Modeling

Background

B.s and B.ti are both used as bio-pesticide. Some researches had shown that Cry4Ba and Mtx1 have synergism when they are working in the same system, which means that if we mix or fuse the two toxins together, the pesticide will be more effective than using them separately. It’s important to find out whether the synergism exist and then estimate the best ratio of cry4Ba and Mtx1 in the mixed system. In order to achieve this goal, we introduced probit model to analyze the data we’ve got in toxicity test and then calculated the best ratio of these two toxin by curve fitting.

Probit model

In statistics, probit model is a type of regression that the dependent variable can take only two values like “dead or not dead”. The purpose of the model is to estimate the probability of an observation with particular characteristics falling into a specific one of the categories.

Let’s suppose a response variable Y is binary, which means that it can only have two possible outcomes, 1 and 0. We also have a vector of regressors X, which are assumed to influence the outcome Y. Specifically, we assume that the model takes the form , where Pr denotes probability, and Φ is the Cumulative Distribution Function (CDF) of the standard normal distribution. The parameters β are typically estimated by maximum likelihood.

Estimation of toxicity intensity

There are several index to demonstrate the intensity of toxin. Here we calculated KT_50 to measure the intensity. KT_50 means the time a kind of toxin cost to kill half of the total number of insects in an independent system. Lower KT_50 indicates higher toxicity.

After succeeded in expressing cry4Ba in E.coli, we designed our experiment for modeling. In our experiment, every independent system contains 50mL ddH₂O, 2.5mL bacterial(induced cry4Ba E.coli and Bs ) and 15 subjects of 2^nd-3^rd instar larvae. The ratio of cry4Ba in the mixed toxin ranged from 0 to 100%. Each group had three repetitions, and all the groups were observed for 48 hours. We recorded the number of death in 2h, 4h, 6h, 8h, 12h, 24h, and 48h. Based on the data we got in the experiment, we used probit model to estimate KT_50.

Table 1. The groups of toxicity test.

Group	Volume of cry4Ba E.coli (mL)	Volume of Bs (mL)	Ratio of cry4Ba (%)
1	0	2.5	0
2	0.4	2.1	16
3	0.8	1.7	32
4	1.2	1.3	48
5	1.6	0.9	64
6	2.0	0.5	80
7	2.5	0	100

Table 2. Data record

Time	2 hr	4 hr	6 hr	8 hr	10 hr	12 hr	24 hr	48 hr
1	0	0	0	2	3	6.67	10.33	15(ALL)
2	0	0	0	1.33	2.33	3.33	5.33	6.33
3	0	0	0	0.33	1.67	2.67	7.33	12
4	0	0.33	0.67	1.33	3	4.33	8.67	12.33
5	0	0	0	1.67	2	3.67	11	13.33
6	0	0	0	0.33	1	2.33	10	14.67
7	0	0	0	0	3	12	15	15(ALL)

KT_50

We used SPSS to calculate KT_50. Response frequency is number of death. Covariate is time. Total observed is 15. When probability=0.5, the corresponding “Time” is KT_50. Data is shown in figures below.

Figure 1. Probit regression model and KT_50 data from the model.

Curve fitting

We’ve got KT_50 data in probit model, so the next step is to figure out whether there is a best ratio of the two toxins when they’re used to kill Aedes albopictus larvae .

Percent of cry4Ba(%)	KT50(h)
16.00	45.53
32.00	25.22
48.00	21.03
64.00	18.86
80.00	19.54
100	20.43

Because the KT_50 of group 1 (ratio of cry4Ba is 0% and ratio of Bs is 100%) is too high and may interfere our fitting curve, we decided to abandon this data. Data we used to do the curve fitting is shown in table 3.

Discussion

Our modeling can illustrate two significant points:

• Mtx1 and cry4Ba actually have synergism.
• The mixture could kill mosquito species Aedes albopictus. Also the ideal ratio of cry4Ba E.coli in the mixed toxin with Bs is 73.13 % (target to Aedes albopictus).

Based on such information, we decided to design a fusion protein of cry4Ba and Mtx1 and meanwhile we are optimizing the product of our entrepreneurship.