FREQUENCY AND MARKOV CHAIN ANALYSIS OF THE AMINO-ACID SEQUENCE OF HUMAN ALCOHOL DEHYDROGENASE {alpha}-CHAIN

GUANG WU*

Service of Clinical Pharmacology and Toxicology, Medical School, University of Udine, Piazzale Santa Maria della Misericordia, I-33100 Udine, Italy

Received 27 September 1999; in revised form 17 December 1999; accepted 20 December 1999


    ABSTRACT
 TOP
 FOOTNOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 ACKNOWLEDGEMENTS
 REFERENCES
 
The amino-acid sequences of human alcohol dehydrogenase {alpha}-chain (ADH1) were analysed according to two-, three- and four-amino-acid sequences. The measured frequencies and probabilities were compared with the predicted frequencies and probabilities. Of 373 two-amino-acid sequences in the ADH1, 92 (24.665%) and 32 (8.579%) sequences can be explained by the predicted frequencies and probabilities according to a purely random mechanism. Of 191 non-appearing two-amino-acid sequences in the ADH1, 119 (62.304%) and 52 (27.225%) sequences can be explained by the predicted frequencies and probabilities according to a purely random mechanism. Of 373 measured first-order Markov transition probabilities for the second amino acid in two-amino-acid sequences, three (0.804%) probabilities match the predicted conditional probabilities and therefore can be explained by a purely random mechanism. No more-than-two-amino-acid sequences can be explained by a purely random mechanism.


    INTRODUCTION
 TOP
 FOOTNOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 ACKNOWLEDGEMENTS
 REFERENCES
 
The enzymes involved in human alcohol metabolism have been the object of many studies, among which the mathematical approach has frequently been used. However, to the best of the author's knowledge, the cryptological approach has not yet been used, thus this study attempts to use this approach to analyse the human alcohol dehydrogenase {alpha}-chain (ADH1).

The human alcohol dehydrogenase {alpha}-chain (EC 1.1.1.1, ADH1) is composed of 374 amino acids (Ikuta et al., 1986Go; Von Bahr-Lindstrom et al., 1986Go; Matsuo and Yokoyama 1989Go; Yasunami et al., 1990Go); any two amino acids in order can construct a two-amino-acid sequence, thus a total of 373 sequences can be constructed, i.e. the first and second, the second and third, etc. Furthermore, any three amino acids in order can construct a three-amino-acid sequence, thus a total of 372 sequences can be constructed, i.e. the first, second and third; the second, third and fourth; etc. These considerations can be extended to more-than-three-amino-acid sequences, and equal opportunity is given to all amino acids.

The rationale for adopting such a treatment of amino-acid sequences is because we do not know whether (1) an amino acid ‘word' can be constructed by three amino acids, as it is in DNA by three elements, so at this stage an amino acid ‘word’ may be constructed by any number of amino acids; (2) we do not know whether there is ‘punctuation’ and ‘space’ in the amino acid sequence of a protein, so we do not know where an amino acid ‘word’ begins and finishes and, as far as is known, an amino acid ‘word’ can begin and finish anywhere. Therefore this is the first step for the attempted understanding of protein wording with cryptological methods.

In the ideally random situation, in which two amino acids in a two-amino-acid sequence could be constructed from any one of 20 amino acids, there would be 400 (20 4;x 20) possible sequences (combinations). Naturally, any two-amino-acid sequence in ADH1 should be one of these 400 possible sequences, and any two-amino-acid sequence that does not appear in ADH1 should also be one of these 400 possible sequences. If each two-amino-acid sequence had the same probability of occurring in ADH1, each two-amino-acid sequence would be expected to appear about 0.933 times (373/400). Similarly, if three amino acids in a three-amino-acid sequence could be randomly constructed from any one of 20 amino acids, there would be 8000 (20 4;x 4;20 4;x 4;20) possible sequences (combinations). Naturally, any three-amino-acid sequence in ADH1 should be one of these 8000 possible sequences and any three-amino-acid sequence that does not appear in ADH1 should also be one of these 8000 possible sequences. If each three-amino-acid sequence had the same probability of appearing in ADH1, this probability would be about 0.047 (372/8000). Similar reasoning can also be applied for more-than-three-amino-acid sequences.

