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05/09/2012: Brief review of the literature concerning ecosystem modelling with a focus on evolution and viruses

Classic NPZD models

Population-based model (PBM). No diversity within each class. Only one species of P with a unique (and fixed) biogeochemical function (growth rates, etc). No evolutionary mechanisms included.

Functional Group Models (FGMs)

Basically, same principle as NPZD except that there is a diversity of P (typically 2 to 6), each within their own (fixed) biogeochemical functions. No evolutionary mechanisms included.

A la Follows et al. (2007)

Same as FGMs except that there are way more diversity introduced (up to about 70 for instance in Follows et al. 2007). Each simulation is seeded with a group, the biogeochemical functions and constants of which are chosen randomly from a realistic (observed) ranges of values with the additional constraints of trade-offs. For instance, we cannot have a species that is both fast to reproduce and can store large amount of nutrient: either the species is fast to reproduce but has a low nutrient cell quota (K-strategists) or is slow to reproduce but has a large nutrient cell quota (r-strategists). These constraints enable in particular to prevent the dominance of a single species, a “Darwinian demon”, that is well adapted to all types of environment.

Natural selection then selects the species that is the best adapted to their environment. To do that, nothing more than running the equations (similar to the equations in NPZD’s model) needs to be done.

Bruggeman and Kooijman (2007) have performed a similar work except that 1) only two traits are studied but 2) they have an “infinite diversity” as all possibilities, given the trade-off, in the 2D space of traits are considered.

For a theoretical discussion of trade-offs, see Huisman and Weissing (2001). For a discussion based on numerical simulation, see Huisman et al. (2008). In this latter work, the authors show trade-offs are needed to reproduce a large diversity of species (more species than there are resources).

The models of this category deal well with diversity but they do not consider actual evolutionary mechanisms.

Population-based models that consider viruses and/or evolutionary mechanisms

Scott Grant (personal communication) has introduced evolutionary mechanisms in the model used in Follows et al. (2007). No viruses though.

Weitz et al. (2005) present a co-evolutionnary (PBM) model of a family of hosts and phages with a single resource. The core of the model are PBM equations for the resource, the family of bacteria and of phages. The growth rate of the bacteria depends on the trait x of the bacteria while the absorption rate for the phage depends on both the bacteria trait x and the phage trait y. This models the trade-off for a host between being resistant and having a fast uptake rate. If, during the course of the simulation, a population goes below a threshold number, it is eliminated from the ecosystem (extinction). In the same time, new species can appear; if the mutation is beneficial, the new species (either of bacteria or phage) is kept and a new equation is added to the ecosystem model. The authors work out analytical solutions in the case of weak mutation rate (following the theory of adaptive dynamics) and also perform numerical simulations.

The authors find that multistrain coexistence is possible as long as the peak of fitness of the bacteria is wider than the peak in fitness of phage times a constant that depends, among other things, on the rate at which the resource is restored. Furthermore, ” [a]lthough the total number of distinct strains is in the hundreds, [...] they are easily clustered into phenotypically distinct quasispecies which persist stably through time”. Say differently, quasispecies are stable while their strains are not, a result that is consistent with the observations of Rodriguez-Brito et al. (2010). They also found that the strains that are the least efficient at the uptake of resources are also the most dominant because they are more resistant to the viruses. That is, K-strategists are the most dominant, in accord with the prediction of Suttle (2007, his Fig. 3) (it would have been nice if Weitz et al. compared the ranked abundance of bacteria and phage). The main result is that, thanks to the presence of viruses, large diversity of bacteria can be obtained even in the presence of a single resource. A similar result has been found by Buckling and Rainey (2002).

The model of Weitz et al. (2005) is one of the most sophisticated coupled ecological-evolutionary model of co-evolutionof between bacteria and bacteriophage that I found. I have since, however, found a lot of variations/extensions of that model that I am not listing here; the interesting papers have been downloaded, though, and they all refer to Weitz et al.’s paper. Possible extension of the model from my point of interest: 1) more than one resource (but this is maybe not too interesting given the work already done by Shoresh et al. 2008, see below), 2) the effect of time-varying input of resources, 3) embed the populations in an eddying regime, 4) consider other parameters (in particular, the width of the fitness peaks) to be evolvable and 5) include two peaks and find under which conditions a bridge between the two peaks appears, as described recently by Thompson (2012; see below). Maybe, the extension that is both achievable given my background and more interesting for me is to study the effect of time-varying resource. Especially, what happens to the diversity when the populations are embedded in a eddying regime.

