Team:Groningen/Model
Mathematical modelling is an integral aspect of any iGEM project. This year we developed two population models that focus on the bacteria and on the nematodes. Our aim for modelling this year was to test the feasibility of our experimental design and to guide the engineered efforts for further optimization of RootPatch. The Bacterial population model was designed to evaluate the robustness of the genetically modified Bacillus mycoides bacteria in the rhizosphere and how the biofilm growth is influenced by the environmental factors. The Nematode population model will estimate whether RootPatch is able to prevent Globodera pallida from critically damaging the potato plant under varying environmental conditions. Combined, these two models will highlight the drawbacks in our primary design and suggest appropriate course of action for the future.
Introduction
The bacteria population model was developed in order to:
Investigate how temperature, water activity and competition influence the population of genetically modified (GM) bacteria in RootPatch.
assess the sensitivity of the GM bacteria to these environmental factors.
To do this, we set up a list of differential equations that describe the growth of bacteria (Wild Type & GM) in the soil throughout the potato season. In order to achieve this, the following assumptions were made:
The potato season is 200 days, from planting the seed potatoes to the harvesting of the new potato tubers.
Wild Type bacteria in the soil have a competitive advantage over GM bacteria because GM bacteria have been found to experience higher metabolic load. (De Leij et al., 1998). This assumption was integrated in the model with the arbitrary competition coefficients α & β from the Lotka-Volterra Model (Gavina et al., 2018).
Wild Type and GM bacteria share the same niche at the potato roots.
The biggest competitor of GM bacteria in the soil is its Wild Type counterpart already present.
Bacteria prefer to move towards the nutritionally rich plant environment.
Only temperature and humidity are taken into account as environmental factors due to their significant impact on bacterial growth.
There is no impact of root location on the growth of the bacteria.
Since there is no empirical data on the growth and dynamics of Bacillus mycoides in the root environment, and on the impact of genetic modifications on Bacillus mycoides, many parameter values in this model were assumed (see Table 1). Therefore, the analysis approach was qualitative rather than quantitative. This model will therefore not tell us for how long and how many the bacteria will survive, but rather which parameters have the biggest impact on the survival of RootPatch and should thus be taken into consideration while developing RootPatch. While impert
Below you will find a description and methodology of the model itself, followed by the most important results extracted from the model. At the end, we will describe the final conclusions of the model and how they impact the RootPatch.
Figure 1: graphical abstract of the bacteria population model. There are two locations populated by two types of bacteria. (1) Near the roots where the bacteria of RootPatch are applied and live together with Wild Type soil bacteria. (2) In the bulk soil, where the bacteria of RootPatch will not be able to suvive due to the safety mechanisms of RootPatch. The competition between Wild Type and GM bacteria is modeled using the two arbitrary values α and β.
Methodology
The model is built on a system of four Ordinary Differential Equations (ODEs) describing four different bacteria populations: GM bacteria near the root which represents RootPatch (GMOp), GM bacteria that diffused away from the root to the bulk soil (GMOs), Wild Type bacteria near the root (WTp) and Wild Type bacteria in the bulk soil (WTs) (see Figure 1 & 2). For both the GM and WT bacteria, Bacillus mycoides was taken as the model organism since we expect that the biggest competitor for the GM bacteria in RootPatch in the soil, is its Wild Type counterpart. The ODEs are composed of 3 types of factors that influence the population: environmental factors that affect the growth, competition between bacteria, and the transfer of bacteria from the root environment to the bulk soil.
Figure 2: ODE equations describing the four bacteria populations. GMOp describes the population of GM bacteria at the plant (RootPatch). GMOs describes the population of GM bacteria in the bulk soil. WTp describes the Wild Type bacteria near the plant. WTs describes the Wild Type bacteria in the bulk soil
▪ Growth population:
Two environmental factors were taken into account for the growth of the population: temperature and water activity.
The temperature throughout the potato season of 2019 (01-04-2019 to 30-09-2019) in the Netherlands was extracted from the Raam soil temperature measurement network dataset and fitted using the MATLAB curve fitting tool (see Figure 3) (Mathworks, Curve Fitting Toolbox, 2020). Only temperature values at 50 cm depth were taken into account because this roughly matches half the depth of the potato root system (Zarzyńska et al., 2017).
