Basic question that we think is interesting etc

Species interact each other in a variety of different ways. The long term survival of a given specie is therefore reliant on the long time survival of other species – the ecology. Said another way, the fitness of a specie is defined by its ecology. Cheetahs have risen in numbers, so mutations that confer faster running speeds to wilderbeasts will be selected. The long term survivability of the ecology is therefore imperative. To again be concrete, variations in an individual specie that confer it a faster growth rate might eventually lead to a collapse in the ecology. At its heart this has to do with nonlinearities and the emergent slow/fast time scales they can give rise to – the time scale over which a beneficial mutation is reflected in species number can be fast compared to the, potentially negative, ecological consequences of having more of a given specie. Additionally, nonlinearities can give rise to bifurcations where small changes in individual growth rates can lead to gross changes in the overall ecology.

This begs the question: What’s the role, or function, of the ecology? Manifestly, it is the fitter ecology that will survive. But what is a fit ecology? Speaking empirically, and from an experimental standpoint, one can ask the following simpler question: What do fit ecologies look like? Are ecological parameters of fit ecologies tuned according to certain principles? To answer this question of course requires defining a measurement protocol for “ecological parameters”. We will address this in later sections. The question of what a fit ecology looks like can be potentially addressed theoretically using ideas adopted from in silico evolution, which we will describe in later sections.

Before we start discussing details it is worth delineating the Leibler grumpiness: “You can’t write down ODEs for ecologies”. Who knows what the Delphic Oracle really means, but here is how we interpret it. The definition of microbial species is unclear owing to their asexual life style, as well as potential high rates of horizontal gene transfer. This juxtaposes humans, for example, where variation clearly exists but the ability to reproduce clearly demarcates us from another species. So what is the dimensionality of a population model for an ecology that is run for timescales long enough for mutations to arise? Accounting for every degree of freedom, defined by their unique role in the overall ecology, will lead to a rapid divergence in the number of variables – a meaningless conceptual framework.

“Ask not for the meaning but the use of a word” – Wittgenstein.

What roles do bacterial populations play in an ecology? A priori this is a difficult thing to ascertain in a natural ecology. Herein potentially lies the benefit of working in a synthetic system where the initial/seed species have distinct and clearly understood ecological roles.

The beginnings of a mathematical model as a conceptual framework

A generic population model that minimally accounts for interacting species is \[\frac{d n_i}{dt}=\alpha_i n_i(M-n_i)+\sum\limits_{j} S_{ij}n_in_j +\eta_i\] \(\alpha_i\) is the isolated growth rate. \(S_{ij}\) are quadratic interaction terms between species i and species j. \(\eta_i\) is a stochastic forcing that introducing fluctuations into the dynamics that could potentially be important close to bifurcations or small numbers. M is the carrying capacity that limits the system. Two extreme models for M are 1) where each species has its own independent food resource and \(M_i\) are single species parameters, and 2) \(M=\sum\limits_i n_i\), where species are competing for a common set of resources. Looking ahead quadratic terms in population models will be sufficient to capture ecological interactions in the ABC system – another advantage of a curated synthetic system.

For the sake of argument let each species carrying capacity be decoupled \[\frac{d n_i}{dt}=\alpha_i n_i(M_i-n_i)+\sum\limits_{j} S_{ij}n_in_j +\eta_i\] In general, the genotype of a clone determines the coefficients of the population model above. As mutations occur, model parameters are altered. In general however, each bacteria will have some set of mutations that makes it distinct in the entire population, endowing it will its very unique set of ecological parameters and its own degree of freedom in an ecological model. That said, genotype space is astronomically larger than the space of model parameters, suggesting that related species might not have dramatically different ecological roles. Said another way, its unlikely that a photosynthetic algae will transition to a predator on the timescale of an evolution experiment. Thats not to say dramatic changes cannot occur, which might well lead to selective sweeps within the population of photosynthetic algae, for example. This suggests that constructing phylogenies of the ABC subpopulations, and contrasting that with changes in ecological parameters and population numbers, could be fascinating. Regardless, the issue of the number of degrees of freedom is a real one and any course graining protocol has to be thought through carefully.

