Team:Lambert GA/Model
MODEL
BACKGROUND & OBJECTIVE
To develop a nitrogen sensor that detects nitrate levels in hydroponics systems, Lambert iGEM utilized the Nar system, an operon native to E. coli that regulates nitrate reactions within the cell. Nar is a two-component system containing the response regulator NarL and its conjugate membrane-bound protein NarX. When NarX detects and later binds to extracellular nitrate, it phosphorylates NarL, which represses the PdcuS promoter, resulting in lower GFP expression. Conversely, low concentrations of nitrate results in higher GFP expression.
Lambert iGEM modified 2015 HKUST-Rice iGEM team’s part BBa_K1682018, which uses the Nar system, by adding the promoter BBa_J23106. This Anderson Promoter constitutively produces TetR, which represses the PtetO promoter that generates NarL. We can then control the expression of NarL by regulating the concentration of aTc, a molecule that inhibits TetR.
Before cloning and characterizing the nitrate biosensor, Lambert iGEM simulated the effects of varying concentrations of nitrate on GFP expression by creating a deterministic Ordinary Differential Equations (ODE) model. The model translates all of the reactions present in the NarL signaling pathway into ordinary differential equations and evaluates them based on an initial input of extracellular nitrate to ultimately predict GFP expression.
DEVELOPMENT
The 2020 Lambert iGEM team created an ODE model to predict GFP expression in response to varying concentrations of extracellular inorganic phosphate in the Pho regulon. This year, the team developed an ODE model for the Nar regulon based on the same principles. Lambert iGEM also reached out to Dr. Mark Styczynski from the Georgia Institute of Technology for guidance on identifying rate constants and protein abundances in the Nar signaling pathway. He provided online databases for rate constants and suggested that protein abundances were likely not necessary for constructing the model. We were able to discover some rate constants using the online database, Datanator; however, our collection was still incomplete. As a result, the team reached out to Megan McSweeney and Yue Han, members of the Styczynski Lab at the Georgia Institute of Technology, to assist us with parameter estimations on MATLAB Simbiology.
EXPERIMENTAL
Diagram of Signaling Pathway
To construct the model, the team began by following the process outlined in A Tutorial on Mathematical Modeling of Biological Signaling Pathways [5] to construct each reaction in the nitrate signaling pathway starting with the introduction of extracellular nitrate into the cell to final GFP expression in MATLAB Simbiology.
Figure 1. Diagram of the reactions of the Nar signaling pathway in a single E. coli cell, created in MATLAB Simbiology software.
Biochemical Reactions
The model contained a total of 12 biochemical reactions, as shown below:
| Number | Reaction | Parameter(s) | Description |
