A Computational Model of Mitochondria Motility in Axons

Abstract

Mitochondria play a critical role in regulating cellular processes including ATP production, intracellular calcium signaling and generation of reactive oxidative species (ROS). Neurons rely on mitochondrial function to perform a range of complex processes, and mitochondrial dysfunctions have been shown to have an impact in pathologies of the nervous system. Yet, neurons contain a finite number of mitochondria, and their location is known to change in response to a number of factors including age and cellular activity, thereby impacting neuronal response. In this paper, we introduce a novel computational model of mitochondria motility that focuses on their movements along the axon. We describe the biological processes involved and the main parameters of the model. We use the model to investigate how some of these parameters affect the ability of mitochondria to position themselves in regions of high energy demand. Finally, we discuss the significance of our work and its downstream applications in further understanding pathologies of the nervous system such as Alzheimer’s disease, and help identify potential novel therapeutic targets.

I. Introduction

The central nervous system is characterized by a very high metabolic rate as it consumes approximately 20% of oxygen inhaled at rest, while representing only 2% of the body weight [1]. This massive metabolic demand is due to neurons being highly differentiated cells that require large quantities of ATP to maintain the resting ionic gradients across the cell membrane and necessary for neurotransmission. Consequently, the mitochondria, power plants of the cell, have been shown to play a critical role in a neuron’s function; similarly, neuronal function and survival has been shown to be highly sensitive to mitochondrial dysfunctions [2], [3]. Functionally, not only do these organelles produce the ATP critical for all energy-dependent cellular processes, they also play an active role in buffering and regulating calcium concentration inside the cell (calcium being an essential molecular species that impacts a myriad of processes, amongst which are learning and memory) and generating reactive oxidative species (ROS) [4].

Yet, the number of mitochondria within neurons is finite, and the influence of a mitochondrion on cellular processes is constrained to a finite region of the subcellular space in its neighborhood. Given their importance, understanding how these organelles are positioned and how they move within the neuron is critical for many aspects of cell biology. A well-positioned mitochondrion may support the energy requirements of an adjacent presynaptic terminal for neurotransmitters release and metabolically sustain proper ionic fluxes and balances. Experimental results indicate that mitochondria location and their movements within the cell depend on a number of factors, amongst which are glucose concentration and calcium concentration [5]. Yet, no computational model of mitochondria motility currently exists to quantitatively investigate the effects of mitochondria motility and distribution within the neuron. We therefore propose to fill this gap and develop a model capable of simulating mitochondria movements within the neuron.

Studies on mitochondria motility have reported that they follow different sets of rules depending on whether they are located in dendrites or in axons, with a larger number of studies investigating axonal distribution. Hence, the model presented here focuses on mitochondria motility in the axon and describes the mitochondria behavior as action potentials propagate down the axon and consequently modify the membrane potential and calcium concentration in the vicinity of presynaptic terminals. This paper briefly describes the overall methodology and the model elaboration and calibration. We then use the model while varying several conditions, discuss our results and outline the applications of this methodology to support a more quantitative understanding of mitochondrial function and its potential effects on pathological dysfunctions of the nervous system, but also as a tool to support target identification and predict therapeutic efficacy.

II. MODEL

A. Computational methodology

All models are coded in Python 3 using the standard Numpy library at time of submission (Anaconda 3 64bits default distribution). For dissemination purposes, the model is being translated into SBML [6], [7], an XML-based representation used for specifying computational models of biological processes that is open and widely adopted. All model parameters are provided in Table I.

TABLE I.

