19 Stochastic Processes: From Single Trajectories to Probability Distributions

The previous chapters focused on deterministic models of biological systems.

In deterministic models, the current state of the system uniquely determines its future. Once the governing equations, parameter values, and initial conditions are known, only one trajectory is possible. Stable equilibria, oscillations, and even deterministic chaos all emerge from this common mathematical framework.

These models have proven remarkably successful for describing many biological processes.

However, they also raise an important question.

Why do genetically identical cells often behave differently?

Modern experimental techniques have revealed a surprising amount of variability among individual cells. Even cells that share the same genome and grow in apparently identical environments may express different levels of the same gene, produce different amounts of protein, or respond differently to the same external signal.

This variability is not simply experimental error.

Instead, it reflects an intrinsic property of many biological systems.

How should such behavior be modelled?

One possibility is that our deterministic models are incomplete. Perhaps an important regulatory interaction has been overlooked.

Another possibility is that biological systems contain genuine randomness at the molecular level.

Distinguishing between these explanations is one of the central challenges of modern systems biology.

In this chapter, we introduce stochastic models, which explicitly incorporate random events into the mathematical description of biological systems.

Unlike deterministic models, stochastic models no longer predict a single future trajectory.

Instead, they predict an ensemble of possible trajectories together with the probability that each one will occur.

This seemingly small change fundamentally alters the way we think about biological models.

The central object is no longer a single trajectory through state space.

Instead, it becomes a probability distribution over many possible outcomes.

Throughout this chapter, we will see that probability distributions contain biological information that is invisible when considering only average behavior. In particular, we will discover that two competing biological mechanisms may produce the same average observation while generating completely different probability distributions.

This insight marks an important conceptual transition.

Deterministic models explain individual trajectories. Stochastic models explain the distributions of many possible trajectories.