| 1 | [Outer Membrane].NarX + [Inner Membrane].[NarX_NO3-] <-> [Outer Membrane].[N03-] | [nitrate to narX_binding.kf]*[Outer Membrane].NarX*[Inner Membrane].[NarX_NO3-]*[Outer Membrane].[NarX_NO3-] - [nitrate to narX_dissociation.kr] | NO3- (Nitrate) binds to the membrane bound protein NarX, forming the bound NarX_NO3- complex. |
| 2 | [Inner Membrane].[NarX_NO3-] + [Inner Membrane].Pi <-> [Inner Membrane].[NarX_NO3-_Pi] | [autophosphorylation.kf]*[Inner Membrane].[NarX_NO3-]*[Inner Membrane].Pi - [NarX_dephosphorylation.kr]*[Inner Membrane].[NarX_NO3-_Pi] | NarX and NO3- complex (NarX_NO3-) is autophosphorylated |
| 3 | [Inner Membrane].[NarX_NO3-_Pi] + [Inner Membrane].NarL -> [Inner Membrane].active_NarL | [narXcomplex to NarL_phosphorylation.kf]*[Inner Membrane].[NarX_NO3-_Pi]*[Inner Membrane].NarL | After transporting Pi, NarX_NO3- dissociates from Pi and phosphorylates NarL |
| 4 | [Outer Membrane].ATC + [Outer Membrane].TetR <-> [Inner Membrane].inactive_TetR | [ATC to TetR_binding.kf]*[Outer Membrane].ATC*[Outer Membrane].TetR - [ATC to TetR_dissociation.kr]*[Inner Membrane].inactive_TetR | ATC binds to TetR, an Anderson Promoter |
| 5 | [Outer Membrane].TetR <-> [Inner Membrane].inactive_Pteto | [TetR to Pteto_inhibit.kf]*[Outer Membrane]TetR - [TetR to Pteto_dissociation.kr]*[Inner Membrane].inactive_Pteto | TetR inhibits Pteto |
| 6 | [Inner Membrane].inactive_TetR -> [Inner Membrane].Pteto | [Pteto_activation.kf]*[Inner Membrane].inactive_TetR | In the presence of ATC, TetR is inactive |
| 7 | [Inner Membrane].Pteto -> [Inner Membrane].NarL | [Pteto to NarL_transcribes.kf]*[Inner Membrane].Pteto | Active Pteto transcibres NarL, Nar Regulon transcription factor |
| 8 | [Inner Membrane].active NarL <-> [Inner Membrane].Pdcus | [NarL to Pdcus _binding.kf]*[Inner Membrane].active_NarL - [Nar to Pdcus_dissociation.kr]*[Inner Membrane].Pdcus | NarL promoter of the Nar Regulon genes, activating the promoter. |
| 9 | [Inner Membrane].Pdcus -> [Inner Membrane].mRNA | [Pdcus to mRNA_transcription.kf]*[Inner Membrane].Pdcus | Transcription of the downstream genes occurs |
| 10 | [Inner Membrane].mRNA -> [Inner Membrane].GFP | [mRNA to GFPexpression.kf]*[Inner Membrane].mRNA | Translation of GFP occurs. |
| 11 | [Inner Membrane].mRNA -> null | [mRNA_degredation.kf]*[Inner Membrane].mRNA | mRNA degradation occurs. |
| 12 | [Inner Membrane].GFP -> null | [GFP_degredation.kf]*[Inner Membrane].GFP | GFP degradation occurs |
Table 1. Model reactions of the Nar Regulon signaling pathway.
Initial Values of Species
The initial value of each species is shown below. Protein abundance data was gathered from Fundamental Constraints on the Abundances of Chemotaxis Proteins [1], Protein abundance profiling of the Escherichia coli cytosol [2], Inorganic Polyphosphate in Escherichia coli: the Phosphate Regulon and the Stringent Response [4].
| Species | Initial Value |
| NO3- | 0 |
| NarX | 1.3100 |
| ATC | 20 ng/ml |
| TetR | 500 |
| NarX_NO3- | 0 |
| Pi | 20 millimole |
| NarX_NO3-_Pi | 0 |
| inactive_Pteto | 0 |
| inactive_TetR | 0 |
| Pteto | 600 |
| NarL | 229 | active_NarL | 0 |
| Pdcus | 0 |
| mRNA | 0 |
| GFP | 0 |
Table 2. Initial values of model species.
