Mathematical modeling is a core part of Waterloo iGEM: we have nearly as many team members typing furiously away in our dry lab as we do wrangling transformations in our wet lab. This year, we created models in Python, MATLAB and NetLogo that examine the feasibility of our design and provide tools for assessing future designs. The code for each model is available on our GitHub and details on the formulation of each model may be found in the pages linked below.

Cas9 Frameshift Dynamics

We modelled the accumulation of mutations in a target genome and eventual deactivation of target genes after cutting by CRISPR/Cas9 and repair by Non-Homologous End Joining (NHEJ). A stochastic and probabilistic model of the effect of CRISPR/Cas9 on a single genome allowed us to investigate the accumulation of indels in a genome with multiple targets over time. Our Python model provides a tool for other teams looking to investigate multiplexed Cas9 targeting for deactivation of target genomes.
Image of agent-based model of spread of viral infection among plant cells sorted into several leaves.

PAM Structural Bioinformatics

A key limitation of the current use of S. pyogenes Cas9 for gene editing is the inability to target sequences which are not adjacent to a NGG PAM sequence. To overcome this issue, we developed a pipeline to probe Cas9 mutants with altered PAM specificity using simulation tools in PyRosetta and Python. A detailed description of the analysis and software used in this work can be found on the Modeling Engineered Cas9 PAM Flexibility.

Cas9 with a mutated residues indicated
A PyMOL generated image of the three mutations found in the VQR spCas9 variant.
The docking scorings of wildtype spCas9 in PyRosetta to the 64 PAM variants
The docking scorings of wildtype spCas9 in PyRosetta to the 64 PAM variants.

The simulation managed to successfully predict the PAM specificity of wild type spCAs9. However, validation against known Cas9 mutants was less successful, predicting a mix of correct and incorrect, though biochemically similar, PAM sequences.

CRISPR Plant Defense

Mathematical models are not limited by the same time constraints as growing plants, so they were used to validate the CRISPR Plant Defense aspect of our project. The team looked at the antiviral effects of CRISPR/Cas9 targeting on three scales: CaMV genomes, plant cells and plant leaves. A background primer on Cauliflower Mosaic Virus (CaMV) genetics, replication and spread can be found on the CaMV Biology page.

Stylized viral genome
CaMV Genomes
Stylized plant cell
Plant Cells
Stylized plant leaves
Plant Leaves

On the scale of individual genomes, we used our dynamics model of CRISPR/Cas9 to model viral genomes becoming non-functional over time as frameshift mutations were introduced. The results of running 1000 simulations using our sgRNA target design to deactivate the CaMV P6 gene predict the following pattern of P6 functionality over time.

Exponential decay of functional viral genomes to non-functional genomes
CRISPR effect parameter (t1/2) is derived from an exponential fit to the fraction of CaMV genomes active at each timestep across 1000 stochastic genome simulations.

The exponential decay fit shown in the graph above was then passed to a model on the scale of plant cells. We modelled CaMV Replication in a single infected cell using ordinary differential equations. The ODE replication model allowed us to see the effect that the CRISPR/Cas9 decay of P6 would have on virion production. We also examined the interaction bewteen intracellular plant defenses (RNA interference) and the CRISPR/Cas9 defense and conducted a sensitivity analysis on our parameters.

Time series of functional and non-functional virions in the cell in the presence of CRISPR and CRISPieR systems.
Replication model shows total CaMV virions within a cell over time with and without a CRISPR/CRISPieR defense system. Non-functional virion production is lowered because fewer functional proteins that protect against RNAi are produced.

The ODE model was then scaled up to the level of plant leaves. We modeled the spread of infection between plant cells using an agent-based framework, which is described in detail on the Intercellular Viral Spread page. The agent-based model allowed us to explore the effect of intercellular defense signalling (which leads plant cells to resist to infection and become suseptible to apoptosis).

Overall, our multi-scale model of CRISPR Plant Defense allowed us to investigate the extent to which we could slow CaMV spread using CRISPR/Cas9 and its interaction with host defense systems such as RNA interference and intercellular defense signalling.