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Parikh vectors
3PAY_1 3KQY_1 8TKM_1 Letter Amino acid
12 32 18 A Alanine
16 9 14 F Phenylalanine
20 11 17 T Threonine
24 24 26 G Glycine
11 7 24 K Lycine
12 20 17 P Proline
15 8 12 Y Tyrosine
15 20 17 R Arginine
20 7 10 N Asparagine
20 14 17 D Aspartic acid
31 20 28 V Valine
23 19 22 E Glutamic acid
13 8 15 I Isoleucine
7 6 1 W Tryptophan
31 31 24 L Leucine
7 4 5 M Methionine
20 17 16 S Serine
2 6 7 C Cysteine
10 14 14 Q Glutamine
5 12 8 H Histidine 

3PAY_1|Chains A, B, C, D|putative adhesin|Bacteroides ovatus (411476)
>3KQY_1|Chains A, B|Phenylethanolamine N-methyltransferase|Homo sapiens (9606)
>8TKM_1|Chains A, B|Nuclear factor NF-kappa-B p50 subunit|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 \(w=f(c)\)
3PAY , Knot 140 314 0.85 40 203 305 
GSCDSIREDLPRCELWLEFVFDYNMEYADAFNPQVKSVDVLVFDSDDKLLFTKSVKVAALVGGNRMSLTDELDFGSYKVLTVGSLSDRFRLSDNAGNKLVPGTTTLQQVIVSLKRETGGVNFEFQHLYFGEVVEVDHLPSNTNHKIYPVNLIRDTNRFNLALMGYEENKVDGTQYTFEIQAPENAVYSWENEPTGQGPITYVPYYTGPGEISDVVMSARLNTMRLLNRSGWDYKFIIRDANTEAEVWSYNLMTLLSIARPVSRYDGTELPFQEYLDRQSEWNLVFTVVEKNGGGFLQIGIVVGTWIHWLHGMEV
3KQY , Knot 129 289 0.84 40 177 283 
MSGADRSPNAGAAPDSAPGQAAVASAYQRFEPRAYLRNNYAPPRGDLCNPNGVGPWKLRCLAQTFATGEVSGRTLIDIGSGPTVYQLLSACSHFEDITMTDFLEVNRQELGRWLQEEPGAFNWSMYSQHACLIEGKGECWQDKERQLRARVKRVLPIDVHQPQPLGAGSPAPLPADALVSAFCLEAVSPDLASFQRALDHITTLLRPGGHLLLIGALEESWYLAGEARLTVVPVSEEEVREALVRSGYKVRDLRTYIMPAHLQTGVDDVKGVFFAWAQKVGLEHHHHHH
8TKM , Knot 140 312 0.86 40 197 305 
GPYLQILEQPKQRGFRFRYVCEGPSHGGLPGASSEKNKKSYPQVKICNYVGPAKVIVQLVTNGKNIHLHAHSLVGKHCEDGVCTVTAGPKDMVVGFANLGILHVTKKKVFETLEARMTEACIRGYNPGLLVHSDLAYLQAEGGGDRQLTDREKEIIRQAAVQQTKEMDLSVVRLMFTAFLPDSTGSFTRRLEPVVSDAIYDSKAPNASNLKIVRMDRTAGCVTGGEEIYLLCDKVQKDDIQIRFYEEEENGGVWEGFGDFSPTDVHRQFAIVFKTPKYKDVNITKPASVFVQLRRKSDLETSEPKPFLYYPE

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 \(c\).

\(|P_{f(3PAY_1)}(2) \setminus P_{f(3KQY_1)}(2)|=101\), \(|P_{f(3KQY_1)}(2) \setminus P_{f(3PAY_1)}(2)|=75\). 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 1:10000100011000111011100010010110101001011110000011100010111111100101000101100011011010001010001100111100010011101000011101010010110110100110000001011011000001011111000001010000101011001100100010101110011000111010011101010010110001100011100100010110001101101101100001001110001000001011101100011111011111101101101101
Pair \(Z_2\) Length of longest common subsequence
3PAY_1,3KQY_1 176 3
3PAY_1,8TKM_1 160 4
3KQY_1,8TKM_1 182 4

Newick tree

 
[
	3KQY_1:92.46,
	[
		3PAY_1:80,8TKM_1:80
	]:12.46
]

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 \((0.85)(0.8)(\frac{603 }{\log_{20} 603}-\frac{289}{\log_{20}289})=87.9\)
Status Protein1 Protein2 d d1/2
Query variables 3PAY_1 3KQY_1 115 108.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]