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A new model of carbon and phosphorus transfers in
arbuscular mycorrhizas
Blackwell
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10.1111/j.1469-8137.2007.02268.x
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phosphorus
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transfers
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arbuscular mycorrhizas

F. C. Landis & L. H. Fraser

Frank C. Landis1 and Lauchlan H. Fraser2
1Botany Department, University of Wisconsin-Madison, Madison, WI, USA; 2Department of Natural Resource Science, Thompson Rivers University,

Kamloops, BC, Canada

Summary
Author for correspondence:
Frank C. Landis
Tel/Fax: +011 310472 0624
Email: franklandis03@yahoo.com
Received: 30 March 2007
Accepted: 20 August 2007

• Existing models of nutrient transfer in arbuscular mycorrhizal (AM) symbioses are
inadequate as they do not explain the range of real responses seen experimentally.
A computer simulation model was used to evaluate the novel hypotheses that
mycorrhizal nutrient transfers were based solely on symbionts’ internal needs, and
that carbon and phosphorus transfers were quantitatively unlinked. To be plausible,
simulated mycorrhizal plants would show a ±50% variation in weight vs nonmycorrhizal controls, with a normal response distribution (mimicking a real data set).
• One plant and one arbuscular mycorrhizal fungus (AMF) growing in a soil volume
were simulated, using C, P and nitrogen nutrient cycling and stoichiometry. C- and
P-exchange rates were independent and could be varied at will. The model was
tested at realistic nutrient concentrations and a full range of nutrient exchange rates.
• The model showed –20% to +55% range in mycorrhizal plant weight distributed
close to normal, suggesting that the hypotheses were plausible.
• The model suggests that theoretical assumptions about mycorrhizas should be
reassessed. The model worked only because the symbionts possessed incomplete
information on their partner and environmental conditions. Conventional cost–benefit
models do not work under these circumstances, but both mutualistic and parasitic
interactions were successfully simulated.
Key words: arbuscular mycorrhizas, ecological stoichiometry, modeling mycorrhizal
interactions, plant–arbuscular mycorrhizal fungi (AMF) interactions.
New Phytologist (2008) 177: 466–479
© The Authors (2007). Journal compilation © New Phytologist (2007)
doi: 10.1111/j.1469-8137.2007.02268.x

Introduction
The ubiquitous arbuscular mycorrhizas (AM) provide a
number of services to the plants and arbuscular mycorrhizal
fungi (AMF) that form them. However, most research has
focused on the symbioses’ role in nutrient exchange: the plant
provides carbohydrates in return for soil nutrients, especially
phosphorus (Smith & Read, 1997 and many others). The
cellular, molecular and genetic processes involved in these
nutrient transfers are still being elucidated (Harrison, 2005;
Javot et al., 2007), and there are substantial questions about
the nutrient fluxes between the symbionts. While we do not
yet have an experimental system in which these fluxes can be
studied directly, we can model nutrient transfers based on

466

information available from studies of organismal growth and
nutrient uptake.
Although the model described here focuses on physiological
interactions, it was stimulated by two independent research
lines suggesting that the conventional view of nutrient fluxes
– where nutrients are exchanged, and the costs and benefits of
exchange are explored – might be inadequate. First, at high
soil P levels, either in the soil or in the plant, some plants have
been shown to exclude AMF from their roots (summarized by
Smith & Read, 1997), presumably because the cost of obtaining
P from the AMF outweighs the benefit of adding to the surplus
P available, and the plant’s resources are better spent obtaining
limiting nutrients. Second, AMF can suppress plant growth
(relative to nonmycorrhizal plants) (Johnson et al., 1997 and

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many others), and Klironomos (2003) demonstrated that
such suppressive interactions occurred frequently among
many combinations of plant and AMF species and ecotypes
growing in monocultures at low but typical soil P concentrations. In this paper, the relative suppression of one symbiont’s
growth through the mycorrhizal interaction will be termed
parasitism.
Together, these studies present a conundrum: if a plant can
expel an AMF when it has sufficient P to forgo the carbon
costs of the fungus, why does the plant not eliminate AMF
under lower soil P conditions when the cost of the fungus
outweighs its benefits to the plant? This does not seem to
follow conventional cost–benefit logic, and no current model
of nutrient fluxes explains it.
As an example, Fitter (2006) proposed a model for C and
P fluxes to explain mutualistic nutrient exchanges wherein
both symbionts benefit. His model is that P secreted by the
AMF inside a root provokes a corresponding surge in growth
and hexose secretion by the plant, which the fungus captures,
and that, while AMF parasites exist, ‘they would have to
scavenge for sugars at the normal and typically low concentrations in the apoplast’ (Fitter, 2006). This model may well be
a good description of the process of nutrient exchange within
a root when both P and hexose are available for exchange.
However, it does not explain the conundrum described above.
While this model can explain why AMF are excluded from
roots at high soil P, it does not explain the prevalence of
parasitism found by Klironomos (2003) and other studies.
Clearly, another model is needed.
In creating a new model, one place to start is with the
nature of P itself. It is weakly soluble, and is generally immobile
and patchily distributed in soil (Brady & Weil, 2001). While
both plants and AMF are able to obtain soil P in most soils,
both organisms typically develop P depletion zones around
roots or hyphae (Nye & Tinker, 1977; Clarkson, 1981). This
contrasts with N and other soluble nutrients, which diffuse
quickly (in some forms), and concentrations of which are
therefore less patchy (Brady & Weil, 2001).
The immobility and patchiness of soil P has two implications
for modeling. First, plants and AMF that need P cannot
simply abide passively while P diffuses to them; rather, they
have to grow actively through the soil to find P patches
(through root extension), and take up P where they find it
(through proliferation of fine roots). An organism that waits
until P is found to expend C in acquiring it is doomed,
because it will never grow beyond its initial P-depletion zone.
Fitter’s (2006) model misses this, because it focuses solely on
the exchange of nutrients already obtained, not on the investment necessary to find and take up the nutrients. Under
Fitter’s model, a plant that is P-deficient, and that is not
receiving P from an AMF, will expel that AMF, then presumably attempt to obtain P solely through root elongation
out of the depletion zone. If this interpretation of Fitter’s
model is correct, it apparently runs counter to the basic

