Question

Suppose that a one-celled organism can be in one of two states-either \(A\) or \(B\). An individual in state \(A\) will change to state \(B\) at an exponential rate \(\alpha ;\) an individual in state \(B\) divides into two new individuals of type \(A\) at an exponential rate \(\beta .\) Define an appropriate continuous- time Markov chain for a population of such organisms and determine the appropriate parameters for this model.

Short answer

The state space of the continuous-time Markov chain (CTMC) is defined as a vector \((n_A, n_B)\), representing the number of individuals in state \(A\) and \(B\). The transition rates are \(q( (n_A, n_B), (n_A-1, n_B+1) ) = \alpha n_A\) for an \(A\) organism changing to \(B\), and \(q( (n_A, n_B), (n_A+2, n_B-1) ) = \beta n_B\) for a \(B\) organism dividing into two \(A\) organisms. Transition probabilities can be calculated as \[P_{(n_A, n_B) \to (n_A-1, n_B+1)}(t) = e^{(-\alpha n_A \cdot t)}\] and \[P_{(n_A, n_B) \to (n_A+2, n_B-1)}(t) = e^{(-\beta n_B \cdot t)}\] for the respective transitions. This CTMC model can be used to analyze various properties of the one-celled organism population.

Step-by-step solution

Define the state space of the Markov chain

We will define the state of the system as a vector \((n_A, n_B)\), where \(n_A\) indicates the number of individuals in state \(A\), and \(n_B\) indicates the number of individuals in state \(B\). The state space of the continuous-time Markov chain (CTMC) will be all possible combinations of \((n_A, n_B)\). For example, if there are 2 organisms in state \(A\) and 3 organisms in state \(B\), the state of the system would be represented as \((2, 3)\).

Determine the transition rates between states

Since the individuals in state \(A\) transition to state \(B\) and the individuals in state \(B\) divide into two new organisms of type \(A\), there are two types of transitions that can occur: 1. An organism in state \(A\) transitions to state \(B\) at rate \(\alpha\), which decreases the number of organisms in state \(A\) by 1 and increases the number of organisms in state \(B\) by 1. 2. An organism in state \(B\) divides into two new organisms of type \(A\) at rate \(\beta\), which increases the number of organisms in state \(A\) by 2 and decreases the number of organisms in state \(B\) by 1. Therefore, given a state \((n_A, n_B)\), the transitions rates are: - For the first type of transition: \(q( (n_A, n_B), (n_A-1, n_B+1) ) = \alpha n_A\) - For the second type of transition: \(q( (n_A, n_B), (n_A+2, n_B-1) ) = \beta n_B\)

Calculate the transition probabilities of the Markov chain

Now that we have the transition rates, we can find the transition probabilities for the CTMC. The transition probabilities are given by the following formula: \[P_{ij}(t) = e^{(q(i, j) \cdot t)}\] Using this formula, we can compute the transition probabilities between states: 1. The probability of going from state \((n_A, n_B)\) to \((n_A-1, n_B+1)\) is: \[P_{(n_A, n_B) \to (n_A-1, n_B+1)}(t) = e^{(-\alpha n_A \cdot t)}\] 2. The probability of going from state \((n_A, n_B)\) to \((n_A+2, n_B-1)\) is: \[P_{(n_A, n_B) \to (n_A+2, n_B-1)}(t) = e^{(-\beta n_B \cdot t)}\] The continuous-time Markov chain has been defined and the appropriate parameters have been determined. The CTMC represents the behavior of the population of one-celled organisms and can be used to analyze various properties of the population, such as the expected number of organisms in each state over time, equilibrium states, and more.