_{1}

Elegans are one of the best model organisms in neural researches, and tropism movement is a typical learning and memorizing activity. Based on one imaging technique called Fast Track-Capturing Microscope (FTCM), we investigated the movement regulation. Two movement patterns are extracted from various trajectories through analysis on turning angle. Then we applied this classification on trajectory regulation on the compound gradient field, and theoretical results corresponded with experiments well, which can initially verify the conclusion. Our breakthrough is performed computational geometric analysis on trajectories. Several independent features were combined to describe movement properties by principal composition analysis (PCA) and support vector machine (SVM). After normalizing all data sets, no-supervising machine learning was processed along with some training under certain supervision. The final classification results performed perfectly, which indicates the further application of such computational analysis in biology researches combining with machine learning.

When an animal experiences an environment at one point, it can form a kind of memory for its future behaviors. This movement pattern requires the animal a mechanism of navigating, along with conditions memorizing and decision-making based on its previous experiences. It is a process that includes learning, memorizing, and performing [

We take the standard cultivating technology for all elegans [

Our analytical pipeline for computational geometry began with the collection of 2D trajectories. We obtained 2D-FTCM images of elegans trajectories (_{1}, y_{1}), (x_{2}, y_{2}). Especially, (x_{n}, y_{n}) represented the endpoint of one trajectory and (x_{0}, y_{0}) represented the original point of one trajectory, namely (0, 0). Unit length in this system is reset, not according to millimeter nor any other normal length unit. We next used programs written in MATLAB to remove self-intersection in small scales (smaller than 50 × 50 scale) (

Before further analysis, we should point out that although we expect a definite direction for elegans to move toward, there were still lots of elegans in the hesitation and sleepy states within twenty-five minutes (

To describe these tracks, we firstly regarded movement as Brownian motion [

θ = arccos ( Δ x Δ x 2 + Δ y 2 ) = arcsin ( Δ y Δ x 2 + Δ y 2 ) (1)

Δ x = x i + 1 − x i (2)

Δ y = y i + 1 − y i (3)

However, different description methods came to different results (

The compound field is composed of a horizontal thermal gradient field and a vertical salinity gradient field. The temperature gradient and salinity gradient in the compound field are consistent with the previous single gradient field. The cool side is set at the left edge and the suitable salinity side is set at the lower edge. It is obvious that the elegans may be inclined to the third quadrant (

For the convenience of following expressions, we first define the final trajectories angle in the elegans trajectories. The angle, which we record as θ, is defined as the following:

θ = { arctan ( y n x n ) The final point falls in the first quadrant arctan ( y n x n + π ) The final point falls in the second , third quadrant arctan ( y n x n + 2 π ) The final point falls in the fourth quadrant (4)

Under this definition, we can guarantee all angles are in the range [0, pi/2]. We then define N (θ), which equals to the ratio of the number of trajectories which the final angle is less than θ to the total number. In this method, we can determine at which angle the elegans are most inclined to.

d N ( θ ) θ = 0 when θ = θ e s t (5)

θ_{est} the most movement angle, can describe the general orientation of all trajectories. We applied this method to trajectories in the compound field. Instead of 225˚, the middle line in the third quadrant as our common sense, θ_{est} equals to 237˚ (

The most reasonable explanation of deviation of θ_{est} from our common sense might be different weight of intention which movement strategy elegans, HM or LM, may take in thermal and salinity gradient field. After specific statistics, the ratio of the number of elegans which take HM strategies to the left which take LM strategy is 1.425 in salinity gradient, while the number is 0.543 in the thermal gradient. The ratio to total trajectories number is (N (HM), N (LM)) = (35.19%, 64.81%) for thermal gradient and (N (HM), N (LM)) = (58.76%, 41.24%) for salinity gradient. This quantity strongly means elegans could prefer to take LM in thermal gradient and HM in salinity gradient. What’s more, we also calculated the average coordinate value for final points respectively for trajectories with HM pattern and LM pattern, which is written as ( 〈 x H n 〉 , 〈 y H n 〉 ) for HM and ( 〈 x L n 〉 , 〈 y L n 〉 ) for LM. As the symbol of general movement orientation vector, ( x ¯ n , y ¯ n ) is certainly different for elegans in different gradient.

