4. Ring modelling results and discussion 105
4.2. Modifying the ring structure and dimensionality 113
Since the publication of the original model, super-‐resolution microscopy of the S. pombe ring has provided strong evidence that the precursor nodes that are present during ring formation (approx. 140 of them) also persist throughout ring maturation and contraction [65]. Based on previous measurements of protein concentrations in the S. pombe ring, it is estimated there is an average of one formin Cdc12 dimer and ~10 Myo2 dimers per node [64,65]. Therefore, we first modified the model to reflect these observations, with formin and myosins both placed in the same node structures. The simulated rings contained an average of 150 nodes, the same as the number of formin dimers in the previous simulation, and this meant that each node nucleated a single actin filament. To model the presence of Myo2 in these nodes, they were also given the ability to capture and pull any nearby actin filaments. This was largely implemented in the same way as for the myosin clusters in the original model, with the exception that a node could not interact with the filament that it nucleates.
Nodes were given a drag coefficient of 1.5 nN s/μm, which is between the values used previously for the formin dimers and myosin clusters. A previous model, examining ring formation, used a node drag coefficient of 400 pN s/μm [164], significantly less than was used in our
simulations, and in the original simulations using this model. However, in terms of the general behaviour of the ring, the precise value used should not matter too much, as long as it is high enough to limit myosin clusters to the low velocity/high force end of their F-‐v relationship, which allows us to assume that the magnitude of the pulling forces is a constant. We used the formin dimer unbinding rate as the unbinding rate for nodes in the simulation, and when a node unbinds from the ring the filament that it nucleates is also removed from the simulation. Pairs of nodes also experience excluded volume interactions when they get too close to each other, in the same manner as the myosin clusters in the original model, with the same parameter values used calculate the magnitude of this force.
Because we doubled the number of myosin-‐containing entities in the model, going from 75 Myo2 clusters in the original model to 150 nodes, in order to keep the total pulling force in this model consistent with the previous one, we needed to do the equivalent of halving the number of Myo2 molecules in each node. To do this, we reduced the value of maxInt from 10 down to 5.
Additionally, we increased the length of the ring from 10 μm to 12 μm (change in diameter from 3.2 μm to 3.8 μm), as it was previously observed that adf1-‐M2 and adf1-‐M3 cells have a slightly increased cell diameter when compared to WT cells [66]. Finally, we also decreased the ring width from 0.2 μm to 0.1 μm, to reflect the results of recent super resolution observations of the ring [92].
Next, the model was made to be 3-‐dimensional, by allowing for height (z coordinate) above the membrane (defined as the x-‐y plane, where z = 0), and we set the volume above the membrane, i.e. z > 0, as being inside the cell, while the volume below the membrane, i.e. z < 0, was defined as being outside of the cell. The nucleation angle of actin filaments above the membrane was set to 8°, as this was the angle that was previously measured for actin filaments during the process of ring formation [62]. The nucleation angle in the x-‐y plane was chosen randomly, as was the case in the 2D version of the model. The motion of
nodes was constrained to only take place in the x-‐y plane (i.e. no z motion), and for any actin filament beads with a z-‐coordinate below the membrane height, a constant force of 5 pN was exerted in the +z direction, to account for the excluded volume below the membrane, and to try and return the actin beads to the ‘allowed’ volume in the simulation.
Simulations with this model produced the same basic behaviour as observed previously, with the nodes undergoing bidirectional motion, and with most actin filaments being successfully captured into the ring (Figure 4.2A, blue circles represent nodes, grey lines are actin filaments, and green circles are crosslinkers). However, the measured tension was reduced slightly, from 340 ± 57 pN to a value of 273 ± 22 pN (Figure 4.2B). This is most likely because our simulated ring had a length of 12 μm, rather than 10 μm, as was previously used. As we effectively kept the total amount of myosin in the ring the same (by halving the value of maxInt), this means there is a slightly lower linear density of myosin in our simulated ring, which would logically lead to a slightly lower ring tension in our simulations [35]. Changing the ring length from 10 μm to 12 μm will decrease the myosin density by a factor of 1.2, so correcting for this by multiplying our measured tension by 1.2 produces a value of 327 ± 26 pN, which is closer to the original value of 340 ± 57 pN (this assumes that the relationship between myosin density and ring tension is linear, at least over the range of values used here).
0 100 200 300 400 500 Time (s) 0 50 100 150 200 250 300 Contractile force (pN) 2 T Fc F c = 235 ± 15 pN 2 T = 243 ± 17 pN 0 50 100 150 200 250 300 350 400 450 500 Time (s) -10 0 10 20 30 40 50 Tension (pN) T = 41.7 ± 3.0 pN Figure'M1.'Making'the'model'35dimensional,'and'u8lising'different'geometries' For$A$&$C,$all$data$shown$is$from$simula5ons$using$a$maxInt'value$of$5.$All$data$in$D$–$F$is$from$simula5ons$using$maxInt$=$1.$ All$scale$bars$are$2$μm.$ (A) Montage$of$snapshots$from$a$simulated$3D$ring,$using$the$same$flat$geometry$that$was$used$in$simula5ons$from$the$ original$model.$$ (B) Ring$tension$vs.$5me$for$the$simulated$ring$shown$in$A.$The$average$tension$is$calculated$from$aMer$100$s$of$simulated$ 5me,$un5l$the$end$of$the$simula5on.$ (C) Snapshots$from$a$simula5on$using$a$3D$cylindrical$geometry,$with$the$same$parameter$values$as$the$simula5on$shown$ in$A.$ (D) Snapshots$from$another$simula5on$using$the$cylindrical$geometry,$but$with$maxInt$reduced$to$1.$ (E) Contrac5le$force$vs.$5me$for$the$simula5on$shown$in$D.$The$contrac5le$force$is$measured$using$two$methods,$and$the$ results$of$both$of$these$are$ploQed.$ (F) Tension$vs.$5me$for$a$simula5on$using$flat$3D$geometry$(images$not$shown),$similar$to$A,$but$with$maxInt$=$1,$as$was$ used$to$successfully$simulate$rings$with$cylindrical$geometry.$ 0 100 200 300 400 500 600 Time (s) 0 50 100 150 200 250 300 350 Tension (pN) T = 273 ± 22 pN (A)' (B)' (C)' maxInt$=$5$ Δt$=$4s$ Δt$=$12s$ (D)' (E)' (F)' maxInt$=$1$ Δt$=$12s$
Figure 4.2: Making the model 3D, and utilising different geometries. For A -‐ C, all data shown is from simulations using a maxInt value of 5. All data in D – F is from simulations using maxInt = 1. All scale bars are 2 μm.
(A) Montage of snapshots from a simulated 3D ring, using the same flat geometry that was used in simulations from the original model.
(B) Ring tension vs. time for the simulated ring shown in A. The average tension is calculated from after 100 s of simulated time, until the end of the simulation. (C) Snapshots from a simulation using a 3D cylindrical geometry, with the same
parameter values as the simulation shown in A.
(D) Snapshots from another simulation using the cylindrical geometry, but with maxInt reduced to 1.
(E) Contractile force vs. time for the simulation shown in D. The contractile force is measured using two methods, and the results of both of these are plotted.
(F) Tension vs. time for a simulation using flat 3D geometry (images not shown), similar to A, but with maxInt = 1, as was used to successfully simulate rings with cylindrical geometry.
4.3. Altering the model geometry from flat to cylindrical