Implementation Model

General objective

To generate a computational model that simulates the most relevant relationships between our engineered system and the plant pathogens inside the appropriate habitat for the Rust control.

Specific Objectives

- To limit the multifactorial ecological problem in a way that a simple mathematical model may be proposed. Such model should be able to answer relevant questions regarding the implementation method.

- To find the populational proportions between our organism and the plant pathogens that optimize our biological control.

- To generate hypotheses for future experimental confirmations.

Biological Panorama

Coffee Rust dispersion is based on the generation of [http://botanydictionary.org/uredospore.html uredospores]. These are dispersed by wind and water predominantly, as well as by active animal or human dispersion. These spores require about 24 to 48 hours of free continuous humidity, so the infection process usually occur only during rainy seasons. The fungus grows as a [http://en.wikipedia.org/wiki/Mycelium mycelium] on the leaves of the plant, and the generation of new spores takes about 10 to 14 days. Since leaves drop prematurely, this effectively removes important quantities of epidemic potential inoculum; nevertheless, a few green leaves will survive through the dry season. Dry uredospores may live for about 6 weeks. In this way, there is always a viable inoculum capable of infecting new leaves ath the beginning of the next rainy season.

In this year's iGEM, our main goal is to significatively reduce the mycelial form of the fungus in order to control inocula from a season to the next. The way this works is by spraying bacteria on top of the leaves of the plants, however, the amount and concentration of bacteria are not known. Thanks to a population control system by toxin-antitoxin modules, a small fraction (near 15%) of the bacterial population will live in a persistant state. Persister cells have very low metabolic rates. Non-persister active cells, even though more sensitive to environmental hazards, readily detect fungal infections. If a determined chitin profile (based on our molecular mathematical models) is detected, active bacteria are stimulated in a way that they are capable of secreting a plant hormone to induce its natural defense responses.

Mathematical Model Description

Let's begin with the expected dynamics for the inoculated bacteria in absence of a fungal infection (R variable). An initial number of bacteria (B variable) are sprayed on the leaf. As mentioned earlier, these may be in a persistant (I variable) or active (A^-- variable) state in a 15:85 ratio. By assuming persistence as a static metabolic state, persister death rate is neglected. On the other hand, active bacteria die with a given rate (delta_A parameter per active bacterium). However, these populations are maintained through a dynamic equilibrium with a persistance transition rate (gamma_1 parameter per active bacterium), and another one in the reverse direction (alpha(R) parameter per persister bacterium). The alpha(R) parameter should, in principle, have a term independant of R in order to maintain the described equilibrium. If this were not true, A^- would have no population inputs and would decay to zero in steady state.

In the presence of fungi, cells should wake up more often (which should be included in the alpha(R) parameter). Additionally, the A^-- population should generate a stimulated cell population (A^+ variable) at a certain rate (sigma(R) parameter per inactive bacterium). Stimulated bacteria are capable of producing salycilic acid, a plant hormone that induces plant defense mechanisms that should decrease fungal populations at a given rate (delta_R(A^+) per fungus). The only fungi relevant to our model are those who already germinated from the uredospores and are infecting the plant (i.e., that are in a mycelial form). Taking this into account, their random removal and natural death rates are neglected. In the same fashion as with the active cell population, stimulated once are capable of returning to a persister state with a certain rate (gamma_2 parameter per stimulated bacterium) and also eventually die at a given rate (which we approximated to be comparable to the active one's). Persister state stimulating toxins act at a intercellular level, so cell cross-activation/inactivation phenomena are discarded. The following schematic represents the expected population dynamics for this model for a single infection cycle. Subsequent cycles should work in a similar fashion, where the next cycle's inputs are the previous cycle's outputs.

The following table indicates the different parameters and variables of our system, together with its units and explanation.

Differential Equations

From the schematic above the following ordinary differential equations were constructed:

As well as the following initial conditions:

Inferences from the Molecular Mathematical Model

First of all we had to find our parameters' values, as well as define some of those more thoroughly.

- alpha(R): As mentioned earlier, this parameter should have both R dependent and independent terms. The independent term was searched for in literature, where we found it to be 0.1 h^-1 (Balaban et al, 2004). For the R dependent term, we thought of two possibilities. The first one that it may be aproximated as a line in the form of beta*R, and the second one as a heaviside function (step function). In order to answer this, we went back to our original mathematical molecular model and plotted chitin concentration against the difference between toxin and antitoxin concentrations. This should give us an idea of the shape of the function we are looking for. The following figure shows how alpha(R) heavily resembles a line, so we went for the linear option (beta slope = 0.103562).

- sigma(R): Because of the way that we defined our bacterial states, there is no way that there are intermediate states between our activated and stimulated populations. With this in mindo we decided that the stimulation transition state was to be described with a heaviside step function. This function's value is zero until a certain criterion is met. In our case, that is that the R value surpasses a given threshold. Since we were not able to measure how much chitin in a Coffee Rust sample, we decided to transform our R function to a chitin function. This should not be a problem since their relationship should behave linearly. As a way to define an Rnot, that is, the chitin threshold for successful stimulation, we went back to our molecular mathematical model and plotted chitin concentration against salycilic acid. The chitin concentration that gave us half the maximum production of salycilic acid would be the value chosen for Rnot. We successfully estimated Rnot = 0.19124 mM from the following figure.

References

1. Balaban, N. Q., Merrin, J., Chait, R., Kowalik, L., & Leibler, S. (2004). Bacterial persistence as a phenotypic switch. Science (New York, N.Y.), 305(5690), 1622–5. doi:10.1126/science.1099390