Water activity of the soil is the ratio between the vapor pressure of the soil, when in a completely undisturbed balance with the surrounding air media (FDA, 2014). In other words, water activity is the unbound water available for bacterial growth. The survivable range of most bacteria is between a water activity of 0.9 and 1.0, but generally, bacteria grow best at higher water activity values (Reykdal 2018). We could not find an exact value for the water activity of Bacillus mycoides. So we tightened our water activity range to be between 0.95 and 1 to ensure we were studying RootPatch in a range that would be likely survivable. To simplify the model, we assumed a linear relationship between the water activity of the soil and time during the potato season. Benninga et al., 2018 studied moisture levels in the Netherlands and found little fluctuations throughout the season, so for our model, we decided to increase the water activity gradually with a step size of 0.0001 per day since water activity rises slowly with increasing temperature (see Figure 3).
The growth of the bacteria is defined in terms of the soil temperature and water activity using the Ratkowsky square root equation (Ratkowsky et al., 1983). This model is based on the observation that at lower temperatures, the square root of the specific growth rate is linear with temperature (see Figure 3). Since the root environment is higher in nutritional content, we assumed that the growth of the bacteria population is doubled (Growths & Growthp). Moreover, for the kill switch of RootPatch, an arbitrary value was chosen that reduces the growth rate of GM bacteria in the bulk soil to zero.
Figure 3: Growth equations for bacteria populations dependent on water activity (Aw) and temperature (T) in the soil. Growth rate of bacteria in the bulk soil (Growths) is assumed to be half of the growth rate for bacteria near the root (Growthp) due to the relatively low nutrient concentrations in bulk soil.
▪ Competition between bacteria: Since bacterial competition can be the biggest challenge for RootPatch to survive, we included a relatively simple competitional relationship between the different bacteria populations based on the competitive Lotka-Volterra model (Gavina et al., 2018). This model does not include a complex nutritional or spatial competition but simply models the competition between GM bacteria and WT bacteria using two arbitrary values between 0 and 1: α (competition of Wild Type bacteria on GM bacteria) and β (competition of GM bacteria on Wild Type bacteria) (see Figure 2).
Since WT bacteria are assumed to have a competitive advantage over GM bacteria, α was always assumed to be larger than β. The carrying capacity (Kp) in this model is shared between the GM and WT bacteria.
▪ For diffusion between bacteria at the roots and in the bulk soil, we designed diffusion relationships of bacteria moving towards the potato plant, and away from the plant towards the soil. We define two types of diffusion: active diffusion and passive diffusion (Figure 4).
Passive diffusion is a constant diffusion of bacteria from the root environment to the bulk soil, and vice versa. The rate of passive diffusion from the bulk soil to the plant environment (D2) is assumed to be always higher than the rate of passive diffusion from the plant to the bulk soil (D1) due to the nutrient-rich environment of the roots.
Active diffusion is based on the relatively low carrying capacity at the roots. Higher bacterial populations increase the force of diffusion due to the limited space. The function of active diffusion is dependent on the fraction of bacteria at the roots over the carrying capacity of the roots multiplied by the active diffusion parameter (s).
Figure 4: Equations describing the diffusion of bacteria between the root environment and the bulk soil. ToSoilGMO describes the active and passive diffusion of GM bacteria from the root environment to the bulk soil. ToSoilWT describes the active and passive diffusion of WT bacteria from the root environment to the bulk soil. ToPlantGMO describes the passive diffusion of GM bacteria from the bulk soil to the root environment. ToPlantWT describes the passive diffusion from the bulk soil to the root environment.
