Intercellular Variability in Protein Levels from Stochastic Expression and Noisy Cell Cycle Processes

Abstract

Inside individual cells, expression of genes is inherently stochastic and manifests as cell-to-cell variability or noise in protein copy numbers. Since proteins half-lives can be comparable to the cell-cycle length, randomness in cell-division times generates additional intercellular variability in protein levels. Moreover, as many mRNA/protein species are expressed at low-copy numbers, errors incurred in partitioning of molecules between two daughter cells are significant. We derive analytical formulas for the total noise in protein levels when the cell-cycle duration follows a general class of probability distributions. Using a novel hybrid approach the total noise is decomposed into components arising from i) stochastic expression; ii) partitioning errors at the time of cell division and iii) random cell-division events. These formulas reveal that random cell-division times not only generate additional extrinsic noise, but also critically affect the mean protein copy numbers and intrinsic noise components. Counter intuitively, in some parameter regimes, noise in protein levels can decrease as cell-division times become more stochastic. Computations are extended to consider genome duplication, where transcription rate is increased at a random point in the cell cycle. We systematically investigate how the timing of genome duplication influences different protein noise components. Intriguingly, results show that noise contribution from stochastic expression is minimized at an optimal genome-duplication time. Our theoretical results motivate new experimental methods for decomposing protein noise levels from synchronized and asynchronized single-cell expression data. Characterizing the contributions of individual noise mechanisms will lead to precise estimates of gene expression parameters and techniques for altering stochasticity to change phenotype of individual cells.

Fig 1. Sample trajectory of the protein level in a single cell with different sources of noise.

Stochastically expressed proteins accumulate within the cell at a certain rate. At a random point in the cell cycle, gene duplication results in an increase in production rate. Stochastic cell-division events lead to random partitioning of protein molecules between two daughter cells with each cell receiving, on average, half the number of proteins in the mother cell just before division. The steady-state protein copy number distribution obtained from a large number of trajectories is shown on the right. The total noise in the protein level, as measured by the squared coefficient of variation (CV²) can be broken into contributions from individual noise mechanisms.

Fig 2. Stochastic models of gene expression with cell division.

Arrows denote stochastic events that change the protein level by discrete jumps as shown in Eqs (1) and (4). The differential equation within the circle represents the time evolution of x(t) in between events. A) Model with all the different sources of noise: proteins are expressed in stochastic bursts, cell division occurs at random times, and molecules are partitioned between the two daughter cells based on Eq (5). The trivial dynamics $\dot{x}=0$ signifies that the protein level is constant in-between stochastic events. B) Hybrid model where randomness in cell-division events is the only source of noise. Protein production is modeled deterministic through a differential equation and partitioning errors are absent, i.e., α = 0 in Eq (5). C) Hybrid model where noise comes from both cell-division events and partitioning errors. Protein production is considered to be deterministic as in Fig 2B. Since x(t) is continuous here, x₊(t_s) has a positively-valued continuous distribution with same mean and variance as in Eq (5)

Fig 3. A continuous-time Markov chain model for the cell-cycle time.

Left: The cell-cycle time is assumed to follow a mixture of Erlang distributions. At the start of cell cycle, a state G_i1, i = {1, …, n} is chosen with probability p_i. The cell cycle transitions through states G_ij, j = {1, …, i} residing for an exponentially distributed time with mean 1/ik in each state. Cell division occurs after exit from G_ii and the above process is repeated.

Fig 4. Scaling of noise as a function of the mean protein level for different mechanisms.

The contribution of random cell-division events to the noise in protein copy numbers (extrinsic noise) is invariant of the mean. In contrast, contributions from partitioning errors at the time of cell division (partitioning noise) and stochastic expression (production noise) scale inversely with the mean. The scaling factors are shown as a function of the protein random burst size B, noise in cell-cycle time ($C{V}_T^2$) and magnitude of partitioning errors quantified by α (see Eq (5)). With increasing mean level the total noise first decreases and then reaches a baseline that corresponds to extrinsic noise. For this plot, B is assumed to be geometrically-distributed with mean 〈B〉 = 1.5, $C{V}_T^2=0.05$ and α = 1 (i.e., binomial partitioning).

Fig 5. Model illustrating stochastic expression together with random gene-duplication and cell-division events.

At the start of cell cycle, protein production occurs in stochastic bursts with rate k_x. Genome duplication occurs at a random point T₁ within the cell cycle and increases the burst arrival rate to fk_x (f > 1). Cell division occurs after time T₂ from genome duplication, at which point the burst arrival rate reverts back to k_x and proteins are randomly partitioned between cells based on Eq (4).

Fig 6. Contributions from different noise sources as a function of the timing of genome duplication for CVT12=CVT22=0.05.

Different noise components in Eq (46) are plotted as a function of β, which represents the fraction of time within the cell cycle at which gene duplication occurs. The mean protein level is held constant by simultaneously changing the transcription rate k_x. Noise levels are normalized by their respective value at β = 0. The noise contribution from partitioning errors is maximized at β ≈ 0.6. In contrast, the contribution from stochastic expression is minimum at β ≈ 0.6. The extrinsic noise contribution from random gene-duplication and cell-division events is maximum at β ≈ 0.2 and minimum at β ≈ 0.8. For this plot, the mean of the protein is 170 molecules per cell; and the bursts are geometrically distributed with 〈B〉 = 10.