Population Dynamics

For the final product, BeeT, we intend to use toxins produced by Bacillus thuringiensis, also called Cry toxins. However, these toxins are also harmful to our chassis, resulting in a reduction of toxin production when it is needed to kill the mites. To counteract this effect we envision the use of quorum sensing that activates Cry toxin production only when there is a large quantity of BeeT present. Ideally, BeeT is able to produce Cry toxin over a long period to improve its effectiveness against mites. However, if an entire population of BeeT is synchronized, we hypothesize that only a single burst of Cry toxin will take place before both the BeeT and mites are killed. Thus, this system will not be maximally effective over long time periods. To accomplish this we need multiple subpopulations of BeeT: some producing BeeT while others are recuperating. To help understand this complex system we use dynamic modelling.

What is Quorum Sensing?

Quorum sensing is a cell-cell communication system. The detection of chemical molecules allows the bacteria to sense the presence of other bacteria. In this way the bacteria control gene expression in response to changes in cell number ^1,4. This process is achieved through the production and release of an Acylated Homoserine Lactone (AHL) autoinducer An autoinducer is a molecule that can diffuse through the cell membrane. . The AHL can diffuse from one cell to the other. There are many different types of autoinducers in quorum sensing systems. When sensed, the autoinducer can trigger other cells to produce more autoinducers.

Genetic circuit

The quorum sensing system consists of two proteins: LuxI and LuxR. LuxR is constitutively expressed, together with AHL it forms a complex which activates the transcription of a toxin. In our case we use the detection protein Green Fluorescence Protein (GFP) to follow the behaviour of the system ². LuxI encodes an AHL synthase. AHL is a molecule that can diffuse freely through the cell membrane, and in this way travels from cell to cell. LuxR encodes for a protein and together they form a complex. In a natural system the LuxR-AHL complex controls transcription of LuxI, forming a positive feedback loop that increases the amount of AHL in the system ³. When there is more AHL in the system, AHL is more likely to bind to the LuxR protein. In Figure 1 you can see a schematic representation of the system.

According to the 2011 team from Davidson College and Missouri Western State University⁵, LuxR protein represses transcription of the luxR gene in the absence of AHL, establishing a negative feedback. We investigated whether this negative feedback can be used to create different populations. As you can see in Figure 2 different states of GFP responses can occur when the production rate of luxR and rate of complex formation are changed. However, when we remove the feedback in the system and change the same parameters, we get similar responses of the system as shown in Figure 3.

This shows that the feedback has a minimal influence on the system to create different subpopulations. Also when there is no negative feedback in the system, we found more parameter sets that give a high amount of GFP (see Figures 4 and 5).

The figures are simulated with the parameter sets obtained from the quorum sensing model. We are using the parameter sets that give the best response according to the confidence intervals (see Method section). In the presence and absence of negative feedback, the system behaves similarly by assuming the production rate of luxR and the formation rate of the complex are different in each cell. The quorum sensing system can simulate different responses with the same cells and different parameters, if we could control the LuxR production rate. However, we could not envision a method of doing this experimentally. We aim to make cells that are genetically identical but still can respond differently. Therefore, we designed another system that could help us create this response downstream of the quorum sensing mechanism. The two systems were tested in the lab parallel with the modeling.

Why include a subpopulation system?

Since the quorum sensing mechanism could not provide the subpopulations we desired using genetically identical cells, we created a system that could act downstream of quorum signalling to provide the response we needed. Population-wide Cry toxin overexpression if likely to kill all E. coli cells in the first moment of toxin expression. Since this would only affect one of the subpopulations, the second subpopulation would be able to initiate a new growth phase after death of the toxin-producing cells. The critical requirement for this is that cells respond at different times to the quorum stimuli despite being genetically identical. A small part of the population that acts differently from the rest of the population is called a subpopulation. The subpopulation system consist of two genes; the first encodes for the protein that inhibits the systems expression, the other encodes for the corresponding activation protein of the system. The 434-cI-LVA inactivates the λ-cI directly, or prevents the translation of the λ-cI protein. 434-cI-LVA has a higher turnover rate than λ-cI. This subpopulation system is based on a natural system for persister cell formation. ⁶.

As you can see in Figure 6, glucose has a suppressing function on the system. Arabinose has a activating function on the system. We used the model to predict what will happen when we add glucose in different amounts. The assumptions we made for the subpopulation system are the following; 434-cI-LVA degrades faster than λ-cI, cell division happens when the cell size has doubled, and two cell populations exist with one growing differently than another. We tested a number of parameter sets and picked the parameter set that gave the biggest RFP response based on the strength of glucose. To see how the model behaved when we changed the ratios of the initial amounts of λ-cI and 434-cI-LVA we investigated the effect of starting conditions on the outcome of the system. The initial conditions were varied between 1 and 10. Within the heat map you can see in which ratios the initial amounts of λ-cI and 434-cI-LVA in the system are needed to get high RFP production (here glucose and arabinose have a fixed concentration).

Shown in Figure 7 when λ-cI and 434-cI-LVA are present in similar amounts we have a subpopulation system. But if they are present in different amounts than there will be no subpopulations. This can be expected when you look at the subpopulation system. The system is inhibited by 434-cI-LVA, which represses the RFP production, and λ-cI activates the RFP production. In figure 4 you can see balance between the 434-cI-LVA and λ-cI amounts that are present for the output of RFP. This means that the initial conditions do not have much influence on the λ-cI and 434-cI-LVA. With this particular parameter set, we can conclude that the translation rates are more important for the RFP response than the initial conditions. For other parameter sets the initial conditions are important. To understand how changing the Ribosomal Binding Site (RBS) impacts RFP responses, we simulated the system for different combinations of transcription rates. From these simulations we found:

To confirm that our model represents Biological behaviour we analysed the RBS library. We can conclude out of these results that the RFP quantitatively corresponds to the Figure 7.

