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Parikh vectors
7VLE_1 3ZKX_1 2JHH_1 Letter Amino acid
14 31 18 A Alanine
7 27 3 I Isoleucine
9 36 18 S Serine
4 26 9 T Threonine
12 17 17 D Aspartic acid
5 20 8 E Glutamic acid
3 1 6 H Histidine
5 7 4 M Methionine
4 20 5 P Proline
9 24 10 V Valine
3 6 6 C Cysteine
12 36 23 G Glycine
12 32 17 L Leucine
10 13 13 K Lycine
12 23 12 F Phenylalanine
7 12 8 R Arginine
7 17 14 N Asparagine
5 13 8 Q Glutamine
3 7 7 W Tryptophan
3 18 12 Y Tyrosine 

7VLE_1|Chains A, E|Extracellular A1 globin|Lamellibrachia satsuma (104711)
>3ZKX_1|Chain A|BETA-SECRETASE 2|HOMO SAPIENS (9606)
>2JHH_1|Chains A[auth C], B[auth F]|FICOLIN-1|HOMO SAPIENS (9606)
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 \(w=f(c)\)
7VLE , Knot 73 146 0.83 40 111 140 
DCNILQRLKVKMQWAKAYGFGTERAKFGNSLWTSIFNYAPDARDLFKSVKSEDMRSPQFKAHIARVIGGLDRVISMFDNEDALNADLEHLKSQHDPRGLDALNFVVFGKALFATVGGQFGVCFDLPAWESCYKVIAMGITGNDMFS
3ZKX , Knot 164 386 0.84 40 219 370 
ANFLAMVDNLQGDSGRGYYLEMLIGTPPQKLQILVDTGSSNFAVAGTPHSYIDTYFDTERSSTYRSKGFDVTVKYTQGSWTGFVGEDLVTIPKGFNTSFLVNIATIFESENFFLPGIKWNGILGLAYATLAKPSSSLETFFDSLVTQANIPNVFSMQMCGAGLPVAGSGTNGGSLVLGGIEPSLYKGDIWYTPIKEEWYYQIEILKLEIGGQSLNLDCREYNADKAIVDSGTTLLRLPQKVFDAVVEAVARASLIPAFSDGFWTGSQLACWTNSETPWSYFPKISIYLRDENSSRSFRITILPQLYIQPMMGAGLNYECYRFGISPSTNALVIGATVMEGFYVIFDRAQKRVGFAASPCAEIAGAAVSEISGPFSTEDVASNCVPA
2JHH , Knot 102 218 0.84 40 157 213 
QSCATGPRNCKDLLDRGYFLSGWHTIYLPDCRPLTVLCDMDTDGGGWTVFQRRMDGSVDFYRDWAAYKQGFGSQLGEFWLGNDNIHALTAQGSSELRTDLVDFEGNHQFAKYKSFKVADEAEKYKLVLGAFVGGSAGNSLTGHNNNFFSTKDQDNDVSSSNCAEKFQGAWWYADCHASNLNGLYLMGPHESYANGINWSAAKGYKYSYKVSEMKVRPA

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 \(c\).

\(|P_{f(7VLE_1)}(2) \setminus P_{f(3ZKX_1)}(2)|=36\), \(|P_{f(3ZKX_1)}(2) \setminus P_{f(7VLE_1)}(2)|=144\). 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 1:00011001010101101011100010110011001100110100110010000100101010110111110011011000011010100100000101101101111101111011101110101111000001111110100110
Pair \(Z_2\) Length of longest common subsequence
7VLE_1,3ZKX_1 180 3
7VLE_1,2JHH_1 162 4
3ZKX_1,2JHH_1 182 4

Newick tree

 
[
	3ZKX_1:93.45,
	[
		7VLE_1:81,2JHH_1:81
	]:12.45
]

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 \((0.85)(0.8)(\frac{532 }{\log_{20} 532}-\frac{146}{\log_{20}146})=112.\)
Status Protein1 Protein2 d d1/2
Query variables 7VLE_1 3ZKX_1 145 98.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]