Mitochondria Motility Model - Parameter Values

Parameter	Value	Unit
Axonal length	12	mm
Duration of rise in [Ca2+] [11]	1.0	s
Frequency of action potentials	Variable from 0.01 to 1.0	Hz
Mitochondria velocity	0.3–1.0	μm/s
Length of presynaptic terminal	1.0	μm
Interbouton interval	6.0	μm
Time during which mitochondria are immobile	40–60	min

B. Neuron and Ultrastructure

We consider a 12 millimeters-long axon with a density of 280 mitochondria per millimeter. Although mitochondria density in axons has been reported to vary as a function of cellular health, typical healthy neurons have been reported to contain about 28 mitochondria per 100 micrometers of axonal length [8], yielding a total of 3360 mitochondria in our model. Regions of high calcium concentration associated with presynaptic terminals are defined as being 1 micrometer wide. Although the interbouton interval, i.e., the space between two adjacent presynaptic terminals has been observed to be highly variable on a given axon varying from a few to dozens of micrometers, with a median of 3 – 10 micrometers [9]; we consider it uniform with a value of 6 micrometers (see Fig. 1). The axon model is terminated by the soma on one end and a terminal presynaptic bouton on the other. At the soma end, mitochondria simply reverse their direction and move away from the soma; at the synapse end, mitochondria become immobile when an action potential (AP) is triggered; if no AP takes place while on the terminal presynaptic bouton, they simply reverse their directions and move toward the soma (retrograde motion). The somatic end is 1.0 micrometer long; the other end has the same length as all other presynaptic terminals along the axon, i.e., 1.0 micrometer; notably, mitochondria will reverse their directions when they reach the axonal and somatic ends. All presynaptic terminal regions along the axon are regions in which calcium concentration rises in response to an action potential. Notably, on the somatic side, old mitochondria moving away from the axon are recycled while new mitochondria are produced in the soma and moving toward the axon; in our current model, this process is simplified as old mitochondria simply reverse their direction.

Figure 1.

Diagram illustrating a portion of the axon modeled. The model contains presynaptic terminals evenly distributed along its length. Mobile and stationary mitochondria are distributed along the axon; when an action potential reaches presynaptic terminals (grey), voltage-dependent calcium channels open and let calcium inside the cell, thereby leading to a local increase in calcium concentration. Mobile mitochondria are transported along the axon via proteins linked to microtubules. When they encounter regions with elevated calcium concentrations, mobile mitochondria detach from the microtubule and become stationary.

C. Motility

The functional significance of mitochondrial motility is that mitochondria are recruited to synapses in response to elevated intracellular Ca²⁺ during sustained synaptic activity: elevated calcium influx, either by activating voltage-dependent calcium channels or NMDA receptors halt mitochondrial movement in their vicinity. Long-range mitochondria transport relies on microtubules and associated proteins, while actin skeleton has been shown to participate in the short-range transport. The mechanisms underlying transport are attributed to a number of protein complexes that attach or detach mitochondria from the microtubules present in the axon. When mitochondria are attached to the microtubules, they are mobile and transported either away from the soma (anterograde, considered positive direction) or toward the soma (retrograde or negative direction). Mitochondria velocity in typical neurons is reported to be equal to 0.3–1 micrometers/second [10].

D. Calcium-dependent processes

When motile mitochondria reach a presynaptic terminal during an action potential, the elevated calcium concentration has been reported to cleave the link between the microtubules and the mitochondria, which becomes stationary. The stationary mitochondria have been observed to remain in this state for at least 15 minutes after immobilization [11]. Here, we assume that stationary mitochondria become motile after 40 to 60 minutes of immobilization (in either anterograde or retrograde direction). When presynaptic terminals receive an action potential, we assume that the concentration of calcium instantly rises. We assume the duration of the rise to last one second.

III. RESULTS

To validate the model, we used the results obtained in [12] in which the authors measure the ratio of mitochondria that have moved during 15-minute time windows when the frequency of action potentials is 0.05Hz. Using similar simulation conditions, we obtained comparable ratios of motile mitochondria (Fig. 2). An example of result obtained by our model is provided in Fig. 3, where we show the path taken by five mitochondria taken randomly on the axon during a 24hr-long simulation. In addition, we complemented our analysis by further simulating the model with varying AP frequencies and examined (a) the distribution of mitochondria (Fig. 4) and (b) the ratio of motile versus stationary mitochondria as a function of AP frequency (Fig. 5). The results obtained indicate (i) a clear increase in the number of stationary mitochondria at synaptic terminals (Fig. 4C. through 4E.) and (ii) a sizeable decrease in the number of motile mitochondria as the frequency of AP increases (Fig. 5).