ODE Equations
Lambert iGEM selected the Mass Action kinetic law to represent the model's reactions and generated 15 ordinary differential equations, as shown below:
| d([NO3-])/dt = -([nitrate to narX_binding.kf]*NarX*[NarX_NO3-]*[NO3-]-[nitrate to narX_dissociation.kr]) |
| d(NarX)/dt = 1/[Outer Membrane]*(-([nitrate to narX_binding.kf]*NarX*[NarX_NO3-]*[NO3-]-[nitrate to narX_dissociation.kr])) |
| d(ATC)/dt = 1/[Outer Membrane]*(-([ATC to TetR_binding.kf]*ATC*TetR-[ATC to TetR_dissociation.kr]*inactive_TetR)) |
| d(TetR)/dt = 1/[Outer Membrane]*(-([ATC to TetR_binding.kf]*ATC*TetR-[ATC to TetR_dissociation.kr]*inactive_TetR) - (([TetR to Pteto_inhibit.kf]*TetR)*[Outer Membrane]-([TetR to Pteto_dissociation.kr]*inactive_Pteto)*[Inner Membrane])) |
| d(Pi)/dt = -([autophosphorylation.kf]*[NarX_NO3-]*Pi-[narX_dephosphorlation.kr]*[NarX_NO3-_Pi]) |
| d([NarX_NO3-])/dt = 1/[Inner Membrane]*(-([nitrate to narX_binding.kf]*NarX*[NarX_NO3-]*[NO3-]-[nitrate to narX_dissociation.kr]) - ([autophosphorylation.kf]*[NarX_NO3-]*Pi-[narX_dephosphorlation.kr]*[NarX_NO3-_Pi])) |
| d([NarX_NO3-_Pi])/dt = 1/[Inner Membrane]*(([autophosphorylation.kf]*[NarX_NO3-]*Pi-[narX_dephosphorlation.kr]*[NarX_NO3-_Pi]) - ([narXcomplex to NarL_phosphorylation.kf]*[NarX_NO3-_Pi]*NarL)) |
| d(inactive_Pteto)/dt = 1/[Inner Membrane]*((([TetR to Pteto_inhibit.kf]*TetR)*[Outer Membrane]-([TetR to Pteto_dissociation.kr]*inactive_Pteto)*[Inner Membrane])) |
| d(inactive_TetR)/dt = 1/[Inner Membrane]*(([ATC to TetR_binding.kf]*ATC*TetR-[ATC to TetR_dissociation.kr]*inactive_TetR) - (([Pteto_activation.kf]*inactive_TetR)*[Inner Membrane])) |
| d(Pteto)/dt = 1/[Inner Membrane]*((([Pteto_activation.kf]*inactive_TetR)*[Inner Membrane]) - (([Pteto to NarL_transcribes.kf]*Pteto)*[Inner Membrane])) |
| d(NarL)/dt = 1/[Inner Membrane]*(-([narXcomplex to NarL_phosphorylation.kf]*[NarX_NO3-_Pi]*NarL) + (([Pteto to NarL_transcribes.kf]*Pteto)*[Inner Membrane])) |
| d(active_NarL)/dt = 1/[Inner Membrane]*(([narXcomplex to NarL_phosphorylation.kf]*[NarX_NO3-_Pi]*NarL) - ([NarL to Pdcus_binding.kf]*active_NarL-[NarL to Pdcus_dissociation.kr]*Pdcus)) |
| d(Pdcus)/dt = 1/[Inner Membrane]*(([NarL to Pdcus_binding.kf]*active_NarL-[NarL to Pdcus_dissociation.kr]*Pdcus) - (([Pdcus to mRNA_transcription.kf]*Pdcus)*[Inner Membrane])) |
| d(mRNA)/dt = 1/[Inner Membrane]*((([Pdcus to mRNA_transcription.kf]*Pdcus)*[Inner Membrane]) - (([mRNA to GFPexpression.kf]*mRNA)*[Inner Membrane]) - (([mRNA_degradation.kf]*mRNA)*[Inner Membrane])) |
| d(GFP)/dt = 1/[Inner Membrane]*((([mRNA to GFPexpression.kf]*mRNA)*[Inner Membrane]) - (([GFP_degredation.kf]*GFP)*[Inner Membrane])) |
Table 3. Ordinary differential equations for the model reactions.
