Model

Introduction

As a part of ARGUS-2440, we aim to convert mobile nitrate present in the soil into ammonium, which is retained in soil, to reduce the nitrogen loss in soil and increase fertilizer efficiency. Although several soil bacteria perform nitrate ammonification in soil, we have designed a novel bacterium to selectively perform this pathway under any soil conditions of oxygen content, pH, soil type, etc.

Why make a model?

We decided to build this kinetic model to get a better idea of how our modified bacterium's efficiency will change according to different initial conditions of initial nitrate & ammonium concentration, initial bacterial population, rate of nitrate & ammonium uptake, rate of leaching, and cellular enzyme concentration.

Given, as input, ideal conditions required for the growth of a certain plant in a certain soil type, over a certain time frame, this model will aid in giving the required amount of fertilizer and initial bacterial volume to be added to give the desired result.

Structure of the model

To simulate the pathway, we used the following backbone in the figure, based on a similar backbone put forth by Huang et al, 2020, in the closely related strain P. putida Y-9, which performs nitrate ammonification.

Image source: Biorender (Created in BioRender.com)

Mathematical formulation

To characterize the metabolite concentrations in reactions, we assumed ODEs based on the assumption of Michaelis-Menten kinetics.

According to Michaelis-Menten kinetics, for a reaction A → B, the rate of product formation:

Where k_cat (s⁻¹) = turnover number, K_m (mol/L) = Michaelis–Menten constant

Since the volume of the cellular compartment affects the concentration of the metabolites, we have to account for the growth of the bacterium in the medium.

Growth of Cell Population

We describe the growth of the bacterial population under 3 assumptions:

• Growth is according to the Monod equation
• Bacterial growth is selectively according to ammonium concentration
• Bacterial growth stops beyond a certain population volume

Why selective growth on ammonium? +

For any given scenario, the extracellular ammonium concentration will be higher than the extracellular nitrate concentration since the bacteria are producing ammonium and consuming nitrate, and the nitrate is also being leached away.

Since utilisation of ammonium is more energy efficient and the external ammonium concentration is higher than that of nitrate, we assume that the bacteria preferentially grow on ammonium. Thus, the growth rate depends solely on the ammonium concentration.

Why stop growth? +

Due to the very long timescales being considered in the model, the growth of ammonium according to the Monod equation gives highly inaccurate values, the extracellular medium always contains ammonium, and would predict growth of bacterial culture to extremely high values.

Graph: Uncontrolled growth of population volume(L) with time(s)

However, this is not possible since the growth of the bacterium is fundamentally dependent on the carbon source, which is present in only a limited amount, and cannot sustain such growth. Instead, the growth would be better characterized by a logistic-like growth.

Hence, for the sake of getting meaningful results, we assumed that the bacterium stops growing past a certain carrying capacity, here arbitrarily assumed to be a total population volume of 10^-4L.

Where V_c(L) = non-segregated population volume, and μ (s^-1) = growth rate

To characterise the growth rate, we used growth on ammonium data from a closely related strain, P. putida PGA1, which was characterized according to the Monod equation as:

Where μ_max (s^-1) = maximum growth rate, K_s (mol/L) = half-velocity constant, and [NH₄⁺]_ex (mol/L) = extracellular concentration of ammonium.

For exchange reactions, we assume reversible Michaelis-Menten-like kinetics, as used by the Wageningen iGEM team Cattlelyst.
For example, A_ex ↔ A_cell (where A_ex represents metabolite A in the medium, A_cell represents A in the cell)

Where n₁ (mol) = moles of A_ex, n₂ (mol) = moles of A_cell, T_m1 (mol/s) = maximum uptake rate of A by cell, T_m2 (mol/s) = maximum excretion rate of A, k_t1 (mol) = binding affinity of A_ex, and k_t2 (mol) = binding affinity of A_cell.

We obtained these values for NH₄⁺, NO₃^-, NO₂^- obtained by team Cattlelyst.

Modifying Ammonium Transport +

Upon testing, we observed that if we consider this transport as is, we start observing accumulation of ammonium inside the cell at extremely high concentrations (up to ~10⁵M), which would never happen in reality, as ammonium concentrations of >10mM are generally toxic for cells.

Figure: variation of cellular concentration(M) of ammonium against time(s)

Upon further investigations in literature, we found out that in several species, beyond certain ammonium concentrations, the import of ammonium by the ammonium transport protein AmtB gets inhibited by a protein GlnK.

Since the kinetics of this inhibition have not been studied well, we assumed that beyond a cellular concentration of 1mM ammonium, the cell stops importing ammonium, ie, T_{m1, NH4} = 0.

