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Parikh vectors
7PXW_1 5ZVD_1 3DAQ_1 Letter Amino acid
2 2 0 C Cysteine
16 25 22 I Isoleucine
1 11 2 M Methionine
22 28 25 A Alanine
6 14 10 F Phenylalanine
20 13 13 S Serine
16 13 25 T Threonine
6 2 0 W Tryptophan
7 23 7 R Arginine
16 15 18 D Aspartic acid
8 9 12 Q Glutamine
5 35 20 E Glutamic acid
18 28 16 G Glycine
11 41 31 L Leucine
17 13 19 N Asparagine
8 5 4 H Histidine
3 45 18 K Lycine
20 21 13 P Proline
10 17 12 Y Tyrosine
23 24 25 V Valine 

7PXW_1|Chain A|Auxiliary activity 9|Lentinus similis (292560)
>5ZVD_1|Chains A, B|389aa long hypothetical nucleolar protein|Pyrococcus horikoshii (strain ATCC 700860 / DSM 12428 / JCM 9974 / NBRC 100139 / OT-3) (70601)
>3DAQ_1|Chains A, B, C, D|Dihydrodipicolinate synthase|Staphylococcus aureus (282458)
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 \(w=f(c)\)
7PXW , Knot 105 235 0.81 40 154 226 
HTLVWGVWVNGVDQGDGRNIYIRSPPNNNPVKNLTSPDMTCNVDNRVVPKSVPVNAGDTLTFEWYHNTRDDDIIASSHHGPIAVYIAPAASNGQGNVWVKLFEDAYNVTNSTWAVDRLITAHGQHSVVVPHVAPGDYLFRAEIIALHEADSLYSQNPIRGAQFYISCAQITINSSDDSTPLPAGVPFPGAYTDSTPGIQFNIYTTPATSYVAPPPSVWSGALGGSIAQVGDASLE
5ZVD , Knot 157 384 0.81 40 205 355 
YLEAFPKELREYYKNLFGKEEANKIMKKLREPVEHYYIRVNTLKISREKLIGELKKEGLKPLRSPYLPEGLYFVREGPNFSDDFEPKLPVVVANKYAAESVYQGAMLYAPGVLKADKNIKEGDEVQIRDPKGLLVGIGIARMDYKEMTEATRGLAVEVTLPKFKLPSLSELKAFEKGYFYPQGLPSMVTARVLEPKEDDVIIDMAAAPGGKTTHIAQLLENKGEIIAIDKSKNRLRKMEENIKRLGVKNVKLVQMDARKLPDLGIKADKILLDAPCTALGVRPKLWEERTLKHIEATARYQRAFIWAAIKSLRRGGVLVYSTCTLSYEENEGNVKFMIRKGMKLEEQSIFIGSPGIGMNKVQRFYPHKHLTQGFFIAKLRKVKD
3DAQ , Knot 123 292 0.79 36 176 278 
THLFEGVGVALTTPFTNNKVNLEALKAHVNFLLENNAQAIIVNGTTAESPTLTTDEKELILKTVIDLVDKRVPVIAGTGTNDTEKSIQASIQAKALGADAIMLITPYYNKTNQRGLVKHFEAIADAVKLPVVLYNVPSRTNMTIEPETVEILSQHPYIVALKDATNDFEYLEEVKKRIDTNSFALYSGNDDNVVEYYQRGGQGVISVIANVIPKEFQALYDAQQSGLDIQDQFKPIGTLLSALSVDINPIPIKALTSYLGFGNYELRLPLVSLEDTDTKVLREAYDTFKAGE

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 \(c\).

\(|P_{f(7PXW_1)}(2) \setminus P_{f(5ZVD_1)}(2)|=68\), \(|P_{f(5ZVD_1)}(2) \setminus P_{f(7PXW_1)}(2)|=119\). 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 1:0011111110110010100101001100011001001010001000111001110110010101000000001110000111110111110010101110110010010000111001101010001111011110011010111100100100001101101010010101000000011111111111000001110101000110001111101101111101101101010
Pair \(Z_2\) Length of longest common subsequence
7PXW_1,5ZVD_1 187 4
7PXW_1,3DAQ_1 152 3
5ZVD_1,3DAQ_1 149 5

Newick tree

 
[
	7PXW_1:88.48,
	[
		3DAQ_1:74.5,5ZVD_1:74.5
	]:13.98
]

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 \((0.85)(0.8)(\frac{619 }{\log_{20} 619}-\frac{235}{\log_{20}235})=108.\)
Status Protein1 Protein2 d d1/2
Query variables 7PXW_1 5ZVD_1 136 109.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]