Not surprisingly, some kinds of amino-acid sequences do not appear at all in ADH1, not only because ADH1 does not have a sufficiently long amino acid structure to have all possible combinations, but also, more importantly, because the evolution process determines the preference of some particular amino-acid sequences, resulting in some of these appearing more frequently.

In the case of the ADH1, there are 31 alanines (A). If a two-amino-acid sequence of ‘AA’ were constructed by a purely random mechanism, ‘AA’ would be expected to occur with a frequency of 2.486 (31/374 4;x 4;30/373 4;x 4;373), i.e. the ‘AA’ would be expected to appear twice. In the real-life situation, ‘AA’ appears five times, so the construction of ‘AA’ does not follow a purely random mechanism, but appears to follow a functional and evolutionary purpose. By contrast, there are 23 serines (S) in ADH1, so the probability of random construction of ‘AS’ would be expected to be 1.906 (31/374 4;x 4;23/373 4;x 4;373), i.e. the ‘AS’ should be expected to appear twice, which is true in the real-life situation. The construction of ‘AS’ can be explained by a purely random mechanism.

It is very easy to say that no amino-acid sequences of a protein are constructed by a purely random mechanism, but it is not easy to deduce, for example, what percentage of two-amino-acid sequences in a protein is constructed by a purely random mechanism and what percentage is not. Thus the first question we should try to answer in this study is what percentage of amino-acid sequences in ADH1 can be explained by a purely random mechanism and what percentage cannot, by comparing predicted probabilities and frequencies with the measured probabilities and frequencies. Following this, we are also interested in what percentage of non-appearing amino-acid sequences in ADH1 can be explained by a purely random mechanism and what percentage cannot, by comparing predicted probabilities and frequencies with the measured probabilities and frequencies.

In an amino-acid sequence, the issue of which amino acid is more likely to follow a preceding amino acid is also interesting. In the ideally random situation, any amino acid could be possible, thus the probability of following a preceding amino acid is 1/20. In the case of the ADH1, there are 36 glycines (G). Thus a ‘G’ would have the probability of 0.097 (36/373) of following a preceding amino acid; for example, an ‘A’, this probability is true in the real-life situation for ‘AS’, so the fact that a ‘G’ follows a preceding ‘A’ can be explained by a purely random mechanism. By contrast, an ‘A’ would have the probability of 0.081 (30/373) to follow a preceding ‘A’ according to a purely random mechanism. But in the real-life situation, ‘A’ has the probability of 0.161 to follow a preceding ‘A’; this real-life probability is what the Markov chain is concerned with (i.e. the first-order Markov chain transition probability). Thus the second question we should try to address in this study is what percentage of the Markov transition probability can be explained by a purely random mechanism and what percentage cannot, by comparing predicted conditional probabilities with measured Markov transition probabilities.


    MATERIALS AND METHODS
 TOP
 FOOTNOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 ACKNOWLEDGEMENTS
 REFERENCES
 
The amino acid sequence of the ADH1 was obtained from the Swiss-Protein, access number P07327 (Bairoch and Apweiler, 1999Go).

Measuring two-, three- and four-amino-acid sequences
The measurement of two-, three- and four-amino-acid sequences in the ADH1 was conducted as stated in the Introduction. In the case of two-amino-acid sequences, the first and second amino acids, the second and third, the third and fourth, until the 376th and 377th were recorded, and their frequencies and probabilities were calculated. No measurement on more-than-four-amino-acid sequences was conducted, because no repetition regarding more-than-four-amino-acid sequences was found; thus each more-than-four-amino-acid sequence is unique in this sequence.

