Model

We have created a complete model of our system involving 22 chemical species participating in 32 biochemical reactions. Our model consists of a toggle switch which is controled by either two ligand-receptor pathways or more directly by the addition of guide RNA, like in our liposome fusion experiments.

ON and OFF and ON and OFF and ON again...

This time course simulation of our model shows that the toggle switch can be switched between two stable states of gene expression ("ON" and "OFF") endlessly by temporarily increasing only a single chemical species as a trigger.

In the ON state a complex of guide RNA ON and dCas9 strongly binds promoters of the gene coding for guide RNA OFF, blocking its production. This repressor ON holds the system in the ON state, until eventually the switch is flipped by adding ligand OFF. This happens because ligand OFF will lead to the production of guide RNA OFF, which can form the second repressor, repressor OFF respectively. Before enough repressor OFF has accumulated, the system finds itself in a short transition period until it reaches the stable OFF state. In the OFF state expression of UGFP mRNA is prohibited because guide RNA OFF has target sites in the promoter region of its gene.

Try switching it ON and OFF yourself!

Detailed model description

               <img class="d-block w-50" src="" alt="First slide">

                   <p>This is a complete representation of the model with all reactions and species involved. It further includes important constants and gives hints at the type of interactions. At first sight this might look rather complex, but it can be broken down into smaller pieces. Our model is divided into 4 Independent sections:</p>

two receptor pathways, which produce a transcription activator,
the reporter pathway, which produces a reporter protein (here Ubiquitin-GFP) that indicates whether the toggle switch is in an ON or OFF position, and,
the toggle switch itself

               <img class="d-block w-50" src="" alt="Second slide">

                       <p>Let's assume our toggle switch is currently in the OFF position, which means there is a lot of repressor OFF (red) in the system. This trimeric repressor reduces the production of gON, short for guide RNA ON, and UGFP mRNA to a minimum, which means we won't see any fluorescence signal.</p>
                       <p>Repression is expressed by the repression function (1). Where the product P is produced over time t dependent on the concentration of the repressor [R] in the system. k represents the strength of the promoter, which we choose to be 0.05 in the case of gON and 0.1 for UGFP mRNA. Furthermore, k_a and k_b stand for the efficiency of the repression.</p>

                       <img class="w-50" src="" alt="Equation One">

               <img class="d-block w-50" src="" alt="Third slide">

                       <img class="w-50" src="" alt="Equation Two">

                       <p>Now that we know what our current state is, let's see what happens if we add ligand ON to our system (indicated by a green arrow). First of all the ligand ON will bind to his receptor ON thereby forming an activator ON.</p>
                       <p>Since this step is reversible, even though favoring the activator: k₂₅ > k_-25, it is indicated by a forward and backward arrow respectively.</p>
                       <p>For a similar reaction A + B ⥂ C the formation of the complex C over time t will follow the rules of simple mass action (2). Where k_i and k_-i are the rate constants for the forward and backward reaction.</p>

               <img class="d-block w-50" src="" alt="Fourth slide">

                   <p>After our activator ON was formed it will bind to its target DNA sequence and will live up to his name, activating (green) the expression of gON. Since we know that our repressor OFF is constantly trying to shutdown gONs production, we need enough activator ON to compete against. This is a critical point of our system. However in reality repressor and activator do not act on the same gene, so gON should always be expressed in the presence of activator ON.</p>
                   <p>The activation is formulated in the activation function (3).  Where the product P is produced over time t dependent on the concentration of the activator [A] in the system.  Just like in the repression function k_c and k_d stand for the efficiency of the activation.</p>

               <img class="d-block w-50" src="" alt="Third slide">

                   <p>Now that we have gON in the system it can bind to dCas9 and form an initial collision complex dCas9.gON^cc, which more slowly then undergoes a transition to the active dCas9.gON complex, like described by Raper et al. (2018). Both reactions are reversible and follow the mass action law. The dCas) in the active complex can then be guided by gON to its DNA target site gtON, which will lead to the trimeric repressor ON complex. Repressor ON is a rather stable complex indicated by k₇ >> k_-7</p>

               <img class="d-block w-50" src="" alt="Third slide">

                   <p>Since the target site lies in the promoter region of a gene coding for gOFF. The repressor ON will hinder the production of gOFF, just like the repressor OFF did to gON.</p>
                   <p>The crucial point here is that if gOFF is not being expressed anymore, then new repressor OFF can not form. Over time repressor OFF will degrade until it reaches its minimal concentration. This allows for both the production of gON, which accelerates the toggling event, and UGFP mRNA, showing us that it actually switched by the translation of UGFP.</p>

