Edited by: Gabriel A. Silva, University of California, San Diego, USA
Reviewed by: Gabriel A. Silva, University of California, San Diego, USA; Gianluigi Mongillo, Paris Descartes University, France
*Correspondence: Robert Sinclair, Mathematical Biology Unit, Okinawa Institute of Science and Technology Graduate University, 1919-1 Tancha, Onna-son, Okinawa 904-0495, Japan e-mail:
This article was submitted to the journal Frontiers in Computational Neuroscience.
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When a given tissue must, to be able to perform its various functions, consist of different cell types, each fairly evenly distributed and with specific probabilities, then there are at least two quite different developmental mechanisms which might achieve the desired result. Let us begin with the case of two cell types, and first imagine that the proportion of numbers of cells of these types should be 1:3. Clearly, a regular structure composed of repeating units of four cells, three of which are of the dominant type, will easily satisfy the requirements, and a deterministic mechanism may lend itself to the task. What if, however, the proportion should be 10:33? The same simple, deterministic approach would now require a structure of repeating units of 43 cells, and this certainly seems to require a far more complex and potentially prohibitive deterministic developmental program. Stochastic development, replacing regular units with random distributions of given densities, might not be evolutionarily competitive in comparison with the deterministic program when the proportions should be 1:3, but it has the property that, whatever developmental mechanism underlies it, its complexity does not need to depend very much upon target cell densities at all. We are immediately led to speculate that proportions which correspond to fractions with large denominators (such as the 33 of 10/33) may be more easily achieved by stochastic developmental programs than by deterministic ones, and this is the core of our thesis: that stochastic development may tend to occur more often in cases involving rational numbers with large denominators. To be imprecise: that simple rationality and determinism belong together, as do irrationality and randomness.
Aristotle tells us that the Pythagoreans believed that the principles of mathematics were the principles of all things (Primavesi,
I wish to propose that these same rational numbers may help us to understand conditions under which a developmental process may tend to appear to be random from some point of view, where I use the word “random” in the sense of something which is not readily predicted. Wu et al. (
Before going into the details of the hypothesis, is there any reason to think that one can infer anything useful at all about living organisms on the basis of what may appear to be a type of numerology? There are two historical cases which illustrate the potential of such an approach, one from chemistry and one from palaeontology.
John Dalton is famous for, among other things, his law of multiple proportions, which is now a part of any basic chemistry course, but was advanced theory 200 years ago, before atomic theory was generally accepted and well before the structures of molecules were known. In a standard modern textbook (Brown et al.,
As my second historical example, I wish to use the case of the “conodont animal” (Sweet and Donoghue,
I do understand that even these examples will still fail to convince some readers that hypotheses of the type I am putting forward can actually be of use, but I will refrain from listing any more (although Mendelian genetics comes to mind). I am aware of the attractiveness of particularism as opposed to broad theory (Gremillon et al.,
The intention of this work is to suggest a novel framework for thinking about the relationship between natural selection and stochasticity in development. I hope that it will be useful in designing and interpreting experiments. I have no intention of falsely elevating my hypothesis to anything approaching a law. There are good reasons to expect exceptions, and, to be specific, competence for DNA uptake in bacteria may be one (see Johnston and Desplan,
The Drosophila compound eye provides excellent examples of both deterministic assembly and stochastic development, in which we can clearly see the proposed relationship of developmental mechanism with rational numbers between 0 and 1.
