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

• Superior document: Lecture Notes in Computational Science and Engineering Series ; v.111
• Author: Tveito, Aslak.
• TeilnehmendeR: Lines, Glenn T.
• Place / Publishing House: Cham : : Springer International Publishing AG,, 2016.; ©2016.
• Year of Publication: 2016
• Edition: 1st ed.
• Language: English
• Series: Lecture Notes in Computational Science and Engineering Series
• Physical Description: 1 online resource (279 pages)

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.