Revision as of 06:59, 17 October 2018

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MODEL

Overview

The models we built included four parts. First, we established a fluorescent protein model to screen out the most suitable fluorescent protein, the main modeling method here is grayscale analysis. Then, for the large amount of measured OD values, we drew the growth curve of yeasts and it fitted logistic model. It described the growth situation of the yeasts after plasmid introduction, and we compare it with yeasts without any foreign plasmid. The growth curve also offers the best measuring point and the best measuring interval. What’s more, we drew the degradation curve of the fluorescent protein, which helps us know different characteristics of the two chosen fluorescent proteins better. Finally, we constructed a model to illustrate the oscillation of KaiA, KaiB and KaiC protein called Mars Model, it explained the reason why the cycle reduced in yeasts nicely. Modeling work integrated with experiments tightly made our project complete and convincing.

Fluorescent Protein Evaluation Model

As a lot of fluorescent proteins and luciferase existing, an evaluation model to evaluate the performance of every fluorescent protein was established to find the most suitable one.

In this model, Stokes shift, quantity yield (QY), brightness, bleaching time, maturing time and acidity and alkalinity were mainly considered. First, with the analytic hierarchy process, the weight of the above index was obtained. Then, an evaluation matrix was obtained comparing the effect of any two factors ai and aj according to the standard below.

relationship	a_i=a_j	a_i>a_j(a little)	a_i>a_j	a_i>a_j(obviously)	a_i>>a_j
score	1	3	5	7	9

There were five factors we should consider. To determine the weight of them, we discussed with Prof. Zhao Kun, school of chemical engineering and technology, Tianjin University and professors in Tianjin Institute of Industrial Biotechnology, Chinese Academy of Sciences, they gave us a suggested order as shown below.

Between every two standards, select numerical value 1, 3, 5, 7 to measure the relative relation between index.

	Stokes	QY*Brightness	Bleaching time	ph	Maturing time
Stokes	1	3	5	1/3	5
QY*Brightness	1/3	1	3	1/5	3
Bleaching time	1/5	1/3	1	1/7	1
ph	3	5	7	1	7
Maturing time	1/5	1/3	1	1/7	1

Construct standard matrix shown as below.

$$A = \begin{bmatrix} 1 & 3 & 5 & 1/3 & 5 \\ 1/3 & 1 & 3 & 1/5 & 3 \\ 1/5 & 1/3 & 1 & 1/7 & 1 \\ 3 & 5 & 7 & 1 & 7 \\ 1/5 & 1/3 & 1 & 1/7 & 1 \end{bmatrix}$$

Next, calculate the concordance index (CI) to judge whether the matrix is reasonable.

$$CI = {\lambda_{max}-n \over n-1} = 0.0340$$

$$\lambda_{max} = 5.1361$$

λ_max -- the largest eigenvalue of the evaluation matrix and n is the dimension of the matrix.

According to the dimension of the matrix, RI value could be found from the form below.

n	1	2	3	4	5	6	7	8	9
RI	0	0	0.58	0.90	1.12	1.24	1.32	1.41	1.45

In our model, because n=5, RI=1.12.

$$CR = {CI \over RI} = 0.0304$$

Because CI<0.1, the matrix passes the consistency check. Therefore, the method we used to find the weight above is suitable.

Calculate the corresponding eigenvector of λ_max and normalize it, W=[0.0762 0.1607 0.3621 0.0389 0.3621].

At last, using the method mentioned in the [1], the score table is shown as below.

Score table
score	25	50	75	100
Stokes	0-0.02	0.02-0.05	0.05-0.1	0.1-
*QYBrightness**	0-0.01	0.01-0.04	0.04-0.06	0.06-
Bleaching time	0-0.01	0.01-0.10	0.10-0.20	0.20-
pH	1.5-2.5 8.5-9.5	2.5-3.5 7.5-8.5	3.5-4.5 6.5-7.5	4.5-6.5
Maturing time	0-0.01	0.01-0.05	0.05-0.30	0.30-

The score of every fluorescent protein can be obtained shown as Figure1.

