Hidden Markov Models and Protein Secondary Structure Prediction

Abstract

Hidden Markov model (HMM) is for inferring hidden states of a Markov model based on observed data. For example, intron and exon are hidden states and need to be inferred from the observed nucleotide sequences. Similarly, secondary structural elements such as alpha helices and beta sheets are hidden states and need to be inferred from observed amino acid sequences. The accuracy of HMM in inferring hidden states depends on the transition probability matrix and emission probability matrix derived from training HMM with representative observations. If different states have very different probability to transit into each other, and if the emission probability matrix of the hidden states are highly different from each other, then HMM can be quite accurate. This chapter details the key algorithms used in HMM, such as Viterbi algorithm for reconstructing the hidden states and the forward algorithm to compute the probability of the observed sequence of events. Both Viterbi and forward algorithms are dynamic programming algorithms that we were first exposed to in the chapter on sequence alignment. HMM is applied to reconstructing protein secondary structure as an illustrative example.

Postscript

We see informal applications of HMM in our daily life. By making a telephone call, parents with their ears trained from many years of caring for their children can often detect hidden troubles of their children based only on the voice of the latter. In contrast, people unfamiliar to each other often find it frustratingly difficult to make sense of each other’s behavior, and misunderstanding ensues. The most agonizing moment for me watching the movie “Waterloo Bridge” is when Lady Margaret Cronin failed to detect the distress experienced by Myra.

I once heard a story about the late Stephen Jay Gould giving a talk on evolution to the congregation of an All Souls Church in New York. When the guest and hosts were having lunch together, someone suggested that they should go around the table to introduce themselves. At that point Gould said something that seemed to be extraordinarily rude, something to the effect that he did not really care who the hosts were as he would never see them again. The name of Gould instantly became synonymous to rudeness among the church members.

However, soon after the incident, the members of the church learned from the newspaper that Gould had died of cancer and that his lecture in the church was in fact Gould’s last public engagement – he reserved all the rest of his time to finish his 1464-page magnum opus entitled “The Structure of Evolutionary Theory.” They realized that, at that moment when the seemingly rude remark erupted, Gould must have felt melancholy, as everyone would, knowing that his days were numbered, and that he was merely stating a heartbreaking truth that he would never see anyone around the table again.

In the HMM parlance, the remark by Gould is the emitted event from which the listeners should ideally be able to infer his hidden melancholy state of mind. An inference they did make, but it is wrong. Worse, they did not realize that it was wrong, otherwise they could have prayed more for Gould.

Stephen Jay Gould had spent all his life fighting two kinds of fundamentalists, the religious fundamentalists who believe that God is a micromanager of everything and that the Bible literally encompasses all truths in nature, and the evolutionary fundamentalists who believe that every bit of biodiversity manifests adaptation and results from natural selection. All Souls Church is perhaps the equivalent of Gould in the religious field. I would have expected Gould to have an easy time with members of this very liberal church. Yet misunderstanding still arose, and the misunderstanding could have lasted for a long time if Gould’s death had not been so well publicized.

It is truly enigmatic and paradoxical that, with the advanced computational algorithms helping us to infer the hidden, we still do not seem to make any progress in understanding each other and in understanding ourselves. The ancient Greek sage, Plato, had discovered the root cause of all misunderstanding and evil. It is called arrogance or the illusion that we are better than others. Plato illustrated his point with his famous allegory of the cave.

Imagine prisoners chained inside a cave since childhood, with their heads immobilized in such a way that their eyes were fixed on a gigantic wall. Immediately behind the prisoners was a road along which men, animals, and other things traveled. Behind the road was an enormous fire that projected the shadow of the travelers to the wall that the prisoners were facing. Also, the voice of the travelers was echoed from the wall in such a way that the prisoners believed that the words came from the shadows. Gradually, the prisoners became quite good at identifying the travelers by their shadows and voices. The shadows and the voices, as well as the interpretation of the shadows and voices by the prisoners, constituted the world of reality in the mind of the prisoners.

Now suppose a prisoner was freed and went outside the cave. Gradually he would comprehend a new reality from what he could sense. Once thus enlightened, he naturally would want to return to the cave to convey the new reality to his fellow prisoners. Unfortunately, once back in the cave, he found himself much less able to identify the travelers by their shadows than his fellow prisoners. Being thus perceived as inferior and stupid by his fellow prisoners, he failed completely in communicating the new reality to his fellow prisoners who believed to know better. The fellow prisoners were too arrogant to listen.

It is the arrogance in the mind of the prisoners that prevents them from comprehending the new reality hidden from them. It is the arrogance in the mind of the religious fundamentalists and the evolutionary fundamentalists that prevents them from understanding each other. It is the arrogance in the mind of the presidents and prime ministers that prolongs the misunderstanding among nations. Arrogance is Satan in Christianity.

I have had the privilege of meeting some of the religious and evolutionary fundamentalists. What is particularly ironical is that they all know Plato’s allegory of the cave quite well, but all point to themselves as the enlightened who has seen the real world and sneer at the other party as the chained prisoners with restricted vision.

None of us is omnipresent and eternal, and our view of the world is consequently the same as that of the chained prisoners. Without grasping this painful but basic truth, we will misinterpret what we see or hear, either with HMM or not.