Team:Michigan/Model
Modelling
Overview
We developed a mathematical model based on Michaelis-Menten kinetics that treats the binding of Cas9 to the target DNA as an irreversible enzymatic reaction and allows for direct competition between two Cas9 enzymes based on their relative dissociation constants. By simply switching which enzyme-substrate complex is able to produce a product, we can mimic the behavior of the comparison experiments. Because we are directly comparing two Cas9 enzymes with the same target sequence, many of the complexities of PAM searching, protein expression, and sequence comparison can be neglected without a serious effect on the overall behavior of the comparison. It is our intent that other teams will be able to use our modeling software to make sense of their own data and have some quantifiable metric to compare between different engineered Cas9 designs. Please reach out to our team’s contact for the Matlab code used to generate these figures.
Dynamic Equations
These dynamic equations represent a very simplified version of what is actually occurring but should give us a basic understanding of relative behavior between the two competing Cas9 enzymes. In this case E1 and E2 represent the first and second Cas9 respectively, S represents the free target DNA concentration, and ES represents the complex between the two. For k values, positive values (k1+ and k2+) are for the forward (i.e. binding to target DNA) while negative values (k1- and k2-) are for the reverse reaction (i.e. dissociation from target DNA). Finally, kpro represents the cleaving of target DNA and it is only applied to the Cas9 with nuclease activity. If we set it up so that the first Cas9 has 10 times greater binding affinity, then the following plots can be produces. Note that the actual values are not important, only the larger behavior and trends within the plots.
Discussion
In these plots we can see that the varied k values of each Cas9 determines the time it takes for the system to reach steady state. This is important in our system since faster binding will result in that Cas9 “winning” that piece of target DNA. While significant experimental standardization will be required to standardize these parameters, it is our intention to show the effects of changing the values of these constants on the overall behavior. We believe that engineering a Cas9 enzyme to have a different sequence and structure will result in variations in binding affinity, PAM search ability, and ease with which Cas9 is able to make the large conformational change required for DNA binding (Jiang, F., & Doudna, J. A. (2017)). We are simplifying all of these modifications down to a simple change in one of the k values. While this simplification is quite extreme, it should still provide an insightful model in terms of relative behavior. Other models have been developed by other groups that consider more complicated behavior (Farasat, I., & Salis, H. M. (2016).) but we can ignore most of the complex behavior due to the scope of our project and the fact that most of this behavior should be similar between both Cas9 enzymes regardless of how they are engineered. In the future, we intend on using experimental data to fit our model more closely to the behavior observed and have constant values that directly correspond to reality.
References
Farasat, I., & Salis, H. M. (2016). A biophysical model of CRISPR/Cas9 activity for rational design of genome editing and gene regulation. PLOS Computational Biology, 12(1), e1004724.
Jiang, F., & Doudna, J. A. (2017). CRISPR–Cas9 structures and mechanisms. Annual Review of Biophysics, 46(1), 505-529. doi:10.1146/annurev-biophys-062215-010822.