Calculating possible two-, three- and four-amino-acid sequences
Because all 20 amino acids exist in the ADH1 and the number of each amino acid is >1, there are 400 (202) possible sequences for two-amino-acid sequences, but 7600 (202 4;x 4;19) possible sequences for three-amino-acid sequences, because of the presence of only two tryptophans (W) in ADH1, and of 152 4;000 (203 4;x 4;19) for four-amino-acid sequences.

Calculating predicted probability and frequency
The predicted probability was calculated according to the random mechanism as stated in the Introduction. In the case of a two-amino-acid sequence, there are 31 alanines (A) and nine arginines (R) in the ADH1. For any position in the ADH1, the predicted probabilities for ‘AA’, ‘AR’, ‘RR’ and ‘RA’ are 31/374 4;x 4;30/373, 31/374 4;x 4;9/373, 9/374 4;x 4;8/373 and 9/374 4;x 4;31/373, respectively. In the case of a three-amino-acid sequence, the predicted probability for ‘AAA’ is 31/374 4;x 4;30/373 4;x 4;29/372. The numbers of predicted probabilities are identical to the numbers of possible two-, three-, and four-amino-acid sequences, e.g. 400 (202) for two-amino-acid sequences.

The predicted frequency is the rounded integral value of the production of predicted probability and the total number of amino-acid sequences, thus the predicted frequency for ‘AA’ is 2 (31/374 4;x 4;30/373 4;x 4;373). Naturally, the predicted frequency is less accurate than the predicted probability; however, the predicted frequency is easier to use for the more-than-two-amino-acid sequences, because the predicted probability is extremely low.

Calculating predicted conditional probability
The predicted conditional probability for an amino acid to follow a preceding amino acid is calculated according to the random mechanism as stated in the Introduction. For example, there 31 alanines (A) and nine arginines (R) in ADH1: the predicted conditional probabilities for ‘AA’ and ‘RA’ are 30/373 and 31/373 for the second amino acid of ‘A’ in two-amino-acid sequences to follow an ‘A’ and an ‘R’, the predicted conditional probabilities for ‘AR’ and ‘RR’ are 9/373 and 8/373 for the second amino acid of ‘R’ to follow an ‘A’ and an ‘R’. The predicted conditional probability of the third amino acid of ‘A’ in a three-amino-acid sequence to follow ‘AA’ is 29/372. The numbers of predicted conditional probabilities are identical to the numbers of possible two-, three- and four-amino-acid sequences, e.g. 400 (202) for two-amino-acid sequences.

Calculating Markov transition probability
The Markov chain allows calculation of the transition probability from one state to another state (Ash, 1965Go; Feller, 1968Go; Csiszár and Körner, 1981Go; van der Lubbe, 1997Go). In the case of a two-amino-acid sequence, an amino acid has a certain probability of following a certain preceding amino acid, which gives a conditional probability (the first-order Markov chain), i.e. the probability of an amino acid occurring in a two-amino-acid sequence given a particular first amino acid is defined as [P(second amino acid/first amino acid)]. In the case of a three-amino-acid sequence, the second-order Markov chain can be defined, i.e. the probability of an amino acid occurring in a three-amino-acid sequence given a particular combination of first two amino acids is [P(third amino acid/first and second amino acids)].


    RESULTS
 TOP
 FOOTNOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 ACKNOWLEDGEMENTS
 REFERENCES
 
Two-amino-acid sequences and their first-order Markov chain transition probabilities
In ADH1, 191 of 400 (47.750%) possible two-amino-acid sequences (combinations) do not exist, 111 (27.750%) sequences appear once, 60 (15.000%) sequences twice, 20 (5.000%) sequences three times, nine (2.250%) sequences four times, eight (2.000%) sequences five times and one (0.250%) sequence six times. These sequences are not randomly distributed.

