Overview
Our experiments on mussels allowed us to determine the killing effect of the FitD toxin. Using this data and some literature-based information, we decided to create a model that would estimate the effect of our product when applied in the real world, specifically in infested water pipes. In particular, we wanted to find out how long it takes for a pipe to become blocked with and without the application of our product. We also wanted to examine the influence of different application regimes on clogging time.
Model
Population dynamics in normal conditions
We assume in our model that the mussel population grows at a constant birthrate and a constant death rate. From day to day, a certain number (depending on the population size) of young mussels are added and a certain number (also depending on the population size) of mussels die.
We can therefore describe the population dynamics as follows:
$$N(1)=N(0)+b\cdot N(0)-d\cdot N(0)$$ $$N(2)=N(1)+b\cdot N(1)-d\cdot N(1)$$ $$N(3)=N(2)+b\cdot N(2)-d\cdot N(2)$$ $$\downarrow$$ $$N(t+1)=N(t)+b\cdot N(t)-d\cdot N(t)$$
Where N is the number of mussels at a given time, b is the birthrate and d is the mortality rate. This model results in the following exponential function:
$$N(t)=N_0\cdot e^{(b-d)\cdot t}$$
One single mussel releases up to 1 million eggs annually. Of course, not all of those survive and become adults: based on literature, we know that a single mussel produces up to 30’000 adult mussels per year (information gained from the Flathead Lake Biological Station, 2019).
If we assume an initial population N(0) of one mussel and a population size N(365) of 30’000 after one year, we can derive the birth rate as follows:
$$N(t)=N_0\cdot e^{b\cdot t}$$ $$30’000=1\cdot e^{b\cdot 365}$$ $$ln(30’000)=b\cdot ln(e^{365})$$ $$b=\frac{ln(30’000)}{365}=0.028\ day^{-1}$$
This calculation is obviously simplified because a mussel alone cannot produce offspring, but the estimate is biologically sound.
Furthermore, we know from existing literature that a mussel has a lifespan of 3-9 years (information gained from the Flathead Lake Biological Station, 2019). In addition to that, quagga mussels are very robust and adaptable, which leads us to assume a very low mortality rate under normal circumstances. This was also confirmed by the control group of our experiments, where no death events were observed for the 45 untreated mussels over a 3 day time window. If an initial population consists of 100 individuals, we assume that after one year 5 individuals have died. Applying the same reasoning as for the birth rate, this corresponds to a mortality rate of d= 0.0001 day-1. We therefore assume a zero-approximating mortality rate for our model when no toxin is applied.
Population dynamics when the FitD toxin is applied
The population dynamics change when the toxin is applied. Right after treatment, the death rate is increased. After a relatively short activity period, the toxin degrades and the population returns to the normal death rate until the next treatment. Ideally, one should measure the protein half-life in solution and consider the exponential decay of the toxin over time. To the best of our knowledge, these data for the FitD toxin are not available in the literature. Besides that, we do not have a way to measure the concentration of the toxin in our solution directly, thereby we decided to simplify the problem and assume a constant concentration of active protein for a 5-day window after the treatment application. For the sake of simplicity, we also assume that the birth rate remains unchanged when the toxin is applied, although it would be likely that the birth rate decreases since the mussels are exposed to stress.
We therefore get a population dynamics whose mortality rate depends on whether the toxin is present or has already been degraded:
$$1)\quad N(t+1)=N(t)+b\cdot N(t)-d_{FitD}\cdot N(t)$$ $$2)\quad N(t+1)=N(t)+b\cdot N(t)-d\cdot N(t)$$
Where equation 1) describes the population dynamics when FitD is applied and equation 2) describes the population dynamics after FitD has been degraded.
The FitD toxin proved to be very effective against mussels in our experiments. Following the application of 8x10 8 cells/ml, 10 out of 10 mussels died within 37 hours. We think a lower concentration is safer and more realistic for a real-life application. The administration of 2x108 cells/ml was sufficient to showcase a statistically significant difference when compared to Pseudomonas protegens wild type, but we think this would be insufficient to contain a mussel invasion in a real-life situation.