observation that plants in low-P environments invest more
rather than less heavily in AMF (Smith & Read, 1997).
The second implication of the nature of P is that plants and
AMF have no way to assess the amount of P available in the
soil beyond their depletion zones, in contrast to their ability
to assess amounts of soil N. A plant in need of P cannot
determine whether it would acquire more P through growing
more roots or through sending more C to a mycorrhiza.
To accommodate the immobility and patchiness of P, a new
conceptual model was created. It starts with the assumption
that a plant with growth limited by P has to invest to acquire
more P. That investment can be in biomass allocated to root
growth, or carbohydrates (simplified here to carbon) given to
an AMF. Because the availability of P is unknown, the plant
cannot make a cost–benefit calculation to determine which
investment will have a better return, so it may invest in both
to ‘hedge its bets’. The AMF similarly needs to invest in
hyphal growth to acquire soil P. However, it also needs to
acquire C through the mycorrhizal interaction, as it has a
negligible ability to acquire C from the soil. Fortunately, the
plant will provide C so long as it needs P, so the fungus can be
supported in acquiring new P sources and passing some to the
plant. Similarly, the P sent by the AMF to its photobiont is
also an investment. An AMF cannot assess the photosynthetic
ability of its photobiont, as it can only monitor the amount
of C available to it at the mycorrhizal interface, rather than
the light environment of the plant. Thus the AMF cannot
make a cost–benefit calculation of the optimal amount of P to
exchange in return for C.
The term ‘investment strategy’ is assumed to be the result
of gene action, not conscious ‘choice’ on the part of the plant
or AMF. However, it is a useful idea. In this model, C and P
fluxes of these two nutrients are decoupled, rather than being
instantaneously exchanged. The organisms are sending surplus
nutrients (plant C, AMF P) to their partner at some rate based
on internal need and, if the relationship is mutualistic, each
partner receives sufficient amounts to live and reproduce.
The model put forward here solves the conundrum. A
plant that has sufficient P will not invest any C in the AMF,
resulting in diminishment or loss of mycorrhizas. However,
under P-limiting conditions the plant will invest in mycorrhizas
and/or more roots, simply because it needs to acquire more
P. AMF parasitism of plants in glasshouse pot conditions
should be common because, with a limited amount of P available
in the system, the amount of P residing in AMF biomass is
unavailable to the plant, resulting in a smaller plant. In the
wild, either a plant or AMF can be parasitized, but this can be
seen as a failed investment. The C sent to a mycorrhiza simply
did not result in limiting P being supplied back.
This model seems plausible, but it needs to be tested. As
there is no experimental system within which to monitor C
and P fluxes across a mycorrhizal interface over time and
under varying soil conditions, we constructed a physiological
model of one plant and one AMF in a defined volume of soil,

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Fig. 1 Model schematic. Arrows show flows (movement or transformation of nutrients); boxes show pools. Labels are explained under
Description. 1, NMin,Rt; 2, CMin,Rt; 3, PMin,Rt; 4, NMin,Sh; 5, PMin,Sh; 6, CMin,Sh; 7, NMin,Hyp; 8, PMin,Hyp; 9, CMin,Hyp; 10, NInflow,Pl; 11, NInflow,A; 12,
PInflow,Pl; 13, PInflow,A.

matching the common experimental setup. C and P fluxes
were independent and partially controlled by initial settings,
and the organism’s decisions about nutrient and biomass
allocations were based solely on internal needs, not cost–benefit
equations. The output from this model was tested against the
range of plant responses to AMF seen by Klironomos (2003)
under similar experimental conditions. As Klironomos’s
data set involved many plant and AMF species, this test also
explored how much of the variation seen could be explained
by variations in nutrient exchange. A poor match would suggest
that the physiological or morphological diversity of the
organisms studied by Klironomos explained his results,
whereas a good match would suggest that the mechanisms of
nutrient transfer played a major role in determining the results.
Description
This model simulates a common experimental setup: one
plant and one AMF growing in a fixed volume of soil. The
plant and fungus are modeled in similar ways. The simulation
uses only three nutrients: C, N and P (Fig. 1). Where the
plant (subscript Pl) takes up C through its shoot (Sh) and N

New Phytologist (2008) 177: 466–479

and P through its roots (Rt), the fungus takes in C through its
‘arbuscules’ (Arb, technically, the intraradical hyphae), and N
and P through its ‘hyphae’ (Hyp, the extraradical hyphae).
Nutrients are taken up by the organisms from the soil and
incorporated into biomass, which in turn is lost by senescence
back into litter. The litter is broken back into its component
nutrients by soil microbes, which also take up nutrients,
grow and senesce. The plant transfers C to the AMF, while the
AMF transfers P to the plant; both compete for soil N, and
compete with soil microbes for soil N and P. In the model, N
and P cycles are closed (total recycling; Chapin, 1991;
Lambers et al., 1998), while the C cycle is open, with C being
assimilated by the plant and respired by plant, AMF and soil
microbes. Nitrogen is not exchanged between the partners,
but it serves two useful purposes: it is important in decomposition, because the role of C : N in litter decomposition
is well known compared with C : P, and because biomass
incorporates N, it serves as a useful mathematical check
throughout the model (for instance, ensuring that total
biomass does exceed the amount possible based on system N).
In this model, variables typically have the form ‘AB,C’
(Fig. 1). ‘A’ is typically, although not exclusively, a nutrient (C,

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N, P or Mass for biomass). Subscript ‘B’ describes the flow or
pool, such as ‘S’ for soil, ‘uptake’, ‘assim’ for assimilation,
‘senes’, for senescence, etc. Finally, subscript ‘C’ denotes the
organism, part or pool, such as ‘Pl’ for Plant, ‘A’ for AMF, or
‘M’ for soil microbes. Thus NUptake,Pl is the N taken up
(‘Uptake’) by the plant (‘Pl’).
The two key features in this model involve the nutrient
exchange between plant and AMF. First, C and P exchange
(CPl-A and PA-Pl) are not coupled. The rates of P and C transfer
are not linked, but rather are governed by the surface area of
the arbuscules, the need for P (plant) or C (AMF) in the
donor, and an exchange-efficiency term. The second feature is
the C- and P-exchange efficiency terms (CExEff and PExEff),
which are unitless coefficients that can be set anywhere from
0.0 to 1.0. The CExEff and PExEff terms make a number of
states possible. At a (CExEff, PExEff) of 1, 1, nutrients are flowing
at their maximum rate, while at (0, 0) no nutrients are flowing.
These are analogous to mycorrhizal and nonmycorrhizal
states in experiments. Furthermore, at (1, 0), C is flowing to
the fungus, but no P is flowing back, simulating the fungus
acting as a parasite. However, the reverse (0, 1) should be
the same as (0, 0). As the fungus obtains its entire C from the
plant, it should fail to grow under these conditions, resulting
in a nonmycorrhizal system. Note that other states, such as
(0.5, 0.5), are also possible.
The key features were deduced from two results of Klironomos
(2003): first, in a system where many plant species were grown
with a single AMF species, the resulting final biomass curve
(relativized to the biomass of nonmycorrhizal species) was
normal with a mean of zero; and second, in an experiment
where different ecotypes of plants and AMF were cross-matched
in a fully replicated block experiment. In this case there was
no pattern to the results. One plant species might grow
relatively larger with one AMF ecotype, but it would show a
growth reduction with a different ecotype of the same AMF
species, and the results could be reversed for another plant
species or ecotype (Klironomos, 2003).
Both results favor decoupling C and P transfers. The fact
that 50% of the plants were smaller than their nonmycorrhizal
conspecifics suggests that C and P were not being exchanged
in any constant C : P ratio. If there was a constant exchange
ratio, it would be difficult to find mycorrhizal plants that were
smaller than their nonmycorrhizal conspecifics, which result
from no nutrient exchange at all. Similarly, the lack of pattern
in the ecotype-based responses, and the frequent parasitic
responses seen there, cannot be explained by direct nutrient
exchange at a constant ratio.
Both results, particularly those from the ecotype experiment,
also favor the C- and P-exchange efficiency terms. The range
of results in the first experiment, coupled with the lack of
pattern in the second, suggest that there was a wide range
of C- and P-transfer rates. Lacking more definite information,
continuously variable exchange coefficients are the simplest
way to model this complexity.