Combined system

Based on the modeling we hypothesized the following: When there are more cells are present in the system, more AHL-LuxR complex is formed. The complex inhibits the subpopulation promoter. When the promoter is inhibited production of 434 will be suppressed and production of λ cl will be activated. At a certain time point λ-cl takes control over the system, because 434 has a higher turnover rate than λ-cl. In this case more λ-cl results in more RFP.

In a later stage, to get the desired response, the quorum sensing system and the subpopulation system were combined. As shown in Figure 8, you can see how we think to generate different cell populations. With this extended part we try to generate a system which predicts the behaviour of the subpopulations. In figure 9 you can see the increasing RFP over time after many cell divisions, indicating an increasing cell population.

Methods

During the research Matlab version R2016a has been used.
Because there was no data from the wet lab we assumed that all parameters exist within a biologically reasonable range numbers between 0 and 1. For the analyses we used 100.000 parameter sets, 10 cells, random initial conditions and 60 timesteps. Tuneable parameters are used, each parameter set can produce dramatically different population dynamics. To determine which of these parameters produce the best system response we used Latin Hypercube Latin hypercube is a statistical method to get random numbers from a box of x by n numbers. For example, if x = 4, where x is the number of divisions within the parameter value range, and n = 2, where n is number of parameters, you will obtain a box with 4 square times 2 square, giving you 24 random numbers. Within each division of the parameter space a single random number is chosen. sampling.

With the parameter numbers of the Latin Hypercube sampling we made parameter sets that are obtained from a lognormal distribution a lognormal distribution assigns probabilities to all positive values. However the distributions is skewed to favour smaller values. Biological reaction rates have been found to follow this distribution 8 with a parameter estimation based on Raue et al ⁷. Below you could see an example;

With these confidence intervals Equation for confidence interval used the best parameter sets could be chosen.

In conclusion

Quorum sensing ensures that the toxin is only produced when the density of bacteria is high enough, this standardizes the amount of toxin produced by the bacteria population. The subpopulation system was coupled to the quorum sensing system. Together, quorum sensing and formation of non-producing subpopulations allow bacteria to produce waves of toxin.
With the results of the combined system we can conclude that waves of toxins are produced. Different parameters in the quorum sensing system elicit different GFP responses. However, we can not expect to be able to make genetically identical cells with varying values for these parameters with the subpopulation system we may be able to predict which RBSs of the library have the highest chance to successfully create subpopulations. The library in further research it could be tested in the lab.

Optogenetic Kill Switch Model

One important aspect of our project is the biocontainment of BeeT to the hive, preventing it from spreading into the environment. In collaboration with the wet-lab, we designed an optogenetic kill switch based on the blue light sensing pDusk and pDawn systems as described by Ohlendorf et al.¹. The cell death is inflicted by the mazEF toxin-antitoxin (TA) system. The latter functions by the antitoxin (mazE) preventing the toxin (mazF) from cleaving mRNAs which ultimately kills the cell. With the underlying biology of the system in mind, we used synthetic bioengineering principles to mathematically describe our design and explore the dynamics of the system.
Our results show that we have a working mathematical model which describes the data of Ohlendorf et al.¹. With this model, we now know where to look for the relative kinetic rates in the parameter space to give suggestions to the wet-lab on how to build a functioning optogenetic kill switch. Our differential equations of pDusk and pDawn can also be used by future iGEM teams to mathematically describe and build new optogenetic tools.

Light On

Modelling The Optogenetic Tool

For the first part of our analysis, we optimized the mathematical models of pDusk and pDawn using published data¹. For now, we ignored the toxin-antitoxin system. This gave us the option to construct our scoring function to evaluate both systems simultaneously with a weighted means approach.
By constructing an accurate model of the optogenetic part of our overall kill switch system, we are confident that we are mathematically representing the behaviour of the microorganism in the lab. Results from the model can then be used as a reference point for building the toxin-antitoxin system into the kill switch model. We will search the parameter space to identify parameter sets which fit the data given by Ohlendorf et al.¹. In addition, we will select the best fit to data based on a scoring function as described by Raue et al.⁴. Lastly, we provide a mathematical model which can be used by future iGEM teams to build new optogenetic tools.

System Design

We first studied the pDusk system in more detail to derive simplified equations which can describe the system behaviour. The genetic switches are based on the interactions between the light sensing protein YF1 and the response regulator FixJ as described by Möglich et al.². In darkness, YF1 phosphorylates FixJ, which in turn activates the promoter pFixK₂. This activation leads to the expression of the desired genes. Under light conditions, YF1 enters an excited light-state, which inhibits the phosphorylation of FixJ. In addition, it prevents the activation of the pFixK₂ promoter and thus represses the target genes in Figure 11 for pDusk.

The pDawn system functions in the same way as the pDusk system, with the addition of cI_λ being expressed depending on the activity of the pFixK₂ promoter. The protein form of cI_λ inhibits the expression of the RFP reporter gene in Figure 11 for pDawn.

Assumptions

To derive mathematical equations describing the above systems, we made some simplifying assumptions.

• We neglect the effects of dilution/ growth. We thus assume the cell’s volume to be constant over time.

• We assumed that the components in our system are neither affected by nor affect other cellular mechanisms.

