Fig 1.
Single-cell and population perspectives for investigating stochastic gene expression with dilution-based feedback regulation.
(A) (top:) For the single-cell perspective, concentration of a given protein is tracked along a single lineage. (bottom:) From a population point of view, the protein concentration distribution is obtained across all descendants of the colony. Different shades of green represent the protein level for each cell. (B) (left:) Schematic of the gene expression model with random bursts of protein synthesis, and concentration dilution in between burst events. In the model without regulation (blue), the dilution rate is constant. In the model with feedback on dilution (green), the dilution rate decreases as the protein concentration increases, according to (3). (right:) Sample trajectories of protein concentration in a single cell, along with the corresponding single-cell protein concentration distributions for both models. The gray lines in the background show protein dilution trajectories; horizontal dashed lines represent the mean concentration in both models. These trajectories are for different single cells with initial values selected from previous simulations such that they start from steady state conditions. Parameter values used for these trajectories are , k = 1/100,
is found using (7) with the mean protein level set to 100. Therefore,
(no feedback),
(with feedback). Time is presented in units such as
.
Table 1.
Model parameters and variables studied in the text.
Fig 2.
Comparison of protein distribution in single-cell and population perspectives as the feedback intensifies.
(A) The region of existence of the steady-state protein distribution in terms of the feedback strength k and relative burst frequency . Green region: stationary distribution p(x) exists in both single cell and population perspectives. Brown region: Only the distribution in the population perspective exists. Yellow region: Distribution does not exist in any of the frameworks. Bold red line: a set of values
, resulting in fixed
as per (7). Black dashed line represents the maximum burst frequency in which the distribution exists for both perspectives regardless the feedback strength. (B) Comparison of protein distribution in single-cell (solid green line) and population perspectives (brown dashed) as feedback increases: (top:) weak feedback, (bottom:) strong feedback. (C) From top to bottom: Mean protein level, protein noise, and distribution asymmetry in the single cell and population frameworks;
is chosen so that
as k increases following the bold red line in panel (A). For all plots, we set
,
.
Fig 3.
Noisy cell-cycle durations amplify protein noise differences between single-cell and population perspectives.
(A) (top:) The cell-cycle duration is a random variable following an arbitrary distribution. Within the cell cycle, the protein concentration evolves deterministically as per the ordinary differential equation (15). (bottom:) During mitosis, protein molecules are randomly segregated among daughters resulting in differences in the inherited concentration. Different shades of green represent different levels of protein concentration. (B) (left:) Trajectories of protein concentration in an expanding cell colony, where jumps represent randomness in protein partitioning among daughters during cell division. The green line: a single-cell trajectory is generated by randomly choosing one of the two daughter cells (red lines) after each division event. The light brown lines represent other descendant cells. The cell-cycle times are assumed to be exponentially distributed in this simulation. (right:) The steady-state probability density functions of the protein concentration in single-cell and population perspectives. Single-cell statistics are estimated over a 5000 independent individuals; population statistics are estimated using all cells of 2000 colonies (including sisters, progenitor and other cells). Statistics were calculated after 6 generations. (C) Effect of noise in the cell cycle time as quantified by its squared coefficient of variation (
) on the noise in the protein concentration (
). The solid line is the analytically predicted noise in the single-cell perspective as given by (18), and the dots represent noise levels computed from simulations. Mean concentrations in both models are identical
. (D) A logarithmic scale representation of the steady-state protein noise level as a function of the mean protein level, highlighting variability differences between single-cell and population perspectives. Parameters used for the plot are
,
,
.
Fig 4.
Increasing randomness in molecular segregation between daughters enhances protein noise differences between single-cell and population perspectives.
(A) Sample trajectories of protein concentration in an expanding cell colony when expression variability is dominated by partitioning noise ( and
in (20)). (B) Comparison of the steady-state protein concentration noise (
) from single-cell (green circles) and population perspectives (brown squares) calculated from simulations of the agent-based model. The solid line represents the analytically-predicted noise level (20). (C) Sample concentration trajectories for the high intrinsic noise scenario (
and
in (20)). Other parameters are taken as
,
). 2000 colonies where simulated for population perspective, 5000 individuals where simulated for single-cell perspective.