Abstract:We propose a mechanism for the homeostatic control of synapses along the ventral cord of Caenorhabditis elegans during development, based on a form of Turing pattern formation on a growing domain. C. elegans is an important animal model for understanding cellular mechanisms underlying learning and memory. Our mathematical model consists of two interacting chemical species, where one is passively diffusing and the other is actively trafficked by molecular motors, which switch between forward and backward moving… Show more
“…In many cases, it is the primary mechanism by which cells distribute or transport particles, both within single cells and throughout tissues. Diffusion-driven morphogen gradients, in conjunction with chemical reactions and active transport, are responsible for directional signaling in development [2,16]; diffusion of signaling molecules from the cell membrane to the nucleus is important for gene regulation [4,10]; and within the nucleus, crowding by chromatin has a profound effect on the diffusion of proteins to specific DNA binding sites [5]. There is a wealth of other biological examples.…”
Diffusion in cell biology is important and complicated. Diffusing particles must contend with a complex environment as they make their way through the cell. We analyze a particular type of complexity that arises when diffusing particles reversibly bind to elastically tethered binding partners. Using asymptotic analysis, we derive effective equations for the transport of both single and multiple particles in the presence of such elastic tethers. We show that for the case of linear elasticity and simple binding kinetics, the elastic tethers have a weak hindering effect on particle motion when only one particle is present, while, remarkably, strongly enhancing particle motion when multiple particles are present. We give a physical interpretation of this result that suggests a similar effect may be present in other biological settings.
“…In many cases, it is the primary mechanism by which cells distribute or transport particles, both within single cells and throughout tissues. Diffusion-driven morphogen gradients, in conjunction with chemical reactions and active transport, are responsible for directional signaling in development [2,16]; diffusion of signaling molecules from the cell membrane to the nucleus is important for gene regulation [4,10]; and within the nucleus, crowding by chromatin has a profound effect on the diffusion of proteins to specific DNA binding sites [5]. There is a wealth of other biological examples.…”
Diffusion in cell biology is important and complicated. Diffusing particles must contend with a complex environment as they make their way through the cell. We analyze a particular type of complexity that arises when diffusing particles reversibly bind to elastically tethered binding partners. Using asymptotic analysis, we derive effective equations for the transport of both single and multiple particles in the presence of such elastic tethers. We show that for the case of linear elasticity and simple binding kinetics, the elastic tethers have a weak hindering effect on particle motion when only one particle is present, while, remarkably, strongly enhancing particle motion when multiple particles are present. We give a physical interpretation of this result that suggests a similar effect may be present in other biological settings.
“…The GM model has been successfully used to study similar patterns in other biological systems [27, 28], and we applied that model here to better understand specification of aperture positioning. A similar model has previously been used to simulate pattern forming processes, where the underlying interaction network is unknown, to great effect [25]. As more information is gained about the proteins involved in aperture specification, more detailed and biologically realistic models can be developed to explore aperture formation in more detail.…”
Section: Discussionmentioning
confidence: 99%
“…To explore biochemical and geometric parameters that may be responsible for the observed aperture patterns but cannot yet be approached experimentally, we turned to mathematical modeling. As with other studies where little is known about the underlying interaction network [25], a pattern-forming model can be used to study broad properties of the biological system. The equally spaced distribution of the INP1-decorated membrane domains and apertures in Arabidopsis is reminiscent of patterns that can be generated with mathematical pattern-formation models.…”
Pollen provides an excellent system to study pattern formation at the single-cell level. Pollen surface is covered by the pollen wall exine, whose deposition is excluded from certain surface areas, the apertures, which vary between the species in their numbers, positions, and morphology. What determines aperture patterns is not understood.
Arabidopsis thaliana
normally develops three apertures, equally spaced along the pollen equator. However, Arabidopsis mutants whose pollen has higher ploidy and larger volume develop four or more apertures. To explore possible mechanisms responsible for aperture patterning, we developed a mathematical model based on the Gierer-Meinhardt system of equations. This model was able to recapitulate aperture patterns observed in the wild-type and higher-ploidy pollen. We then used this model to further explore geometric and kinetic factors that may influence aperture patterns and found that pollen size, as well as certain kinetic parameters, like diffusion and decay of morphogens, could play a role in formation of aperture patterns. In conjunction with mathematical modeling, we also performed a forward genetic screen in Arabidopsis and discovered two mutants with aperture patterns that had not been previously observed in this species but were predicted by our model. The
macaron
mutant develops a single ring-like aperture, matching the unusual ring-like pattern produced by the model. The
doughnut
mutant forms two pore-like apertures at the poles of the pollen grain. Further tests on these novel mutants, motivated by the modeling results, suggested the existence of an area of inhibition around apertures that prevents formation of additional apertures in their vicinity. This work demonstrates the ability of the theoretical model to help focus experimental efforts and to provide fundamental insights into an important biological process.
“…Numerical simulations of the model using experimentally based parameters generated patterns with a wavelength consistent with the synaptic spacing found in C. elegans , after identifying the in-phase CaMKII/GLR-1 concentration peaks as sites of new synapses. Extending the model to the case of a slowly growing one-dimensional compartment, we subsequently showed how the synaptic density can be maintained during C. elegans growth, owing to the insertion of new concentration peaks as the length of the compartment increases [25]. Figure 1( a ) Schematic figure showing the distribution of synaptic punta along the ventral cord of early- and late-stage C. elegans .…”
Section: Introductionmentioning
confidence: 99%
“…The structure of the paper is as follows. The hybrid reaction–transport model of [24,25] is introduced in §2; this is a three-component model consisting of a passively diffusing activator concentration A and a pair of actively transported inhibitor concentrations H ± , which correspond to left-moving and right-moving velocity states, respectively. In steady state, the full model is shown to be equivalent to the classical two-component GM model for ( A , H ), where H = H + + H − .…”
Simulations of classical pattern-forming reaction–diffusion systems indicate that they often operate in the strongly nonlinear regime, with the final steady state consisting of a spatially repeating pattern of localized spikes. In activator–inhibitor systems such as the two-component Gierer–Meinhardt (GM) model, one can consider the singular limit
D
a
≪
D
h
, where
D
a
and
D
h
are the diffusivities of the activator and inhibitor, respectively. Asymptotic analysis can then be used to analyse the existence and linear stability of multi-spike solutions. In this paper, we analyse multi-spike solutions in a hybrid reaction–transport model, consisting of a slowly diffusing activator and an actively transported inhibitor that switches at a rate
α
between right-moving and left-moving velocity states. Such a model was recently introduced to account for the formation and homeostatic regulation of synaptic puncta during larval development in
Caenorhabditis elegans
. We exploit the fact that the hybrid model can be mapped onto the classical GM model in the fast switching limit
α
→ ∞, which establishes the existence of multi-spike solutions. Linearization about the multi-spike solution yields a non-local eigenvalue problem that is used to investigate stability of the multi-spike solution by combining analytical results for
α
→ ∞ with a graphical construction for finite
α
.
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