One question is how the key features relate to current findings
in mycorrhizal genetics. The normal curve results suggest that
many genes could be at work, as single gene controls would be
more likely to produce a large plant/small plant bimodal
response. There is evidence for multiple P transporters in
plants that are uniquely involved in the AM symbiosis
(Rausch et al., 2001; Harrison et al., 2002; Paszkowski et al.,
2002; Versaw et al., 2002; Maeda et al., 2005, 2006; Nagy
et al., 2005, 2006; Javot et al., 2007), so it is plausible to
assume that many genes are responsible. However, all these
papers focus on the effects of having functional vs nonfunctional copies of single genes, and such single-gene responses
are definitely not underlying Klironomos’s (2003) results. For
example, if one ecotype had a defective gene, all mycorrhizas
formed with that ecotype would effectively be nonmycorrhizal.
This was not seen. The important point is that both key
features were deduced from Klironomos (2003), and it will be
fascinating to see whether this model is supported when
researchers determine the biology of nutrient exchange across
the mycorrhizal interface.
As with any simulation, this model embodies a number
of assumptions. The three most important are nutrient
stoichiometry, nutrient cycling and goal-seeking optimality
behavior by the plant and AMF. Even more fundamentally,
the model is built on weights and volumes, rather than
moles and concentrations. While some authors suggest that
concentrations are proper, especially when using stoichiometry
(Sterner & Elser, 2002), in this model, weights were simpler.
Indeed, volumes were only needed for calculating nutrient
uptake from soil.
Nutrient stoichiometry is the assumption that plant and
AMF biomass contain a relatively constant ratio of nutrients.
In model terms, biomass has a constant C : N : P. This is
supported by data for both plants and fungi (Sterner & Elser,
2002), with the major caveat that both are capable of luxury
consumption. Basically, luxury consumption is the uptake of
nutrients beyond the immediate needs for forming new
biomass, and in plants (and probably in fungi) surplus nutrients
appear to be stored mostly in vacuoles (Leigh & Wyn-Jones,
1985; Sterner & Elser, 2002). In the model, nutrients are
handled in a straightforward way. All nutrients are divided
into two pools, ‘nonstructural’ and ‘biomass.’ Nutrients are
assimilated into the nonstructural pool, and biomass is
formed by taking nutrients out of the nonstructural pool in
defined C : N : P ratios. When biomass is lost, it decomposes
into its original nutrients, based again on the underlying
C : N : P ratios. Finally, the maximum amount of the nonstructural pool is assumed to equal to the biomass pool. For
example, if plant biomass contained 10 g N, then the
maximum nonstructural N would be 10 g.
Nutrient cycling is the second assumption of the model.
Available data suggest that 80% of soil nutrients taken up by
plants are recycled within the local ecosystem from the
senesced tissues of that or other organisms (Chapin, 1991;

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Lambers et al., 1998). Arguably, 100% recycling is closer to
80% than is 0%, so in this model, N and P are closed-cycled,
whereas C is open-cycled. While total recycling is not strictly
realistic, it avoids the bigger problem of nutrient gains
and losses from the system unduly influencing the results.
More importantly, closed cycles for N and P meant that the
amounts of these nutrients in the model stayed constant
over time. This characteristic proved vital for error checking
during the model’s development.
The third and final assumption is that the organisms seek
the goal of maximizing growth through allocating biomass to
acquire the nutrients limiting growth at any particular time.
For example, if the plant is limited by N, it would allocate
more to root growth, whereas if it was limited by C, it would
allocate more to shoot growth, and P limitation would
provoke increased C allocation to mycorrhizas and roots.
Mechanistically, the limiting nutrients were determined by
comparing the nutrients in the nonstructural pools to
organismal C : N : P biomass requirements. This part of the
model was inspired by Thornley’s work (especially Thornley,
1995, 1997).
Biomass and volumetric controls
Plant and AMF Plant and AMF biomass are modeled in
similar ways. Plant biomass (MassPl) is subdivided into shoot
and root biomass (MassSh and MassRt), and AMF biomass
(MassA)is composed of arbuscular (MassArb) and extraradical
hyphae (MassHyp) (Table 1). In both organisms, the amount
of biomass added (MassIncr,Pl, MassIncr,A) is the minimum of a
relative growth rate (RGRA, RGR P) and minimum growth
rates (GMin,Pl, GMin,A) based on limiting nutrients and
stoichiometric ratios (Table 1). The plant's biomass : C : N : P
ratio is 1 : 3 : 10 : 100, whereas the AMF ratio is 1 : 2 : 10 : 100
(Verhoeven et al., 1996; Sterner & Elser, 2002). GMin is
constrained by the minimum concentration of C, N or P,
based on biomass stoichiometry. Maximum potential growth
rate (GMax,Pl, GMax,A) is used to determine what nutrient is
most limiting, in order to allocate future growth, as explained
below (Table 1). MassIncr,A is more complex than MassIncr,Pl,
because root mass (MassR) sets an upper limit for arbuscule
mass (MassArb), here arbitrarily modeled as 20% of MassR.
Without the logistic term in MassIncr,A, arbuscular biomass
could exceed the biomass of the root in which it is growing.
Biomass increase depletes nonstructural pools of C, N, and
P. Only the limiting nutrient is depleted at any step, so persistent nonstructural nutrient pools can occur for nonlimiting
nutrients. When any nonstructural pool reaches an upper
limit (see below), uptake of that nutrient stops.
Once biomass has increased, the increase is immediately
and entirely parceled into mass increase in shoot and root
(MassIncr,Sh and MassIncr,Rt), based on stoichiometric growth
ratios (Table 1). Root mass allocation (MassIncr,Rt) is a somewhat complex equation. Where NPl or PPl constrains growth,

