Nitrogen Cycling and Limitation
modelling Level 3

Nitrogen Cycling and Limitation

Nitrogen limits productivity in most ecosystems. This model models the nitrogen cycle: mineralization from organic matter, nitrification, plant uptake, and how N availability constrains photosynthesis and NPP. We'll derive Michaelis- Menten uptake kinetics and couple N to the carbon cycle.

Prerequisites: michaelis menten, nutrient limitation, mineralization, uptake kinetics

Updated 12 min read

1. The Question

Why do farmers add nitrogen fertilizer, and what happens if they add too much?

Nitrogen is essential for:

  • Proteins (enzymes, including Rubisco for photosynthesis)
  • Chlorophyll (light capture)
  • Nucleic acids (DNA, RNA)

Plants need large amounts, but N is often limiting:

  • Atmosphere is 78% N₂, but plants can’t use N₂ directly
  • Soil mineral N (NH₄⁺, NO₃⁻) is often scarce
  • Addition of N fertilizer → dramatic growth increase

The mathematical question: How do we model N cycling through soil, microbes, and plants, and how does N availability limit productivity?

2. The Conceptual Model

Nitrogen Pools

Five major pools:

  1. Organic N (soil organic matter):
    • Locked in proteins, amino acids
    • Not directly available to plants
    • Largest pool
  2. Ammonium (NH₄⁺):
    • Released by mineralization
    • Plant-available
    • Retained on soil particles (positively charged)
  3. Nitrate (NO₃⁻):
    • Produced by nitrification
    • Plant-available
    • Mobile (leaches easily)
  4. Plant N:
    • In leaves, roots, stems
    • High N requirement for photosynthesis
  5. Atmospheric N₂:
    • Only accessible via N fixation (legumes, lightning)

Key Processes

Mineralization:
Organic N → NH₄⁺ (via decomposition)

Nitrification:
NH₄⁺ → NO₃⁻ (via bacteria, aerobic)

Plant uptake:
NH₄⁺, NO₃⁻ → Plant N

Immobilization:
Microbes consume NH₄⁺, NO₃⁻ (compete with plants)

Leaching:
NO₃⁻ washes out of soil (water pollution)

Denitrification:
NO₃⁻ → N₂, N₂O (anaerobic, waterlogged soils)

3. Building the Mathematical Model

Mineralization

Mineralization rate tied to decomposition (Model 26):

\[M = \sum_i k_i C_i / \text{C:N}_i\]

Where:

  • $k_i$ = decay rate of pool $i$
  • $C_i$ = carbon in pool $i$
  • C:N$_i$ = carbon to nitrogen ratio

Example: Decomposing litter with C:N = 50, decay rate 1.0 year⁻¹, carbon 0.4 kg C/m²:

\[M = \frac{1.0 \times 0.4}{50} = 0.008 \text{ kg N/m}^2\text{/year}\]

Net mineralization vs. immobilization:

  • High C:N litter (> 25): Net immobilization (microbes consume more N than they release)
  • Low C:N litter (< 25): Net mineralization (excess N released)

Nitrification

Nitrification rate:

\[N_{\text{nitr}} = k_{\text{nitr}} \times [\text{NH}_4^+] \times f(T) \times f(W)\]

Where:

  • $k_{\text{nitr}} \approx 0.1$–1.0 day⁻¹
  • $f(T)$, $f(W)$ are temperature and moisture factors (as in Model 26)

Inhibited by:

  • Low pH (< 5.5)
  • Anaerobic conditions
  • Low temperature

Plant Uptake

Michaelis-Menten kinetics:

\[U = U_{\max} \frac{[N]}{K_m + [N]}\]

Where:

  • $U$ = uptake rate (kg N/m²/year)
  • $U_{\max}$ = maximum uptake rate
  • $[N]$ = soil N concentration (NH₄⁺ + NO₃⁻)
  • $K_m$ = half-saturation constant (mg N/L)

Shape:

  • Low [N]: Uptake proportional to [N] (linear)
  • High [N]: Uptake saturates at $U_{\max}$ (enzyme-limited)

Typical values:

  • $U_{\max} \approx$ 0.02–0.05 kg N/m²/year (crops)
  • $K_m \approx$ 1–10 mg N/L

N Limitation of NPP

Nitrogen use efficiency:

\[\text{NUE} = \frac{\text{NPP}}{N_{\text{uptake}}}\]

Units: kg C per kg N

Typical: NUE ≈ 40–100 (need ~10–25 g N per kg C produced)

N-limited NPP:

\[\text{NPP}_{\text{actual}} = \min(\text{NPP}_{\text{potential}}, \text{NUE} \times U)\]

If N uptake is low, NPP is reduced below potential (light-saturated) value.

