The dose response curves and probability distribution of the output protein in the 3-layer cascades (denoted by ) as a function of inducing signal , 1-layer in (A), 2-layer (B) and 3-layer (C).

<p>The probability distribution can be directly obtained from Eq. (15) after normalization. The Hill coefficient for each cascade is fitted as 2.00, 3.15 and 4.08 respectively.</p

Constructing the Energy Landscape for Genetic Switching System Driven by Intrinsic Noise

<div><p>Genetic switching driven by noise is a fundamental cellular process in genetic regulatory networks. Quantitatively characterizing this switching and its fluctuation properties is a key problem in computational biology. With an autoregulatory dimer model as a specific example, we design a general methodology to quantitatively understand the metastability of gene regulatory system perturbed by intrinsic noise. Based on the large deviation theory, we develop new analytical techniques to describe and calculate the optimal transition paths between the on and off states. We also construct the global quasi-potential energy landscape for the dimer model. From the obtained quasi-potential, we can extract quantitative results such as the stationary distributions of mRNA, protein and dimer, the noise strength of the expression state, and the mean switching time starting from either stable state. In the final stage, we apply this procedure to a transcriptional cascades model. Our results suggest that the quasi-potential energy landscape and the proposed methodology are general to understand the metastability in other biological systems with intrinsic noise.</p></div

Summary of the schematic quasi-potential energy landscape for the yeast cell cycle network.

<p>(A) When nutrient availability is poor, the system has a global stable state shown as G1, and restrains the cell around this state. (B) When the amount of nutrients becomes sufficient, the system shifts to having a limit cycle, the cell is released and the cell cycle is activated. (C) When the S phase or M phase checkpoint mechanism is activated, a temporal stable state appears and holds the system there until the issue is resolved (the abbreviation of checkpoint is chk). (D) When taking finite volume effect into account, i.e. the noise strength is of the same magnitude as its reaction rates, the area with extremely slow rates in the cell cycle process lowers to form small pits that provide longer duration of stay when the system passes by. These new small pits play the role of metastable states. The black arrows represent the driving force on the landscape, the blue arrows illustrate the deformation of the landscape and the orange arrows show the movement of the system under noise perturbations.</p

The mean switching time (MST) and quasipotential energy landscape as a function of parameters.

<p>(A) and (B): MST as a function of transcription rate . Promoter transition rates , the gMAM results with numerical prefactor of off-to-on transition (red solid line) and on-to-off transition (blue dashed line), compared with MC simulations () and (), respectively. (C) and (D): The gMAM results with different promoter transition rates of off-to-on transition (red) and on-to-off transition (blue), where solid line with is same as (A) and (B), the faster transition rate in dashed line with , the slower transition rate in dotted line with . Other parameters are ; in (A,C), and (B,D) </p

The coefficient of variation (CV) versus mean number of (A) mRNA and (B) protein induced by varying transcription rate with different promoter transition rates.

<p>The lines and discrete dots correspond to analytical results and MC simulations, respectively. The results with fast promoter transition rates are shown in blue dash-dotted line and , medium rates in red solid line and , and slow rates in magenta dashed line and ◊. The parameters here are the same as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0088167#pone-0088167-g004" target="_blank">Fig. 4 (A,C)</a>.</p

The energy landscape of the yeast cell cycle network with finite volume effect.

<p>(A) on the <i>x</i>-<i>y</i> plane with <i>z</i> = 0 and (B) on the <i>y</i>-<i>z</i> plane with <i>x</i> = 0. The moving direction of the system is shown as brown arrows. Here we use “−ln(<i>P</i>)” to define the energy landscape of the system, where <i>P</i> = <i>P</i>(<b><i>x</i></b>) is the stationary probability distribution of the system from simulation. The “G1”, “S”, “early M” and “late M” in bold refer to G1 phase, S phase, early M phase and late M phase respectively.</p

Switching paths (A) from off to on state (purple solid curve) and (B) from on to off state (red solid curve) and MC simulations for both switching trajectories.

<p>We take the two stable fixed points in the deterministic dynamics as the starting and ending points. Darkness of the shading points represents the number of visits for reactive trajectories with smoothing. (C) Averaged switching trajectories from MC simulation. For each number of protein, we average in the mRNA dimension using probability as weight. Here the statistical results around each stable state is not shown because of the restrictions by our MC simulation algorithm (see Text SI:VI-A). The results are obtained from 1000 independent long time MC simulations. The parameters here are , , , , , , , , and </p

The effects of the non-gradient force in the yeast cell cycle trajectory.

<p>(A) The energy landscape with the force strength (gray arrows) on the <i>x</i>-<i>y</i> plane with <i>z</i> = 0. The length of the arrow is positively related to the force strength. (B) The driving force strength (red dashed line) and the fluctuation strength in the vertical direction (black line) along the cell cycle trajectory. For the <i>x</i>-axis, we use the natural coordinates of the cell cycle trajectory, i.e. the evolution distance from the start point <i>P</i>₂. Inset: the evolution trajectory of the yeast cell cycle process where the letters “a”, “b” and “c” mark three points on the ODE trajectory with a large force strength. (C) and (D) The pseudo energy landscape on the <i>x</i>-<i>y</i> plane with <i>z</i> = 0 (C) and <i>z</i> = 0.3 (D). The “S” and “early M” in bold refer to S phase and early M phase respectively.</p

Quasipotential energy landscape of the whole genetic switching system with (A) two and (B) three dimensional view as well as switching paths between two stable fixed points.

<p>Each path passes through the saddle point. Here, the parameters are the same as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0088167#pone-0088167-g002" target="_blank">Fig. 2</a>.</p

Different slices of the global energy landscape of the three-variable yeast cell cycle model.

<p>(A) The landscape on the <i>x</i>-<i>z</i> plane with <i>y</i> = 0 corresponds to the G1/S phase in the cell cycle process. (B) and (C) The landscapes on the <i>x</i>-<i>y</i> plane with <i>z</i> = 0.3 and <i>z</i> = 0.05. (D) The landscape on the <i>x</i>-<i>y</i> plane with <i>z</i> = 0 corresponds to the S phase and the early M phase transition. The “G1”, “S” and “early M” in bold refer to G1 phase, S phase and early M phase respectively.</p