Logic-based models and ODEs - a bidirectional modeling approach

Last modified: 2014-03-31

#### Abstract

We consider two different mathematical formalisms for modeling biological networks: ordinary differential equations (ODEs) and asynchronous logic-based models. ODEs can represent the mechanisms in a more realistic way, but parameter identification is often impeded by the huge number of parameters and the limited availability of quantitative data. On the contrary, logic-based models provide only a coarse qualitative description of the mechanisms, but the finiteness of state and parameter space allows for a comprehensive analysis. In order to exploit their strengths and counteract their respective weaknesses, we consider ODE and logic-based models of the same system in parallel and analyze methods for transforming quantitative and qualitative models into one another.

Utilizing a corresponding logic-based model allows to evaluate hypothesis concerning

the behavior of an ODE system. We apply this idea to a previously developed ODE model of the bovine estrous cycle, which is based on fully coupled feedback mechanisms connecting different parts of the organism. To analyze the observed stable oscillatory behavior more globally, we transform the ODE model into a logic-based model. We use the signs of the Jacobian to derive an interaction graph, and decide on the value ranges of the variables. In the discrete model, the interplay of mechanisms is captured by logical parameters, which we derive directly from the continuous model or by limit considerations.

Model checking is used to validate this discretization procedure in terms of conservation of important trajectories. Our analysis reveals that the discrete model does not have any steady states and only exhibits limit oscillations. Concerning global stability we show the capacity of the system to return to the biologically meaningful oscillatory pattern upon perturbations. Specific model reduction techniques then allow to identify the core dynamics, which are expected to be underlying any larger model of the bovine estrous cycle.

Conversely, we propose to use an ODE model derived from a logical model to resolve more details of the observed discrete dynamics, to weed out modeling artifacts and to allow for a more realistic interpretation in the context of the underlying biological system. For example, it might allow to distinguish between a stable and a damped oscillation, both of which can be represented by a cycle in the discrete model's state transition graph (STG). Mirroring the procedure proposed above, the network structure and the logical functions are converted into a system of ODEs. Parameter values are estimated from biologically meaningful, validated discrete trajectories, i.e., paths in the STG, using Gauss-Newton methods. Comprehensive sensitivity analysis allows us to evaluate the results and highlight mechanisms and parameters that need to be clarified for a better understanding of the underlying system. The methodology was developed for researching regulation of cytokinin signaling in plants, a system where quantitative data is extremely hard to generate.

As our applications show, the dual approach exploiting both logic-based and ODE models in parallel has the potential to further our understanding of the fundamental mechanisms underlying biologically relevant behavior and to pinpoint the details indispensable for fine tuning them.

Utilizing a corresponding logic-based model allows to evaluate hypothesis concerning

the behavior of an ODE system. We apply this idea to a previously developed ODE model of the bovine estrous cycle, which is based on fully coupled feedback mechanisms connecting different parts of the organism. To analyze the observed stable oscillatory behavior more globally, we transform the ODE model into a logic-based model. We use the signs of the Jacobian to derive an interaction graph, and decide on the value ranges of the variables. In the discrete model, the interplay of mechanisms is captured by logical parameters, which we derive directly from the continuous model or by limit considerations.

Model checking is used to validate this discretization procedure in terms of conservation of important trajectories. Our analysis reveals that the discrete model does not have any steady states and only exhibits limit oscillations. Concerning global stability we show the capacity of the system to return to the biologically meaningful oscillatory pattern upon perturbations. Specific model reduction techniques then allow to identify the core dynamics, which are expected to be underlying any larger model of the bovine estrous cycle.

Conversely, we propose to use an ODE model derived from a logical model to resolve more details of the observed discrete dynamics, to weed out modeling artifacts and to allow for a more realistic interpretation in the context of the underlying biological system. For example, it might allow to distinguish between a stable and a damped oscillation, both of which can be represented by a cycle in the discrete model's state transition graph (STG). Mirroring the procedure proposed above, the network structure and the logical functions are converted into a system of ODEs. Parameter values are estimated from biologically meaningful, validated discrete trajectories, i.e., paths in the STG, using Gauss-Newton methods. Comprehensive sensitivity analysis allows us to evaluate the results and highlight mechanisms and parameters that need to be clarified for a better understanding of the underlying system. The methodology was developed for researching regulation of cytokinin signaling in plants, a system where quantitative data is extremely hard to generate.

As our applications show, the dual approach exploiting both logic-based and ODE models in parallel has the potential to further our understanding of the fundamental mechanisms underlying biologically relevant behavior and to pinpoint the details indispensable for fine tuning them.

#### Keywords

finite dynamical systems, ordinary differential equations, model parameters, trajectories,stability