Conic Curve

--Form patterns by chemotaxis

Overview

What means would you like to use to get a mathematical pattern? Draw one with compass and ruler, or type a function in a drawing software? Well, E.coli can help us to achieve our goals. We make the first attempt to introduce pseudotaxis to form patterns in shape of conic section (such as ellipse and hyperbola). Firstly, let’s recall the precise mathematical definition on ellipse, hyperbola and parabola.

Mathematical definition

In mathematics, ellipse is a curve on a plane surrounding two focal points such that a straight line drawn from one of the focal points to any point on the curve and then back to the other focal point has the same length for every point on the curve (Figure 1A). [1] And hyperbola is a curve which point’s absolute value of the difference of the distance to the two focal points is a constant (Figure 1B).

Hyperbola could also be defined as a conic consisting of those points whose distances to some point, called a focus, and some line, called a directrix, are in a fixed ratio (>1) , called the eccentricity (e). (Figure 2A) Parabola is a conic whose eccentricity is equal to 1 (Figure 2B). [2]

Hypotheses:

The focal points, the constant k and the eccentricity ratio e are the key of conic section. Therefore, we can easily get any eclipse by presetting reasonable focal points with the acceptable constant, as well as parabola or hyperbola by a fixed ratio. Based on these, we need to combine those mathematical concept with our pattern formation system.

1. If we spot stimulus on semi-solid plate, it will spread from the spotting center out to the periphery. In the spreading process, the concentration is negative correlation to the distance from center. If we draw a line with stimulus, a concentration gradient with square shape will be formed. And the both concentration gradient will maintain for a long period. Generally, the stimulus spots are defined as focuses while those lines are defined as directrixes.

2. There is a threshold ratio of the concentrations of inducer and repressor. [3] This means that more repressor will cause more repression, hence more inducer is needed to relieve the repression, and vice versa. As the concentration of inducer and repressor are directly connected to the concentration of stimulus sources, thus we can tell the constant k and eccentricity e by combining the relationship between distance and concentration.

Circuit design

Our circuit consists of two parts, of which one is named C (constraint), the other is named M (motile). (Figure 3):

1. We build our circuit in E.coli CL-1 which lacks gene lacI and CheZ (ΔlacI, ΔCheZ). In the absence of CheZ, CL-1 adopts non-motile phenotype.

2. Without any exogenous stimulus, E.coli will produce background amount of AraC to repress pBAD in limit degree. That means even no L-arabinose involves in, promoter pBAD has leakage expression, so that part C will produces repressor LacI which can bind to the operon of promoter pLac and thus repress its transcription. Because L-arabinose could induce pBAD, within certain concentration range, the more L-arabinose involves in, the more repressor LacI part C could produce resulting the inhibition to chemotaxis. Because of its ability to constrain chemotaxis, this part is named C.

3. When IPTG involves in, it can relieve the repression from repressor LacI, therefore protein CheZ is produced to make our engineering bacteria (CL-1) regain motile ability. Within certain L-arabinose concentration range which means certain constraint condition, the more IPTG involved in, the more CheZ is produced leading to stronger motile ability. Because of its ability to make CL-1 motile, this part is named M.

Characterization of circuit

We sequenced the circuit above and characterized it in E.coli CL-1. As CL-1 lacks LacI gene, promoter pLac won’t be repressed by background repressor LacI.

We apply gradient test to find out which influence would be made on reprogrammed chemotaxis under the following parameters: the concentration of chloramphenicol, IPTG and L-arabinose.

1. Characterization of backbone effect:

To begin with, we test for the best chloramphenicol concentration. We test gradient concentration of chloramphenicol at semi-solid medium culture as Table 1 show. We find that the activity of chemotaxis doesn’t have overt linear relationship to chloramphenicol. Interestingly, 50μg/ml of chloramphenicol gives CL-1 the best chemotaxis. So we apply that to our following characterization.

2. Characterization of IPTG effect

As promoter pBAD leads to a certain level of expression leakage of LacI, CL-1 has the worst chemotaxis without any stimulus. We added IPTG at gradient concentration and got the results (Table 2). We find that the activity of chemotaxis keeps increasing when the concentration of IPTG increases from 0 to 0.02μM and gets the best performance with the IPTG range from 0.02 to 0.025μM. We apply 0.025μM IPTG for our following characterization.

