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Parikh vectors
6UIE_1 9RCR_1 7DXA_1 Letter Amino acid
3 37 6 N Asparagine
9 47 7 D Aspartic acid
39 59 3 L Leucine
7 46 6 S Serine
7 7 1 W Tryptophan
26 46 9 R Arginine
14 60 10 G Glycine
1 36 3 K Lycine
4 39 5 T Threonine
5 54 9 V Valine
12 56 11 E Glutamic acid
4 44 8 I Isoleucine
3 17 7 M Methionine
8 42 7 F Phenylalanine
4 33 4 Y Tyrosine
8 16 0 C Cysteine
14 44 6 Q Glutamine
4 25 1 H Histidine
13 35 4 P Proline
25 73 9 A Alanine 

6UIE_1|Chains A, B|Type III export protein PscK|Pseudomonas aeruginosa (287)
>9RCR_1|Chains A, B, C, D|Glycyl radical protein|Raoultella planticola (575)
>7DXA_1|Chain A|Photosystem II reaction center Psb28 protein|Thermosynechococcus vulcanus (32053)
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 \(w=f(c)\)
6UIE , Knot 95 210 0.80 40 122 195 
GSMPLTAYQLRFCPARYIHESHLPAVLLRLLPALPDWRRQSVLNAWLLEQLELDCAFRMPAQLGGLALYPQAALERTLGWLGALLHGQALRQVLDGARVRRIRAQIGEQGQRFCLEQLDLLIGRWPPGWQRALPENPEEGYFRRCGLAFWLAACSDADCGFSRRLRLRLRLEAMPAPADWTFDEQRRSLARTLCLKVARQASDECFHLLN
9RCR , Knot 314 816 0.86 40 317 744 
MGSSHHHHHHSQDHENLYFQGSMGLEKGFTHPTDRVVRLKNMILHAKPYVESERAVLATEAYKENEQLPAIMRRAKVVEKIFNELPVTIRPDELIVGAVTINPRSTEICPEFSYDWVEKEFDTMAHRVADPFIIEKKTADELHQAFKYWPGKTTSSLAASYMSEGTKESMSNGVFTVGNYFFGGIGHVSVDYGKVLKIGFRGIIDEVTRALENLDRSTSDYIKKEQFYNAVILSYQAAINFAHRYAQEASRLAREERDPTRQRELEQIASNCTRVPEYGATTFWEACQTFWFIQSMLQIESSGHSISPGRFDQYMYPYLAADTAISSEFAQELVDCCWIKLNDINKTRDEISAQAFAGYAVFQNLCVGGQTEDGRDATNPLTYMCMEATAHVRLPQPSFSIRVWQGTPDEFLYRACELVRLGLGVPAMYNDEVIIPALQNRGVSLHDARDYCIIGCVEPQAPHRTEGWHDAAFFNVAKVLEITLNNGRSGNKQLGPMTGEMTQYAGMDDFYAAFKKQMAHFVHQLVEACNSVDIAHGERCPLPFLSALVDDCIGRGKSLQEGGAIYNFTGPQAFGIADTGDSVYAIQKQVFEDRRLTLQELKGALDANFGYRSGNPRYEEIRHILENSPCFGNDIDDVDLVARQCALIYCQEVEKYTNPRGGRFQAGIYPVSANVLFGKDVSALPDGRLAKEPLADGVSPRQGKDTLGPTAAANSVAKLDHFIASNGTLYNQKFLPSSLAGEKGLRNFGGLVRNYFDKKGMHVQFNVIDRNTLLEAQKNPQQHQDLVVRVAGYSAQFVVLAKEVQDDIISRTEQQF
7DXA , Knot 61 116 0.83 38 97 113 
MGAMAEIQFIRGINEEVVPDVRLTRARDGSSGQAMFYFDNPKIVQEGNLEVTGMYMVDEEGEIVTRDVNAKFINGQPVAIEATYTMRSPQEWDRFIRFMDRYAASHGLGFQKSENS

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 \(c\).

\(|P_{f(6UIE_1)}(2) \setminus P_{f(9RCR_1)}(2)|=17\), \(|P_{f(9RCR_1)}(2) \setminus P_{f(6UIE_1)}(2)|=212\). 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 1:101110100101011001000011111101111110100001101111001010011011101111110101110001111111101011001101101001010110010010100101111011111001110010010100011111111000100110001010101011111101010000001100101011001000010110
Pair \(Z_2\) Length of longest common subsequence
6UIE_1,9RCR_1 229 4
6UIE_1,7DXA_1 147 3
9RCR_1,7DXA_1 238 3

Newick tree

 
[
	9RCR_1:12.98,
	[
		6UIE_1:73.5,7DXA_1:73.5
	]:54.48
]

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 \((0.85)(0.8)(\frac{1026 }{\log_{20} 1026}-\frac{210}{\log_{20}210})=221.\)
Status Protein1 Protein2 d d1/2
Query variables 6UIE_1 9RCR_1 285 176
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]