Box 2: Adaptive dynamics analysis.
Interaction between resident and mutant strains. To model the
competition effect of a resident phenotype, φ res,
on the population growth of a mutant phenotype,φ mut, we extend the ecosystem model written for a
single type (equation (1c) in Box 1). To account for the local nature of
the interaction between rare mutant and common resident cells, we
introduce a function (hereafter denoted by c ) of the difference
between φ res and φ mut to
measure how local decomposition by mutant and resident cells differ from
‘mean field’ (average) decomposition by resident cells. Thus, for givenC , D , Z , the growth of the mutant population is
governed by
(5)\(\frac{dM_{\text{mut}}}{\text{dt}}=\left(1-\varphi\right)\gamma_{M}\frac{v_{\max}^{U}\left(1+c\left(\varphi_{\text{mut}}-\varphi_{\text{res}}\right)\right)D_{\text{res}}}{K_{m}^{U}+\left(1+c\left(\varphi_{\text{mut}}-\varphi_{\text{res}}\right)\right)D_{\text{res}}}M_{\text{mut}}-d_{M}M_{\text{mut}}\)
where D res is the equilibrium D predicted
by the ecosystem model for the resident phenotypeφ res. Here function c satisfiesc (0) = 0, c (z ) > 0 if z> 0 and c (z ) < 0 if z< 0.
The underlying assumption is that each microbe has access to DOC partly
as a public good and partly as a private good
(Driscoll & Pepper 2010).
The public good part results from the diffusion of exoenzymes. The
private good part results from local decomposition at the microscopic
scale of cells and exoenzymes that they produce themselves. A mutant
cell that invests more (resp. less) in exoenzyme has access to more
(less) DOC than the average resident cell because the cell’s private
good is greater (smaller) whereas all cells share the same public good.
In a spatially implicit model like ours, diffusion is not directly
modeled, but its effect on the accessibility of DOC to a mutant strain
can be phenomenologically accounted for by a parameterization that puts
mutant cells at a competitive advantage for DOC if the mutant phenotype
invests more in exoenzyme production than the resident phenotype, or at
a competitive disadvantage if the mutant phenotype invests less. This
parameterization is achieved with the function c in equation (5),
where c < 1 when φ mut< φ res and c > 1
when φ mut >φ res. This phenomenological approach is
consistent with the mathematical construction and numerical analysis of
a spatially explicit model of resident-mutant local interaction that
accounts for soil diffusion, which will be reported elsewhere.
Invasion fitness and selection gradient. Mutant fitness\(s\left(\varphi_{\text{mut}},\varphi_{\text{res}}\right)\) is given
by the mutant population growth rate per unit biomass:
(6)\(s\left(\varphi_{\text{mut}},\varphi_{\text{res}}\right)=\left(1-\varphi\right)\gamma_{M}\frac{v_{\max}^{U}\left(1+c\left(\varphi_{\text{mut}}-\varphi_{\text{res}}\right)\right)D_{\text{res}}}{K_{m}^{U}+\left(1+c\left(\varphi_{\text{mut}}-\varphi_{\text{res}}\right)\right)D_{\text{res}}}-d_{M}\)
The selection gradient then obtains by taking the first order derivative
of the invasion fitness with respect to the mutant trait:
(7)\(\nabla s\left(\varphi\right)=\frac{d_{M}}{1-\varphi}\left(\left(1-\varphi-\frac{d_{M}}{v_{\max}^{U}\gamma_{M}}\right)c_{0}-1\right)\)
where \(c_{0}=c^{\prime}(0)\) measures the local competitive advantage to
stronger exoenzyme producers, which we call ‘competition asymmetry’.
Note that by definition of function c , we always have\(c_{0}>0\). Variation in \(c_{0}\) may be caused by different soil
diffusion properties, due to e.g. physical texture or moisture.
Evolutionary singularity. Trait values that nullify the selection
gradient are called ‘evolutionary singularities’. An evolutionary
singularity can be attractive or repelling, and invadable or
non-invadable. Evolutionary singularities that are attractive and
non-invadable represent potential end-points of evolutionary adaptation.
Evolutionary singularities that are attractive and invadable can lead to
evolutionary branching
(Geritz et al.1998).
In a given environment (fixed parameters, constant temperature) there is
at most one evolutionary singularity given by defining \(\varphi\)* as
the value of \(\varphi\)that makes\(\nabla s\left(\varphi\right)\)= 0
in equation (7):
(8)\(\varphi^{*}=1-\frac{d_{M}}{v_{\max}^{U}\gamma_{M}}-\frac{1}{c_{0}}\).
Existence of φ * > 0 requires\(\frac{d_{M}}{v_{\max}^{U}\gamma_{M}}<1\) and\(c_{0}>\frac{1}{\left(1-\frac{d_{M}}{v_{\max}^{U}\gamma_{M}}\right)}\).
Thus, the (cooperative) trait φ can evolve above zero only if the
local competition advantage to stronger enzyme producers is large
enough. The condition for φ * to be evolutionarily stable is\(c^{\prime\prime}\left(0\right)<2\ c_{0}^{2}\) and no other condition than
existence is required for φ * to be always convergent. Here we
assume that function c is such that φ * is evolutionarily
stable and attractive.
Equation (8) shows that more cooperation (larger φ *) should
evolve in microbial populations with lower mortality, greater nutrient
uptake, and/or higher MGE. When comparing microbial populations with
similar life-history traits γ M, \(v_{\max}^{U}\)and d M, stronger competitive advantage to
exoenzyme producers (i.e. higher c 0) selects for
larger φ *.