Shoresh *et al*. (2008) present also a PBM model with evolutionary mechanisms, with or without trade-offs ,but without viruses. Unlike Weitz et al.’s paper, though, they consider multiple resources and study the effect of evolution on diversity. Their procedure is slightly different from that in Weitz et al. (2005). Mutations do not happen any time but only once the ecological model has reached a statistically steady state. In their simulations, they find that evolution can drive species to extinction so that the final number of species that survive is below the total number of resources, contrary to the expectation of resource competition theory. Unlike in Huissman and Weissing 2001, they do not find that trade-offs enable diversity to rise. Their overall conclusion is that evolution exacerbates the paradox of the plankton.

Shoresh et al.’s results have to be compared to the results of Weitz et al. (2005). Thus, although evolution without viruses tend to decrease the diversity (even with a large number of resources), evolution with viruses tend to increase it, even in the presence of a single resource.

The above results should be thought in the context of sympatric speciation due to adaptive diversification (see below).

Other population-based models that consider viruses

Models by Levin et al. (1977) and Bohannan and Lenski (2000) inspired the work of Weitz et al. (2005).

Hoffman et al. (2004) use a PBM of pairs of virus-host that do not interact. Parameters are used so that, for each pair, the population of the host undergoes a bloom that subsequently collapses because its phage’s blooming. Such model gives a power-law distribution of the phage population which is, according to them, an important characteristics of observed phage population. According to this model, the dominant phage is never the same. As for the model of Rodriguez-Valera et al. (2009; see below), there is nothing that can emerge from the model and the model has more an illustrative purpose. No evolutionary mechanisms are considered.

Rodriguez-Valera et al. (2009) have constructed a model to illustrate how viruses can explain the high diversity of hosts (the “kill-the-winner” hypothesis). The model is a PBM of six species of microbes, each being attacked by a single and different viruses. In this case, the diversity is an inherent, and not emergent, property of the model. The model is also too simple because no virus attacks more than one microbial species. No evolutionary mechanisms are considered.

Individual-based models (IBMS)

Another way to consider evolution is to consider models of individual agents or individual-based models (IBMs). IBMs have been developed for organisms that do not organize well into homogenized populations in space and time, like animals at the top of the food chain. But, as Clark et al. (2011) recently argue, IBMs can also be used to represent microorganisms. In this case, a single entity (agent) is a superorganism and represents several microbes. The main advantage of IBMs is that it is relatively straightforward and higlhly intuitive to introduce evolutionary mechanisms with IBMs. Evolutionary mechanisms are introduced as followed: a DNA code (a string) is assigned to each agent, each character of the string corresponding to the strength of a trait. An agent can replicate and new species can appear because of mutations. The main disadvantage of IBMs is that one has to limit ourselves to zero or one-dimensional space due to computational limitation.

The study of Clark *et al*. (2011) considers an IBM for a microbial population with one single resource (that can be time-varying), with or without trade-offs, and without viruses. The authors show how an IBM without evolution can reproduce the results of a PBM and then study the behavior of the model when evolution is allowed, with or without nutrient pulses and with or without trade-offs. Their results are the following. With only one trait allowed to evolve per simulation, without trade-off and without nutrient pulses, they find that competitive exclusion drives the species toward either end of the permissible range of the trait values (and thus toward low diversity), in accord with resource competition theory. Results are the same when nutrient pulses are included except when the trait that evolves is the maximum cell nutrient quota. In the latter case, even if cells with high quota reproduce slowly are not selected without pulses, in the presence of pulses, they are able to store enough nutrient which compensates for their slow growth and to dominate the ecosystem.

Another interesting result concerns the inclusion of trade-offs between the maximum cell nutrient quota and the maximum growth rate allowed. They find that organisms with low quota and maximum growth rate are selected (low diversity) as long as pulses are inexistent or too frequent. Once the pulses become less frequent, however, a diversity of species appears. Here, we see that the inclusion of trade-off combined with time-varying nutrient resource can give rise to diversity and violate the expected result from resource competition theory.

Viruses have yet to be considered with IBMs.

