Introduction

There have been well-established fundamental concepts in the physical sciences that have seen the day of light in the biological world. However, the integration between concepts in biological circuitry (e.g. gene circuits) and fundamental concepts in the field of electronics (e.g. electric circuitry) are very much at a nascent stage. Enter in the concept of logic gates, an idealized tool to model and represent logical Boolean operations in integrated circuits and programmable devices, to model and represent genetic circuits!

Given the complexities of modelling complex genetic circuits involving multiple variables and parameters ever-changing, use of a well-established technique used in electrical engineering may provide further insight into biological modelling. It is with this hope that Dry Lab pursues the technique of logic gate based modelling to simulate our biological hypothesis. By the end of the project, modelling will allow for understanding the modularization of gene circuit components done by logic gates, while also providing valuable insight from simulations for biological system design implementing our biological hypothesis.

Project Plan

For our project, a 4-step plan was conceived and implemented:

Ordinary Differential Equation (ODE) system model for predictive parameters
A NOR gate program parameter design based on #1
Implementation of singular NOR gate design for simulation
Combination of NOR gate simulation results

Modelling

Systems of ODEs allows for modelling of key dynamic parameters that change with respect to a certain variable (typically time) and in relation to one another. For our modelling, systems of ODEs will be required to model ligand-receptor binding of sgRNA/dCas9/ribozymes to gene sequences via Hill’s equation. Later on, mathematical models will be derived from these previous equations and experimental results in order to decipher mathematical models for repression and activation cascade in the biological NOR gate setting. The advantage to using math models here is that it allows for regulatory parameters to be set and modelled with dynamic mathematical equations that can be numerically solved, but also retooled as necessary to accommodate for modifications from experimental data. Furthermore, assumptions (both mathematical and biological) can be made in order to simplify the mathematical model; the most common one being the steady-state assumption.

Having understood that binding of a sgRNA-dCas9 complex to various gene sequences will not be constant throughout, an ODE system can be devised to model the temporal responses of sgRNA, dCas9, and the resulting sgRNA-dCas9 complex to the gene sequence at various times, based on whether simple activation, simple repression, or competition between the two is induced. In order to mathematically model this, use of a pre-existing, mainstream biochemical model for mapping ligand binding probability to a “receptor” known as the Hill’s equation will be used. Use of Hill’s equation will allow for modelling of the gRNA (and by extension, the mRNA) transcription rates in relation to the interactions between sgRNA, dCas9, and the resulting sgRNA-dCas9 complex.

Based off the above paragraph, manipulating standard derivations of Hill’s equation for transcription factors can yield the probability of activating and inhibiting complexes respectively. Seen below, pa represents “probability of activation” and pi represents “probability of inhibition”:

$$p_a = \frac{\frac{K_ac}{1+K_ac}([\text{dCas9-gRNA}_a])}{1+K_a[\text{dCas9-gRNA}_i]}$$

$$p_i = \frac{\frac{K_ac}{1+K_ac}([\text{dCas9-gRNA}_i])}{1+K_a[\text{dCas9-gRNA}_a]}$$

Converting the mathematics back into biology, let us translate the meanings of these equations briefly. The logic gate of our choice, the NOR gate, requires 2 inputs based upon which a single output is generated. For our inputs, simple activation/repression or competitive activation/repression of sgRNA-dCas9 ligands/complexes are the possibilities. Should the sgRNA-dCas9 combination enhance expression of the output, activation is at play, likewise, the converse result indicates repression all relative to a “control” expression (known as basal expression). In order to determine the effect at play, the output of choice is fluorescent proteins (e.g. GFP). By quantitative measurement of the fluorescence levels, we can work backwards to determine which effect is happening and at what probability it is to occur in the future.

Biologically however, wet lab data had been suggestive of repression being favoured significantly more than activation. These results have been backed up in journal correlation in a multitude of journal results, indicating that this will be a significant factor to account for likely via a weighted probability.