• As there is no additional information gained from modelling transcription and translation for the dark state of Yf1(y_DD) and the inactive form of FixJ (j_i) we lumped these processes and assumed that they can be described by a single parameter.

• We also assumed the rates of production, binding and degradation of mRNAs and proteins to be linear which resulted in a first order differential equations system

To model the influence of light on the system, we assumed light activation to occur similar to the model of phytochrome B dimerization described in Klose et al.³ ($N\cdot k_2$). In addition, we consider in our system the different stages of Yf1 (y_DD, y_DL/LD, and y_LL) as described by Möglich et al.². Translated to our system, this means that with increasing light intensities (N), the pool of activated Yf1 (y_DL/LD and y_LL) would deplete the dark state of Yf1 (y_DD), which is responsible for the phosphorylation and thus activation from inactive FixJ (j_i) to active FixJ (j_a). As all stages of Yf1 form a dimer, we assumed dimerization to occur quickly and be the dominant and relevant form for our system. Lastly, we assumed a quasi steady-state approximation (Michaelis-Menten kinetics) for the effect of activated FixJ (j_a) on the expression downstream of the RFP (RFP_m) reporter gene. A detailed system design is provided here.

Results

With our mathematical model we can describe the behaviour, as shown in Figure 12, of both systems as tested in the lab by Ohlendorf et al.¹. We can thus say that we built a simplified mathematical representation of the pDusk and pDawn system which can describe the systems’ behaviour.

Preliminary results from the lab show that the RFP_p response after 17 hours incubation time is weak using the pDusk system. The experiment was conducted in darkness and under $1.41\cdot {10}^4[\mu mol\cdot m^{-2}\cdot h^{-1}]$ light intensity and the results are shown in Table 2.

The data from the lab suggests that there is not a high change in RFP for the pDusk system between dark and light conditions. When plotting the computed response of RFP for increasing light intensity in Figure 13 we see, however, that pDusk has a steep decline in RFP levels from complete darkness to very low light intensities. There could be two potential reasons for such a result. First, despite all efforts in creating dark conditions, the experiment could have been contaminated with low light intensities. This would explain why there is no significant change visible in the experimental data visible between different conditions. To confirm this, additional experiments would need to be conducted. Additionally, a more in-depth parameter sensitivity and robustness analysis could also reveal potential causes of the discrepancies between model simulations and experiments conducted under darkness.

Methods

After studying the pDusk and pDawn system, we constructed a parameter estimation procedure by picking random parameter sets based on the latin hypercubeLatin hypercube sampling is a statistical method to sample the parameter space equally. It divides the parameter space according to the number of parameters and sample size and picks a random number from each parameter subspace. sampling (LHS) method in Matlab 2015b (used for the whole study). We simulated 10,000 parameter sets and assumed the initial conditions of only y_DD and j_a being in the on state, i.e. we assigned the value one. All other initial conditions were kept at 0. In addition we needed to transform those parameters to a more biologically feasible distribution as LHS generates uniformly distributed parameters. For those parameters where there was no information available, we assumed a lognormal distribution that is representative of measured biological rates⁷. We transformed the two parameters (k₂ and k₃) for which experimental data were available according to their known distribution. We then loaded the RFP response data ± standard deviation from Ohlendorf et al.¹ for both pDusk and pDawn and transformed the light intensities from $[\mu W\cdot c{m}^{-2}]$ to $[\mu \mathrm{mol}\cdot m^{-2}\cdot h^{-1}]$. For a more detailed explanation see the notebook.

We then simulated both systems according to our ordinary differential equations (ODEs) with the same parameter (sub)-sets and stored the response at 17 hours for later evaluation in the scoring function. We chose to simulate the system until 17 hours, as this is the duration Ohlendorf et al.¹ used in their study. As a next step, we normalised the computed RFP_p response with the measured response at the highest light intensity used by Ohlendorf et al.¹.

To evaluate how well a certain parameter set describes the response of the Ohlendorf et al.¹ system, we used the sum of squared residuals to score each parameter set as described in Raue et al.⁴.

${\chi }^2\left(\theta \right)=\sum _{k=1}^m\sum _{l=1}^d{\left(\frac{y_{kl}^d-y_k(\theta ,t_l)}{{\sigma }_{kl}^d}\right)}^2$

where $y_{kl}^d$ represents $d$ data points for each observable $k$ at time points $t_l$. ${\sigma }_{kl}^d$ are the corresponding measurement errors and yk(θ,tl) the $k$^th observable as predicted by parameters $\theta$ for time point $t_l$.

To construct a corresponding score from the same parameter set for pDusk and pDawn we used the weighted means approach accordingly:

$\frac{{\zeta }_u+{\zeta }_a}{2}$

where ${\zeta }_u$ represents the score of a parameter set for pDusk and ${\zeta }_a$ the corresponding score of the same parameter set for pDawn. To put the equations and parameters into context, we created a more detailed graphical representation of the pDusk and pDawn system and provide the ODEs for both systems:

Light On, BeeT Off

Modelling The Optogenetic Kill Switch

After building the mathematical system for pDusk and pDawn, we were now able to simulate dark and light conditions. We have an understanding of how the optogenetic tool functions and can now explore the parameter space to find parameter sets which describe the wanted behaviour and regulation of the toxin-antitoxin system. As there was no experimental data available in literature to score the combined system, we used conservative constraints which are known from literature^{5 6} to construct our model for the MazE and MazF kinetics.
In the following the objective is to find sets of parameters which give a response within the conservative constraints and thus show us where in parameter space to search in further analysis to describe the toxin-antitoxin system. As the Biobrick BBa_K081007 is not only used in pDawn, but also in the quorum sensing system, we wanted to have a tool to decide whether we can also use the less sensitiveBased on data from Ohlendorf et al. (2012), pDusk shows a smaller change in RFP_p ratio than pDawn. For the kill switch this translates into a smaller change in antitoxin over toxin (A/T) ratio and thus a smaller likelihood to kill the cell effectively., but theoretically feasible pDusk system in the kill switch design.

