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Parikh vectors
2BFD_1 7OQN_1 3UGX_1 Letter Amino acid
27 19 33 E Glutamic acid
26 18 30 S Serine
19 19 41 V Valine
23 12 25 D Aspartic acid
23 8 17 Q Glutamine
22 16 20 I Isoleucine
14 9 25 T Threonine
38 25 27 A Alanine
11 7 8 H Histidine
18 14 36 K Lycine
21 8 26 P Proline
4 2 2 W Tryptophan
5 4 3 C Cysteine
28 23 19 G Glycine
26 32 34 L Leucine
11 7 8 M Methionine
18 9 17 F Phenylalanine
21 5 14 Y Tyrosine
28 14 18 R Arginine
17 6 11 N Asparagine 

2BFD_1|Chain A|2-OXOISOVALERATE DEHYDROGENASE ALPHA SUBUNIT|HOMO SAPIENS (9606)
>7OQN_1|Chain A|Uridine 5'-monophosphate synthase|Homo sapiens (9606)
>3UGX_1|Chains A, B, C, D|S-arrestin|Bos taurus (9913)
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 \(w=f(c)\)
2BFD , Knot 170 400 0.85 40 239 391 
SSLDDKPQFPGASAEFIDKLEFIQPNVISGIPIYRVMDRQGQIINPSEDPHLPKEKVLKLYKSMTLLNTMDRILYESQRQGRISFYMTNYGEEGTHVGSAAALDNTDLVFGQAREAGVLMYRDYPLELFMAQCYGNISDLGKGRQMPVHYGCKERHFVTISSPLATQIPQAVGAAYAAKRANANRVVICYFGEGAASEGDAHAGFNFAATLECPIIFFCRNNGYAISTPTSEQYRGDGIAARGPGYGIMSIRVDGNDVFAVYNATKEARRRAVAENQPFLIEAMTYRIGHASTSDDSSAFRSVDEVNYWDKQDHPISRLRHYLLSQGWWDEEQEKAWRKQSRRKVMEAFEQAERKPKPNPNLLFSDVYQEMPAQLRKQQESLARHLQTYGEHYPLDHFDK
7OQN , Knot 116 257 0.83 40 172 250 
ELSFGARAELPRIHPVASKLLRLMQKKETNLCLSADVSLARELLQLADALGPSICMLKTHVDILNDFTLDVMKELITLAKCHEFLIFEDRKFADIGNTVKKQYEGGIFKIASWADLVNAHVVPGSGVVKGLQEVGLPLHRGCLLIAEMSSTGSLATGDYTRAAVRMAEEHSEFVVGFISGSRVSMKPEFLHLTPGVQLEAGGDNLGQQYNSPQEVIGKRGSDIIIVGRGIISAADRLEAAEMYRKAAWEAYLSRLGV
3UGX , Knot 175 414 0.85 40 214 394 
ASWSHPQFEKMKANKPAPNHVIFKKISRDKSVTIYLGKRDYIDHVERVEPVDGVVLVDPELVKGKRVYVSLTCAFRYGQEDIDVMGLSFRRDLYFSQVQVFPPVGASGATTRLQESLIKKLGANTYPFLLTFPDYLPCSVMLQPAPQDVGKSCGVDFEIKAFATHSTDVEEDKIPKKSSVRLLIRKVQHAPRDMGPQPRAEASWQFFMSDKPLRLAVSLSKEIYYHGEPIPVTVAVTNSTEKTVKKIKVLVEQVTNVVLYSSDYYIKTVAAEEAQEKVPPNSSLTKTLTLVPLLANNRERRGIALDGKIKHEDTNLASSTIIKEGIDKTVMGILVSYQIKVKLTVSGLLGELTSSEVATEVPFRLMHPQPEDPDTAKESFQDENFVFEEFARQNLKDAGEYKEEKTDQEAAMDE

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 \(c\).

\(|P_{f(2BFD_1)}(2) \setminus P_{f(7OQN_1)}(2)|=113\), \(|P_{f(7OQN_1)}(2) \setminus P_{f(2BFD_1)}(2)|=46\). 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 1:0010001011110101100101101011011110011000101101000101100011010001011001001100000010101010001001001101111000011110100111110000110111100010100110100111001000001101001110011011111011001010011100110111001010111011101001111100001011001000000101111011101110101010011110010001000111000111101100011010000000110010010010000011001000110011100000011000000011011001000101010111001000111010000001100100010001100100
Pair \(Z_2\) Length of longest common subsequence
2BFD_1,7OQN_1 159 4
2BFD_1,3UGX_1 159 4
7OQN_1,3UGX_1 166 4

Newick tree

 
[
	3UGX_1:81.84,
	[
		2BFD_1:79.5,7OQN_1:79.5
	]:2.34
]

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 \((0.85)(0.8)(\frac{657 }{\log_{20} 657}-\frac{257}{\log_{20}257})=111.\)
Status Protein1 Protein2 d d1/2
Query variables 2BFD_1 7OQN_1 145 116.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]