Population Modeling / Enzyme modeling

A large part of our project was modeling the effectiveness of our product

Modelling helps in the interpretation, prediction, and improvement of scientific research. For our project, we use biological modelling to create a theoretical impact and validate our research with the help of interactive bee population and kinetic modelling. Should you want to adapt these model to your own projects you're entirely free to do so, all the code needed to run the model is available for download right on the webpage, right click, inspect element and have fun editing!

Honey bees are social insects that makeup colonies with a single queen bee that lays thousands of eggs, fertile males and sterile adult females that work in different sectors of bee hive maintenance, brood rearing, foraging, etc.

Many factors are linked to the population decline of the bee colony like change in climate, agrochemicals, loss of habitat, and nutritional stressors, still, it remains unclear how the entire adult population declines within such a short period. One of the primary reasons for this is a decline in the forager population that leads to the dramatic collapse of the hives. 1 The death of foragers stresses hive dynamics in many ways 1) loss of food supply required for sustenance 2) precocious foraging that increases the risk of death among the forager population 3) unstable division of labour among the hive population. The additive effects of all these factors lead to colony collapse. The maintenance of the hive under such stressful conditions becomes difficult as the observation period between a healthy state and a hive collapse is very short.

Aims

The model is an extension of our project 2 and explores the colony dynamic in a bee hive by simulating a population of bees in a steady state. With an average egg laying rate by the queen bee in the hive we can predict the number of bees in the upcoming 55 days (a full life cycle of honey bees). Moreover, the model predicts if the bee hive can survive in these 55 days by taking the forager population, stored bee food content and the effects of pesticide, Glyphosate, when it comes in direct contact with forager bees. This model aims to provide a possible instability of the hive and gives a margin of possible mitigation thus saving the colony from collapse.

Model structure

The model is based on the steady-state model that used geometrical equations simulating different classes of the bee hive. E, L, P, H, and F0 represent the all classes in the bee hive, i.e., eggs, larvae, pupae, hive bees and foragers respectively. Sn and Nn refer to survivability rates and the number of days spend in each hive class respectively and n refers to the different hive classes. For example, survival rates and the number of days spent by each bee caste in each hive class3 is a reference value that is given by SE i.e., the survival rate of eggs and NE is the number of days spent as an egg.

Result and analysis

Dependence on Food Supply for Hive Sustenance

The food supply that is restored by foragers is one of the major factors that stabilises the hive and with a declining population in the forager class, it is severely affected. In the case studies, we provide some ‘sensitive zones’ that are zones of possible hive collapse. The X-axis represents the number of days in the life cycle of bees and Y-axis represents the food supply and consumption of the hive.

Scenario 1: Abundant Food

The supply of food in general declines as the bee population increases with time. But as the foragers and hive bee population also decline, the consumption of food also decreases. Even when the pesticides are applied, the consumption is always less than the supply, proving that hive will be stabilised even if the pesticides are applied.

Scenario 2: Adequate Food

The general trend remains the same as the previous scenario, but due to less stored food and constant forager decline the hive barely manages to survive. On taking the effects of pesticides into account that leads to forager population decline, even more, the consumption surpasses the supply, and that can be considered a sensitive zone.

Scenario 3: Food Shortage

The food shortage is considered when the consumption is more than the supply from the start. When the effects of pesticides are taken into consideration, the collapse is observed even earlier. The earlier collapse does not allow the development of the hive and stability cannot be achieved.

The winter to summer transition months are very sensitive in terms of food supply as most of the stored food is used up during this period. If the forager decline is more during this period the hive might not be able to sustain the pressure and collapse. Hence there is a need for an alternative method to protect the bees (specially foragers bees) from the effects of pesticides and preserve the forager population. The pollen substitutes used widely come with huge amounts of side effects and create stress on the bee population that might add on to the population decline, and hence pollen substitutes are not a permanent solution to bee population decline. 4 The probiotic is an easy and effective way to deal with the constant declining rates of the forager bees.

Contribution to society and future iGEM teams

iGEM teams

    Society

      Methods

      Equations

      The geometric equations give the distribution of bees in all the bee castes, creating a steady state model. Many models use equations like these to characterise the survival rates of the bees. Here E, L, P, H, F0 represent the total number of bees stages that will be observed over 55 days.

      $$ E = E_{0}\cdot {\frac{1-SE^{NE}}{1-SE}} $$ $$ L = E_{0}\cdot SE^{NE}\cdot{\frac{1-SL^{NL}}{1-SL}} $$ $$ P = E_{0}\cdot SE^{NE} \cdot SL^{NL}\cdot{\frac{1-SP^{NP}}{1-SP}} $$ $$ H = E_{0}\cdot SE^{NE} \cdot SL^{NL} \cdot SP^{NP}\cdot{\frac{1-SH^{NH}}{1-SH}} $$ $$ F_{0} = E_{0}\cdot SE^{NE} \cdot SL^{NL} \cdot SP^{NP} \cdot SH^{NH} \cdot{\frac{1-SF^{NF}}{1-SF}} $$

      Assumptions

        Parameters

        Default values are taken reference from Schmickl et al.3.

