Patch Clamp & Ion Channels

Patch Clamp Electrophysiology

Patch clamp is the gold-standard technique for measuring electrical currents through individual ion channels. Developed by Neher and Sakmann (Nobel Prize 1991), it allows recording of single-channel events in the picoampere range.

Single-Channel Current

Each open ion channel carries a current i determined by Ohm's law:

$i = \gamma \cdot (V_m - E_{\text{rev}})$

• γ = single-channel conductance (picosiemens, pS = 10⁻¹² S)
• V_m = membrane potential (mV)
• E_rev = reversal potential — the voltage at which no net current flows (mV)

The reversal potential is given by the Nernst equation. For K⁺ it is typically −90 mV; for Na⁺ it is typically +60 mV.

Boltzmann Open Probability

Voltage-gated channels open and close with a probability that depends on membrane potential. The open probability P_open follows a Boltzmann distribution:

$P_{\text{open}} = \frac{1}{1 + \exp\left(-\dfrac{V_m - V_{1/2}}{k_{\text{slope}}}\right)}$

• V_{1/2} = half-activation voltage (mV) — membrane potential at which P_open = 0.5
• k_slope = slope factor (mV) — steepness of voltage dependence (~10 mV)

Whole-Cell Current

In whole-cell mode, the total macroscopic current from N identical channels is:

$I = N \cdot P_{\text{open}} \cdot i$

Your Task

Implement three functions:

1. single_channel_current_pA(gamma_pS, V_m_mV, E_rev_mV) — Single-channel current in pA
2. open_probability(V_m_mV, V_half_mV, k_slope_mV=10) — Channel open probability (0–1)
3. whole_cell_current_pA(N_channels, gamma_pS, V_m_mV, E_rev_mV, V_half_mV, k_slope_mV=10) — Total whole-cell current in pA