Parameter sensitivity analysis
There are a variety of methods to determine how sensitive bacterial growth is to environmental parameters (D.M. Hamby 1994). To quantify the sensitivity index, we used a parameter sensitivity equation that was introduced to us by Prof. Sander van Doorn (see Human Practices page). This equation identifies the most important parameters by quantifying how much the model’s outcome changes after a small change in the parameter value. After taking the derivative of a small change in the outcome divided by a small change in the parameter, the simplification results in the equation in Figure 5. “S” is a dimensionless ratio which refers to the sensitivity value of “X” the parameter, to “a” the outcome. “New” refers to the values after the small change compared to the “Old”, or original value. In our model, we applied a 2% decrease to the parameters and outcome and assessed the sensitivity of the parameters on the population of RootPatch at 150 days into the season. We found that regardless of a small increase or small decrease as a change, we obtained the same results. We chose to perform this analysis at 150 days because this equation needs to be used far from a stability boundary, and our stability boundary is at about 50 days, after which the population reaches equilibrium.
Figure 5: The Sensitivity Parameter Index Equation
Open Table 1 to watch the values used to set up the model. Most values for the GM and WT bacteria are taken for Bacillus mycoides as it is the primary host organism of RootPatch.
| Parameter | Description | Assumption | Initial value sensitivity analysis | Units | Reference |
|---|---|---|---|---|---|
| Ks | Carrying capacity of the bulk soil | Significantly higher than Kp | 3000 (no sensitivity analysis) | Number of bacteria | - |
| Kp | Carrying capacity of the root environment | Significantly lower than Ks | 1000 (no sensitivity analysis) | Number of bacteria | - |
| α | Competition of Wild Type bacteria on GM bacteria | Higher than β | 0.5 | Arbitrary value | - |
| β | Competition of GM bacteria on Wild Type bacteria | Lower than α | 0.5 | Arbitrary value | - |
| Tmin | Minimum temperature at which bacteria can proliferate | - | 10 | ℃ | Lechner et al., 1998 |
| AWmin | Minimum water activity at which bacteria can proliferate | - | 0.95 | Arbitrary value | Fontana, 2007 |
| Kill | Kill switch that ensure that GM bacteria in the bulk soil die | Taken by studying the behavior of the growth curve of the bacteria | 10 (no sensitivity analysis) | day-1 | - |
| s | Active diffusion from root environment to bulk soil | Assumed by studying the behavior of the model | 0.001 | % of total bacteria population in the root environment | - |
| D1 | Passive diffusion from root environment to the bulk soil | Assumed by studying the behavior of the model. Lower than D2 | 0.01 | % of total bacteria population in the root environment | - |
| D2 | Passive diffusion from the bulk soil to the root environment | Assumed by studying the behavior of the model. Higher than D1 | 0.05 | % of total bacteria population in the root environment | - |
| GMOs(0) | Starting population of GM bacteria in bulk soil | Assumed by logical assumptions and studying the behavior of the model. | 5 | Number of bacteria | - |
| GMOp(0) | Starting population of GM bacteria in plant environment | Assumed by logical assumptions and studying the behavior of the model. | 500 | Number of bacteria | - |
| WTs(0) | Starting population of WT bacteria in bulk soil | Assumed by logical assumptions and studying the behavior of the model. | 2000 | Number of bacteria | - |
| WTp(0) | Starting population of WT bacteria in plant environment | Assumed by logical assumptions and studying the behavior of the model. | 500 | Number of bacteria | - |
Results
Exploring the model
After modeling the populations over time, the populations show to reach an equilibrium state after about 70 days (Figure 6). As expected, the WT population in the bulk soil shows the highest population size due to a high carrying capacity. The WT bacteria at the roots and in the bulk soil are always more abundant than in the GM bacteria, probably due to the competitive advantage. Moreover, the graph confirms that the theoretical kill switch implemented in the model works, keeping the GM bacteria population in the bulk soil constant.
Figure 6: population dynamics of the 4 populations. After a while (t~70 days), the populations reach an equilibrium state with WTs showing the biggest population size.
The starting population can affect the outcome a population model. For RootPatch, it is interesting to explore whether the starting population size (i.e. the number of bacteria in the inoculant formulation) would influence the population size throughout the season. As indicated by the results in Figure 7, this does not appear to be the case. Different starting populations (in this case 10, 300, and 800) of the GM bacteria at the roots always end up in the same equilibrium state ( t ~ 90). However, the moment of reaching equilibrium is affected; the smaller the initial population, the longer it takes before the population reaches a stable size.
