Overview

Modeling is usually used to make sense of the experimental discovery in traditional biological studies. In this synthetic biology project, we believe that carefully carried out modeling will be critical for the experimental design and data analysis at different stages of the project. We hope to demonstrate that the modeling is especially helpful to finalize the microfluidics chip design, and determine the distribution of the C.elegans in microfliudics, as well as the transfection of the MiniMos in our experiment.

1. Microfluidic

In our project, we are required to insert a new gene into the C.elegans’ genome and express the two channelrhodopsins in the olfactory receptor neuron pair. Because of the exogenous genes, we need to test if those exogenous genes will influence the olfactory receptor neuron pair. Therefore, we designed the Gaussian plate (Fig.1.1) a microfluidic chip.^[1] Using this chip we can know the C.elegans’ favors and repulsions through their distribution in the chip.

Fig.1.1 The Gaussian plate

In order to simulate the distribution of the C.elegans we need to know the time that the chemicals diffuse to a specific position, for example ‘a’ (Fig.1.1) the chemicals need to go through 3.1mm of the PDMS. We need to know the diffusion of diacetyl (C.elegans prefers it) and 2-nonanone (C.elegans detests it^[2]) in Polydimethylsiloxane (PDMS), a kind of transparent and air permeable material used in microfluidics (Fig.1.2). Therefore, we came up with following modeling.

Fig.1.2 Schematic representations of interstitial diffusion.

1.1 Diffusion

For nonsteady-state diffusion, there is an equation named Fick’s second law^[3]: \frac{\partial C}{\partial t}=D \frac{\partial^2 C}{\partial x^2}

The constant of proportionality D is called the diffusion coefficient, which is expressed in square meters per second. Concentration C is plotted versus position (or distance) within the solid x and the time t.

For t=0, C=C₀ at 0≤x≤∞;

For t>0, C=C_s (the constant surface concentration) at x=0;

C=C₀ at x=∞. And then there is a general solution: \frac{C_x-C_0}{C_s-C_0}=1-erf(\frac{x}{2 \sqrt{Dt}}) Where C_x represents the concentration at distance x after time t.

\frac{x}{2 \sqrt{Dt}}=constant, so if this constant can be calculated, the equation of distance x and time t can also be gotten.

In our model, C₀=0, so the equation is changed to \frac{C_x}{C_s}=1-erf(\frac{x}{2 \sqrt{Dt}}) C_x represents the concentration at distance x that C.elegans can feel. We set that the C_x = 1uM (for diacetyl) ,the optimum concentration can be sensed by the C.elegans.

C_s is the concentration set by our own and it is a constant. We set that the C_s = 2.3uM (for diacetyl), x is from 0mm to 7.4mm.

According to the equation we can get that the erf (\frac{x}{2 \sqrt{Dt}})=0.56 and then referred to the Table 1.1 we can get that the \frac{x}{2 \sqrt{Dt}}=0.55, and D is the diffusion coefficient for 2-nonanone or diacetyl in PDMS. We need to calculate the D to get the time t according to the equation. We used the force-field method to get the D of those two chemicals (the concrete operational process is shown in 1.3).

D_(2-nonanone) = 6.78*10^-6cm²/s

D_(diacetyl) = 6.78*10^-6cm²/s

Finally, we can get the time t = 1.5 hours, on the other words, the distribution of the C.elegans going through the chip during 1.5 hours will shift to the chemicals (for diacetyl).

1.2 The Calculation of Diffusion Coefficient(D)

The diffusivity of a gas in an organic solvent, polymer, or zeolite can be calculated by running a molecular dynamics simulation and determining the mean square displacement of the gas in the material. This allows us to calculate the self-diffusivity coefficient of the gas and gives an insight into the overall diffusivity. As we are performing a molecular dynamics calculation, we can analyze the effect of temperature, pressure, density, and penetrant size and structure on diffusion. We used force-field method to calculate the D in PDMS by Material Studio (MS) a software for material calculation.^[4][5]

1.2.1 Set up the initial structures

First, we set up the initial structure of the PDMS and the chemicals (Fig.1.3 and Fig.1.4).

Fig.1.3 The structure of PDMS

Fig.1.4 The structure of diacetyl

1.2.2 Build an amorphous cell

Then, we put the two molecules in to an amorphous cell (Fig.1.5).

