Trans aconitic acid (TAA) is a small molecule acid. In recent years, it has been discovered that it can effectively reduce the harm of nematodes to crops as a nematode killing agent. Our engineered strain, based on Bacillus velezensis, can colonize plant roots and continuously secrete TAA, enhancing plant resistance to nematode infestations.

To guide precise application strategies during planting and to evaluate the nematode control efficacy of our engineered bacteria, we developed a diffusion model based on 3D convection-diffusion equation. This model simulates the diffusion process of TAA in soil, helping predict its effectiveness in real-world applications. By studying the model, we aim to gain a deeper understanding of TAA behavior under actual conditions, which is crucial for optimizing application strategies.

We first conducted a simulation for a single diffusion source. This simulation helped us assess the area TAA can cover from a single application point, providing insight into the optimal layout of application points. Subsequently, we performed a field-scale simulation, which considered interactions between multiple diffusion sources and various environmental factors, ensuring the model's accuracy and reliability in practical applications.

The simulation results show that the diffusion range of TAA, upon reaching a steady state, is approximately double the radius of the primary root zone, and the effective killing area of nematodes was between 10cm and 15cm. Based on this data, we decided to apply our engineered bacteria via seed coating. This method ensures uniform distribution around the crop root zone, establishing an effective barrier against nematodes. Seed coating also increases TAA efficiency while minimizing unnecessary exposure to non-target areas, thereby reducing its environmental impact on the soil.

Given the complexity of soil environments and the influence of uncontrollable factors, we introduced several simplifying assumptions to make the model practical and manageable:

Simulation Conditions

 • Soil Homogeneity: We assume uniform physical and chemical properties of the soil in all directions, with no changes in pore structure during the simulation.

 • TAA Stability: We assume TAA remains chemically stable during diffusion, disregarding any interactions with plant roots or soil microbiota, as well as its degradation.

 • Constant Water Content: We neglect the sink term, assuming constant soil moisture content during TAA diffusion, without considering irrigation, evaporation, or water loss.

 • Root System Stability: We assume the root system and microbial community are fully developed and stable at the start of the simulation, providing a well-defined initial environment.

Diffusion Source

Our engineered bacteria primarily colonize the plant root zone, secreting TAA continuously. Bacteria dispersed in the soil often fail to germinate due to insufficient nutrients or remain in spore form, thus not contributing as active diffusion sources.

To simulate the diffusion of TAA in soil, we simplified the model by considering the plant root system as a perfect sphere, where radius 𝑟 represents the spatial distribution of the primary root zone. This spherical model provides a straightforward way to understand TAA diffusion dynamics.

Based on the root density 𝜌, we randomly selected points within the sphere as the initial diffusion sources. The entire sphere serves as the starting point for TAA diffusion in the case of a single plant.

We take soybeans as a reference, taking 𝑟 as 10 cm and 𝜌 as 0.55[1].

The convection-diffusion equation describes the diffusion and convection of substances in a physical system, and is widely used to simulate solute transport in soil.It accounts for both convective flow, driven by fluid movement, and diffusion, caused by concentration gradients[2].

The general form of the unsteady convection-diffusion equation is:

When solving the three-dimensional convection-diffusion equation, its analytical solution is often difficult to obtain, we often use finite difference method, finite element method, finite volume method and other numerical methods to solve. Here we choose the finite difference method.

The core idea of finite difference method is to use difference instead of differential, to discrete space and time, it discretes continuous partial differential equation into algebraic equation, and approximates the solution of partial differential equation through discrete grid.

Spatial Grid Discretization

We simulated a finite field area, dividing it into a 3D grid with spatial intervals Δx,Δy,Δz, representing the grid spacing in each direction.

Discretization of Governing Equations

Due to the different physical properties of convection and diffusion, we usually use different difference schemes to discrete the convection and diffusion terms to ensure the stability and accuracy of the numerical solutions.

