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Computing Characterizations of Drugs for Ion Channels and Receptors Using Markov Models.

Bibliographic Details
Main Author: Tveito, Aslak.
Other Authors: Lines, Glenn T.
Format: eBook
Language:English
Published: Cham : Springer International Publishing AG, 2016.
Edition:1st ed.
Series:Lecture Notes in Computational Science and Engineering Series
Subjects:
Electronic books.
Online Access:Click to View
Table of Contents:
  • Intro
  • Preface
  • Acknowledgments
  • Contents
  • 1 Background: Problem and Methods
  • 1.1 Action Potentials
  • 1.2 Markov Models
  • 1.2.1 The Master Equation
  • 1.2.2 The Master Equation of a Three-State Model
  • 1.2.3 Monte Carlo Simulations Based on the Markov Model
  • 1.2.4 Comparison of Monte Carlo Simulations and Solutions of the Master Equation
  • 1.2.5 Equilibrium Probabilities
  • 1.2.6 Detailed Balance
  • 1.3 The Master Equation and the Equilibrium Solution
  • 1.3.1 Linear Algebra Approach to Finding the Equilibrium Solution
  • 1.4 Stochastic Simulations and Probability Density Functions
  • 1.5 Markov Models of Calcium Release
  • 1.6 Markov Models of Ion Channels
  • 1.7 Mutations Described by Markov Models
  • 1.8 The Problem and Steps Toward Solutions
  • 1.8.1 Markov Models for Drugs: Open State and Closed State Blockers
  • 1.8.2 Closed to Open Mutations (CO-Mutations)
  • 1.8.3 Open to Closed Mutations (OC-Mutations)
  • 1.9 Theoretical Drugs
  • 1.10 Results
  • 1.11 Other Possible Applications
  • 1.12 Disclaimer
  • 1.13 Notes
  • 2 One-Dimensional Calcium Release
  • 2.1 Stochastic Model of Calcium Release
  • 2.1.1 Bounds of the Concentration
  • 2.1.2 An Invariant Region for the Solution
  • 2.1.3 A Numerical Scheme
  • 2.1.4 An Invariant Region for the Numerical Solution
  • 2.1.5 Stochastic Simulations
  • 2.2 Deterministic Systems of PDEs Governing the Probability Density Functions
  • 2.2.1 Probability Density Functions
  • 2.2.2 Dynamics of the Probability Density Functions
  • 2.2.3 Advection of Probability Density
  • 2.2.3.1 Advection in a Very Special Case: The Channel Is Kept Open for All Time
  • 2.2.3.2 Advection in Another Very Special Case: The Channel Is Kept Closed for All Time
  • 2.2.3.3 Advection: The General Case
  • 2.2.4 Changing States: The Effect of the Markov Model
  • 2.2.5 The Closed State.
  • 2.2.6 The System Governing the Probability Density Functions
  • 2.2.6.1 Boundary Conditions
  • 2.3 Numerical Scheme for the PDF System
  • 2.4 Rapid Convergence to Steady State Solutions
  • 2.5 Comparison of Monte Carlo Simulations and Probability Density Functions
  • 2.6 Analytical Solutions in the Stationary Case
  • 2.7 Numerical Solution Accuracy
  • 2.7.1 Stationary Solutions Computed by the Numerical Scheme
  • 2.7.2 Comparison with the Analytical Solution: The Stationary Solution
  • 2.8 Increasing the Reaction Rate from Open to Closed
  • 2.9 Advection Revisited
  • 2.10 Appendix: Solving the System of Partial Differential Equations
  • 2.10.1 Operator Splitting
  • 2.10.2 The Hyperbolic Part
  • 2.10.3 The Courant-Friedrichs-Lewy Condition
  • 2.11 Notes
  • 3 Models of Open and Closed State Blockers
  • 3.1 Markov Models of Closed State Blockers for CO-Mutations
  • 3.1.1 Equilibrium Probabilities for Wild Type
  • 3.1.2 Equilibrium Probabilities for the Mutant Case
  • 3.1.3 Equilibrium Probabilities for Mutants with a Closed State Drug
  • 3.2 Probability Density Functions in the Presence of a Closed State Blocker
  • 3.2.1 Numerical Simulations with the Theoretical Closed State Blocker
  • 3.3 Asymptotic Optimality for Closed State Blockers in the Stationary Case
  • 3.4 Markov Models for Open State Blockers
  • 3.4.1 Probability Density Functions in the Presence of an Open State Blocker
  • 3.5 Open Blocker Versus Closed Blocker
  • 3.6 CO-Mutations Does Not Change the Mean Open Time
  • 3.7 Notes
  • 4 Properties of Probability Density Functions
  • 4.1 Probability Density Functions
  • 4.2 Statistical Characteristics
  • 4.3 Constant Rate Functions
  • 4.3.1 Equilibrium Probabilities
  • 4.3.2 Dynamics of the Probabilities
  • 4.3.3 Expected Concentrations
  • 4.3.4 Numerical Experiments
  • 4.3.5 Expected Concentrations in Equilibrium.
