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Several mathematical models from theretical biology deal with various forms of cell motility (for example, cancer growth) or evolution of populations.

The equations that make up such models usually contain (among others) a Fickian diffusion term of the form $$- \delta \Delta u(t,x),$$ where $u$ is the cell (or population) density.


To fix some ideas, consider the following examples.

  1. A chemotaxis model:

$$\partial_t u(t,x) = \delta \Delta u(t,x) - \nabla \cdot(uK\star u),$$ where $K \star u = \int_\Omega K(x,y)u(t,y)dy$, for a certain kernel $K$.

  1. A model arising in population dynamics:

$$\partial_t u(t,x) = \delta\Delta u(t,x) + u(t,x) \left(\lambda - a\int_\Omega u(t,y) dy\right).$$


  • Biologically, what is the meaning of a situation where $\delta = 0$ (that is, where there is not Fickian diffusion governed by the Laplacian term)?

  • Is there any interest in studying such situations?

  • Can you point out some references on this topic?

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    $\begingroup$ I'm not familiar with these models, but is it even proposed that there could be situations where that constant is zero? Note that biology is not math, and much like physics, sometimes certain assumptions aren't bothered to be stated, because there is no biological interest in equation results outside those naturally observed. (also links to papers are always nice for people to follow the situation you are describing) $\endgroup$ – Bryan Krause Sep 25 '17 at 21:40
  • $\begingroup$ @BryanKrause For example, there are many models governed by equations of the kind $$\partial_t u = \delta\partial_{xx} u + \mathcal{I}[u],$$ where $\mathcal{I}$ is an integral term that accounts for nonlocal interactions and nonlocal diffusion of particles (cells). My general question is: does it ever make sense (biologically) for particles to not mode randomly, that is, to have $\delta = 0$? $\endgroup$ – user36814 Sep 26 '17 at 12:09
  • $\begingroup$ No, it doesn't make sense. You would be talking of drift in a common direction with no interactions or offsets, all cells in the field moving in synchrony. $\endgroup$ – Bryan Krause Sep 26 '17 at 15:43
  • $\begingroup$ @Jay I think this needs more context before you can get a good answer, i.e. a more specific model. I could imagine, e.g. a model where cells are not motile on their own (e.g. no drift or diffusion) but interact at a distance (via chemotaxis, etc.). This could lead to diffusion-like terms that aren't the simple Laplacian here, and thus delta = 0. I think Hans Othmer and others have worked on deriving these models from single-cell starting points, which would be a useful guide. $\endgroup$ – AJK Sep 26 '17 at 18:39
  • $\begingroup$ @AJK I edited my question. $\endgroup$ – user36814 Sep 27 '17 at 20:38
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Just so we're on the same page..

  • $u(x,t)$ is a concentration/density function that describes the number of particulate species (bacteria, gas molecules, etc.) at point-in-space $x$ and time $t$. And although it may be an obvious statement to make.. with $u$ having the parameters $(x,t)$, this indicates that particulate concentration values are dependent upon both position and time.

  • $\delta$ is the diffusion coefficient of the particulate concentration, defined as a proportionality constant between particulate diffusion and the overall particulate concentration gradient. Particulate diffusion can be regarded as local behavior, whereas the overall particulate concentration gradient describes the behavior of distribution for the entire particulate concentration, and of which may be affected by forces other than the particulate's response to an attractant/repellant gradient, including temperature, pressure, and/or other environmental variables. An in depth understanding of this coefficient can be obtained by reading sections 2.1 & 2.2.


Your questions

  • Biologically, what is the meaning of a situation where $\delta = 0$?

Well, if you did in fact read the recommended sections, you may already have realized that it's just not possible for $\delta = 0$, unless you're considering concentrations of mass that simply can't diffuse throughout one another, but even then $\delta$ would be considered as $DNE$.

To reference a portion of 2.1 & 2.2:

Molecular diffusion, strictly speaking, cannot occur under conditions in which both the net flux and the [concentration gradient] are simultaneously zero. If the [concentration gradient] is uniform, then in general fluxes are different for different species, and the net flux is not zero. If the net flux is zero, a small [concentration] gradient must exist in order to counter the tendency for the different species fluxes to be different.

That being said, the diffusion coefficient can be an extremely small value - of which is based on the nature of the particulate(s) undergoing diffusion, and the medium for which they travel throughout - but it's never zero. See here for a series of tables that contain diffusion coefficients for common substances at standard conditions.

And as for the biological meaning to $\delta = DNE$.. the only conclusion that could be made is that the mass concentrations under consideration can't diffuse with one another, for whatever reason. Whether or not there's meaning beyond this, I believe that would be dependent upon what's specifically being studied.


  • .. that is, where there is not Fickian diffusion governed by the Laplacian term?