Figure 2.

Evolution of mitochondria pools as observed experimentally with 1000-second sampling windows (about 16 minutes) when axons are stimulated by action potentials at a frequency of 0.05Hz. Total simulation time is 24 hours. Mitochondria are considered mobile if their position changed during the 1000-second sampling window; results are evaluated by increments of 100 seconds. Results are reported as a function of time. All mitochondria are originally motile and equally distributed along the axon. As the motile mitochondria reach presynaptic terminals with high concentration of calcium (i.e., receiving action potentials), they may dissociate from the microtubule and become stationary.

Figure 3.

Illustration of the paths of five random mitochondria in the model; they are originally equally distributed along the axon and are subjected to action potentials at a frequency of f(AP)=0.05Hz. As they travel along the axon and encounter presynaptic terminals with high concentration of calcium, the mitochondria become stationary for some time; they may become mobile again after an immobilization period (see Table I).

Figure 4.

Distribution of mobile (green) and stationary (red) mitochondria for different frequencies of action potential propagated along the axon. A. Position of the presynaptic terminals along the axon. B. Initial distribution of mitochondria at the beginning of the simulation; mitochondria are uniformly distributed along the axon. C, D and E. Distributions of mobile and stationary mitochondria after 24 hours of simulation with AP frequencies of 0.005, 0.05 and 1.0Hz respectively. Note how the number of mobile (green) mitochondria decreases as AP frequency increases.

Figure 5.

Motility as a function of action potential frequency. The solid blue surface indicates the short-term variations in the ratio of motile mitochondria (phenomenon clearly visible as oscillations in Fig. 2).

IV. DISCUSSION

Mitochondria are critical organelles that are known to actively participate and shape neuronal response through exchanges with their direct surrounding environment via ATP production, calcium buffering and ROS production. Mitochondria positioning within the cell is therefore critical to address the local metabolic needs. The present model aims at shedding some light on the parameters that affect motility of mitochondria in the axon and presynaptic terminals, i.e., their ability to move and stop in regions where their contribution is needed. This work is in accordance with experimental observations that indicate that axonal mitochondria may contribute to variations in presynaptic strength [12] through calcium buffering and ATP production, as mitochondria tend to accumulate in these locations. Results of our simulation indicate that higher frequencies of AP result in a larger number of stationary mitochondria. Notably, the work presented is limited to the use of uniform AP frequencies and more biologically realistic situations need to be investigated, e.g., though the Miro-Calcium dependent system is the main regulator of mitochondria motility, some other system also participate in the regulation process [13]. Nonetheless, the larger number of stationary mitochondria in periods of high activity leads to two important features; (1) the activity of pumps and exchangers that attempt to maintain intracellular ionic balances are highly dependent upon local availability of ATP and the stationary mitochondria are stationed where they are most needed to best sustain the energetic needs of local cellular processes; (2) on the other hand, the number of mitochondria that remains motile is very low; if a new region on the axon needs mitochondria after a period of generally high activity, no mobile mitochondria will be available to accommodate these needs, thereby creating a temporary shortage.

Of importance, aging and pathological changes in calcium homeostasis have been reported to affect mitochondrial motility [14]. Future work will hence consist in extending the proposed model to incorporate the downstream effects of changes in mitochondria motility on neurotransmission and neuronal health through (i) diminished ATP availability, (ii) reduced mitochondrial calcium buffering and (iii) detrimental ROS production, whether due to normal ageing or neurodegeneration.

The work presented here is an important step that enables us to quantitatively examine the downstream effects of mitochondrial motility; the computational approach outlined constitutes an insightful instrument to examine the downstream consequences of disruptions in mitochondrial motility and detect their potential pathogenic consequences on higher level observables, thereby setting the path for insilico discovery and testing of therapeutic agents.