        Parameter Description Default Values Units
        E0 Initial egg count 10,000
        NE Number of days spent in egg stage 5 days
        SE Survival rate of eggs 0.97
        E Total eggs -
        NL Number of days spent in larvae stage 5 days
        SL Survival rate of larvae 0.99
        L Total larvae -
        NP Number of days spent in pupae stage 12 days
        SP Survival rate of pupae 0.99
        P Total pupae -
        NH Number of days spent in hive stage 21 days
        SH Survival rate of hive bees 0.98
        H Total hive bees -
        NF Number of days spent in forager stage 14 days
        SF Survival rate of foragers 0.95
        F0 Total Foragers -
        CL Food consumption of larvae 0.018 g/day
        CH Food consumption of hive bees 0.007 g/day
        CF Food consumption of foragers 0.007 g/day
        S Stored food 20 g/day
        SFP Survival rate of foragers under influence of pesticide 0.90
        F0P Number of foragers with pesticide effects -
        supply Food supply in the hive - g/day
        consumption Consumption of food by all hive castes - g/day
        supplyP supply of food with pesticide - g/day
        consumptionP Consumption of food by all hive castes with pesticide - g/day

        E0: Eggs laid by queen bee
        (initial egg count)

        SE: Survival rate of eggs

        SL: Survival rate of larvae

        SP: Survival rate of Pupae

        SH: Survival rate of Hivebees

        SF: Survival rate of foragers

        SFP: Survival rate of foragers under pesticide

        S: Stored food in the hive

        The graph includes two scenarios: effect on food supply without pesticides and with pesticides. With the use of pesticides, the supply decreases drastically as the forager population responsible for supply of food in hive decreases but there are almost no observable changes in the consumption. The reason being that foragers are responsible for most of the supply of food in hive but consume a very less part of it in comparison. As the supply is more when there is no pesticide usage, the hive is stable for longer period as compared to case when pesticide is used. Whenever the consumption of food goes beyond the supply, instability in hive is predicted.

        References

        1. Kuan AC, DeGrandi-Hoffman G, Curry RJ, Garber KV, Kanarek AR, Snyder MN, Wolfe KL, Purucker ST.
          2018,May. Sensitivity analyses for simulating pesticide impacts on honey bee colonies (Ecol Modell. 2018 May 24;376:15-27. ).
          Full text
        2. Torres DJ, Ricoy UM, Roybal S .
          (2015). Modeling Honey Bee Populations. PLOS ONE 10(7): e0130966.(7567), 68–72.
          Full text
        3. Thomas Schmickl, Karl Crailsheim
          2007 HoPoMo: A model of honeybee intracolonial population dynamics and resource management,Ecological Modelling,Volume 204, Issues 1–2, (US5463175A).
          Full text
        4. Emily R. Noordyke and James D. Ellis
          November 2021 Reviewing the Efficacy of Pollen Substitutes as a Management Tool for Improving the Health and Productivity of Western Honey Bee (Apis mellifera) Colonies
          Full text

        With our kinetic model, we want to describe the degradation of Glyphosate over time depending on the kinetic properties of the Enzymes and concentrations used. Glyphosate oxidoreductase (goxA and goxB) catalyses the following reaction:

        $$ 2\ Glyphosate + O_{2} \rightarrow 2\ Aminomethylphosphonate\ +\ 2\ glyoxylate $$

        The CP-Lyase catalyses a different reaction as it cleaves the C-P bond.

        $$ Glyphosate + H_{2}O \rightarrow Sarcosine\ +\ P_{i} $$

        The kinetics of the latter reaction is not very well characterised as it likely involves multiple steps in the catalytic subunits of the C-P-Lyase1. GoxA and goxB share similarities with glycin oxidoreductases and have been described to have a low substrate specificity for Glycin or Glyphosate respectively2. Hence, we assume the concentration of Glyphosate is the limiting factor for the reaction kinetics and consider it a first-order reaction. The change of glyphosate over time thus can be described by Michaelis-Menten kinetics as follows.

        $$ \frac{dc}{dt} = \ -v_{max} \frac{c}{K_{m}+c} $$

        The concentration of Glyphosate c is described by the maximum rate the system can achieve Vmax, the Michaelis constant Km and the time t. This integration leads to a form where the concentration can be determined at any given time point knowing the kinetic parameters.

        $$ c = K_{m}\cdot W(\frac{c_{0}\cdot e^{(\frac{c_{0}}{K_{m}} - \frac{t\cdot v_{max}}{K_{m}})}}{K_{m}}) $$

        Knowing the kinetic parameters of the optimised goxA being v_max = 0.6 mmol/s and K_m = 2.6 mmol/L2 the concentration of glyphosate over time can be calculated as seen in the interactive tool below. From experimental data obtained in the lab, the unknown kinetic parameters of the degrading enzyme could be determined. For that, the concentration of Glyphosate at different time points as well as the concentration of Cells at those times to estimate a growth curve is needed. Then the best-fit algorithm can be used to determine the corresponding kinetic parameters.

        In the following you will find an interactive tool where you are able to modify input values like starting concentration of glyphosate and cell biomass as well as the kinetic parameters vmax and Km. The last parameter gives the intervall in minutes in which you want to see the degradation. For the amount of active protein we assumed that 2% of the total biomass of our probiotic will be glyphosateoxidoreductase.

        Glyphosate start concentration
        (mM : mg/L)

        Km (mmol/L)

        Vmax(U/mg)

        Biomass (g/L)

        Time intervall (min)

        References

        1. Seweryn, P., Van, L. B., Kjeldgaard, M., Russo, C. J., Passmore, L. A., Hove-Jensen, B., Jochimsen, B., & Brodersen, D. E.
          (2015, August 17). Structural insights into the bacterial carbon–phosphorus lyase machinery. Nature, 525(7567), 68–72.
          Full text
        2. Barry, G. F., & Kishore, G. M.
          (1995, October 31). GLYPHOSATE TOLERANT PLANTS (US5463175A).
          Full text