Team:TUDelft/Model/Rate Kinetics


Model - Rate kinetics

To find out how the concentration of TorA-GFP evolves over time for bacteria that produce vesicles and those that do not, we made a kinetic model. The kinetics of the system can be described by the flow diagram shown in Figure 1. In this model the transcription of the mRNA ($m(t) = [mRNA]$) depends on the promoter strength, $P$, and the copy number, $C_n$. The mRNA is translated into immature TorA-GFP, $n(t)$, with rate $k_t$. The immature protein is folded with rate km to become cytoplasmic mature TorA-GFP with concentration $c(t)$. Mature TorA-GFP is translocated to the periplasm with a rate kp, where it is in turn transported into vesicles with rate $k_v$. As explained in the design page, TorA will be cleaved off GFP once the fusion protein is present in the periplasm. We denote the concentration GFP in the periplasm by $p(t)$. In addition to being produced, the mRNA and protein are also degraded, with degradation rates $\gamma_m$ and $\gamma_{GFP}$ respectively, and diluted due to the growth of the cell (with growth rate $\mu(t)$). The dynamics of the whole system is given by Eqs. 1, 2, 3 and 4.

Figure 1: Flow diagram. Flow diagram of kinetic model of TorA-GFP.

$$\frac{dm}{dt} = C_nP - ( \mu(t) + \gamma_m) m(t) \tag{1}$$ $$\frac{dn}{dt} = k_tm(t) - (\mu(t) + \gamma_{GFP} + k_m)n(t) \tag{2}$$ $$\frac{dc}{dt} = k_mn(t) - (\mu(t) + \gamma_{GFP} + k_p)c(t) \tag{3}$$ $$\frac{dp}{dt} = k_pc(t) - (\mu(t) + \gamma_{GFP} + k_v)p(t) \tag{4}$$

In this model we assume that the only way for TorA-GFP to leave the periplasm is by being transported into vesicles.

Growth rate

For our kinetic model we need the growth rate of the bacteria, which changes over time. The bacterial colony starts in a lag phase, in which the bacteria grow slowly. Next, the colony grows exponentially as the bacteria double in number every cycle. At some point however, they run out of resources and enter a steady phase. This growing behavior, in terms of the colony’s $OD_{600}$ value, is given by Equation 5 (Koseki et al. 2012).

$$\frac{dOD(t)}{dt} = \mu_m \frac{OD(t)(1 - \frac{OD(t)}{OD_m})}{1 + \frac{1}{q_0}e^{-\mu_mt}} \tag{5}$$

In Equation 5 $\mu_m$ is the maximal growth rate ($sec^{-1}$), $q_0$ is a dimensionless quantity that describes the physiological state of the cell at $t=0$ and $OD_m$ is the maximal optical density. Equation 5 can be solved for the optical density as a function of time (Eq. 6).

$$ OD(t) = \frac{OD_mOD_0 (q_0 e^{\mu_mt} + 1) }{OD_0q_0(e^{\mu_mt} - 1) + OD_m(q_0 + 1)} \tag{6}$$

where $OD_0$ is the optical density at $t=0$. In the lab the OD curves of the WT and KEIO strain with TorA-GFP were measured. We fitted Equation 6 to the measured optical densities of the WT and KEIO strain (see Figure 2). The variables of the fit are shown in table 1.

Figure 2: OD curves. The optical density ($OD_{600}$) of a growing bacterial colony as a function of time (blue circles) and the fit of Equation 6 (orange line) for (a) the WT strain and (b) the KEIO strain. Fit values are given in table 1.

Table 1: Values obtained by fit The values for the parameters obtained from fitting the optical density for the WT and the KEIO strain.

$OD_0$ [-] 0.2073 0.0690
$OD_m$ [-] 0.7315 0.5167
$\mu_m$ [$hr^{-1}$] 1.3922 0.9144
$q_0$ [-] -2.9041E3 2.9391E3

The growth rate depends on the OD by Eq. 7 (Leveau et al. 2001). From this equation we can determine growth rate using Eq. 6 and the values obtained from fitting the OD, see Eq. 8.

$$ \mu(t) = \frac{1}{OD(t)} \frac{dOD}{dt} \tag{7}$$ $$\mu(t) = \frac{q_0 \mu_m}{q_0 + e^{-\mu_m t}} \frac{(OD_m - OD_0)(1 + q_0)}{OD_0(1 + q_0e^{\mu_mt}) + (OD_m - OD_0)(1 + q_0) } \tag{8}$$

However, we can also integrate Eq. 5 and use this to estimate the growth rate of the measured data (Eq. 9).

$$ \mu(i) =\frac{ \ln\left[\frac{OD(i+1)}{OD(i)}\right] }{t(i+1) - t(i)\tag{9}} $$

In Figure 3 the estimated growth rate using Eq. 9 and the fitted growth rate using Eq. 8 are shown for both the WT and KEIO strain. We find that in both cases the graph using the determined constants, and those estimating mu from the measured data, have the same shape.

Figure 3: Growth curves. The growth rate of (a) the WT strain and (b) the KEIO strain determined in two different ways: estimates from the measured data (Eq. 9, blue circles) and analytical expression (Eq. 8 ) using the constants found when fitting the OD (continuous orange lines).