Furthermore, with much assistance from our graduate student advisor Danny Salem and the wet-lab team, more differential equations were generated with the purpose of finding out the dissociation, transcription, and decay/maturation rates based on the promoter gene sequence upon which sgRNA-dCas9 results would be calculated.

\begin{align} [\dot{\text{GEV}}_m] &= p_1-\gamma_m[\text{GEV}_m] \\ [\dot{\text{GEV}}] &= p_2[\text{GEV}_m]-k_1[\text{GEV}][\text{est}] \\ & \qquad +k_{-1}[\text{GEV}/\text{est}]+\gamma_p[\text{GEV}] \end{align} \begin{align} [\dot{\text{tetR}_m}] &= p_1-\gamma_m[\text{tetR}_m] \\ [\dot{\text{tetR}]} &= p_2[\text{tetR}_m]-k_2[\text{tetR}][\text{dox}]+k_{-2}[\text{tetR}/\text{dox}] \\ & \qquad -\omega_2[\text{tetR}]+\omega_{-2}[\text{tetR}]_{nuc}-\gamma_p[\text{tetR}] \end{align} \begin{align} \dot{[\text{Timer}_m]} &= b_0 \\ & \quad +b_1\left(\frac{1}{1+[\text{tetR}/\text{dox}]^n_{nuc}}\right)\left(\frac{[\text{GEV}/\text{est}]^n_{nuc}}{1+[\text{GEV}/\text{est}]^n_{nuc}}\right) \\ & \quad -\gamma_m[\text{Timer}_m] \end{align}

Based off these derived differential equations of binding parameters to the different sequences on the gene sequences, assumptions of steady state and constitutive expression of gene sequences can be made. Therefore, the 1st and 3rd equations simplify to:

$[\text{GEV}_m]^*=\frac{p_1}{\gamma_m}$ and $[\text{tetR}_m]^*=\frac{p_1}{\gamma_m}$

With tetRm and tetR also at steady state (where N is a newly defined term to represent the simplification),

$$N=[\text{tetR}]+[\text{tetR/dox}]-[\text{tetR/dox}]_{nuc}$$

Therefore, the final ODEs with regard to the gene sequence become:

\begin{align} [\dot{\text{GEV}}_m] &= p_2[\text{GEV}_m]-k_1[\text{GEV}][\text{est}] \\ & \qquad +k_{-1}[\text{GEV}/\text{est}]+\gamma_p[\text{GEV}] \\ [\dot{\text{GEV}/\text{est}}] &= k_1[\text{GEV}][\text{est}]-k_{-1}[\text{GEV}/\text{est}]-\omega_1[\text{GEV}/\text{est}] \\ & \qquad +\omega_{-1}[\text{GEV}/\text{est}]_{nuc}-\gamma_p[\text{GEV}/\text{est}] \end{align} \begin{align} [\dot{\text{tetR}}] &= p_2[\text{tetR}_m]-k_2[\text{tetR}][\text{dox}]+k_{-2}[\text{tetR}/\text{dox}] \\ & \qquad -\omega_2[\text{tetR}]+\omega_{-2}N-\gamma_p[\text{tetR}] \\ [\dot{\text{tetR}/\text{dox}]} &= k_2[\text{tetR}][\text{dox}]-k_{-2}[\text{tetR}/\text{dox}]-\omega_2[\text{tetR}/\text{dox}] \\ & \qquad +\omega_{-2}[\text{tetR}/\text{dox}]_{nuc}-\gamma_p[\text{tetR}/\text{dox}] \end{align}

For the aforementioned ODEs, specific constants and experimental values had to be found (both from journals and wet-lab – special thanks again to our graduate student advisor Danny Salem for his assistance and results). The list of parameters are listed below:

Note that logic gate design involves codifying some of the above parameters into a mathematical software, MATLAB in our case, in order to continue numerical solving of equations and aggregating simulation data for analysis. As mentioned above, oftentimes the steady state assumption is used to simplify the modelling given adequate correlation between wet-lab results and modelling data. This allows for the implementation of a NOR logic gate as a modularized unit in simulation capable of being used independently or chained to create circuits.

