Alan Turing's "Chemical basis of morphogenesis"

[Swapnam]

Ana suggested me to read this seminal paper by Turing, that uses reaction-diffusion theory for describing how chemical processes can lead to the spontaneous formation of biological patterns like stripes, spots, and other morphologies in organisms. As she said, "Turing provides a framework for understanding how patterns emerge in nature, bridging the gap between simple mathematical rules and the complex forms we observe." Its historical importance and the subsequent work that it has inspired can be found in Philip Ball's wonderful book Shapes, first in his trilogy on Nature' patterns.

The paper itself is freely available online. I am posting a paraphrase below that I jot down as I read the paper, trying to capture the essence under the mathematical intricacies.

[Swapnam]

Turing identifies the mechanical and chemical modes of analysis and focuses upon the latter, with the unit of attention as morphogens. The examples he provides hint at a functionalist approach - genes, hormones, skin pigments all fall under the category as long as they satisfy the relational properties of a morphogen as an evocator.

Straight off he makes one ponder over an epistemological issue - how do different "layers of science" interact i.e. strong vs. weak emergence - Can everything eventually be reduced to the dynamical laws operating over quantum fields (or whatever the lowest level might be), or do different conceptual classes like atoms -> molecules -> cells -> humans have independent power of explanation that contributes something original and not derivable, even in principle, from the other levels?

Morphogens diffuse into a tissue and react together, at times catalyzing the processes, which in turn lead to other morphogens, eventually resulting in a substance with a novel function. This development proceeds through diffusion from regions of higher to lower concentrations at a rate influenced by the properties of the substance (diffusability) and the environment (gradient). Additionally, the presence of the cell walls acts as a filter for some molecules over others.

So our model is that of N cells and M morphogens reacting in them and diffusing across, and the state of the system is described by MN numbers i.e. concentration of each morphogen in each cell, evolving as per the underlying law. Note that if the concentrations move towards an extreme value that is not sustainable biologically, that means we will simply not observe it in nature.

We come to the central problem - how does an (almost) spherically symmetrical blastula evolve into a highly differentiated organism? Turing contends that there can a high number of slight irregularities that get nudged from the initial unstable equilibrium into a few stable regions sans the initial homogeneity. The cause can be random disturbances such as temperature fluctuations or cell growth.

This reminded me of "spontaneous symmetry breaking" in particle physics that is used to explain the present form of forces e.g. electroweak unification at higher energies that we observe separately as electromagnetism and weak nuclear force.

In our particular example, the rate at which different morphogens are produced, destroyed, transformed into one another and diffuse across cells represent this drifting. He sets up a ring model of cells where diffusion is limited to a cell's left and right neighbors alongside intra-cell reactions, and proceeds to solve the differential equations for their concentrations under some simplifying assumptions (e.g. linearity of chemical reaction rate so that the system doesn't move too far from initial homogeneity, no external disturbances once the system has been provoked out of its stability).

This leads to an oscillating wave solution, which can be stationary or travelling, based on parameter values. He analyzes various possibilities and gives biological examples that can be posited to emerge from these, such as dappled patterns, the recursive development of hydra's tentacles, woodruff's whorl of leaves, polygonal symmetry of flowers etc. Computational simulations are also provided as supporting evidence.

Finally he generalizes from the ring to a spherical model which is relevant for the blastula gastrulation. The concentration solutions then follow surface harmonics. The breakdown of homogeneity comes out to be axially symmetrical, and the growth can be amplified along one pole over another, leading to organ formation.