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Bacterial chemotaxis, which is universal in E.coli, is defined as bacteria cells migration in response to a chemical stimulus. The natural E.coli chemotaxis has limited receptor proteins which can bind to only six kinds of amino acid. Nevertheless, the reprogrammed chemotaxis named pseudotaxis makes the bacteria able to respond to molecules, whose receptor proteins do not exist in classical E.coli, such as IPTG and L-arabinose, etc.

 

Figure 1 Chemotaxis   mechanism of E.coli. The direction   of rotation of the flagellar motor is controlled by the protein CheY. If the   CheY is phosphorylated (CheY-P), it can bind to the flagellar motor protein   FliM, causing the cell to tumble. While CheY is not phosphorylated, the   flagellar motor rotates counterclockwise (CCW). [1]

 

E.coli have several flagella per cell (4–10 typically), which can rotate in two ways: counterclockwise (CCW) and clockwise (CW). [2] The former aligns the flagella into a single rotating bundle, causing the bacterium to swim in line, while the later breaks the flagella bundle apart such that each flagellum points in a different direction, causing the bacterium to tumble. The motility is determined by the phosphorylation state of CheY protein governed by CheZ protein. In the presence of CheZ protein, CheY-P is dephosphorylated and produce CheY, and the flagellar motor rotates CCW resulting in swimming. In the absence of CheZ, CheY is phosphorylated to CheY-P which binds to the flagellar switch protein FliM resulting in tumbling (Figure 1). [1] Therefore, we are able to control the bacterial motility by knocking out the CheZ gene of the wild-type then transfecting circuit containing CheZ gene into a CheZ knockout (ΔCheZ) strain. Besides, we introduce aptamers responding to a mass of specific molecules which can be applied to regulate gene expression, in our project, CheZ (Figure 2).

 

Figure 2  Mechanism of how aptamers controls the   translation of CheZ protein. In the absence of target molecules (theophylline   as an example). The mRNA’s ribosome binding site is paired, which inhibits   the translation of CheZ protein. In the absence of CheZ, CheY-P will remain   phosphorylated and the cells tumble in place. While in the presence of   theophylline, the mRNA’s ribosome binding site will expose and the CheZ can   be expressed, allowing the cells to run and tumble. [1]

 

Characterizing the circuit we constructed, we combine mathematical modeling with experiments, using modeling to guide experiments and to explain experimental phenomena. .

 

And we can characterize the efficiency of RBS and promoter via migration distance positively associated with the expression strength of CheZ.

 

, Mathematic is the simplest and clearest language, whose value to the development of human civilization is now widely recognized because its extensive application of science, society and daily life. However, the mathematical laws in life sciences is still unclear and even in chaos. Luckily, synthetic biology can overcome these shortcomings on some level. Based on this, we design a gene circuit and expect mathematical regularities to realize the regulation and control of life activities. We hope our work can inspire people's interests to combine mathematic with synthetic biology.

 

参考文献

[1] Topp, Shana, and Justin P. Gallivan. "Guiding bacteria with small molecules and RNA." Journal of the American Chemical Society 129.21 (2007): 6807-6811.

[2] http://en.wikipedia.org/wiki/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 at introducing pseudotaxis of bacteria to form patterns in shape of conic section (such as ellipse and hyperbola). Firstly, let’s make precise mathematical definition on ellipse, hyperbola and parabola.

 

What are ellipse, hyperbola and parabola?

 

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 1).

Figure 1 Schematic of   ellipse. Point F1, F2 are the two focal points and Point A is on the ellipse   curve. The sum of the distance AF1 and AF2 is equal to the constant k:   AF1+AF2=k.

And Hyperbola is 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. (Figure 2B)

Parabola is a conic whose eccentricity is equal to 1 (Figure 2C).

   

A

B

Figure 2A, Schematic of   hyperbola. Point F1, F2 are the two focal points and Point A is on one of the   hyperbola branches. The absolute value of the difference of the distance AF1,   AF2 is equal to the constant k: |AF1-AF2|=k. 2B, one branch of hyperbola can be defined by directrix and   eccentricity. Point A is a spot on the curve, point F is the focus. AB is the   distance between A and directrix. The eccentricity e equal to TF/AB, and   e>1.