Figure 7: The effect of different starting populations on Rootpatch’s population throughout the potato growing season. Purple: N(0) = 10, Magenta: N(0) = 300, Pink: N(0) = 800.
Sensitivity analysis
To assess the sensitivity of the different parameters on the survival of the GM bacteria at the roots (RootPatch), a sensitivity analysis was performed. In short, this analysis fluctuates the parameters by 2% and measures the sensitivity of GM bacteria population size at the roots over time or at a fixed time point.
When plotting the sensitivity of RootPatch over time for the different parameters, it becomes clear that it is the initial phase of the potato season before the population reaches a stable equilibrium, that is the most sensitive (Figure 8). We can especially identify this relationship for the parameter describing competition of WT on GM bacteria (α), active diffusion (s), and the minimum water activity (Awmin). Whereas active diffusion appears to play a strong part very early in the season, water activity and competition are more important in the phase where RootPatch is growing towards its equilibrium.
Figure 8: sensitivity of RootPach's GM bacteria population on different parameters throughout the potato season.
Moreover, when we assess the sensitivity at a moment of a stable equilibrium (t = 150), we can investigate to which parameters the bacterial population of RootPatch is the most sensitive ( Table 2). RootPatch appeared to be the most sensitive to water activity, whereas other parameters such as temperature, competition and diffusion show a much smaller impact. Striking is the sensitivity difference between alpha and beta, which is also represented in Figure 9. Apparently, the impact of WT bacteria on GM bacteria is much greater than the impact of GM bacteria on WT.
| Parameter | Sensitivity at t = 150 days |
|---|---|
| Water Activity in soil (Aw) | 0.197 |
| Minimum Water Activity (Awmin) | 0.0624 |
| Competition WT on GM bacteria (α) | 0.0258 |
| Temperature in soil (T) | 0.0159 |
| Competition GM on WT bacteria (β) | 0.00802 |
| Minimum temperature (Tmin) | 0.00802 |
| Active transfer to bulk soil (s) | 0.0062 |
| Passive diffusion to the root environment (D2) | 0.0032 |
| Passive diffusion to the bulk soil(D1) | 0.0001 |
Bifurcation Sensitivity Analysis
To further explore the sensitivity of RootPatch on different parameters, we utilized bifurcation plots. In short, bifurcation plots show how the population at the end of the potato season is affected by the parameters in the model. The results of this analysis are only shown for the parameters with the most significant impact that can be controlled by the engineering of RootPatch such as minimum water activity, competition and diffusion.
Figure 9: bifurcation plot for minimal water activity and active diffusion to the bulk soil at t=150 days. (Green = WTp, Purple = GMOp, Blue = GMOs)
The parameter analysis for the minimal water activity shows clearly that all of the bacterial populations (GM bacteria and Wild Type bacteria) drop whenever the minimum water activity is bigger than the water activity in the soil (Figure 9). We set the water activity in the soil to 0.985 at 150 days. This was done to ensure that the minimal water activity never became higher than the water activity in the soil for the sensitivity analysis, thereby preventing negative growth in the model. Lowering the minimum water activity of both the WT and GM bacteria seems to improve the survival of the GM bacteria in the soil at expense of the WT bacteria (Figure 9).
Active diffusion (s) shows to be important as well. The higher the pressure of bacteria to leave the root environment due to overpopulation at the roots, the lower the survival of the GM bacteria of RootPatch, but also the more likely the development of a GM bacteria in the bulk soil, despite the implemented kill switch (Figure 9).
Figure 10: bifurcation plot for the two competition parameteters α and β at t=150 days. (Green = WTp, Purple = GMOp, Blue = GMOs)
The competition of WT bacteria on GM bacteria (α) showed to be important for the model. This is confirmed by the bifurcation plot, which shows that at a high value for α, the GM bacteria population can go extinct (Figure 10). β appears to be less important but the bifurcation plot still shows a clear relationship between the value of this parameter and the GM population size. Moreover, both plots in Figure 10 confirm that a low α and/or a high β value ensures the dominance of the GM bacteria over the WT bacteria in the root environment.
Conclusion
With this model, we were able to explore the robustness of RootPatch to several factors: competition between bacteria, environmental factors and the diffusion of bacteria in the soil. While this model is far from the actual situation in nature, it did help us to guide our attention to factors in our experimental design and in future experimental plans.