To facilitate a discussion on course graining and the construction of a well defined effective model let us tailor the above general population dynamics model to the ABC system, where A is algae (producer), B is bacteria (decomposer), and C is ciliate (predator).

\[\frac{dn_a}{dt}=\rho n_a - \alpha n_a^2 -Sn_an_c=\rho n_a(1-\alpha/\rho n_a)-Sn_an_c\]

\[\frac{dn_b}{dt}=\beta n_c^2+\alpha n_a^2 - \gamma n_b\]

\[\frac{dn_c}{dt}=Sn_an_c-\beta n_c^2 = Sn_c(n_a-\beta/Sn_c)\]

  1. \(n_a\): There is a maximum photosynthetic rate that limits growth, \(\rho\). Death rate is quadratic that gives a carrying capacity \(n_a*=\rho/\alpha\). Predating of a’s by b is captured through \(-Sn_an_c\) where \(S>0\).

  2. \(n_b\): The death terms in the \(n_a\) and \(n_c\) terms must appear in the decomposers growth terms. Death rates can be assumed to be linear or quadratic.

  3. \(n_c\): \(Sn_an_c\) must appear to appropriately account for the mass in the system. Death rate is again assumed quadratic to give a carry capacity.

How many parameters are there in the above model? To answer this question we must first non-dimensionalize the 3 equations of motion, according to the following rescalings

  1. \(t'=\rho t\)

  2. \(n_a'=\frac{\alpha}{\rho}n_a\)

  3. \(n_b'=\frac{\alpha}{\rho}n_b\)

  4. \(n_c'=\frac{\alpha}{\rho}n_c\)

giving us

\[\frac{dn_a'}{dt'}= n_a'(1-n_a')-S'n_a'n_c'\]

\[\frac{dn_b'}{dt'}=\beta' n_c'^2+n_a'^2 - \gamma' n_b\]

\[\frac{dn_c'}{dt'}=S'n_a'n_c'-\beta' n_c^2\]

Just three parameters are left in the model: 1) \(S'=S/\alpha\), the dimensionless predation term, 2) \(\beta'=\beta/\alpha\), the dimensionless death rate of c, and 3) \(\gamma'=\gamma/\rho\), the dimensionless death rate of b. [Karna, can you check the nondimensionalization that I did?]

Dropping all the primes we finally have

\[\frac{dn_a}{dt}= n_a(1-n_a)-Sn_an_c\]

\[\frac{dn_b}{dt}=n_a^2+\beta n_c^2 - \gamma n_b\]

\[\frac{dn_c}{dt}=Sn_an_c-\beta n_c^2\]

We can now start to be precise about what we meet by species. The qualitative nature of the terms in the above three equations define the roles of the three species, while the quantitative values of parameters are allowed to drift. Were a mutation, or set of mutations, to confer a new ecological role captured through a qualitatively new term in the above equations then a new species (degree of freedom) needs to be introduced. Mutations that lead to quantitative changes in the 3 parameters can be considered to still retain the essence of the 3 species ABC system.

[Seppe, as it stands the model above doesn’t complete the rock paper scissor setup. How does a care about c? Does it get some nutrients from the decomposers? As it stands, if c were to go to 0 no other specie would care]

–SK writing below –
The short answer to whether A cares at all about C is “no”. I’ve done simple co-culture experiments and seen only small changes in A abundances due to the presence of C (with the absence of C as a control). This would seem to suggest that there is not a strong metabolic linkage between the two – e.g. C does not excrete something that A must have to grow.

A few other comments.

  • I think the sign of \(S\) in the equation for \(n_a\) is positive. That is the interaction between A and B is mutualism. I modified the equations below to reflect this.

  • This model has carbon fixation by algae as the ONLY source of carbon for B and C. That is there is no exogenous source of carbon that B and C can use to grow in the absence of A at \(t=0\), (e.g. terms like \(\rho n_b\), or \(\rho n_c\)). This makes sense after the initial growth phase where the food we supply is consumed. However, it makes me realize that a much cleaner experiment would be to initiate the system without any carbon source in the medium. In this case all biomass in the form of B or C must come from carbon fixation by A. I think this could actually work! (starting in a medium with no carbon and forcing all of the carbon to come in through A).

  • Don’t the death terms in the equation for \(\dot{n}_c\) need to include a factor of \(n_c\) to ensure no spontaneous generation? Is there a reason to have these death terms be quadratic?

\[\frac{dn_a}{dt}=\rho n_a - \alpha n_a^2 +Sn_an_b=\rho n_a(1-\alpha/\rho n_a)+Sn_an_b\]

\[\frac{dn_b}{dt}=Sn_an_b-\beta n_b^2\]

\[\frac{dn_c}{dt}=\beta n_c n_b^2+\alpha n_c n_a^2 - \gamma n_c\]