For elegans in thermal gradient

{ ( 〈 x H n 〉 , 〈 y H n 〉 ) = ( − 174.33 , 29.87 ) ( 〈 x L n 〉 , 〈 y L n 〉 ) = ( − 56.28 , 21.84 )

For elegans in salinity gradient

{ ( 〈 x H n 〉 , 〈 y H n 〉 ) = ( − 187.86 , − 9.23 ) ( 〈 x L n 〉 , 〈 y L n 〉 ) = ( − 53.02 , − 16.15 )

Considering different weight in HM and LM, we can combine these two vectors in HM and LM into one general orientation vector.

For elegans in thermal gradient

{ x thermal = − 174.33 × 35.19 % − 56.28 × 64.81 % = − 97.82 y thermal = 29.87 × 35.19 % + 21.84 × 64.81 % = 54.67

For elegans in salinity gradient

{ x salinity = − 187.86 × 58.76 % − 53.02 × 41.24 % = − 132.25 y salinity = − 9.23 × 58.76 % − 16.15 × 41.24 % = − 12.08

Then we introduced influence factor due to different impact from thermal and salinity gradient. The influence factor is defined as following

β = η HM inthermal η LM inthermal = 58.76 % 35.19 % = 1.67 (6)

Combining two general vectors in two fields when the influence factor taken into considering, we can get a standard vector, (−0.82, −1.42), to measure movement orientation in compound field. Angle of this vector is 239.995˚, which is according with 237˚ to a great extend.

The method aims to sort these trajectories automatically according to criteria based on their morphological features (

We present our method in datasets of salinity trajectories, while the result is similar to datasets under other gradients. Principal component analysis (PCA) was performed with these five descriptors to obtain trajectory distribution in the new space. The first third features (principal components 1-3, PC 1-3)covered about 85% of the variance in the data (

We then investigated the distribution of trajectories in the feature space with axes corresponding to PC1, PC2 and PC3 (

We selected two trajectories, on the thermal gradient field, one of which is LM pattern and another is HM pattern, with dimension more than 2 × 200, which suggests it had a long period of morphological evolution. We then used the combination of features obtained by PCA to describe their transition. The first pair of features is PC1 and PC2 (

The next step is to utilize automatically classification by machine learning. and the first step should be normalization. There are two main methods of normalization dealing with trajectories. One is transforming all trajectories into

standard vectors, for example, with a dimension of 2 × 400. The longest dimension of datasets of one trajectory is no more than 2 × 350, so the size 2 × 400 can guarantee we will not lose any information. Lagrangian interpolation was used when replenishing the vector. Every three adjacent points in the datasets were selected to construct a parabola to predict coordinate information except for these known points. Another method is to use those certain computational geometric features as well as PCA combinations. Similarly, every value should be processed with max-min normalization. In this way, we can apply unsupervised machine learning to each processed data set. However, there were always some special trajectories which belong to a category of their own. There is no meaning when one category only contains one example. Naturally, we first classified these trajectories artificially and then took these examples as the test set, while using the classified trajectories as the training set to supervise and train a simple linear model.

The results are shown as follows (

above problem about these trajectories automatically. The second and third classifications have definite different movement directions. One is moving up along the y-axis to a great extend while another is moving down approximately. The results of four groups are shown below (

In this paper, we mainly invest the movement regulation of C.elegans on the experimental platform with a salinity and temperature gradient. We perform computational geometric analysis on thousands of trajectories. Different statistical methods show obviously different distributions on the turning angles of trajectories, and we extract two movement modes according to such results. We then verify this classification mainly from two viewpoints. One is its application on the prediction of movement direction on the compound field, which matched with the experiment well. The other verification is performed by auto-classification by clustering algorithms. Although there exist various advantages and disadvantages in each method, all final results show an apparent classification, which can correspond with the previous assumption. On the one hand, our conclusion can be a solid basement for further elegans fluorescence imaging, which aims to invest more on elegans’ neural activities. On the other hand, it suggests a potential application of machine learning in biological researches.

The author declares no conflicts of interest regarding the publication of this paper.

Chu, Y.T. (2020) Computational Geometric Analysis for C. elegans Trajectories on Thermal and Salinity Gradient. American Journal of Computational Mathematics, 10, 578-590. https://doi.org/10.4236/ajcm.2020.104033