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Parikh vectors
3VFI_1 6ACL_1 3LIW_1 Letter Amino acid
5 15 20 T Threonine
5 8 7 N Asparagine
8 7 11 D Aspartic acid
4 7 9 Q Glutamine
5 19 6 P Proline
6 18 14 R Arginine
8 10 13 I Isoleucine
8 3 5 M Methionine
2 4 4 W Tryptophan
8 15 11 S Serine
2 9 9 C Cysteine
1 9 5 H Histidine
8 18 13 L Leucine
5 12 17 K Lycine
2 5 8 Y Tyrosine
3 21 16 V Valine
6 30 13 A Alanine
8 20 19 E Glutamic acid
4 24 22 G Glycine
6 13 12 F Phenylalanine 

3VFI_1|Chain A|thioredoxin|Silicibacter phage DSS3phi2 (490912)
>6ACL_1|Chain A|NAD-dependent protein deacylase sirtuin-5, mitochondrial|Homo sapiens (9606)
>3LIW_1|Chain A|Activated factor Xa heavy chain|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 \(w=f(c)\)
3VFI , Knot 54 104 0.80 40 88 101 
LRSLSDSDFQLEVRQHPDPIIIMFTGSWCQPCKKMKPTFEEMASQMEGDIRFAYMDAEDAEKTMAELNIRTLPSLALFVDGMIREVFSGTMNKSDLRYWINNNI
6ACL , Knot 121 267 0.84 40 181 262 
PSSSMADFRKFFAKAKHIVIISGAGVSAESGVPTFRGAGGYWRKWQAQDLATPLAFAHNPSRVWEFYHYRREVMGSKEPNAGHRAIAECETRLGKQGRRVVVITQNIDELHRKAGTKNLLEIHGSLFKTRCTSCGVVAENYKSPICPALSGKGAPEPGTQDASIPVEKLPRCEEAGCGGLLRPHVVWFGENLDPAILEEVDRELAHCDLCLVVGTSSVVYPAAMFAPQVAARGVPVAEFNTETTPATNRFRFHFQGPCGTTLPEALA
3LIW , Knot 109 234 0.84 40 171 226 
IVGGQECKDGECPWQALLINEENEGFCGGTILSEFYILTAAHCLYQAKRFKVRVGDRNTEQEEGGEAVHEVEVVIKHNRFTKETYDFDIAVLRLKTPITFRMNVAPACLPERDWAESTLMTQKTGIVSGFGRTHEKGRQSTRLKMLEVPYVDRNSCKLSSSFIITQNMFCAGYDTKQEDACQGDSGGPHVTRFKDTYFVTGIVSWGEGCARKGKYGIYTKVTAFLKWIDRSMKT

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 \(c\).

\(|P_{f(3VFI_1)}(2) \setminus P_{f(6ACL_1)}(2)|=49\), \(|P_{f(6ACL_1)}(2) \setminus P_{f(3VFI_1)}(2)|=142\). 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 1:10010000101010001011111101010010001010100110010101011010100100011010100110111110111001101010000100110001
Pair \(Z_2\) Length of longest common subsequence
3VFI_1,6ACL_1 191 3
3VFI_1,3LIW_1 175 3
6ACL_1,3LIW_1 170 4

Newick tree

 
[
	3VFI_1:93.68,
	[
		3LIW_1:85,6ACL_1:85
	]:8.68
]

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 \((0.85)(0.8)(\frac{371 }{\log_{20} 371}-\frac{104}{\log_{20}104})=82.1\)
Status Protein1 Protein2 d d1/2
Query variables 3VFI_1 6ACL_1 106 73
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]