Promoter strength and transport rate into vesicles

For most of the constants in our model, we have an estimate from the literature. However, two constants are unknown, the promoter strength ($P$) and the transport rate into vesicles ($k_v$). To get values for these constants we used the fact that in steady state the concentrations do not change over time, which gives:

$$ m_{ss} = \frac{C_nP}{\mu_{ss} + \gamma_m} \tag{10}$$ $$ n_{ss} = \frac{k_t m_{ss}}{\mu_{ss} + \gamma_{GFP} + k_m}\tag{11} $$ $$ c_{ss} = \frac{k_m n_{ss}}{\mu_{ss} + \gamma_{GFP} + k_p} \tag{12}$$ $$ p_{ss} = \frac{k_p c_{ss}}{\mu_{ss} + \gamma_{GFP} + k_v} \tag{13}$$

In these equations $m_{ss}$, $n_{ss}$, $c_{ss}$ and $p_{ss}$ are the concentrations of mRNA, immature TorA-GFP, mature TorA-GFP in cytoplasm and GFP in periplasm in steady state in and $\mu_{ss}$ is the growth rate in steady state. We cannot measure the concentration of mRNA and immature TorA-GFP. Therefore, we first eliminate $m_{ss}$ in Eq 11 and then eliminate $n_{ss}$ in Eq. 12. From the first equation for the promoter strength follows (Eq. 14). Furthermore, we eliminate $c_{ss}$ in Eq. 13, giving Equation 15.

$$ P = \frac{c_{ss}}{C_n k_t k_m} (\mu_{ss} + \gamma_{GFP} + k_p) (\mu_{ss} + \gamma_{GFP} + k_m ) (\mu_{ss} + \gamma_m)\tag{14}$$ $$ P = \frac{c_{ss}}{C_n k_t k_m k_p} (\mu_{ss} + \gamma_{GFP} + k_v) (\mu_{ss} + \gamma_{GFP} + k_p) (\mu_{ss} + \gamma_{GFP} + k_m ) (\mu_{ss} + \gamma_m) \tag{15}$$

As mentioned in Design and Result we measured the GFP intensity after approximately 20 hours in the cytoplasm and periplasm. This was done for both the KEIO and WT strain. The intensity values are shown in table 2. We assume $ c_{ss} = \frac{F_c}{OD_m} $ and $ p_{ss} = \frac{F_p}{OD_m} $, where $F_c$ is the fluorescence in the cytoplasm, $F_p$ the fluorescence in the periplasm and $OD_m$ the maximal optical density measured at 600nm (Leveau et al. 2001). With these assumptions we estimate $P$ to be

$$ P = \frac{F_p}{OD_mC_n k_t k_m k_p} (\mu_{ss} + \gamma_{GFP} + k_v) (\mu_{ss} + \gamma_{GFP} + k_p) (\mu_{ss} + \gamma_{GFP} + k_m ) (\mu_{ss} + \gamma_m)\tag{16}$$

At the same time we can calculate $k_v$ in the KEIO strain with the following equation.

$$ k_v = \frac{k_p F_c}{F_p} - \mu_{ss} - \gamma_{GFP} \tag{17}$$

We assume that the WT strain does not produce any vesicles, which gives $k_v = 0$.

Table 2: Fluorescence osmo-shock experiment TorA-GFP intensity in cytoplasm and periplasm form the osmo-shock experiment of the K12 BW25133 strain with (KEIO) and without (WT) a deletion of TolA.

$F_p$ [-] 6.5702E4 0.3254
$F_c$ [-] 1.5497E4 0.4398E4

Determining concentration of GFP in the periplasm and cytoplasm

We numerically solve the ordinary differential Equations 1 to 4 with the ODE45 function of Matlab. The constants we use are given in table 3. For all the constants we assume that the addition of TorA to GFP does not change the rates. For the growth rate we use Eq. 8 together with the determined values of $OD_0$, $OD_m$, $\mu_m$ and $q_0$ (Table 1). Figure 4 shows how the concentrations evolve over time in the WT strain (4a) and KEIO strain (4b). In both cases the concentration of TorA-GFP in the cytoplasm is low in the steady state. However, the concentration in the periplasm is different in the two strains. In the KEIO strain the periplasmic concentration is low, which is due to the TorA-GFP going into the vesicles. In the WT strain the amount of TorA-GFP in the periplasm keeps growing. The reason behind this continued growth is two assumptions we make in the model. Firstly, the only way that TorA-GFP can leave the periplasm is by going into the vesicles. Secondly, the size of the periplasm is not taken into account, which means that no limit is set on the amount of TorA-GFP present in the periplasm. From Figure 4b we can read off that the system reaches the steady state after approximately 25 minutes. Therefore, the best moment to harvest vesicles is after 25 minutes of induction. Therefore, in experiments measuring the GFP fluorescence in vesicles, the vesicles were harvested after 25 minutes.

Figure 4: Concentrations over time. Concentration over time of (a, b & c) the WT strain and (d, e & f) the KEIO strain. The concentrayions are shown for mRNA ($m(t)$), immature TorA-GFP ($n(t)$), mature TorA-GFP in cytoplasm ($c(t)$) and GFP in periplasm ($p(t)$).

Table 3: Constants Constants used in solving the kinetic model.

WT KEIO Reference
$C_n$ [-] 100 100 pSB1C3
$P$ [-] 5.1246 4.8E-3 Calculated
$\gamma_m$ [-] 4.8E-2 4.8E-3 Kelly et al. 2009
$k_t$ [$sec^{-1}$] 0.2 0.2 Kelly et al. 2009
$\gamma_{GFP}$ [$sec^{-1}$] 5.833E-6 5.833E-6 Kelly et al. 2009
$k_m$ [$sec^{-1}$] 1.8E-3 1.8E-3 Kelly et al. 2009
$k_p$ [$sec^{-1}$] 10 10 Estimated
$k_v$ [$sec^{-1}$] 0 13.5157 Calculated