Question

In the chapter we derived the Nernst equation (eqn 17.3 ) using the Boltzmann distribution. Derive the same result. but this time by using the fact that in equilibrium the net flux of tons across the membrane is zero. Use eqn 13.56 (p. 503 ) to derive the contribution of the electrical force to the overall flux and show that by balancing this flux with the diffusive contribution you recover the Nernst equation.

Short answer

The Nernst Equation was derived b applying the condition that at equilibrium the net flux of ions across the membrane is zero, and by splitting the net flux into its electrical and diffusive components. After balancing these components, it was found that $\mathrm{\Delta }V=\frac{RT}{zF}\ast \ln (\frac{[C{]}_{out}}{[C{]}_{in}})$, which is the Nernst Equation.

Step-by-step solution

Equation 13.56

Recognize that Equation 13.56 (from p.503) is given by $J=P(zF\mathrm{\Delta }V-RT\ln (\frac{[C{]}_{in}}{[C{]}_{out}}))$, where J represents the net flux, P is the permeability, z is the charge of the ions, F is the Faraday constant, $\mathrm{\Delta }V$ is the electrical potential difference, R is the ideal gas constant, T is the absolute temperature, and $[C]$ is the concentration of ions. This equation gives the net flux of ions across the membrane.

Calculate Electro-chemical Flux

Split the equation into two parts, representing two different forces. The first term represents the flux due to the electrical force, $J_{elec}=P\ast zF\ast \mathrm{\Delta }V$, and the second term represents the flux due to the concentration difference, $J_{diff}=P\ast (-RT\ln (\frac{[C{]}_{in}}{[C{]}_{out}}))$.

Balancing Electrical and Diffusive Flux

In equilibrium, the net flux is zero, so the flux due to electrical forces must balance the diffusive flux: $P\ast zF\ast \mathrm{\Delta }V=P\ast (-RT\ln (\frac{[C{]}_{in}}{[C{]}_{out}}))$. The permeability P cancels out.

Derive Nernst Equation

Rearranging and simplifying gives the Nernst Equation, $\mathrm{\Delta }V=\frac{RT}{zF}\ast \ln (\frac{[C{]}_{out}}{[C{]}_{in}})$. This is the desired equation, showing that at equilibrium, the membrane potential is related to the concentrations of the ions on each side of the membrane.

Key concepts

Electrochemical Gradients

Understanding electrochemical gradients is key to grasping the mechanisms that underpin various biological processes, including the generation of membrane potentials. An electrochemical gradient consists of two components: a concentration gradient and an electrical gradient. The concentration gradient arises when there is a difference in the number of ions on each side of a biological membrane. The electrical gradient is created by a difference in charge across the membrane. Together, these gradients drive the movement of ions across the membrane, a process essential for functions such as nerve signal transmission and muscle contraction.

The Nernst equation, which can be derived by considering that in equilibrium the net flux of ions across a membrane is zero, provides a way to calculate the membrane potential that balances these two gradients. This potential, also known as the equilibrium potential, is vital for maintaining cell homeostasis. The movement of ions 'down' their electrochemical gradient, from high to low, releases energy, which cells can harness to do work, such as transporting other substances against their own gradients.

Ion Permeability

Ion permeability refers to the ease with which ions can pass through a membrane. In biological systems, cell membranes typically have selective permeability, allowing some ions to pass more easily than others. This selectivity is achieved through protein channels and pumps embedded in the lipid bilayer. Each channel or pump is specific to certain ion types, and this specificity plays a crucial role in maintaining the cell's electrochemical environment.

When deriving the Nernst equation, permeability factors into the calculation of ion flux across a membrane. It is represented by the variable 'P' in the equation presented in the textbook solution. Permeability directly influences the magnitude of ion movement due to both concentration differences and electrical forces. However, in the context of the Nernst equation, which describes the situation at equilibrium, permeability itself cancels out, highlighting that the equilibrium potential is not dependent on how easily ions can cross the membrane but rather on the ratio of ion concentrations and their charges.

Membrane Potential

Membrane potential is the voltage difference across a cell's membrane. It is crucial for many cellular activities, including the transmission of nerve impulses and muscle contractions. The membrane potential results from the ion concentration differences between the inside and outside of the cell, as well as the selectivity of the cell membrane to certain ions. This electric potential is used by the cell to store energy and transmit signals.

The Nernst equation shows us how to calculate the membrane potential for a particular ion, under the condition that its flux across the membrane is in equilibrium. This state of equilibrium implies that the electrochemical driving forces across the membrane are balanced and there is no net movement of ions. The derived Nernst potential, therefore, reflects the voltage required to prevent ions of a specific type from diffusing across the membrane, given the existing concentration gradient.

Boltzmann Distribution

The Boltzmann distribution provides a statistical view of the distribution of particles, such as ions, across different states, characterized by varying energy levels. In the context of membrane potentials, it explains how ion distribution is affected by both concentration gradients and electric potentials. At thermal equilibrium, the distribution of ions is governed by factors that include temperature and the presence of electric fields.

The Nernst equation can also be derived using the concept of the Boltzmann distribution, which considers the likelihood of ions being in a certain energy state on either side of the membrane. The step-by-step solution in the textbook hinted at this approach, implying that the equilibrium membrane potential can be understood as the point at which the energy distribution of ions across the membrane due to thermal motion (described by the Boltzmann distribution) matches the energy distribution caused by the electric potential. In essence, Boltzmann's insights enable us to comprehend how ions distribute themselves at equilibrium, providing the fundamental basis for the Nernst equation.