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1. Suppose that protein X, which plays an important role in cell sig- nalling, can be in any of three states, consisting of the native form (N), a phosphorylated form (P) and an acetylated form (A). If the protein is in the native form the probability that it will next be in the phospho- rylated state is 0.7, while the probability that it will become acetylated is 0.3. The phosphorylated form cannot be acetylated without first be- ing dephosphorylated (i.e. returning to the N state) and, similarly, the acetylated form cannot become phosphorylated without first returning to the N state. (a) Provide a definition of a discrete Markov chain. а [5 marks] (b) Assuming that the above process can be described by a discrete Markov chain, write down the transition probability matrix of the chain and draw a diagram showing the three states, with arrows labelling the possible transitions between the states. [5 marks] (c) If the protein is currently phosphorylated, what is the probability that after two steps along the chain it will be acetylated? Relate your answer to the Chapman-Kolmogorov equations. [5 marks] (d) Define the stationary distribution of a Markov chain [5 marks] (e) Is the Markov chain in this question irreducible? Justify your answer by reference to the definition of irreducibility.