Computing Characterizations of Drugs for Ion Channels and Receptors Using Markov Models
Flow of ions through voltage gatedchannels can be represented theoretically using stochastic differentialequations where the gating mechanism is represented by a Markov model. The flow through achannel can be manipulated using various drugs, and the effect of a given drugcan be reflected bychanging...
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| Main Authors: | , |
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| Format: | eBook Book |
| Language: | English |
| Published: |
Cham
Springer Nature
2016
Springer Open Springer International Publishing AG SpringerOpen |
| Edition: | 1 |
| Series: | Lecture Notes in Computational Science and Engineering |
| Subjects: | |
| ISBN: | 9783319300306, 331930030X, 9783319300290, 3319300296, 3319398881, 9783319398884 |
| Online Access: | Get full text |
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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
- 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
- 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
- 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
- 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
- 12.5 A Theoretical Drug Repairing the Sodium Channel Mutation