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Parikh vectors
6NKK_1 2RCW_1 7XOI_1 Letter Amino acid
15 25 26 I Isoleucine
12 33 17 K Lycine
17 16 41 P Proline
17 14 40 T Threonine
17 25 49 D Aspartic acid
6 9 13 H Histidine
8 18 31 E Glutamic acid
22 23 40 G Glycine
1 1 9 W Tryptophan
9 11 17 R Arginine
7 16 33 N Asparagine
8 9 7 M Methionine
2 10 31 F Phenylalanine
22 29 25 S Serine
6 16 34 Y Tyrosine
27 22 33 V Valine
5 2 4 C Cysteine
8 14 21 Q Glutamine
32 19 39 A Alanine
24 38 38 L Leucine 

6NKK_1|Chains A, B, C, D, E, F|Short chain dehydrogenase|Penicillium fellutanum (70095)
>2RCW_1|Chain A|Poly [ADP-ribose] polymerase 1|Homo sapiens (9606)
>7XOI_1|Chains A, C, E, G, I, K, M, O|alpha-glucosidase N-terminal polypeptide|Aspergillus sojae NBRC 4239 (927772)
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 \(w=f(c)\)
6NKK , Knot 122 265 0.85 40 169 257 
MTPAPTPRTDQLHGSRVLVIGGTSGIGFAVCAAALGHGAIVTIVGSNAQKLKDSVARLKSSFPSTDPDDIVAVRCDLSNSDTVEQDIEKALQLAAGNSKINHIVITAADMTAPPPLEDLTVDSVQRPGIIRLVAPLMVAKHLPKYMNKCPQSSLTLTSGAHCLRPDPGWTVISGYCGAVEAMSRGLAIDLKPLRVNVVAPGAVLTEAVKDILGDAYDAAVEMAEAKSTVGQTGSPESVAQAYIYLMKDHYASGSVVSTNGGMLLV
2RCW , Knot 150 350 0.83 40 205 333 
KSKLPKPVQDLIKMIFDVESMKKAMVEYEIDLQKMPLGKLSKRQIQAAYSILSEVQQAVSQGSSDSQILDLSNRFYTLIPHDFGMKKPPLLNNADSVQAKVEMLDNLLDIEVAYSLLRGGSDDSSKDPIDVNYEKLKTDIKVVDRDSEEAEIIRKYVKNTHATTHNAYDLEVIDIFKIEREGECQRYKPFKQLHNRRLLWHGSRTTNFAGILSQGLRIAPPEAPVTGYMFGKGIYFADMVSKSANYCHTSQGDPIGLILLGEVALGNMYELKHASHISKLPKGKHSVKGLGKTTPDPSANISLDGVDVPLGTGISSGVNDTSLLYNEYIVYDIAQVNLKYLLKLKFNFKT
7XOI , Knot 221 548 0.84 40 271 516 
APVNTTTEDETAQIPAEAVIGYSDLEGDFDVAVLPFSNSTNNGLLFINTTIASIAAKEEGVSLEKREAEAEFQSKDGVEDLDGPGDDLYVKDLSGCPGYKATKHWQTRSGFYADLTLAGPACNVFGTDLPDLKLEVEYQTSDRLHVKILDTNNTVYQVPDSVFPRPGFGEWCSPKDSKLKFDFQADPFSFTVSRTDTGEVLFDTTGNKLVFESQYVYLKTHLPQNPHLYGLGEHSDAFMLNTTNYTRTIYTRDAYGTPQGENLYGAHPIYFDHRQTGTHGVFLLNSNGMDIFIDNNATQYLEYNIIGGVLDFYFIAGPSPRDVAIQYAEITQTPLMTPYWGLGYHQCKYGYQDVYEVAAVVANYSTNNIPLETIWTDIDYMDRRRIFTIDPERFPANLYKDLVDTIHARDQHYIVMVDPAVYYKESNPALDEGLRYDIFMKENNGSEYQGVVWAGPSHFPDWFHPDSQQYWSEQFLAFFDGTNGPDIDALWIDMNEPANFYNRPYPGNNTTPENFAEVDGDPPAAPAVRDGPDAPIPGFPASLQPNWV

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 \(c\).

\(|P_{f(6NKK_1)}(2) \setminus P_{f(2RCW_1)}(2)|=61\), \(|P_{f(2RCW_1)}(2) \setminus P_{f(6NKK_1)}(2)|=97\). 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 1:1011101000010100111111001111110111110111101110010010001101000110001001111000100000100010011011110001001110110101111100101001001111011111111001100100010001010011001010111011010011101100111101011010111111110011001110100111011010001100101001101010110000101011000111111
Pair \(Z_2\) Length of longest common subsequence
6NKK_1,2RCW_1 158 4
6NKK_1,7XOI_1 170 3
2RCW_1,7XOI_1 156 4

Newick tree

 
[
	6NKK_1:83.36,
	[
		2RCW_1:78,7XOI_1:78
	]:5.36
]

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 \((0.85)(0.8)(\frac{615 }{\log_{20} 615}-\frac{265}{\log_{20}265})=98.3\)
Status Protein1 Protein2 d d1/2
Query variables 6NKK_1 2RCW_1 126 109.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]