  • 4.4 Markov Model of a Mutation
  • 4.4.1 How Does the Mutation Severity Index Influence the Probability Density Function of the Open State?
  • 4.4.2 Boundary Layers
  • 4.5 Statistical Properties as Functions of the Mutation Severity Index
  • 4.5.1 Probabilities
  • 4.5.2 Expected Calcium Concentrations
  • 4.5.3 Expected Calcium Concentrations in Equilibrium
  • 4.5.4 What Happens as μ-3mu→∞?
  • 4.6 Statistical Properties of Open and Closed State Blockers
  • 4.7 Stochastic Simulations Using Optimal Drugs
  • 4.8 Notes
  • 5 Two-Dimensional Calcium Release
  • 5.1 2D Calcium Release
  • 5.1.1 The 1D Case Revisited: Invariant Regions of Concentration
  • 5.1.2 Stability of Linear Systems
  • 5.1.3 Convergence Toward Two Equilibrium Solutions
  • 5.1.3.1 Equilibrium Solution for Closed Channels
  • 5.1.3.2 Equilibrium Solution for Open Channels
  • 5.1.3.3 Stability of the Equilibrium Solution
  • 5.1.4 Properties of the Solution of the Stochastic Release Model
  • 5.1.5 Numerical Scheme for the 2D Release Model
  • 5.1.5.1 Simulations Using the 2D Stochastic Model
  • 5.1.6 Invariant Region for the 2D Case
  • 5.2 Probability Density Functions in 2D
  • 5.2.1 Numerical Method for Computing the Probability Density Functions in 2D
  • 5.2.2 Rapid Decay to Steady State Solutions in 2D
  • 5.2.3 Comparison of Monte Carlo Simulations and Probability Density Functions in 2D
  • 5.2.4 Increasing the Open to Closed Reaction Rate in 2D
  • 5.3 Notes
  • 6 Computing Theoretical Drugs in the Two-Dimensional Case
  • 6.1 Effect of the Mutation in the Two-Dimensional Case
  • 6.2 A Closed State Drug
  • 6.2.1 Convergence as kbc Increases
  • 6.3 An Open State Drug
  • 6.3.1 Probability Density Model for Open State Blockers in 2D
  • 6.3.1.1 Does the Optimal Theoretical Drug Change with the Severity of the Mutation?
  • 6.4 Statistical Properties of the Open and Closed State Blockers in 2D.