The starting population did not appear to be important on the outcome of the end population size. For RootPatch this implies that the number of bacteria in the inoculant formulation can fluctuate, and still lead to the same outcome. However, the moment when a stable population is reached, is dependent on the starting population and this may possibly have an effect on the protection in the beginning of the potato season.
The sensitivity analysis indicated that the first period of the potato season appears to be the most critical for RootPatch’s survival. The minimum water activity of the bacterium, active soil diffusion, and the competition of the WT bacteria on the GM bacteria show a significant impact on the model in this phase. After the initial period is over, and the equilibrium population is reached, RootPatch appears quite robust. For our project, this would mean that the initial phase after application, requires the biggest care and perhaps additional help to establish a stable equilibrium population at the roots.
Competition is important and choosing the right host organism is therefore important. Especially, the effect of WT bacteria on the GM bacteria appears to be important. By choosing a bacterium that is more robust, RootPatch may be better suited to overcome the initial sensitive period. Afterwards, sensitivity to competition decreases as a stable equilibrium population is reached.
Introduction
The nematode population model was developed in order to:
Evaluate the RootPatch efficacy necessary to halve the damage from Globodera pallida
Evaluate the influence of important environmental factors on the required RootPatch efficacy from (1)
With these goals in mind, we defined a number of differential equations to represent transitions between populations of the different life stages of G. pallida and the damage dealt to the root based on G. pallida population size. We made key assumptions to simplify the design of our model:
The growing season is 150 days. The potato plant starts fully grown with RootPatch formed already.
NLP forms a dense area close to the root. NLP concentration is constant within 10 cm around the root.
In real life, cysts cannot hatch for about 2 weeks due to a natural lag phase, called diapause. Our model does not take diapause into account.
In the following sections, we describe the overview and methodology of the model itself, primary validation by considering the results of field experiments, sensitivity analysis of environmental factors, and, finally, the simulated effect of RootPatch. Lastly, we present our conclusions and the impact of this model on the project in general.
Figure 1: Overview of the nematode population model, based on the life cycle of G. pallida. To the regular life cycle (egg→ juvenile→ infective juvenile→ adult) two things were added to make the model work: a variable reflecting the damage to the root and the new population of the nematodes repelled after taking up NLP.
Methodology
Seven state variables were defined: eggs (E), second stage juveniles far (>10 cm) from the root (S), second stage juveniles close (<10 cm) to the root (C), repelled second-stage juveniles (R), juveniles inside the root (P) and adult females inside the root (A). In addition, both P and A cause damage to the root (represented by a fraction of damaged root tissue - X). The model consists of a series of ODEs regulating the transitions of G. pallida nematodes between these distinct populations. The ODEs were constructed to depend on the population density, other state variables, environmental variables, and a rate constant. Whenever a mathematical relationship could not be found, a simple 1:1 dependency was assumed. Rate constants were either (1) taken from the literature, (2) adapted from the literature data, or (3) their values were assumed within an accepted range. An overview of the model and the rate equations is presented in Figure 1. Also, the list of rate constants and parameters can be found in Table 1.