New Phytologist (2008) 177: 466–479

especially where both are low, the second term in MassIncr,Rt
will be close to 1, and most of the mass increase will be allocated to the root. Where C is limiting, the second term of
MassIncr,Rt will be closer to zero, and most of the biomass
will be allocated to the shoot, because shoot mass allocation
(MassIncr,Sh) is merely the difference between total mass
increase (MassIncr,P) and root mass increase. The equations
for hyphal mass increase (MassIncr,Hyp) and arbuscular mass
(MassIncr,Arb) increase are equivalent (Table 1).
Roots and hyphae are modeled as cylinders of fixed radius
and variable length, which are used to calculate surface areas
(SARt, SAHyp; Table 1). All these equations ignore the complex
geometry of real roots, hyphae and arbuscules, along with
the related questions of how these structures grow, branch and
increase in radius. Experiments in modeling these structures
more realistically demonstrated that more realistic modeling
of root and AMF geometry also required more realistic
models of soil heterogeneity and nutrient patchiness in
order to achieve believable results. That experience suggested
that such complex models should be reserved for experimental
systems in which the benefits of complexity justify the costs
of obtaining the data to support it. In this proof-of-concept
model, such complexity was counterproductive.
All plant and fungal organs have a lifespan, after which they
senesce and are lost. For shoots and arbuscules, N and P are
recycled back into the nonstructural pool, either partially or
wholly. There is no evidence for nutrient recycling in roots or
extraradical hyphae, so all nutrients are lost to the soil when
these organs senesce (Lambers et al., 1998). In the current
model, senescence is simulated with a delay function. The
mass increase to an organ on a day is lost through senescence
(MassSenes,Sh, MassSenes,Rt, MassSenes,Arb, MassSenes,Hyp) at a set
period later (Table 1). While this is an imperfect solution, it
allows organ lifespans to be varied at will. In this model, shoot
and root senescence were set at 180 d, arbuscular and hyphal
senescence at 30 d.
Biomass from shoots, roots and hyphae senesces into litter
(MassLitter,Sh, MassLitter,Rt, MassLitter,Hyp) (Table 1). Senesced
arbuscule biomass is recycled into nonstructural nutrients
in the fungus, as the arbuscules are inside the roots. Litter is
mineralized (MassMin) at a constant rate (DepolyRate) as a
function of microbial biomass (MassM) (Table 1). The C, N
and P are released to soil pools (CS, NS, PS) as a function of
mass stoichiometry. For example, C input to soil from shoot
litter is 1/3 of MassSenes,Sh. This is detailed below.
Soil microbial biomass Soil microbial biomass (MassM;
Table 1) has a C : N : P ratio of 120 : 10 : 1. Biomass increase
(MassIncr,M) is based on a C : N ratio of 12 : 1 (Brady & Weil,
2001; Table 1). The persistence of MassM depends entirely on
microbial C (CM). So long as C comes in, the microbes
grow. However, when C is inadequate to maintain microbial
respiration (CResp,M), the microbial pool cannibalizes itself,
losing biomass and respiring the C made available from this

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Table 1 Plant, arbuscular mycorrhizal fungal
(AMF) biomass and volumetric equations,
variables, and constants

Equation

Units

CNRatioM = (CM + (12MassM))/(NM + MassM)
DensityArb = 0.2
DensityHyp = 0.2
DensityRt = 0.2
DepolyRate = 0.8
GMax,A = (2CA,100NA,1000PA)
GMax,Pl = max(3CPl,100NPl,1000PPl)
GMin,A = min(2CA,100NA,1000PA)
LengthArb = (MassArb/DensityArb)/πRadius 2Arb
LengthHyp = (MassHyp/DensityHyp)/πRadius2Hyp
GMin,Pl = min(3CPl,100NPl,1000PPl)
MassA = MassArb + MassHyp (initially 0.002)
MassArb(t) = MassArb(t – dt) + (MassIncr,Arb – MassSenes,Arb)dt (initially 0.001)
MassHyp(t) = MassHyp(t – dt) + (MassIncr,Hyp – MassSenes,Hyp)dt (initially 0.001)
MassIncr,A(t) = MassIncr,A(t – dt) + (MassIncr,A – MassIncr,Arb – MassIncr,Hyp)dt
MassIncr,Arb = MassIncr,A – MassIncr,Hyp
MassIncr,Hyp = MassIncr,A((100NA + 1000PA)/(2CA + 100NA + 1000PA))
MassIncr,M = min((CM – CResp,M),12NM,120PM)
MassIncr,Rt = MassIncr,Pl(1 – (100NPl + 1000PPl)/(3CPl + 100NPl + 1000PPl))
MassIncr,Sh = MassIncr,Pl – MassIncr,Rt
MassLitter,Hyp(t) = MassLitter,Hyp(t – dt) + (MassSenes,Hyp – MassMin,Hyp)dt (initially 0)
MassLitter,Rt(t) = MassLitter,Rt(t – dt) + (MassSenes,Rt – MassMin,Rt)dt (initially 0)
MassLitter,Sh(t) = MassLitter,Sh(t – dt) + (MassSenes,Sh – MassMin,Sh)dt (initially 0)
MassM(t) = MassM(t – dt) + (MassIncr,M – MassSenes,M)dt (initially 0.012)
MassMin,Hyp = MassLit,HypMassMDepolyRate
MassMin,Rt = MassLit,RtMassMDepolyRate
MassMin,Sh = MassLit,ShMassMDepolyRate
MassPl = MassRt + MassSh (initially 0.2)
MassRt(t) = MassRt(t – dt) + (MassIncr,Rt – MassSenes,Rt)dt (initially 0.1)
MassSenes,Arb(t) = 0.75MassIncr,Arb(t – 30) + Max(0,(MassArb – 0.2MassRt)) (initially 0)
MassSenes,Hyp(t) = 0.75MassIncr,Hyp(t – 30) (initially 0)
MassSenes,Rt(t) = MassIncr,Rt(t – 180) (initially 0)
MassSenes,Sh(t) = MassIncr,Sh(t – 180) (initially 0)
MassSenes,M = Max((0.5MassM – 0.0012 – CResp,M),0)
MassSh(t) = MassSh(t – dt) + (MassIncr,Sh – MassSenes,Sh)dt (initially 0.1)
RGRA = 0.1
RGRPl = 0.1
SAArb = 2MassArb/(DensityArbRadiusArb)
SAHyp = 2MassHyp/(DensityHypRadiusHyp)
SARt = 2MassRt/(DensityRtRadiusRt)
2
VDepN,A = min(πLengthHyp((1.3 + RadiusHyp)2 – RadiusHyp
),Vs)
2
VDepN,Pl = min(πLengthRt((1.3 + RadiusRt)2 – RadiusRt
),Vs)
2
VDepP,A = min(πLengthHyp[(0.0018 + RadiusHyp)2 – RadiusHyp
],Vs)
2
2
VDepP,Pl = min(πLengthRt[(0.0018 + RadiusRt) – RadiusRt ],Vs)
VS = 10 000

g cm−3
g cm−3
g cm−3
g g−1
g g−1
g g−1
g g−1
cm
cm
g g−1
g
g
g
g
g d−1
g d−1
g d−1
g d−1
g d−1
g
g
g
g
g d−1
g d−1
g d−1
g
g
g
g
g
g
g
g
g g−1
g g−1
cm−2
cm−2
cm−2
cm3
cm3
cm3
cm3
cm3

loss. N and P lost in this way return to the soil pool (NS and
PS), where they may be taken up by either the microbes or the
plant or fungus.
Soil volume and nutrient concentrations While nutrient
cycling is done almost entirely on a mass basis, nutrient-

uptake equations are based on nutrient concentrations in
the soil. Here the soil volume (VolS) is divided into five
overlapping compartments: the predefined soil volume
(10 000 cm3), and nutrient depletion zones for N and P
(VolDep,N,Pl, VolDep,P,Pl, VolDep,N,A, VolDep,P,A) for plant roots
and fungal hyphae (Table 1). Each of the four depletion zones