Coupled C-N Model

Plant C:N ratio varies by tissue:

  • Leaves: C:N ≈ 20–40
  • Wood: C:N ≈ 200–500
  • Roots: C:N ≈ 40–80

N requirement for growth:

\[N_{\text{demand}} = \frac{\text{NPP}}{\text{C:N}_{\text{plant}}}\]

If $U < N_{\text{demand}}$, growth is N-limited.

4. Worked Example by Hand

Problem: A crop field has:

  • Soil organic N: 5 kg N/m² with C:N = 12
  • Decomposition rate: $k = 0.05$ year⁻¹
  • Soil mineral N: [NH₄⁺] = 10 mg/L, [NO₃⁻] = 20 mg/L
  • Plant uptake: $U_{\max} = 0.03$ kg N/m²/year, $K_m = 5$ mg N/L
  • Plant C:N = 25

(a) Calculate mineralization rate
(b) Calculate plant N uptake
(c) Calculate maximum NPP supported by N
(d) Is the crop N-limited?

Solution

(a) Mineralization

Assuming SOM has C:N = 12 and total SOM carbon is:

\[C_{\text{SOM}} = N_{\text{SOM}} \times \text{C:N} = 5 \times 12 = 60 \text{ kg C/m}^2\] \[M = \frac{k \times C_{\text{SOM}}}{\text{C:N}} = \frac{0.05 \times 60}{12} = 0.25 \text{ kg N/m}^2\text{/year}\]

(b) Plant uptake

Total mineral N concentration:

\[[N] = 10 + 20 = 30 \text{ mg N/L}\] \[U = U_{\max} \frac{[N]}{K_m + [N]} = 0.03 \times \frac{30}{5 + 30}\] \[= 0.03 \times \frac{30}{35} = 0.03 \times 0.857 = 0.026 \text{ kg N/m}^2\text{/year}\]

(c) Maximum NPP from N

\[\text{NPP}_{\max} = U \times \text{C:N} = 0.026 \times 25 = 0.65 \text{ kg C/m}^2\text{/year}\]

(d) N limitation

If light-saturated NPP potential is > 0.65 kg C/m²/year, crop is N-limited.

Typical crop potential: 1–2 kg C/m²/year → Yes, N-limited (can only achieve ~50% of potential).

Adding fertilizer (increase [N] to 60 mg/L):

\[U = 0.03 \times \frac{60}{5 + 60} = 0.028 \text{ kg N/m}^2\text{/year}\]

Small increase because already near saturation ($K_m = 5$ is low).

5. Computational Implementation

Below is an interactive nitrogen cycle simulator.

Mineral N: mg N/L

Plant N uptake: kg N/m²/year

NPP (N-limited): kg C/m²/year

Leaching loss: kg N/m²/year

Try this:

  • Add fertilizer: Mineral N increases → uptake increases → NPP increases
  • Natural forest: Low decomposition → low mineralization → N-limited
  • Fertilized crop: High mineral N → near-maximum uptake → high NPP
  • High C:N ratio: Less N released per unit C decomposed → lower mineralization
  • Notice: NPP tracks mineral N availability with Michaelis-Menten saturation

Key insight: N availability controls productivity in most terrestrial ecosystems. Fertilizer works by relieving N limitation.

6. Interpretation

Why Ecosystems Are N-Limited

N₂ is abundant (78% of atmosphere) but unavailable to most organisms.

