Dynamic-energy-budget-driven fruiting-body formation in myxobacteria
M. Hendrata
1,
*
and B. Birnir
2
1
Department of Mathematics, California State University, 5151 State University Drive, Los Angeles, California 90032, USA
2
Center for Complex and Nonlinear Science and Department of Mathematics, University of California,
Santa Barbara, California 93106, USA
Received 15 June 2008; revised manuscript received 11 March 2010; published 1 June 2010
We develop an interacting particle model to simulate the life cycle of myxobacteria, which consists of two
main stages—the swarming stage and the development fruiting body formation stage. As experiments have
shown that the phase transition from swarming to development stage is triggered by starvation, we incorporate
into the simulation a system of ordinary differential equations ODEs called the dynamic energy budget,
which controls the uptake and use of energy by individuals. This inclusion successfully automates the phase
transition in our simulation. Only one parameter, namely, the food density, controls the entire simulation of the
life cycle.
DOI: 10.1103/PhysRevE.81.061902 PACS numbers: 87.18.Fx, 87.17.Jj, 87.18.Ed, 87.18.Gh
I. INTRODUCTION
Myxobacteria Myxococcus xanthus change their life
cycle according to food availability in their environment. In
an ideal condition, myxobacteria grow as swarms that spread
away from the center of a colony to search for nutrients from
the medium and oxygen from above. However, when nutri-
ents are depleted, myxobacteria undergo a phase transition in
which they stop growing individually, but instead they merge
and build a complex structure, called the fruiting body. The
stage of fruiting body development is thus initiated by star-
vation and built by cell movements and interactions. The
substages of fruiting body development that are observed in
the experiments consist of the formation of traffic jams and
initial aggregates, streaming, formation of three-dimensional
hemispherical mounds, formation of toroidal mounds, and
sporulation within the fruiting body 1.
In addition to starvation condition, there are other two
crucial factors required for the cells to proceed through the
stages in their life cycle. The first one is the cell motility.
Experiments have shown that there are two motility engines
in myxobacteria, namely, the social S motility and the ad-
venturous A motility. Nonmotile cells, which lack both of
these motility systems, are unable to form fruiting bodies 2.
S motility is driven by type IV pili that are found on the
cell’s leading pole 3,4. The cell can shoot its pili and attach
them to other cell or group of cells nearby. When the pili
retract, the cell body gets pulled forward toward the group. A
motility is driven by slime secretion from the cell’s lagging
pole. This slime secretion generates thrust that pushes the
cell forward 5. Cells have the tendency to turn at acute
angle and follow the slime trails secreted by other cells 6.
A wild-type A + S+ cell possesses both A and S motility
engines. The A + S- strain has only the A motility, while the
A - S+ strain has only the S motility. These three strains exist
and can be studied in a laboratory environment 3.
The other factor that is important in fruiting body devel-
opment is the cell signaling. Among many types of cell sig-
naling that occur during the development, C-signaling is the
one that controls the initial aggregation and also the transi-
tion between the substages during the fruiting body forma-
tion. C-signal is a 17-kDa cell-surface protein that is trans-
mitted by end-to-end contact between two cells 2,7. Once
C-signal molecule is inserted onto the cell surface, the ex-
pression of the csgA gene increases and this in return creates
a positive feedback loop that increases the number of
C-signal molecules on the signaling cells at an exponential
rate. As increasing C-signal reaches different levels, it pro-
vides the thresholds that trigger the cell to proceed through
the substages of the fruiting body development in a proper
temporal order 7,8. The cells within the fruiting body con-
tinue to C-signal until their individual C-signal level has
reached the final threshold for differentiation into spores.
Several models have been proposed to explain myxobac-
terial swarming and fruiting body formation. Two continuous
models analyze the spreading rates of myxobacteria swarms
on both short and long time scales 9. Alber’s group devel-
oped two types of discrete models: lattice and off-lattice.
Both discrete models are based on nonchemotactic cell-cell
interactions. Their early lattice gas cellular automaton
LGCA model succeeds in modeling the initial aggregation
and stream formation and 3D stochastic LGCA model simu-
lates the two stages of cell aggregation 10. Later on, their
unified 3D LGCA model successfully produces all stages of
the fruiting body formation 11. They next developed an
off-lattice model to minimize the geometric constraint inher-
ited in the lattice model. Their off-lattice model simulates
myxobacterial swarming and quantifies the contributions of
A and S motilities to swarming 12,13.
In this paper we extend Alber’s off-lattice model by add-
ing cell growth and cell division mechanism and further de-
velop an algorithm to simulate the entire life cycle of myxo-
bacteria, which includes the swarming and the fruiting body
development. Our model consists of four main components,
namely, the off-lattice cell representation, the motility algo-
rithm, a logistic equation, and a dynamic energy budget
DEB equations. The off-lattice cell representation and mo-
tility algorithm are described in Secs. II and VII, respec-
tively. A logistic equation is an ODE that we use to describe
the behavior of C-signal level during the development. DEB
is a system of ODEs that describes the acquisition and use of
*
mhendra@calstatela.edu
PHYSICAL REVIEW E 81, 061902 2010
1539-3755/2010/816/06190212 ©2010 The American Physical Society 061902-1