Team:NAWI-Graz/Model
Team:NAWI-Graz
Goal
We wanted to predict what concentration of phosphorus (P) in the soil was necessary to activate the sender cell and reach sufficient signal strength in the receiver cell. The model was built before the experiments started and would be refined after the wet lab with empirical data. The input variables were stated in a manner that anyone who has access to a wet lab would be able to measure them and use our model.
Questions we wanted to answer with the model:
How are the concentration of P and the amount of GFP produced in the receiver cell quantitatively related? And how efficient is this synthetic system?
How does the strength of a promoter affect the end amount of GFP?
Are the experimentally obtained values similar to the model values? Further on do they aid in the development of a detailed model to predict AHL-based signaling dynamics?
Method
We used Vensim as a means to generate a simple signal transfer black box model that shows how the different parameters influence the size of the transferred signal. The inflow constitutes the concentration of P in the soil. The first loss occurs when P is diffusing from the environment into our sender cell. The soil and the membrane of the cell are attributed to this loss. The repressor and promoter in the cell have conversion rates less than 100% which also have to be accounted for. During the transfer of the AHL 3-oxo-C6 HSL molecule (here after AHL molecule!) from the sender to the receiver cell a part of the information is lost in the soil and membranes again. The strength of the output signal in the receiver cell is highly volatile and subject to all these uncertain rates and processes. To try and predict this we developed an equation based model in Python.
The model generates values for each rate randomly in range of the standard deviation for every iteration. The time in the model was chosen as discrete. Every one time step is representative of one level of relative signal strength.
The mean was set to the values found in literature and a normal distribution was chosen. In one iteration, this algorithm is called 1000 times, with the same starting level of P. The different values that the function generates are stored in a csv, which can be imported into every visualisation program.
The rates were estimated using the cited sources and given a standard deviation of 10-20% of the value, depending on the perceived volatility of the process. The starting parameter, the concentration of P in the soil, was changed before each simulation run.
Assumptions:
homogeneous environment, with constant:
pH,
soil matric potential (retention of water),
tortuosity,
porosity
The diffusion coefficient and effective diffusion coefficient of phosphate and AHL are negligible
the phosphate is spreading one-dimensional
strength of promoters is arbitrarily determined on a range between 0 and 1, where 0 = no expression and 1 = maximal expression
Schematic representation of the model we used. The formulas are hidden.
Regarding the loss of available phosphorus in soil, a wide variety of factors influence the amount of mass which is either immobilized or lost through diffusion. One of the most important parameters affecting the phosphorus loss from soils is known as “Tortuosity", a natural characteristic of the ground. It characterizes the specific degree of unevenness in the transportation paths of porous materials like soil. It is used to describe the difficulty of various physical transport processes, which we have characterized as “loss in soil” and, along with porosity and permeability, is a parameter for describing the properties of porous materials. In our case it is the primary factor in which our “loss in soil” parameter derives its value from. (The diffusion through the soil is influenced by the tortuosity)
The factors influencing loss of P in soil are the amount of water currently present in the soil, its porosity and the effective diffusion coefficient of the solute (in our case phosphate) . The amount of water held in the soil, also known as soil matric potential, refers to water held in the ground against gravity, while adhering to pores and cavities of porous soil and rock that are smaller than 10 µm. This effect comes about due to the surface tension of water and implies that rougher and more coarse ground will lead to a higher loss of phosphorus in soil. In soil sciences this value can be characterized as “porosity”. Since pores or capillaries usually fill with air and/or water it stands in direct relation to the amount of retention water held within the soil. The effective diffusion coefficient described the relationship between the solute and an adsorbing environment, in our case we neglected the effect of the adsorbing surrounding. Influence factors on this are the content of water in the soil and the porosity. [1] [2]
When determining the strength of the phoA promoter and the subsequent expression of LacI we had to determine the most influential factors. Those were the concentration of phosphate, the phoB/phoA binding relationship and the degradation constant of LacI. The concentration of phosphate is essential for the activation or repression of the system, since it regulates the seven component P signaling pathway. As suggested in theory, for P concentrations higher than 4 uM the phoB is phosphorylated and activates the expression of LacI repressor, which then represses the production of the AHL synthesizing enzyme. Oppositely, at concentrations lower than 4 uM the phoB is inactive, and thus LacI expression is inhibited and hence the AHL synthase is produced. The protein phoB/phoA promoter binding and the consequential LacI expression was estimated using the Hill equation. [3]
We decided to split the promoter strength of the LuxI gene into 3 discrete categories to see if the elevated expression of AHL synthase changes the output. We simulated a weak, a medium and a strong promoter under the assumption that the weakest promoter has a relative signal transduction between 0 and 0.3, the medium between 0.31 and 0.5 and the strongest on the range between 0.61 and 1.0. The value 0 is equal to no expression (no signal transduction), whereas 1 is considered to be maximal expression (no loss of signal). In this case maximal expression relates to the limit of the sender’s signal production. This means the amount of AHL synthase produced and further on the limit of AHL production per cell. Since the expression of the AHL gene is regulated by the LacI repressor, we included the degradation halftime of the LacI repressor and it’s concentration in the equation. The diffusion loss of AHL during the transport to the receiver cell was modeled estimating based on values obtained via the cited sources. [4]
The AHL molecule activates the transcription of the GFP in the receiver cell. The GFP expression amount was proposed to be in a linear relationship with the amount of AHL molecules reaching the cell. The amount of molecules which reach the receiver cell depends on the produced amount of AHLs, the degradation rate of AHL, the diffusion coefficient of AHL and the distance from the sender. The value, depicting the loss of the initially produced amount of AHL is based on theory, is 95%.
