A Growth-Fragmentation Approach for Modeling Microtubule Dynamic Instability

Abstract

Microtubules (MTs) are protein filaments found in all eukaryotic cells which are crucial for many cellular processes including cell movement, cell differentiation, and cell division. Due to their role in cell division, they are often used as targets for chemotherapy drugs used in cancer treatment. Experimental studies of MT dynamics have played an important role in the development and administration of many novel cancer drugs; however, a complete description of MT dynamics is lacking. Here, we propose a new mathematical model for MT dynamics, that can be used to study the effects of chemotherapy drugs on MT dynamics. Our model consists of a growth-fragmentation equation describing the dynamics of a length distribution of MTs, coupled with two ODEs that describe the dynamics of free GTP- and GDP-tubulin concentrations (the individual dimers that comprise of MTs). Here, we prove the well-posedness of our system and perform a numerical exploration of the influence of certain model parameters on the systems dynamics. In particular, we focus on a qualitative description for how a certain class of destabilizing drugs, the vinca alkaloids, alter MT dynamics. Through variation of certain model parameters which we know are altered by these drugs, we make comparisons between simulation results and what is observed in in vitro studies.

Appendix: Advantages of Our Modeling Approach Over Existing Ones

Here, we illustrate a comparison between the model described here and the model of White et al. (2017). Both models describe MT dynamics in terms of simple advective growth, and incorporate a fragmentation-type term for MT shortening. The model (White et al. 2017) further incorporates an advection term to describe the growth and shortening of the MT cap region, as well as a system of ODEs to describe the binding/unbinding of end-binding proteins (EBs) to the cap.

Fig. 10

Left: Full model results from simulation of model in White et al. (2017). Parameters are: $\alpha = 2$, $\gamma ^\mathrm{h} = 5$, $\beta _{\infty }=20$, $x_0 = 4$, $\mu = 5.9 \times 10^3$, $m = 2$, $x_{\mathrm{min}}$ = 0.5, $\kappa = 2$, $p_\mathrm{c} = 2$, $p_\mathrm{N} = 12$, $p_0 = 15$, $\lambda = 0.136$. Right: Simplified model of this paper. Parameters are (units given in Table 1): $\alpha = 2$, $\gamma ^\mathrm{h} = 5$, $\beta _{\infty }=20$, $x_0 = 2$, $\mu = 0.1$, $m = 2$, $x_{\mathrm{min}}$ = 0.8, $\kappa = 2$, $p_\mathrm{c} = 2$, $p_\mathrm{N} = 12$, $p_0 = 15$

Figure 10 describes the long-term dynamics of the “full” model of White et al. (2017) (left) and the “mini” model described in this paper (right). From this figure, it is clear that the long-term (averaged) tubulin concentrations are very similar. In particular, we make a comparison of the mean tubulin concentrations after t = 3 minutes (when the concentrations (roughly) oscillate about a mean). The steady-state values (calculated numerically) are recorded in Table 2. Here, all model parameters used in the simulation of the “mini” model are the same as those for the “full” model in White et al. (2017), where the full model also has a few extra parameters—all parameter values are recorded in the figure caption.

Table 2 Table of long-term tubulin concentrations

In the full model of White et al. (2017), we incorporate a compartment for shrinking MTs. In particular, MTs that undergo fragmentation (a catastrophe) do so in a separate shrinking compartment. From there, MTs that undergo fragmentation are completely depolymerized into free GDP-tubulin. For our mini model, both growth and fragmentation are described by a single equation. That is, there is no separate equation to describe MT shortening. Thus, the slight differences in concentrations could be due to the fact that, in the full model, as a MT shortens, polymerized GDP-tubulin remains in the shortening compartment for some period of time before entering the free GDP-tubulin compartment. Also, in the full model, the shortening compartment has a term for rescue, where some of the MTs that are shrinking enter back into the growing compartment.