Adaptive metabolic strategies explain diauxic shifts and promote species coexistence

Competitive systems are most commonly described mathematically using MacArthur's consumer-resource model, leading to the "competitive exclusion principle" which limits the number of coexisting competing species to the number of available resources. Nevertheless, several empirical evidences - in particular bacterial community cultures - show that this principle is violated in real ecosystems. Another experimental evidence that cannot be explained in this framework is the existence of diauxic (or polyauxic) shifts in microbial growth curves: bacteria consume resources sequentially, using first the one that ensures the highest growth rate and then, after a lag phase, they start growing slower using the second one. By introducing adaptive metabolic strategies whose dynamics tends to maximize species' relative fitness, we are able to explain both these empirical evidences, thus setting the paradigm for adaptive consumer-resource models.

Competitive ecosystems are generally described mathematically using MacArthur's 33 consumer-resource model [1,4], which prescribes that for a system of m species and p 34 resources the population n σ (t) of species σ evolves following (we omit for simplicity the time 35 dependence on both n σ and c i from now on)

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where c i represents the abundance of resource of type i (whose dynamics will be soon in-38 troduced), δ σ is the death rate of species σ, r i (c i ) is the uptake rate of resource i (which 39 we assume to have the form of a Monod function r i (c i ) = c i /(K i + c i )), and α σi are the 40 "metabolic strategies", i.e. the rates at which species σ absorbs resource i; the parameters 41 v i are often called "resource values" and give a measure of how much efficiently a resource the vector δ · does not lie in the subspace spanned by the p vectors α ·i , which occurs with 50 zero probability if the vectors are drawn randomly. 51 In this setting the metabolic strategies α σi are treated as fixed parameters instead of being 52 considered as dynamical (see however [16] for recent work, where microbes can instanta-53 neously switch from using one nutrient to another); there is however incontrovertible evi-54 dence that microbes' metabolic strategies do change over time according to their surrounding 55 environmental conditions. As early as the '40s Jacques Monod [17,18] observed for the first 56 time that Escherichia coli and Bacillus subtilis exhibit a particular growth curve, which he 57 called "diauxie", when exposed to an environment containing only two different sugars. In 58 particular, instead of metabolizing the two sugars simultaneously it turned out that bacteria 59 consume them sequentially, using first their "favorite" resource (i.e. the one that ensures 60 the highest growth rate) and then, after a lag phase, they start growing slower using their 61 "least favorite" one. Since then diauxic growth has always been the subject of thorough 3 certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not this version posted August 6, 2018. . https://doi.org/10.1101/385724 doi: bioRxiv preprint violation of the competitive exclusion principle, have always been considered as separate 83 and independent. In this work we will show that they are actually related, since they can 84 be solved together by allowing the metabolic strategies, α σi , of a MacArthur's consumer-85 resource model to be dynamical variables evolving according to an appropriately defined 86 differential equation that tends to increase the relative fitness of each species, measured as 87 its growth rate. In this letter we show that by introducing an adaptive dynamics regulating 88 the evolution of the metabolic strategies α σi so that to increase the growth rate of each 89 species, we can explain both the existence of diauxic shifts and the coexistence of any  We first consider the equation for the resources' dynamics, coupled with Eq. (1):

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where s i is a constant nutrient supply rate (we are assuming that the resources are being 95 supplied constantly from the outside world), and the second term in this equation represents 96 the action of the consumers on the resources, which depends on the metabolic strategies α σi .

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In the above equation we have not considered a term accounting for the resource degrada-98 tion; we assume that it is sufficiently small and its effect is felt on timescales larger than the 99 one considered here. 100 We now introduce our adaptive mechanism: we require that each metabolic strategy α σ 101 evolves in order to maximize its own species' relative fitness, that is typically [41, 42] mea-102 sured by the growth rate Supp. Mat.
[43]) that the metabolic strategies evolve in time following a simple "gradient 108 ascent" equation: 110 where τ α is the characteristic timescale over which metabolic strategies evolve. For microbes 111 this timescale is comparable with the one of the resource dynamics, and therefore we can set 112 4 certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not this version posted August 6, 2018. . https://doi.org/10.1101/385724 doi: bioRxiv preprint but they occur based on the available resources (as in the case of diauxic shifts). Time scales 114 will be discussed more accurately later. The values of the metabolic strategies, however, 115 cannot be completely unbounded because microbes have limited amounts of energy they can 116 use for metabolism: we must therefore introduce a trade-off in the utilization of resources. 117 We translate this constraint by requiring that each species has a maximum amount of energy 118 to be allocated for metabolism, i.e.
"resource costs", and take into account the fact that the energetic cost for the metabolization 120 of different resources can be different. In our setting E σ /δ σ represents the "efficiency" of 121 species σ, and a priori we assume it to be a species-independent property, i.e. E σ /δ σ = Q ∀σ.

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In other words, in a neutral environment (i.e. before specifying s) all species have the 123 same potential to grow (but this could be no longer true once the resources are supplied   Figure 1 shows the result of a simulation of the model in the case 135 c 1 (0) > c 2 (0) and v 1 < v 2 : in this example, therefore, resource 2 is the "favorite" one and is 136 scarce, while resource 1 is the "less preferred" one and is highly abundant. As we can see,   metabolic strategies are all "squeezed" near to or exactly on that same side. This means 162 that species do not waste energy producing enzymes needed to metabolize scarce or totally 163 absent resources, and will focus their efforts on the more easily available ones. The copyright holder for this preprint (which was not this version posted August 6, 2018. . https://doi.org/10.1101/385724 doi: bioRxiv preprint 169 i.e. the system is time-scale invariant. We can thus fix the time-scale using the following 170 constraint: 177 Notice also that at variance of Eq. (4) the time scale of the time evolution of the α σ,i is set 178 by δ σ , the death rate of species σ. In figure 4 we show the outcome obtained by simulating 179 the species' abundance dynamics using adaptive strategies following Eq. (7). As we can see, Time evolution of the values of the ratios E σ (t)/δ σ . As we can see, the system self-organizes again in order to find the conditions for coexistence.
10 certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.
The copyright holder for this preprint (which was not this version posted August 6, 2018. . https://doi.org/10.1101/385724 doi: bioRxiv preprint acknowledges Cariparo Foundation, S.S. acknowledges the University of Padova for SID2017 198 and STARS2018 grants. 199 11 certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.