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Parikh vectors
4KFQ_1 1GAV_1 2DLJ_1 Letter Amino acid
20 9 0 T Threonine
10 5 0 Y Tyrosine
13 16 2 A Alanine
19 8 0 L Leucine
10 0 0 M Methionine
12 5 0 P Proline
15 5 0 R Arginine
19 4 0 E Glutamic acid
20 14 0 S Serine
7 0 2 C Cysteine
12 3 0 Q Glutamine
17 8 0 I Isoleucine
22 7 0 K Lycine
15 4 0 F Phenylalanine
4 2 0 W Tryptophan
15 8 0 N Asparagine
16 5 0 D Aspartic acid
19 9 2 G Glycine
7 0 0 H Histidine
20 17 0 V Valine 

4KFQ_1|Chains A, B|Glutamate receptor ionotropic, NMDA 1|Rattus norvegicus (10116)
>1GAV_1|Chains AA[auth 1], A[auth 0], BA[auth 3], B[auth 2], CA[auth 5], C[auth 4], DA[auth 7], D[auth 6], EA[auth 9], E[auth 8], FA[auth B], F[auth A], GA[auth D], G[auth C], HA[auth F], H[auth E], IA[auth H], I[auth G], JA[auth J], J[auth I], K, KA[auth L], LA[auth N], L[auth M], MA[auth P], M[auth O], NA[auth R], N[auth Q], OA[auth T], O[auth S], PA[auth V], P[auth U], QA[auth X], Q[auth W], RA[auth Z], R[auth Y], SA[auth b], S[auth a], T[auth c], U[auth d], V[auth e], W[auth f], X[auth g], Y[auth h], Z[auth i]|BACTERIOPHAGE GA PROTEIN CAPSID|Enterobacteria phage GA (12018)
>2DLJ_1|Chain A|5'-D(*GP*(UMS)P*GP*(BRU)P*AP*CP*AP*C)-3'|null
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 \(w=f(c)\)
4KFQ , Knot 135 292 0.87 40 207 287 
GMSTRLKIVTIHQEPFVYVKPTMSDGTCKEEFTVNGDPVKKVICTGPNDTSPGSPRHTVPQCCYGFCIDLLIKLARTMNFTYEVHLVADGKFGTQERVNNSNKKEWNGMMGELLSGQADMIVAPLTINNERAQYIEFSKPFKYQGLTILVKKGTRITGINDPRLRNPSDKFIYATVKQSSVDIYFRRQVELSTMYRHMEKHNYESAAEAIQAVRDNKLHAFIWDSAVLEFEASQKCDLVTTGELFFRSGFGIGMRKDSPWKQNVSLSILKSHENGFMEDLDKTWVRYQECDS
1GAV , Knot 67 129 0.84 34 102 124 
ATLRSFVLVDNGGTGNVTVVPVSNANGVAEWLSNNSRSQAYRVTASYRASGADKRKYTIKLEVPKIVTQVVNGVELPVSAWKAYASIDLTIPIFAATDDVTVISKSLAGLFKVGNPIAEAISSQSGFYA
2DLJ , Knot 5 8 0.43 8 5 6 
GUGUACAC

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 \(c\).

\(|P_{f(4KFQ_1)}(2) \setminus P_{f(1GAV_1)}(2)|=137\), \(|P_{f(1GAV_1)}(2) \setminus P_{f(4KFQ_1)}(2)|=32\). 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 1:1100010110100011101010100100000101010110011001100001101000110000110101110110010100010111010110000100000001011110110101011111101000010010100110001101110010010110010100100011010100001010100010100100010000000110110110000101111001110101000001100101110011111100001100010101100000111001000110000000
Pair \(Z_2\) Length of longest common subsequence
4KFQ_1,1GAV_1 169 4
4KFQ_1,2DLJ_1 212 1
1GAV_1,2DLJ_1 107 1

Newick tree

 
[
	4KFQ_1:10.28,
	[
		1GAV_1:53.5,2DLJ_1:53.5
	]:52.78
]

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 \((0.85)(0.8)(\frac{421 }{\log_{20} 421}-\frac{129}{\log_{20}129})=87.8\)
Status Protein1 Protein2 d d1/2
Query variables 4KFQ_1 1GAV_1 114 79
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]