Rate Constants
The team was able to find reaction rate constants for a few biochemical reactions but was unable to identify most other values using available literature on the novel NarL signaling pathway. Thus, the team used the Simbiology Model Analyzer feature to run parameter estimations based on available experimental data from the 2015 HKUST-Rice iGEM team, with guidance from Ms. Yue Han of Styczynski Research Group. Since the characterization data simply reflects nitrate concentration to final GFP expression without data for GFP expression over time, it was difficult to estimate a global set of parameters that would simulate GFP expression dependent on nitrate concentration. Instead, Lambert iGEM selected one characterization data point (one nitrate concentration and one GFP expression value) at a time (8 hours) to estimate parameters with a local solver, Isqnonlin, and then analyzed the relationship between each set of parameters.
| Reaction Number | Name(s) | Estimated Values |
| 1 | [nitrate to narX_binding.kf] | 0.0217 |
| 2 | [nitrate to narX_dissociation.kr] | 91.8189 |
| 3 | [autophosphorylation.kf] | 0.00005 |
| 4 | [narX_dephosphorylation.kr] | 0.0300 |
| 5 | [narXcomplex to NarL_phosphorylation.kf] | 1.0026 |
| 6 | [ATC to TetR_binding.kf] | 1 |
| 7 | [ATC to TetR_dissociation.kr] | 1 |
| 8 | [TetR to Pteto_inhibit.kf] | 1 |
| 9 | [TetR to Pteto_dissociation.kr] | 1 |
| 10 | [Pteto_activation.kf] | 1 |
| 11 | [Pteto to NarL_transcribes.kf] | 1.0001 |
| 12 | [NarL to Pdcus_binding.kf] | 4.8473 |
| 13 | [NarL to Pdcus_dissociation.kr] | 0.5630 |
| 14 | [Pdcus to mRNA_transcription.kf] | 4.8473 |
| 15 | [mRNA to GFPexpression.kf] | 3.6354 |
| 16 | [mRNA_degradation.kf] | 0.00771982196224858 |
| 17 | [GFP_degredation.kf] | 0.0013 |
| 18 | [translation_GFP.kf] | 1063100 |
Table 4. Estimated rate constants for each biochemical reaction.
Assumptions
The ODE model assumes that all rate constants remain constant and are not affected by environmental factors other than GFP translation. It also assumes that the input nitrate and aTc concentration remains constant throughout the simulation of GFP expression.
RESULTS
With the set of ODE equations, we utilized Simbiology Model Analyzer to simulate the GFP expression over time corresponding to the various nitrate concentration. Our input data consists of characterization data from 2015 HKUST-RICE.
Figure 2. Graph of the model’s simulated GFP expression to nitrate concentration.
The model results confirm that nitrate concentrations between the range 40ppm to 160ppm will result in GFP expression. Thus, cloning part BBa_K1682018 for the use of a nitrate biosensor is sensitive enough to detect nitrate concentrations that are typically found in hydroponics.
The model also follows the same log relationship as the original characterization data from 2015 HKUST-RICE, shown below:
Figure 3. Graph of the original characterization data from 2015 HKUST-RICE.
While the model predictions follow the same trend as HKUST-RICE, Lambert iGEM’s model estimates smaller values of GFP expression because the model represents the GFP expression of one single E. coli cell, whereas HKUST-RICE’s data is from a culture of cells.
PPB MODEL
Objective
With a focus on employing the toehold switch as our pathogen biosensor’s basis, Lambert iGEM researched this mechanism to further optimize and understand the biosensor. As current estimates on the specifics of toehold biosensor expressions were vague and nonspecific, we constructed the Plant Pathogen Biosensor (PPB) model to accurately predict the time and expression of our sensors.
Design
Similar to Lambert iGEM’s Nar and Pho models, the PPB model was developed as a deterministic ODE model with the exception that Green Fluorescent Protein (GFP) expression response prediction was correlated to the time elapsed. The design took significant insight from A Tutorial on Mathematical Modeling of Biological Signaling Pathways by Zhike Zi to form the equations to simulate the biochemical processes that occur in the pathway. These equations primarily consist of the law of mass action and Hill equations which would employ rates found from literature for their parameter values.
Diagram
To lay out a basis for this model, we constructed a diagram of the signaling pathway, starting from the root sample extraction to the GFP response, in MATLAB Simbiology. However, the prior species in the model, concerning extraction, would not affect the reaction and, thus, have their boundary condition set to true—serving solely as visual representations.
