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Team:XMU-China/Project Modelling sdmodel
From 2014.igem.org
STIMULUS DIFFUSION MODEL
Pigment Diffusion Experiment
Why Pigment?
In our characterization experiments, we desire to get concentration gradient of stimulus on semi-solid culture medium. However, the work is difficult because of measurement obstacle. So we practice pigment diffusion experiment to explore the regularity of the concentration changing versus distance.
Experimental Process and Results
Spotting 5μL pigment on the center of semisolid culture medium. By measuring the pigment diffusion diameters at different time, we got the following plot (Figure 1). We found that the diffusion rate almost keeps constant.
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Figure 1. The plot of pigment diffusion diameter versus time. |
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After that, we apply grey processing to measure pigment concentration at different distance away from pigment center (Figure 2A). We set 1.0 as the center concentration, then we got the plot of relative concentration versus relative distance (Figure 2B). From the plot, concentration keeps decreasing slowly near the center. Then the concentration decreases linearly with the distance away from center increasing (see further explanation in
Presupposition).
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Figure 2A. Pigment diffusion experiment applied on semisolid culture medium. 2B. The plot of relative concentration versus relative distance. |
We use two kinds of pigments, Ponceau 4R and Sunset Yellow,(Figure 1) to repeat experiments above. Although both pigments defer significantly in molecular formula, we got almost the same result. Thus, it’s reasonable to assume that stimuli (IPTG and L-arabinose) will perform similarly as those pigments.
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Figure 3A. Molecular formula of Ponceau 4R 3B. Molecular formula of Sunset Yellow |
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Presupposition
According to the diffusion equation, if the diffusion coefficient depends on the density then the equation is nonlinear, otherwise it is linear. [1] The results of pigment diffusion experiment are analyzed by the software, ImageJ. On account of the data, we write a program by matlab and get the best exponent, 1, to fit the polynomial, i.e. the diffusion is linear (see igem_fit.m).
Identify the stimulus effect on E.coli is SE. SE of inducers (eg. IPTG) is positive and SE of repressors (eg. L-arabinose) is negative. And the effects of inducers and repressors reach equilibrium at SE=0. SE is proportional to the concentration as the following equation.
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\(\frac{{{\rm{S}}{{\rm{E}}_1}}}{{{\rm{S}}{{\rm{E}}_2}}} = \frac{{{{\rm{K}}_1}{{\rm{C}}_1}}}{{{{\rm{K}}_2}{{\rm{C}}_2}}}\) |
The diffusion of the concentration is linear versus distance, so is SE.
Representation Model
Eclipse
Add L-arabinose to the culture medium evenly. Assume that the effect of the L-arabinose is SE1 (fixed). Draw two spots of IPTG at certain concentrations apart at proper positions. Each of their effect is SE2 (changing with diffusion). According to the diffusion model, the gradient of the concentration is simplified to linear trend, so is SE. A three-dimensional model is built with x, y and SE of IPTG on certain position spreading from the single-point sources. Then we get flanks of two cones. Add up effect of IPTG spreading from the double-point source to SE. At SE=0, the effects of IPTG as inducer and L-arabinose as repressor reach equilibrium and a critical line is formed. We can draw isolines on the three-dimensional graph and simulate the edge of the pattern formed by the colony on the semi-solid culture medium plate.
(see igem_ellipse.m)
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Figure 4A. The three-dimensional model built with x, y and SE of IPTG on certain position spreading from two single-point sources -- flanks of two cones. 4B. Add up the effect of IPTG spreading from the double-point source to SE and get the final model. 4C. Isolines of the graphics in 4B. 4D. The vertical view of the isolines in 4C. The edges are eclipses. |
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Hyperbola
Draw a line of L-arabinose at an effect of SE1 and a spot of the mixture of IPTG and bacteria solution at an effect of SE2 apart at proper positions. According to the diffusion model, the gradient of SE is simplified to linear trend. Similarly, a three-dimensional model is built in which we can get a half-plane and a flank of a cone. The tilt of the plane represent the changing extent of SE versus distance. The half-plane and the flank intersect at a line where SE=0, and the effects of IPTG as inducer and L-arabinose as repressor reach equilibrium. The vertical view of the line is a branch of hyperbola, which is the edge of the colony on the semi-solid culture medium plate. (see igem_hyperbola.m)
Adjust the tilt of the plane until the shape of the branch matches the colony edge. According to the tilt and the original concentrations of the stimuli, we can work out the ratio of the stimulus concentrations at SE=0. And it is the reciprocal of the SE of the stimuli of the same concentration.
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(when \({\rm{S}}{{\rm{E}}_1}\)= \({\rm{S}}{{\rm{E}}_2}\)) |
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\(\frac{{{{\rm{C}}_1}}}{{{{\rm{C}}_2}}} = \frac{{{{\rm{K}}_2}}}{{{{\rm{K}}_1}}}\) |
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(when \({{\rm{C}}_1}\)= \({{\rm{C}}_2}\)) |
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\(\frac{{{\rm{S}}{{\rm{E}}_2}}}{{{\rm{S}}{{\rm{E}}_1}}} = \frac{{{{\rm{K}}_2}}}{{{{\rm{K}}_1}}}\) |
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Figure 5A. The three-dimensional model built with x, y and SE of stimuli spreading from L-arabinose line and IPTG point—a half-plane and a flank of ta cone. 5B. Add up SE of the stimuli and get the final model. 5C. Isolines of the graphs in 5B. 5D. The vertical views of the isolines in 5C are in the shape of hyperbola, one of which is the edge of the colony. |
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Quasi-hyperbola
Draw a spot of L-arabinose at an effect of SE1 and a spot of the mixture of IPTG and bacteria solution at an effect of SE2 apart at proper positions. Similarly, a three-dimensional model is built in which we can get flanks of two cones. The tilt of the flanks represent the changing extent of SE versus distance. The flanks intersect at a line where SE=0, and the effects of IPTG as inducer and L-arabinose as repressor reach equilibrium. The vertical view of the line is a branch of quasi-hyperbola, which is the edge of the colony on the semi-solid culture medium plate. (see igem_quasihyperbola.m)
Adjust he tilt of the flanks until the shape of the branch matches the colony edge. According to the tilt and the original concentrations of the stimuli, we can work out the ratio of the stimulus concentrations at SE=0. And it is the reciprocal of the SE of the stimuli of the same concentration.
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Figure 6A. The three-dimensional model built with x, y and SE of stimuli spreading from L-arabinose point and IPTG point -- flanks of two cones. 6B. Add up SE of the stimuli and get the final model. 6C. Isolines of the graphs in 6B. 6D. The vertical views of the isolines in 6C are in the shape of quasi-hyperbola, one of which is the edge of the colony. |
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Reference
1. http://en.wikipedia.org/wiki/Diffusion_equation