Another example of a model is that of Williams and Lenton (2007; 2007b) which is similar in spirit to the model presented by Clark et al. (2011). Read the more recent Williams (2013).

See Boyle et al. (2012) for a simplified version of the model of Williams and Lenton in order to explore the effect of a parasite species on a recycling loop between a source and a mutualist in a spatial metacommunity. The authors show that spatial variations in productivity is necessary for the mutualists to survive and not being eliminated by the parasites.

Which model for which question?

The rise of viruses (macro-evolution issue)

A cool thing to do would be to actually witness the rise of viruses from evolution in a microbial ecosystem. The works of Williams and Lenton (2007; 2007b) and Clark et al. (2011) do not include viruses but there have been work (work by Ray; see book on “Self-organization in complex ecosystems” by Sole and Bascompte) on digital organisms (organisms that fight for cpu time) that shows that virus-like organisms arise naturally from evolution. A current code that we could use is the Avida sofware that is maintained by the BEACON center.

About diversity and evolutionary mechanisms

Viruses and their hosts appear to evolve quickly and it has been argued that this co-evolution is responsible for the diversity at the level of the genes and species (see Bohannan and Lenski 2000; Forde et al. 2008; Rodriguez-Valera et al. 2009; Anesio and Bellas 2011).

One mechanism that has been developed is an arm race between the phage and its host (Buckling and Rainey 2002; see Weitz et al. 2005 for a model of co-evolution; Paterson et al. 2010; Kashiwagi and Yomo 2011; Marston et al. 2012 for observations of marine host and phage).

Recently, Thompson (2012) explains how co-evolution between a host and a phage also increases the chance for the phage to evolve toward another local fitness maximum. The resistance of the host to the phage favored two mutations in the phage, which then, favored two more mutations; these four mutations are then needed for the final mutation that permits the phage to attach to the host using a different locus (and thus for the phage to be on a different local fitness maximum). Results show that this is path-dependent, however; only in a subset of the experiments, was the final mutation observed. The author argues that this mechanism goes contrary to the arm race view of co-evolution; for the author, if all that was happening was an arm race, then extinctions would have occurred a long time ago.

Viruses are also carrier of pieces of genomes (“DNA dealers” see Breitbart 2011) that are central to the physiological functionning of their hosts, which prompts the idea that viruses actually control the whole ecosystem (Rohwer and Vega Thurber 2009).

Another mechanism of eco-evolution recently discribed by Duffy et al. (2012) is that the co-evolution depends on the size of the epidemy. “[W]hen hosts face a resistance-fecundity trade-off, they might evolve increased resistance to infection during larger epidemics but increased susceptibility during smaller ones.”

Thus, an idea would be to study these different effects using either a variation of the model used by Weitz et al. (2005) or an IBM (restricted to zero dimension).

There is a large body of theory of sympatric speciation (speciation from non-isolated population; see also the Wiki page on this) due to adaptive diversification (the case where there is a feedback between the environment and evolution, that is, when phenotype can modify the environment and thus their own fitness; see the introduction of Herron and Doebeli 2013 for a nice summary). See Dieckmann and Law (1996), Doebeli and Dieckmann (2000), Doebeli and Ispolatove (2010) and the monograph by Doebeli Adaptive Diversification (2011) for the theory. Also, using experiments on E. coli, Herron and Doebeli (2013) were able to show that the speciation process does not occur due to many small steps but instead due to sporadic large steps. Given that the same type of mutations are observed, they also speculate that these mutations might actually be predictable to some extent. These studies need to be considered when we speak of the models of diversity described above, in particular Weitz et al. (2005). (A quick look at Weitz et al. (2005) reveals indeed that their model is based on the theory of adaptive dynamics.)

We need to talk to Gried and Alex in order to see the type of questions we could answer with these types of models.

Effect of time-varying environments

As stated earlier, a relevant question might be how co-evolution changes when the populations are embedded in a time-varying environment such as an eddying regime around station ALOHA.

This idea is confirmed by a comment in the Introduction of Boyle et al. (2012) in which they reference Nowak et al. (2010) and Goodnight (2011) to indicate that space is important in evolutionary processes. It is also confirmed by the results of Boyle et al. (2012) themselves that show that a mutualist can survive against the presence of parasites as long as there is spatial variations in productivity. We could ask the question of where in a varying flow field, are the mutualists more likely to survive?

References

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