System Design

To build the kill switch we first studied the mazEF system and its functionality. The mazEF module is an toxin-antitoxin (TA) system native in E. coli. For the purpose of building a mathematical model, we describe the relevant aspects of the system here. For more background on the mazEF system, please go to 5. The functionality of the system in nature is given by a fast degrading antitoxin and a slower degrading toxin. Once expression of the TA module is terminated, mazE will degrade faster and mazF will start cleaving mRNAs which ultimately leads to cell death. For the kill switch, we will consider either “under”-production of mazE in pDusk or overproduction of mazF in pDawn both under light inducing conditions as shown in Figure 14.

Constraints

For the parameter estimation of the optogenetic kill switch, all the assumptions mentioned in the above sections hold. Further, we assumed a lumped mazEF complex formation, where we disregard potential intermediate stages of complex formation. In addition, we defined the following constraints:

• As we have literature data supporting that mazF degrades faster than mazE⁵, we exploited this as a constraint in our parameter generation procedure.

• Another constraint was added to the system as the ratio of free A/T is assumed to be bigger than one⁶ under non-inducing conditions.

• Lastly, we added a logical constraint which describes the underlying biology of the system. Under the highest simulated light intensity, the A/T ratio has to be smaller than one to show that toxin wins over antitoxin and kills the cell.

Results

We can show that our parameter estimation procedure found two parameter sets, out of 1,000 sampled sets, which satisfy these constraints as shown in Figure 15 and 16.

The two parameter sets give us an indication where in parameter space we will have to focus on for further analysis. These two sets also both describe a functioning pDusk and pDawn based optogenetic kill switch. Although, one might argue, that set number one (blue and red lines) is to be favoured as the change in mazE in pDusk compared to the change of mazE of set two (yellow and purple lines) in pDusk is higher and thus more likely to function and kill the cell. With such information, an indication can be given on which relative kinetic parameters are needed to make the system work.

When plotting the A/T ratio against the increasing light intensities, it becomes more evident that parameter set 1 has a higher change in the A/T ratio within pDusk between different light intensity conditions and is thus to be slightly favoured. To further evaluate these findings, more data from the lab is needed.

Methods

Adding the mazEF module to the pDusk/ pDawn system requires another 10 parameters to describe the system. We constructed our parameter estimation similar to the parameter estimation of the previously described procedure and evaluated the system based on the constraints described above. It should be noticed here, that this is a qualitative evaluation and is not backed by statistical real world data. Nonetheless, it can be used as an indication.
We then took the best parameter set from the pDusk/ pDawn evaluation and added the equations describing the interactions with the MazEF system to the set of ODEs. As we noticed that we have to discard around half of the parameter sets as they do not satisfy the constraint that mazE degrades faster than mazF, we built this constraint already into the parameter sampling procedure. In addition, we considered the native mazEF system of E. coli. To prevent any potential interaction between the engineered and the native system, a mazEF deficient strain was created in the lab. This is, why we do not consider a native mazEF system in the model.

In Silico Kill Switch

With the above mentioned results, we conclude to have built a mathematical model of pDusk and pDawn which describes the system behaviour. Initial results from the lab, did not show a significant change in response for the pDusk system. However, the experimental results are in agreement with the computed data in the pDawn system. This leads us to suspect that either there was an irregularity with the dark conditions in the lab, or the model does not fully reflect the underlying biology of the system in the lower light intensity regions. We suggest that, in future, a parameter sensitivity and robustness analysis could be conducted to find the most likely cause of the mismatch to data. We further conclude, that we achieved a good fit in Figure 12 to the data available in literature ¹.

We then studied the mazEF toxin-antitoxin integration to design our optogenetic kill switch. After analysis of this system, we found two parameter sets which satisfy the constraints. These parameters give us an idea for how to search the parameter space for further analysis to select functioning kinetic rates in collaboration with the wet-lab.
In addition, future work could show the effect of parameter sensitivity such as leaky promoters and their effect on the system to engineer the pDusk system, adjusting it to suit our needs. We also made the code available for future iGEM teams with a head start to build optogenetic tools.

Metabolic Modeling

In order to assess the real world viability of BeeT, we evaluated the proposed system of application by making a model of the entire system. To do this we used Flux Balance Analysis (FBA) to model the chassis. The chassis The chassis-organism is the framework that is modified for use in synthetic biology experiments. of BeeT is a variant of ​ Escherichia coli, for which it is known that it does not grow overnight in high osmotic pressure environments of 1 mol sucrose / liter or above. ¹ Supplementing an Apis mellifera (honey bee) colony with sugar water is a well established practice amongst beekeepers. ⁴ This is usually done with about 1 kg of table sugar for each liter of table sugarChemically speaking this is pure sucrose: C₁₂H₂₂O₁₁. After heating and stirring, this ends up as a concentration of 625 grams of sucrose per liter of water. This concentration can also be defined as 1.82 mol sucrose per liter, which is almost twice the threshold value at which E. coli would no longer grow. However, in this project we only need the ​E. coli to survive in the sugar water. This only needs to be as long as the worker bees require to pick the BeeT-cells up and bring them to where the Varroa destructor are. The question we try to answer is: does our chassis survive in the sugar water, and if so, for how long?

