One can average over the cells at a given distance r and time t to construct a plot of the average directionality cos in space and time. When the concentration at ri is equal to the threshold Cth=c(ri,tinit) cells outside the initiating colony begin to participate in the relay and the wave is initiated. Examining Equation (7) as we did Equation (4) reveals that every concentration in a thick extracellular medium is proportional to a. The initial signaling colony is of size ri=4D/v (dashed vertical line). For v2/D, Equation (41) gives the decay-free relationship. Nonetheless, other details signaling molecule decay, pulsed emission, discreteness of cells can alter these robust scaling laws (Keener, 2000; Dieterle Pand Amir A, 2020. An explicit implementation of this method is provided at github./pdieterle/diffWavePropAndInit (Dieterle, 2020; copy archived at swh:1:rev:f8d9feffd57d05f47c8c14c6d9850643b2858d0a). In this limit, with =1/d, Equation (60a) becomes Cth=aD/h2v2, the one-dimensional analog of Equation (24); similarly, Equation (60b) simplifies to Cth=2a/hv, the one-dimensional analog of Equation (29).
Here, we revisit the propagation and initiation of diffusive waves in the context of cell signaling.
With cells sitting on a substrate, signaling molecules can only diffusive in the upper half of the plane, and we have a semi-infinite environment which accounts for an extra factor of 2 in the emission term, yielding: Instead of working directly with (z), we consider the cells to be of a thickness H such that 2(z)1H2exp(-z2/2H2) and. We see that the wave speed is indeed D-independent over two orders of magnitude in the diffusion constant. Thus, we are validated in using a continuum model with a thick extracellular medium, as for this experiment the extracellular medium thickness h=2 mm D/v and the mean distance between neutrophils, d=50 m, satisfies vd/4D0.171. While we focus on positive activation functions in this work, we emphasize that if signaling molecule degradation is dominated by cell-induced processes like uptake, then signaling molecule degradation is also proportional to the cell density and the resulting (presumably bistable) production curve will yield dynamics that are also beholden to this scaling law. This is relevant in, forexample, quorum sensing models in which the local presence of a signaling molecule can both upregulate (at relatively low concentrations) and downregulate (at relatively high concentrations) release of the same signaling molecule (Parkin and Murray, 2018). For cells and diffusion in 3D (right), signaling waves do not initiate for vri/D<3. Otherwise, they move around with no sustained directionality. It should not go without saying that there are dozens of candidate signaling molecules which govern accumulation of neutrophils and short-range recruitment, as outlined in Reategui et al. Thus, as long as vD/r, we can say that vc/r~ dominates terms like D(c/r~)/r and we can ignore the latter (Tanaka et al., 2017). will help to orient readers. With cells at a density of =(1/50 m)2 and a 3 mm thick extracellular medium, this corresponds to a production rate of a40 molecules/second/cell. By tracking the neutrophils in space and time, they observe highly directed motion of the neutrophils towards the target (pink) starting around t=200 s. There is a clear boundary in space and time the information wave front between the regions where cells migrate toward the target (pink) and jostle around with no particular direction (white and light blue). There, we note that: The same dynamics hold for cells on a curved surface (such as epithelia) as long as the length scale of the curvature and the thickness of the extracellular medium are both large compared to D/v. If one wishes to find the information front for a simple diffusive theory, one performs the same integral as above, but truncates the integration over R at ri. Such a configuration is relevant for signaling in bacterial consortia atop thick, permeable substrates (Parkin and Murray, 2018) or anywhere that a lower dimensional tissue abuts a thick and permeable extracellular environment as can be found, forexample, in the retina.
Here, cells on the target (within ri) signal distant neighbors by continuously emitting a single signaling molecule. In this limit, it is smaller than c0 by roughly a factor of n a very small correction. Thus we can see that, as in the continuum theory of the main text, the wave will always initiate for m=1,2 but has a critical radius for m=3. The same dynamics hold for cells on a curved surface (such as epithelia) as long as the length scale of the curvature and the thickness of the extracellular medium are both large compared to D/v. With this choice, Equation (57) becomes: where in Equation (58b) we have taken the f0 limit. 5) Including the neutrophil swarming data in the main text succeeds in its purpose, which is to motivate the formal analysis and to help illustrate the benefits of relay propagation over simple diffusion that the immune system might exploit. As shown in Appendix 5figure 1, for n2, the wave speeds are very close to v with significant deviation only for n=1. We begin by considering a static group of cells uniformly distributed in two dimensions forexample, atop a solid surface and described by an area density (Figure 1A). Moreover, the information wave fronts of simple diffusive theories are inconsistent (Appendix 9figure 1C) with the observed chemotactic index dynamics catalogued by Retegui et al., 2017 for neutrophils with blocked LTB4 (BLT1/2) receptors. We consider cells and diffusion in the same number of dimensions, m. In an m-dimensional diffusive environment, the concentration created by this source which emits at a rate a at a radius ri and time t will be given by: For m=3, this integral is bounded from above by a/4Dri as t while for m=1,2, the concentration diverges as t. For cells in 1D with diffusion in 1D, the scenario described above can be described with the following equation of motion: One can non-dimensionalize Equation (79) by dividing all the concentration scales by Cth, dividing all the length scales by lc=h2CthD/a, and dividing all the time scales by c=h2Cth/a, thusly arriving at. Here, we consider the effects of each independently. This heuristic also holds for cells in two and three dimensions, and for cells scattered randomly according to a Poisson process. We can also see that the concentration profile beyond the wave front is exponential, not Gaussian as for simple diffusion.