N fixation pathways:

  1. Biological: Legumes with Rhizobium bacteria (~100 kg N/ha/year)
  2. Industrial: Haber-Bosch process for fertilizer
  3. Lightning: Converts N₂ to NO₃⁻ (minor)

N losses:

  • Leaching: NO₃⁻ washes to groundwater
  • Denitrification: Anaerobic bacteria convert NO₃⁻ → N₂, N₂O
  • Volatilization: NH₃ gas loss

Result: Inputs < Outputs in many ecosystems → chronic N limitation.

Fertilizer and Eutrophication

Excess fertilizer leads to:

  • Runoff → rivers, lakes
  • Eutrophication: Algal blooms, oxygen depletion, fish kills
  • Dead zones: Gulf of Mexico, Baltic Sea

Nitrous oxide (N₂O):

  • Greenhouse gas (300× more potent than CO₂)
  • Produced by nitrification and denitrification
  • Agricultural soils are major source

C:N Stoichiometry

Redfield ratio (aquatic): C:N:P = 106:16:1

Terrestrial plants: C:N ≈ 20–50 (higher than aquatic)

Herbivore constraint:

  • Plant C:N = 40
  • Herbivore C:N = 6
  • Must excrete excess C or retain N

Decomposer constraint:

  • Microbe C:N = 8
  • High C:N litter → immobilize soil N
  • Low C:N litter → release N

7. What Could Go Wrong?

Assuming Unlimited N

Models without N often overestimate NPP in:

  • Boreal forests (cold, slow mineralization)
  • Tropical forests on old soils (N leached over millennia)
  • Grasslands (fire volatilizes N)

Including N can reduce predicted NPP by 20–50%.

Ignoring N₂O Emissions

Nitrification and denitrification produce N₂O (greenhouse gas).

Emission factor: ~1–2% of applied fertilizer N becomes N₂O.

Global warming potential of agricultural N₂O is significant.

Constant C:N Ratios

Real plant C:N varies with:

  • N availability: High N → lower C:N (more protein)
  • Tissue type: Leaves < roots < wood
  • Species: Legumes lower than grasses

Better models: Allow flexible C:N based on N supply.

Neglecting Phosphorus

Tropical soils are often P-limited, not N-limited.

Old, weathered soils: P leached or bound to iron/aluminum oxides.

Full model needs both N and P cycles.

8. Extension: N Saturation

Chronic N deposition (from air pollution) can lead to N saturation:

Symptoms:

  • Nitrate leaching increases
  • Soil acidification (nitrification produces H⁺)
  • Aluminum toxicity
  • Forest decline

Threshold: ~10–20 kg N/ha/year deposition

Regions affected: Europe, eastern US, parts of China

9. Math Refresher: Michaelis-Menten Kinetics

Derivation from Enzyme Kinetics

Enzyme-substrate reaction:

\[E + S \xrightarrow{k_1} ES \xrightarrow{k_2} E + P\]

At steady state:

\[V = \frac{V_{\max}[S]}{K_m + [S]}\]

Where:

  • $V_{\max} = k_2[E]_{\text{total}}$ (maximum rate)
  • $K_m = (k_{-1} + k_2)/k_1$ (half-saturation)

Same form for nutrient uptake:

  • $S$ → nutrient concentration
  • $E$ → uptake transporters in roots

Properties

Low [S]: $V \approx \frac{V_{\max}}{K_m}[S]$ (linear, first-order)

High [S]: $V \approx V_{\max}$ (saturated, zero-order)

At $[S] = K_m$: $V = V_{\max}/2$ (half-maximum rate)

Summary

  • Nitrogen limits productivity in most terrestrial ecosystems
  • Key processes: Mineralization (organic → NH₄⁺), Nitrification (NH₄⁺ → NO₃⁻), Plant uptake, Leaching, Denitrification
  • Michaelis-Menten uptake: $U = U_{\max}[N]/(K_m + [N])$
  • N requirement for growth: NPP / C:N ratio
  • Low soil C:N → net mineralization; high C:N → immobilization
  • Fertilizer increases mineral N → increases uptake → increases NPP
  • Excess N causes eutrophication and N₂O emissions
  • Most ecosystems are N-limited; tropical old soils may be P-limited
  • N cycling couples tightly to carbon cycle (C:N stoichiometry)

References