Figure 4. Toehold switch biological signaling pathway in MATLAB Simbiology following the lysis of Cells to DNA to GFP.
ODE Equation
Lambert iGEM selected a combined expression of the mass action kinetic and Hill equation to serve as the model’s parameters. As the dissociation constant and the binding of ligands significantly guide interactions in the pathway, Hill equations are converted into a percentage and are multiplied to the mass action kinetic rate constant to predict the interactions between the reactant collisions [2]. Equation (1) was the derived culmination of the two kinetics, with k representing the mass action kinetic rate constant and the expression in the parentheses representing the Hill equation. It serves as the first parameter setting for yielding the results.
(1)
Lambert iGEM chose to utilize this Hill kinetic to accommodate the chemical representation of the toe-hold switch biological signaling pathway. The chemical equilibrium expression of the binding of the ligand (Trigger RNA) and the toehold biosensor has been written such that n is the number of binding sites of the ligand, ka is the forward reaction, and kd is the reverse reaction as seen in (2).
(2)
From this expression, the impact from this reverse reaction on the ultimate products can be solved into the aforementioned Hill kinetic through a series of utilizing theories, rearranging equations, and substituting resulting expression and values. First, the principles of the collision theory helped derive the dissociation constant which is generalized by dividing the reverse with the forward reaction as shown in (3).
(3)
Equation (4) can be rearranged from (2) and (3).
(4)
Not all of the binding sites will be occupied as a result of the dissociation constant which reverses the reactions, thus an expression of the proportion of occupied binding sites to total binding sites can be derived as seen in (5).
(5)
Substituting the numerator with a previously found equation that solves for the bound ligands provides the Hill Kinetic seen in (6). Equation (6) removes the unnecessary Toehold Biosensor aspect as toehold biosensor concentrations seldom affect binding with its proposed abundance in a paper strip.
(6)
Rate Constants
Online literature yielded the majority of the rate constants needed in the first iteration of the model. Below is the table of rate constants and the parameters they participate in.
| Name | Symbol | Value | Parameter | Source(s) | Additional Comments |
| Toehold Switch Leakage Rate | kf | 1 | Off State Toehold Switch > GFP | Estimation | Little data online |
| Binding of DNA to Toehold Switch Mass Action Kinetic Rate | kf_2 | 1 | DNA > Activated Toehold Switch | Estimation | Little data online |
| Toehold Switch Binding Dissociation Constant | KA | 10 | DNA > Activated Toehold Switch | [2] | In Hill Kinetic |
| Toehold Switch Ligand Hill Constant | n | 2.78 | DNA > Activated Toehold Switch | [2] | In Hill Kinetic |
| GFP Production Rate | kf_3 | 200/230 | Activated Toehold Switch > GFP | [3] | Ribosomes produce an average of 200 amino acids per second while a GFP molecule is 230 amino acids |
| GFP Production Hill Constant | n_2 | 2 | Activated Toehold Switch > GFP | [4] | In Hill Kinetic |
| GFP Degradation Rate | 𝛅 | -0.25 | GFP > 𝛅 | [5] | To accommodate for the degradation of GFP |
Table 5. Rate constants for each biochemical reaction.
Assumptions
This model is under the premise that all rate constants remain constant regardless of environmental factors. It also assumes that ribosomes utilize all types of amino acids at the same rate and that all cells get lysed into DNA at a constant proportion.
Results
Figure 5. Simulation results of time vs. GFP response.
Figure 6. Simulation results of concentration vs. GFP response.
The immediate reaction of the toehold switch in response to pathogen presence is thus shown in Figure 5. Within minutes, multitudes of GFP procured from an ample presence of pathogen cells. Additionally, there is a correlation between concentration and time upon the output with a direct relationship between the concentration and output. Time correlates with an S curve to output while concentration has a linear relationship with output as seen in Figure 6. From this, the relative concentration of the pathogen can be extrapolated from the brightness of GFP on the paper strip.
