Difference between revisions of "Team:KU Leuven/Modeling/Hybrid"

The hybrid model represents an intermediate level of detail in between the colony level model and internal model. Bacteria are treated as individual agents that behave according to Keller-Segel type discretized stochastic differential equations, while chemical species are modeled using partial differential equations.

Introduction

Partial differential equations Spatial reaction-diffusion models that rely on Partial Differential Equations (PDEs) are based on the assumption that the collective behavior of individual entities, such as molecules or bacteria, can be abstracted to the behavior of a continuous field that represents the density of those entities. The brownian motion of molecules, for instance, is the result of inherently stochastic processes that take place at the individual molecule level, but is modeled at the density level by Fick’s laws of diffusion. These PDE-based models provide a robust method to predict the evolution of large-scale systems, but are only valid when the spatiotemporal scale is sufficiently large to neglect small-scale stochastic fluctuations and physical granularity. Moreover, such a continuous field approximation can only be made if the behavior of the individual entities is well described.

Agent-based models

Agent-based models on the other hand explicitly treat the entities as individual “agents” that behave according to a set of “agent rules”. An agent is an object that acts independently from other agents and is influenced only by its local environment. The goal in agent-based models is to study the emergent systems-level properties of a collection of individual agents that follow relatively simple rules. In smoothed particle hydrodynamics for example, fluids are simulated by calculating the trajectory of each individual fluid particle at every timestep. Fluid properties such as the momentum at a certain point can then be sampled by taking a weighted sum of the momenta of the surrounding fluid particles. A large advantage of agent-based models is that the agent rules are arbitrarily complex and thus they allow us to model systems that do not correspond to any existing or easily derivable PDE model. However, because every agent is stored in memory and needs to be processed individually, simulating agent-based models can be computationally intensive.

Hybrid modelling framework

In our system, there are both bacteria and chemical species that spread out and interact on a petri dish to form patterns. On the one hand, the bacteria are rather complex entities that move along chemical gradients and interact with one another. Therefore they are ideally modeled using an agent-based model. On the other hand, the diffusion and dynamics of the chemicals leucine and AHL are easily described by well-established PDEs. To make use of the advantages of each modelling approach, we decided to combine these two different types of modeling in a hybrid modelling framework. In this framework we modeled the bacteria as agents, while the chemical species were modeled using PDEs. There were two challenges to our hybrid approach, namely coupling the models and matching them. Coupling refers to the transfer of information between the models and matching refers to dealing with different spatial and temporal scales to achieve accurate, yet computationally tractable simulations.
In the following paragraphs we first introduce our hybrid model and its coupling. Once the basic framework is established, the agent-based module and PDE module are discussed in more depth and the issue of matching is highlighted. We also expand on important aspects of the model and its implementation such as boundary conditions and choice of timesteps. Then the results for the 1-D model and 2-D model simulations are shown and summarized. Finally, the incorporation of the internal model into the hybrid model is discussed and a proof of concept is demonstrated.