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Parikh vectors
2DYI_1 4LAL_1 2BHJ_1 Letter Amino acid
17 18 31 E Glutamic acid
19 26 26 G Glycine
17 17 23 R Arginine
0 8 18 N Asparagine
4 14 16 H Histidine
8 19 26 I Isoleucine
21 43 38 L Leucine
1 14 18 K Lycine
4 17 19 F Phenylalanine
9 21 19 D Aspartic acid
3 9 21 Q Glutamine
1 12 12 M Methionine
9 27 23 P Proline
2 14 27 T Threonine
9 9 17 Y Tyrosine
13 48 26 A Alanine
0 3 9 C Cysteine
1 23 19 S Serine
1 3 13 W Tryptophan
23 31 21 V Valine 

2DYI_1|Chain A|Probable 16S rRNA-processing protein rimM|Thermus thermophilus (300852)
>4LAL_1|Chains A, B, C, D|Uracil-5-carboxylate decarboxylase|Cordyceps militaris (983644)
>2BHJ_1|Chain A|NITRIC OXIDE SYNTHASE|MUS MUSCULUS (10090)
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 \(w=f(c)\)
2DYI , Knot 73 162 0.76 36 97 148 
MRLVEIGRFGAPYALKGGLRFRGEPVVLHLERVYVEGHGWRAIEDLYRVGEELVVHLAGVTDRTLAEALVGLRVYAEVADLPPLEEGRYYYFALIGLPVYVEGRQVGEVVDILDAGAQDVLIIRGVGERLRDRAERLVPLQAPYVRVEEGSIHVDPIPGLFD
4LAL , Knot 162 376 0.85 40 211 359 
MAASTPVVVDIHTHMYPPSYIAMLEKRQTIPLVRTFPQADEPRLILLSSELAALDAALADPAAKLPGRPLSTHFASLAQKMHFMDTNGIRVSVISLANPWFDFLAPDEAPGIADAVNAEFSDMCAQHVGRLFFFAALPLSAPVDAVKASIERVKNLKYCRGIILGTSGLGKGLDDPHLLPVFEAVADAKLLVFLHPHYGLPNEVYGPRSEEYGHVLPLALGFPMETTIAVARMYMAGVFDHVRNLQMLLAHSGGTLPFLAGRIESCIVHDGHLVKTGKVPKDRRTIWTVLKEQIYLDAVIYSEVGLQAAIASSGADRLMFGTAHPFFPPIEEDVQGPWDSSRLNAQAVIKAVGEGSSDAAAVMGLNAVRVLSLKAE
2BHJ , Knot 180 422 0.86 40 251 409 
QYVRIKNWGSGEILHDTLHHKATSDFTCKSKSCLGSIMNPKSLTRGPRDKPTPLEELLPHAIEFINQYYGSFKEAKIEEHLARLEAVTKEIETTGTYQLTLDELIFATKMAWRNAPRCIGRIQWSNLQVFDARNCSTAQEMFQHICRHILYATNNGNIRSAITVFPQRSDGKHDFRLWNSQLIRYAGYQMPDGTIRGDAATLEFTQLCIDLGWKPRYGRFDVLPLVLQADGQDPEVFEIPPDLVLEVTMEHPKYEWFQELGLKWYALPAVANMLLEVGGLEFPACPFNGWYMGTEIGVRDFCDTQRYNILEEVGRRMGLETHTLASLWKDRAVTEINVAVLHSFQKQNVTIMDHHTASESFMKHMQNEYRARGGCPADWIWLVPPVSGSITPVFHQEMLNYVLSPFYYYQIEPWKTHIWQNE

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 \(c\).

\(|P_{f(2DYI_1)}(2) \setminus P_{f(4LAL_1)}(2)|=25\), \(|P_{f(4LAL_1)}(2) \setminus P_{f(2DYI_1)}(2)|=139\). 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 1:101101101111011011101010111101001010101101100100110011101111000011011111010101101111001000011111111010100110110110111001111011100100010011110110101001010101111110
Pair \(Z_2\) Length of longest common subsequence
2DYI_1,4LAL_1 164 4
2DYI_1,2BHJ_1 212 4
4LAL_1,2BHJ_1 182 4

Newick tree

 
[
	2BHJ_1:10.77,
	[
		2DYI_1:82,4LAL_1:82
	]:21.77
]

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 \((0.85)(0.8)(\frac{538 }{\log_{20} 538}-\frac{162}{\log_{20}162})=109.\)
Status Protein1 Protein2 d d1/2
Query variables 2DYI_1 4LAL_1 138 96
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]