Team:Northwestern/Model

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Model objectives

This model is an extension of the work carried out by the 2013 Dundee team and is based on a TatA polymerization mechanism (1,2). By modeling Cas9 translocation across the different bacterial compartments, this kinetic model aims to estimate the quantity of Cas9 exported to the periplasm, and ultimately Outer Membrane Vesicles, as a function of time. This mathematical model could provide answers to the following questions:

• What should the rate of Cas9 production be to package a certain protein “dose” in each OMV?
• How do the levels of inducer present affect the amount of Cas9 packaged as a function of time?
• What is the optimal time to isolate vesicles to attain a certain protein concentration?
• How many OMVs should be administered in total to achieve the effective dose if OMVs were to be given as a therapeutic?
• What fraction of the vesicles is occupied by protein?

Assumptions

• All processes are reversible with two exceptions: protein movement from the cytoplasm to the periplasm and protein export in OMVs
• The protein is homogenously distributed in the cell’s periplasm once released from Tat machinery
• The inducer remains abundant; this model does not consider inducer depletion
• TatA assemblies are pre-formed from TatA proteins and TatBC is present in the cell from t = 0
• OMVs are spherical in shape and their size (diameter = 100nm) is independent on a bacterium’s life-cycle.
• The volume of the periplasm does not change as OMVs pinch off
• Vesicle production rate remains constant
• The protein is expressed in a high copy plasmid (~200 copies)
• Protein is bound to a Tat signal peptide (TorA, YcbK etc)

Model breakdown

Step 1 DNA Transcription and Step 2 mRNA degradation

Step 3 mRNA translation and Step 4 Cas9 degradation

Step 5 TatBC complex binds the signal peptide of the protein in an energy-independent step. The RR consensus motif in the signal peptide is specifically recognized by a site in TatC.

Step 6 TatA protomers are recruited to the TatBC complex and polymerized. Passenger domain of the substrate protein crosses the membrane via the polymerized Tat component. Then, the signal peptide is proteolytically removed by a signal peptidase at the periplasmic face of the membrane and Tat dissociates from TatBC and depolymerizes back to free protomers.

Step 7 Cas9 is exported in the cells periplasm and Step 8 Cas9 is exported in OMVs

Mathematical representation

The processes outlined above were expressed as a set of 9 differential equations.

Model parameters

Parameters were selected from reported literature values. In combination, they provided outcomes for Cas9 export that match the expected time scale.

Simulation results

First, protein production under a constitutive promoter was considered for a range of protein and mRNA degradation rates. Initially, export was ignored. This way, we estimated a baseline for the number of Cas9 molecules in the cell's cytoplasm and were able to observe approximately what protein value corresponded to the degradation rate selected.

Figure 1. - mRNA concentrations for various mRNA degradation rates

Next, expression under an inducible promoter was modeled using a first order Hill function. From hereafter the full set of ODEs was used. Induction was added to the model to examine the possibility of modifying protein export levels by tuning this value. In the model, K is the concentration for half saturation of the ligand (averaged dissociation constant), r(t) represents the levels of inducer (constant in this case) and n is the Hill coefficient describing the switch-like character of the induction process (n = 1). β is the maximal expression rate. Figure 3 allows visualization of Cas9 trajectories during export.

Figure 3. - Cas9 concentrations across different cellular compartments

The amount of Cas9 packaged in individual vesicles (Figure 5) was derived from Figure 4 following normalization by the vesicle formation rate, estimated to be 10 seconds.

Figure 4. - Cas9 exported in OMVs (cumulative) as a function of time

The number of Cas9 molecules packaged in each vesicle at steady state was calculated to be 3.58 molecules. This value is reached after approximately 4.5 hours

Figure 5. - Cas9 molecules packaged in each vesicle as a function of time

Sensitivity analysis

Sensitivity analysis was carried out to investigate the influence of the chosen parameters on the state variables. Since the main objective of this model is to approximate the number of molecules of Cas9 incorporated in OMVs as a function of time, we focused our analysis on predicted trajectories for the incorporation of Cas9 in vesicles (i.e. the very last ODE of the system)(3). We examined sensitivity for the time varying rate of change of Cas9 by computing a sensitivity matrix which encapsulates how sensitive the concentration of the output variable is to parameter changes (Figures 6-7).

We concluded that initially export (Figure 6) is more strongly influenced by parameter values, however, as steady state is reached this effect is attenuated (Figure 7). By plotting sensitivity for the steady state (Figure 8) we compared the significance of each of the 10 variables on the system's output.

Figure 8 - Steady state sensitivity values for the 10 model parameters

By performing sensitivity analysis on the model, we have identified key parameters that play the biggest role in this delivery system. In particular, k2, the assembly of the TatABC complex, had the highest sensitivity value and thus appears to be a good target when attempting to optimize this pathway.

Combination analysis

Lastly, we examined how modifying the levels of inducer and the expression of TatA and TatBC complexes could influence the amount of protein exported. Using values initially selected for the number of TatA and TatB complexes present in an E.coli cell as a baseline, we looked at how a 10-fold increase or decrease could affect the output. By testing different combinations we observed that a decrease in the inducer present would lead to a dramatical decrease in Cas9 packaging, while an increase would not lead to a significant increase unless one of the two complexes is also expressed at higher amounts. Furthermore, higher expression of both TatA and TatBC did not lead to significantly higher expression when compared to TatA overexpression alone.