A model of E. coli survival

In order to make this model a technique called Flux Balance Analysis (FBA) was used to describe the relationship between the metabolism of the E. coli and the osmotic pressure of the sugar water. FBA is a mathematical method that can be used for predicting reaction fluxes and optimal growth rates of species using genome-scale reconstructions of their metabolic networks. In this case we used it to specifically predict the effect of water efflux on the total amount of ATP the cell has available for maintenance. The model used as a base for this analysis was iJO1366 ³from the BIGG database.

The relationship between maximum ATP available for survival and water efflux is shown in Figure 1 the results from the metabolic model suggest that there is a linear relationship between ATP availability and water efflux. This implies that if no water is available for ATP used for maintenance, the cell will die. When the model runs without any modification, i.e. in an environment where it is in the exponential growth phase, an ATP Maintenance flux of 3.15 mmol⋅gram Dry Weight^-1⋅hour^-1 is given as output by the model.

We do not know the exact amount of ATP needed for survival in sugar water conditions, but because of the results from Figure 1 we can start looking at the relationship between survival time and water efflux.

In Figure 2 we can see not only the relationship of survival time against max ATP available for survival, but also how different thresholds of minimal cell-water tolerance would affect this relationship. The minimal cell-water tolerance threshold gives the value at which percentage of the remaining cell-water reaches a state that will cause cell-death. This has a drastic effect on survival time, changing 20 minutes of maximum survival time to a mere ~90 seconds in the worst case scenario.

Up to 90 minutes

Figure 1 shows us that osmotic pressure alone can indeed have an effect on cell regulation and cell death, and from Figure 2 it appears that the minimal water allowance threshold has a high impact on range of possible times. Our model is limited in that it predicts an infinite survival time beyond 90 minutes. This suggests that our model may be missing some form of regulation that allows for longer survival times. To understand this effect, we conducted an experiment to test this prediction. This experiment can be found here.

What we can say is that if the cells can survive for longer outside of this period, then they must have enough ATP available for basic maintenance. If cell death occurs, other processes than pure water-efflux must be the cause of that. This could be due to a combination of lacking nutrients and water-efflux, or over production of osmolytes to keep the balance.

Beehave

Due to regulatory and experimental hurdles it is difficult to test the effectiveness of BeeT in combating Varroa destructor. We would still like to be able to give advice to local beekeepers on the ideal application strategy of BeeT, based on several scenarios. To accomplish this, the previously published model BEEHAVE was adapted to include the effect of BeeT on Varroa mite and virus dynamics in simulated colonies. BEEHAVE is an open-source agent-based model, which can be used to examine the multifactorial causes of Colony Collapse Disorder ^{1 2} . It consists of several modules which control aspects like foraging, Varroa mite dynamics, and colony growth (Figure 1), and was extensively tested for robustness and realism ¹ .

BeeT can be transported to the hive by adding it in sugar water or bee bread. Both these sources have their advantages and disadvantages. In this project we have used Escherichia coli as our model chassis, which can be applied to sugar water [link to Ronald]. Sugar water is supplemented to support a colony during honey harvest before winter, as a substitute for nectar ^{3 4} . Supplementing Apis mellifera,honey bee, with sugar water is a well established and familiar practice amongst beekeepers ³ . Alternatively, reconstructing the same system in Lactobacillus species would allow us to use BeeT in bee bread. Bee bread has an advantage over sugar water, as it is more frequently transported to larvae. Bee bread is a combination of pollen, regurgitated nectar, and glandular secretions. To protect against harmful micro-organisms honey bees also inoculate bee bread with fermenting bacteria. ⁵ . The fermentation process reduces the pH .

BeeT Module

BEEHAVE is an open-source GNU licensed agent-based model utilizing NetLogo, and consists of several interlocking modules which each model different aspects of the beehive. It models the wide variety of stresses affecting honey bees. ¹ . As such, it is the ideal basis for our investigation into the effects of BeeT on Varroa mites and honey bee dynamics. BEEHAVE has several modules covering colony dynamics, foraging, and a Varroa mite model as depicted in Figure 1. Two viruses are also included in the model: deformed wing virus (DMV), and acute paralysis virus (APV), both for which Varroa mites are a vector. Our BeeT module, which runs parallel to BEEHAVE, is capable of modeling transport of BeeT into the hive using sugar water or bee bread. It also calculates how much BeeT is transported to larvae based on consumption of honey and pollen stores. The Varroa mite mortality is determined by the amount of BeeT near larvae when Varroa mites emerge from brood cells. This in turn affects Varroa mite population levels in the hive, reducing virus loads in the hive, and allowing colony survival.

