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Parikh vectors
3VOY_1 2EMV_1 4UXH_1 Letter Amino acid
12 1 6 Y Tyrosine
27 1 15 V Valine
18 1 3 N Asparagine
9 2 10 Q Glutamine
28 7 10 G Glycine
20 1 9 F Phenylalanine
25 10 7 S Serine
12 2 10 T Threonine
19 1 16 A Alanine
17 0 13 D Aspartic acid
9 2 4 H Histidine
12 2 9 I Isoleucine
25 1 12 L Leucine
15 4 12 E Glutamic acid
19 2 9 K Lycine
15 2 6 P Proline
2 0 1 W Tryptophan
7 3 16 R Arginine
11 2 9 C Cysteine
13 0 7 M Methionine 

3VOY_1|Chain A|Glutaminase kidney isoform, mitochondrial|Homo sapiens (9606)
>2EMV_1|Chain A|Zinc finger protein 268|Homo sapiens (9606)
>4UXH_1|Chains A, B|THYMIDINE KINASE|LEISHMANIA MAJOR (5664)
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 \(w=f(c)\)
3VOY , Knot 145 315 0.88 40 207 308 
SMIPDFMSFTSHIDELYESAKKQSGGKVADYIPQLAKFSPDLWGVSVCTVDGQRHSTGDTKVPFCLQSCVKPLKYAIAVNDLGTEYVHRYVGKEPSGLRFNKLFLNEDDKPHNPMVNAGAIVVTSLIKQGVNNAEKFDYVMQFLNKMAGNEYVGFSNATFQSERESGDRNFAIGYYLKEKKCFPEGTDMVGILDFYFQLCSIEVTCESASVMAATLANGGFCPITGERVLSPEAVRNTLSLMHSCGMYDFSGQFAFHVGLPAKSGVAGGILLVVPNVMGMMCWSPPLDKMGNSVKGIHFCHDLVSLCNFHNYDNL
2EMV , Knot 25 44 0.71 34 35 39 
GSSGSSGTREKPYECSECGKAFIRNSQLIVHQRTHSGESGPSSG
4UXH , Knot 90 184 0.85 40 145 179 
GPFRGRIELIIGPMFAGKTTELMRRVKREIHARRSCFVIKYSKDTRYDEHNVASHDQLMLRAQAAVSQLTEVRDTWKRFDVLAIDEGQFFSDLVDFCNTAADAGKVVMVSALDGDYRRKPFGQICELVPYCEAVDKLTAVCMMCHEQPACFTRRTVNVEQQELIGGADMYIATCRECYSKQQLP

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 \(c\).

\(|P_{f(3VOY_1)}(2) \setminus P_{f(2EMV_1)}(2)|=185\), \(|P_{f(2EMV_1)}(2) \setminus P_{f(3VOY_1)}(2)|=13\). 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 1:011101101000100100010000110110011011010101111010010100000100011101000101100111100110001000110010110100111000001001110111111001100110010010011011001110001110010100000010001111001000001101001111101010100101000010111101101110110100110101100010110001100101011101111100111111111110111110101110011001011010001101001000001
Pair \(Z_2\) Length of longest common subsequence
3VOY_1,2EMV_1 198 3
3VOY_1,4UXH_1 178 3
2EMV_1,4UXH_1 158 3

Newick tree

 
[
	3VOY_1:98.66,
	[
		4UXH_1:79,2EMV_1:79
	]:19.66
]

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 \((0.85)(0.8)(\frac{359 }{\log_{20} 359}-\frac{44}{\log_{20}44})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 3VOY_1 2EMV_1 133 74
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]