| Parameter | Description | Value | Units | Reference |
|---|---|---|---|---|
| K | Carrying capacity of the root system | 4.000.000 | Number of nematodes | Trudgill, 1967; Zarzynska, 2017; |
| b | Relative nutritional needs of parasitic juveniles compared to adults | 0.5 | - | Value assumed |
| eggs_per_cyst | Number of eggs per cyst | 300 | - | Stone and Evans, 1977 |
| Rae | Rate of development from adults into cysts | 0.00826 | day-1C-1 | Langeslag et al., 1982 |
| Res | Rate of hatching | 0.00128 | day-1C-1 | Kaczmarek et al., 2019 |
| Rsc | Rate of movement to the root | 0.001 | day-1C-1 | Kaczmarek et al., 2019 |
| Rcp | Rate of invasion | 0.002 | day-1C-1 | Kaczmarek et al., 2019 |
| Ds | Juvenile death rate | 0.0099 | day-1 | Ward, 1985 |
| Rpa | Development rate from J2 juvenile to adult | 0.00215 | day-1C-1 | Mugniery et al., 1978;Schans, 1993 |
| Da | Adult death rate | 0.0099 | day-1 | Ward, 1985 |
| N | NLP concentration | No set value | mmol g-1 | N/A |
| Rcr | Repelling activity of NLP | 1 | day-1 mmol g-1 | Value assumed |
| Rrs | Recovery rate of juveniles from the effects of NLP | 0.5 | day-1 | Value assumed |
We took the environmental conditions and changes into account to make our model as accurate as possible. Soil temperature and moisture were approximated by sine functions based on the climatic conditions, regional difference, and soil type. The model works for four general soil types (sand, clay, silt, loam) and the common combinations of those. The propagation of second-stage juveniles through the soil (i.e. the S→C transition) depends on the motility index. The motility index is defined by the average distance a nematode will traverse in 2 days (Townsend and Webber, 1971). The motility index is calculated by constructing a parabolic equation based on the soil type and moisture. The motility index is dependent on the water retention curve of each soil type. Each soil type has a sigmoidal water retention curve and the nematodes have the highest motility index at the moisture levels where this sigmoidal curve is the steepest (Robinson, 1986). The motility index curve for nematodes in sandy soil from Robinson (1986) was fitted to the water retention curve of the other soil types to obtain the motility index equations for these soil types. The environmental variables and their equations can be found in Table 2.
| Environmental variable | Equation | Units | Reference |
|---|---|---|---|
| Dutch temperature | 4.75 + 14.09 * sin(0.01057 * t + 0.3018) | ℃ | Benninga et al., 2018 |
| Dutch soil moisture | 30 + 10 * sin(2 * π / 365 * t) | % | Benninga et al., 2018 |
| Scottish temperature | 10 + 5 * sin(2 * π / 365 * t)) | ℃ | Metoffice.gov.uk, 2020 |
| Scottish soil moisture | 17.5 - 7.5 * sin(2 * π / 365 * t) | % | COSMOS-UK, 2019 |
| Motility index sand | -0.0029 * (moisture%)2 + 0.1115 * moisture% + 0.5281 | - | Robinson, 1986 |
| Motility index silt | -0.0017 * (moisture%)2 + 0.075 * moisture% + 0.6667 | - | Likar et al., 2017; Robinson, 1986 |
| Motility index clay | -0.0017 * (moisture%)2 + 0.1083 * moisture% - 0.25 | - | Tuller & Or, 2005; Robinson, 1986 |
| Motility index loam | -0.0033 * (moisture%)2 + 0.1833 * moisture% - 1 | - | Hlavacikova & Novak, 2018; Robinson, 1986 |
Figure 2: Validation of the nematode population model by comparing output to the field trial experimental findings. The multiplication ratio is the ratio between the current egg density and the initial egg density. The red lines show the data points obtained from the model and are overlaid on the graphs of experimental values from Kaczmarek et al. (2019). The plots show (A) the change in multiplication ratio along the growing season and (B) the final end-season multiplication ratio at the different initial egg densities.
Validation of the model
To make the model as accurate as possible, we compared our results to the experimental findings of the field experiments (V. C. Blok, James Hutton Institute, Scotland). They performed the field experiments with G. pallida in Scotland and monitored the egg density every month by measuring the DNA content of eggs in the soil samples with qPCR. To this end, we adapted the environmental conditions to those found in Scotland (Table 2). The soil type was chosen based on the findings of Aitkenhead and Coull (2019). In Figure 2A, the multiplication ratio is low at the beginning of the season but peaks towards week 16 and then drops. Our results closely resemble this behavior. However, the multiplication ratio in our model increases earlier but less steeply than the experimental multiplication ratio because our model does not incorporate diapause, but rather has a constant rate of hatching. Despite these differences, the key features of the graph (i.e. peak height and timing, and the subsequent drop) are well replicated by our model. We assume that the potato plant is completely vulnerable to G. pallida. So the fact that the model data most closely replicates the results from a G. pallida non-resistant cultivar, Maris Piper is a further validation of our model (“Maris Piper”, 2019).