FSIR Model
Overview
With an objective of creating biosensors for hydroponic diseases, Lambert iGEM chose to target two of the most common root rot diseases. The diseases chosen were specifically Fusarium and Phytophthora. Despite their prevalence and known lethality, data on their infection characteristics, such as the incubation time, remained vague.
To solidify our understanding of the target plant pathogens’ epidemiology (Fusarium Oxysporum in this case), Lambert iGEM constructed a predictive model that would simulate the spread of this disease in a hydroponics farm.
Design
The Suspected-Infected-Removed (SIR) method, a technique utilizing the progression of plants through three main states of disease infection, served as the basis of the model. The model employs four main components: Fusarium’s basic reproductive number, incubation period, symptomatic period, and fatality proportion. To gather these parameter values, Lambert iGEM utilized online literature rates and equations. These four variables formed the summation equation which graphs the spread of Fusarium in a hydroponics system. Equation (1) is for procuring infected plants from a “patient zero” in a period of time [1].
(1)
The number of cases of infected plants at a certain time was depicted as “C”. The “I” represented the total Infected cases at the number of incubation periods elapsed, “t”, as per the Fusarium incubation period in hours, indicated by “i”. This equation used the basic reproductive number, “R0”, as the base of the exponent that the equation summed its ultimate output from.
The remaining key variables provided the total number of removed plants from the previous equation, multiplying to the previous cases of infected plants at a certain time and adding to the total incubation period in hours, forming the total disease infection time to death/recovery in hours, as seen in (2) [1]. Hence F represents the fatality proportion of these Fusarium cases, and d represents the symptomatic period in hours.
(2)
This was all then translated into MATLAB code and then ran through a live editor to be fitted on the curve fitter app—yielding the model’s result after inputting the parameter values. These values were derived from various online sources with the fatality proportion and symptomatic period taken from a study on the pathogenicity of Fusarium on hydroponically grown lettuce [2]. The incubation period and basic reproductive number values, however, needed an equation or matrix of its own as the effects of the characteristics of these farms affected the epidemiology of this disease [3]. For example, the growth conditions to Fusarium differed in hydroponics with varying temperatures, plant species (affecting general film thickness), and water stress/vapor pressures; therefore, the incubation period was characterized and developed to mathematically quantify Fusarium growth to percent visual infection as modeled in (3) [4].
(3)
Solving the equation for i with a graphing calculator yielded the incubation period in hours while the other constants were input according to natural values of Fusarium epidemiology and the characteristics of Sweetwater Urban Farms which Lambert iGEM collaborated with to further tailor this model towards predicting accurately (see Table 6).
This similar approach of sourcing from Sweetwater Urban Farms and online literature applied to acquiring the basic reproductive number of Fusarium. These constant values, seen in Table 6, solved for the dominant (first) eigenvalue, equal to the basic reproductive number, in the matrix below (see Figure 7) [3]. However, as the determinant of the matrix with the parameters included did not equal zero, meaning it could not be solved by hand, this eigenvalue was derived utilizing an online calculator designed to find eigenvalues analytically.
Figure 7. Basic reproductive number matrix.
With a complex dynamic to simulate such as the cross-contamination combination between towers (physically separate structures upon which the plants are situated in the hydroponics), a matrix was employed to accommodate for the interdependence of these towers to gather one solution. From the linear transformation (as it is a matrix now) of three separate rate of infection equations, as each row represents a separate equation modeling the infection from a different starting location, the eigenvector of the matrix can be said to be the solution of these three equations. In order to get the basic reproductive number, which is the rate of infection, the eigenvalue of this system is solved for. The eigenvalue can be stated as the rate multiplied to the matrix in order to retrieve that eigenvector, the solution of the matrix. Hence, the first eigenvalue seen was taken as the basic reproductive number.
