L2: Distributions in Biochemical Systems

Alien Spaceship

Sound

What are the characteristics of the sound?

background (white) noise, change in frequency

What does the shape of the histogram tell you about the signal's source?

amplitude up, then down - passes by, gets closer and farther

How can you connect these patterns to real-world biological processes, like protein diffusion or fluorescence emission?

chemical signatures - similarly can change over time

How would changes in the signal (e.g., added noise or decay rates) affect the histogram?

more uncertainity

Probability in Biophysics

Molecular interactions occur in noisy, stochastic environments.

Biological processes with inherent randomness: → Diffusion of molecules in cells. → Ligand binding to receptors. → Photobleaching of fluorescent molecules

Boltzmann definition of entropy

S = k_B ln W

large number of microstates

Binomial distribution

Describes the number of successes in 𝑛 independent trials.

Parameters: Number of trials (𝑛) and success probability (𝑝).

1. There is a fixed, finite number of n trials or observations.

2. The trials are independent.

3. The trials end in one of two possible outcomes: Success (S) or Failure (F).

4. The probability of success, p, is the same for all trials.

Practical examples: Coin toss, Stephen Curry shooting 3 pointers, Getting doubles upon throwing 2 dice

Anti-examples: radioactive decay, allosteric inhibition

Biochemical applications: ligand-receptor binding events, probability of ion channel opening

Gaussian Distribution

If the probability of success is finite and number of trials is very large, n.p → ∞ as n → ∞ GAUSSIAN distribution

Biochemical applications: localisation precision in microscopy, diffusion of particles (Brownian motion), thermal fluctuations

Prevalence in Biochemical Processes

Central Limit Theorem (CLT): When independent random variables are added, their sum tends toward a Gaussian distribution, regardless of the original distribution of the variables.

[picture on phone - MatLab]

Biological Systems with Many Contributions:

• Thermal Motion: Atomic vibrations in macromolecules result from countless small energy transfers.

• Diffusion: The random walk of a particle is driven by numerous independent collisions with solvent molecules.

Poisson Distribution

If the probability of success is very small such that n.p remains finite as n → ∞ POISSON distribution

describes the probability of a given number of discrete events happening in a fixed interval of time or space, provided the events occur independently and at a constant average rate.

k = number of times event is observed

Biochemical applications: photon emission in fluorescence microscopy, ion channel openings, molecular collisions

Understanding Randomness

Predictive Power

Practical Application of Statistical Tools