In Figure 2B, the multiplication ratio at the end of the season ranges from 0 to 5 for initial densities greater than 10 eggs/g soil but rapidly increases when the initial egg densities fall below 10 eggs/g soil. Our model reproduces this relationship accurately for a range of initial densities from 5 eggs/g soil to 50 eggs/g soil. Thus, our model not only reproduces the real-life experimental findings with sufficient accuracy but can predict the asymptotic behavior as the initial density approaches 0 eggs/g soil. However, our model predictions will become less accurate as the initial egg density decreases, since we do not regard the nematodes as discrete units, but rather use population averages to describe them. This method is only valid if there are enough nematodes to dampen the noise of individual behavior.
Results
RootPatch implementation
An important question, of course, is how the application of RootPatch to the plant affects the damage from G. pallida. In Figure 3, X (damage) at the end of the growing season is plotted against RootPatch efficacy (defined as [NLP] * Rcr), modeled for 4 soil types. Altogether, at RootPatch efficacy over 0.9 day-1, RootPatch starts to have an effect on the root damage. The greatest marginal benefit is achieved in the range of 0.9 to 2.0 day-1, while a further increase of NLP production is not as efficient anymore in reducing the damage. Since the rate with which the NLP repels nematodes is constant, the only way to increase RootPatch efficacy is to increase the NLP concentration. This can be done by increasing the RootPatch bacteria density or by increasing the bacterial NLP production and secretion.
Figure 3: Sensitivity of total damage inflicted to the RootPatch efficacy. Damaged root fraction at the end of a growing season is plotted against the varying RootPatch efficacy for the four different soil types. RootPatch efficacy is the product of the NLP concentration and the repulsion rate of the NLP. The average moisture content is 20%, the temperature ranges from 12 to 19 ℃, and the initial egg density is 10 eggs/g.
We set the goal for RootPatch to reduce the damage by 50%. In this section, we predict the necessary efficacy of RootPatch to halve the damage, and how this value would change in response to different environmental conditions. However, it is unknown how fast the nematodes recover from the elevated NLP levels and revert to their normal behavior (Rrecovery, see Figure 1). Therefore, we plotted the recovery rate against the required RootPatch efficacy at the corresponding rate of recovery. The shape of the resulting graphs (Figure 4) suggests that the recovery rate only makes a significant difference at values lower than 0.2, with the required RootPatch efficacy plateauing at higher recovery rates. This is good because we can easily predict the threshold for the required minimal NLP concentration in soil. Therefore, the goal should be to achieve an NLP concentration above the curve for any given rate of restoration.
Three plots were made in total using the standard Dutch conditions (loamy soil, 30% average moisture, a range of soil temperatures from 12 to 19 ℃, and 10 eggs/g soil initial egg density). In each plot, one of the three relevant parameters was varied: initial egg density, temperature, and soil moisture (Fig. 4).
Figure 4: The RootPatch efficacy required to reduce the damage by 50% in response to the recovery rate of nematodes. Bifurcation plots were made using standard Dutch conditions (30% average soil moisture, soil temperature in the range of 12-19 ℃, initial egg density is 10 eggs/g soil). In each of the plots one condition was changed to evaluate its effect on the RootPatch efficiency: (Left) initial egg density, (Middle) soil moisture, (Right) average seasonal soil temperature.
The initial density is a major determinant of the required RootPatch efficacy (Figure 4 (Left)). An increase in initial egg density by a factor of 10 increases the required RootPatch efficacy at the plateau (high recovery rate) by a factor of around 6. However, when the nematodes are permanently repelled (no recovery), this factor is lower, about 10:5. As this discrepancy increases with higher initial egg densities, the recovery rate has a more significant effect on the required RootPatch efficacy value. Since G. pallida field infectivity in the Netherlands is roughly 10 times higher than in the UK, finding the rate of recovery is even more crucial for the local Dutch situation.