Leaching of Nitrate and N-Uptake by Plants

To incorporate the leaching and uptake, we assumed:

• Constant rates of leaching and uptake as long as the extracellular concentration is > 0
• Equal rates of uptake of ammonium and nitrate
• Only nitrate gets leached

Assimilation

The ammonium inside the cells gets incorporated into the N assimilation GS/GOGAT pathways. Taking the reference of team Cattlelyst that modeled a similar pathway in the same bacterial species, we assumed the assimilation step to be an irreversible first-order reaction with a rate constant of .003 mol/min = .00005 mol/s, i.e.

Enzyme Concentration

Since we are using constitutive promoters to express the nap and nrf genes in the bacterium, we assume that the enzyme concentration will remain constant with time.

In the future, we will incorporate the promoters induced by plant exudates, and get the different concentrations of enzymes depending on concentration of exudate.

k_m and k_cat values of enzymes:

We obtained the k_cat and k_m values for the enzymes we inserted into P. putida, Nap and Nrf, based on characterisations from literature, since obtaining isolated data on DNRA by bacteria proved to be difficult, since most bacterial species performing DNRA also perform pathways like nitrification and denitrification, which further change the ammonium and nitrate concentrations.

Results

Running the initial model, which did not account for nitrate leaching and N uptake by plant root, after about ~30 days, about 98% of nitrate is transformed into ammonium which can be taken up by the soil, and stabilizes at this value (the remaining was assimilated into GS/GOGAT cycles of the bacterium).

Figure: Variation of nitrate and ammonium concentration(M) in the medium with time

Figure: Particle numbers vs time (s). Extracellular ammonium (nh4_ex) stabilizes at ~7 × 10¹⁹ particles, assimilated ammonium (nh4_a) stabilizes at ~7 × 10¹⁷ particles.

Checking for Nitrite Toxicity

Throughout this timeframe, the nitrite concentration was always very low (< 10^-5 M). This reveals no threat of nitrite toxicity buildup in the cell for nitrite reductase and nitrate reductase enzymes expressed with the same constitutive promoter, as verified for (equal) enzyme concentrations of various orders of magnitude from 10^-3 to 10^-9 M enzyme.

Figure: Y-axis shows the concentration of cellular nitrite; X-axis represents time.

Figure: Concentration of nitrite (M) in the cell vs time(s) with enzyme concentrations of 10^-3 M (left) and 10^-9 M (right)

Test Study Using Final Model

We considered the following test case: 5mg of NH₄NO₃ fertilizer was added to a selected volume of medium (0.125L). By mass, this is equivalent to 1.125 mg NH₄⁺ and 3.875 mg NO₃^-, or 0.5 mM each. The scenario assumes a nutrient absorption rate of 10^-11 mol/s and a leaching rate of 9 × 10^-11 mol/s.

Scenario 1: Without Our Bacterium

Without our bacterium in the soil, all the ammonium is taken up by the plant after 1.58 years (5 × 10⁷s). However, only one-tenth of the nitrate is taken up, resulting in a 45% loss of the total applied nitrogen.

Scenario 2: With Our Bacterium

With our bacterium present, the nutrient uptake after 1.92 years is as follows:

• 0.88 mmol NH₄⁺ uptaken
• 0.01 mmol NO₃^- uptaken
• 0.1 mmol NO₃^- leached
• 0.01 mmol N assimilated by the bacterium

Figure: Particle numbers vs. time in the presence of our engineered bacterium.

In this scenario, all extracellular ammonium is taken up by the plant, and only 0.11 mM of NO₃^- is lost. This represents a total nitrogen loss of only 11%, a significant improvement.

Future Work

• Accounting for N loss through volatilization of ammonia: Track and model ammonia volatilization to minimize nitrogen loss from the soil.
• Measure constitutive promoter expressed concentrations: Quantify expression levels of nitrogen-related promoters to better predict nitrogen flux in soil.
• Make leaching periodic rather than constant: In reality, nitrogen leaching occurs in intervals (e.g., during rainfall, soil flooding). The values should be adjusted to reflect soil type, precipitation, and other environmental factors.
• Periodic application of fertilizer: Increase nitrogen concentrations in intervals to replenish soil N rather than a single bulk dose.
• Expand model to different soil volumes: Account for variable root mass and nitrogen content across different soil zones to make predictions more accurate.
• Model enzyme concentration based on plant root exudates: For enzymes whose promoters are induced by root exudates, consider spatial variation: expression is highest near roots and decreases with distance.

How our Model can be used by Farmers

Imagine a farmer who wants to use our genetically engineered bacterium in soil to reduce the N loss in their field and decrease the damage done to the nearby ecosystems.

For a given soil type and plant species, our model would be able to give the optimal amounts of fertilizer and bacterial volume to be added, for the plant to be able to uptake an amount of N that would be tailored to its growth needs, while minimizing loss by leaching.

How would it do this? We vary the inputs across a range of possible values, for given rates of uptake and leaching. Then, we will find the optimal condition (i.e., the one with least N loss and optimal N uptake) using flux balance analysis (steady state assumed to occur when all the ammonium and nitrate in medium has been depleted) to give the ideal input.