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Parikh vectors
1MFA_1 6GVJ_1 2MYD_1 Letter Amino acid
10 31 18 L Leucine
5 12 6 F Phenylalanine
9 28 4 P Proline
13 39 6 S Serine
4 23 7 D Aspartic acid
2 9 0 C Cysteine
6 21 14 E Glutamic acid
3 18 12 H Histidine
8 23 8 V Valine
9 21 17 A Alanine
6 21 1 N Asparagine
6 35 9 I Isoleucine
16 20 5 T Threonine
4 36 19 K Lycine
3 6 2 W Tryptophan
1 17 3 Y Tyrosine
3 9 4 R Arginine
5 33 5 Q Glutamine
14 16 11 G Glycine
0 11 2 M Methionine 

1MFA_1|Chain A[auth L]|IGG1-LAMBDA SE155-4 FAB (LIGHT CHAIN)|Mus musculus (10090)
>6GVJ_1|Chain A|Dual specificity protein kinase TTK|Homo sapiens (9606)
>2MYD_1|Chain A|MYOGLOBIN (N-PROPYL ISOCYANIDE)|Physeter catodon (9755)
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 \(w=f(c)\)
1MFA , Knot 60 127 0.76 38 96 120 
QIVVTQESALTTSPGETVTLTCRSSTGTVTSGNHANWVQEKPDHLFTGLIGDTNNRAPGVPARFSGSLIGDKAALTITGAQPEDEAIYFCALWSNNHWIFGGGTKLTVLGQPKSSPSVTLFPPSSEG
6GVJ , Knot 179 429 0.84 40 233 400 
MAHHHHHHSAALEVLFQGPGPAASSNHWQIPELARKVNTEQKHTTFEQPVFSVSKQSPPISTSKWFDPKSICKTPSSNTLDDYMSCFRTPVVKNDFPPACQLSTPYGQPACFQQQQHQILATPLQNLQVLASSSANECISVKGRIYSILKQIGSGGSSKVFQVLNEKKQIYAIKYVNLEEADNQTLDSYRNEIAYLNKLQQHSDKIIRLYDYEITDQYIYMVMECGNIDLNSWLKKKKSIDPWERKSYWKNMLEAVHTIHQHGIVHSDLKPANFLIVDGMLKLIDFGIANQMQPDTTSVVKDSQVGTVNYMPPEAIKDMSSSRENGKSKSKISPKSDVWSLGCILYYMTYGKTPFQQIINQISKLHAIIDPNHEIEFPDIPEKDLQDVLKCCLKRDPKQRISIPELLAHPYVQIQTHPVNQMAKGTTEE
2MYD , Knot 73 153 0.80 38 111 148 
VLSEGEWQLVLHVWAKVEADVAGHGQDILIRLFKSHPETLEKFDRFKHLKTEAEMKASEDLKKHGVTVLTALGAILKKKGHHEAELKPLAQSHATKHKIPIKYLEFISEAIIHVLHSRHPGDFGADAQGAMNKALELFRKDIAAKYKELGYQG

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 \(c\).

\(|P_{f(1MFA_1)}(2) \setminus P_{f(6GVJ_1)}(2)|=40\), \(|P_{f(6GVJ_1)}(2) \setminus P_{f(1MFA_1)}(2)|=177\). 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 1:0111000011000110010100000010100100101100010011011110000011111101010111001110101101000110101110000111111001011101000101011110001
Pair \(Z_2\) Length of longest common subsequence
1MFA_1,6GVJ_1 217 3
1MFA_1,2MYD_1 141 3
6GVJ_1,2MYD_1 204 4

Newick tree

 
[
	6GVJ_1:11.57,
	[
		1MFA_1:70.5,2MYD_1:70.5
	]:44.07
]

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 \((0.85)(0.8)(\frac{556 }{\log_{20} 556}-\frac{127}{\log_{20}127})=125.\)
Status Protein1 Protein2 d d1/2
Query variables 1MFA_1 6GVJ_1 161 103.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]