Question

Ion-channel events. A biological membrane contains \(N\) ion-channel proteins. The fraction of time that any one protein is open to allow ions to flow through is q. Express the probability \(P(m, N)\) that \(m\) of the channels will be open at any given time.

Short answer

The probability \( P(m, N) \) is given by \[ P(m, N) = \frac{N!}{m!(N-m)!} q^m (1-q)^{N-m} \]

Step-by-step solution

Understand the Binomial Distribution

This problem involves finding the probability of a certain number of successes (open ion channels) in a fixed number of trials (total ion-channel proteins). This can be modeled using the binomial distribution.

Formula for Binomial Probability

The probability of exactly \(m\) successes in \(N\) independent Bernoulli trials, each with probability \(q\) of success, is given by the binomial formula: \[ P(m, N) = \binom{N}{m} q^m (1-q)^{N-m} \] where \(\binom{N}{m}\) is the binomial coefficient.

Calculate the Binomial Coefficient

The binomial coefficient \(\binom{N}{m}\) represents the number of ways to choose \(m\) successes out of \(N\) trials and is calculated using: \[ \binom{N}{m} = \frac{N!}{m!(N-m)!} \]

Combine All Elements

Substitute the binomial coefficient into the binomial formula to express the full probability: \[ P(m, N) = \frac{N!}{m!(N-m)!} q^m (1-q)^{N-m} \]

Key concepts

Ion-channel proteins

Ion-channel proteins are essential components of cell membranes.
They control the flow of ions such as sodium, potassium, calcium, and chloride in and out of cells.
These channels are vital for numerous physiological processes, including nerve impulse transmission, muscle contraction, and maintaining the cell's resting potential.

Each ion-channel can be in one of two states: open or closed.
The 'open' state allows ions to flow through, while the 'closed' state prevents this flow.
In the context of this problem, we are interested in the fraction of time these channels remain open.
This fraction is denoted by the symbol 'q'.
Understanding whether these channels are open or closed at any given time has practical implications in fields such as neurobiology and pharmacology.

Binomial probability

Binomial probability refers to the likelihood of a specific number of successes in a sequence of independent trials.
In this case, 'success' means an ion-channel protein is open.

For a fixed number of trials, N (the number of ion-channel proteins), and a success probability of q for each trial, the binomial distribution can be used to calculate the probability of obtaining exactly m successes.
This is represented by the equation: \[ P(m, N) = \binom{N}{m}q^m (1-q)^{N-m} \]

The binomial probability formula combines the likelihood of each individual outcome with the number of ways to achieve those outcomes (via the binomial coefficient).

Bernoulli trials

Bernoulli trials are experiments or tests where the outcome is binary: success or failure.
In the context of ion-channel proteins, a single Bernoulli trial examines whether a particular ion channel is open (success) or closed (failure).

Each trial is independent of others, meaning the state of one ion channel does not affect the state of another.
The probability of success (an open ion channel) is consistent across all trials and is denoted by 'q'.
Bernoulli trials are crucial for applying the binomial probability formula, as each ion channel's state can be treated as a separate Bernoulli trial.

Binomial coefficient

The binomial coefficient, denoted by \(\binom{N}{m}\), plays a critical role in binomial probability.
It represents the number of ways to choose m successes out of N trials, without considering the order of outcomes.

The binomial coefficient is calculated using the formula: \[ \binom{N}{m} = \frac{N!}{m!(N-m)!} \]

Here, N! (N factorial) is the product of all positive integers up to N.
For example, 5! = 5 × 4 × 3 × 2 × 1.
The factorial operation also applies to m and (N-m).
This coefficient is a combinatorial concept that ensures the binomial probability formula accounts for all possible ways m successes can occur in N trials.