  • 6.5 Numerical Comparison of Optimal Open and Closed State Blockers
  • 6.6 Stochastic Simulations in 2D Using Optimal Drugs
  • 6.7 Notes
  • 7 Generalized Systems Governing Probability Density Functions
  • 7.1 Two-Dimensional Calcium Release Revisited
  • 7.2 Four-State Model
  • 7.3 Nine-State Model
  • 8 Calcium-Induced Calcium Release
  • 8.1 Stochastic Release Model Parameterized by the Transmembrane Potential
  • 8.1.1 Electrochemical Goldman-Hodgkin-Katz (GHK) Flux
  • 8.1.2 Assumptions
  • 8.1.3 Equilibrium Potential
  • 8.1.4 Linear Version of the Flux
  • 8.1.5 Markov Models for CICR
  • 8.1.6 Numerical Scheme for the Stochastic CICR Model
  • 8.1.7 Monte Carlo Simulations of CICR
  • 8.2 Invariant Region for the CICR Model
  • 8.2.1 A Numerical Scheme
  • 8.3 Probability Density Model Parameterized by the Transmembrane Potential
  • 8.4 Computing Probability Density Representations of CICR
  • 8.5 Effects of LCC and RyR Mutations
  • 8.5.1 Effect of Mutations Measured in a Norm
  • 8.5.2 Mutations Increase the Open Probability of Both the LCC and RyR Channels
  • 8.5.3 Mutations Change the Expected Valuesof Concentrations
  • 8.6 Notes
  • 9 Numerical Drugs for Calcium-Induced Calcium Release
  • 9.1 Markov Models for CICR, Including Drugs
  • 9.1.1 Theoretical Blockers for the RyR
  • 9.1.2 Theoretical Blockers for the LCC
  • 9.1.3 Combined Theoretical Blockers for the LCC and the RyR
  • 9.2 Probability Density Functions Associated with the 16-State Model
  • 9.3 RyR Mutations Under a Varying Transmembrane Potential
  • 9.3.1 Theoretical Closed State Blocker Repairs the Open Probabilities of the RyR CO-Mutation
  • 9.3.2 The Open State Blocker Does Not Work as Well as the Closed State Blocker for CO-Mutations in RyR
  • 9.4 LCC Mutations Under a Varying Transmembrane Potential
  • 9.4.1 The Closed State Blocker Repairs the Open Probabilities of the LCC Mutant.
  • 10 A Prototypical Model of an Ion Channel
  • 10.1 Stochastic Model of the Transmembrane Potential
  • 10.1.1 A Numerical Scheme
  • 10.1.2 An Invariant Region
  • 10.2 Probability Density Functions for the Voltage-Gated Channel
  • 10.3 Analytical Solution of the Stationary Case
  • 10.4 Comparison of Monte Carlo Simulationsand Probability Density Functions
  • 10.5 Mutations and Theoretical Drugs
  • 10.5.1 Theoretical Open State Blocker
  • 10.5.2 Theoretical Closed State Blocker
  • 10.5.3 Numerical Computations Using the Theoretical Blockers
  • 10.5.4 Statistical Properties of the Theoretical Drugs
  • 10.6 Notes
  • 11 Inactivated Ion Channels: Extending the Prototype Model
  • 11.1 Three-State Markov Model
  • 11.1.1 Equilibrium Probabilities
  • 11.2 Probability Density Functions in the Presence of the Inactivated State
  • 11.2.1 Numerical Simulations
  • 11.3 Mutations Affecting the Inactivated State of the Ion Channel
  • 11.4 A Theoretical Drug for Mutations Affecting the Inactivation
  • 11.4.1 Open Probability in the Mutant Case
  • 11.4.2 The Open Probability in the Presence of the Theoretical Drug
  • 11.5 Probability Density Functions Using the Blocker of the Inactivated State
  • 12 A Simple Model of the Sodium Channel
  • 12.1 Markov Model of a Wild Type Sodium Channel
  • 12.1.1 The Equilibrium Solution
  • 12.2 Modeling the Effect of a Mutation Impairing the Inactivated State
  • 12.2.1 The Equilibrium Probabilities
  • 12.3 Stochastic Model of the Sodium Channel
  • 12.3.1 A Numerical Scheme with an Invariant Region
  • 12.4 Probability Density Functions for the Voltage-Gated Channel
  • 12.4.1 Model Parameterization
  • 12.4.2 Numerical Experiments Comparing the Properties of the Wild Type and the Mutant Sodium Channel
  • 12.4.3 Stochastic Simulations Illustrating the Late Sodium Current in the Mutant Case.
  • 12.5 A Theoretical Drug Repairing the Sodium Channel Mutation.

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