Figure1 Selection of report genes

Because the fluorescent protein we wanted to choose had to be optimized in yeast and had to be given by parts, the following fluorescent proteins in Figure2 were all what we could choose from.

Figure2 Fluorescent proteins could be chose from

It shows that mCherry, mOrange and EYFP ranks top 3. However, after our pre-experiment, it was found that the signal of mOrange was too weak. The reason maybe was that we sequenced the part of mOrange and found that there were some mutations. Therefore, we finally chose mCherry(66.0336 ranking 12/478) and EYFP(34.9807 ranking 82/478) as our report genes.

Degradation Curve

The degradation of the fluorescent protein is also of great importance to our experiment. Therefore, the change of fluorescence intensity with time was measured, and here are the results.

Figure3 EYFP Degradation Curve

Figure4 mCherry Degradation Curve

To better explain the degradation of the fluorescent protein, we consulted a large number of documents^[2,3,4]. It shows that the degradation of the fluorescent protein is exponential.

The EYFP degradation curve can be described by function (1):

$$f(x) = a·e^{bx} + c·e^{dx} (1) $$

Coefficients (with 95% confidence bounds):

a = 133, b =-0.005066, c =-44.38, d =-0.02168

Figure5 Fitted EYFP Degradation Curve

Goodness of fit:

SSE: 76.77

R-square: 0.9443

Adjusted R-square: 0.9363

RMSE: 1.912

The mCherry degradation curve also can be described by function (1), but the coefficients are different.

Coefficients (with 95% confidence bounds):

a = 613.8, b = -0.0003886, c = 0.0003207, d =0.06852

Figure6 Fitted mCherry Degradation Curve

Goodness of fit:

SSE: 82.27

R-square: 0.9482

Adjusted R-square: 0.9404

RMSE: 2.028

The fitted degradation curve of mCherry approximates a straight line, so it can be simplified as a linear function:

$$ f(x) = p_1·x + p_2 (2)$$

Coefficients (with 95% confidence bounds):

$$ p_1 = -0.2287 , p_2 = 613.6$$

Figure7 Linear mCherry Degradation Curve

Goodness of fit:

SSE: 82.67

R-square: 0.9479

Adjusted R-square: 0.9455

RMSE: 1.938

It can be seen that the degradation rate of different fluorescent protein is very different, which made it clearer for us to know about the characteristics of the two fluorescent protein and offered important information to our experiment.

Growth Curve

For the OD₆₀₀ values we got, we did some processing and modeling work. And here are our steps and results.

There were three groups in our experiment. They were blank control group, partial control group and experimental group. After getting all the data, first, we drew a histogram and a scattergram of time and maximum OD₆₀₀ values (Figure8, 9). These results were very instructive to experiments that these results told us the best measuring point and the best measuring interval.

Figure8 histogram of Time-Maximum OD Value

Figure9 Scatter gram of Time-Maximum OD Value

From the beginning to the maximum OD₆₀₀ value, it fits the logistic model. The block effect of resource and environment for the growth of yeasts is reflected in the growth rate r, which makes r decrease with the increase in the number of yeasts x. Express r as a function r(x) of x, and take a simple and convenient linear reduction function r(x)=a+bx. In order to give a real meaning to the coefficients a and b in the growth rate function, we introduced two parameters:
(1)Intrinsic growth rate r : r is the growth rate when x=0 (in theory);
(2)Population capacity x_m : x_m is the largest yeast amount that can be accommodated by resources and the When x=x_m, the quantity of yeasts is no longer increasing, that is r(x_m)=r+bx_m=0, then b=-r/x_m.
r and x_m values in our experiments are shown in the chart below.

