Description

The eye's ability to separate two distinct colors depends on the hue, saturation and brightness of the observed color.
In this model, we want to design a procedure that take the desired image as input and the image needed to be project on the bacteria as output.

Approach

RGB is a convenient color model for computer graphics because the human visual system works in a way that is similar — though not quite identical — to an RGB color space.
We first find the RGB tristimulus values of the image. Then, it is possible to deduce corresponding fluorescent protein quantity that is needed to produce this visual effect. After that, Hill function was used to describe the relationship between gene expression output and light intensity input. While going through all these procedures, previous matrices that represent RGB values are transformed at each step. Finally, three transferred matrices were compound to one picture as output.

Assumptions

1. Emission peaks of three kinds of fluorescent proteins(red, green, blue) located closely to the primary wavelengths which form the three axes for RGB color space respectively. To be specific, the RGB color space created by the Commission Internationale de L'Eclairage defined monochromatic primaries the following wavelength: red----700.0 nm, green---546.1 nm, blue---435.8 nm.
2. From observation, the green light sensing system is much more sensitive to green light than the sensitivities of red and blue system to red and blue light. However, we can modify the green light response to make the three color systems coherent with each other. It has already be shown that in the CcaS-CcaR system, red light competitively inhibits activation by green light. So growing under red light reduce the sensitivity of CcaS-CcaR to green light while preserving the full output range. Here, we assume that the only effect of an additional background illumination of red light is to reduce the sensitivities of green light without much interference to other members in this scenario.
3. The Cph8-OmpR and YF1-Fixj system are both light–intensity sensors so that we simulate their expression by simply increasing the light intensity input.
4. Steady-state response with respect to light intensity of these light sensors can be described by Hill function. Hill function describes the fraction of the macromolecule saturated by ligand as a function of the ligand concentration. In our project, it is used in determining the degree of cooperativeness of the ligand (CcaR, OmpR) binding to the promoters
5. Fluorescent protein quantity is perceptually uniform which means the lightness can be perceived by human’s eye is proportional to the quantity of fluorescent protein thus can be obtained from the RGB values through linear transformation.

RGB components

Here we works with The CIE 1931 color space standard which consisted of monochromatic primaries red, green and blue.
The tristimulus value extracted from sample image can be associated with a RGB color space where axes range for 0 to 255.
We use Matlab to perform this color-protein-color transformation.
By importing the sample image into Matlab, we get its RGB matrix from which we extract each primaries separately so that we have three two-dimensional matrices for red, green and blue. Labeled as r, g, b.

Protein expression output

As mentioned in assumption, we assume that two colors that are equally distant in the color space are equally different in the amount of fluorescent protein, which means relationship between the RGB values and required protein quantity can be characterized by a linear function.

We believe the Hill function can describe the ligand-binding situation in our light-sensing system by determining the degree of cooperativeness of the ligand binding to receptors. This has also been demonstrated in previous papers.
$$P=b+\frac{aI^n}{I^n+k^n}$$
I-light intensity input.
P-protein quantity produced by sensing light.
For green and blue sensor, required light intensity is thus
$$I_g=k(\frac{a}{P-b} – 1)^{-\frac{1}{n}} + r_0$$
$$Ib=k{\frac{a}{(P-b)-1}^{(-1)/n}$$

However the green sensor needs to grow under red light to perform a similar response as the red one.
For red sensor
$$Ir=k{\frac{a}{(P-b)-1}}^{(-1)/n}+r_0$$
a, b, k- Hill coefficients. From data provided in paper [1]
bg=19.9;ag=65.3;ng=2.49;kg=0.138;
br=10.6;ar=97.9;nr=1.42;kr=0.0239;
bb=16.0;ab=75.3;nb=1.58;kb=0.03;

(Hill coefficients for blue sensor was assumed)

We select a proper area for linear transformation.
Light intensity 0-0.4
Protein quantity 20-80
Thus RGB (0-255) will be transfer to protein (20-80) following the equation:
protein=0.2353R/G/B +20
I=k(a/(0.2353r+20-b)-1)^((-1)/n)*255/0.4(+r0)
Constant r0=150

elements in r, g, b matrices (tristimulus values of desired image)

Elements in transferred r, g, b matrices which we simplify by calling matrix R, G and B. Here we assume that the red light doesn't prohibit the growth and protein expression of blue bacteria. The only effect of the additional background red light is to manipulate the sensitivity of green light inducible bacteria.

To illustrate this point, we test their growth in tubes.

We observed cell populations containing the blue light inducible system grown in four tubes under same conditions, however are differently illuminated with red light, green light and blue light and in dark.

Their growing curves are approximately the same with acceptable deviation. The growth of dark exposed bacteria slightly surpassed others since they eat a lot and produce less, which should has nothing to do with repress from the light.

We may conclude that neither red nor green light has repressive effects on blue inducible bacteria.

matrix R plot in 3D space
matrix G plot in 3D space
below is the image we finally get through this whole procedures.