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Equation

Units

CA(t) = CA(t – dt) + (CPl-A + CRecyc,A – CResp,A – CStruct,A)dt
(initially 0.1MassA)
CAssim,Pl = NAR SLA(LMR MassSh)0.75(1 – 3CPl)/MassPl
CExEff varies from 0 to 1
CM(t) = CM(t – dt) + (CUptake,M + CSenes,M – CResp,M – CStruct,M)dt
(initially 0.0012)
CMin,Hyp = MassMin,Hyp/2
CMin,Rt = MassMin,Rt/3
CMin,Sh = MassMin,Sh/3
CP(t) = Cpl(t – dt) + (CAssim,Pl – CResp,Pl – CPl-A – CStruct,Pl)dt
(initially 0.1MassPl)
CPl-A = 0.000018SAArbCExEff((GMax,Pl – 1000PPl)/
(GMax,Pl – GMin,Pl))(1 – (2CA/MassA))
CRecyc,Arb = MassSenes,Arb/2
CResp,A = 0.014MassA + (CA – 0.2MassA)
CResp,M = 0.2MassM
CResp,Pl = 0.014MassPl
CS(t) = CS(t – dt) + (CMin,Sh + CMin,Rt + CMin,Hyp – CUpdake,M)dt
(initially 1)
CSenes,M = MassSenes,M
CStruct,A = 0.5min(RGRA, MassA, GMin,A)((1 – MassArb)/(0.2MassRt))
CStruct,M = MassIncr,M
CStruct,Pl = 1/3min(RGRPl, MassPl, GMin,Pl)
CUptake,M = 1.2MassM
LMR = 0.49
NAR = 9.0
SAArb = 2MassArb/(DensityArbRadiusArb)
SLA = 0.0407

g

is a cylindrical torus: the inner edge is the surface of the
root or hypha, the radius is defined as the distance that an
atom of N or P can travel in one day (1.3 and 0.0018 cm,
respectively), and the length of each torus is the length of the
root or hypha. The depletion zone equations specify the
amount of nutrients available to the organisms each day, and
limit the maximum depletion zone to the entire soil volume
(VolS) (Table 1).
Carbon
Plant carbon The initial nonstructural C pool for the plant
is CPl = 0.1 MassP. The plant takes in C through photosynthesis
(CAssim,Pl), using net assimilation rate (NAR), specific leaf area
(SLA), leaf mass ratio (LMR, ratio of leaf to shoot), MassSh
and MassP (Table 2). The assimilation equation is based on a
standard plant growth equation (Lambers et al., 1996), modified
by a 0.75 scaling exponent (following Brown et al., 2004) that
scales photosynthesis to shoot mass without explicitly modeling
leaf and stem growth and architecture. The final logistic term
is based on plant mass stoichiometry, and its function is to
shut off assimilation if the plant contains a C surplus.

New Phytologist (2008) 177: 466–479

Table 2 Plant, arbuscular mycorrhizal fungal
(AMF), and microbial carbon equations,
variables, and constants

g d−1
unitless coefficient
g
g d−1
g d−1
g d−1
g
g d−1
g d−1
g d−1
g d−1
g d−1
g
g d−1
g d−1
g d−1
g d−1
g d−1
g g−1
g m−2 d−1
cm−2
m2 g−1

Assimilated C (CPl) is allocated, in order, to respiration
(CResp,Pl), exchange to AMF (CPl-A), and finally to biomass
(CStruct,Pl) (Table 2). The C transferred as CPl-A is a function
of arbuscule surface area (SAArb), C-exchange efficiency CExEff,
and the plant growth rates GMax,Pl and GMin,Pl. Some C
(CStruct,Pl) is incorporated into biomass based on minimum,
maximum and relative growth rates, and any C not used
remains in the nonstructural pool (CP) (Table 2).
AMF carbon The initial nonstructural C pool for the
fungus is CA = 0.1 MassA, as with the plant (Table 2). The
fungus obtains all of its C (CA) from the plant through
CPl-A, and allocates it in order to respiration (CResp,A) and
biomass (CStruct,Pl) (Table 2). Additionally, half of the C
in the arbuscules is recycled (CRecyc,Arb), mimicking the
presumed function of vesicles for nonstructural C storage
(Smith & Read, 1997; Table 2). Fungal respiration (CResp,A)
uses the same equation as plant respiration (Table 2).
Structural C (CStruct,A) is also similar, but it is limited by
up to 20% of root mass (Table 2), so that the intraradical
hyphae will not grow larger than the root containing
them.

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Research

Table 3 Plant, arbuscular mycorrhizal fungal
(AMF), microbial, and soil phosphorus
equations, variables, and constants

Equation

Units

IMaxP,A =1.21e-7
IMaxP,Pl = 1.21e-7
KMinP,A =1.4e-5
KMinP,Pl = 1.4e-5
PA(t) = PA(t – dt) + (PUptake,A + PRecyc,Arb – PA-Pl – PStruct,A)dt
(initially 0.001MassA)
PA-Pl = min{(0.0000003SAArb)[(GMax,A – 2CA)/(GMax,A – GMin,A)],
(0.5PUptake,A)}PExEff
PDep,A(t) = PDep,A(t – dt) + (PInflow,A – PUptake,A)dt
(initially (PS/VolS)VolDep,P,A)
PDep,Pl(t) = PDep,Pl(t – dt) + (PInflow,P – PUptake,P)dt
(initially (PS/VolS)VolDep,P,Pl)
PExEff varies from 0 to 1
PInflow,A = PS/VSVDep,P,A – PDep,A
PInflow,Pl = ((PS – PDep,A – PUptake,M)/VSVDep,P,Pl – PDep,Pl)/TS
PM(t) = PM(t – dt) + (PUptake,M – PStruct,M)dt (initially 0.0001)
[P]Min,A = 2.8e−10
[P]Min,Pl = 2.8e−10
PMin,Hyp = MassMin,Hyp/1000
PMin,M = MassMin,M/120
PMin,Rt = MassMin,Rt/1000
PMin,Sh = MassMin,Sh/2000
PPl(t) = PPl(t – dt) + (PUptake,Pl + PA-Pl + PRecyc,Pl – PStruct,Pl)dt
(initially 0.001MassPl)
PRecyc,Arb = MassSenes,Arb/2000
PRecyc,Sh = MassSenes,Sh/2000
PSoil(t) = PSoil(t – dt) + (PLitter,Sh + PLitter,Rt + PLitter,Hyp + PSenes,M –
PUptake,Pl – PUptake,A – PUptake,M)dt
PStruct,A = 0.001MassRateIncr,A
PStruct,M = MassRateIncr,M/120
PStruct,Pl = 0.001MassRateIncr,Pl
PUptake,A = SAHypIMaxP,A((PDep,A/VPDep,A) – [P]Min,A)/(KMinP,A +
((PDep,A/VPDep,A) – [P]Min,A)) × (1 – (1000PA/MassA))
PUptake,M = If(CNRatioM < 120) then 0 else MassM/120
PUptake,Pl = SARtIMaxP,Pl((PDep,Pl/VPDep,Pl) – [P]Min,Pl)/(KMinP,P +
((PDep,Pl/VPDep,Pl) – [P]Min,Pl)) × (1 – (1000PPl/MassPl))
TortS = 1.0

