Question

When using ordinary differential equations to model gene regulatory networks, what physical limitation prevents the model from accurately predicting expression levels at extremely low protein copy numbers?

Accepted Answer

The physical limitation is the intrinsic stochasticity, or randomness, of molecular interactions at low copy numbers. Ordinary differential equations are deterministic models, meaning they assume that concentrations of proteins change in a smooth, continuous way as if they were fluids flowing in a tank. This approach relies on the law of mass action, which effectively averages out the behavior of millions of molecules. However, in a living cell, genes are often present in only one or two copies, and the proteins they produce may exist in very small numbers. At these low counts, the timing of molecular events becomes discrete and probabilistic rather than continuous. For example, a single protein molecule binding to a promoter site is an event governed by random collisions rather than a steady, predictable flow. Because these events occur as individual, unpredictable arrivals, the system exhibits significant fluctuations, or noise, that ordinary differential equations cannot capture. These fluctuations mean that the actual number of protein molecules in a cell will frequently deviate from the average predicted by a deterministic equation. Consequently, while ordinary differential equations are useful for predicting the average behavior of a large population of cells, they fail to account for the cell-to-cell variability and random bursts of gene expression that characterize systems with very few molecules.

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When using ordinary differential equations to model gene regulatory networks, what physical limitation prevents the model from accurately predicting expression levels at extremely low protein copy numbers?