Parameter Explanations
All of the rates used in the model are listed below. Additionally, some of the equations that gather parameter values for the previous three equations are inscribed. See Table 6.
| Name | Symbol | Value | Original Equation(s) | Source(s) | Additional Comments |
| Susceptible | S | 248 | N/A | Sweetwater Urban Farms | The total plants in the system that is being simulated; All of the plants are located on three towers |
| Basic Reproductive Number | R0 | 6.752 | (1); (2) | [3]; Basic reproductive number matrix | One of main components |
| Incubation Period (Hours) | i | 142.5 | (1); (2) | [4]; Equation (3) | One of main components |
| Fatality Proportion | F | 1; 100% | (2) | [2] | One of main components |
| Symptomatic Period (Days) | d | 2 | (2) | [2] | One of main components; Extrapolated a symptomatic period of two days based on article’s description of symptoms appearing from 5-7 days |
| Fusarium Growth as Seen by Percent Visual Infection (%) | G | 100 | (3) | [4] | This indicates the plant is fully infected and nearing its death |
| Plant Growth Coefficient | c1 | 0.00413 | (3) | [4] | The study’s [4] used value to approximate an average plant growth rate; Future plans with Sweetwater to gather their plant growth rates |
| Water Stress | a5 | 1 | (3) | [4] | Coefficient |
| Temperature Function | TF | 1 | (3) | [4]; Equation (5) | Coefficient |
| Wetness Function | WF | 2.3573125 | (3) | [4]; Equation (4) | Coefficient |
| Time Step | k | null | (3) | [4] | Input |
| Wetness Response | a3 | 1.71 |  (4) |
[4] | Coefficient |
| Film Thickness (of Plant) | RF | 0.79375 | (4) |
[4] | Approximated from study’s [4] values |
| Scale Factor (of Temperature Function) | A | 1 |  (5) |
In equation (5) [4] | |
| Shape Factor (Temperature Function) | 𝛼 | 3 | (5) |
In equation (5) [4] | |
| Additional Shape Factor (Temperature Function) | 𝛃 | 1.058823529 | (5) |
In Equation (5) from study [4]; Source is Equation (6) | |
| Hydroponics Hourly Temperature (℃) | Thourly | 28.89 | (5) |
Value utilized from Sweetwater parameters; Source from study [4] | |
| Minimum Hydroponics Temperature | Tmin | 19.5 | (5) |
Value utilized from Sweetwater parameters; Source from study [4] | |
| Maximum Temperature for Fungal Growth | Tmax | 31 | (5) |
Maximum temperature Fusarium could growth at according to study [4] | |
| Optimum Temperature for Fungal Growth | Topt | 28 |  (6) |
[4] | Source of the additional shape factor |
| Infected Plant’s Infectious Propagule Production Rate | 𝛼1 | 4.25 | Basic reproductive number matrix | [3]; [5]; Converted from article | Coefficient |
| Time Plant is Infectious (Hours) | 𝛕 | 190.5 | Basic reproductive number matrix | [3]; [4]; used incubation period value | Coefficient |
| Fraction of Infectious Propagules Spread through Water and Farmer’s contact | p | 0.228758 | Basic reproductive number matrix | [3]; [6]; Utilized spore rate of closest Fusarium oxysporum relative | Coefficient |
| Connectivity Between Towers (All Tower Configurations) | cxy | 1 | Basic reproductive number matrix | [3]; Sweetwater Farms | Coefficient; Sweetwater towers were shown to be directly connected at one point through the water system |
| Crop Density of Tower One (Plants per Inch Squared) | H1 | 28/1972 | Basic reproductive number matrix | [3]; Sweetwater Farms | Coefficient; Measured by Sweetwater Farms |
| Crop Density of Tower Two (Plants per Inch Squared) | H2 | 44/2523 | Basic reproductive number matrix | [3]; Sweetwater Farms | Coefficient; Measured by Sweetwater Farms |
| Crop Density of Tower Three (Plants per Inch Squared) | H3 | 176/2523 | Basic reproductive number matrix | [3]; Sweetwater Farms | Coefficient; Measured by Sweetwater Farms |
| Infection Parameter | 𝛾 | 1 | Basic reproductive number equation | [3] | Coefficient; Estimation |
Table 6. Fusarium epidemiological model utilized parameter tables.