 

Figure 2C Schematic of   Parabola. Point A is a spot on the curve, point F is the focus. AB is the   distance between A and directrix. The eccentricity e equals to AF/AB, and   e=1.

 

 

Hypothesis

 

We find that the focal points, the constant and the ratio 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 put forward the following hypothesis:

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. And the concentration gradient of the stimulus will be formed and maintained for a long period.

2. There is a threshold ratio of the concentrations of inducer and repressor. This means that more repressor will cause more repression, hence more inducer is needed to relieve the repression, and vice versa.

 

Design of circuit

 

Our circuit consists of two parts, of which one is named C (constraint), the other is named M (motile). (Figure 4):

 

1. We build our circuit in E.coli CL-1 which lacks gene lacI and CheZ (ΔlacI, ΔCheZ). At 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. Even when L-arabinose isn’t involved in, promoter pBAD has expression leakage, so that part C will produces protein 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, more L-arabinose involved in means that part C could produce more LacI to repress the expression of pLac. Because of its ability to constrain chemotaxis, this part is named C (abbreviate from constraint).

3. When IPTG is involved in, it can relieve the repression from 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, more cheZ is produced with more IPTG involved in leading to stronger motile ability. Because of its ability to make CL-1 motile, this part is named M (abbreviate from motile).

Figure 4 Part C produces LacI to repress the   expression of part M. Part M could produce CheZ to make CL-1 regain motile ability.

 

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 amount of LacI protein.

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.

At first, we need to find out at which chloramphenicol concentration we could get the best chemotaxis performance. We try gradient concentration of chloramphenicol at semi-solid medium culture as Table 1 show. We find that the chemotaxis performance doesn’t have overt linear relationship to chloramphenicol. Interestingly, 50μg/ml of chloramphenicol gives CL-1 the best chemotaxis. So we applied that to our following characterization.

 

Characterization of circuit

We sequence the circuit above and characterize it in E.coli CL-1. As CL-1 lacks LacI gene, promoter pLac won’t be repressed by background amount of LacI protein.

 

We design gradient tests of the concentration of chloramphenicol, IPTG and L-arabinose to find out their influence on reprogrammed chemotaxis. 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.

 

 

Table 1 Curve of chemotaxis diameter under gradient concentration of   Cm (chloramphenicol).

 

As promoter pBAD leads to a certain level of expression leakage of LacI, CL-1 has the worst chemotaxis. We added IPTG at gradient concentration and got the results (Table 2). We find that the chemotaxis performance keeps increasing when the concentration of IPTG increases from 0 and 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.

 

Table 2 Curve of chemotaxis diameter over time under gradient   concentration of IPTG.

 

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, chemotaxis performance 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.

 

Table 3 Curve   of chemotaxis diameter under gradient concentration of L-arabinose.

 

 

Eclipse

 

If we spot IPTG on the center of the semi-solid culture medium, concentration gradient will be formed as Figure 3A. Larger circle represents lower concentration with a lower number labeled.

A	B
Figure 3 Schematic of concentration   gradient. A, single point   spreading model. Larger number represents higher concentration while smaller   number represents lower concentration. B,   double points spreading model. Letter A~J represent equal concentration   points on the ellipse with two labeled number added up to 6. Red curve   represents the ellipse with two focal points on the IPTG spots.

If we spot two IPTG simultaneously, two spots’ concentration gradient will interact with each other to form the ellipse boundary as image in Figure 3B.

 

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 the image in 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 while the bacteria density inside the critical line will decline. Thus, an ellipse boundary is formed.

Figure 5 Schematic of   Critical Line model for ellipse. Critical line represents the IPTG   concentration which can just relieve the repression from part C. So that CL-1 is motile inside the ellipse while non-motile outside the   ellipse.

 

 

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 r