# Chaperones Role in Ageing

In this section we will reproduce the results in the paper Modelling the actions of chaperones and their role in ageing by Proctor CJ, Soti C, Boys RJ, et al. Mechanisms of Ageing and Development. 2005. 126, 119-131. We will detail the process of importing and editing the model as well as running the simulations. If, in following this tutorial, your results differ from these presented, you can open the file `examples/cain/Proctor2005_Hsp90.xml`. Here the models and simulation methods are already defined.

Go to the BioModels Database and search for Proctor. Select the model BIOMD0000000091-Proctor2005_Hsp90. From the "Download SBML" pull-down menu, download the model in any of the offered SBML versions. Then open the SBML file in Cain. You wil get some warnings that Cain ignored the tags relating to unit definitions. However, because the model uses substance units (number of molecules), the model is imported correctly.

## Unstressed cell

First we will study the model for the unstressed cell. In the model list change the name of the imported model from "Proctor2005_Hsp90" to "Unstressed." In Figure 2 on page 125 shows simulation results for the unstressed cell. We now create a simulation method to reproduce these results. Click the add button in the method list and name the new method "Direct1e4." We will use the default simulation method, which you may see in the method editor has the following properties:

- It works on time homogeneous models. (None of the reaction propensities in this model are time-dependent.)
- It generates uniformly-spaced, time series data. That is, it records the species populations and reaction counts at uniformly-spaced intervals in time.
- It uses Gillespie's direct method.
- It uses a 2-D search algorithm to generate discrete deviates.

In the method editor, set the recording time to 10,000. In order to record the state every 10 seconds, set the number of frames to 1001. In the launcher panel, click the launch button to generate a single trajectory. The simulation will take a minute or two to complete.

Click the plot button in the output panel. In the plot configuration window first right-click on the "Show" column head to deselect all species. In order to match the first plot, select the following species: Hsp90, MisP, MCom, NatP, and AggP. Next left-click on the "Line Color" column header to color these species in contrasting hues. Change the legend location to center right. Enter a title and axes labels. The plot configuration window is shown below.

Click the "New plot" button to generate the following plot.

Because NatP is present in relatively large numbers, it is difficult to see the behavior of the other species. Deselect NatP and click the "New plot" button to get a better view of the dynamics for Hsp90 and MCom.

As noted in the paper, the system quickly reaches a steady state with about 95% of the total protein in its native form and the remaining 5% being misfolded and complexed to Hsp90.

The second plot in Figure 2, shows ROS, ATP, and ADP. Since the copy numbers for these species differ greatly, we use the same trick we introduced in the Lac Operon section. We plot each species in a long, narrow window. We see that, after equilibrating, the populations of these species remain roughly constant throughout the simulation.

## Cell exposed to transient stress

Now we consider a cell exposed to a transient stress. First the system is simulated for 8 minutes. Then the amount of ROS is doubled, and the simulation advanced for 10 minutes. Next the amount of ROS is halved and the simulation continues to a total time of 2000 seconds.

We could modify the model by adding events to double and then halve the population of ROS. However, the simulation is expensive and the solvers that support events (written in Python) are much slower than the C++ solvers that do not support events. Since we are only generating one trajectory, we will use a hack to use the optimized solvers and finish the simulation in a reasonable amount of time. We will run the simulation for 8 minutes and then note the species populations at the end time. Then we will clone the model and replace the initial amounts with the amounts at 8 minutes on the partial trajectory, with the exception of doubling the amount of ROS. We then simulate this model for 10 minutes and again transfer the final populations to a cloned model with these as initial conditions. Thus we will generate the trajectory using three models, which differ only in their initial conditions, and three methods, which differ only in the recording time.

In the methods list, clone the "Direct1e4" method and name the clone "Stage1." For this method set the recording time to 480 seconds and the number of frames to 49. Then click the launch button to simulate the equilibration in the unstressed cell. In the output panel, click the table button and then choose to display "Populations" and then "Ensemble showing the last frame." The table of final amounts is shown below.

In the model list, clone "Unstressed" and name the result "Doubled". Set the initial amounts for the species to be the final amounts from the first stage, except for doubling the amount of ROS. In the method list, clone "Stage1" to obtain "Stage2." Set the start tim to 480, the recording time to 600, and the number of frames to 61. Simulate the second stage of the trajectory and bring up a table of the final populations.

In the model list, clone "Doubled" and name the result "Halved". Set the initial amounts for the species to be the final amounts from the second stage, except for halving the amount of ROS. In the method list, clone "Stage2" to obtain "Stage3." Set the start tim to 1080, the recording time to 920, and the number of frames to 93. Simulate the third stage of the trajectory.

Now that we have the three stages of the trajectory, we will plot them together. Click the plot button in the simulation output panel. Select "Unstressed, Stage1" at the top of the plot configuration window. Select Hsp90, MisP, MCom, NatP, and AggP, and color them with contrasting hues. Then click the New plot button to start a new plotting window. Now do the same for "Doubled, Stage2" and "Halved, Stage3", except deselect the legend and use the Plot button to add the plots to the current window. You may do the same for plotting ROS, ATP, and ADP. The results are shown below.

## Increase in ROS with time

Next we decrease the rate of ROS removal, which results in an increase of ROS over time. This causes a decline in native protein and a corresponding increase in the denatured protein. In the model list, clone the "Unstressed" model and name the result "IncreaseROS." In the parameter editor change the value of k21 from 0.001 to 0.00001. Since the interesting dynamics occur over a longer time scale, we will run this simulation ten times longer than for the unstressed cell. In the method list, clone the "Direct1e4" method and name the result "Direct1e5." Set the recording time to 100,000 seconds and launch the simulation. It will probably take about one and a half hours to complete the simulation. The results are shown below.

Since this simulation is expensive, it is worth considering approximate methods to generate trajectories. However, this model is a poor candidate for tau-leaping. It has fast reactions involving species with low populations. Thus tau-leaping is actually much slower than the direct method for this problem.