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Model

Abstract

Verticillium dahliae is a parasitic, filamentous fungus that affects more than 300 species, cotton being one of the most commonly affected.1 It attacks the plant’s vascular system, using it to spread through the whole plant. After a while, the mycelium clogs the vascular system and causes the whole plant to wilt. This is known as Verticillium wilt, and it is responsible for annual losses from 0.5% up to 5% in some states of the U.S.2

Our project’s purpose is to create a product using three different peptides (WAMP1b, AtPFN1, and PsDef1) that attack V. dahliae at different stages of its life cycle as an alternative to the current roundabout methods used to treat this fungus, since there is no fungicide produced specifically for V. dahliae. Those methods result harmful to the plants that they are trying to protect, the soil beneath them, and the land around them.3

Since our end goal is to develop a product, we needed to know the optimal concentrations of our peptides, since an oversaturated concentration of peptides would be costly and potentially detrimental to plant health, and and insufficient concentration could do no effect on fungal growth inhibition. From this problem, we designed our mathematical model.

Using a 96 well protocol and Gompertz equation, we were able to plot V. dahliae growth curve to determine its maximal growth rate and lag time. We applied a similar methodology to to perform an inhibition assay with a known antifungal agent in order to develop the steps necessary to determine IC50. This process will be applied to our peptides once they are purified to estimate the optimal concentration in our product to reduce costs and assure effectiveness.

Our Model

Our model seeks to find an optimal concentration of our peptides to effectively and efficiently inhibit V. dahliae’s growth. To do this we first needed information on our fungus’s growth kinetics. However, not much has been documented about V. dahliae’s growth. Since we needed that information, we first had to define V. dahliae’s growth curve. Once we had that data, we could differentiate it to its growth with different concentrations of our peptides. After that evaluation of Relative IC50 of each of our peptides was planned to get an adequate concentration for WAMP1b, AtPFN1, and PsDef1. Due to the complexity of the project and the limited amount of time, we knew that purification of our peptides was possibly unattainable. Therefore, we designed a methodology to generate a model that could work as the basis for the determination of the optimal concentration of our peptides. To do so we decided to use a general fungicide with benzimidazole as its active ingredient.

This gives our model two goals: Modelling the growth of Verticillium dahliae, and Modelling benzimidazole inhibition.

Modelling the growth of Verticillium dahliae

There are many different ways to model the growth of an organism. A popular method and the fist one we found was Monod’s Equation . It expresses rate of growth (μ) as a function of substrate concentration (S) in which max states that there is a point where growth stops accelerating and reaches a maximum growth speed and a point midway through the exponential growth phase (KS).4 Though the equation can be applied to growth models, we realized it was too simple for what we needed, and the concentration of growth substrate wasn’t relevant for our experiment. Thus, we started searching for a more specific equation that could describe fungal growth.

We found many morphologically structured models that had parameters representing different morphological properties of a fungus like average hyphae length, hyphae count, diameter. However, we discovered that rate of growth from dry mass was one of the most commonly used techniques for measuring fungal growth in these models, but we realized that this methodology although accurate, is tedious, time consuming and was not applicable to large-scale, multivariable, multireplicate experimentation 5 and since our sample size would be too small to measure with traditional methods, obtaining that parameter would also be impossible.

Considering that we couldn’t measure relevant parameters directly, our Mathematical Model advisor suggested an abstraction of data. Instead of measuring any morphological parameters, we could measure optical density with a spectrophotometric assay. Also using microtiter plates in combination with a microplate reader is more efficient for growth experiments with filamentous fungi because it is extremely fast, reliable, and requires as little as 75 ~ul of total culture volume. Furthermore, this measurement helps with screening for antifungal agents, antifungal drug susceptibility testing, and physiological growth experiments.6

Taking into account that the best idea was to work with optical density, we decided to use spore suspension as inoculum due to the relation that culture absorbance has with fungal biomass and also that relative standard deviation of absorbance measurements are usually lower when spores are used as inoculum.7

After electing the way we would estimate our data and while trying to come up with a new equation to fit them, we found a study where many models that worked with OD were compared, one of them seemed to be our best bet for our own implementation, the Gompertz Equation . 8

Gompertz Equation

The Gompertz Growth Curve was first published in 1825 in the Philosophical Transactions of the Royal Society. Although its original purpose was to detail a law of human mortality where a man’s resistance to death diminished as time increased, it has since been used as a growth curve for biological and economic phenomena9.

