Modeling

In our mathematical model we tried to reproduce the ideal dynamics of the system species. The model was realized by means of ordinary differential equations in which the variables correspond to the cellular species while the parameters represent actual biochemical constants. Assumptions are made in order to make the model clear and intuitive; each of them will be discussed and justified in the following sections. Since our systems literally “counts” yeast cell cycles, we decided to describe the model results following the natural progression of the cell replicative lifespan, i.e., we start by describing the cellular events in the first cell cycle and then we move on to describe the actual counting system in the following two cycles.

First cell cycle - the daughter cell resetter

Our system was designed in order to “reset” the age counter in newly born daughter cells. To accomplish this, one of the key components of the age counter, dCas9, is not produced in the very first cell cycle of daughter cells. As a consequence, the gRNA molecules that leak into the daughter cell during cell division won’t be able to induce the production of any fluorescent protein. As can be seen in fig. 1 we expect only Csy4 to be produced in the first cell cycle. To be noted, the G1-specific degradation tag inserted in both Csy4 and dCas9 is expected to trigger the respective degradation of the species at the end of the G1 phase. This feature can also be seen in fig1, in which Csy4 is rapidly depleted after the end of G1 phase.

Our daughter resetter not only impedes the production of fluorescent proteins induced by gRNA-dependent promoter but it also produces YFP, which becomes the characteristic color of the first cell cycle, together with the signal needed for the expression of CFP during the next cell cycle. As showed in fig. 2, during the first cycle YFP is constitutively produced by the endogenous daughter-cell specific promoter PDSE4. By design, together with the mRNA for YFP, also the 28-gRNA1-28 is produced. This is the signal the will last until the second cell cycle and, after being processed by CSY4 and bound to dCas9, will be able to induce the production of CFP. One non-ideality included in the model is the fact that CSY4 is present during the first cell cycle (fig. 1); as a consequence some 28-gRNA1-28 are turned into gRNA1 at the very moment of their creation. We had no way to test experimentally how long either form of gRNA lasts in the intracellular environment; this is the main source of uncertainty in our model. For the sake of simplicity, we chose a parametrization that would allow for both gRNA molelcules to survive degradation until the next cell cycle.

Second cycle: the age counting system

Starting from the second cell cycle, the real age counter is activated. The activation signal is represented by the gRNA1 molecules that are still present in the nucleus from the previous cell cycle. As can be seen in fig. 3, in the G1 phase of the second cell cycle both Cas9 and Csy4 are expressed; Csy4 processes the remaining 28-gRNA1-28 molecules while Cas9 binds with gRNA1 to form the mature transcription factor. The newly formed transcription factor then binds to the promoter specific for the production of CFP, which rightfully becomes the characteristic color of the second cycle. In fig. 3 the dynamics of CFP and the newly produced 28-gRNA2-28 are also reported. As in the previous cell cycle the 28-gRNA2-28 molecules constitute the signal for the next cell cycle.

Our daughter resetter not only impedes the production of fluorescent proteins induced by gRNA-dependent promoter but it also produces YFP, which becomes the characteristic color of the first cell cycle, together with the signal needed for the expression of CFP during the next cell cycle. As showed in fig. 3, during the first cycle YFP is constitutively produced by the endogenous daughter-cell specific promoter PDSE4. By design, together with the mRNA for YFP, also the 28-gRNA1-28 is produced. This is the signal the will last until the second cell cycle and, after being processed by CSY4 and bound to dCas9, will be able to induce the production of CFP. One non-ideality included in the model is the fact that CSY4 is present during the first cell cycle (fig. 1); as a consequence some 28-gRNA1-28 are turned into gRNA1 at the very moment of their creation. We had no way to test experimentally how long either form of gRNA lasts in the intracellular environment; this is the main source of uncertainty in our model. For the sake of simplicity, we chose a parametrization that would allow for both gRNA molelcules to survive degradation until the next cell cycle.

Figure 3. Dynamics of gRNA molecules, dCas9 and Csy4 during the first two cell cycles.

The third cell cycle

By now it should be clear how the counting system works. As in the previous cycle dCas9 and Csy4 are produced in the early G1 phase. Csy4 turns 28-gRNA2-28 into gRNA2 and dCas9 binds to gRNA2 and forms the mature transcription factor. Unfortunately we didn’t get so far as to implement the counting until the third cell cycle. However, in figs 6 it is shown how the expression of the third fluorescent protein would be if we did. Similarly, fig 6 also shows the dynamics of the newly produced gRNA molecules, which in a hypothetically extended counter would constitute the signal for the next cycle.

Figure 4. Dynamics of gRNA molecules, dCas9 and Csy4 during the first three cell cycles.