Computing Characterizations of Drugs for Ion Channels and Receptors Using Markov Models pp 193-221 | Cite as
Mutations Affecting the Mean Open Time
Abstract
The core difficulty here is that a CO-mutation does not change the mean open time of the channel. A closed state blocker is therefore well suited because such a blocker does not affect the mean open time. However, for an OC-mutation, an increased mean open time is part of the problem and a closed state blocker is not the solution, simply because it cannot affect the mean open time. Rather, an open state blocker must be used.
In this chapter, we will explain the notion of mean open time and study mutations that lead to an increased open probability and an increased mean open time. We will show that open state blockers are optimal for such mutations.
13.1 The Mean Open Time
13.1.1 Mean Open Time for More Than One Open State
13.1.1.1 Special Cases
13.2 Numerical Experiments
It is useful to have a look at the mean open time computed in specific numerical experiments to determine how well it is represented by the theoretical value derived above. Similarly, it is useful to consider how well the theoretical equilibrium open probability represents the data we observe in actual computations. In this section, we will present experiments that hopefully clarify these matters.
13.2.1 Mean Open Time and Equilibrium Open Probability: Theoretical Values Versus Sample Mean Values
13.2.2 The Closed to Open Rate k_{ co } Does Not Affect the Mean Open Time
13.2.3 The Mean Open Time in the Presence of Two Open States
13.2.4 Changing the Mean Open Time Affects the Dynamics of the Transmembrane Potential
13.3 Changing the Mean Open Time Affects the Probability Density Functions
13.4 Theoretical Drugs for OC-Mutations
We have seen earlier that when mutations increase the open probability by increasing the reaction rate from C to O (k_{ co }), the effect of the mutation can be completely repaired by using an optimal closed state blocker. Now we are interested in a mutation that increases the open probability by reducing the reaction rate from O to C \(\left (k_{oc}\right ).\) Such a mutation increases both the open probability and the mean open time and we will observe that a closed state blocker is unable to repair the effect of such a mutation.
13.4.1 The Theoretical Closed State Blocker Does Not Work for the OC-Mutation
13.4.2 The Theoretical Open State Blocker Repairs the Effect of the OC-Mutation
13.4.3 The Theoretical Open State Blocker Is Optimal
13.4.3.1 The Probability Density Function of the Blocked State Is Proportional to the Probability Density Function of the Wild Type Closed State
In the right panel of the figure, we show the graph of ρ_{ c } for the wild type (solid line) and for the mutant case in the presence of the open blocker. We show both ρ_{ c } and ρ_{ b }. We note that these graphs seem to have the same shape and we will show that they indeed differ only by a constant.
13.4.4 Stochastic Simulations Using the Optimal Open State Blocker
The graphs show that the effect of the mutation is repaired using the drug (13.25); the solutions are not identical and this is reasonable, since a random number generator is involved in updating the state of the Markov model and therefore two computed solutions will not be identical (not even two wild type solutions). However, we note that the qualitative properties of the upper and lower solutions are similar, whereas the mutant case is different due to the increased open probability and prolonged mean open time.
13.5 Inactivated States and Mean Open Time
13.5.1 A Theoretical Open State Blocker
We observed above that to repair the effect of changes in the mean open time, it is necessary to use an open state blocker. The reason for this is that neither a closed blocker nor an inactivated blocker has any effect on the mean open time and, therefore, it is inconceivable that such blockers can repair the effect of a mutation on the mean open time. An open state blocker directly affects the mean open time and the drug must be tuned to repair the effect of the mutation.
13.5.2 Probability Density Functions Using the Open State Blocker
Statistical properties of ρ_{ o } for the cases shown in Fig. 13.13
k = 1 | k_{ co } = 10 | k_{ io } = 0. 1 | |||||
---|---|---|---|---|---|---|---|
π _{ o } | E _{ o } | π _{ o } | E _{ o } | π _{ o } | E _{ o } | ||
WT | 0.333 | 16.366 | 0.476 | 22.995 | 0.083 | -12.867 | |
MT | 0.476 | 23.272 | 0.833 | 31.074 | 0.333 | 17.702 | |
MT+OB | 0.333 | 16.366 | 0.476 | 23.169 | 0.083 | -9.225 |
13.5.3 Stochastic Simulations Using the Open State Blocker
13.6 Notes
Footnotes
- 1.
The index s here is used to indicate sample, since these are values for a specific computation and not the theoretical value computed above.
References
- 42.J. Keener, J. Sneyd, Mathematical Physiology (Springer, New York, 2009)Google Scholar
- 85.G.D. Smith, Modeling the stochastic gating of ion channels. In Computational Cell Biology, vol. 20 of Interdisciplinary Applied Mathematics, chapter 11, pp. 285–319, ed. by C.P. Fall, E.S. Marland, J.M. Wagner, J.J. Tyson (Springer, New York, 2002)Google Scholar
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