Of 373 two-amino-acid sequences in the ADH1, 92 (24.665%) and 32 (8.579%) sequences can be explained by their predicted frequencies and probabilities, respectively, according to a purely random mechanism. Of 191 non-appearing two-amino-acid sequences in the ADH1, 119 (62.304%) and 52 (27.225%) sequences can be explained by their predicted frequencies and probabilities respectively, according to a purely random mechanism.

The two-amino-acid sequences which do not match the predicted frequency are particularly interesting, especially when the difference between measured and predicted frequencies is equal to or larger than two, because the predicted frequency is the rounded value from the predicted probability and a difference being equal to 1 may be due to the rounding error. For example, the predicted frequency of ‘AA’ is 2, whereas the measured frequency of ‘AA’ is 5, and thus this difference should have some non-random underlying reason. Table I shows these two-amino-acid sequences in the ADH1. For example, the ‘AA’ has the measured probability of 0.013 (5/373 sequences).

Three (0.804%) first-order Markov chain transition probabilities match the predicted conditional probabilities and therefore can be explained by a purely random mechanism. The measured first-order Markov chain transition probabilities for the difference between measured and predicted frequencies are >=2 and are presented in Table 1Go; for example, if the first amino acid in a two-amino-acid sequence is ‘A’, then the probability for the second amino acid is ‘A’ is 0.161.


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Table 1. The measured frequency (MF), predicted frequency (PF) and the first-order Markov chain transition probabilities (MP) of two-amino-acid sequences, which have a difference of >=2 between measured and predicted frequencies in the alcohol dehydrogenase {alpha}-chain (ADH1)
 
Three-amino-acid sequences and their second-order Markov chain transition probabilities
7243 out of 7600 (95.303%) possible three-amino-acid sequences do not exist in the ADH1, 342 (4.500%) sequences appear once and 15 (0.197%) sequences twice. These sequences are not randomly distributed.

The possible maximum predicted probability and frequency of three-amino-acid sequences are 0.001 (36/374 4;x 4;36/373 4; x 4;35/372, for example ‘VGV’) and 0 (40/377 4;x 4;39/376 4;x 4; 38/375 4;x 4;375), respectively. Thus no three-amino-acid sequence in the ADH1 can be explained by a purely random mechanism. Table 2Go shows the occurrence of more-than-once sequences, for example, ‘AAG’ has the measured probability of 0.005 (2/372).


View this table:
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Table 2. The measured probability (MF) and predicted probability (PF) and the second-order Markov chain transition probabilities (MP) of three-amino-acid sequences in alcohol dehydrogenase {alpha}-chain
 
More-than-three-amino-acid sequences and their Markov chain transition probabilities
In fact, the possible maximum predicted probability and frequency for more-than-three-amino-acid sequence go to zero. The multiple appearance of more-than-three-amino-acid sequences in ADH1 cannot be attributed to the purely random mechanism. There is only one four-amino-acid sequence of ‘CKAA’ appearing twice, with a third-order Markov transition probability of 1.000.

No repetition was found regarding any more-than-four-amino-acid sequence, thus the human ADH1 has no mechanism to favour any specific combinations of them.


    DISCUSSION
 TOP
 FOOTNOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 ACKNOWLEDGEMENTS
 REFERENCES
 
This study attempted to answer two questions, which constitute the first step for understanding the wording of ADH1 by cryptological methods (Sinkov, 1966Go; van Tilborg, 1989Go; van der Lubbe, 1997Go). A perfect code developed by cryptography can resist any statistical test for random distribution, whereas the ADH1 sequence cannot do so. This allows the possibility to be deciphered.

