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Parikh vectors
4XKS_1 4WBG_1 8SFZ_1 Letter Amino acid
6 27 10 V Valine
0 0 2 C Cysteine
11 38 17 G Glycine
9 19 4 K Lycine
4 36 12 S Serine
2 3 3 W Tryptophan
24 33 11 L Leucine
1 23 3 P Proline
6 18 7 Y Tyrosine
8 46 12 A Alanine
6 16 8 Q Glutamine
18 16 7 E Glutamic acid
5 17 7 H Histidine
12 14 2 I Isoleucine
7 12 3 M Methionine
9 13 8 R Arginine
10 12 4 N Asparagine
14 12 8 D Aspartic acid
4 10 7 F Phenylalanine
2 15 10 T Threonine 

4XKS_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|Bacterioferritin|Escherichia coli K-12 (83333)
>4WBG_1|Chain A|Beta-lactamase|Klebsiella pneumoniae (573)
>8SFZ_1|Chains A, B, C|LaG35|Lama glama (9844)
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 \(w=f(c)\)
4XKS , Knot 74 158 0.79 38 116 152 
MKGDTKVINYLNKLLGNELVAINQYFLHARMFKNWGLKRLNDVEFHESIDEMKHADRYIERILFLEGLPNLQDLGKLNIGEDVEEMLRSDLALELDGAKNLREAIGYADSVHDYVSRDMMIEILRDEEGHIDWLETELDLIQKMGLQNYLQAQIREEG
4WBG , Knot 158 380 0.82 38 191 347 
MGSSHHHHHHSSGLVPRGSHMGEASPVDPLRPVVDASIQPLLKEHRIPGMAVAVLKDGKAHYFNYGVANRESGASVSEQTLFEIGSVSKTLTATLGAYAVVKGAMQLDDKASRHAPWLKGSVFDSITMGELATYSAGGLPLQFPEEVDSSEKMRAYYRQWAPVYSPGSHRQYSNPSIGLFGHLAASSLKQPFAQLMEQTLLPGLGMHHTYVNVPKQAMASYAYGYSKEDKPIRVNPGMLADEAYGIKTSSADLLAFVKANIGGVDDKALQQAISLTHKGHYSVGGMTQGLGWESYAYPVTEQTLLAGNSAKVILEANPTAAPRESGSQVLFNKTGSSNGFGAYVAFVPARGIGIVMLANRNYPIPARVKAAHAILAQLAG
8SFZ , Knot 70 145 0.80 40 115 137 
GSMADVQLVESGGGLVQAGGSLRLSCTVSGRTFSNYAMGWFRQAPGKEREFVAGISWTGGHTLYTDSVKGRFTISRDNAKNTVYLQMNSLKPEDTALYYCAADRAADFFAQRDEYDYWGQGTQVTVSGLEVLFQGPSLEHHHHHH

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 \(c\).

\(|P_{f(4XKS_1)}(2) \setminus P_{f(4WBG_1)}(2)|=54\), \(|P_{f(4WBG_1)}(2) \setminus P_{f(4XKS_1)}(2)|=129\). 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 1:10100011001001110011110001101011001110010010100010010010001001111011101001101011001001100011101011001001110100100010001110110000101011000101100111000101010001
Pair \(Z_2\) Length of longest common subsequence
4XKS_1,4WBG_1 183 3
4XKS_1,8SFZ_1 153 3
4WBG_1,8SFZ_1 174 6

Newick tree

 
[
	4WBG_1:93.14,
	[
		4XKS_1:76.5,8SFZ_1:76.5
	]:16.64
]

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 \((0.85)(0.8)(\frac{538 }{\log_{20} 538}-\frac{158}{\log_{20}158})=110.\)
Status Protein1 Protein2 d d1/2
Query variables 4XKS_1 4WBG_1 139 99
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]