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Improving our design

Regulatory region design

Selection of a constitutive promoter

We are using only constitutive promoters in our design of the toggle switch, which means these gene will be constantly transcribed without needing any kind of activation. Those are 3 genes coding for: UGFP, guide RNA ON (gON), and guide RNA OFF (gOFF). In order to ensure the genetic integrity of our final construct containing these 3 genes we had to use different promoters and terminators for each of them. Our PI's suggested 3 constitutive promoters for P. pastoris (GAP, RPP1B and TEF), which are very close in strength:

GAP = 1 (reference)
TEF = 0.5---0.8 (≈ 0.65)
RPP1B = 0.5

Predictions from our model show that using GAP as a promoter for either of the 2 guide RNAs accounts for having to use at least a 32'500-times higher concentration of one ligand relative to the other in order to switch it! If we were to use GAP in combination with RPP1B, that number would grow to over 10¹²-times the concentration, which is simply impossible to be done in reality. This is why we choose to use TEF for gOFF, because we wanted the OFF state to be more stable, and RPP1B for gON. For that the model predicts an equal concentration of ligand to be used.

Using multiple target sites

There are 35 different parameters influencing the behaviour of our model, but not all of them play an important role for the functionality of it. Two of the most import ones k~a~ and k~b~ represent the strength of repression mediated by an active dCas9 complex binding to a promoter.

k~a~ is the first parameter of the repression function (1) explained earlier. Its original value in the model is 100. As we can see in this graph k~a~ can take a lot of different values, without destroying the switching behaviour. However it has to be at least 75. Since it reduces the transition time between the two states, giving us a much sharper UGFP signal, we want it to be as big as possible in reality.

k~b~ being the second parameter of the repression function (1) has a similar effect as k~a~. Going up in value it produces a sharp switching curve, so we too want k~b~ to be high. Its original value is 2, which also turn out to be the minimal value it can take for the system to function properly.

We interpreted the meaning of k~a~ to be the general repressive strength of a dCas9 complex binding to a promoter. As described by Qi et al. (2013) one dCas9 will decrease the transcription of a gene about 100-fold, that is why assigned the value 100 to k~a~ in the first place. The binding of a second dCas9 to the same promoter surely increases the repressive effect, but we do not know how much. We think that multiple binding has a nonlinear, rather exponential, effect and might therefore be linked to k~b~. Since k~b~ starts giving us a relatively sharp curve at 4, we decided to use 4 target sites for the same guide RNA at our promoter regions.

Selecting ligand-receptor pathways

When we first designed our system we had 3 possible ligand-receptor pathways in mind, that could be used in both yeast and plants. One of them uses copper(I) as a ligand that can bind to its receptor CUP2 (ACE1) further activating transcription. However Fürst and Hamer (1989) suggested that binding of 4 Cu(I) molecules to CUP2 happens cooperative, which means that binding of one ligand increases the affinity of the receptor for a second ligand and so on. In comparison to the other 2 ligand-receptor interactions, this would mean a non-linear response to ligand application.

Giving one of the two ligand-receptor pathways cooperative binding properties in our model showed that this would make it impossible to switch it with the other ligand. That gave us a clear hint for using the other two ligands (estradiol and ethanol) for switching.

Model vs. Reality

Influence of Mitosis

During mitosis the genome is nearly transcriptionally silent, which means our system will be almost shutdown for some time. As we can see here in this graph our model suggests that even if transcription is completely halted for some time, it can fully recover its previous state. However if that period is very long, the system will always return to the slightly more stable OFF state.

Here you can see yeast cells with our toggle switch. A small fraction of them is expressing UGFP, which means they are in the ON state. It seems as these are always clusterd together, which suggests they originated from a common mother cell that switched to the ON state in the first place. That shows that our assumtions taken from the model might be correct, cells seem to be able to maintain their state over mitosis.

R reference Raper, Austin T.; Stephenson, Anthony A.; Suo, Zucai (2018): Functional Insights Revealed by the Kinetic Mechanism of CRISPR/Cas9. In: Journal of the American Chemical Society 140 (8), S. 2971–2984. DOI: 10.1021/jacs.7b13047

Qi, Lei S.; Larson, Matthew H.; Gilbert, Luke A.; Doudna, Jennifer A.; Weissman, Jonathan S.; Arkin, Adam P.; Lim, Wendell A. (2013): Repurposing CRISPR as an RNA-Guided platform for sequence-specific control of gene expression. In: Cell 152 (5), S. 1173–1183. DOI: 10.1016/j.cell.2013.02.02

Fürst, P.; Hamer, D. (1989): Cooperative activation of a eukaryotic transcription factor: interaction between Cu(I) and yeast ACE1 protein. In: Proceedings of the National Academy of Sciences of the United States of America 86 (14), S. 5267–5271

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