By comparing the predicted frequency/probability with the measured frequency/probability, one can know which amino acid ‘word’ is favoured by a protein. Also, the most frequently appearing amino-acid sequences may serve as the potential targets of new drugs, because the drug would have more chances to interact with them. This study also suggests that the amino-acid sequences which have the biggest difference between predicted and measured frequencies/probabilities could serve as potential targets of new drugs, because these sequences are highly evolved for the difference between predicted and measured frequencies/probabilities. Moreover, as the Markov chain transition probability increases from lower order to higher order, the random chance for an amino acid to follow an arbitrary amino acid decreases as the length of amino-acid sequence increases. Hence, a mutation is unlikely to occur at the amino acid with a high Markov chain transition probability, and the amino acids with high Markov transition probabilities may serve as potential targets of new drugs, because they are unlikely to change into other amino acids. Nevertheless, some sequences may play a role somewhat similar to ‘grammar structure' or ‘vowels' or ‘auxiliary verbs' in languages.

It is somewhat unexpected that 24.665% and 8.579% of two-amino-acid sequences can be explained by the predicted frequencies and probability according to a purely random mechanism, and 0.804% of measured first-order Markov chain transition probabilities can be explained by a purely random mechanism. This is different from the view that no amino-acid sequences in a protein are constructed by a purely random mechanism, but it may explain why some mutations have no effect on protein function.

If a possible amino-acid sequence does not appear in a protein, naturally it is not needed for its function. For example, 191 two-amino-acid sequences (combinations) do not exist in the ADH1, so they may not be needed. Of these ‘useless' sequences, some can be explained by a purely random mechanism and some cannot; for example, the predicted frequency of ‘AE' is 2, whereas the measured frequency is 0, so the lack of ‘AE' cannot be explained by a purely random mechanism. By contrast, both predicted and measured frequencies of ‘AW' are 0, thus the lack of ‘AW’ can be explained by a purely random mechanism. This also differs from the traditional view. It appears likely from such an analysis that the protein may not be as highly structured as was previously thought. This also leaves an interesting question of whether the inclusion and exclusion of non-random sequences in a protein may lead to functional changes; this, however, requires scanning a total family of proteins to draw a firm conclusion.

For more-than-two-amino-acid sequences, no sequence occurring in the ADH1 can be explained by a purely random mechanism, but, on the other hand, all the sequences which do not appear in the ADH1 can be explained by a purely random mechanism. Thus, we may argue that the reason that most more-than-two-amino-acid sequences are not selected for the construction of the protein is due to a purely random mechanism.

As amino-acid sequences are functionally and evolutionarily biased, it would be interesting to know why ADH1 favours some two-, three- and four-amino-acid sequences leading to the difference between the predicted and measured frequency/probability, especially the repeated four-amino-acid sequence, where the chance of repetition is extremely low. Unfortunately, current knowledge does not provide a clear answer to this.

The reason for the deliberate differences between the measured and predicted probabilities/frequencies may be attributed to numerous factors, such as the high conservation pressure, and the comparison between different species would be particularly important for this line of analysis. An interesting research line to follow this clue, related to alcohol metabolism, would be to correlate the relationship between ADH function and amino-acid sequences which have the biggest difference between measured and predicted frequency/probability, for example, the relationship between amino-acid sequences and Michaelis–Menten constants of different ADH.

In this study, we used the probabilistic method to analyse only the ADH1 sequence, which seems somewhat different from the statistical method, which requires a huge amount of data and then making statistical inference. In most real-life situations, we can know the difference between two parameters without employing statistics, such as the speeds of aircraft and bicycle. When we have a difficulty in distinguishing the difference, we can employ statistical methods, and the statistical power increases as the sample size increases. In fact, the difference between predicted and measured probabilities in this study is more significant than that using the criterion of P 4;< 4;0.05. In the general case of this type of analysis, the probability of a repetition of two-amino-acid sequences is 0.0025 (1/20 4;x 4;1/20). In the case of ADH1, the probabilities for the first, second, and third repetitions of ‘AA’ are 0.0059 (29/372 4;x 4;28/371), 0.0051 (27/370 4;x 4;26/369) and 0.0044 (25/368 x 24/367), respectively. All these probabilities are much less than 0.05.