	YPD	SC	BY4741	d-three
r			0.0164	0.0172
x_m			0.8523	0.8034
	pABaC＋p1m	pABaC＋p1E	pABaC＋p2N	pABaC＋p1F
r	-0.002364402	0.001746617	-0.002826764	-0.001905785
x_m	0.402523944	0.508816901	0.424323944	0.542298592
	pCiRbS＋p1m	pCiRbS＋p2N	pCiRbS＋p1F	pbCiRS＋p1m
r	-0.006367923	-0.007098618	-0.007176452	-0.007853975
x_m	0.410507042	0.254873239	0.446169014	0.315098592
	pbCiRS＋p1E	pbCiRS＋p2N	pbCiRS＋p1F	pABaC+pCiRbS＋p1m
r	-0.024143608	-0.012145451	0.002428334	-0.006280764
x_m	0.413985915	0.458239437	0.270442254	0.337278873
	pABaC+pCiRbS+p1E	pABaC+pCiRbS＋p2N	pABaC+pCiRbS＋p1F	pABaC+pbCiRS＋p1m
r	0.002305512	-0.00217225	0.002272595	0.002039534
x_m	0.33171831	0.293661972	0.303701408	0.289346479
	pABaC+pbCiRS+P1e	pABaC+pbCiRS＋p2N	pABaC+pbCiRS＋p1F
r	0.001894111	-0.003848457	-0.007151104
x_m	0.301574648	0.345819718	0.329769014

The resulting growth rate function is $$r(x) = r(1 - {x \over x_m}) $$ Replacing intrinsic growth rate with r(x), get $${dx \over dt} = rx(1 - {x \over x_m}) , x(0) = x_0 (3)$$

Factor rx in the function shows the growth trend of yeast amount itself, while factor (1 - x/x_m) reflects the block effects of resources and environment to the yeast quantity growth. Obviously, the bigger x is, the former factor is bigger and the latter factor is smaller. The growth of yeast amount is the result of the two factors.

Take x as the horizontal axis and dx/dt as the vertical axis, we obtained a parabola (Figure 10), when x = x_m/2, dx/dt reaches the maximum. As shown in Figure 10, dx/dt changes with the increasing x, and we can do the following analysis to the curve x(t).

Setting when t = 0 x₀ < x_m/2, with the increase of t, dx/dt increases, so x is growing faster and faster and the curve x(t) is raised downward; dx/dt decreases when x₀ > x_m/2, x grows slower and slower and the curve x(t) is raised upward. x = x_m/2 is the inflection point of the curve. When x → x_m , dx/dt → 0, so x = x_m is the asymptote of x(t). From the above analysis, we can draw the figure as shown in Figure 11.

Actually, equation (1) can be solved by separation of a variable method as

$$x(t) = {x_m \over {1+({x_m \over x_0}-1)e^{-rt}}} (4) $$

Figure10 example x-dx/dt curve

Figure11 example t-x curve

By Analyzing our data, following results were obtained similar to the above. Figure12,13 shows the t-x curve and x-dx/dt curve of our blank control group, partial control group and experimental group. To make the results clearer, we enlarged two of them.

Figure12 t-x curve

Figure13 x-dx/dt curve

Mars Model

Model Construction

Oscillation in KaiC phosphorylation is the best-observed parameter in this system and represents a key state variable for the clock in vivo. Thus we have sought to closely mimic this output in our project. Nakajima et al. ^[6] suggest, given the dual function of KaiC and ‘‘cooperation between KaiA and KaiB,’’ that autonomous oscillation of KaiC phosphorylation might be achieved. We established a model based on known biological and biochemical observations and our experiments that did not involve transcription or translation. In Figure14, we summarized the key steps of three Kai proteins oscillation when ATP is provided in excess. It was well established that we used three circles to represent all possible combinations of three Kai proteins, just like Mars and its two satellites. This was also why we call it Mars Model.

