Computational simulation of the horsetail movement

This method has been described in [1]. The simulation was implemented in R. The default values of the parameters used in the simulations are summarized in Table 1.

1. Definitions

In this simulation, the SPB–nucleus complex moves in the range of $\left[-\frac{L}{2},\frac{L}{2}\right]$, where $L$ is the longitudinal length of the cell. The position of the SPB–nucleus at time $t$ is designated as $X\left(t\right)$. $n$ microtubules emanate from the SPB towards each direction along the axis and behave independently of each other. The position of the plus end of $i$-th microtubule $\left(i=1,\dots ,2n\right)$ at time $t$, designated as $x_i\left(t\right)$, is given by: $\begin{array}{}\text{(1.1)}& x_i\left(t\right)=X\left(t\right)+e_i\cdot l_i\left(t\right),\end{array}$ where $\begin{array}{}\text{(1.2)}& e_i=\left\{\begin{array}{c}1\left(i=1,3,\dots ,2n-1\right)\\ -1\left(i=2,4,\dots ,2n\right)\end{array}\right.,\end{array}$ which gives the direction of $i$-th microtubule, and $l_i\left(t\right)$ is the length of $i$-th microtubule. When $\left|x_i\left(t\right)\right|\ge \frac{L}{2}$, $i$-th microtubule contacts the cell-end cortex. Microtubules can extend over a range of cell $\left[-\frac{L}{2},\frac{L}{2}\right]$, as they can curl and extend at the tips of a cell.

2. Dynamic instability of microtubules

We measured the frequency of catastrophe and rescue in the cytoplasm or at the cell end cortex, designated as $f_{\text{cat}\left(\text{cyt}\right)}$, $f_{\text{res}\left(\text{cyt}\right)}$, $f_{\text{cat}\left(\text{ctx}\right)}$ and $f_{\text{res}\left(\text{ctx}\right)}$. These values are listed in Appendix Table S1. Consider these frequency values as the probability of each event occurring in a unit of time. We define two modes of $i$-th microtubule, ${\phi }_i=\left\{G,S\right\}$, where $G$ is an elongating mode and $S$ is a shrinking mode. Mode transitions designated as $G\to S$ and $S\to G$ correspond to catastrophe and rescue events of a microtubule, respectively. In case $\left|x_i\left(t\right)\right|<\frac{L}{2}$, $G\to S$ and $S\to G$ transitions occur according to the probabilities $f_{\text{cat}\left(\text{cyt}\right)}$ and $f_{\text{res}\left(\text{cyt}\right)}$, respectively. In case $\left|x_i\left(t\right)\right|\ge \frac{L}{2}$, $G\to S$ and $S\to G$ transitions occur according to the probabilities $f_{\text{cat}\left(\text{ctx}\right)}$ and $f_{\text{res}\left(\text{ctx}\right)}$, respectively. To keep the number of microtubules constant, $S\to G$ transition occurs when a microtubule becomes very short. Hence, when $l_i\left(t\right)<0.1$, $S\to G$ transition occurs at a probability of $1$.

We measured velocity of microtubule growth and shrinkage in cytoplasm or cortex, designated as $v_{\text{plus}\left(\text{cyt}\right)}$, $v_{\text{plus}\left(\text{ctx}\right)}$, $v_{\text{minus}\left(\text{cyt}\right)}$ and $v_{\text{minus}\left(\text{ctx}\right)}$, and these values are given by Appendix Table S1. In the case ${\phi }_i=S$, the shrinkage velocity of $i$-th microtubule is given by $\begin{array}{}\text{(2.1)}& \frac{d{l}_i}{dt}=\left\{\begin{array}{c}-v_{\text{minus}\left(\text{cyt}\right)}\left(\left|x_i\left(t\right)\right|<\frac{L}{2}\right)\\ -v_{\text{minus}\left(\text{ctx}\right)}\left(\left|x_i\left(t\right)\right|\ge \frac{L}{2}\right)\end{array}\right..\end{array}$

In the case where ${\phi }_i=G$, the elongation velocity of $i$-th microtubule is given by the following: In the model that considers only the pulling force, but not the pushing force, $\begin{array}{}\text{(2.2)}& \frac{d{l}_i}{dt}=\left\{\begin{array}{c}{v}_{\text{plus}\left(\text{cyt}\right)}\left(\left|x_i\left(t\right)\right|<\frac{L}{2}\right)\\ v_{\text{plus}\left(\text{ctx}\right)}\left(\left|x_i\left(t\right)\right|\ge \frac{L}{2}\right)\end{array}\right..\end{array}$ In a model that considers the pushing force, a microtubule does not elongate when it generates a pushing force. Therefore, if $\left|x_i\left(t\right)\right|\ge \frac{L}{2}$ and $\frac{C_{\text{buckle}}}{{l_i\left(t\right)}^2}>F_{\text{push}}$, the elongation velocity is given by: $\begin{array}{}\text{(2.3)}& \frac{d{l}_i}{dt}=0.\end{array}$ Otherwise, elongation velocity is given by (2.2).