Assumptions

Not everything is known about the BeeT model, and as such it was necessary to make certain assumptions. Some of these are related to BeeT in general, while others are specific to the sugar water or bee bread applications. We will first discuss general limitations of our model, and then address each of the applications separately.
General assumptions
Several properties of BeeT and its chassis are currently unknown. This includes the degradation rate of BeeT both inside and outside the hive. With both treatments BeeT encounters different environments while being transported to larvae. For sugar water, BeeT is exposed to the honey stomach of bees, honey stores, and brood cells, whereas bee bread encounters pollen and brood cells. We do not know the effects of these different environments on degradation rates of BeeT. For this reason we assumed that degradation rates are stable inside and outside the hive.
Another uncertainty is what the effect of BeeT is on Varroa mite mortality. We modelled this by using a saturating function. We also assumed that the effect of BeeT on Varroa mite mortality is entirely determined by the amount of BeeT present at larvae when a brood cell is capped.
Finally, the BEEHAVE model is only able to model a single virus: DMV or APV. We are unable to model the combined effects of both viruses on a bee colony. For all analyses we used the DWV virus, as it is more harmful to honey bee colony survival than APV 8 .
Sugar water
We know that, based on the experimental results [hyperlink] obtained by metabolic modelling of E. coli, and it is able to survive in sugar water for at least 48 hours. Additionally, modeling predicts that if E. coli is capable of surviving 90 minutes, it can survive indefinitely[hyperlink].
Additionally, we estimate that in sugar water there are approximately 1 * 10^6 cells*ml^-1. This estimate is based on the assumption that we add saturated medium (1 * 10 ^ 8 cells*ml^-1) containing BeeT to sugar water and diluted this by a factor of 100. The dilution is performed to avoid rejection of sugar water by bees, as it would be too contaminated to consume.
We based the hourly uptake rate of sugar water on a paper by Avni et al. ⁹ , but it is possible that this is not the maximal uptake rate of sugar water. In the measured period all of the sugar water was taken up by the colony, meaning a higher theoretical uptake rate.
Bee bread
We assume that the amount of Lactobacillus cells in bee bread is roughly equal to the number of CFU*ml^-1 in yoghurt ¹⁰ . Artificial bee breads can be made using yoghurt, and are readily accepted by bees ¹⁰ . Additionally, we assume that the removal rate of artificial bee bread is equal to 22.7 g*day^{-1 4} . Based on Avni et al. we know that the uptake rate depends on the manner in which bee bread is applied ⁹ . We chose to base the uptake rate of bee bread on Brodschneider et al., since their experimental setup is similar to Avni et al. ^{4 10}

Key results

The functionality of the BeeT module is primarily based on literature research, and as such requires a relatively low amount of parametrization. The main unknown quality is how BeeT will behave, namely degradation and its effectiveness at combating Varroa mites nearby larvae. We are interested in how BeeT can best be applied given certain assumptions [hyperlink previous section]. As such we are interested in the best period to apply BeeT, either before winter or during spring, so we can give recommendations to beekeepers. We also examine the difference in effectiveness between BeeT in sugar water and in bee bread for both periods. This can inform future work on whether it is beneficial to adapt BeeT for functionality in bee bread. There are three additional parameters related to BeeT that can be varied, and upon which effectiveness relies. These are: degradation of BeeT in the hive, outside the hive, and the effect of BeeT on Varroa mite mortality. Each is varied across a range of values. Additionally, for every combination we have five replicates to reduce the variance in our results. We divided the analyses into four treatments, each with the same three parameter ranges and five replicates per set of parameters. This can be seen in Table 1.

Each combination of period and treatment has the same ranges. We compared the winter period, from with the spring period for the different ranges and treatments. In each case the spring period was significantly more effective in combating Varroa mites than the winter period. For further analysis we continued with the spring period.

Three scenarios were identified: the colony dies due to virus load, the colony survives at a lower population level, or the colony thrives. Every year the BEEHAVE model checks the population after overwintering. If it is below 4000 worker bees the colony is presumed dead. To differentiate between the scenarios we looked at the mean population of worker bees after winter. If this number is below 3000, we assume that the colony is dead. If the mean population ofbees after winter is between 3000 and 5000, the colony is capable of surviving. Above 5000 worker bees, the colony is thriving and the Varroa mite infestation is under control. Each parameter combination for sugar water and bee bread falls within one of these three scenarios. For both treatments we chose a parameter set which is representative for a certain scenario. From this we have 6 representative parameter sets: three for sugar water (death, survival and thriving) and three for bee bread (death, survival and thriving). As can be seen in Table 2, bee bread is significantly more effective than sugar water. Some BeeT is consumed by worker bees instead of being deposited at the larvae. This is represented by the fraction of total BeeT moved to the hive that is consumed by larvae and also depicted in Table 2.

In the case of sugar water a far lower fraction of BeeT ends up at larvae, and most is consumed by worker bees. This is likely the main reason for the difference in effectiveness for sugar water and bee bread.

Each of the 6 parameter sets was further analysed by examining the population dynamics of worker bees and Varroa mites over time. In Figure 2 and Figure 3 these are shown, respectively, for a colony barely surviving a Varro mite infestation (Figure 2), and a thriving colony (Figure 3), regardless of the presence of Varroa mites at the start of the simulations. When the colony is barely surviving we can see that worker bee and Varroa mite populations reach a dynamic equilibrium.

When the colony is thriving, the starting Varroa mite population rapidly declines and disappears.

This indicates that the treatment is effective for a low population of initial Varroa mites. We further analysed a situation with a starting population of 10.000 Varroa mites which is shown in Figure 4.

Even with a higher starting population and a high initial virus load, we can see that the treatment is successful in reducing Varroa mite populations and that the colony recovers.

Model analysis

The BEEHAVE model utilizes multiple pseudo-random number generators to calculate mortality amongst Varroa mites and bees. These generators increase the variance inherent in the model and, as such, it is advisable to use multiple replicates per parameter set. To determine the error bars we ran a simulation where we varied the Varroa mite mortality, and plot the mean worker bee population over 10 years with 10 replicates per value of Varroa mite mortality. This was done for the first version of the BEEHAVE model, which governs year round Varroa mite mortality when they emerge from a brood cell.

