Team:TU Delft-Leiden/Modeling/Curli/Gene

Gene Level Modeling

We will start with the modeling of the expression of curli on the gene level. Proteins that are dedicated to the curli formation are CsgA/B/D/E/F/G [1]. CsgA is the main building block of the curli. When produced, this protein is secreted out of the cell by the CsgEFG complex. In the absence of CsgB, there is no curli formation, since the CsgA proteins remain unpolymerized. CsgB is the starting block of the curli fibrils and connect the cell membrane to the first CsgA protein in the curli fibril. Once CsgB is located on the outside of the cell surface, the CsgA can polymerize onto the starting curli fibril.
In the constructs we made in the wet lab, CsgA is continuously being produced. However, in our constructs the CsgB gene is placed under the control of a landmine promoter, activated by either TNT or DNT reference to landmine. So, when the cells get induced by TNT or DNT, CsgB protein production will get started and CsgA will already be present in the system, as CsgA is continuously being produced.

summary of the conclusions

Extensive Gene Level Modeling

to be written, low priority

Simplified Gene Level Modeling

Though the model described above, providing that all rates are known, has a more accurate (though still simplified) representation of the curli assembly system, we have chosen to decrease the complexity further to the bare essentials, as most of the production rates cannot be found in literature. Measuring the accurate rates in the wet lab is, within the scope of this project, infeasible and therefore, we constructed a model that only includes the rate limiting step of the system as this will mostly determine the dynamics of the system.
First of all, we investigated if the diffusion of the CsgA and CsgB proteins to their final destination is the rate limiting step in curli formation. From the literature and the wet lab, we know that the system response to the induction by TNT or DNT is in the order of hours [reference]. If diffusion is the rate limiting step, it would mean that CsgA and CsgB proteins would pile up inside and outside the cell, because it takes a long time for them to travel to their final destination, the end of a growing curli fibril and the outer membrane, respectively. A quick calculation shows that after one second, the displacement of a spherical particle with radius \(r = \ 10 \ nm\) is 6.6 μm due to Brownian motion in liquid water at room temperature using equation 1; many times the bacterial radius! Hence, we conclude that diffusion is not rate limiting [4].

$$ \bar{x}^2 = \ \frac{k_b T t}{3 \pi \eta r} \tag{1}$$

What we do expect to be the rate limiting step for curli formation is the large amount of CsgA and CsgB proteins that have to be produced. Hence, we expect the production rate of one of these proteins to be the rate limiting step. Instead of including the intermediate steps, we have implemented the production of the CsgA and CsgB proteins with one reaction and associated production rate each. These rates have to be measured in the lab. We will use the following system of equations:

$$ \emptyset \xrightarrow{p_{A}} \ CsgA_{free} \tag{2} $$ $$ \emptyset \xrightarrow{p_{B}} \ CsgB \tag{3} $$ $$ CsgA_{free} + \ CsgB \xrightarrow{k} \ CsgA_{curli} + \ CsgB \tag{4} $$

Reactions 2 and 3 represent the production of CsgA and CsgB proteins, respectively. Equation 4 represents the growing of a curli fibril, where a curli fibril reacts with a free CsgA protein to become part of the curli. In reality, this reaction only happens at the end of the curli fibrils. In our model, we assume a homogeneous concentration of all the substances and we cannot discriminate between curli subunits. It is theoretically possible to model the system as an infinite amount of possible reactions that can take place to increase a curli fibril with length i to length i+1 at rate k [7]. However, we are merely interested in the growth rates of the curli, since the distribution of the curli length will follow from the model at the cell level. Therefore, we decided to model the growing of curli at the gene level as reaction 4. We assume that each CsgB protein is the start of a curli fibrils, thus the concentration of CsgB equals the concentration of curli. We can do this, because we showed that the diffusion of CsgA and CsgB proteins to their final destination is not the rate limiting step. Therefore, nearly all the CsgB proteins will be the beginning of a curli fibril in reality and our assumption is valid.
So, in reaction 4 we let a free CsgA protein react with a curli fibril to a CsgA protein that is part of that curli and the curli itself again, as it is immediatily again availible for the next reaction with a free CsgA protein to grow even more. Therefore, curli growth is dependent on the rate k and the concentration of \(CsgA_{free}\) and CsgB.

Writing reactions 2-4 into differential equations results in:

$$ \frac{d}{dt} [CsgA_{free}] = \ p_{A} - \ k [CsgA_{free}][CsgB] \tag{5.1} $$ $$ \frac{d}{dt} [CsgB] = \ p_{B} \tag{5.2} $$ $$ \frac{d}{dt} [CsgA_{curli}] = \ k [CsgA_{free}][CsgB] \tag{5.3} $$

Fortunately, this system can be solved analytically. To do this, we need the initial conditions. Say the CsgB promoter is activated at \(t= \ 0\). At this time there are no curli present, so \([CsgB]|_{t=0} = \ [CsgA_{curli}]|_{t=0}= \ 0\). However, the CsgA promoter is continuously active, so we expect to have an initial concentration \(A_0\) of free CsgA proteins at time \(t= \ 0\).