The equation in its most general form is:

Where:
y: Absorbance (OD)
y0: Initial Absorbance (OD)
C: Range (ymax - y0)
μ: Maximum Growth Rate (OD/h) = (dy/dt = 0)
λ: Lag Time (h) = (d2/dt=0)
t: Time (h)

Fitting Gompertz to V. dahliae

To find our parameters we first needed to plot V. dahliae’s growth curve, so we designed a methodology to measure optical density in a 96 well plate. Six wells were filled with 100 μL of a spore suspension and 100 μL of 2x liquid media to complete a final volume of 200 μL. A final concentration of 2x104 spores/mL per well was assured by counting spores in a haemocytometer chamber and taking the necessary dilutions into consideration. Plates were incubated at 25°C and absorbance readings were performed at a wavelength of 405 nm on a Varioskan Lux 3020-231 microplate reader twice a day for 5 days. For more information visit our Experiments page.

After measuring our fungus’ growths, we had to fit the equation to our data. We could easily determine two of our parameters since they were our first reading (y0) and our range (C). However, we were still missing Maximum Growth Rate (μ) and Lag Time (λ). To find these parameters, we developed a MATLAB program that would take two initial estimates (we could infer these by looking at the data points), graphing the equation, and solving by a least squares regression method. After running the program a few times, changing our estimates with what the program returned, we were able to determine our growth curve:

In Figure 1, we can appreciate the growth curve of V. dahliae. Each point represents one of the ten absorbance measurements done through five days. The curve presents the characteristic sigmoidal shape of microbial growth where the lag, exponential, and stationary phases can be clearly visualized. As determined by the Gompertz equation, the lag phase λ was calculated to be 33.467 h and the maximum growth rate μ was calculated to be 0.104 OD/h. These points can be observed on the graph, and they represent the first inflection point, and the point of steepest incline, respectively.

Figure 1. Verticillium dahliae growth curve in OD over time.

Inhibition Experiment Design

As previously mentioned, our end goal is to develop a product with optimal concentrations of our peptides. Therefore as a proof of concept and to develop a more complex mathematical model that would allow us to mathematically analyze the inhibition of each of our peptides in a near future, a growth inhibition assay against V. dahliae was carried out using a chemical with known activity against filamentous fungi. Its active ingredient is benzimidazole. An initial antifungal solution was prepared with a concentration of 250 μg/mL, as recommended by the provider in order to produce complete inhibition, and nine further 1:2 serial dilutions were prepared. Solutions were transferred to a 96 well plate, as well as a spore suspension of V. dahliae and liquid media to complete a final volume of 200 μL. Conditions used in the growth curve protocol were replicated; therefore, six replicates were made for each dilution, a final concentration of 2x104 spores/mL was used, plates were incubated at 25°C, and absorbance readings at 405 nm on a Varioskan Lux 3020-231 microplate reader were performed twice a day for five days. For more information about this protocol visit our Experiments page.

Determining Inhibition

To calculate our inhibition the only parameter used was maximum velocity (μ), so we compared the velocity from the normal growth of Verticillium dahliae against the ones with different concentrations of antifungal agent to get inhibition percentage with the following formula:

Where:
I: Inhibition percentage (%)
μp: Maximum Growth Rate with inhibitor (OD/h)
μVd: Maximum Growth Rate of free Verticillium dahliae (OD/h)

Determining IC50

During our research we found that when optimal inhibition was required, a measure of effectiveness called IC50 was commonly used. This is used to evaluate the performance of substances when working with inhibition. For our final equation we needed to determine relative IC50 which is the midway concentration from the most inhibition (upper plateaus) to the least inhibition (lower plateaus).10

The best way to determine IC50 is through a model that includes this resource as a parameter, for example the 4-parameter logistic model. This model is capable of describing the sigmoidal pattern generally seen in dose response and antimicrobial screening assays.10 Thus, we plotted the inhibition percentages with their respective concentration to adjust the following formula:

