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Parikh vectors
3VPD_1 8UHJ_1 2RHR_1 Letter Amino acid
5 1 4 M Methionine
7 7 5 F Phenylalanine
9 8 6 Y Tyrosine
28 7 23 E Glutamic acid
24 24 35 G Glycine
13 5 10 I Isoleucine
30 18 23 L Leucine
37 16 33 A Alanine
6 6 5 Q Glutamine
5 4 10 H Histidine
8 33 13 S Serine
12 23 19 T Threonine
3 5 4 C Cysteine
6 8 7 N Asparagine
8 5 8 D Aspartic acid
10 13 7 K Lycine
14 11 9 P Proline
4 4 2 W Tryptophan
29 20 35 V Valine
23 7 19 R Arginine 

3VPD_1|Chains A, B|Ribosomal protein S6 modification protein|Thermus thermophilus (274)
>8UHJ_1|Chain A[auth H]|hSC44h.ck.20.N32F Fab heavy chain|Oryctolagus cuniculus (9986)
>2RHR_1|Chains A, B|Actinorhodin Polyketide Ketoreductase|Streptomyces coelicolor (1902)
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 \(w=f(c)\)
3VPD , Knot 118 281 0.79 40 164 256 
MLAILYDRIRPDERMLFERAEALGLPYKKVYVPALPMVLGERPEALEGVTVALERCVSQSRGLAAARYLTALGIPVVNRPEVIEACGDKWATSVALAKAGLPQPKTALATDREEALRLMEAFGYPVVLKPVIGSWGRLLAKVTDRAAAEALLEHKEVLGGFQHQLFYIQEYVEKPGRDIRVFVVGERAIAAIYRRSAHWITNTARGGQAENCPLTEEIARLSVGAAEAVGGGVVAVDLFESERGLLVNEVNHTMEFKNSVHTTGVDIPGEILRYAWEVARG
8UHJ , Knot 99 225 0.79 40 146 214 
EVQLVESGGGLVQPGGSLRLSCTASGFSIDSYGFSWVRQAPGKGLEHIGYLTAGGRAFYASWAKSRSTITRNTNENTVTLKMNSLRAEDTAVYYCAKLGTGSRFAIWGQGTLVTVSSASTKGPSVFPLAPSSKSTSGGTAALGCLVKDYFPEPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKKVEPKSCDKTHT
2RHR , Knot 115 277 0.77 40 151 259 
HHHHHHSSGLVPRGSHMATQDSEVALVTGATSGIGLEIARRLGKEGLRVFVCARGEEGLRTTLKELREAGVEADGRTCDVRSVPEIEALVAAVVERYGPVDVLVNNAGRLGGGATAELADELWLDVVETNLTGVFRVTKQVLKAGGMLERGTGRIVNIASTGGKQGVVHAAPYSASKHGVVGFTKALGLELARTGITVNAVCPGFVETPMAASVREHYSDIWEVSTEEAFDRITARVPIGRYVQPSEVAEMVAYLIGPGAAAVTAQALNVCGGLGNY

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 \(c\).

\(|P_{f(3VPD_1)}(2) \setminus P_{f(8UHJ_1)}(2)|=82\), \(|P_{f(8UHJ_1)}(2) \setminus P_{f(3VPD_1)}(2)|=64\). 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 1:11111000101000111001011111000101111111110010110110111000100001111100101111111001011010100110011110111101001110000011011011101111011110110111010001110111000011111000110100010011001011111001111100001011000101101000110001101011110111111111011000011110010001010001000110111011001101101
Pair \(Z_2\) Length of longest common subsequence
3VPD_1,8UHJ_1 146 4
3VPD_1,2RHR_1 145 4
8UHJ_1,2RHR_1 153 4

Newick tree

 
[
	8UHJ_1:75.51,
	[
		3VPD_1:72.5,2RHR_1:72.5
	]:3.01
]

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 \((0.85)(0.8)(\frac{506 }{\log_{20} 506}-\frac{225}{\log_{20}225})=80.9\)
Status Protein1 Protein2 d d1/2
Query variables 3VPD_1 8UHJ_1 102 91.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]