g cm−2 d−1
g cm−2 d−1
g cm3
g cm3
g

Soil and microbial carbon Carbon enters the C soil pool
(CS) via mineralization of shoot, root and hyphal litter
(CMin,Sh, CMin,Rt, CMin,Hyp; Table 2). From there, C is taken
up by the soil microbes (CUptake,M), which goes to microbial
biomass and respiration. Ultimately, soil C is lost through
microbial respiration (CResp,M). Microbial C uptake (CUptake,M)
is 1.2 times the microbial biomass, to account for growth
and respiratory needs (Table 2). Microbial C feeds to the
nonstructural microbial C pool (CM) (Table 2). In addition
to the CUptake,M inflow, CM also receives recycled C from
consuming microbial biomass for respiration (CSenes,M) when
CS is insufficient (Table 2). Microbes lose nonstructural C
to microbial respiration (CResp,M) and to incorporation in
microbial biomass (CStruct,M) (Table 2).

g d−1
g
g
unitless coefficient
g d−1
g d−1
g
g cm−3
g cm−3
g d−1
g d−1
g d−1
g d−1
g
g d−1
g d−1
g
g d−1
g d−1
g d−1
g d−1
g d−1
g d−1
unitless coefficient

Phosphorus and nitrogen
The nutrient cycles for N and P use similar equations. Because
of this, the P cycle is discussed in some detail, while the N
cycle is described based on its differences from the P cycle.
Plant phosphorus For simplicity’s sake, the movement of
nutrients from the soil volume into the depletion zone is
based on ratios. The soil nutrients (such as soil P, PS) are
assumed to be uniformly distributed within VolS, and this
uniform concentration simply goes up or down as nutrients
are added to or removed from the soil. Nutrients move into
the depletion zone (for the plant, PDep,Pl; Table 3), based on
volumetric concentration and soil tortuosity (TortS, Table 3).

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TortS is a soil tortuosity term that can be used to slow down
nutrient movement and simulate a clay soil, but in this paper
TortS = 1, so that part of the equation can be ignored.
The plant obtains P (PPl) through uptake from the soil P
pool (PUptake,Pl), and as transfer from AMF (PA-PL) (Table 3).
It recycles half of the P bound in shoot biomass when it
senesces (PRecyc,Pl), while the nonstructural P pool decreases
through incorporation to plant biomass (PStruct,Pl) (Table 3).
The P transferred from the fungus to the plant is described in
the next section, and the other three equations are described
below.
The equation for P uptake from the soil (PUptake,Pl) is a
variation of a Michaelis–Menten equation (Lambers et al.,
1998; Table 3). In the first half of the equation, SARoot is root
surface area, IMaxP,Pl is the maximum P-inflow rate for plant
roots, PDep,P is the amount of P in the depletion zone around
the root, VDep,P is the volume of the P depletion zone, [P]Min,Pl
is the P concentration at which uptake is zero, and KMinP,Pl is
the P concentration at which the uptake rate is half maximum
(Table 3). In the second half of the equation, PPl is the amount
of nonstructural P in the plant, and MassPl is plant mass. This
second half is a logistic equation that acts as an uptake cut-off
when nonstructural P becomes equal to P in plant biomass
(Table 3).
Plants are known to recycle a substantial portion of P and
N from their leaves before shedding them (Lambers et al.,
1998). In this model, half the P from senescing shoots is
recycled (PRecyc,Pl; Table 3). There is no evidence that roots
similarly recycle their nutrients back into the plant, so all root
nutrients cycle instead to the soil.
Plant structural P (PStruct,Pl) and structural C (PStruct,C)
equations are similar (Tables 2, 3). Any P not used to make
biomass remains in the nonstructural pool (PPl).
AMF phosphorus The AMF P-cycling equations (PA, PUptake,A,
PStruct,A, PRecyc,Arb; Table 3) are very similar to those for plant
P cycling. However, transfer of P from the fungus to the plant
(PA-PL) (Table 3) differs from the C-exchange term (CPL-A).
There is a difference between P and C exchange because AMF
are obligate biotrophs, acquiring the entire C from their
phototrophic partners (Smith & Read, 1997). Unlike the
plant, the fungus cannot avoid transferring P unless PExEff is
zero. Because of this, the model assumes that up to half the P
taken from the soil can be transferred to the plant, limited by
the surface area of the arbuscules through which the P is
flowing and measured maximum transfer rates for the P.
Soil and microbial phosphorus The flow of P in the soil is
similar to C, except that the P cycle is closed in the model,
whereas the C cycle is open. As with CS, inputs to the P soil
pool (PS) are P from depolymerized plant, AMF and microbial
litter (PMin,Sh, PMin,R, PMin,Hyp, PMin,M, respectively), while
plant, fungus and microbes take P out (PUptake,Pl, PUptake,A,
PUptake,M; Table 3).

New Phytologist (2008) 177: 466–479

Microbial P is lost into the soil (PMin,Pl) as 1/120th of
senesced biomass (MassSenes,M) (Table 3). Whereas CSenes,M
rejoins the microbial C pool for respiration, PSenes,M is shed
into the soil (Table 3). Unlike plants, microbes have no
mechanism for recycling P or N in this model.
Finally, PUptake,M is modeled with a logical statement
(Table 3). It is based on the assumption that microbial growth
occurs only when the microbial C : N ratio > 12, otherwise
growth is 0.
Plant and AMF nitrogen Fundamentally, N flows and pools
in plant and AMF follow the same equations, which are very
similar to those for P. Nitrogen is taken up from the soil into
a nonstructural pool, from which biomass is formed. When
the shoot or arbuscular biomass senesces, half is recycled back
into the nonstructural pool, while none of the root or soil
hyphal biomass is recycled. The N-recycling terms (NRecycle,Pl,
NRecycle,A) differ slightly from the P-recycling term, because N
is 10 times more abundant than P in biomass, for both plant
and AMF. The only major difference between N and P
models for plant and AMF is that both compete for the soil
N pool, and do not exchange this element (Table 4).
Soil and microbial nitrogen Nitrogen in the soil (NS) is
very similar to P in the soil: it is shed from depolymerized
litter and taken up by plant, AMF and microbes. As with
microbial P uptake, NUptake,M is simulated with a logical
statement: microbes have to take up substantial C to survive.
If N gets relatively too high (C : N < 12), then the microbes
stop taking up N until the C : N reaches 12 again. Otherwise,
N uptake is a function of microbial mass (MassM). As with P,
the N in senescing microbial biomass is shed back into the
soil (as MassSenes,M/12) rather than being recycled within the
microbes (Table 4).