To interpret the results we introduced two arbitrary units. The first describes the GFP amount and the second describes the amount of cell growth, as seen in the following root function. This is only an approximation (not an empirically determined relationship) which is of use only for the reader so he/she can intuitively better understand the relationship.
Discussion: Relation between AHL and GFP a.k.a. cell growth
Bacterial quorum-sensing molecules, like N-acyl homoserine lactones (AHL), are one of the most prevalent and efficient means of crosstalk between bacterial cells and even entire bacterial populations. Rhizobial bacteria, which include symbiotes living near or on plant roots, perceive and respond to those molecules in a way that has been extensively studied. Most of these studied interactions however are being assessed in a manner, which rarely occur, if at all, in nature. Reducing the interaction between the quorum signalling molecule and the bacteria to a bilateral system without considering in-vivo influences such as the aforementioned soil condition and water retention status will often lead to inaccurate results and therefore wrong assumptions in some cases.
With this model, we wanted to create a mathematical approach to involving as many environmental factors into the equation and create a logical relationship between the phosphorus in the soil that activates the quorum sensing signalling through to the effect it will have in the rhizobial cell (we call it receiver cell) responding/reacting to the signal. Our main goal was to characterize the dynamics of the AHL-production in the rhizosphere bacterium using our sender as the producer of the molecule and phosphorus as the trigger. The model should give a clearer picture as to how the amount of phosphorus would impact the production of AHL and signal strength and loss there-of while reaching the receiver cell.
Regarding the loss of phosphorus in soil
The first step to finding a correct initial situation for our model was to determine the factors which reduce the average amount of phosphorus, which is found in soil. This parameter proves important, since establishing a link between soil quality and phosphate loss in soil will be an essential part of the model.
Mentioned in the methodology part, the term Tortuosity was explained and split up into certain characteristics of the soil. A large amount of the phosphates in soil is absorbed by soil particles and therefore incorporated into the soil, making up the amount of retention water. Factors such as precipitation, hydrological condition, air temperature and pH-value also play a role when determining loss of phosphorus in soil [5]. Since we needed to work in a realizable framework, our mathematical model had to rely on several degrees of freedom. The amount of soil parameters would prove to be too many to handle and focussing on these variables specifically was not the primary purpose of this model. Assumption about the pH-value, the amount of retention water, tortuosity and porosity of the soil has been made homogenous and constant.
Therefore the value of phosphorus diffusion was introduced to our mathematical model as modifiers between 1.5 and 3.0, with 1.5 being soil with lower tortuosity and lower water retention. The other end of the spectrum represents soil that is high in tortuosity and high in water retention, therefore trapping more of the free phosphate and making it unavailable for plants. [6]
Regarding the strength of the promoter
As already mentioned in the methodology part the assumption of promoter strength division into 3 parts (weak, medium and strong) was based on covering most of the possible phosphorus concentrations in soil. Since the AHL expression is strongly associated with environmental phosphorus we deemed the implementation of 3 different promoter strengths as essential. In the final equation we could condense the functions of promoter strength into its own framework and reduce the value to percentages of promoter strength, with each represented by values between 0 and 1 (0 meaning no promoter activity and 1 meaning 100% activity).
The results, shown by the output of our model, show direct correlation between the promoter strength and AHL-expression, as we expected.
Regarding AHL-expression and diffusion losses
The loss of signal in the rhizosphere is significant, especially since the whole input is affected. The different promoter strengths gave us AHL expressions in range from approximately 1 - 5% of the input signal.
Representation of the results of the model.
Dependence of GFP signal production on concentration of soil phosphate and the phosphate in the cell.
Regarding the conclusive statement for the relationship between phosphorus and AHL
We have presented a simple model of quorum sensing using autoinduction by different amounts of environmental phosphorus based on known parameters in soil and cells. Using this model we demonstrate that the amount of phosphorus in the soil directly influences the rate of production of the quorum sensing molecule N-acyl homoserine lactone and thus the activity of cells receiving this signal and reaction to it accordingly. With high phosphate concentrations in soil the auto-induced promoter will switch to higher expression of the AHL-signalling molecule and therefore stimulate nearby phosphate-solubilizing bacteria to a higher degree.
Consequently, on a theoretical level, this model supports our claim that introducing a modified bacteria into soil that induces phosphate solubility in certain rhizobial bacteria.
References
- [1] https://acsess.onlinelibrary.wiley.com/doi/epdf/10.2136/sssaj1972.03615995003600010012x
- [2] [3] https://2013.igem.org/Team:Tokyo_Tech/Experiment/phoA_Promoter_Assay#5._Modeling
- [4] https://bionumbers.hms.harvard.edu/search.aspx
- [5],[6] https://www.aces.edu/blog/topics/crop-production/understanding-phosphorus-forms-and-their-cycling-in-the-soil/?cn-reloaded=1