The model (Figure 14) contained twelve processes (R1-R12) describing all the protein-protein interactions and phosphorylation-dephosphorylation between the Kai proteins. KaiXY represents KaiX and KaiY compound and KaiC* represents fully phosphorylated KaiC. Process R1, R2 and R3 are six aggregations of KaiC protein, two aggregations of KaiA protein and four aggregations of KaiB protein respectively. In process R4, KaiC₆ binds KaiA₂, forming KaiA₂C₆ compound. Since KaiA₂ facilitates the autokinase activity of KaiC₆, KaiA₂C₆ first converts to partial phosphorylated form, KaiA₂C₆C₆*, by process R5, and then rapidly converts to fully phosphorylated form, KaiA₂C₆*, by process R10. Then, fully phosphorylated protein KaiA₂C₆* combines with KaiB₄, forming KaiA₂B₄C₆*, by process R6. In process R7, KaiA₂ is displaced from KaiA₂B₄C₆*. When KaiA₂ no longer exists in KaiA₂B₄C₆*, KaiB₄ dissociates from KaiB₄C₆*, by process R8. Process R9, R11, and R12 are depolymerization of KaiC₆, KaiA₂ and KaiB₄ protein, respectively^[7].

Figure14 A dynamic model of KaiABC proteins oscillation.See text for description

We established rate equation to every process (Figure 15) and the corresponding reaction rate constants are k₁-k₁₂.

Figure15 Rate equations of every reaction

Input reaction rate constants k₁-k₁₂ and initial concentration of every protein, oscillatory curve of every protein could be obtained as shown in Figure16.

Figure16 Oscillatory curve of every protein

From Figure16 it was known that although the peak time of each protein varies, the oscillation period of every protein is the same. Therefore, in the following analysis, we take KaiC as an example to show the change of periods.

The effect of temperature change

(1)Assume the reaction rate constants change proportionally with temperature changing, the period of protein oscillation shortens as shown in Figure17.
(Note: All the blue curves represent the initial data and red curves represent the revised data.)

(a)Period shortens with temperature rising temperature falling

(b) Period prolongs with

Figure17 Period changes with temperature changing(k changes proportionally)

(2)Assume the reaction rate constants change slightly and equally with temperature changing, the period also shortens with the temperature rising and prolongs with the temperature falling as shown in Figure18.

(a) Period shortens with temperature rising temperature falling

(b) Period prolongs with

Figure18 Period changes with temperature change(k changes slightly)

When the reaction rate constants change slightly with temperature changing, the period shortens while amplitude shortens too. Therefore, if k changes disproportionately, when the temperature increases, the cycle is shortened and the oscillation is unsteady. The curve tends to be gentle with time, which means the oscillation disappears shown asFigure 19.

Figure19 The disappearance of oscillation with temperature changing

The effect of phosphorylation rate

The involved process of phosphorylation is R4 and the relating reaction rate constant isk₄.

Phosphorylase in yeasts may have promoting effect to the phosphorylation of protein and yeasts offer enough ATP/ADP in vivo, which increase the rate of phosphorylation. Therefore, k₄may increases in yeasts, which makes oscillation cycle shortens shown as Figure20.

Figure20 Period changes with phosphorylation rate changing

When the oscillation system is transplanted into yeasts, the supply rate of KaiA , KaiB and KaiC may increase and the relating reaction rate constants are k₂, k₃ and k₁. They will increase with the supply rate of three Kai proteins increasing and the result is shown as Figure21.

(a)Period change with the supply rate of KaiA increasing

(b)Period change with the supply rate of KaiB increasing

(c)Period change with the supply rate of KaiC increasing

Figure21 Period changes with oscillation environment changing

Analysis of other process

The above analysis involves some reaction rate constants that has real biological meaning. Besides, other reaction rate constants were also analyzed and we found them had an impact to the oscillation.

In the following table, we list the impact of k₅-k₁₂ changing to the oscillation. It can be seen in Figure22 that k₅ and k₆ have great influence on the disappearance of the oscillation. The impact of k₅-k₈ may relates to phosphorylation and temperature changing as well as other factors. Therefore, figuring out biological factors related to these reaction rate constants is one of our future work.

