# Modeling

## Why did we model?

Our goal of this part was to develop the dynamic model of the expression of our outputs, to precisely describe, predict and control the expression of the proteins and the generation of our colors. What’s more, our modelling also provided instructions for our experiments.

## What have we done?

Although various actuators were used in our project, we finally chose fluorescent proteins to build our models because fluorescence could be measured easily by ELIASA (microplate reader) and flow cytometry to get quantitative results, and the expression period of fluorescent proteins is much shorter than those of chromoproteins and enzymes. Besides the modelling of the expression of proteins, we also modelled the light intensity distribution in our hardware, to further optimize our hardware to get evener light on the plates and 96-well plates.

Our model consisted of six parts. In part 1, we established model about free growth of bacteria. In part 2, we discussed the influence of light on the growth of bacteria. In part 3, the expression of fluorescent proteins over time was described. In part 4, the effect of illuminance on the expression of fluorescence was shown. In part 5, we combined the models in part 3 and part 4, drawing a general view about how the expression of fluorescence changed with time and illuminance. In part 6, we introduced how we built models about our hardware and optimize the design of our hardware.

## How did the models improve our project?

Our model was tightly combined with other parts of our project, especially our experiment and hardware. The part 2 of model provided methods for experiment to make the growth rate of bacterium on same plate even. The part 4 of model revealed how to get wanted R/G/B of color by changing the wavelength of projected light. The part 5 of model shew how to get wanted fluorescence intensity by adjusting the time and illuminance. The part 6 of model gave evidence on the feasibility of hardware improvement.

## Part1 Dynamics of Free Growth

We created a model to simulate the process of bacteria’s free growth.

Questions to answer:

1. How fast the bacteria grow?

2. Is there growth difference between bacteria carrying different plasmids of outputs?

According to Logistic function [1]:

$$\frac{dN}{dt} = \frac{rN(K-N)}{K}$$

##### N refers to the number of bacteria,r refers to the growth rates,K refers to the carrying capacity.

The value of OD600 is proportional to the number of bacteria in a certain interval, so it was used to present the number of bacteria.

Parameters can be obtained by least square method to fitting the curve into the experiment data.

##### Figure 1 Experimental OD absorption fitting curve.

We considered the growth of bacteria carrying different plasmid, which corresponding to two of our outputs. It can be seen that different plasmids bring different metabolism burden to bacterium. The total expression of outputs can toughly be considered to be proportional to the number of bacteria. The model of free growth is important to predict the total expression, and is also basic for other models.

## Part 2 Light’s influence to the growth of bacteria

We developed a model to assess the influence of illuminance to the growth of bacteria.

Questions to answer:

1. How does the illuminance change with distance?

2. What is influence of illuminance to the growth of bacteria?

3. In which illuminance the change rate of growth is lowest?

We measure the illuminance in different distance and get an experience formula describing how illuminance change with distance. Let I(lux) be the illuminance, let x(cm) be the distance, we get:

##### Figure 2 Illuminance of different distance.

Let h(cm) be the vertical distance from the plate to the light, and d(cm) be the horizontal distance from the light. We get:

$$x^{2}=d^{2}+h^{2}$$

And by measuring d, h, OD600, we get the curve.

##### Figure 3 The influence of illuminance to growth of bacteria.

##### Figure 4 Change rate of OD600.

It can be seen from the figure 3 that the light has a negative effect to the growth of bacteria. And larger the illuminance is, slower the bacteria grow. Figure 4 is obtained by the curve from Figure 3, which describes the absolute value of change rate of OD600. And the decrease rate is lowest when the illuminance is around 2200lux, in which condition the growth of bacteria is even。

## Part 3 How expression of fluorescent changes with time.

We created a kinetic model to simulate the dynamics of the fluorescent expression system we sued.

Questions to answer:

1. What are the reactions happened in this process?

2. How does the concentrate of outputs change with time?

The following reactions were modelled:

1.$$\varnothing\xrightarrow[k_1]{Light} OmpR1$$ 2.$$\varnothing\xrightarrow[k_2]{Light} OmpR2$$ 3.$$\varnothing\xrightarrow[k_3]{Light} OmpR3$$ 4.$$DNA_{CI}\xrightarrow[k_4]{OmpR1,RNAP}mRNA_{CI}$$ 5.$$DNA_{CGG}\xrightarrow[k_5]{OmpR2,RNAP}mRNA_{CGG}$$ 6.$$DNA_{phIF}\xrightarrow[k_6]{OmpR3,RNAP}mRNA_{phIF}$$ 7.$$mRNA_{CI}\xrightarrow[k_7]{Rib}CI$$ 8.$$mRNA_{CGG}\xrightarrow[k_8]{Rib}CGG$$ 9.$$mRNA_{phIF}\xrightarrow[k_9]{Rib}phIF$$ 10.$$DNA_{K1F}\xrightarrow[k_{10}]{CI RNAP}mRNA_{K1F}$$ 11.$$DNA_{T3}\xrightarrow[k_{11}]{phIF RNAP}mRNA_{T3}$$ 12.$$Core+mRNA_{K1F}\xrightarrow{k_{12}}Core:mRNA_{K1F}$$ 13.$$Core+mRNA_{CGG}\xrightarrow{k_{13}}Core:mRNA_{CGG}$$ 14.$$Core+mRNA_{T3}\xrightarrow{k_{14}}Core:mRNA_{T3}$$ 15.$$DNA_{RFP}\xrightarrow[k_{15}]{Core:mRNA_{K1F}}mRNA_{RFP}$$ 16.$$DNA_{GFP}\xrightarrow[k_{16}]{Core:mRNA_{CGG}}mRNA_{GFP}$$ 17.$$DNA_{BFP}\xrightarrow[k_{17}]{Core:mRNA_{T3}}mRNA_{BFP}$$ 18.$$mRNA_{RFP}\xrightarrow[k_{18}]{Rib}RFP$$ 19.$$mRNA_{GFP}\xrightarrow[k_{19}]{Rib}GFP$$ 20.$$mRNA_{BFP}\xrightarrow[k_{20}]{Rib}BFP$$ 21.$$mRNA_{CI}\xrightarrow{k_{21}}\varnothing$$ 22.$$mRNA_{CGG}\xrightarrow{k_{22}}\varnothing$$ 23.$$mRNA_{phIF}\xrightarrow{k_{23}}\varnothing$$ 24.$$CI\xrightarrow{k_{24}}\varnothing$$ 25.$$CGG\xrightarrow{{k_25}}\varnothing$$ 26.$$phIF\xrightarrow{k_{26}}\varnothing$$ 27.$$mRNA_{K1F}\xrightarrow{k_{27}}\varnothing$$ 28.$$mRNA_{T3}\xrightarrow{k_{28}}\varnothing$$ 29.$$RFP\xrightarrow[k_{29}]{Rib}\varnothing$$ 30.$$GFP\xrightarrow[k_{30}]{Rib}\varnothing$$ 31.$$BFP\xrightarrow[k_{31}]{Rib}\varnothing$$ 32.$$OmpR1\xrightarrow{k_{32}}\varnothing $$ 33.$$OmpR2\xrightarrow{k_{33}}\varnothing $$ 34.$$OmpR3\xrightarrow{k_{34}}\varnothing $$ 35.$$mRNA_{RFP}\xrightarrow{k_{35}}\varnothing $$ 36.$$mRNA_{GFP}\xrightarrow{k_{36}}\varnothing $$ 37.$$mRNA_{BFP}\xrightarrow{k_{37}}\varnothing $$

The process of expression of RFP and GFP are similar, the process of GFP are different.

We consider the expression of RFP first.

1.$$\varnothing\xrightarrow[k_1]{Light} OmpR1$$ 4.$$DNA_{CI}\xrightarrow[k_4]{OmpR1,RNAP}mRNA_{CI}$$ 7.$$mRNA_{CI}\xrightarrow[k_7]{Rib}CI$$ 10.$$DNA_{K1F}\xrightarrow[k_{10}]{CI RNAP}mRNA_{K1F}$$ 12.$$Core+mRNA_{K1F}\xrightarrow{k_{12}}Core:mRNA_{K1F}$$ 15.$$DNA_{RFP}\xrightarrow[k_{15}]{Core:mRNA_{K1F}}mRNA_{RFP}$$ 18.$$mRNA_{RFP}\xrightarrow[k_{18}]{Rib}RFP$$ 21.$$mRNA_{CI}\xrightarrow{k_{21}}\varnothing$$ 24.$$CI\xrightarrow{k_{24}}\varnothing$$ 27.$$mRNA_{K1F}\xrightarrow{k_{27}}\varnothing$$ 29.$$RFP\xrightarrow[k_{29}]{Rib}\varnothing$$ 32.$$OmpR1\xrightarrow{k_{32}}\varnothing $$

[X] refers to the concentration of X in equations appearing behind.

The sensor sense light and influence the product of OmpR1.

