*Christopher L-H. Huang*

- Published in print:
- 1993
- Published Online:
- March 2012
- ISBN:
- 9780198577492
- eISBN:
- 9780191724190
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198577492.003.0003
- Subject:
- Neuroscience, Molecular and Cellular Systems

This chapter explores the capacitative properties of biological membranes and provides ways to measure the membrane capacitance in striated muscle. The capacitance of a bilayer is inversely related ...
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This chapter explores the capacitative properties of biological membranes and provides ways to measure the membrane capacitance in striated muscle. The capacitance of a bilayer is inversely related to the chain length of its constituent hydrocarbons, and the latter is in turn proportional to bilayer thickness. The nature of the polar head groups does not greatly affect membrane capacitance. Skeletal muscle membranes form a network of 50-nm diameter branching tubules penetrating into the fibre whose lumina are continuous with extracellular fluid. Such a system would intrinsically have particular capacitative properties. Using the Laplace transform, a square-root relationship between membrane impedance properties and the input parameters is found. The ‘lumped’ four-component model offers the simplest available electrical description of striated muscle membrane geometry. In contrast, distributed models represent the transverse tubules as a cable system that allows radial voltage differences. Lattice models have proven useful in both the interpretation of the measurements of linear capacitances and the localization of non-linear charge to different regions of membrane. The chapter also discusses voltage-clamp methods for studying the dielectric properties of skeletal muscle. Each voltage-clamp method has characteristic advantages and limitations. The most appropriate equivalent circuit to represent the transverse tubular system remains incompletely resolved. The cable properties of tubular membrane necessitate an operational definition of measured effective capacitance arising from an application of the properties of the Laplace transform to a general circuit network. The studies suggest that, at least through the frequency range over which charge movements have been measured, the effective capacitance accurately reflects the properties of the actual electrical elements of surface or tubular membrane, given appropriate bathing solutions and voltage-clamp geometry.Less

This chapter explores the capacitative properties of biological membranes and provides ways to measure the membrane capacitance in striated muscle. The capacitance of a bilayer is inversely related to the chain length of its constituent hydrocarbons, and the latter is in turn proportional to bilayer thickness. The nature of the polar head groups does not greatly affect membrane capacitance. Skeletal muscle membranes form a network of 50-nm diameter branching tubules penetrating into the fibre whose lumina are continuous with extracellular fluid. Such a system would intrinsically have particular capacitative properties. Using the Laplace transform, a square-root relationship between membrane impedance properties and the input parameters is found. The ‘lumped’ four-component model offers the simplest available electrical description of striated muscle membrane geometry. In contrast, distributed models represent the transverse tubules as a cable system that allows radial voltage differences. Lattice models have proven useful in both the interpretation of the measurements of linear capacitances and the localization of non-linear charge to different regions of membrane. The chapter also discusses voltage-clamp methods for studying the dielectric properties of skeletal muscle. Each voltage-clamp method has characteristic advantages and limitations. The most appropriate equivalent circuit to represent the transverse tubular system remains incompletely resolved. The cable properties of tubular membrane necessitate an operational definition of measured effective capacitance arising from an application of the properties of the Laplace transform to a general circuit network. The studies suggest that, at least through the frequency range over which charge movements have been measured, the effective capacitance accurately reflects the properties of the actual electrical elements of surface or tubular membrane, given appropriate bathing solutions and voltage-clamp geometry.

*Peter Mann*

- Published in print:
- 2018
- Published Online:
- August 2018
- ISBN:
- 9780198822370
- eISBN:
- 9780191861253
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198822370.003.0008
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter builds on the previous two chapters to tackle constrained systems, using Lagrangian mechanics and constrained variations. The first section deals with holonomic constraint equations ...
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This chapter builds on the previous two chapters to tackle constrained systems, using Lagrangian mechanics and constrained variations. The first section deals with holonomic constraint equations using Lagrange multipliers; these can be used to reduce the number of coordinates until a linearly independent minimal set is obtained that describes a constraint surface within configuration space, so that Lagrange equations can be set up and solved. Motion is understood to be confined to a constraint submanifold. The variational formulation of non-holonomic constraints is then discussed to derive the vakonomic formulation. These erroneous equations are then compared to the central Lagrange equation, and the precise nature of the variations used in each formulation is investigated. The vakonomic equations are then presented in their Suslov form (Suslov–vakonomic form) in an attempt to reconcile the two approaches. In addition, the structure of biological membranes is framed as a constrained optimisation problem.Less

This chapter builds on the previous two chapters to tackle constrained systems, using Lagrangian mechanics and constrained variations. The first section deals with holonomic constraint equations using Lagrange multipliers; these can be used to reduce the number of coordinates until a linearly independent minimal set is obtained that describes a constraint surface within configuration space, so that Lagrange equations can be set up and solved. Motion is understood to be confined to a constraint submanifold. The variational formulation of non-holonomic constraints is then discussed to derive the vakonomic formulation. These erroneous equations are then compared to the central Lagrange equation, and the precise nature of the variations used in each formulation is investigated. The vakonomic equations are then presented in their Suslov form (Suslov–vakonomic form) in an attempt to reconcile the two approaches. In addition, the structure of biological membranes is framed as a constrained optimisation problem.