A normal Drosophila ommatidium contains exactly eight photoreceptor neurons. Only photoreceptor neurons R1 to R6 express the major Drosophila rhodopsin Rh1. Therefore, the fraction of photoreceptor neurons in a Drosophila compound eye which express Rh1 is a rational number: 6/8 = 3/4 = 0.75 = 75%. If we include the fact that there is a single R8 founder cell, being one out of eight, we find the lowest common denominator is 8. The simplicity of ommatidial composition is reflected in the simplicity of the rational numbers 3/4 and 1/8 with their small lowest common denominator (i.e., 8). Assembly essentially always follows the same pattern within an ommatidium. On the other hand, R7 photoreceptors express either Rh3 or Rh4 rhodopsin, but here the fraction is not easily written down as a rational number, instead being a value in the range of 60% (Bell et al.,
However, a stochastic mechanism can be tuned (Wu et al.,
The hypothesis is, that
Moving away from the compound eye of Drosphila, let us return to the theoretically more basic problem of tissue consisting of different types of cells which are not strictly arranged in units (such as an ommatidium). Here, the relationship between properties of the “simple” rational numbers and the proposed tendency towards stochastic development can be made more directly. The fact that rational numbers with bounded denominators are distributed unevenly in any large interval is useful in understanding the hypothesis, and is illustrated in Table
Simple Fraction | Percentage | Difference from previous |
---|---|---|
1 | 100.0% | |
9/10 | 90.0% | 100.0–90.0 = 10.0% |
8/9 | 88.9% | 90.0–88.9 = 1.1% |
7/8 | 87.5% | 88.9–87.5 = 1.4% |
6/7 | 85.7% | 87.5–85.7 = 1.8% |
5/6 | 83.3% | 85.7–83.3 = 2.4% |
4/5 = 8/10 | 80.0% | 83.3–80.0 = 3.3% |
7/9 | 77.8% | 80.0–77.8 = 2.2% |
3/4 = 6/8 | 75.0% | 77.8–75.0 = 2.8% |
5/7 | 71.4% | 75.0–71.4 = 3.6% |
7/10 | 70.0% | 71.4–70.0 = 1.4% |
2/3 = 4/6 | 66.7% | 70.0–66.7 = 3.3% |
5/8 | 62.5% | 66.7–62.5 = 4.2% |
3/5 = 6/10 | 60.0% | 62.5–60.0 = 2.5% |
4/7 | 57.1% | 60.0–57.1 = 2.9% |
5/9 | 55.6% | 57.1–55.6 = 1.5% |
1/2 = 2/4 = 4/8 = 5/10 | 50.0% | 55.6–50.0 = 5.6% |
The gaps in Table
The retina is a relatively easily accessible brain tissue belonging to the vertebrate central nervous system. It is composed of a number of different cell types, and the mechanism by which the appropriate proportions of retinal neurons are derived from retinal precursor cells has been studied in a number of different vertebrate model organisms (see the review by Johnston and Desplan,
The experiments and analysis of Gomes et al. (
The largest gap in Table
The hypothesis is, by its nature, not restricted to neurons. To test whether it might apply to other contexts, it is interesting to examine some examples from other areas of biology.
It is known that an extremely small fraction of wild-type bacterial cells will be found to be in a dormant state, and that these cells are responsible for resistance to antibiotic treatment, since their reduced metabolism allows them to survive. Johnston and Desplan (
This is to be contrasted with the case of heterocyst formation, a response to nitrogen starvation, along cyanobacterial filaments. Heterocysts are differentiated cells which differ from all the other cells in the filament, and are found approximately in the proportion 1:10 (Yoon and Golden,
A further example is provided by the worm
It is commonly believed that the human ratio of male to female live births is 1:1. If that were the case, our hypothesis would predict a deterministic mechanism for sex determination, presumably in terms of pairs. In other words, if the sex ratio were under very strong selection to be identically 1, then one might predict that humans would almost always bear twins, one male and one female. Clearly, this is not the case!
Interestingly, the human sex ratio is in fact stably greater than 1, usually around 1.05 (Mathews and Hamilton,
From an evolutionary perspective, it is clear that so potentially dramatic a change from a deterministic to a stochastic developmental mechanism, or the reverse, should be a rare event. This implies that there is no reason to expect that all tissues in all organisms, to which this hypothesis could apply, must automatically be assumed to be using stochastic developmental mechanisms according to a simple numerical assay of cell type abundances. Also, there are many possible selective forces which can and do influence the choice between determinism and stochasticity, and alternative ways to interpret this choice, a recent example of which is provided by Fisek and Wilson (
I would like to finish with a question: Should we expect programmed cell death under stochastic control (Spencer et al.,
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.