To do so, we write down the Greens functions Gn,m(r,t;R,T) describing diffusion of molecules in m dimensions released by cells in n dimensions at (R,T) and measured by cells at (r,t). (A) Information fronts for relay (black) and simple diffusion (gray) models. We explicitly discuss only the asymptotics of cells in 2D with diffusion in 2D (equivalent to cells in 1D with diffusion in 1D or cells in 3D with diffusion in 3D, as shown previously), although we quote the results for cells in 2D with diffusion in 3D (equivalent to cells in 1D with diffusion in 2D) which are obtained using the Fourier transform machinery in Appendix 2: Asymptotic wave ansatz. Adding signaling molecule decay to Equation (4) would yield a model first considered by McKean, 1970 in the context of nerve impulse propagation. Here, we explicitly calculate some of the properties of a simple diffusion model. In essence, we wish to examine the wave from the perspective of an observer moving at the wave front. Cells (pink with purple nucleus) release a signaling molecule that diffuses (blue clouds). Finally, it would also be interesting to leverage the design principles we have discussed for engineering synthetic relays, a field with a rich history (Parkin and Murray, 2018; Brenner et al., 2008; Brenner et al., 2007; Basu et al., 2005). As with 1D cells/diffusion and 2D cells/diffusion, we recover.
With these definitions, c/r~=c/r and c/t=-vc/r~.
This gives: which is another pair of piecewise, straightforward-to-solve, linear ODEs.
In vitro experiments (Retegui et al., 2017) indicate that small colonies of a pathogen can indeed fail to incite a swarm. However, there is another potential reason to use diffusive relays: they create strong gradients that may help cells chemotax effectively. With a time delay of 250 s, the information wave fronts of simple diffusive models can be made consistent with the observed information wave fronts. (B) Same as A, but for cells in 2D and diffusion in 2D. We additionally account for other phenomena molecule decay, pulsed emission, and the discreteness of cells that do affect the asymptotic wave dynamics; in so doing, we provide an intuitive rubric for determining under what conditions these effects alter the wave propagation speed. This effect and its analogs have long been studied in the context of excitable media (Keener and Sneyd, 2009; Keener, 1987; Muratov, 2000) and observed in biological phenomena as diverse as natural cell signaling circuits (Noorbakhsh et al., 2015; Plsson and Cox, 1996; Kessler and Levine, 1993; Gelens et al., 2014), synthetic cell signaling circuits (Parkin and Murray, 2018), apoptosis (Cheng and Ferrell, 2018), range expansions (Tanaka et al., 2017; Fisher, 1937; Kolmogorov et al., 1937; Barton and Turelli, 2011; Gandhi et al., 2016; Birzu et al., 2018), and development (Chang and Ferrell, 2013; Vergassola et al., 2018; Muratov and Shvartsman, 2004; Nolet et al., 2020). For cells in 1D and diffusion in 1D, we assume a narrow channel of width h in both dimensions perpendicular to the channel. The referenced data source includes an inhibited relay experiment, more data from the heatmap shown in Figure 4, and quantification of the signaling molecule concentrations at different time points. But first, we note that the right sides of Equations (28a) and Equations (28b) have no support when kv/D. We see that even low-order (n=2,3) Hill functions exhibit relatively large (compared to c) initiation times for rilc. We consider the integral in Equation (62), but take tinit which gives us a maximum concentration Cmax,2,3 at r=ri of: Thus, by Equation (29), if ri
Does the response profile appear consistent as the wave moves? We plot the resulting gradients in this thin extracellular medium limit against the gradients from a comparable relay model in Appendix 10figure 1B. We can solve for the bi by applying boundary conditions. The full dependence of tinit on ri is pictured in Figure 3C, where we show that the above limits are valid approximations.
In the interests of transparency, eLife publishes the most substantive revision requests and the accompanying author responses. The details of the simulation are described in Appendix 5: Asymptotic wave dynamics with Hill function activation. A full solution of Equation (7), obtained in Appendix 2: Asymptotic wave ansatz by combining a partial Fourier transform in the z-dimension and the methods used to solve Equation (4), yields. On a mechanistic level, although a relay mechanism would allow neutrophils to quickly coordinate their response, it remains unclear how inflammatory response is modulated in such a scenario. (A) Wave speed v for square pulse emission by cells in 2D with diffusion in 2D as a function of pulse width . Here, the dynamics are governed by the following equation of motion: Now that we are considering the initiation dynamics, we must include terms like D(c/r)/r, which we could previously neglect in our asymptotic analysis of cells in 2D with diffusion in 2D. (In the n limit of Heaviside activation, c=D/v2 and lc=D/v.) In the case of neutrophils, this means that we are ignoring the possibility that a cell initially located off the target randomly encounters the target and starts signaling. We thank the reviewers for noticing this gap in our Discussion. Here, in the context of cell signaling, we show that system dimensionality the shape of the extracellular medium and the distribution of cells within it can dramatically affect the wave dynamics, but that these dynamics are insensitive to details of cellular activation.
The wave speed relationship given in Equation (5) is analogous to the Fisher-Kolmogorov wave speed (Fisher, 1937; Kolmogorov et al., 1937; Gelens et al., 2014) with hCth/a replacing the doubling time as the characteristic time scale in the problem and has been discussed in beautiful previous work (Gelens et al., 2014; Meyer, 1991), starting with Luther, 1906.
what is contact-dependent signaling used for
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what is contact-dependent signaling used for