Soil moisture does not have such a linear relationship with the required efficacy. The highest required RootPatch efficacy occurs at 20% average moisture content, closely followed by 30% (Figure 4 (Middle)). Both lower (10%) and higher (40%) moisture content leads to a decrease in the required RootPatch efficacy. The nematodes need a film of water on the soil particles to move but also require sufficient oxygen levels. Low moisture content hampers their ability to move because the water films are too thin. In contrast, too much water, while facilitating their movement, deprives nematodes of oxygen by filling the air bubbles in the ground. For loamy soils, the inflection point between complete waterlogging and dry soil lies between 20% and 30%, which explains why the required RootPatch efficacy is highest at 20-30% moisture content.
An increase in the mean soil temperature results in an upwards shift of the required RootPatch efficacy (Figure 4 (Right)). This is since the life cycle of the nematodes speeds up with increasing temperature, resulting in more nematodes, which makes it more challenging for RootPatch to protect the plant. Thus, a higher RootPatch efficacy will be required in hotter regions (i.e., South America). However, this relationship only holds up to 25 ℃. The nematodes cannot survive at temperatures above 25 ℃, an effect not considered in our model. Therefore, our results are only accurate when peak temperatures stay below 25 ℃.
Current climate change trends suggest an increase in temperature and dryness of the soil in the next 5-10 years (B. Kurnik, 2019). With our model, we can argue that such changes will collectively generate better conditions for the survival of G. pallida nematodes. As a result, the RootPatch efficacy currently predicted may not be enough at the moment of the actual RootPatch implementation. We have to consider these possible future changes to adapt our project for a higher chance of fighting nematodes later.
Conclusion
With our model, we are now able to quantitatively predict the population dynamics of G. pallida and the action of RootPatch, given the environmental conditions. We paid a lot of attention to reproducing the environmental factors and validating the model, which allows us to make these predictions more accurate. According to our model, higher seasonal soil temperature and slightly drier conditions facilitate the nematode life cycle and their movement towards the plant. Moreover, the required efficacy of RootPatch is proportional to the initial field infectivity at the time of RootPatch's application.
The main reason to develop this model is to guide engineering efforts. Therefore, we argue that experimentally determining (1) the number of NLP14a molecules necessary to repel a single nematode and (2) the rate of G. pallida recovery in the soil will greatly improve the integrity of the nematode model. With this important information in hand, a more valid estimate of the necessary production rate of NLP could be set as a target for the future engineering of B. mycoides. As a result, we have set up an efficient feedback system of coordinated modeling and experimental efforts to make RootPatch as efficient as possible.
Water activity and temperature are two important parameters for the interaction between the bacteria and the nematodes. To our computational prediction, periods of high heat and low soil irrigation are advantageous to the nematodes and less beneficial to the bacterial growth. We hereby propose two strategies that could improve RootPatch:
Ensure that the bacteria can survive in drier conditions
These considerations are especially important given that climate change could impact the agriculture in the EU by heat waves, resulting in drier soils. It is important for us to ensure that the GM Bacillus mycoides is competitive against other soil bacteria, not just now, but in the long-term too. Improvement of minimal water activity through directed mutagenesis or choosing a host organism with better suited water activity characteristics are good strategies to enhance RootPatch’s robustness.
If experiments indicate that RootPatch must be reapplied, it should be in a period of rainfall or after field irrigation by a farmer.
We want to provide farmers with the best guidelines for a possible reapplication of RootPatch. It will likely be during the summer, a hot and dry period, when the nematodes come into full force. As dry soil conditions make it harder for RootPatch to initially establish on the roots, we suggest reapplying RootPatch after rainfall or after irrigating the field, where the chances of finding a niche for the RootPatch's bacteria is increased.
However, survival of RootPatch is only a prerequisite for a good production of NLP. Despite being able to estimate the necessary production rate of NLP, we rely on additional experimental findings to make our estimate more accurate. Still, nothing can be said with certainty until the field trials, as there are many other factors involved (e.g., NLP degradation by proteases or diffusion in the soil).
Overall, our models complement each other to predict what future experiments should be done to enhance the robustness of RootPatch and to improve its efficacy. With this laboratory-modelling cycle we can use models to guide experimental efforts to obtain results that could improve the models back. This creates a bigger potential for RootPatch in the future, which is what makes modelling so important for us this year. Look at the Engineering page to learn more about how we used the models to improve our experimental design and develop experiments.
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