The diffusion term involves the dispersion of the second-order reciprocal, and we choose the symmetric central difference method to deal with it. The convection term is related to the direction of fluid flow, and the central difference method may cause numerical oscillation, so the upwind difference method is chosen to deal with it.

 • Time term

  To introduce changes in time, we need to discretize time, using an explicit finite difference scheme:

 • Convective term

  Discrete using upwind difference scheme:

 • Diffusion term

  Discretization using a central difference scheme:

 • The discretized governing equation

Boundary Conditions

Boundary conditions are used to define the behavior trend of solute at the regional boundary, which can simulate the flow of solute in and out, the water barrier of underground soil, etc., and will affect the accuracy of the final numerical solution of solute transport in the region.

Here we use Dirichlet boundary conditions and assume that the boundary is zero concentration.

In the simulation of solute transport, the average pore velocity and hydrodynamic dispersion coefficient are more important, which are adjusted and calculated by literature search.

The average pore velocity is used to describe the actual velocity of water flow through pores in porous media, and is calculated by soil water flux and soil water content:

In the process of solute transport in unsaturated soil, the hydrodynamic dispersion coefficient is the result of the interaction of molecular diffusion and mechanical dispersion, and is also affected by many factors such as water content and porosity. According to the literature, the calculation formula of hydrodynamic dispersion coefficient is as follows[5]:

In the above equation:

Based on the actual situation of the simulation, we adjusted the parameters to make our model more stable.

Single Diffusion Source Simulation

We assume a single diffusion source simulation in a 2 m × 2 m × 2 m field for 7 consecutive days. The radius of the diffusion source is 0.1m, we selecte the central part for visualization, and the simulation results on the seventh day are as Figure 2.

The partial cross-section visualization results are as Figure 3.

A

B

Over time, the engineered bacteria continuously secreted TAA at the central diffusion source, forming a stable and highly concentrated TAA diffusion zone in the main root region. Due to the influence of gravity, water flow direction and other factors, TAA migrates faster in the vertical direction, so its distribution is not perfectly spherical when it finally reaches equilibrium. In the soil layer where nematodes are active, the effective protection area of TAA may be mainly concentrated near the root system.

Subsequently, in order to accurately quantify the effective protection range of TAA, we drew a line chart showing the variation of TAA concentration in soil with the distance from the root center. The curves in different colors represented different simulation times, providing us with an intuitive view of the variation of TAA concentration. According to the existing literature, the semi-lethal concentration of TAA for nematodes is 235.5 μg/mL[6], and we set the standard of effective killing concentration of TAA accordingly. The chart data showed that with the extension of simulation time, the TAA concentration gradually maintained stability, and the effective killing radius also stabilized in the range of 10 cm to 15 cm. This result may change as initial conditions change.

Field Simulation

In order to further verify the final distribution of TAA concentration under the condition of multiple diffusion sources and to further ensure the appropriate application measures, we conducted a multi-diffusion source simulation in the field range of 4 m × 4 m × 4 m. In the simulation process, we took into account the planting spacing of crops, and set the simulated application spacing to 20 cm to simulate the actual field conditions. The simulation lasted for 7 days, and the cross-sectional visualization results at 10 cm underground revealed the distribution characteristics of TAA in the soil.

The results shows that when TAA diffusion reaches a stable state, its spreading range in the field don't affect the neighboring plants too much. Therefore, the choice of seed dressing application can ensure a stable and lasting preventive effect. At the same time, the results also proved that the TAA secreted by the engineered bacteria would be controlled in a limited range and would not have too much impact on the overall ecological environment of the farmland. In addition, seed dressing and application can also reduce the waste of drugs and improve the efficiency of application, which is also of great significance for the sustainable use of resources.

However, our model still has some limitations. The parameters used in this model are mostly derived from empirical values, but there are differences in physical and chemical properties and hydrological parameters of farmland soil in different regions, which may affect the simulation results.