3. Characterization of L-arabinose effect

As more L-arabinose added in, the expression from promoter pBAD will be stronger which leads to more LacI produced resulting in the inhibition to chemotaxis. As our expectation, the activity of chemotaxis keeps going down as the concentration of L-arabinose increases (Table 3). We find that 0.2% of L-arabinose has the best inhibitory effect on chemotaxis with 0.025μM of IPTG added in.

Results

Eclipse

If we spot IPTG on the center of the semi-solid culture medium, concentration gradient will be formed as Figure 4A. Larger circle represents lower concentration with a lower number labeled.

If we spot two IPTG simultaneously, two spots’ concentration gradient will interact with each other to form the ellipse boundary as image in Figure 4B.

If we spot E.coli CL-1 with circuit above transformed in at the center between two IPTG spots on semi-solid culture plate as Figure 5. The concentration of L-Arabinose added in the culture medium determines the degree of constraint from part C. Each red ellipse curve represents equal IPTG concentration, and there are a series of such ellipses from inside to outside represent IPTG concentrations from highest to lowest. One of the ellipses is the critical line indicating that the constraint from part C can just be relieved by the certain concentration of IPTG. Initially, as the concentration of IPTG is enough to relieve the constraint, bacteria can swim from the center out to periphery. When the bacteria swim out of the critical line, the concentration of IPTG can’t relieve the constraint, so the

bacteria adopt non-motile phenotype. On the contrary, when the bacteria are inside of the critical line, they adopt motile phenotype. When the bacteria swim from inside to outside, motile bacteria become non-motile, so the bacteria will aggregate outside the critical line thus the bacteria density inside the critical line will decline. Thus, an ellipse boundary is formed.

Parabola and hyperbola

We got the optimum concentrations of IPTG and L-arabinose as inducer and repressor through preliminary experiments. Because the concentration of stimulus will decrease during spreading, so does their effect, we use IPTG and L-arabinose of which concentrations are a little bit higher than the optimum values for our experiments.

We draw a straight line with L-arabinose on the semi-solid culture medium, and a spot with the mixture of IPTG and CL-1 on one side of the line. In the area around the spot, the induction of IPTG is stronger than the repression of L-arabinose, CheZ is expressed and the bacteria adopt motile phenotype. However, when they approach the line where the repressor have a greater effect on the motility, they will lost their motile phenotype and stop.

According to the hypothesis 2, on the threshold ratio of the concentration of inducer and repressor, their effects are offset and a critical line is formed. Distances of the points on the critical line to the IPTG spot (focus), and the L-arabinose line (directrix) are in a fixed ratio (eccentricity). If the ratio is equal to 1, the critical line is a parabola. If the ratio is larger than 1, it is a branch of a hyperbola. Then we conducted experiment to verify the mechanism (Figure 6B). Left boundary of the colony is regarded as the critical line. We found that cells has the tendency to swim away from L-arabinose line which is an expected performance of the circuit.

Other function curves-- Quasi-hyperbola

After explorations, we tried different ways to arrange bacteria and stimulus on semisolid culture medium. We got some interesting results.

We draw two spots on the semi-solid culture medium, one with L-arabinose (Figure 7A) and the other with the mixture of IPTG and CL-1. Similarly, on the threshold ratio of the concentration of inducer and repressor, their effects are offset and a critical line is formed. Distances of the points on the right side of colony boundary to the spot A and the spot B are in a very narrow ratio (Figure 7B). Actually, as the critical line is quiet similar to hyperbola, we name it quasi-hyperbola.

Other interesting ways to spot bacteria and stimulus are waiting to be discovered, and the idea can be extended to other function curves and patterns.

We took two kinds of stimulus to construct a square (Figure 8A). Two opposite sides were paved by one kind of stimulus to form thin paths. Programmed cells were spotted on the center of the square, we got two oval rings after 24 hours culturing (Figure 8B). We find that cells rings are stretched by IPTG sides while squeezed by L-arabinose sides.

References

1. http://en.wikipedia.org/wiki/Ellipse

2. http://en.wikipedia.org/wiki/Hyperbola

3. O’Gorman, R. B., Rosenberg, J. M., Kallai, O. B., Dickerson, R. E., Itakura, K., Riggs, A. D., & Matthews, K. S. (1980). Equilibrium binding of inducer to lac repressor.operator DNA complex. Journal of Biological Chemistry, 255, 10107–10114.

http://www.jbc.org/content/255/21/10107.abstract