Pattern formation and traveling waves in myxobacteria: Theory and modeling

Abstract

Recent experiments have provided new quantitative measurements of the rippling phenomenon in fields of developing myxobacteria cells. These measurements have enabled us to develop a mathematical model for the ripple phenomenon on the basis of the biochemistry of the C-signaling system, whereby individuals signal by direct cell contact. The model quantitatively reproduces all of the experimental observations and illustrates how intracellular dynamics, contact-mediated intercellular communication, and cell motility can coordinate to produce collective behavior. This pattern of waves is qualitatively different from that observed in other social organisms, especially Dictyostelium discoideum, which depend on diffusible morphogens. Myxobacteria are common components of soil, but their life cycle is far from common. Although they are prokaryotes, their life, in some respects, is similar to that of multicellular organisms (1, 2). Under starvation conditions, a population of myxobacterial cells aggregates by streaming into a number of central foci, eventually forming at the focus a multicellular fruiting body. During this aggregation phase, the cells may pass through a period where the surface is swept by a complex pattern of waves, called the ''ripple phase.'' These waves are composed of bacteria moving in concert in such a way that colliding waves appear to pass through one another (3). This is quite unlike the seemingly similar phenomenon observed in Dictyostelium discoideum and in chemical waves where colliding wave fronts annihilate one another (4, 5). Here we present a quantitative model for the ripple phase in Myxococcus xanthus that reproduces most of the observed phenomena. A distinguishing feature of this model is that it depends only on intercellular communication by direct cell contact, without any diffusible morphogen signaling.We shall base our model on the following consequences of experimental observations on Myxobacteria.(i) Contact Signaling. Myxobacteria signal via the C-signaling system, which operates only when two cells contact one another nearly end to end (3, 6, 7). The ripple patterns can be altered significantly, or even abolished, by manipulation of external C-signal protein concentration or dilution of wild-type cells by mutants that can receive, but not send, C-signal (3). Therefore, we shall base the model on signaling that depends entirely on direct cell contacts, with no diffusible signaling molecule.(ii) Reversal Cycle. Experiments on individual prerippling bacteria under various conditions show that they glide back and forth, reversing their direction spontaneously about every 5-10 min with a variance much smaller than the mean (see table 1 of ref. 8 and table 2 of ref. 9). Thus the times between reversals are not exponentially distributed, i.e., not Markovian. We interpret this to mean that the internal biochemical circuit controlling reversals contains a delay or cycle time for completion.(iii) Density Dependence. Measurements show that reversal frequencies depend on the amount of C...

Stochastic models of intracellular transport

“…Several stochastic simulation packages have also been developed including MCell (Franks et al, 2001(Franks et al, , 2002 and smoldyn (Andrews, 2012). In addition to the example of phase separation during cell polarity, macroscopic reaction-diffusion systems can exhibit complex spatiotemporal dynamics including coherent oscillations, wave propagation and Turing pattern formation (Falcke, 2003;Igoshin et al, 2001;Keener and Sneyd, 2009;Lenz and Sogaard-Andersen, 2011;Loose et al, 2011). These are thought to play an important role in a variety of cellular processes including morphogenesis, cell division, and embryogenesis (Murray, 2002).…”

A biochemical oscillator explains several aspects ofMyxococcus xanthusbehavior during development

“…Counterpropagating crests appear to pass through one another unaffected, unlike developmental waves in Dictyostelium or Turing instability patterns in diffusion-reaction systems, which annihilate upon collision (28)(29)(30)(31). A mathematical model of the myxobacteria traveling waves explains most of their characteristics (3,4). In particular, the model shows that the waves do not pass through one another; rather, colliding waves reflect off one another, so that a crest oscillates back and forth between neighboring crests.…”

“…This is an illusion: the waves actually reflect off one another, with every participating bacterium simply oscillating back and forth. A mathematical model for these waves and for the swirling aggregations that follow has been published (3,4). Alternative models of rippling sharing similar ingredients have been developed (5)(6)(7).…”

“…With this assumption, the model successfully reproduces observed spatial patterns. However, the treatment of the reversal clock was abstract and did not involve any biochemical modeling (3,4). Here we provide a biochemical model for this clock that explains both the cellular oscillations and the developmental progression of myxobacteria morphogenesis from vegetative swarming to the ripple phase and thence to the swirling aggregation phase that leads to fruiting body formation.…”

Mutations of the Act Promoter in Myxococcus xanthus

“…The low initial level of C signal induces traveling wave behavior (20,37,55,70), a higher level produces streaming behavior, and the highest level induces spores to form. Cells in a stream are arranged end to end, all following the same trajectory, which leads to frequent end-to-end contacts (22)(23)(24).…”