The impact of k₅-k₁₂ changing
k₅	Oscillation reduces largely with it increasing slightly; oscillation disappears with it increasing by order of magnitude.
k₆	The similar effect as k₅.
k₇	Oscillation reduces slightly with it increasing.
k₈	The similar effect as k₇
k₉	Moderate impact on the oscillation. Oscillation disappears with it increasing by one order of magnitude.
k₁₀	Has little impact on the oscillation. Phase changes with it increasing.
k₁₁	The oscillation is fine when k₁₁ <0.1; KaiA and KaiB curve oscillates when the order of magnitude is 10^-2; KaiA and KaiB oscillation disappear gradually with the order of magnitude decreasing
k₁₂	The similar effect as k₁₁

Figure22 Period changes with k₅ -k₁₀ changing

Figure23 Period changes with k₁₁ and k₁₂ changing

Summary

We propose some assumptions in combination with the model based on our experimental results.
The following are the three main factors:

(1)Temperature: There are two possible patterns that temperature may effects. If the reaction rate constants k changes proportionately, the period will shorten. If the reaction rate constants k changes slightly and equally, the period will shorten too while the oscillation disappears gradually.

(2)Phosphorylation: We have demonstrated through model tests that the oscillations decay rapidly with accelerated phosphorylation. Therefore, we have two conjectures: one is that phosphorylase in yeast plays a better role in promoting phosphorylation, and the other is that yeast provides sufficient or excess ATP/ADP to accelerate the rate of phosphorylation. This direction is also the focus of our further research.

(3)Concentrations of KaiA, KaiB and KaiC: Unlike the envisaged results, the concentrations of KaiA, KaiB and KaiC did not have much effect on our model in the testing of mathematical models.

Besides the above three factors, other reaction rate constants were also analyzed and their biological significance needs to be figured out in the future.

References

[1]Ester M, Kriegel H P, Sander J et al. A density-based algorithm for discovering clusters in large spatial databases. In: Simondis E, Han J W, Fayyad U M eds. Proceedings of the 2^nd International Conference on Data Mining (KDD-96). Portland: Oregon, 1996. 226~231

[2]Zhou Lei. Effects of different soil conditions on the degradation of BT protein. Diss.Harbin Institute of Technology, 2018

[3]李珍. Bt蛋白在土壤中降解的影响因素研究[D]. 华中农业大学, 2009.

[4]张美俊, 杨武德, 冯美臣,等. 土壤含水量、温度对Bt棉间苗叶、蕾Bt蛋白降解的影响[J]. 中国生态农业学报, 2014, 22(3):300-307.

[5]Nakajima M, Imai K, Ito H, Nishiwaki T, Murayama Y, et al. (2005) Reconstitution of circadian oscillation of cyanobacterial KaiC phosphorylation in vitro. Science 308: 414–415.

[6]Pattanayek R, Williams DR, Pattanayek S, Xu Y, Mori T, et al. (2006) Analysis of KaiA-KaiC protein interactions in the cyano-bacterial circadian clock using hybrid structural methods. EMBO J 3: 2017–2028.

[7]A. Mehra, C.I. Hong, M. Shi, J.J. Loros, J.C. Dunlap, P. Ruoff, Circadian rhythmicity by autocatalysis, PLoS Comput. Biol., 2 (2006) 816-823.

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                                      Figure14 A dynamic model of KaiABC proteins oscillation.See text for description</p>
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-                                         When the reaction rate constants change slightly with temperature changing, the period shortens while amplitude shortens too. Therefore, if <em>k</em> changes disproportionately, when the temperature increases, the cycle is shortened and the oscillation is unsteady. The curve tends to be gentle with time, which means the oscillation disappears shown as<a href="#18">Figure 18</a>.
+                                         When the reaction rate constants change slightly with temperature changing, the period shortens while amplitude shortens too. Therefore, if <em>k</em> changes disproportionately, when the temperature increases, the cycle is shortened and the oscillation is unsteady. The curve tends to be gentle with time, which means the oscillation disappears shown as<a href="#19">Figure 19</a>.
                                      </p>
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References