This statement however, does have biological meaning to it. When $\delta \Delta u(t,x) = 0$, this means there is a zero net flux occurring at point-in-space $x$ at time $t$. Which is to say that - the number of particulates of a given species entering a given region of space over a given time interval is the exact same as the number of particulates of the same species leaving that exact same region of space over the exact same time interval. When this is true for all regions of space within the [biological] system, it means that the system is in a steady-state, and/or has reached equilibrium.

Note: Equilibrium and steady-state systems are not the same, however, the differences involved aren't relevant to the scope of this discussion, and so nothing further will be said about this.

When a biological system is in a steady-state, and/or is in equilibrium, that can carry significant implications. Perhaps the most ubiquitous example that also first comes to mind are the steady-states of ion channels, with the effects of cell activation/non-activation being the implications. Another example could be pretty much any chemical reaction that reaches equilibrium.


  • Is there any interest in studying such situations? Can you point out some references on this topic?

Yes, there are many occurrences where these states are studied, of which you may already be aware of. Regardless, here are a few that seem worthy enough for me to provide in this response:


In summary

The biological meaning of $\delta \Delta u(t,x) = 0$ usually deals with scenarios where the region(s) in space experiences a flow of mass that has equal incoming & outgoing rates for a given particulate species. Most of the time, this is indicative of a steady-state, and/or equilibrium, and this can be confined to local regions, or globally throughout the system.

And I'd also like to offer a slightly different perspective. Normally, the diffusion coefficient is a temperature & pressure dependent function. However, if you're considering situations with STP/NTP, the diffusion coefficient can truly be treated as a constant. When this is the case, the Laplacian term is analogous to the heat equation. By taking this perspective, the Laplacian term [being zero] is stating that there is no loss or gain in heat (species-specific particulate motion) when considering a region of space over a defined time interval, and the overall model would describe the change(s) in motion (heat) for the entire system. From there, biology could explain as to why the "heat" behaves the way it does, of which would involve the mechanisms & behaviors of an organism with respect to its reaction(s) to an attractant/repellent.

And lastly, for the sake of inclusivity, the other terms in these models generally represent behaviors that are external to local interactions, and each may carry differing effects, such as birth/death rates in population dynamics, or particulate concentration movement that's perpendicular to the axis to which the concentration gradient moves upon, as is the case with the second term in the chemotaxis model. So then, when the Laplacian term is not contributing to any change, i.e., when there is a zero net flux, these other terms will then be the only candidate contributors to any potential changes in the system, whatever those changes may be.


References

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  • $\begingroup$ I think this ignores a central aspect of the question, which is that there is a second term, which is not just the Keller-Segel chemotaxis term. Secondly, I think it's confusing to claim delta can't be zero. delta = 0 in KS corresponds to a perfectly useful (if somewhat unrealistic) limit - cells without random motility! Third, in general, student presentations aren't a great reference. $\endgroup$ – AJK Oct 3 '17 at 23:52
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    $\begingroup$ "I think this ignores a central aspect of the question, which is that there is a second term, which is not just the Keller-Segel chemotaxis term." - the OP never once brings into question the second term. If you want a brief explanation though: the second term is the total amount of cell concentration movement that's perpendicular to the axes that the concentration gradient moves on. Because of this, there's zero movement along the axis of the concentration gradient, and therefore that cell concentration quantity must be subtracted from the total change in cell concentration. $\endgroup$ – Charles Oct 4 '17 at 13:19
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    $\begingroup$ Secondly, $\delta$ can only exist if a concentration gradient between species exists. And there can only be a concentration gradient if the two mass concentrations can diffuse with one another. Hence, there can be no diffusion coefficient if the two mass concentrations can't diffuse. This would be analogous to a non permeable membrane separating two mass concentrations entirely. In this case, it wouldn't be valid to model motility with diffusion coefficient between the two particulate species because there's literally no chance of them diffusing with one another, and so, again, $\delta = DNE.$ $\endgroup$ – Charles Oct 4 '17 at 13:26
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    $\begingroup$ @AJK And then, for the linked sources for 2.1 & 2.2 -- I apologize for that; it would seem that those were mislinked. Given this, I understand your unwilling to accept that as a source for an in depth understanding of diffusion coefficient. I have updated that portion of my response accordingly. Thank you for pointing this out. FWIW, the content in their presentation paper is still accurate, and is congruent with all other information I've encountered about this topic thus far. $\endgroup$ – Charles Oct 4 '17 at 13:28
  • $\begingroup$ @AJK I've edited my response again to briefly address the other terms in the models, however I don't discuss them in depthly. I do hope though that this is enough to resolve the concerns you had expressed, as I don't believe I'll modify my response any further. $\endgroup$ – Charles Oct 5 '17 at 2:03

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