## Modeling

We were confused with many problems during the experiment. For bird diagnosis, we wanted to prove the feasibility of our new method of BV detection mathematically and calculate the concentration of BV accurately. For intestine bacteria, we needed to evaluate our bacteria before we injected the bacteria to mice intestines and estimate the optimal concentration of bacteria and BV rapidly. Sometimes we wanted to keep the fluorescence bright enough for long time, so we also desired of a way of fluorescence control.

### I. The feasibility and BV accounting method of bird diagnosis

We are confident that our new method for measuring the concentration of BV in bird blood is not only accurate but also convenient. This model is built to prove its feasibility and analysis the detection result.

### 1.1. Feasibility

#### 1.1.1. Sensitivity

Figure 1 Sensitivity of BV measure

After analysis our data, we found that the sensitivity of our BV detection system was pretty good, which meant the RFD would change gigantically if the concentration of BV changed slightly. In another word, we could find barely BV in bird blood.

#### 1.1.2. Nonlinear-error

Figure 2 Nonlinear-error of BV measure

The nonlinear-error of our new method was very limited，which meant that it would not bring extra nonlinear-error to the detection system.

### 1.2. BV accounting method

We determined an empirical formula with test method to describe the relationship between the concentration of BV(μmol/L) and relative fluorescence units（RFU）and got a formula:

For anyone who want to know the concentration of BV, what they need to do is just measuring the fluorescence in standard way and calculating the concentration of BV by our formula.

### II. Evaluation and estimation of intestinal bacteria experiment

### 2.1. Experiment evaluation

Confocal is widely used in our experiment, but we cannot use the precision instrument whenever we want as undergraduate students. So, we need to come up with a convenience model to calculate how much smURFPs were displayed on the cell and evaluate the experiment as soon as we finish culturing.

The assumptions of the model are as follows:

The assumptions of the model are as follows:

• The solution of bacteria is stable;

• The smURFPs move slightly;

• Every bacteria produces the same amount of smURFPs;

• The size of linker could be ignored compared with smURFP;

• Every bacteria is at the same size.

• The smURFPs move slightly;

• Every bacteria produces the same amount of smURFPs;

• The size of linker could be ignored compared with smURFP;

• Every bacteria is at the same size.

#### 2.1.1. First step: calculate the concentration of smURFP

As we all know that the production of smURFP in the cell is uncontrollable, but the result of the production of smURFP is obvious that almost the whole cell is covered by smURFP. Thus, we can calculate the amount of smURFP by following steps:

##### a. Calculate the surface area of the cell

Although the shape of bacteria is various, we can abstract it to regular shapes. For example, the surface area of E.coli is:

Figure 3 Size of E.coli

##### b. Calculate the minimum area that a smURFP taking on the surface

The smURPFs are locked on the surface of cells by anchored proteins and their orientations are uncontrollable either. But in the extreme situation, all the proteins stood straightly like soldiers and their minimum surfaces were parallel to the cell. As we clearly know the structure of smURFP, we can easily know the minimum area that a smURFP taking on the surface.

Figure 4 The structure of smURFP

Figure 5 The abstraction of the structure of smURFP

##### c. Calculate the maximal concentration of smURFP

Consider the smURFP may concentrate in a particular area, we come up with a correction factor k. set the area that covered by smURFP as s

_{covered}:Thus, the concentration of smURFP could be described by following equation:

#### 2.1.2. Second step: calculate the expectation of relative fluorescence identity

We analyzed all the data about the relationship among smURFP, BV and RFD, and found the optimal empirical formulas to describe it.

Case 1

If the concentration of E.coli is between 1.47×10

^{13}CFU/L and 7.37×10^{13}CFU/L, which means the concentration of smURFP is between 1μmol/L to 5μmol/L, we can describe the relationship among smURFP, BV and RFU by following equation:
.

.

As it may waste too much time and energy to solve the equations, we came up with a table that contains 5 concentrations of bacteria and their corresponding coefficients. If the concentration of cells is between 2 of those 5 concentrations, we can figure out the approximate solution of coefficients by linear interpolation.

c_{E.coli} (CFU/L) |
a | b | c |
---|---|---|---|

1.47×10^{13} |
1885 | 374.6 | -389.8 |

2.21×10^{13} |
2689 | 445.4 | -534.8 |

2.94×10^{13} |
3811 | 288.3 | -862.5 |

3.68×10^{13} |
4443 | 605.1 | -1265 |

7.37×10^{13} |
8574 | -2758 | -2585 |

Table 1 Five concentrations of bacteria and their corresponding coefficients

Case 2

If the concentration of bacteria is not between 1.47×10

^{13}CFU/L and 7.37×10^{13}CFU/L, we have another equation:We are confident about the relative fluorescence intensity when x≤5y.

#### 2.1.3. Last step: compare the relative fluorescence identity test result with the result of computation

If the test result is far less than the computation result, we should stop and analyze the possible mistakes of our experiment instead of continuing the work.

### 2.2. Experiment estimation

Before we injected bacteria into mice intestines, we needed to estimate the concentration of bacteria and the concentration of BV. BV was expensive and culturing bacteria took long time. So we wanted to find the optimal combination of BV and bacteria that not only took less BV and bacteria but also sent out brighter light.

We analyzed the data and piecewise linearized the nonlinear curves. We obtained a clever way to find the optimal concentration of BV and bacteria.

We have already known that the concentration of smURFP could be describe by following equation:

And we came up the linearized relationship between c

_{P}, c_{bv}and RFU (Z):Thus, we can solve the least BV that meet our needs:

Figure 4 Four possible combinations of bacteria and BV for target FU. Yellow line: the combination that waste BV. Blue line and gray line: the combinations that waste bacteria. The pink line: the optimal combination

### 2.3. The control of fluorescence

When we use the fluorescence bacteria to trace other objects, we want the fluorescence to be stable. However，could the fluorescence in intestine be stable？ In most case，the fluorescence will die out gradually because the lack of plasmids replication and the degradation smURFP. So the model of keeping the fluorescence stable is very essential for us.

The assumptions of the model:

• The bacteria grow exponentially.

• The time of the combination of smURFP and BV could be ignored compared with its cell circle.

• The time for bacteria to product smURFP could be ignored compared with its cell circle.

• The plasmids do not replicate inside cells.

• The time of the combination of smURFP and BV could be ignored compared with its cell circle.

• The time for bacteria to product smURFP could be ignored compared with its cell circle.

• The plasmids do not replicate inside cells.

We came up with following equation to describe the change of RFU（Z）.

.

We analyzed the data of our experiment, and found out how A, B and N change by time:

Thus,

Set

In most cases, （a+b+c）<0，from phase-plane analysis, we know that the FU will converge to the origin.

Figure 6 the trajectory of FU

If there is no supply of BV and bacteria, the fluorescence would die out after 4τ.

Figure 7 Without supply of BV and bacteria, the FU changes by time

Form our experiment we know that if the maximal FU is more than 25000， τ≈4h. So we suggest that the period of supply is 0.5τ so that we can make sure that the RFU is more than 0.6RFU

_{max}.Figure 8 If the period of supply is 0.5τ,the FU changes by time