Team:TU Delft-Leiden/Modeling/Curli

Curli Model

Gene Level Modeling

We will start with modelling the expression of Curli on the gene level. Proteins that are dedicated to curli formation are the CsgA/B/D/E/F/G [1]. CsgA is the main building block of the curli. When produced this 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, likes piles under a house, that connect the cell membrane to the first CsgA. Once this is placed on the cell surfaced, the CsgA can polymerize onto the starting curli fibril. In the constructs we produce, CsgA is continuously produced. The CsgB promoter, on the other hand is in our design activated by a landmine promoter, activated by either TNT or DNT. In our design, the curli are first grown in a medium. During this time, they produce CsgA. Hence, there is an initial concentration CsgA present in the system when the CsgB promoter is activated. The csgEFG operon encodes four accessory proteins required for the curli assembly. Overexpression of these genes is not needed [bron?] for curli formation.

Extended gene level modeling

to be written

Analytical gene level modeling

Though the model as described above, providing that all rates are known, has a more accurate (though still simplified) representation of the curli assembly system, we have chosen decrease the complexity further to the bare essentials. Most of the production rates and kinetic constants cannot be found in literature. Measuring the accurate rates is, within the scope of this project, unfeasible for many reasons. First of all, we think that the secretion of the proteins is not a rate limiting step. We are interested in the CsgA/B production and curli formation. The scope in which we expect the system to respond is in the order of hours. If secretion were the rate limiting step, it would mean that many CsgA/B units are piled up inside the bacteria. One might raise the question whether it is justified that the growth of new subunits is independent of the length of the curli, since the CsgA is secreted at the edge of the cell distance and has to ‘travel’ towards the end of the curli fibrils. This would suggest that the assumption used in deterministic modelling that all concentrations are homogeneous is not justified. A quick calculation shows that after one second, the displacement of a spherical particle with radius r=10nm due to Brownian motion in liquid water at room temperature using equation 0 is 6.6 μm; many times the bacterial radius! Hence the diffusion is not rate limiting. [4]

$$ \bar{x}^2 = \frac{k_b T t}{3 \pi \eta r} \tag{0}$$

What we do expect to be rate limiting for biofilm formation is that a large amount of curli subunits has to be created, hence the expression rate of the proteins. Instead of including the intermediate steps, we have replaced the production rate of the CsgA and CsgB with one expression rate each. These rates have to be measured in the lab. What we did not include is the maturation time needed for the production. Luckily, for CsgB speeds up the CsgA formation, the lag phase of CsgA is decreased to the same lag phase as CsgB when they’re both expressed. [2][3]

We will use the following system:

$$ \emptyset \xrightarrow{p_{A}} \ CsgA \tag{1} $$ $$ \emptyset \xrightarrow{p_{B}} \ CsgB \tag{2} $$ $$ CsgA + CsgB \xrightarrow{K} C_{sub} + CsgB \tag{3} $$

\(C_{sub}\) is a folded CsgA protein that has connected to one of the curli fibrils. \(P_{A}\) and \(P_{B}\) are the production rates of CsgA and CsgB respectively. Equation (3) may seem confusing, since in reality the growing of curli takes place between a \(C_{sub}\) and a CsgA. However, this happens only at the end of the curli fibrils. In deterministic modelling, the system is a homogeneous batch and we cannot discriminate between the outer and the other curli subunits. It is theoretically possible to model the system, as an infinite amount of possible curli reactions that can take place that increase a curli 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 of the cell level. The reactions will only take place at the end points of the curli fibrils. The reaction rate is therefore proportional to the fibril concentration, hence the amount of CsgB units.

Writing equations 1-3 into differential equations results in:

$$ \frac{d}{dt} [CsgA] = p_{A} - K [CsgA][CsgB] \tag{4.1} $$ $$ \frac{d}{dt} [CsgB] = p_{B} \tag{4.2} $$ $$ \frac{d}{dt} [C_{sub}] = K [CsgA][CsgB] \tag{4.3} $$

Fortunately, this system can be solved analytically. To do this, we need the initial conditions. Say the CsgB promoter is activated at \(t=0\) (or maybe at \(t=-T_{lagtime}\) if you want to include this as well). At this time there are no fibrils present, so \([CsgB]|_{t=0} =[Csub]|_{t=0}=0\). However, the CsgA promoter is continuously active, so we expect that at time \(t_{0}\), there is an initial concentration \(A_0\) of CsgA.

The solution to equation 4.2 is trivial:

$$ [CsgB] = p_B t \tag{5}$$

Substituting this into equation 4.1 results in:

$$ \frac{d}{dt} [CsgA] = p_{A} - K p_B [CsgA]t \tag{6} $$

It can easily be proven that a first order differential equation of the form \(y(t)'+f(t)y(t)=g(t)\) has solution of the form \(y(t)=e^{-F(t) }\int{g(t) e^{F(t)} dt}+y_0 e^{-F(t) }\), where \(F(t)= \int{f(t) dt}\). Replacing \(f(t)\) with \(K p_B t\) and \(g(t)\) with \(p_A\) yields equation 7

$$ [csgA]= p_A e^{\frac{-Kp_B t^2}{2}}\int{e^{\frac{Kp_B t^2}{2} } dt} + C e^{\frac{-K p_B t^2}{2}}=p_A e^{\frac{-Kp_B t^2}{2}}\int_{\tau=0}^{t}{e^{\frac{Kp_B \tau^2}{2} } d\tau}+\tilde{C} ̃e^{\frac{-Kp_B t^2}{2} } \tag{7} $$

One with a keen eye may recognize the Dawson function (equation 8):

$$ D_+ (x)= e^{-x^2 } \int_{y=0}^x{e^{y^2} dy} \tag{8} $$

Substituting \(x^2\) and \(y^2\) with \(K p_B t^2\) and \(K p_B \tau^2\), equation 9 obtained.

$$ [csgA]=\frac{p_A D_+ (t\sqrt{\frac{K p_B}{2}})}{\sqrt{\frac{K p_B}{2}}} + \tilde{C} e^{\frac{-K p_B t^2}{2 } } \tag{9}$$

Using the boundary condition \([CsgB]|_{t=0}=A_0\), the concentration becomes:

$$ [csgA]=\frac{p_A D_+ (t\sqrt{\frac{K p_B}{2}})}{\sqrt{\frac{K p_B}{2}}} + A_0 e^{\frac{-K p_B t^2}{2 } } \tag{10}$$

Since we’re mainly interested in the curli production rate, rather than the total amount of curli, we can fill in equations 10 and 5 into equation 4.3, which gives us equation 11.

$$ \frac{d}{dt} [C_{sub}] = K p_B t \frac{p_A D_+ (t\sqrt{\frac{K p_B}{2}})}{\sqrt{\frac{K p_B}{2}}} + A_0 e^{\frac{-K p_B t^2}{2 } } \tag{11} $$

For the parameters in equation 11, 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
\(\boldsymbol{P_{A}}\)	\(1.0 10^{-10}\)	\(\frac{1}{Ms}\)
\(\boldsymbol{P_{B}}\)	\(3.4 10^{-12}\)	\(\frac{M}{s}\)
\(\boldsymbol{K}\)	4.0 10^{4}	<\(\frac{1}{Ms}\)
\(\boldsymbol{A_0}\)	6.0 10^{-6}	<\(M\)