Fig.1.5 The amorphous cell

1.2.3 Relax the cell

When we generate an amorphous cell, the molecules may not be equally distributed throughout the cell, creating areas of vacuum. To correct this, we must perform a short energy minimization to optimize the cell. After the minimization, we should run a short molecular dynamics simulation to equilibrate the cell.

Fig.1.6 The cell after relaxing

1.2.4 Run and analyze molecular dynamics

1.2.5 Export data and calculate the diffusivity

Then we can got the D of 2-nonanone and diacetyl:

D_(2-nonanone) = 6.78*10^-6cm²/s

D_(diacetyl) = 6.78*10^-6cm²/s

Fig.1.7 The diffusion coefficient of 2-nonanone in PDMS

Fig.1.8 The diffusion coefficient of diacetyl in PDMS

2. miniMos Transfection

In our program, we aim to change the physical behaviour of C. elegans using the two specific optogenetic traits.Through microinjection and selection, we are able to get two strains with two phenotypes of the preference to blue light and the aversion to red light , and the next step is to obtain the single worm with the combination ofe these 2 different traits. Therefore, we build a miniMos transfection model to simulate and estimate the final results of our transfection.

Fig 2.1 The overview of miniMos Transfection Model

2.21 The MiniMos successful rate

First considering the successl rate of miniMos system. The microinjection needs acurate operation to correctly inject plasmids into C. elegans gonads, which allows plasmids have chances to enter the embryos of offsprings and maintain the phenotypes. Therefore, the more accurate manual operations can lead to a more promising results. According to the miniMos protocol, the probability of correct injection is approximately 60% due to manual errors, and about 10% of F1 offsprings are all rescued, meaning that the plasmid expression rate(transposon transfer target gene into the chromosome) is about 10%.^[6] So the successful rate is:

       accurate rate of manual operation * plasmid expression rate = 60% * 10% = 6%.

Fig 2.2. The MiniMos

2.22 The stable transfection rate

After got the heterozygote worms via miniMos transfection, the next step is to get the stable expression worms in F2. During the F2 generation, based on the Mendel's laws, if two genes are located on different chromosomes, their alleles will segregate independently during the formation of gametes. And Mendal calculated the possiblity of getting stable expression offsprings, the ideal rate is 1/16. Therefore, in our experiment, the prospecting result of getting target strains is also 1/16.

2.23 The ideal result

Therefore, the overall probability to get the target strain is: accurate rate of manual operation * plasmid expression rate * F2 stable expression rate * 100% = 60% * 10% * (1/16) *100% = 0.375%.

Fig 2.1 The possibility of target strains by Mendel's laws

2.24 The effect of crossing over

However, this probability does not consider the chance of crossing over. Crossing over is the exchange of genetic material between homologous chromosomes and results in recombinant chromosomes during sexual reproduction.^[7] Therefore, ceossing over may affect the rate of stable transfection by exchanging genetic materials. Accroding to the formula, the rate of crossing over is: (the number of recombinant genotype offsprings)/(the number of total offsprings)*100% Therefore, the possible rate of crossing over in our experiment is 10/354 = 2.8%. And the final rate is 0.375% * (1 - 2.8%) = 0.3645%.

References

1. ↑ Albrecht, D. R. and C. I. Bargmann (2011). “High-content behavioral analysis of Caenorhabditis elegans in precise spatiotemporal chemical environments.” Nature Methods 8(7): 599-605.
2. ↑ Troemel, E. R., Kimmel, B. E., & Bargmann, C. I. (1997). Reprogramming chemotaxis responses: sensory neurons define olfactory preferences in c. elegans. Cell, 91(2), 161-9.
3. ↑ Callister, W. D., & Rethwisch, D. G. (2004). Fundamentals of Materials Science and Engineering. John Wiley and Sons Ltd.
4. ↑ S. G. Charati† and, & Stern, S. A. (1998). Diffusion of gases in silicone polymers:  molecular dynamics simulations. Macromolecules, 31(16), 5529-5535.
5. ↑ Hofmann, D., Fritz, L., Ulbrich, J., Schepers, C., & Böhning, M. (2000). Detailed‐atomistic molecular modeling of small molecule diffusion and solution processes in polymeric membrane materials. Macromolecular Theory & Simulations, 9(6), 293–327.
6. ↑ [6]http://www.wormbuilder.org/minimos/commentsfaq/
7. ↑ [7]Griffiths, AJF; Gelbart, WM; Miller, JH; et al. (1999). "Modern Genetic Analysis: Mitotic Crossing-Over". New York: W. H. Freeman.