Modeling the Nonlinear Dynamics of Intracellular Signaling Networks

Equipment

1. Laptop computer

   Proposed protocols can be implemented on a laptop computer; however, parameter sampling procedure is time-consuming, and using a computational cluster is advantageous. For example, sampling parameter space of the model (Bolado-Carrancio et al., 2020) to obtain a comprehensive list of different regimes took approximately 5 days at a workstation with 4-core Intel^® Core^TM i7 Processor. The high-quality scan of 2D parametric plane takes approximately 5 hours given the same computational power. Because the sampling procedures are very well parallelized, the computational time is inversely proportional to the number of available cores.

   Since most computational clusters use Linux-based operating systems, we further refer to software packages mainly developed for Linux.

Software

1. Python and python libraries, including matplotlib (Hunter, 2007), NumPy (Harris et al., 2020), and scipy (Virtanen et al., 2020)

2. Python version of DYVIPAC software package (Nguyen et al., 2015)

3. Software for parameter fitting of ODE systems, e.g., BioNetFit (Thomas et al., 2016; Mitra et al., 2019), PEPSSBI (Degasperi et al., 2017), etc.

4. Software for performing spatiotemporal calculations of reaction-diffusion-convection models, e.g., FiPy (Guyer et al., 2009), OpenFOAM (Jasak, 2009; OpenCFD, 2009) or Virtual Cell (Loew and Schaff, 2001)

Procedure

1. Build an ODE model of the signaling network based on the available topology of interactions between nodes. Implement the model in a software package that supports export to the SBML format, such as COPASI (Hoops et al., 2006) or BioNetGen (Blinov et al., 2004; Harris et al., 2016).

   1. If the kinetics of interactions between two nodes is known, this network connection is modeled mechanistically.

   2. For other connections, the interaction kinetics might not yet be exact mechanistically characterized, or these connections operate via intermediate species that are not included in the network under study. These connections are modeled using hyperbolic multipliers α_ij that specify the negative or positive influence of the activity (x_j) of node j on the activity (x_i) of node i (Tsyganov et al., 2012; Bolado-Carrancio et al., 2020),

      
      

      

      
      

      The coefficient γ_ij>1 indicates activation; γ_ij<1 inhibition; and γ_ij=1 denotes the absence of regulatory interactions, in which case the modifying multiplier α_ij equals 1. K_ij is the activation or inhibition constant.

   3. Specify the parameter ranges using the data on the protein abundances and kinetic constants and assign numerical values to model parameters.

   4. Non-dimensionalize the model to reduce the number of parameters (Barenblatt, 2003).

   5. Export the model to SBML format.

      
      

2. Determine dynamical regimes that can be observed in the system.

   1. Import the sbml-file of the model into the DYVIPAC package. Sample parameter space of the model using the DYVIPAC package, which determines the number and stability of steady states for a given set of parameters.

   2. Rank protein abundances in a model in descending order and pick two proteins (nodes) with the highest abundance. Using the QSS approximation for the other protein activities (Sauro and Kholodenko, 2004; Eilertsen and Schnell, 2020), derive a 2D model from the initial ODE model.

   3. Based on the specification of the steady states and the analysis of the shapes of nullclines (Figure 2, for example), determine the dynamic regimes and validate these regimes by the integration of ODEs.

      
      

      
      Figure 2. Examples of the nullcline analysis. Nullclines and vector fields calculated for 2D ODE system are derived from a five-dimensional ODE system using a quasi-steady-state approximation. Circles show stable steady states; triangles represent unstable steady states. Red and blue curves are nullclines for variables x₁ and x₂, respectively. Green line represents trajectories of limit cycles projected from the original five-dimensional system to 2D space of x₁ and x₂ [see Bolado-Carrancio et al. (2020) for details].

      
      

   4. If more than two proteins have high and comparable abundances, make a list of pairs of these proteins and apply the above point c) to the ODE system for each pair.

3. Scan 2D parametric planes and determine the types of bifurcations that can occur in the system.

   1. Scan the parameter planes and determine the borders that separate different dynamic regimes (see Figure 3, for example). This step can predict the previously unobserved dynamic regimes, which must be validated in the experiments.

      
      

      
      Figure 3. Examples of 2D parametric scans. Different 2D parametric diagrams obtained using scanning of 2-parameter planes. Different colors indicate different dynamical regimes. Black lines represent borders between these regimes where bifurcations happen (see Bolado-Carrancio et al., 2020 for details and Figure 2 there).

      
      

   2. The analysis of the changes in the number and stability of steady states can reveal local bifurcations, such as the Andronov-Hopf or saddle-node bifurcations (Kuznetsov, 2004). However, non-local bifurcations might exist in a 2D system where a limit cycle can appear or vanish (Nekorkin, 2015). Scan the parameters for dynamic regimes that contain focuses to determine the appearance or annihilation of limit cycles.

4. If there are time-series data, fine-tune model parameters to reproduce the system trajectories for a biologically relevant regime. This step uses standard software packages for parameter optimization (Glover et al., 2000; Maiwald and Timmer, 2008; Kreutz and Timmer, 2009; Penas et al., 2015; Thomas et al., 2016; Degasperi et al., 2017; Mitra et al., 2018 and 2019).

5. Derive a PDE system by adding the terms describing mass transfer processes such as diffusion and convection to the ODE system studied (Tsyganov et al., 2012; Bolado-Carrancio et al., 2020). Perform spatiotemporal simulations using specific software packages, such as OpenFOAM, FiPy, or Virtual Cell.

A set of examples that illustrate the analysis of the dynamics of the RhoA-Rac1 signaling network model and python scripts can be found in “examples.zip” archive.

Acknowledgments

This protocol was adapted from Tsyganov et al. (2012) and Bolado-carrancio et al. (2020). This work was supported by NIH/NCI grant R01CA244660 and EU grant NanoCommons (grant no. 731032).

Competing interests

The authors declare no competing interests.

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