A.3.3. Aim 1(iii): Mathematical Modeling of the Nonlinear Dynamics of Cellular Networks

Computational tools for modeling biological processes, at the cellular scale using a series of differential equations: XPPAUT

Numerous approaches have been used in the mathematical modeling of cellular and biochemical processes starting from the work of Jacob & Monod (100;101).  On one end is to include as many details as possible. Some of the packages now available for these types of simulations are listed in the first three rows of Table A.2. This massive modeling approach has the advantage that every known reaction is included in the system thus giving some hope for comparisons to experiments. However, in many cases, the rates of reactions and the original concentrations are not known, nor are all the intermediate states and connectivities (state/block diagram); and the complexity of the system makes it difficult to study sensitivity to parameters and initial conditions. On the other extreme is the abstract approach taken by Glass and Kaufmann (102); the individual components are taken to be Boolean variables (either on or off) and the behavior is completely determined by the topology of the interactions and the switching rules. While this greatly simplifies the models, the graded nature of responses is often important.

We aim for an intermediate regime of modeling between these two extreme cases, which will take into account the known biology but then simplifies it through a series of mathematical steps. A description of the methods for simplifying and reducing complex models is presented by Ermentrout (103). These methods essentially exploit differences in time and spatial scales, when these scales are separable. Many of the systems we intend to model are inherently stochastic and we will utilize a master equation formalism (104) where transition (or jump) probabilities between states control the probabilistic evolution of states (see for example the analysis of voltage-gated or ligand-gated ion channels (105)). Closely associated are the hybrid models that involve stochastically forced differential equations (Langevin dynamics), or the time evolution of probability density functions (Fokker-Planck formalism). These approaches have been successfully used in other disciplines (see for example the review by Bahar and coworkers (106)), but have not yet been exploited by theoretical biologists (107). Deterministic differential equations for mean concentrations can be derived directly from the stochastic formulations. Associated approaches for obtaining differential equation descriptions include the mean-field approximation, which is appropriate for interactions that are sufficiently fast and pre-equilibrated within the time scale of an examined slower process, and/or spatial averaging for movements or paths that are again fast. The differential equation description can be further reduced to produce simpler models that still capture the essential properties of the system. An example is the pseudo-steady state approximation for the reactants and products of the fast steps in serial reactions (e.g. Michealis-Menten mechanism). The solution of averaged differential equations in terms of the slow variables, as used in neural networks or weakly coupled oscillators, is another mathematical tool for model reduction.

Therefore, we will consider a hierarchy of models from discrete representation all the way up to continuous levels for proper reduction of our models. In addition to these reductions, equally important is to develop efficient methods for output analysis and interpretation. Mathematical tools such as PCA for decomposing the dynamics into its different modes, filtering out the noise or reconstructing the dominant pathways will be utilized to this aim. For example, the eigenvector corresponding to the zero eigenvalue of the transition matrix in the master equation formalism yields the steady state probabilities of the individual components of the system, while the eigenvector associated with the smallest eigenvalue extracts the slowest (or least probable) passage.

These approaches should help in developing models that are (i) more amenable to understanding, (ii) connected to experimental parameters, and (iii) have far fewer free parameters.  Methods of dynamical systems or control theory can then be used to study how these systems change with changes of parameters. In such a way, the different qualitative regimes of behavior can be mapped out explicitly.

Some of the underlying ideas of using simplified models and methods of dynamical systems have been applied in our previous work. We modeled and analyzed the spatio-temporal distribution of F actin in the cell in the presence of choppers and initiators (9;10). Other investigators used a simplified model to determine the point in the cell cycle when apoptosis is triggered, also showing the utility of such simple models (108). Even though the latter model was lacking in many details, there was a clear distinction between six possible mechanisms for cell death, which could be compared to experimental data. Simple systems of differential equations employing mass-action kinetics were the basis for the design of a bistable genetic switch (109) and a genetic oscillator (110), lending hope that reduced descriptions can capture natural cellular mechanisms. Finally, in the absence of sufficient data on kinetic and transport parameters, or initial concentrations, we will exploit cellular automata approaches (11;111).

We have also been involved in the last few years in other modern techniques for simplifying complex models (112;113) in the presence of multiple scale effects. Its relevance for biological systems is apparent when considering that even a small subset of the metabolic reactions may evolve into a chaotic behavior. Simple averaging techniques may not be applicable in these cases, and more advanced mathematical concepts, such as “measures” are needed, which we will use when appropriate.

Two challenges are, however, to ensure that the important details are not being neglected, and to be able to construct simplified models that are quantitative, rather than just qualitative. For example, the mean-field approximation is appropriate only if the system has intrinsic randomness and separable fast processes; and it is usually a major task to optimize the parameters representative of the mean-field effects in different environments. Likewise the network connectivity and pseudo-steady state conditions for individual steps must be validated by supporting data, either from experiments or simulations with more detailed models. More importantly, the continuous, deterministic approaches may not be adequate in most cases, and in particular in cell signaling and regulatory processes, due to the intrinsic discrete and stochastic nature of molecular interactions. Specific Aim 2 indeed focuses on the integration of the results from simulations and analytical methods at different levels for defining the reduced models and parameters that contain the ‘important details’.