Assumptions
This model assumes no other external factors, such as countermeasures against the disease, affect the pathology of Fusarium. It also assumes that the plant remains infectious and alive for longer after reaching 100% visual infection.
Results
With the two summation equations and their parameter values, Lambert iGEM gathered resulting output values and fitted them into a graph, using MATLAB’s Curve Fitter app, against the correlated time. Infected and Removed curves were created showing Fusarium’s virulence in three towers of Sweetwater Urban Farms.
Figure 8. Simulation results of time vs. infected.
Figure 9. Simulation results of time vs. removed (also the total deaths).
The Fusarium epidemiological model exemplifies the information delay stemming from utilizing traditional methods. With a week for lab results to come in, an incubation period passes allowing for significantly more plants to become infected as the initially infected plant transitions into its symptomatic period [7]. Waiting for results from traditional testing methods after sending in samples from a dead plant allows for up to an additional 100 plants to become contaminated which is an increase of more than 50 cases from our PPB method.
Future Plans
Lambert iGEM plans to continue the collaboration with Sweetwater Urban Farms to incorporate the Infected and Removed models into a hydroponics data app developed by the farm. These models will be coded on the app to extrapolate cases of infections from confirmed cases of plant deaths. It will also provide the general pathology of Fusarium in the farm along with additional measurements including the plant growth coefficient.
PHO IMPLEMENTATION
Model From Previous Team
Before beginning experimental testing of the phosphate biosensor, last season our team developed a deterministic Ordinary Differential Equation (ODE) model of the Pho Regulon utilizing MATLAB Simbiology (Figure 9). The model utilizes parameters within the Pho signaling pathway to predict the relationship between GFP expression and initial extracellular phosphate concentration for the phosphate sensitive biosensor (Figure 11). Through the process of researching the Pho Regulon signaling pathway, finding different parameters such as rate constants, and inputting values, the team was able to create a model that represents the expected GFP values of the biosensor. The model served as a guide for characterization as the predicted GFP expression values were used to confirm the experimental results.
Figure 10. Diagram of the reactions of the Pho Regulon signaling pathway in a single E. coli cell, created in MATLAB Simbiology software.
Figure 11. Prediction of relationship between GFP expression and phosphate concentrations ranging from 0 to 100uM made by the deterministic ODE model.
Experimental Design
Upon completion of the model, our team created a procedure for experimental testing. After running multiple trials following the protocol, we created a characterization curve using the data obtained from the plate reader. However, the data did not match the trend shown in the predictive curve from the ODE model. As a result, we had to modify our protocol accordingly. The addition of a wash step using MOPS media was critical as the original media used to grow cells, Luria Broth, had pre-existing concentrations of phosphate. To negate incorrect readings from the plate reader, the cells were washed in MOPS. In addition, a vortex step was included to allow for an equal distribution of cells when measuring optical density. Due to the edits in the protocol, the characterization curve resembled the same trends as the ODE model.
References
Nitrate Model References
[1] Bitbol, A. F., & Wingreen, N. (2015). Fundamental Constraints on the Abundances of Chemotaxis Proteins. Biophysical Journal, 108(5), 1293–1305. https://doi.org/10.1016/j.bpj.2015.01.024
[2] Ishihama, Y., Schmidt, T., Rappsilber, J., Mann, M., Hartl, F. U., Kerner, M. J., & Frishman, D. (2008). Protein abundance profiling of the Escherichia coli cytosol. BMC Genomics, 9(1), 102. https://doi.org/10.1186/1471-2164-9-102
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PPB Model References
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Fusarium Epidemiological Model References
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