Difference between revisions of "Team:Northwestern/Model"

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?

• 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 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
• Protein is bound to a Tat signal peptide (TorA, YcbK etc)

Step 1: DNA Transcription (and mRNA degradation)

Step 2: mRNA translation (and Cas9 degradation)

Step 3: 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 4: 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 5: Cas9 is exported in outer membrane vesicles.

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

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

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.

Next, expression under an inducible promoter was modeled using a first order Hill function and the full set of ODEs was considered. Induction was added to the model in order to examine the possibility of modifying protein export levels by tuning critical parameters. 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 allowed the visualization of Cas9 trajectories during export.

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.

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

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 each OMV as a function of time, we focused our analysis on predicted trajectories for the incorporation of Cas9 in each vesicle.

By performing sensitivity analysis on the modeled system, we have identified key parameters that could have the biggest effect on this delivery system.

what do these results mean and how do they affect/could they impact our project.