With 10 replicates the variance is relatively low, particularly when the mean worker bee population is high.
To analyse the full BeeT module we chose to use 5 replicates. This is a compromise between simulation time and reduction of variance.

We first separated the results of the analyses of the full BeeT module based on the three scenarios. The results can be seen in Figure 6. Within this 3d graph the axis are degradation in the hive, degradation outside the hive, and impact of BeeT on Varroa mite mortality. Additionally, the size of each set of parameters indicates the mean population of worker bees surviving winter.

Discussion and Conclusions

Currently it is unclear how effective BeeT will be at combating Varroa mites since, primarily, toxicity is difficult to measure in the lab and deploying BeeT in the field has many regulatory issues. As such we built a BeeT module for the well known BEEHAVE model to assess the minimum effectiveness for certain treatments. We also examined two different periods for administering the BeeT:; before winter and during spring. Before winter was chosen to reduce the Varroa mite population and support honey bees during their most fragile period in the year. Additionally, it is normal for current beekeeper practices to supplement hives with sugar water before winter and, thus, applying BeeT at this time would provide a minimal amount of disturbance for beekeepers.
The advantage of administering BeeT during spring is that this is the period at which most of the Varroa mite population is present at larvae. By administering during this period we can ensure that BeeT is able to affect a large percentage of the Varroa mite population. Beekeepers also visit hives regularly during this period as they harvest honey in spring. Recent papers indicate that feeding of artificial bee bread during this period can have significant positive effects on honey bee health ^{6 7} . For both sugar water and bee bread the ideal period for administering BeeT is during the spring, due to the large difference in effectiveness between the two periods. In a future application of BeeT this can be given as a recommendation to beekeepers. This is a practice which can be easily adopted since beekeepers already visit hives regularly in the spring period. In particular for supplementation of sugar water, as this is already accepted practice amongst beekeepers.

The main difference between sugar water and bee bread is that a larger fraction of total in-hive BeeT is given to larvae for bee bread. This is not surprising as honey stores are primarily used to support energetic needs of worker bees. ¹ and as such only a small fraction is transported to larvae. On the other hand, the protein contained in bee bread is crucial for the proper development of larvae and as such is mainly given to larvae to support their growth ¹ . Based on our results we expect that BeeT will be effective using sugar water but greater effectiveness can be accomplished by engineering BeeT for compatibility with bee bread.

Modeling methods

In version one of the model the following code was added to BEEHAVE:

This calculates the mortality of Varroa mites when they emerge from the brood cell (mortalityMites). In BEEHAVE the mortality of Varroa mites is encapsulated in the fecundity of female Varroa mites. This means that mortalityNoBeeT is zero. When ScalingVariable is large, the mortalityMites approaches one.
The amount of Varroa mites emerging from a cell is calculated with the variable ‘healthyMitesInSingleCell’. Using a poisson distribution, some of the Varroa mites emerging from the brood cell die. Both versions of the BeeT module only affect the Varroa mite population when Varroa mites emerge from a brood cell. This was done to reduce the chance of introducing bugs into the original BEEHAVE model.

In version 2 of the BeeT module, the mortalityMites was changed to:

so that it is dependent on the concentration of BeeT in the cell at the time of capping. Additionally, we included a procedure BeeTProc which is called daily. It handles BeeT moving from outside to inside the hive at a set rate per hour. It also handles degradation: both hourly degradation outside the hive and daily degradation inside the hive. When Varroa mites invade, the amount of BeeT present at larvae is saved for each invading Varroa mite in the variable BeeTLarvaeInvasion. Whenever a Varroa mite emerges from a cell its mortality is calculated based on BeeTcell, which is dependent on BeeTLarvaeInvasion. The total amount of BeeT moving to larvae is based on the amount of sugar water and pollen consumed by larvae. If BeeT is consumed by worker bees it is considered lost and is removed from the BeeT in-hive store.

All variables added to BEEHAVE can be found in Table 3, including how they are calculated, if applicable what their initial value is, units, in what procedure they are used, and any comments or references.

References

References for Population Dynamics

1. Eri Nasuno, Nobutada Kimura, Masaki J. Fujita, Cindy H. Nakatsu, Yoichi Kamagata, and Satoshi Hanada (2012). Phylogenetically Novel LuxI/LuxR-Type Quorum Sensing Systems Isolated Using a Metagenomic Approach Vol, 78, number 22. ↩

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3. Pearson, B. Bacterial Hash Function Using DNA-Based XOR Logic Reveals Unexpected Behavior of the LuxR Promoter, IBC 2011, vol. 3, article no. 10, pp. 1-10 ↩

4.Sergi Abadal Ian F. Akyildiz (2011). Automata modeling of Quorum Sensing for nanocommunication networks, Volume 2, Issue 1, Pages 74–83 ↩

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6. Gregory Bokinsky, Edward E. K. Baidoo, Swetha Akella, Helcio Burd, Daniel Weaver, Jorge Alonso-Gutierrez, Héctor García-Martín, Taek Soon Lee, and Jay D. Keasling(2011).HipA-Triggered Growth Arrest and β-Lactam Tolerance in Escherichia coli Are Mediated by RelA-Dependent ppGpp Synthesis 2013 Jul; 195(14): 3173–3182. ↩

7. A. Raue et al. Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Vol. 25 no. 15 2009, pages 1923–1929 ↩

8. Martin Bengtsson et al. Gene expression profiling in single cells from the pancreatic islets of Langerhans reveals lognormal distribution of mRNA levels volume 15 page 1388–1392(2005) ↩