1.$$\varnothing\xrightarrow[k_1]{Light} OmpR1$$ 32.$$OmpR1\xrightarrow{k_{32}}\varnothing $$

k1 refers to generation rate of OmpR1, and is related to the type of light. For a certain light, k1 is a const.

k32 refers to the decay rate of OmpR1, and is related to the concentration of OmpR1.

$$\frac{d[OmpR1]}{dt}=k_1-k_{32}[OmpR1]$$

The transcription of protein CI is regulated by OmpR1.

4.$$DNA_{CI}\xrightarrow[k_4]{OmpR1,RNAP}mRNA_{CI}$$

We get

$$\frac{d[OmpR1]}{dt}=k_1-k_{32}[OmpR1]$$

By the assumption that the decay rate of mRNA is large, the concentration change of mRNA is nearly zero.

$$\frac{d[mRNA_{CI}]}{dt}=k_4[DNA_{CI}]-k_{21}[mRNA_{CI}]=0$$

k4 is a function of [OmpR1] and [RNAP]. The concentration of RNAP can be considered as a const. The regulation of OmpR1 can be expressed as a form of Hill equation [2]. We get:

$k_4=k_4([OmpR1],[RNAP])=\frac{V_4[OmpR1]^{n_1}}{K_4+[OmpR1]^{n1}}$

The translation of CI:

7.$$mRNA_{CI}\xrightarrow[k_7]{Rib}CI$$ 24.$$CI\xrightarrow{k_{24}}\varnothing$$

The concentration of CI can be presented as:

$$\frac{d[CI]}{dt}=k_{7}[mRNA_{CI}]-k_{24}[CI]$$

The expression σKIF is regulated by CI:

10.$$DNA_{K1F}\xrightarrow[k_{10}]{CI RNAP}mRNA_{K1F}$$ 27.$$mRNA_{K1F}\xrightarrow{k_{27}}\varnothing$$

We get

$$frac{dmRNA_{KIF}}{dt}=k_{10}[DNA_{KIF}]-k_{27}[mRNA_{KIF}]=0$$ $$[mRNA_{KIF}]=\frac{[DNA_{KIF}]}{k_{27}}*k_{10}([CI])$$

k10 is a function of [OmpR1] and [RNAP]. The concentration of RNAP can be considered as a const. The repression of CI can be expressed as a form of a transformative Hill equation [2].

$$k_{10}=k10([CI],[RNAP])=\frac{V_{10}}{K_{10}+[CI]^{n_2}}$$

Then σKIF combine with the core, forming Core:mRNAKIF, a complete RNAP. We consider the concentration of Core:mRNAKIF is in direct proportion to the concentration of mRNAKIF[3].The combination is then involved in the transcription of RFP:

15.$DNA_{RFP}\xrightarrow[k_{15}]{Core:mRNA_{K1F}}mRNA_{RFP}$ 35.$$mRNA_{RFP}\xrightarrow{k_{35}}\varnothing $$

We get

$$[Core:mRNA_{KIF}]=\frac{N*[mRNA_{KIF}]}{[mRNA_{KIF}]+[mRNA_{CGG}]+[mRNA_{T3}]}$$ $$\frac{dmRNA_{RFP}}{dt}=k_{15}-k_{35}[mRNA_{RFP}]=0$$ $$[mRNA_{RFP}]=\frac{[DNA_{RFP}]}{k_{35}}*k_{15}$$

k15 is a function of [Core:mRNAKIF], and according to Michaelis-Menten equation[4], the function is in form of:

$$k_{15}=k_{15}[Core:mRNA_{KIF}]=\frac{V_{15}*Core:mRNA_{KIF}}{K_{15}+Core:mRNA_{KIF}}$$

And then is the translation of RFP:

18.$$mRNA_{RFP}\xrightarrow[k_{18}]{Rib}RFP$$ 29.$$RFP\xrightarrow[k_{29}]{Rib}\varnothing$$

We get the last equation:

$$\frac{d[RFP]}{dt}=k_{18}[mRNA_{RFP}]-k_{29}[RFP]$$

Similarly, we can get the equations BFP’s expression:

The progress of GFP’s expression is different, the equations are:

We take expression of fluorescent in red light as example.

##### Figure 5 The concentration of three kinds of fluorescent proteins in red light.

##### Figure 6 The relative concentration of three kinds of fluorescent proteins in red light.

##### Figure 7 The change of blue and green fluorescence when the concentration of mRNAKIF changes.

The data and the curve obtained by equations have similar value and trend, which proves that the simulation is reliable.

We can acquire a better understanding of the light sensor and RNAP system[4] through the model. The relationship between concentration of three flourescence and time are shown in figure5 and figure6.

