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 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+(ts) 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 Gi1, i = {1, …, n} is chosen with probability pi. The cell cycle transitions through states Gij, j = {1, …, i} residing for an exponentially distributed time with mean 1/ik in each state. Cell division occurs after exit from Gii 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 () 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,
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 kx. Genome duplication occurs at a random point T1 within the cell cycle and increases the burst arrival rate to fkx (f > 1). Cell division occurs after time T2 from genome duplication, at which point the burst arrival rate reverts back to kx 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 .
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 kx. 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.