References for Optogenetic Kill Switch

1. Ohlendorf, Robert, Roee R. Vidavski, Avigdor Eldar, Keith Moffat, and Andreas Möglich. 2012. “From Dusk till Dawn: One-Plasmid Systems for Light-Regulated Gene Expression." Journal of Molecular Biology 416(4):534–42. Retrieved (http://dx.doi.org/10.1016/j.jmb.2012.01.001) ↩

2. Möglich, Andreas, Rebecca A. Ayers, and Keith Moffat. 2009. “Design and Signaling Mechanism of Light-Regulated Histidine Kinases." Journal of Molecular Biology 385(5):1433–44. Retrieved (http://dx.doi.org/10.1016/j.jmb.2008.12.017). ↩

3. Klose, Cornelia et al. 2015. “Systematic Analysis of How Phytochrome B Dimerization Determines Its Specificity." Nature Plants 1(July):15090.↩

4. Rausenberger, Julia et al. 2010. “An Integrative Model for Phytochrome B Mediated Photomorphogenesis: From Protein Dynamics to Physiology." PLoS ONE 5(5).↩

5. Engelberg-Kulka, Hanna, Ronen Hazan, and Shahar Amitai. 2005. “mazEF: A Chromosomal Toxin-Antitoxin Module That Triggers Programmed Cell Death in Bacteria." J. Cell Sci. 118:4327–32.↩

6. Limpert, Eckhard, Werner a. Stahel, and Markus Abbt. 2001. “Log-Normal Distributions across the Sciences: Keys and Clues." BioScience 51(5):341.↩

7. Fasani, Rick a and Michael a Savageau. 2013. “Molecular Mechanisms of Multiple Toxin-Antitoxin Systems Are Coordinated to Govern the Persister Phenotype." Proceedings of the National Academy of Sciences 110(27):E2528–37.↩

8. Mancinelli, A. L. (1994) The physiology of phytochrome action. In Photomorphogenesis in Plants, 2nd ed. (Edited by R. E.Kendrick and G. H. M.Kronenberg), pp. 211–269. Kluwer Academic Publishers, Dordrecht, The Netherlands.↩



References for Metabolic Modeling

1. Cheng, Y. L., Hwang, J., & Liu, L. (2011). The Effect of Sucrose-induced Osmotic Stress on the Intracellular Level of cAMP in Escherichia coli using Lac Operon as an Indicator. Journal of Experimental Microbiology and Immunology (JEMI) Vol, 15, 15-21. ↩

2. Neidhardt F.C. Escherichia coli and Salmonella: Cellular and Molecular Biology. Vol 1. pp. 15, ASM Press 1996 ↩

3. Orth, J. D., Conrad, T. M., Na, J., Lerman, J. A., Nam, H., Feist, A. M., & Palsson, B. Ø. (2011). A comprehensive genome‐scale reconstruction of Escherichia coli metabolism—2011. Molecular systems biology, 7(1), 535. ↩

4. DeGrandi-Hoffman, G., & Hagler, J. (2000). The flow of incoming nectar through a honey bee (Apis mellifera L.) colony as revealed by a protein marker. Insectes Sociaux, 47(4), 302-306. ↩



References for Beehave

1. M. A. Becher, V. Grimm, P. Thorbek, J. Horn, P. J. Kennedy, and J. L. Osborne, (2014). BEEHAVE: A systems model of honeybee colony dynamics and foraging to explore multifactorial causes of colony failure, Journal of Applied Ecology., vol. 51, no. 2, 470–482. ↩

2. J. Horn, M. A. Becher, P. J. Kennedy, J. L. Osborne, and V. Grimm (2015), “Multiple stressors: Using the honeybee model BEEHAVE to explore how spatial and temporal forage stress affects colony resilience," Oikos, no. September 2015, 1001–1016. ↩

3. Fasasi, K A (2011) "Cumulative effect of sugar syrup on colony size of honeybees, Apis mellifera adansonii (Hymenoptera : apidae) in artificial beehives " Journal of Natural Sciences, Engineering and Technology ed. 10: 33-43. ↩

4. Robert Brodschneider, Karl Crailsheim (2010) "Nutrition and health in honey bees" Apidologie ed. 41: 278-294 ↩

5. Ellis, Amanda M. Hayes, Jr, G. W. (2009) "An evaluation of fresh versus fermented diets for honey bees (Apis mellifera)." Journal of Apicultural Research 48: 215-216 ↩

6. Basualdo, Marina Barragán, Sergio Antúnez, Karina (2014) "Bee bread increases honeybee haemolymph protein and promote better survival despite of causing higher Nosema ceranae abundance in honeybees" Environmental Microbiology Reports 6: 396-400 ↩

7. De Jong, David (2009) "Pollen substitutes increase honey bee haemolymph protein levels as much as or more than does pollen" Journal of Apicultural Research Reports 48: 34-37 ↩

8. Sumpter, D. J T, Martin, S. J., (2004). The dynamics of virus epidemics in Varroa-infested honey bee colonies, Journal of Animal Ecology., vol. 73, is. 1, 51-63. ↩

9. Sumpter, D. J T, Martin, S. J., (2004). The effect of surface area of pollen patties fed to honey bee (Apis mellifera) colonies on their consumption, brood production and honey yields., Journal of Apicultural Research., vol. 48, is. 1, 23-28. ↩

10. Ellis, Amanda M. Hayes, Jr, G. W., (2009). An evaluation of fresh versus fermented diets for honey bees (Apis mellifera)., Journal of Apicultural Research., vol. 48, is. 3, 215-216. ↩

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