Mathematically modelling the effects of counting factor in Dictyostelium discoideum

First-order agent-based models of emergent behaviour of Dictyostelium discoideum and their inspiration for swarm robotics

Collective behaviour in nature provides a source of inspiration to engineer artificial collective adaptive systems, due to their mechanisms favouring adaptation to environmental changes and enabling complex emergent behaviour to arise from a relatively simple behaviour of individual entities. As part of our ongoing research, we study the social amoeba Dictyostelium discoideum to derive agent-based models and mechanisms that we can then exploit in artificial systems, in particular in swarm robotics. In this paper, we present a selection of agent-based models of the aggregation phase of D. discoideum, their corresponding biological illustrations and how we used them as an inspiration for transposing this behaviour into swarms of Kilobots. We focus on the stream-breaking phenomenon occurring during the aggregation phase of the life cycle of D. discoideum. Results show that the breakup of aggregation streams depends on cell density, motility, motive force and the concentration of cAMP and CF. The breakup also comes with the appearance of late centres. Our computational results show similar behaviour to our biological experiments, using Ax2(ka) strain. For swarm robotics experiments, we focus on signalling and aggregation towards a centre.

Directional sensing and streaming in Dictyostelium aggregation

We merge the Kessler-Levine simple discrete model for Dictyostelium cyclic adenosine monophosphate (cAMP) production and diffusion with the Dilão-Hauser directional sensing aggregation mechanism. The resulting compound model describes all the known transient patterns that emerge during Dictyostelium aggregation, which include the spontaneous formation of cAMP self-sustained target and spiral waves and streaming. We show that the streaming patterns depend on the speed of the amoebae, on the relaxation time for the production of cAMP, on the cAMP degradation rate, and on directional sensing. Moreover, we show that different signaling centers emerge during Dictyostelium aggregation.

A cell number counting factor alters cell metabolism

It is still not clear how organisms regulate the size of appendages or organs during development. During development, Dictyostelium discoideum cells form groups of approximately 2 x 10(4) cells. The cells secrete a protein complex called counting factor (CF) that allows them to sense the local cell density. If there are too many cells in a group, as indicated by high extracellular concentrations of CF, the cells break up the group by decreasing cell-cell adhesion and increasing random cell motility. As a part of the signal transduction pathway, CF decreases the activity of glucose-6-phosphatase to decrease internal glucose levels. CF also decreases the levels of fructose-1,6-bisphosphate and increases the levels of glucose-6-phosphate and fructose-6-phosphate. In this report, we focus on how a secreted signal used to regulate the size of a group of cells regulates many basic aspects of cell metabolism, including the levels of pyruvate, lactate, and ATP, and oxygen consumption.

Cell density sensing and size determination

The social amoeba Dictyostelium discoideum is one of the leading model systems used to study how cells count themselves to determine the number and/or density of cells. In this review, we describe work on three different cell-density sensing systems used by Dictyostelium. The first involves a negative feedback loop in which two secreted signals inhibit cell proliferation during the growth phase. As the cell density increases, the concentrations of the secreted factors concomitantly increase, allowing the cells to sense their density. The two signals act as message authenticators for each other, and the existence of two different signals that require each other for activity may explain why previous efforts to identify autocrine proliferation-inhibiting signals in higher eukaryotes have generally failed. The second system involves a signal made by growing cells that is secreted only when they starve. This then allows cells to sense the density of just the starving cells, and is an example of a mechanism that allows cells in a tissue to sense the density of one specific cell type. The third cell density counting system involves cells in aggregation streams secreting a signal that limits the size of fruiting bodies. Computer simulations predicted, and experiments then showed, that the factor increases random cell motility and decreases cell-cell adhesion to cause streams to break up if there are too many cells in the stream. Together, studies on Dictyostelium cell density counting systems will help elucidate how higher eukaryotes regulate the size and composition of tissues.

Understanding streaming in Dictyostelium discoideum: theory versus experiments

Recent experimental work involving Dictyostelium discoideum seems to contradict several theoretical models. Experiments suggest that localization of the release of the chemoattractant cyclic adenosine monophosphate to the uropod of the cell is important for stream formation during aggregation. Yet several mathematical models are able to reproduce streaming as the cells aggregate without taking into account localization of the chemoattractant. A careful analysis of the experiments and the theory suggests the two major features of the system which are important to stream formation are random cell motion and chemotaxis to regions of higher cell density. Random cell motion acts to reduce streaming, whereas chemotaxis to regions of higher cell density reinforces streaming. With this understanding, the experimental results can be explained in a manner consistent with the theoretical results. In all the experiments, alterations in the two main factors of random motion and chemotaxis to regions of higher cell density, not the localization of the release of the chemoattractant, can explain the results as they relate to streaming. Additionally, a comparison of results from a mathematical model that simulates cells which localize the chemoattractant and cells which do not shows little difference in the streaming patterns.

The virtual cell--a candidate co-ordinator for 'middle-out' modelling of biological systems

Understanding the functioning of biological systems depends on tackling complexity spanning spatial scales from genome to organ to whole organism. The basic unit of life, the cell, acts to co-ordinate information received across these scales and processes the myriad of signals to produce an integrated cellular response. Cells interact with and respond to other cells through direct or indirect contact, resulting in emergent structure and function of tissues and organs. Systems biology has traditionally used either a 'top-down' or 'bottom-up' approach. However, neither approach takes account of heterogeneity or 'noise', which is an inherent feature of cellular behaviour and may have significant impact on system level behaviour. We review existing approaches to modelling that use cellular automata or agent-based methodologies, where individual cells are represented as equivalent virtual entities governed by simple rules. These paradigms allow a direct one-to-one mapping between real and virtual cells that can be exploited in terms of acquiring parameters from experimental systems, or for model validation. Such models are inherently extensible and can be integrated with other modelling modalities (e.g. partial or ordinary differential equations) to model multi-scale phenomena. Alternatively, hierarchical agent models may be used to explore the functions of biological systems across temporal and spatial scales. This review examines individual-based models and the application of the paradigm to explore multi-scale phenomena in biology. In so doing, it demonstrates how cellular-based models have begun to play an important role in the development of 'middle-out' models, but with considerable potential for future development.

Combining experiments and modelling to understand size regulation in Dictyostelium discoideum

Little is known about how the sizes of specific organs and tissues are regulated. To try to understand these mechanisms, we have been using a combination of modelling and experiments to study the simple system Dictyostelium discoideum, which forms approximately 20000 cell groups. We found that cells secrete a factor, and as the number of cells increases, the concentration of the factor increases. Diffusion calculations indicated that this lets cells sense the local cell density. Computer simulations predicted, and experiments then showed, that this factor decreases cell-cell adhesion and increases random cell motility. In a group, adhesion forces keep cells together, while random motility forces cause cells to pull apart and separate from each other. As the group size increases above a threshold, the factor concentration goes above a threshold and the cells switch from an adhered state to a separated state. This causes excessively large groups to break apart and/or dissipate, creating an upper limit to group size. In this review, we focus on how computer simulations made testable predictions that led the way to understanding the size regulation mechanism mediated by this factor.