In this study, ADH1 has been chosen for analysis. Such a choice is mainly due to mathematical simplicity, i.e. the number of the least-appearing amino acid in ADH. ADH4, ADH5 and ADH6 have three, four and three tryptophans (W) respectively, thus the possible combinations for three-amino-acid sequences are 8000; whereas ADH1, ADH2, ADH3 and ADH7 have two tryptophans (W), thus the possible combinations for three-amino-acid sequences are 7600. Among ADH1, ADH2, ADH3 and ADH7, it is likely that ADH1 would be the first choice, although any of them could well have served for this study.

At this stage, we are still unable to apply this analysis to the DNA sequence, although it will certainly be our aim. Technically, the DNA sequence is much longer than its corresponding protein sequence, so this needs another algorithm. However, the comparison in predicted and measured frequencies/probabilities between DNA and its corresponding protein sequences will doubtless give more meaningful insights.

Although we have used various mathematical models in our previous studies on ethanol (Wu 1997aGo,bGo, 1998aGo,Wu bGo, 1999Go, 2000Go), as the first step in using the cryptological method, this first approach over-simplifies the situation, restricting the value of the conclusion that may be drawn from it. Nevertheless, it should form the basis for the development of more sophisticated models to assess structures in the future.


    ACKNOWLEDGEMENTS
 TOP
 FOOTNOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 ACKNOWLEDGEMENTS
 REFERENCES
 
The author wishes to thank S.-M. Yan, MD, PhD at Department of Pathology, University of Udine for helpful discussion. The help of Electronic Engineer P. Cossettini at the Center for Advanced Research in Space Optics, Trieste, Italy is kindly acknowledged. Special thanks go to Professor P. Garrett at University of Minnesota, USA for providing his lecture notes on cryptology. The author is very grateful to the Editor and Referees for their valuable comments and correcting the English version of this typescript.


    FOOTNOTES
 TOP
 FOOTNOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 ACKNOWLEDGEMENTS
 REFERENCES
 
* Author to whom correspondence should be addressed at: Laboratorie de Toxicocinétique et de Pharmacocinétique, Faculté de Pharmacie, Université de la Mediterranée (Aix-Marseille II), 27 Boulevard Jean Moulin, 13385 Marseille Cedex 05, France. Back


    REFERENCES
 TOP
 FOOTNOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 ACKNOWLEDGEMENTS
 REFERENCES
 
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Csiszár, I. and Körner, J. (1981) Information Theory. Academic Press, New York.

Feller, W. (1968) An Introduction to Probability Theory and its Applications, 3rd edn, Vol. I. Wiley, New York.

Ikuta, T., Szeto, S. and Yoshida, A. (1986) Three human alcohol dehydrogenase subunits: cDNA structure and molecular and evolutionary divergence. Proceedings of the National Academy of Sciences of the USA 83, 634–638.[Abstract]

Matsuo, Y. and Yokoyama, S. (1989) Molecular structure of the human alcohol dehydrogenase 1 gene. FEBS Letters 243, 57–60.[ISI][Medline]

Sinkov, A. (1966) Elementary Cryptanalysis — A Mathematical Approach. Mathematical Association of America, Yale University Press, New Haven.

van der Lubbe, J. C. A. (1997) Information Theory. Cambridge University Press, Cambridge.

van Tilborg, H. C. A. (1989) An Introduction to Cryptology. Kluwer, Boston.

Von Bahr-Lindstrom, H., Hoeoeg, J.-O., Heden, L.-O., Kaiser, R., Fleetwood, L., Larsson, K., Lake, M., Holmquist, B., Holmgren, A., Hempel, J., Vallee, B. L. and Joernvall, H. (1986) cDNA and protein structure for the alpha subunit of human liver alcohol dehydrogenase. Biochemistry 25, 2465–2470.[ISI][Medline]

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Yasunami, M., Kikuchi, I., Sarapata, D. and Yoshida, A. (1990) The human class I alcohol dehydrogenase gene cluster: three genes are tandemly organized in an 80-kb-long segment of the genome. Genomics 7, 152–158.[ISI][Medline]





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