Modeling stochasticity and robustness in gene regulatory networks

Abstract

Motivation: Understanding gene regulation in biological processes and modeling the robustness of underlying regulatory networks is an important problem that is currently being addressed by computational systems biologists. Lately, there has been a renewed interest in Boolean modeling techniques for gene regulatory networks (GRNs). However, due to their deterministic nature, it is often difficult to identify whether these modeling approaches are robust to the addition of stochastic noise that is widespread in gene regulatory processes. Stochasticity in Boolean models of GRNs has been addressed relatively sparingly in the past, mainly by flipping the expression of genes between different expression levels with a predefined probability. This stochasticity in nodes (SIN) model leads to over representation of noise in GRNs and hence non-correspondence with biological observations.

Results: In this article, we introduce the stochasticity in functions (SIF) model for simulating stochasticity in Boolean models of GRNs. By providing biological motivation behind the use of the SIF model and applying it to the T-helper and T-cell activation networks, we show that the SIF model provides more biologically robust results than the existing SIN model of stochasticity in GRNs.

Availability: Algorithms are made available under our Boolean modeling toolbox, GenYsis. The software binaries can be downloaded from http://si2.epfl.ch/ approximately garg/genysis.html.

Fig. 1.

(a) A GRN. (b) The GRN mapped to Boolean functions (gates). The labels next to the gates represent the output genes/proteins. The gates susceptible to stochasticity are colored dark.

Fig. 2.

Biological functions categorized into three different classes of stochasticity and error probability. From left to right, we can broadly classify different biological processes from very stable structures to highly stochastic systems involving scaffold proteins.

Fig. 3.

T-helper GRN (Mendoza and Xenarios, 2006).

Fig. 4.

Simulation results showing the effect of noise on T-helper cell differentiation process with an external stimulus of IFNγ. Each small circle is representative of a T-helper cell and each cell is modeled to behave independent of the neighboring cells. Red cells represent the naïve undifferentiated Th0 cells, green cells represent Th1 cell state and blue cells represent Th2 cell state. Ratio of number of red (green or blue) cells to total number of cells in a panel is representative of the probability of differentiating into Th0 (Th1 or Th2) cell state. (a) Cell culture maintained in Th0 state. (b) In absence of any stochasticity all Th0 cells differentiate to Th1 cell state on receiving IFNγ. (c) Th0 cells differentiate into Th1 and Th2 under the SIN model of stochasticity. Few cells revert to Th0 state as seen by the few patches of red color. (d) SIF model of stochasticity shows that Th0 cells differentiate into Th1 cells while some cells cannot differentiate on receiving IFNγ and revert to Th0 cell state. None of the cells differentiate into Th2 cell state. The probability of failure (i.e. ϵ_i) is 0.5 for all the nodes (functions) in the SIN model (SIF model).

Fig. 5.

Simulation results showing the robustness of cellular steady states of T-helper cell differentiation network under SIN and SIF stochasticity models. Red cells represent the naïve undifferentiated Th0 cells, green cells represent Th1 cell state and blue cells represent Th2 cell state. In the first row, ratio of number of red cells to total number of cells in a panel represent the robustness of Th0 cell state. Similarly, second and third row represent the robustness of Th1 and Th2. Number of faults n in the network are modeled from n=0 to 5. Multiple faults are injected sequentially (as further discussed in Section 3.2). (a) Robustness under no stochasticity. All the cells remain in their steady states. (b) Robustness under SIN model of stochasticity. (c) Robustness under SIF model of stochasticity. The probability of failure (i.e. ϵ_i) is 0.5 for all the nodes (functions) in the SIN model (SIF model).

Fig. 6.

Simulation results showing the transition probability among the 10 attractors of the T-cell activation network. (a)–(d) Transition probabilities in the SIN model as the number of faults n in the network is increased from n=0 to 4. (e)–(h) Transition probabilities in the SIF model. In each figure, the intensity of yellow color in the entry i−j corresponds to the probability of transition from the attractor i to the attractor j. The colorbar in the rightmost column indicates the color-probability encoding. The probability of failure (i.e. ϵ_i) is 0.5 for all the nodes (functions) in the SIN model (SIF model).