This model describes how microfluidics Gaussian distribution plate works when we use this device to test C. elegans’(Caenorhabditis elegans) preference in the plate.If the plate works, the model can further describe how much preference the C. elegans’ show when attraction or repellent factor is added.

Different from classical Galton board(Fig.1), it is worms rather than balls running on the plate. We use worms as the “ball”. However, when a worm is choosing a direction at the crossing, its previous choice may influence current choice due to worm’s relative long body.

Fig. 1 The simulation of Galton board

Details:

Gaussian distribution is very common in nature and important in statistics. This distribution is tied to many natural phenomena. In our experiment, we use the Gaussian Plate to see C. elegans’’ distribution. Then we use it to see the change of C. elegans’’ distribution after adding attraction or repellent factor (Fig.2).

Fig. 2 The Gaussian distribution A) The ideal Gaussian distribution before adding chemicals. B) The changed Gaussian distribution after adding chemicals by using the first diffusion methods, which means diacetyl diffuse in the same plate as the Gaussian Plate .

Firstly, we introduced a parameter ‘k_a’ (0<=k_a<=1) to describe how much the previous choice influences the current choice

The ‘k_a’ is defined as the absolute value of the difference between the probability of two direction choices.

Fig 3. Diagram of parameter ‘k_a’. ‘k_a’equals to absolute value of difference between the probability of two direction choices.

It means the greater the ‘k_a’ is the greater the influence of previous choice is. Especially, when ‘k_a’ is zero, the simulation model is more like Gaussian distribution.

We counted the number of C. elegans in each channel and calculated the parameter ‘k_a’ at first (Fig.4).

Fig. 4 the Gaussian Plate to study locomotion on-chip

k=\frac{same turning-different turning}{total turning}=\frac{(20+10-12-9)}{(20+10+12+9)}=0.18

Assume that ‘k_a’ doesn’t change when we add attraction or repellent factors. Then this model can be used to stimulate C. elegans preference considering ‘k_a’.

We can analyze how much the factor attracts or repels the C. elegans by introducing another parameter ‘k_perf’ (0<=k_perf<=1).

Both parameters set above are dependent on the assumption that parameters values are proportional to the probability they affect the C. elegans’ choice.

For example, if ‘k_a’ is to ‘0’, previous choice doesn’t affect current choice at all. If ‘’k_a is ‘1’, the worm’s next choice is totally determined by the previous choice and the final result will show polarized distribution( C. elegans only pass through the rightmost or leftmost channels). When ‘k_a’ is 0.5, a worm has 25% probability of turning right/left and 75% probability of turning left/right at a crossing. ‘k_b’ (attraction or repellent parameter) is the same as ‘k_a’

Result:

Simulation results are shown in two steps:

1.Determining ‘k_a’: To determine ‘k_a’, we did a experiment without adding factors‘k_a’ equates to 0.18, according to the table shown above.

2. a. Compare Stimulated 100000 worms with experimental data.

b.Compare Preference stimulated data preference with experimental data after adding diacetyl

’k_a’ is 0.18 determined previously. However, stimulated data is not fit the experimental well. Because there are some bubbles in the Gaussian Plate . These bubbles block the crossings and affect the C. elegans’ choice.

There till some evidence shows the C. elegans’ preference by comparing experimental data to stimulation data with different ‘k_perf’.

‘k_perf’ equals to 0:

‘k_perf’ equals to 0.1:

‘k_perf’ equals to 0.2:

Conclusion

The result shows that ‘k_perf’ is between 0.1 and 0.2. If bubbles don’t block crossings in plate, the stimulation data will fit experimental data better.