We therefore decided to move forward with the parameter estimated from the experiment with intermediate concentration (4x108 cells/ml): within 3 days, 6 out of 45 mussels died, thus N(0)=45 and N(3)=39. From these figures, we can derive the mussel's death rate during the activity period of FitD as follows:
$$N(t)=N_0\cdot e^{-d_{FitD}\cdot t}$$ $$39=45\cdot e^{-d_{FitD}\cdot 3}$$ $$\frac{39}{45}=e^{-d_{FitD}\cdot 3}$$ $$d_{FitD}=\frac{-1}{3}\cdot ln(\frac{39}{45})=0.048\ day^{-1}$$
This death rate is already much higher than the normal death rate of 0.0001 day-1. Interestingly, our experiments showed that the death rate increases not linearly with the cell count, which means that small increments in the concentration of the product translate into a substantial increase in the death rate. Therefore, we can consider 0.048 day-1 as a lower bound, and a higher dFitD would be plausible.
Clogging-time
Now that we can express the population sizes as a function of time, we can estimate the time it takes to clog a pipe with a certain radius and length. Since mussels tend to spread and attach to free surfaces rather than growing on each other, we decided to consider a situation where mussels grow concentrically, and a new layer starts only when the inferior one is complete. Of course, in a real-world situation, it is possible that mussels could grow on each other and rapidly fill a cross-section of the pipe, thereby blocking the water flow, even though the channel is not completely clogged. This scenario is probable, especially in the presence of bends and bottlenecks in the conduit, but for simplicity, we consider a symmetrical pipe. In our model, a pipe is clogged when quagga mussels fill its entire volume. We assume that the pipe would remain functional until it is completely filled, even though the flux could be dramatically affected by the reduction in the cross-section (which could still be compensated by adjusting the pressure).
In our model, the clogging time corresponds to the time point where the pipe volume is equal to the mussel volume:
At tclog :
Where the mussels volume is simply the number of shells multiplied by the volume of a single shell.
To calculate the volume of a mussel, let's assume it has a cubic shape and an edge length of x= 0.8 cm (measured experimentally):
$$v=x^3=0.51\ cm^3$$
We assume a cubic shape, even though a sphere would be more biologically sound, to simplify the calculation (there remains some space between two adjacent spheres, while there is no residual space between two adjacent cubes). Edge effects are negligible and cancel each other out due to the symmetry of the system considered.
Implementation of the model
We can now predict how a mussel population develops without FitD toxin and with FitD toxin. The pink line in Figure 1 shows the maximum capacity of the pipe: if this capacity is crossed, the pipe is completely blocked. The spikes of the blue line showing toxin application are caused by the fact that the toxin is not active indefinitely but is degraded after a certain time (5 days), bringing the population back to its normal death rate until the next application of the toxin, which is indicated by the next spike.
The following parameters were used to generate Figure 1:
- initial size of the mussel population: 10
- pipe radius= 20 cm
- pipe length= 3 m
- application period= 60 days
With these parameters, it takes 68 days longer to clog the same pipe when the FitD toxin is applied: we can consider this as “saved time”, which directly translates into a reduction of disinfestation treatments with the associated costs.
Do you possess or carry out the maintenance of a pipe? If you want to know how much time you would save by using the FitD toxin, you can enter the radius and length of your pipe in the box below. Assuming that the application interval is 60 days, the time saved is calculated directly.
Effect of different application periods
When using the FitD toxin, the growth of the mussel population is strongly influenced by the application period. Figure 2 shows the population growth with different application periods. The pink line marks the maximum capacity of the pipe: if this is crossed, the pipe is completely blocked.
The following parameters were used to generate Figure 2:
- initial size of mussel population= 100
- pipe radius= 10 cm
- pipe length= 1 m
Compared with FitD toxin application, the untreated population size reaches faster the maximum number of mussels the given pipe can hold (pink). With decreasing application-period, the pipe is clogged slower. Even with a long application period of 1 year, some time is saved by applying the toxin.
Differences of clogging time for different pipe structural features
Finally, thanks to the predictions of our model, we can claim that the application of our product is of great interest, especially in large pipes. In large-volume pipes, the time for complete blockage can be delayed for several months by applying FitD toxin, resulting in a considerable reduction in maintenance time and cost.
References
- Flathead Lake Biological Station. (2019).https://flbs.umt.edu/newflbs/media/1837/ais-lesson_4_invasive-mussel-life-cycle-poster.pdf