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## Research Hypotheses

This page contains derivations of analytical solutions to current working hypotheses by Dr. Erik Hobbie. The greek letters lower case delta and upper case delta used in these equations may appear as a d or D, respectively, on some browsers.

Nitrogen isotope patterns in mycorrhizal symbioses.

Analytical solutions to fungal and plant d15N with growth on ammonium and nitrate. For this derivation, we assume that all plant N uptake is mediated through mycorrhizal fungi. Nitrate and ammonium with signatures dnit and damm are supplied at proportions N and A. Following fungal uptake of nitrate, a proportion (TN) is reduced with an isotopic fractionation of DN.

(1) dreduced nitrate = dnit - DN * (1 - TN)

The remainder (1 - TN) is transferred without fractionation to the plant.

(2)dplant nitrate = dnit + D * TN

Of the reduced nitrate in the fungus (TN), a proportion (Tr) will be transferred to the plant, with an isotopic fractionation (Df) during the creation of transfer compounds.

(3)dreduced nitrate (transferred) = dnit- DN * (1 - TN) - (1 - Tr) * Df

The remainder (1 - Tr) is kept by the fungus.

(4)dreduced nitrate (kept by fungus) = dnit- DN * (1 - TN) + Tr * Df

Of the original nitrate, a fraction (1 - TN) was transferred to the plant as nitrate, and an additional fraction (Tr o TN) was transferred to the plant as organic N. The average signature of nitrate-derived plant N can now be calculated.

(5)dPnit = {(1 - TN) * (dnit+ DN * TN) + (Tr * TN) * [dnit- DN * (1 - TN) - (1 - Tr) * Df] }/
[TN * (Tr - 1) + 1 ]

Of the total nitrate supplied, a fraction [TN * (Tr - 1) + 1] ultimately ends up in the plant, and a fraction [TN * (1 - Tr)] ends up in the fungus.

Fortunately, the calculation for ammonium is considerably less onerous, being essentially identical to equations (1) and (2) in the text. Of the total ammonium supplied, a fraction Tr is transferred by the fungus to the plant, and a fraction (1 - Tr) is retained.

(6) d15NPamm = d15Namm - (1 - Tr) * Df

(7) d15NFamm = d15Namm + Tr * Df

With ammonium supplied at a rate A and nitrate supplied at a rate N, the isotopic signatures for plant and fungus are as follows.

(8) d15NP= {A * Tr * dPamm + N *[TN * (Tr-1)+1] * dPnit }/
{A*Tr +N * [TN*(Tr-1) + 1] }

Substituting equations (5) and (6) into equation (8) yields equation (9).

(9) d15NP= A*Tr* [damm-(1-Tr)*Df ] / {A*Tr +N*[TN*(Tr-1)+1] } + N*[TN*(Tr-1)+1] * {(1-TN)*(dnit+DN*TN) +(Tr*TN) * [dnit-DN*(1-TN)-(1- Tr)*Df]}/
[{TN*(Tr - 1)+1 }* {A*Tr +N*[TN*(Tr-1)+1]}]

The pattern in fungi is simpler.

(10) d15NF = A*(d15Namm+Tr*Df) +N * TN*[dnit- DN *(1-TN)+Tr * Df] /
(A+N*TN )

Estimating Carbon Allocation to Mycorrhizal Fungi from Nitrogen Isotope Measurements

Below-ground carbon flux to mycorrhizal fungi has been difficult to estimate because of the rapid turnover of mycorrhizal root tips and hyphae, and difficulties in translating any static measurement of hyphal mass into a carbon flux. The close coupling between nitrogen and carbon cycling in plants (Ågren and Bosatta 1996) suggests that a theoretical treatment of plant-mycorrhizal carbon and nitrogen cycles may prove useful in determining carbon flux to mycorrhizal fungi. I start from the basic premise that uptake of available soil nitrogen by fungi (and plants) is proportional to the growth of hyphae (and roots) into previously unexploited areas (Clarkson 1985; Ingestad and Ågren 1988). In other words, because of the limited mobility of available nitrogen in the soil, the flux of nitrogen to the hyphal or root surface is proportional to growth and not mass. Below I've outlined the mathematical framework by which this approach could be used to estimate carbon allocation to mycorrhizal fungi.

Observations:

• Carbon flux to mycorrhizal fungi results in acquisition of nitrogen.
• Carbon allocation below-ground is proportional to N availability (Ågren and Bosatta 1996).
• Plant d15N declines with lower N availability (Hobbie et al. 1999, 2000).
• What is the relationship among nitrogen isotopes, nitrogen acquisition and carbon flux to mycorrhizal fungi?

Carbon allocation to mycorrhizal fungi can be estimated:

(1) Cdemand = (1/Tr -1) * Np * c/n * 1/e

Cdemand = kg C ha-1 yr-1 to fungus from plant
Tr = transfer ratio (unitless)
Np = N passed to plant (kg ha-1 yr-1)
c = %C of fungus (g C/g biomass)
n = %N of fungus (g N/g biomass)
e = fungal growth efficiency

Derivation of equation (1)

Definitions:
Tr = transfer ratio (unitless)
Np = N passed to plant (kg ha-1 yr-1)
Ntu =N taken up by fungus (kg ha-1 yr-1)
Nk = N retained by fungus (kg ha-1 yr-1)
G = growth of fungus (kg biomass ha-1 yr-1)
n = %N of fungus (g N/g biomass)
Cdemand = kg C ha-1 yr-1 to fungus from plant
e = fungal growth efficiency
c = %C of fungus (g C/g biomass)

Equations:

(2) Cdemand = G * c * 1/e

(3) Tr = Np / Ntu
(4) Ntu = NP / Tr

(5) Nk = (1-Tr ) * Ntu
(6) Nk=G * n

Setting (5) and (6) equal to each other:

(7) G * n = (1-Tr) * Ntu

Substituting (4) into (7) and dividing by n:

(8) G = (1/ Tr -1) * NP /n

Substituting (8) into (2):

(1) Cdemand = (1/ Tr -1) * NP * c/n * 1/e

Because both equation (1) and equations of factors controlling nitrogen isotope patterns in plants use the transfer ratio (Tr) as a key variable (Hobbie et al. 2000), it is possible to relate nitrogen isotope patterns to carbon demand of mycorrhizal fungi.