3. Force generation

3-a. Pulling-pushing and pulling-only models

In the case where ${\phi }_i=S$, force on $i$-th microtubule is given by: $\begin{array}{}\text{(3.1)}& F_i\left(t\right)=\left\{\begin{array}{c}0\left(\left|x_i\left(t\right)\right|<\frac{L}{2}\right)\\ F_{\text{pull}}\left(\left|x_i\left(t\right)\right|\ge \frac{L}{2}\right)\end{array}\right..\end{array}$

In the case of ${\phi }_i=G$, the force on $i$-th microtubule is given by the following: In the model considering the pushing force, a microtubule contacting the cell cortex generates a pushing force if it is sufficiently short to resist buckling. Then, the force on $i$-th microtubule is given by: $\begin{array}{}\text{(3.2)}& F_i\left(t\right)=\left\{\begin{array}{c}{F}_{\text{pull}}\left(\frac{C_{\text{buckle}}}{{l_i\left(t\right)}^2}>F_{\text{push}},\left|x_i\left(t\right)\right|\ge \frac{L}{2}\right)\\ 0\left(\text{otherwise}\right)\end{array}\right..\end{array}$ In the model considering only the pulling force, the force on $i$-th microtubule is equal to $0$.

3-b. Length-dependent pulling model

In the model considering that the pulling force is generated along the microtubule lattice, force on i-th microtubule is given by: $\begin{array}{}\text{(3.3)}& F_i\left(t\right)=C_{\text{pull}}{l}_i\left(t\right),\end{array}$ where $C_{\text{pull}}$ is a coefficient.

3-c. Total force to the SPB–nucleus

Total force to the SPB–nucleus by all microtubules is given by: $\begin{array}{}\text{(3.4)}& F_{\text{total}}\left(t\right)=\sum _{i=1}^{2n}{e}_i\cdot F_i\left(t\right).\end{array}$

4. Migration of the SPB–nucleus

Suppose that the SPB–nucleus has a spherical shape. In case $\left|X\left(t\right)\right|<\frac{L}{2}$ or in case of $\left|X\left(t\right)\right|=\frac{L}{2}$ and $X\left(t\right)\cdot F_{\text{total}}<0$, according to Stokes' law, the force of viscosity on the nucleus dragged through a viscous cytosol is given by: $\begin{array}{}\text{(4.1)}& F_{\text{drag}}=6\pi R\eta \cdot V,\end{array}$ where $R$ is the Stokes radius of the nucleus, $\eta$ is the cytosol viscosity, and $V$ is the flow velocity relative to the nucleus. Requiring the force balance $F_{\text{total}}\left(t\right)=F_{\text{drag}}$ gives the terminal velocity of the SPB–nucleus equal to $V$, and the velocity of the SPB–nucleus is given by: $\begin{array}{}\text{(4.2)}& \frac{dX\left(t\right)}{dt}=\frac{F_{\text{total}}\left(t\right)}{6\pi R\eta }.\end{array}$

In the case of $\left|X\left(t\right)\right|=\frac{L}{2}$ and $X\left(t\right)\cdot F_{\text{total}}\ge 0$, because forces by microtubules are balanced with opposing forces induced by the collision of the nucleus with the cortex, the velocity of the SPB–nucleus is equal to $0$.

Table 1. Parameters for the simulation

Notes

1. [6] reported that the number of microtubules emanating from the SPB was 21.3 ± 4.8 by electron microscopy. [7] showed that 5 or 6 microtubules were visible using GFP-labeled α-tubulin. In our previous study [2], we observed approximately 4–9 EB1 dots (microtubules in the growing phase). We set the default value of the microtubule number to 6, a nearly minimum of the observed values.
2. In vitro studies reported that stall force of a budding yeast cytoplasmic dynein is ∼4 pN [8, 9], and “end-on” pulling force by the dynein at the cortex is ∼2 pN [10]. Stall force of mammalian dynein is in a range between 0.8–7 pN [9, 11, 12]. Considering these studies, we set the default value of the pulling force for a microtubule to 2 pN for simplicity. We confirmed that changing the pulling force in the range 2–10 pN did not affect the oscillation frequency in the simulation [1].
3. We tested simulations of the length-dependent pulling model using a wide range of $C_{\text{pull}}$ values (0–10 pN/μm) and found that 0.4 pN/μm was a nearly minimal value to generate an oscillation between two cell ends.

Codes

The codes are available at git repositories ikumi-fujita/horsetail and ikumi-fujita/horsetailpy implemented in R and Python, respectively.

References

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