David Holloway
Mathematics, British Columbia Institute of Technology
Presented in the Embryo Physics Course, November 30, 2011
Abstract
Chemical pattern formation is critical for putting differentiation factors and growth catalysts in the correct locations during plant morphogenesis. Turing-type reaction-diffusion theory provides a framework for understanding patterning dynamics. The great challenge for understanding chemical patterning in plant morphogenesis is that plants are continually growing. To understand the degree to which chemical patterning drives morphological changes, models must take into account the full feedback cycle between pattern, growth, and pattern (since changes in size and geometry affect pattern). In developing reaction-diffusion models of growth catalysis, we have found that the mechanism for coupling chemical pattern to growth is critical for controlling shape. We show how this control gives rise to pattern selection in 3D morphogenesis, and present a new self-contained model for dynamic control of the patterning boundaries between fast and slow growing regions of the plant.
Presentation
/files/presentations/Holloway11.pdf
David,
my last question was not about pure mathematics vs. physico-chemical mechanism. Rather it is of philosophical nature. Imagine that we have developed a formalism based on the Schrödinger equation directly that describes the growth nicely. Still the question remains, how the Nature computes this? Do you have any idea? Evgenii
Hi Evgenii,
I suppose my attitude is that scientific, mathematical theory is our best human attempt to describe Nature, combining human logic and human observation (hopefully with some objectivity, in that my cat experiences ‘gravity’ too). So, if, as you say, we had a comprehensive theory of plant morphogenesis from the Schrodinger equation up, we would have to solve it for our understanding. And I see computing as a method for solving the mathematical theory. I’m not sure Nature has to solve it the same way. My definition of computing is narrower than, e.g. Wolfram’s ideas of a universal algorithm – but I don’t see a fundamental differnce between a universal algorithm and a comprehensive scientific theory. I do appreciate the need for information storage and processing in biology (genetics), and I think e.g. DNA computing is fascinating in this regard. We need computation to solve our descriptions of how Nature’s processes unfold – I’m not sure computation is needed for those processes themselves to unfold.
I am going to post here all the questions I had during my talk yesterday, and my (written) responses:
[14:27] Paleo Darwin: Would vibrations on a sphere have same harmonics?
For hemispheres (~plant tips) we use the spherical harmonics with Dirichlet boundary conditions of 0 at the equator.
[14:30] Paleo Darwin: Is the bending realistic?
I think the bending is realistic for no gravitropism or phototropism in the model. Model growth only depends on the chemical growth catalysis. Any slight deviations in the orientation of the chemical pattern on the tip cause deviations. (And the bending/short-range deviations aren’t due to the mesh resolution, we’ve tested this.)
[14:31] Paleo Darwin: Consistent with clinostat observations?
[14:31] Paleo Darwin: Approximates zero gravity
[14:32] Paleo Darwin: Clinostat is a poor man’s space ship: stays on ground
I’d thought in terms of how a vine lying on the ground might deviate, but I should look into zero gravity growth experiments.
[14:34] Evgenii Rasa: Is there a correlation between your equations and physico-chemical processes in the real thing?
For algae, such as Acetabularia, there are Harrison’s 1980’s physico-chemical experiments showing pretty good circumstantial evidence that a reaction-diffusion mechanism is patterning proteins in the cell surface. Combined with the cell biology of Kiermayer and Meindl in Austria (Micrasterias), the idea is that the patterned surface proteins are docking proteins, directing where vesicles, carrying wall material from the cell interior, attach and deliver new wall material, causing localized outgrowth. In higher plants, I think the same fundamental applies, that you have to get growth catalysts localized to the areas where localized outgrowth occurs in morphogenesis. But the molecular biology is more likely to do with wall relaxation, e.g. via expansin-class proteins. Reaction-diffusion dynamics have been shown in the molecular biology of trichome (hair) patterning in Arabidopsis (Digiuni et al., Mol. Syst. Biol. 2008), but the experimental connection between RD and growth is still a bit nascent for higher plants.
[14:38] Paleo Darwin: No membranes needed to separate Brusselators?
The coupled Brusselators are in the same spatial continuum, no membranes. In the earlier model (1981), the coupling was strictly feedforward – mechanism 1 controlling where mechanism 2 formed pattern. Now, we are using full feedback to be able to do branching with growth.
[14:39] Evgenii Rasa: Your equations describe only the growing surface? Or the complete concentration profile (2D/3D) in the volume as well?
Yes, the equations are only on the surface. The ‘2D’ computations solve the RD equations on a curve in 2D space; the ‘3D’ computations solve the RD equations on triangular finite elements, no thickness, in 3D space. Our approach is that most of the pattern formation is occurring at the cell surface (single cells) or the tunica layer (higher plants).
