Table 1.
Parameters used in our model, with their definition and units (see S1 Text for the detailed dimensional analysis of the model).
Fig 1.
Comparison between the best fits of MacArthur’s consumer-resource model (dashed lines) and experimental measures of the growth of S. cerevisiae on galactose as the primary carbon source and ethanol as a byproduct of fermentation, in the case of adaptive (A) and fixed (B) metabolic strategies.
Shown are the mean (black lines) and the standard error (gray bands) across n = 8 replicate populations. In (A) the model is not only capable to reproduce very well the experimental data, but the best fit returns parameters whose values are biologically reasonable when contrasted with experimentally-measured ones found in the literature (see Table A in S1 Text). On the other hand, the fit in (B) cannot reproduce a diauxic behavior when the parameters are constrained to vary within a few orders of magnitude away from biologically reasonable values (see Table A in S1 Text). See S1 Text for details on how the fits were performed and the resulting values of the best fit parameters.
Fig 2.
Time of first (orange) and seventh (purple) extinction in the consumer-resource model with adaptive metabolic strategies and with drawn independently of δσ.
We used m = 10, p = 3 and with
drawn from a normal distribution with mean
and standard deviation Σ; see S1 Text for more details on the parameters used. The extinction times were computed as the instants at which the densities of the species fell below 1 cell/mL. Both axes are in logarithmic scale, the error bars represent one standard deviation across 50 iterations of the model and the dashed lines are the best power-law fits. The behavior of the extinction times suggests that if Σ = 0 then all species could coexist indefinitely. Indeed, it is possible to show analytically that when
, all species coexist at the stationary state of the system (see S1 Text for details). The green points show, for comparison, the time of first extinction for a system with the same parameters but where metabolic strategies are fixed. As we can see, even when each species has its own CTR,
, using dynamic metabolic strategies increases by several orders of magnitude the length of the time interval over which species manage to coexist. The results shown do not change noticeably if the initial conditions on the populations are increased, even if by some orders of magnitude.
Fig 3.
Comparison between the initial (orange) and final (purple) convex hull of the rescaled metabolic strategies (colored dots) when they are allowed to adapt according to (4).
These results have been obtained for a system with m = 10 species and p = 3 resources using the graphical representation method introduced by Posfai et al. [10] and using a common value of the CTR for all species. In particular, in this case this method prescribes that the rescaled metabolic strategies and nutrient supply rate vector (black star) all lie on a 2-dimensional simplex (i.e. the triangle in the figure), where each vertex corresponds to one of the resources; for details on the parameters used, and for the plots of the temporal dynamics of the population densities and metabolic strategies, see Figure D in S1 Text. In the final state, the
have incorporated
in their convex hull.
Fig 4.
Comparison between the temporal dynamics of species’ population densities (each color represents a different species) in the consumer-resource models with fixed metabolic strategies, when the resource supply rate vector varies with time.
Here, we simulated a system with m = 20 species, p = 3 resources, and with the nutrient supply rate vector switching at regular intervals between the two values shown (black star and diamond) in (A). Specifically, in (B) we made alternate periodically between
for τin = 12 h and
for τout = 48 h, with
chosen within the convex hull of the initial rescaled metabolic strategies and
chosen outside of it (see Figure G in S1 Text for more information on the parameters used). Panel (C) shows the same quantities, with τin = τout = 48 h. See Figure G in S1 Text for the dynamics of the species’ populations in the consumer-resource model with adaptive metabolic strategies.
Fig 5.
Rank distribution of the (decimal) logarithm of the stationary population densities for different values of the adaptation velocity d (see Figure J in S1 Text for more information on the parameters used).
The lines represent the average value over 100 iterations, while the opaque bands outline the standard error of the mean. For d = 0 (blue line) the rank distribution is very steep and only the first few species have a population density over 1 cell/mL (corresponding to ), while as d increases the distribution becomes more even. Setting
as the extinction threshold, approximately two thirds of the species in the system go extinct with d = 10−7 (yellow line), while all of them survive with d = 10−5.