A viscoelastic model for axonal microtubule rupture
Introduction
The main goal of this study is to develop an appropriate model for studying the behavior of axonal microtubules in response to suddenly applied forces. Axons are one of the most important parts of the neuronal cells and axonal microtubules are fibers in axons. The main function of microtubules is to reinforce the neuronal axons. Microtubule bundles that are located in the inner part of the axons have polar orientation (Fadic et al., 1985). Microtubules have many cross links with their neighboring microtubules using tau proteins. These tau proteins are considered as one of the main elements for the axonal strength in response to the mechanical forces (Drechsel et al., 1992). Since microtubules can resist mechanical forces, these structures have been studied widely from biomechanical point of view. The size of microtubules and the spatial position of these bundles within axons have also been investigated widely (Fadic et al., 1985, Yu and Baas, 1994). There are other microstructural components within axons like microfilaments and different organelles. Microtubules can facilitate the transport of organelles within the axons since these organelles can move along microtubule bundles. Microfilaments are one of the other important parts of the neuronal cell cytoskeleton that in coordination with microtubules, keep the shape and integrity of axons.
Axonal injury is one of the most common central nervous system injuries (Smith et al., 2003, Sanjith, 2011). Most patients who are suffered from axonal injuries, after a period of time, experience deficits like eye problems, problems with executive functions, behavior changes, language deficits etc. (Adams et al., 1989, Sanjith, 2011, Blumbergs et al., 1989, Williams et al., 1990). The injuries of microtubules are shown to be one of the main consequences of the traumatic brain injuries happening during the damage to axons (Peter and Mofrad, 2012, Gennarelli et al., 1982). Some studies have revealed that the most axonal injuries that occur in specific areas of the brain are caused by the extensive strain in axonal microtubules (Margulies et al., 1990). Some experiments have also shown that the rupture of microtubules can cause the most severe damage in traumatic brain injuries (Tang-Schomer et al., 2010). Microtubules are shown to be ruptured under the axonal strain of about 50% (Janmey et al., 1991).
Some in-vitro models have studied the mechanical injuries of neurons (Kilinc et al., 2008, Kilinc et al., 2009, Meaney et al., 1994). The most common techniques to make brain injuries in experimental studies include cortical impact, weight drop and central or lateral fluid percussion (Meaney et al., 1994). Fluid percussion injuries have been found to bring about lesions in the lower brainstem (Dixon et al., 1988, Shima and Marmarou, 1991). Similar to the fluid percussion, cortical impact generates lesions and associated axonal injuries (Dixon et al., 1991). In contrast, it is shown that weight drop can create contusion around the impact region (Feeney et al., 1981).
To study the mechanism of damage to the neuronal axons, computer simulation can be very helpful in determining the effect of mechanical forces applied to the constitutive parts of the axons. There are several models developed by researchers for modeling microtubules, axons or neurons (Erickson and O’Brien, 1992, Flyvbjerg et al., 1994). In 2010, a mathematical model was used to study the dynamics of microtubules (Buxton et al., 2010). In this model, microtubule bundles were assumed to be composed of discrete masses. They posed linear springs between the masses, although in reality, this assumption may not be correct. On the other hand, microtubule bundles are shown to behave like viscoelastic materials. The viscoelastic behavior of axons has been verified in all experiments performed on axons.
Microtubule bundles have been recently studied widely (Shahinnejad et al., 2013, Teixeira et al., 2014). One of the recent studies used finite element method to model axons under tension and torsion (Shahinnejad et al., 2013). One of the other models for axon was suggested by Peter and Mofrad (2012). In their study, it was endeavored to model the axon under tension. In this model, microtubules were assumed to be positioned in a hexagonal form and each microtubule bundle was considered to have one discontinuity. Another model in 2014, studied the behavior of axonal microtubules and their connected tau proteins for different microtubule lengths when specific amounts of strain were applied to the network of microtubules (Ahmadzadeh et al., 2014). They showed that the applied strain rate to the axons plays an important role in microtubular break. They also showed that the stretch of tau proteins is related to their position relative to the ends of the microtubule bundles, the overall length of microtubules and the applied strain rate.
In our model, the dynamic response of microtubule bundles under the action of varying stresses was studied. It was intended to distinguish the critical areas of axons and microtubules that may be disconnected under tension. We also determined the steady state behavior of axons for different force rates and different force magnitudes. Microtubules were connected to their neighboring microtubules using tau proteins as one of the main features providing axonal stiffness. In Fig. 1, an axon within a neuronal cell is shown and axonal microtubules are positioned within the interior part of the axons. We employed a two dimensional (2D) computational model to investigate the dynamic response of axonal microtubules.
Section snippets
Geometry
As discussed earlier, the interior of an axon can be roughly thought as a network of relatively short microtubules cross-linked with tau proteins. The orientation of microtubules is in fact random but preferably lengthwise in the axon. The simplified microtubular geometry proposed by Peter and Mofrad (2012), consists of a hexagonal array of 19 rows. The cross-sectional view of the geometry reconstructed in our computational platform is presented in Fig. 2a. Since it is easier to work with
Viscoelastic model
As shown in supplemental Fig. 1, SLS model that was used in this study consists of a spring that is in parallel with a series of dashpot-spring pair. The right arm consists of a spring, km and a damper, c. This arm is typically called Maxwell arm. The left arm is simply a linear spring, k.
As can be seen in Fig. 3, a discrete point mass, ‘M’, is linked to a stationary wall through a SLS unit. As the mass M is acted upon by a given force, ‘’, the governing equations for its motion are written
Results
Supplemental Table 1 shows the material parameters used in this study (Peter and Mofrad, 2012). Now if the system shown in Fig. 3, is acted upon on both sides by external forces that are in opposite directions, it shows a time-varying deformation. The overall deformation of the system is defined according to Eq. (15). The first and second summations are over the rightmost and leftmost points. ‘’ and ‘’ are the number of microtubules and the initial non-dimensional length of the system.
Conclusion
Investigating the behavior of axonal microtubules using computational modeling is very helpful in studying the damage to the central nervous system. We assumed a microtubule bundle as a series of discrete masses connected to their neighboring masses with a SLS unit while each microtubule bundle had a point of discontinuity. A specific force rate was acted at the end of each bundle and the dynamic response of axonal microtubules was studied. We could predict the rupture and failure of axonal
Conflict of interest statement
No conflict of interest exists for this research study.
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