Implementation of the program for a single NOR gate unit was initially conducted. A Poisson probability distribution was iterated for a number of trials in order to determine the rate of activation vs. repression. Note, given the two possible outcomes (i.e. activation or repression), a bimodal distribution was the result that supported both journal results and wet-lab results that sgRNA-dCas9 repression was strongly favoured over activation. Over 10000 trials (n = 10000), the number of times repression was indicated was nr = 7824, leaving activation to have occurred na = 2176. Another factor to consider however, is the logic gate model used i.e. the NOR gate, is only expressed should both inputs be inhibited. As such, the theoretical expectation should a 75:25 ratio and these results are similar to journal results obtained in Gander et al., 2017.

Furthermore, a Gillespie algorithm was run in order to detail the repression results as a function of the fluorescence levels. The choice for a Gillespie algorithm comes from a need to represent the sgRNA-dCas9 system binding to the promoter sequences in a stochastic manner, involving nothing more than the system and the promoters as molecules. While the results of the Gillespie algorithm varied greatly from multiple simulations (run up to n=100 trials each for 10 simulation total), the general observable trend was a weak logarithmic association between fluorescence levels over time. However, since there was significant fluctuation between the trends of the simulations, an accurate figure representation could not be portrayed, as parameters needed to be modified in order to attain more consistent results. Due to time constraints though, that was not made possible.

Given that a general structure was modeled on MATLAB, the possibility of chaining gates was also considered. While, we were cognizant that there were obvious disparities between expected results and experimental simulated results, the program was still put through its’ paces in order to determine if the sgRNA-dCas9 complexes were still capable of producing results in line with those of the corresponding logic gates. Gander et. al had published results of a similar experiment with results seen below.

Figure 1 – Expected NOR gate circuit results. Varying logic circuits created using solely NOR gates. The associated truth tables and fluorescence results are depicted on the right-hand side. Figure taken from Gander et al., 2017. Grey region represents a “dynamic” zone between activation and repression parameters.

Experimental results from our simulations were not nearly as ideal. Oftentimes, only certain pairings in a truth table correlated with theoretical results and literature results seen above. While “normal” distributions were attained, the distributions would favour the side of repression/activation to be expected, however, often centred much closer to the dynamic zone than was expected. This “centering” of results was further indication that the designed model required tweaking. Possible considerations are influence of steric hindrance possibly being greater than anticipated, missing binding parameters/inability within the coding of the model, and increased leakiness as chained circuit grow. A potential consideration for the future would be to work backwards and design truth tables and pre-set modulated logic gates and evaluate the generated circuit using a “sandwich approach” of experimental results with pre-set, calibrated logic gate designs.

In short, further evaluation will be required for conclusivity of results. However, the data is weekly suggestive of sgRNA-dCas9 complexes to be capable of modulating repression much like a transcription factor on varying promoters (e.g. tet and Gev).

References

Articles

Gander, Miles W., et al. “Digital logic circuits in yeast with CRISPR-dCas9 NOR gates.” Nature Communications, vol. 8, 2017, p. 15459., doi:10.1038/ncomms15459.
Ay, A., & Arnosti, D. N. (2011). Mathematical modeling of gene expression: a guide for the perplexed biologist. Critical Reviews in Biochemistry and Molecular Biology, 46(2), 137-151. doi:10.3109/10409238.2011.556597
Goñi-Moreno, A., & Amos, M. (2012). A reconfigurable NAND/NOR genetic logic gate. BMC Systems Biology,6(1), 126. doi:10.1186/1752-0509-6-126
Ma, D., Peng, S., & Xie, Z. (2016). Integration and exchange of split dCas9 domains for transcriptional controls in mammalian cells. Nature Communications, 7, 13056. doi:10.1038/ncomms13056
Weinberg, B. H., Pham, N. T., Caraballo, L. D., Lozanoski, T., Engel, A., Bhatia, S., & Wong, W. W. (2017). Large-scale design of robust genetic circuits with multiple inputs and outputs for mammalian cells. Nature Biotechnology, 35(5), 453-462. doi:10.1038/nbt.3805

Previous iGEM teams