Methods
Parameterizing and testing the model
Parameterizing the model Some variables were determined
through running the model, but many of the numbers were
built in. One of the challenges in creating this model was that
there is no single data set containing complete physiological
information on an integrated plant/AMF/soil microbial system.
Thus the simulation was a chimera of data from unrelated
sets. Photosynthetic variables (including RGRPl) came from
Poorter et al. (1990), while IMax, KMin and other nutrient-uptake
parameters were taken from Craine et al. (2005). The latter
were also used for the AMF, as they were roughly consistent
with observations (Ezawa et al., 2002). AMF hyphal diameter,
arbuscule surface area and other data were taken from Smith
& Read (1997 and references cited therein), and nutrient
movement in the soil was from Nye & Tinker (1977). Other
variables were determined arbitrarily for lack of data. Plant

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Research

Table 4 Plant, arbuscular mycorrhizal fungal
(AMF), microbial, and soil nitrogen
equations, variables, and constants

Equation

Units

IMaxN,A = 1.21e-5
IMaxN,Pl = 1.21e-5
KMinN,A = 1.4e-5
KMinN,Pl = 1.4e-5
NA(t) = NA(t – dt) + (NUptake,A – NRecyc,Arb – NStruct,A)dt (initially set
at 0.01MassAMF)
NDep,A(t) = NDep,A(t – dt) + (NInflow,A – NUptake,A)dt (initially NS/VolS)VolDep,N,A
NDep,Pl(t) = NDep,Pl(t – dt) + (NInflow,Pl – NUptake,Pl)dt (initially NS/VolS)VolDep,N,P
NInflow,A = (NS/VS)VDep,N,A – NDep,A
NInflow,Pl = ((NS – NDep,A – NInflow,M)/VSVDep,N,Pl – NDep,Pl)TortS
NM(t) = Nm(t – dt) + (NUptake,M – NStruct,M)dt (initially set at 0.0001)
[N]Min,A = 2.8e-8
[N]Min,Hyp = MassMin,Hyp/100
NMin,M = MassMin,Mt/12
[N]Min,Pl = 2.8e-8
NMin,Rt = MassMin,Rt/100
NMin,Sh = MassMin,Sh/200
NPl(t) = NPl(t – dt) + (NUptake,Pl + NRecyc,Sh – NStruct,Pl)dt (initially 0.01MassPl)
NRecyc,Arb = MassSenes,Arb/200
NRecyc,Sh = MassSenes,Sh/200
NSoil(t) = NSoil(t – dt) + (NLitter,Sh + NLitter,Rt + NLitter,Hyp + NSenes,M – NUptake,Pl –
NUptake,A – NUptake,M)dt
NStruct,A = 0.01MassIncr,A
NStruct,M = MassIncr,M/12
NStruct,Pl = 0.01MassIncr,P
NUptake,A = SAArbIMaxN,A((NDep,A/VNDep,A) – [N]Min,A)/(KMinN,A +
((NDep,A/VDep,A) – [N]Min,A)) × (1 – (100NA/MassA))
NUptake,M = If(CNRatioM < 12) then 0 else MassM/12
NUptake,Pl = SARtIMaxN,Pl((NDep,Pl/VNDep,Pl) – [N]Min,Pl)/(KMinN,Pl + ((NDep,Pl/
VDep,Pl) – [N]Min,Pl)) × (1 – (100NPl/MassPl))

g cm−2 d−1
g cm−2 d−1
g cm3
g cm3
g

initial biomass was set at 0.2 g; for AMF it was 0.002 g.
Microbial growth and decomposition rates were set arbitrarily
at 0.2 and 0.8 g g−1. The amount of N and P in the soil was
determined empirically, as described in the following section.
Note that none of these parameters was taken from
Klironomos (2003), to prevent the test from being circular.
Testing the model The model was tested in two ways. First,
NS and PS were varied systematically to determine whether
plant growth would be nutrient-limited. From these results,
an NS and PS were chosen that most closely matched
Klironomos’s (2003) experiments. Second, using this (NS,
PS), we systematically varied CExEff and PExEff to determine if
the simulation would show the variation in mycorrhizal
interactions discovered by Klironomos (2003).
Results from (CExEff, PExEff) runs of (1, 1) and (0, 0) at a
particular (NS, PS) were used to simulate the response of
mycorrhizal (1, 1) and nonmycorrhizal (0, 0) systems to the
same environmental variables. One fundamental hurdle for
the model was for the (1, 1) plant to grow larger than the
(0, 0) plant under some conditions. Initial NS and PS were

g
g
g d−1
g d−1
g
g cm−3
g d−1
g d−1
g cm−3
g d−1
g d−1
g
g d−1
g d−1
g
g d−1
g d−1
g d−1
g d−1
g d−1
g d−1

systematically varied exponentially by six orders of magnitude,
from 0.1 to 10 000 g (N) and from 0.01 to 1000 g (P) in
10 000 cm3 soil. All 36 combinations of N and P were tested,
each combination being one order of magnitude (0.1, 0.01).
The proportional difference between mycorrhizal and
nonmycorrhizal response to (NS, PS) was calculated as ((1,
1) – (0, 0))/(0, 0). Models were run for 365 d, using one day
as the time step.
The criteria for choosing NS and PS for the second test were
that the proportional difference should be as close to 50% as
possible (Klironomos, 2003). Second, if possible NS should
be c. 1–100 mg kg−1 and PS around 0.1–10 mg kg−1, to
match Klironomos’s system. Once NS and PS had been
chosen, these values were used for the next test.
To test the model plausibility, CExEff and PExEff were varied
independently from 0 to 1 in 0.1 increments, for 121 separate
runs. Plant and AMF weight after 365 d were plotted, and
proportional differences between these values and the weights
for (0, 0) control were calculated and tallied. This resulted in
growth surfaces showing biomass as a function of particular
(CExEff, PExEff) combinations. The proportional differences

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Table 5 Plant proportional weight differences under different initial
nitrogen (N) and phosphorus (P) amounts. Proportional weight
differences, shown as percentages, were calculated as explained
in the text
Initial N (g)
Initial
P (g)

0.1

1

10

100

1000

10 000

0.01
0.1
1
10
100
1000

1.4%
40.2%
6.6%
–0.7%
–0.8%
–1.1%

1.3%
–99.4%
–3.0%
–8.8%
–10.7%
–9.2%

1.3%
38.1%
43.6%
4.5%
–3.2%
–3.4%

1.3%
38.1%
43.6%
29.3%
27.9%
27.7%

1.3%
38.1%
43.6%
29.3%
27.9%
27.7%

1.3%
38.1%
43.6%
29.3%
27.9%
27.7%

were compared with those of Klironomos (2003), which show
a proportional biomass response with a mean of 0 ± 50%; in
other words, mycorrhizal plants were, on average, the same
size as nonmycorrhizal ones, and could be anywhere from half
to double that size (Klironomos, 2003).