[14:39] VasilyO: How large are your cells? And is the small number of molecules a problem?
The Acetabularia become ~5cm long, but the tip diameter is a few hundred micrometres. Similarly the Micrasterias diameter is about 500 micrometres. The wavelengths are on the order of 10’s of micrometres, which agrees with reaction rate constants and diffusivities in normal biochemical ranges.
Low numbers of molecules is a very interesting question – I do a lot of stochastic modeling with respect to pattern formation in the fruit fly. I did do some Langevin type noise computations on the 2D Micrasterias model, and was able to simulate some of the defects that are seen at high temperature or in old cultures. The relevant molecules may well be at low concentration, but reaction-diffusion mechanisms can be very strong filters of noise. This is one of my big interests in flies – where it is looking like the relevant RNAs are at copy numbers in the hundreds/cell, but wild-type segmentation is rarely disrupted. RD may be a way of conferring robustness, among other mechansisms.
[14:44] Paleo Darwin: Is time lapse of these pattern formations available?
[14:45] Paleo Darwin: In living organism?
Thurston Lacalli published time lapses of Micrasterias development. For the conifers, we don’t have it. We do follow individual embryos over the week(s) of cotyledon formation, but their orientation changes a lot as the cell mass they’re sitting in grows. It would be possible to stitch together pictures, but we haven’t done it yet.
[14:47] Paleo Darwin: So you have to guess on what morphogen molecules might be?
See above also – I think the higher plant biochemistry is going to be a bit of a mess. Patrick von Aderkas has tried disrupting cotyledon formation with a range of plant hormones. Disruption of auxin transport seems the most promising at this point, but no firm conclusions.
[14:47] Paleo Darwin: Docking proteins?
So, and see above too, I think the algae question is more solved, with a pretty clear picture of how wall material is delivered by vesicles to the cell surface, with patterned membrane-bound docking proteins determining where the vesicles deliver the new wall material.
[15:01] Paleo Darwin: Can you go another level of branching with same 2 Brusselators?
Certainly in principle, but we need to get mesh proliferation working with the double-brusselator mechanism to get that degree of growth.
[15:01] Paleo Darwin: Because of fixed size scaling, seems you cannot get fractal branching
That’s right – Micrasterias do have decreasing lobe size for subsequent branches, somewhat fractal, and we modeled that via a decreasing diffusivity (i.e. decreasing wavelength) in time.
[15:08] Paleo Darwin: David: Is there a formal mapping between mechanical and reaction-iffusion equations?
[15:09] Evgenii Rasa: I have an odd question. Let us imagine that we have a nice function that for example reproduces the growth well. How Nature computes this function?
[15:09] Paleo Darwin: All have same harmonics, so how to distinguish?
Reaction-diffusion, mechanochemical and mechanical dynamic mechanisms have solutions which can be treated as combinations (Fourier picture) of harmonic solutions. In principle, a number of such mechanisms can produce a given growth trajectory. So, I think it’s a matter of both fully exploring what the particular mechanisms can do or tend to do (for instance, in the RD world, we know a lot about how different kinetics tend to give different patterns, i.e. different combinations of harmonics), and experimentally getting a better picture of the molecular biology. For growth, there are probably a number of cases where geometric constraints and internal pressure produce a particular shape – i.e entirely mechanically – and there are models and experiments backing up cases like this. But, I think many cases of morphogenesis require localized wall relaxation/growth catalysis – and this is where you need chemical patterning. Here, we have the whole slate of hormone (i.e. ‘plant growth regulator’) experiments, among others. Ultimately, ‘computing’ a whole plant is going to take a combined chemical and mechanical approach. My attitude, with the 3D computations, is to use normal growth to see how far chemistry can take us. I think the main strength of the Turing models is the ability to break symmetry and select patterns. But, for detailed shape changes, mechanics is going to be needed, especially for cases where the surface shrinks, or has strong anisotropies (local growth preferentially in one direction).
[15:11] Paleo Darwin: Are there chiral effects in these organisms?
Not the ones I’ve been looking at.
[15:11] Ezekiel Zepp: The mechan ics are probably related to tensegrity mechanics and there will be no shear in triangulated nodes
Agreed – proper mechanics probably requires finite elements with thickness. This is on the ‘to do’ list with some code from Dr. Prusinkiewicz in Calgary.
[15:12] Paleo Darwin: Note that Turing himself discussed mechanics in 1952 paper
Yes, just saw that again looking over the 1952 paper a few weeks ago. One of the not-highly-sung-enough giants of 20th century science.