Figure 7 shows the change of BFP and GFP concentration when the concentration of mRNAKIF is twice, five, and ten times than the concentration in figure 6.

The relationship between concentration of three flourescence and time are simulized, which can help us get assumed flourescence concentration by altering the time.

## Part 4 The light’s influence to the expression of fluorescent protein.

We established models to assess how illuminance and wavelength effect the expression of fluorescent.

Questions to answer:

1. How does illuminance effect the expression of fluorescent?

2. How does wavelength effect the expression of fluorescent?

3. What is the color difference among difference color acquired in different light of wavelength?

We consider the influence of intensity first.

Bacteria are put in different illuminance of light, and it is shown that the illuminance of light has a negative effect to the expression of fluorescent. The expression of RFP gradually increases at beginning several hours, and reaches a plateau.

##### Figure 8 Fluorescence in different illuminance of red light.

##### Figure 9 The RFP value of plateau in different illuminance of red light.

The expression of fluorescent decreases when the illuminance increases though the light can motivate the light sensor, indicating that at a certain illuminance, the switch is open and will not change even the illuminance is larger. As the increase of illuminance, the growth and expression of fluorescent are slower, causing the decrease of fluorescent.

Except illustrating the influence between wavelength and fluorescent, we choose another way to show the influence of wavelength more directly. We project 11 kinds of different monochromatic light to plate with our bacteria in a certain intensity, and motivate three kinds of fluorescent separately, take the photo in a same condition, and compound the three pictures. As expected, the compound pictures are nearly in one color. We get the color’s RGB value, and get a fitting curve.

##### Figure 10 The relationship between wavelength and expression of fluorescent.

##### Figure 11 The RGB value of color obtained from plate and fitting curve.

##### Figure 12 The relative RGB value of color obtained from plate and fitting curve.

This curve illustrates how our system responses to different excitation wavelength, which meets the wave band of visible.

Then we analyze the color difference based on human’s vision among different colors.

##### Figure 13 The color difference among different colors.

In Figure 13, the color in diagonal line is dark blue, manifesting the color difference is zero. Smaller the color difference is, closer the color in cross point is to blue. Huger the color difference is, closer the color in cross point is to yellow.

The color obtained in similar wavelength of light has similar R/G/B value, and generally, huger interval of wavelength brings huger difference of the R/G/B of plate’s color.

This model describes the influence of light to fluorescent. The model of illuminance helps us to acquire wanted intensity, and the model of wavelength helps us to acquire wanted R/G/B value.

Part 5 The influence of illuminance and time to the expression of fluorescent protein.

In this part, we combine the result 3 with 4, and get an isogram of expression of fluorescent, which reflect the influence of illuminance and time to the expression of fluorescent.

Question to answer:

1. What is influence of illuminance and time to the expression of fluorescent?

##### Figure 14 The expression of fluorescent in different illuminance and time.

Generally, if the time is longer, the illuminance is lower, the more fluorescent will be produced. This model considers the coeffect of time and illuminance to the concentration of fluorescent, and wet get the relationship between wavelength and R/G/B value of color. Thus, we can get wanted color as long as we project light of proper wavelength, time and illuminance.

## Part 6 Model about hardware.

Except parts mentioned above, in our experiment of projecting light to plate, we also established a model to help us design and remold our hardware to make the light distribution on one plate is nearly uniform.

Question to answer:

1.Is the design of hardware improvement feasible?

Assumptions

1. The luminous flux of LED is the same at all angles above the surface.

2.Only specular reflection occurs on the side face and no reflection on other surfaces. Only reflections for the first time is considered.

3.Simplify the container to a cylinder.

We use the relation of illuminance and distance in model 2 to calculate illuminance at the bottom.

##### Figure 15 The distribution of illuminance at the bottom

The distribution of light intensity is almost even, which proves that our improvement of hardware is effective.

## Reference

[1] Berkson, J. (1944). Application of the logistic function to bio-assay. Publications of the American Statistical Association, 39(227), 357-365.

[2] Hill, A. V. (1910). The possible effects of the aggregation of the molecules of hemoglobin on its dissociation curves. The Journal of Physiology, 40, i--vii.

[3] Segallshapiro, T. H., Meyer, A. J., Ellington, A. D., Sontag, E. D., & Voigt, C. A. (2014). A 'resource allocator' for transcription based on a highly fragmented t7 rna polymerase. Molecular Systems Biology, 10(7), 742.

[4] Thiel, A. (1903). Lois générales de l'action des diastases, par v ictor, h enri. viii und 129 seiten. (paris, a. h ermann, 1903.). Zeitschrift Für Anorganische Chemie, 35(1), 382-382.