Where:
I: Inhibition percentage (%)
X: concentration (μg/mL)
a: Lowest inhibition or lower asymptote (%)
d: Highest inhibition or upper asymptote (%)
b: Steepness constant
C: Relative IC50

To find our missing our missing parameters, first we ran our Gompertz Optimization program for each dilution, got their respective μ’s and plotted them in respect to the inhibitor concentration. We plotted these data in order to analyze it, and yet again, we had two known parameters (a,d), our independent variable (X), and two unknown parameters (b,c). Since the problem was the same, we did the same procedure we had done to find Gompertz’s μ and λ, changing the equation from the optimization program for our inhibition equation (4). We did this for a few cycles and were able to determine our missing parameters and insert them into our equation:

Analyzing our IC50

To plot our data, we adjusted the X-axis to a logarithmic scale in order to accurately represent our data since it would be too steep otherwise. Once we had found our inhibition parameters, we saw that our IC50 = 13.921 μg/mL. The model worked as expected seeing that the closest measured concentration to produce an inhibition close to 50% was 15.6250 μg/mL with a 51.1795% of inhibition. This point can be visualized in Figure 2 close to the IC50. It is clear that little benzimidazole is needed to inhibit V. dahliae’s growth effectively, compared to the original concentration of 250 μg/mL recommended by the supplier. This exemplifies the practicality of this model to estimate a proper and efficient inhibition concentration.

Figure 2. Benzimidazole inhibition percentage curve.

Conclusion

Our experiments proved the idea we had developed about our model. They served as a proof of concept on how we would determine the IC50 for our peptides once they are purified and extracted. This would be very valuable on other parts of the project, such as Entrepreneurship to reduce production costs and optimize concentrations. Also, determining V. dahliae’s growth kinetics allows us to get a deeper understanding on how it grows, and could lead to other applications for its control.

References

  1. Farr, D., & Rossman.(2014). Fungal Databases, Systemic Mycology and Microbiology Laboratory Retrieved from: http://nt.ars-grin.gov/fungaldatabase
  2. Land, C., Lawrence, K., & Newman, M. (2016). First Report of Verticillium dahliae on cotton in Alabama. APS Publications. https://doi.org/10.1094/PDIS-10-15-1143-PDN

  3. FAO. (2004) Manejo de malezas para países en desarrollo. Retrieved from: http://www.fao.org/3/y5031s/y5031s00.htm#Contents

  4. Cunningham, A., Lennox, J. & Ross, R. (2010). Microbial growth. Montana State University.. Retrieved from: https://www.cs.montana.edu/webworks/projects/stevesbook/contents/chapters/chapter002/section002/black/page001.html

  5. Granade, T. C., Hehmann, M. F., & Artis, W. M. (1985). . Monitoring of filamentous fungal growth by in situ microspectrophotometry, fragmented mycelium absorbance density, and 14C incorporation: alternatives to mycelial dry weight.. Appl. Environ. Microbiol., doi: 49(1), 101-108.

  6. Langvad, F. (1999). An automated quantitative assay for fungal growth inhibition.. doi:10.1016/s0167-7012(99)00053-6

  7. Broekaert, W. (1990). A rapid and efficient method for growth measurement of filamentous fungi. Journal of Microbiological Methods, 37(1), 97–100... FEMS Microbiology Letters, 69(1-2), 55–59. doi:10.1016/0378-1097(90)90412-j

  8. María-Leonor Pla, Sandra Oltra, María-Dolores Esteban, Santiago Andreu, and Alfredo Palop. (2015). Comparison of Primary Models to Predict Microbial Growth by the Plate Count and Absorbance Methods.. BioMed Research International.doi.org/10.1155/2015/365025

  9. Winsor, C. (1932) The Gompertz Curve as a Growth Curve. Proceedings of the National Academy of Sciences of the United States of America, 1–8.. doi:10.1073/pnas.18.1.1

  10. Sebaugh, J. L. (2011). Guidelines for accurate EC50/IC50 estimation. Pharmaceutical Statistics, 10(2), 128–134.. doi:10.1002/pst.426

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