4. Ring modelling results and discussion 105
4.3. Altering the model geometry from flat to cylindrical 117
on the inner surface of a cylinder, with a radius of R = 12/2π μm, which represents the inner surface of the S. pombe plasma membrane. The attachments of the nodes to the membrane were modelled as Hookean springs, so that if the nodes are pulled away from the membrane, or pushed in towards it, they will experience a restoring force proportional to the distance that they have been displaced. This connection was given a high spring constant (i.e. high stiffness) in order to ensure that the extension or compression is always very small.
As before, an excluded volume interaction was included, except now this was applied to any beads in the actin filaments that moved to a radius greater than the radius of the cylinder (i.e. outside of the cytoplasmic volume). For these beads, a constant restoring force of 5 pN was applied towards the cylinder’s axis, in order to push the actin beads back into the allowed volume.
It is important to note that, whilst the model now uses 3D cylindrical ring geometry, the simulated ring is still unable to contract, as the cylinder that it is attached to has a fixed radius. We have already discussed how, in S. pombe cells, the process of ring contraction is closely coupled to the process of septum synthesis, and how neither is able to occur without the other (section 1.11, page 30) [32,89]. Therefore, performing a realistic simulation that allows the ring to contract would also require modelling the process of septum synthesis, which would further increase the complexity of the model.
We performed simulations with this cylindrical model, using the same parameter values that were used for the simulations in our flat model. However, in our initial simulations we observed that many actin filaments did not remain against, or near, the membrane, but were pulled away (Figure 4.2C). Because the previous simulations used a flat geometry, this behaviour was not seen before, and we realised this was
most likely happening because the pulling force on the actin filaments was overcoming the grabbing forces that were holding them near the membrane. Comparison of our 3D cylindrical model to a similar 3D model (aimed at describing ring formation [164]) suggested that this was the case, as for their simulations a maxInt value of 3 was used for each node, with slightly less than half the number of nodes that we used in our simulations (and therefore roughly double the amount of Myo2 per node) [164].
We found that reducing the value of maxInt down to 1 was necessary in order to keep the majority of the filaments contained in the ring (Figure 4.2D). It would theoretically have been possible to increase the strength of the myosin grabbing forces, which keep actin filaments near the membrane. However, the way that myosin grabbing forces are implemented in this model arguably leads to them being overpowered already, as they do not experience the same decrease in force when interacting with multiple filaments that is applied to the pulling forces. We will discuss our attempts to resolve these issues in the next section of this chapter.
Additionally, from a basic consideration of the number of Myo2 molecules in each node, and from knowledge of the processivity of type II myosins, it can be argued that the myosin pulling forces in the simulation are also overpowered, hence we chose to reduce these rather than increase the strength of the grabbing forces. This is because, using the measured number of Myo2 molecules in the S. pombe AMR, and assuming there are 75 Myo2 clusters in the ring (as was the case in the original model), it can be calculated that there would be an average of around 40 Myo2 molecules in each cluster [64]. In the original model, these clusters could interact with up to 10 actin filaments at maximum force, which equates to 4 Myo2 molecules exerting a time-‐averaged force of 4 pN on each filament. Considering that type II myosins are known to be not particularly processive, with Myo2 previously measured to have a duty ratio of approximately 10% [95], then for 4 Myo2 molecules interacting with a single actin filament, a time-‐averaged force of 1.6 pN would be a
better estimate1. Therefore, decreasing the value of maxInt in our simulations also makes sense based on considerations of the mechanochemical properties of type II myosins.
We wondered if increasing the amount of crosslinking in the ring would also help to prevent this single-‐filament peeling that was observed: On the one hand, if the actin filaments were more tightly crosslinked to each other, then this would mean that the pulling forces from the nodes would effectively be shared across multiple filaments, and therefore reduce the force on individual filaments. On the other hand, this would not increase the number of connections between actin filaments and the membrane, so it might not prevent the single-‐filament peeling from happening. We performed some simulations where we used a maxInt value of 5, and increased the crosslinker binding rate, however we found that this did not appear to make a difference to the amount of individual filaments peeling off of the ring. We shall discuss more about the effect that crosslinkers have on the simulation in section 4.8.3.
After reducing the value of maxInt in our simulations, and checking that this produced more WT-‐like behaviour (Figure 4.2D), we also decided to implement a new method to measure the ring tension: Previously, the ring tension was measured by summing all the tension in all the springs in the simulation, in the direction parallel to the ring (i.e. along the x-‐axis). As the direction parallel to ring was no longer constant, this method became more complicated to use in our cylindrical model. Instead, to measure the total contractile force the ring exerts on the inner surface of the cylinder, we summed the tension stored in the springs connecting the nodes to the membrane. It can be shown that this contractile force, Fc, is equal to 2π times the ring tension, T (i.e. Fc =
2πT)2. Using both methods, and plotting 2πT and Fc as a function of time on the same graph, showed that both results are largely in agreement
1 Assuming that the 4 Myo2 molecules exert 4 pN of force on a single actin filament 40% (10%×4) of the time, this leads to a time averaged force of 1.6 pN. This is a very
simplistic approximation, which does not take account of, for example, multiple myosin heads binding at the same time.
(Figure 4.2E). Using our 3D cylindrical ring model, with the reduced value of maxInt, the simulated ring produced a mean 2π×tension of 243 ± 17 pN, and exerted a mean contractile force on the membrane/cell wall of 235 ± 15 pN.
Returning to our flat model, and again using the reduced value of maxInt in our simulations, we measured an average ring tension of 41.7 ± 3 pN (Figure 4.2F), which is more than a 5× reduction compared to the case with a maxInt value of 5 (Figure 4.2B). While this decrease in ring tension would be expected after reducing the value of maxInt, this means that the simulated ring tension is no longer in agreement with the experimentally measured value of 390 pN [35]. However, the reduction in the value of maxInt was necessary in order to keep actin filaments near the membrane in our cylindrical model, an issue which was not apparent when using the flat geometry. Therefore, there must be additional mechanisms which help to make the process of tension generation more efficient, or which increase the number of connections between the AMR and the membrane, and we will discuss some of these in section 4.8.2.
4.4. Making the myosin-‐actin interactions more realistic