Results
Response to variation in NS and PS
Both plant and AMF showed nutrient limitation at low N and
P levels (Fig. 2), and for c. 72% of N and P combinations the
‘mycorrhizal’ (1, 1) plant was larger than the ‘nonmycorrhizal’
(0, 0) plant (Fig. 2). The response surface showed interesting
complexities. For instance, both the nonmycorrhizal plant
and AMF showed the largest weight at an initial N = 10 g and
P > 10 g, while the mycorrhizal plant was largest at N > 10 g
and P > 10 g (Fig. 2). These responses were probably caused
by competition for N among the organisms. In this setup, it
was also interesting that the AMF had a beneficial effect on
plant growth even when N and P were not strongly limiting.
Following the criteria above, the closest fit to Klironomos’s
setup was found at PS = 1 g and NS > 10 g (Table 5), where
there was a 43.6% proportional difference. As NS = 10 g and
PS = 1 g were close to Klironomos’s real running conditions,
these soil nutrient levels were used to test the plausibility of
the model. Note that this is not a circular test, as Klironomos
(2003) was not used for any of the physiological parameters
used to construct the model.
Response to variation in CExEff and PExEff
Fig. 2 Response surfaces showing effects of systematically varying
initial soil nitrogen from 0.1 to 10 000 g and phosphorus from 0.01
to 1000 g. (a) Plant weight after 365 d with (CExEff, PExEff) (1, 1); (b)
plant weight after 365 d at (0, 0); (c) arbuscular mycorrhizal fungal
(AMF) weight after 365 d at (1, 1).

New Phytologist (2008) 177: 466–479

The response of plant and AMF showed a mix of expected,
hoped-for and unexpected responses to variations in CExEff
and PExEff. First, as expected, when no C was exchanged, the
system was effectively nonmycorrhizal. Plant weights differed
by approx. 0.6 g along the C = 0.0 isocline no matter what
PExEff was used, and AMF failed to grow (Fig. 3). However,
any C flow (CExEff > 0) resulted in AMF growth (Fig. 3).

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Research

Fig. 4 Tally of proportional differences in plant weights. Data are
from Fig. 3a, reformatted as proportional difference between a
weight and the (0, 0) point, and tallied in 10% increments.

The major unexpected result was that (1, 1) was not the
largest plant or AMF. For both plant and AMF, the combination
that produced the largest biomass was (0.3, 1), where the plant
weighed 416.86 g and the AMF 25.38 g (Fig. 3). In this model,
it was possible for the plant to act inefficiently, transferring
more C than necessary, and allocating more biomass to shoot
than root. This inefficiency occurred above (0.3, 1). AMF
biomass declined in a similar way, largely because arbuscular
biomass was limited by root size. While this result was
unexpected, it meant that the maximum proportional plant
size was roughly the same as that seen in Klironomos’s work.

Discussion

Fig. 3 Response surfaces showing effects of varying CExEff and PExEff
between 0 and 1, using initial soil nitrogen of 10 g and phosphorus
of 1 g. (a) Plant weight after 365 d; (b) arbuscular mycorrhizal fungal
(AMF) weight after 365 d.

In approx. 34% of cases the fungus effectively parasitized
the plant, reducing plant weight to less than the nonmycorrhizal
state (269 g) in a triangular area stretching from (0, 0) to (1,
0.5) (Fig. 3). The minimum plant weight, 217.8 g, occurred at
(1, 0), where plant C flowed freely, but no P flowed from the
fungus (Fig. 3).
The two hoped-for results were the tests against Klironomos’s
data. First, the range of plant weight responses was 81–155%,
relative to the nonmycorrhizal (0, 0) weight (Fig. 4). Second,
the tally of responses was close to a normal curve, with a mean
response of approx. 10%, rather than 0% as predicted, and
missing 20% of the negative side.

The fit between Klironomos’s data and the model output
was not perfect, but it was sufficiently good to render the
model plausible. The chief defect of the model, in its
current formulation, was that the fungus could not parasitize
the plant sufficiently. To replicate Klironomos’s results
perfectly, the fungus would have had to reduce plant weight
by 50% under some conditions. Given that Klironomos’s data
were from dozens of plant and AMF species, while this model
was a chimera of numerical values from multiple sources, the
match between model and reality was surprisingly good.
In this study, it is important to remember that the model
was parameterized using (CExEff , PExEff) of only (1, 1) and (0,
0). Neither parameter was a maximum nor a minimum, so the
maxima, minima, response curves and AMF parasitism truly
tested the model, and the test was not circular. While it is
possible that the match between model and data is coincidence, it is more likely this model of nutrient transfers has
some biological validity.
If we accept the model as plausible, it has implications both
for mycorrhizal biology and for the theory of symbioses. For
mycorrhizologists, the model predicts that C and P transfers are unlinked, and this will be tested as the molecular

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mechanisms of AM nutrient transfer are described in detail.
Additionally this model suggests that the ecotypic variation
described by Klironomos (2003) could be caused by allelic
diversity in the genes underlying mycorrhiza formation, and
to the mismatch between the genes products of different
symbiont ecotypes. Researchers are elucidating the plant
genes responsible for AM nutrient transfer (Harrison, 2005;
Javot et al., 2007). Their work is exciting, but it is still in an
early stage, describing the genes necessary in the symbiosis
rather than the diversity of interactions. However, it is
heartening to note that the model appears to produce results
consistent with those of Javot et al. (2007) and perhaps
others (unpublished data).
Second, and more importantly, the model demonstrates
that investment in the search for nutrients is an important
conceptual model for understanding mycorrhizal nutrient
exchanges. Currently, most models (such as Fitter, 2006) are
predicated on the notion that the instantaneous exchange of
C and P is the important frame of analysis, and questions have
focused on how partners may benefit or be cheated in an
exchange. In contrast, we suggest that C and P fluxes within
mycorrhizas are as much investments as instantaneous
exchanges, and that the economic mathematics of risk, rate of
return, and return on investment apply to mycorrhizal interactions. This opens up a rich new country for theorists, as the
mathematics of investments are well developed but have not
been applied to mycorrhizal interactions in any comprehensive way.
In summary, it is possible to explain much, but not all, of
the variation seen in plant–AMF interactions through a simple model of nutrient exchange. In this model, nutrient transfers are independent of each other, and it is possible for
nutrient flows to be nonexistent, adequate or excessive, resulting in either mutualistic or parasitic interactions. These results
came from only one iteration of this particular model, and the
model can easily be adapted to study other phenomena, such
as variation in organ lifespan, N exchange by mycorrhizas, litter exploitation (reimagining the AMF as an ectomycorrhizal
fungus), or interactions among multiple plants and fungi with
varying properties. This model is an adaptable tool, and we
hope that others will use it freely.

Acknowledgements
This research was funded in part by the US Department of the
Interior. The authors wish to thank many researchers for their
comments and information, including John Klironomos, Louise
Egerton-Warburton, Steve Finkelman, Tony Golubski, Eric
Lilleskov, and others at ESA 2005, and eight anonymous reviewers.

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