2 8 Q Glutamine
1 4 2 W Tryptophan
13 9 39 V Valine
5 4 4 Y Tyrosine
23 2 30 K Lycine
5 4 18 F Phenylalanine
27 19 28 A Alanine
2 0 14 C Cysteine
15 11 21 E Glutamic acid
6 17 38 G Glycine
2 12 7 H Histidine
13 3 24 I Isoleucine
10 11 20 P Proline 

5IDR_1|Chains A, B, C, D, E, F|DsbA-like protein|Proteus mirabilis ATCC 29906 (525369)
>7CCD_1|Chains A, D[auth B]|HNHc domain-containing protein|Streptomyces pristinaespiralis (38300)
>7UCU_1|Chains A, B|Alcohol dehydrogenase E chain|Equus caballus (9796)
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 \(w=f(c)\)
5IDR , Knot 101 224 0.81 40 155 220 
SNAALNAAQEKEVRALVRDTLVSNPEILEEAIMALQTKKADEQQAQFRQALASEHDALYNDAASPRIGAKDAKLVLVSFTDYNCPYCKRFDPLLEKITEQYPDVAVIIKPLPFKGESSAKASQAVLSVWKEDPKAFLALHQRLMQKKTMLDNASIEDAMKSTNTSKIKLTDDSLKTLQNNLELSRKLGIQGTPATVIGDTILPGAVDYDQLEIIVKEQLAKVKK
7CCD , Knot 74 169 0.74 36 103 158 
PLTDTDRSEDFLRRVRGLKAARTANGPRLYQPITLLWAVGRARRGEARTLAWADTDEAIGALLKRHGARGERPRPDYPVLALHRAGLWTLEGHVGEVPTAHGDSALRNWFAEQRPVGGLAEPFHDLLHRSGHSRVSVIEALLTTYFAGLDPVPLLEDTGLYDEGHHHHH
7UCU , Knot 158 374 0.83 40 210 358 
STAGKVIKCKAAVLWEEKKPFSIEEVEVAPPKAHEVRIKMVATGICRSDDHVVSGTLVTPLPVIAGHEAAGIVESIGEGVTTVRPGDKVIPLFTPQCGKCRVCKHPEGNFCLKNDLSMPRGTMQDGTSRFTCRGKPIHHFLGTSTFSQYTVVDEISVAKIDAASPLEKVCLIGCGFSTGYGSAVKVAKVTQGSTCAVFGLGGVGLSVIMGCKAAGAARIIGVDINKDKFAKAKEVGATECVNPQDYKKPIQEVLTEMSNGGVDFSFEVIGRLDTMVTALSCCQEAYGVSVIVGVPPDSQNLSMNPMLLLSGRTWKGAIFGGFKSKDSVPKLVADFMAKKFALDPLITHVLPFEKINEGFDLLRSGESIRTILTF

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 \(c\).

\(|P_{f(5IDR_1)}(2) \setminus P_{f(7CCD_1)}(2)|=104\), \(|P_{f(7CCD_1)}(2) \setminus P_{f(5IDR_1)}(2)|=52\). 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 1:00111011000010111000110010110011111000010000101001110000110001101011100101111010000010000101110010000101111101111010001010011101100010111110001100001100101001100000001010000100100010100011101011011100111111000010111000110100
Pair \(Z_2\) Length of longest common subsequence
5IDR_1,7CCD_1 156 4
5IDR_1,7UCU_1 157 5
7CCD_1,7UCU_1 179 4

Newick tree

 
[
	7UCU_1:86.14,
	[
		5IDR_1:78,7CCD_1:78
	]:8.14
]

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 \((0.85)(0.8)(\frac{393 }{\log_{20} 393}-\frac{169}{\log_{20}169})=66.9\)
Status Protein1 Protein2 d d1/2
Query variables 5IDR_1 7CCD_1 86 74
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]