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]:

Where 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:

x²=d²+h²

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:

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

We consider the expression of RFP first.

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

The sensor sense light and influence the product of OmpR1.

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.

The transcription of protein CI is regulated by OmpR1.

We get

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

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:

The translation of CI:

The expression σKIF is regulated by CI:

We get

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].

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:

We get

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

And then is the translation of RFP:

We get the last equation:

This is a summation of mentioned equations:

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

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

Here are a few examples from previous teams:

• 2016 Manchester
• 2016 TU Delft
• 2014 ETH Zurich
• 2014 Waterloo