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

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

The autoregulatory dimer model with positive feedback.

<p>Promoter transitions are regulated by the dimerized transcription factor with rate and . is the transcription rate of active promotor, with a very small transcription rate of inactive promotor . is kinetic rate of translation, and are degradation rates of mRNA and protein, and are the rates of dimerization and de-dimerization. All the processes are modeled as elementary reactions and all reaction rates are rescaled by the protein decay rate (i.e. unless stated otherwise).</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 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

The model of the three-node yeast cell cycle network.

<p>(A) The network structure of the yeast cell cycle, where <i>x</i>, <i>y</i> and <i>z</i> represent key regulators of the G1/S, early M and late M modules, respectively. Different modules are connected by activation (lines end with arrow) and inhibition (line end with bar) interactions. (B) and (C) The evolution trajectory of the yeast cell cycle process with parameter values <i>j</i>₁ = <i>j</i>₂ = <i>j</i>₃ = 0.5, <i>k</i>₁ = <i>k</i>₂ = <i>k</i>₃ = 0.2, <i>k</i>_<i>i</i> = 5.0, <i>k</i>_<i>s</i> = 1.0, and <i>k</i>_<i>a</i>1 = <i>k</i>_<i>a</i>2 = 0.001. The system starts from <i>P</i>₂ and evolves to <i>P</i>₁. In (B), the time evolution of the variables <i>x</i>, <i>y</i> and <i>z</i> is shown with the green, red and blue lines, respectively.</p

The mean switching time (MST) of off-to-on transition as a function of (A) trigger signal strength that transcribes mRNA at constant rate and (B) degradation rate of protein .

<p>(C) and (D): Quasipotential energy landscape with different trigger strength. in (C), and in (D). Other parameters are ; in (A,C,D), and in (B).</p

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

Sequential Dependence Modeling Using Bayesian Theory and D‑Vine Copula and Its Application on Chemical Process Risk Prediction

An emerging kind of prediction model for sequential data with multiple time series is proposed. Because D-vine copula provides more flexibility in dependence modeling, accounting for conditional dependence, asymmetries, and tail dependence, it is employed to describe sequential dependence between variables in the sample data. A D-vine model with the form of a time window is created to fit the correlation of variables well. To describe the randomness dynamically, Bayesian theory is also applied. As an application, a detailed modeling of prediction of abnormal events in a chemical process is given. Statistics (e.g., mean, variance, skewness, kurtosis, confidence interval, etc.) of the posterior predictive distribution are obtained by Markov chain Monte Carlo simulation. It is shown that the model created in this paper achieves a prediction performance better than that of some other system identification methods, e.g., autoregressive moving average model and back propagation neural network