Mathematic Analysis on AlkSensor
Model Objective
To perform a mathematic analysis on AlkSensor, to find out the relationship between AlkSensor’s input and output, to find ways to specifically regulate the function of AlkSensor.
Problem Description
AlkSensor is composed of protein ALKR and promoter alkM. Genes of ALKR and alkM were synthesized and constructed into a plasmid, as shown in figure 1. As mentioned in the introduction, protein ALKR is a transcription factor which can recognize certain alkanes. ALKR is under a constitutive promoter and is constitutively expressed. Alkane molecules is recognized by ALKR and a dimerized ALKR-alkane complex is formed. Subsequently the reporter’s promoter alkM is induced by the complex.
Figure 1. Scheme of AlkSensor’s mechanism
Problem Abstraction
The mechanism of AlkSensor can be represented by a set of chemical reactions, as shown below.
- Reaction 1: the generation of protein ALKR.
- Reaction 2: the degradation of protein ALKR
- Reaction 3: the dimerization of protein ALKR(AR2-A denotes the dimerized ALKR-alkane complex)
- Reaction 4: the combination of dimerized ALKR with alkane(Pm denotes promoter alkM)
- Reaction 5: the combination of dimerized ALKR-alkane complex with promoter alkM(Pm’ denotes the promoter alkM binding with inducer)
- Reaction 6: the generation of reporter(RNAP denote the RNA polymerase, Rp denotes reporter )
- Reaction 7: the the degradation of reporter
Assumption
There are several assumptions in the abstraction of AlkSensor.
- Because the the genes of ALKR is under a constitutive promoter, we assume that the generation rate of ALKR is constant.
- The three binding reactions, i.e, reaction 3, 4 and 5 are fast reversible reactions since DNA binding and unbinding of the repressors dimers and the dimerization itself occur within seconds, whereas the synthesis (transcription, translation, folding) and degradation of monomers takes minutes to sometimes an hour.
- For simplification, we do not take into consideration the combination of dimerization ALKR with promoter alkM.
Model Development
In the set of reactions reaction 3, 4, and 5 are fast reversible reactions because DNA binding and unbinding of the transcription factor dimers and the dimerization itself occur within seconds. However the synthesis and degradation of monomers takes minutes to sometimes an hour. Therefore, the three reactions are in equilibrium and the steady state concentrations are given in terms of the equilibrium constants. The accumulation rates of ALKR and reporter are subject to their generation rate and degradation rate, which is described by two ordinary differential equations(ODE). So the mechanism of AlkSensor can be described by a mathematical model consisting of 2 ordinary differential equations(ODE) and three equilibrium equation. The input of AlkSensor can be defined as the concentration alkane, the output can be defined as the concentration of reporter protein, as shown in figure 2.
Figure 2. Mathematical representation of AlkSensor
In the first equation, there are three terms that determine the accumulation rate of ALKR, the generation rate of ALKR, the degradation rate of ALKR, the dilution rate of ALKR.
In the second equation, there are four terms that determine the accumulation rate of reporter, the generation rate of reporter, the basal expression rate of reporter, the degradation rate of reporter, the dilution rate of reporter. The last 3 equation describe the quantitative relation in reaction 3,4 and 5.
The [Pm’] is still a intermediate variable, we next deduce the value of [Pm’].
The total concentration of promoter alkM is though to be a constant, which is proportional to the copy number of alkM.
So we can get the the expression of [Pm] and [Pm’]
Then we substitute [Pm’] with the terms on the right side of the equation.
For simplification, we substitute K4K3K2 with Kap. Kap, equal to K4K3K2, is the apparent equilibrium constant of reaction 8, .
Finally we get the ODE about the reporter.
When the leakage of ALKR’s promoter is little, the terms r can be ignored. When the system reach a steady state, the accumulation rate of Rp is zero.
It is easy to get the equation that describe the relation between AlkSensor’s input and output.
The equation can be simplified into a very elegant form. Let x represent [Alkane], y represent [Reporter], the equation can be simplified as 
, in which 
,
.(Latex form: )
Analysis & Discussion
What is the response curve of AlkSensor like?
Figure 3 shows twelve different response curves of AlkSensor. Although they are in different shapes, they are in the same pattern——
.
When the input is very low, the response curve is nearly liner and has a maximum value of slope, aK. With the increase of input, the value of slope decrease. Finally the response curve approach a horizontal line y=a.
How to evaluate the response curve of AlkSensor?
The response curve describes how the output is determined by the input. The slope of the curve represent output’s sensitivity to input. The bigger the slope is, the more sensitive output is to input. If the slope is zero, the output does’t change with input. This is when the AlkSensor loses its function.
We can derive two critical parameters from the response curve to evaluate AlkSensor, sensitive range of input(SRI) and response range of output(RRO). SRI is the range in which the output is sensitive enough to the input. RRO is the range of AlkSensor’s signal strength. The crossing zone of AlkSensor’s SRI and RRO is the effective working zone of AlkSensor.
How does a and K influence the shape of the response curve?
We list a set of response of curves with different values of a and K, as shown in figure 3. It is easy to get that the RRO is a. The SRI is subject to K, the lower K is, the wider the SRI is.
Figure 3. AlkSensor’s response curve with different values of a and K
How to regulate the value of a and K of AlkSensor?
To achieve the goal of rationally regulating AlkSensor, we need to know how to adjust the value of a and K. We already have the equation of a and K.
The value of a is determined by five values. They are:
- kt: the combination constant of RNAP with Promoter alkM. It is hardly changeable, so we can consider it to be a constant;
- kd: the degradation constant of protein ALKR, also thought to be a constant;
- P0: the concentration of RNAP, constant;
- [Pt]: the total concentration of promoter alkM, which is proportional to the copy number. This is easily changeable. We can change it through construct alkM on different plasmid with varying copy number.
- μ：the dilution constant caused by the cell cycle, which is hardly changeable.
As mention before, the a represent the response range of the sensor’s out put, so through changing the copy number, we can rationally control AlkSensor’s RRO.
Next comes the K, which influence the sensitive range of the sensor’s input. K equals the product of Kap, and the square of ALKR’s concentration. Kap is the product of three reactions’ equilibrium constant: the dimerization of ALKR, the formation of inducer, the activation of promoter alkM. The structure of ALKR certainly influence the value of K. The value of K also reflect AlkSensor’s specificity towards input. Through altering the structure of ALKR, we can change AlkSensor specificity towards certain alkane. ALKR’s concentration is determined by the generation rate of ALKR, which can easily be regulated by changing the promoter of ALKR. As mentioned before, ALKR is under a constitutive promoter, the strength and copy number of the promoter have a positive relationship with ALKR’s concentration. So, through changing the strength and copy number of ALKR’s promoter, we can rationally regulate the sensitive range of AlkSensor.
Achievement
In this model, we have
- identified the pattern of AlkSensor’s response curve,
- proposed a template for experimental data of AlkSensor,
- found ways to evaluate AlkSensor,
- developed strategies to rationally regulate AlkSensor.
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