a)Write the transition matrix. Is this an Ergodic Markov chain? Explain your answerb)Starting from a group of nAChR's where 34% are in the C1 state, 42% in the C2 state and 24% in the O state, calculate the probability of finding channels in open state after 4 microseconds. Show all of your work.c)Find the fixed vector of this matrix (show your work)d)Starting with the initial probability distribution of C1=0.24; C2=0; O=0.76, predict what fraction of ion channels will be in each state after 1000 time steps (hint: the probability distribution converges after about 100 time steps) 7. Ion channels are trans-membrane proteins with a hollow channel running through the interior that allows polar ionsto traverse the nonpolar lipid bilayer. The channels may exist in several conformations (shapes), which may either allowor block passage of ions through the channel. Transitions between the conformational states are random (as iseverything on the molecular scale), but they may be induced by external events, such as binding of another molecule,or a change in membrane potential. In this section we will analyze a simple model of nicotinic acetylcholine receptor(NACHR), which is an important ion channel involved in transmission of signals between neurons. The opening of thechannel is induced by binding of the neurotransmitter acetylcholine (Ach). The Markov chain model has three states:closed (C1), closed with Ach bound (C2), and open (O). Suppose that at a certain concentration of Ach, the transitionprobabilities between the different states per two microsecond are as follows: 0.04 (from C1 to C2), 0.07 (from C2 toC1), 0.12 (from C2 to O) and 0.02 (from O to C2); the other transition probabilities between different states are 0.

Answered: a)Write the transition matrix. Is this…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

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a)Write the transition matrix. Is this an Ergodic Markov chain? Explain your answer

b)Starting from a group of nAChR's where 34% are in the C1 state, 42% in the C2 state and 24% in the O state, calculate the probability of finding channels in open state after 4 microseconds. Show all of your work.

c)Find the fixed vector of this matrix (show your work)

d)Starting with the initial probability distribution of C1=0.24; C2=0; O=0.76, predict what fraction of ion channels will be in each state after 1000 time steps (hint: the probability distribution converges after about 100 time steps)

7. Ion channels are trans-membrane proteins with a hollow channel running through the interior that allows polar ions
to traverse the nonpolar lipid bilayer. The channels may exist in several conformations (shapes), which may either allow
or block passage of ions through the channel. Transitions between the conformational states are random (as is
everything on the molecular scale), but they may be induced by external events, such as binding of another molecule,
or a change in membrane potential. In this section we will analyze a simple model of nicotinic acetylcholine receptor
(NACHR), which is an important ion channel involved in transmission of signals between neurons. The opening of the
channel is induced by binding of the neurotransmitter acetylcholine (Ach). The Markov chain model has three states:
closed (C1), closed with Ach bound (C2), and open (O). Suppose that at a certain concentration of Ach, the transition
probabilities between the different states per two microsecond are as follows: 0.04 (from C1 to C2), 0.07 (from C2 to
C1), 0.12 (from C2 to O) and 0.02 (from O to C2); the other transition probabilities between different states are 0.

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