The solution to equation 5.2 is trivial:

$$ [CsgB] = \ p_B t \tag{6}$$

Substituting this into equation 5.1 results in:

$$ \frac{d}{dt} [CsgA_{free}] = \ p_{A} - \ K p_B [CsgA]t \tag{7} $$

It can easily be proven that a first order differential equation of the form

$$ y(t)' + \ f(t)y(t) = \ g(t) $$

has a solution of the form

$$ y(t) = \ e^{-F(t)} \int{g(t) e^{F(t)} dt} + \ y_0 e^{-F(t)} $$

where \(F(t)= \int{f(t) dt}\). In our case, \(f(t) = \ k p_B t\) and \(g(t) = \ p_A\). This yields equation 8.

$$ [CsgA_{free}] = \ p_A e^{\frac{-k \ p_B t^2}{2}} \int{e^{\frac{k \ p_B t^2}{2}} dt} + \ C_{1} e^{\frac{-k \ p_B t^2}{2}} = \ p_A e^{\frac{-k \ p_B t^2}{2}} \int_{0}^{t}{e^{\frac{k \ p_B \tau^2}{2}} d\tau} + \ C_{2} e^{\frac{-k \ p_B t^2}{2}} \tag{8} $$

One with a keen eye may recognize the Dawson function (equation 9):

$$ D_+ (x) = \ e^{-x^2 } \int_{0}^x{e^{y^2} dy} \tag{9} $$

As in our case, \(x^2 = \ k p_B t^2 \) and \(y^2 = k p_B \tau^2 \) and equation 10 obtained.

$$ [CsgA_{free}] = \ \frac{p_A D_+ (t\sqrt{\frac{k \ p_B}{2}})}{\sqrt{\frac{k \ p_B}{2}}} + \ C_{2} e^{\frac{-k \ p_B t^2}{2}} \tag{10}$$

Using the boundary condition \([CsgA_{free}]|_{t=0}= \ A_0\), the expression for the concentration of free CsgA proteins becomes:

$$ [CsgA_{free}] = \ \frac{p_A D_+ (t\sqrt{\frac{k \ p_B}{2}})}{\sqrt{\frac{k \ p_B}{2}}} + \ A_0 e^{\frac{-k \ p_B t^2}{2}} \tag{11}$$

Now, we can fill in equations 11 and 6 into equation 5.3, which gives us equation 12.

$$ \frac{d}{dt} [CsgA_{curli}] = \ k p_B t \left( \frac{p_A D_+ (t\sqrt{\frac{k \ p_B}{2}})}{\sqrt{\frac{k \ p_B}{2}}} + \ A_0 e^{\frac{-k \ p_B t^2}{2}} \right) \tag{12} $$

For the parameters \(p_{A}\), \(p_{B}\), \(k\) and \(A_0\), we have estimated the following values explain:

Table 1: Parameters used to obtain quantitative results from the analytical solution for the curli production.

Parameters	Value	Unit
\(\boldsymbol{p_{A}}\)	\(1.0 \cdot 10^{-10}\)	\(\frac{1}{Ms}\)
\(\boldsymbol{p_{B}}\)	\(1.3 \cdot 10^{-13}\)	\(\frac{M}{s}\)
\(\boldsymbol{k}\)	\(4.0 \cdot 10^{4}\)	\(\frac{1}{Ms}\)
\(\boldsymbol{A_0}\)	\(6.0 \cdot 10^{-6}\)	\(M\)

Plotting equation 12 with the parameter values in table 1 yields the graph shown in figure 1. insert caption

Figure 1 shows a steady production of CsgB. \(CsgA_{curli}\) concentration at \(t= \ 0\) is zero as expected, since there is no CsgB at that point. In the next few hours, \(CsgA_{curli}\) concentration peaks. We think that this is due to the high concentration of \(CsgA_{free}\) that is present at \(t= \ 0\). In figure 2, curli growth as function of time is plotted for different initial concentrations of \(CsgA_{free}\).

We conclude the following from figure 2:
Firstly, as expected, curli growth stabilizes to a rate equal to \(p_{A}\) after approximately 2 hours, independent of the initial concentration of \(CsgA_{free}\), \(A_0\).
Secondly, increasing the initial concentration of \(CsgA_{free}\), \(A_0\), increases the height of the peak. Even with zero initial \(CsgA_{free}\) concentration, a small peak can be found at one hour. This is a consequence of \(CsgA_{free}\) build-up when the CsgB concentration is still very small.
Thirdly, during the first two hours, few CsgB proteins are present in the system. We therefore expect that the length of the curli fibrils that started in the first few hours are much longer than the fibrils that started at later times.

References

still has to be made