A mathematical model of mitochondrial swelling

Theoretical approaches used in the modelling of reversible and irreversible mitochondrial swelling in vitro

Existing theoretical approaches were considered that allow modelling of mitochondrial swelling (MS) dynamics. Simple phenomenological kinetic models were reviewed. Simple and extended biophysical and bioenergetic models that ignore mechanical properties of inner mitochondrial membrane (IMM), and similar models that include these mechanical properties were also reviewed. Limitations of these models we considered, as regards correct modelling of MS dynamics. It was found that simple phenomenological kinetic models have significant limitations, due to dependence of the kinetic parameter values estimated by fitting of the experimental data on the experimental conditions. Additionally, such simple models provide no understanding of the detailed mechanisms behind the MS dynamics, nor of the dynamics of various system parameters during MS. Thus, biophysical and bioenergetic models ignoring IMM mechanical properties can't be used to model the transition between reversible and irreversible MS. However, simple and extended biophysical models that include IMM mechanical properties allow modelling the transition to irreversible swelling. These latter models are still limited due to significantly simplified description of biochemistry, compared to those of bioenergetic models. Finally, a strategy of model development is proposed, towards correct interpretation of the mitochondrial life cycle, including the effects of MS dynamics.

Dynamics and stabilization mechanism of mitochondrial cristae morphofunction associated with turgor-driven cardiolipin biosynthesis under salt stress conditions

Maintaining energy production efficiency is of vital importance to plants growing under changing environments. Cardiolipin localized in the inner mitochondrial membrane plays various important roles in mitochondrial function and its activity, although the regulation of mitochondrial morphology to various stress conditions remains obscure, particularly in the context of changes in cellular water relations and metabolisms. By combining single-cell metabolomics with transmission electron microscopy, we have investigated the adaptation mechanism in tomato trichome stalk cells at moderate salt stress to determine the kinetics of cellular parameters and metabolisms. We have found that turgor loss occurred just after the stress conditions, followed by the contrasting volumetric changes in mitochondria and cells, the accumulation of TCA cycle-related metabolites at osmotic adjustment, and a temporal increase in cardiolipin concentration, resulting in a reversible topological modification in the tubulo-vesicular cristae. Because all of these cellular events were dynamically observed in the same single-cells without causing any disturbance for redox states and cytoplasmic streaming, we conclude that turgor pressure might play a regulatory role in the mitochondrial morphological switch throughout the temporal activation of cardiolipin biosynthesis, which sustains mitochondrial respiration and energy conversion even under the salt stress conditions.

Tristability in mitochondrial permeability transition pore opening

Mitochondrial permeability transition pore (PTP), a key regulator of cell life and death processes, is triggered by calcium ions (Ca²⁺) and potentiated by reactive oxygen species (ROS). Although the two modes of PTP opening, i.e., transient and persistent, have been identified for a long time, its dynamical mechanism is still not fully understood. To test a proposed hypothesis that PTP opening acts as a tristable switch, which is characterized by low, medium, and high open probability, we develop a three-variable model that focused on PTP opening caused by Ca²⁺ and ROS. For the system reduced to two differential equations for Ca²⁺ and ROS, both the stability analysis and the potential landscape feature that it exhibits tristability under standard parameters. For the full system, the bifurcation analysis suggests that it can achieve tristability over a wide range of input parameters. Furthermore, parameter sensitivity analysis demonstrates that the existence of tristability is a robust property. In addition, we show how the deterministic tristable property can be understood within a stochastic framework, which also explains the PTP dynamics at the level of a single channel. Overall, this study may yield valuable insights into the intricate regulatory mechanism of PTP opening.

Reversible and irreversible mitochondrial swelling in vitro

Mitochondrial activity as regards ATP production strongly depends on mitochondrial swelling (MS) mode. Therefore, this work analyzes reversible and irreversible MS using a detailed biophysical model. The reported model includes mechanical properties of the inner mitochondrial membrane (IMM). The model describes MS dynamics for spherically symmetric, axisymmetric ellipsoidal and general ellipsoidal mitochondria. Mechanical stretching properties of the IMM were described by a second-rank rigidity tensor. The tensor components were estimated by fitting to the earlier reported results of in vitro experiments. The IMM rigidity constant of ca. 0.008 dyn/nm was obtained for linear deformations. The model also included membrane bending effects, which were small compared to those of membrane stretching. The model was also tested by simulation of the earlier reported experimental data and of the system dynamics at different initial conditions, predicting the system behavior. The transition criteria from reversible to irreversible swelling were determined and tested. The presently developed model is applicable directly to the analysis of in vitro experimental data, while additional improvements are necessary before it could be used to describe mitochondrial swelling in vivo. The reported theoretical model also provides an idea of physically consistent mechanism for the permeability transport pore (PTP) opening, which depends on the IMM stretching stress. In the current study, this idea is discussed briefly, but a detailed theoretical analysis of these ideas will be performed later. The currently developed model provides new understanding of the detailed MS mechanism and of the conditions for the transition between reversible and irreversible MS modes. On the other hand, the current model provides useful mathematical tools, that may be successfully used in mitochondrial biophysics research, and also in other applications, predicting the behavior of mitochondria in different conditions of the surrounding media in vitro or cellular cyto(sarco)plasm in vivo. These mathematical tools are based on real biophysical processes occurring in mitochondria. Thus, we note a significant progress in the theoretical approach, which may be used in real biological systems, compared to the earlier reported models. Significance of this study derives from inclusion of IMM mechanical properties, which directly impact the reversible and irreversible mitochondrial swelling dynamics. Reversible swelling corresponds to reversible IMM deformations, while irreversible swelling corresponds to irreversible deformations, with eventual membrane disruption. The IMM mechanical properties are directly dependent on the membrane biochemical composition and structure. The IMM deformationas are induced by osmotic pressure created by the ionic/neutral solute imbalance between the mitochondrial matrix media and the bulk solution in vitro, or cyto(sarco)plasm in vivo. The novelty of the reported model is in the biophysical mechanism detailing ionic and neutral solute transport for a large number of solutes, which were not taken into account in the earlier reported biophysical models of MS. Therefore, the reported model allows understanding response of mitochondria to the changes of initial concentration(s) of any of the solute(s) included in the model. Note that the values of all of the model parameters and kinetic constants have been estimated and the resulting complete model may be used for quantitative analysis of mitochondrial swelling dynamics in conditions of real in vitro experiments.

Dual dynamics of mitochondrial permeability transition pore opening

Mitochondria play an essential role in bioenergetics and cellular Ca\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{2+}$$\end{document}2+ handling. The mitochondrial permeability transition pore (mPTP) is a non-specific channel located in the inner mitochondrial membrane. Long-lasting openings of the pore allow the rapid passage of ions and large molecules, which can result in cell death. The mPTP also exhibits transient, low conductance openings that contribute to Ca\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{2+}$$\end{document}2+ homeostasis. Although many regulators of the pore have been identified, none of them uniquely governs the passage between the two operating modes, which thus probably relies on a still unidentified network of interactions. By developing a core computational model for mPTP opening under the control of mitochondrial voltage and Ca\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{2+}$$\end{document}2+, we uncovered the existence of a positive feedback loop leading to bistability. The characteristics of the two stable steady-states correspond to those of the two opening states. When inserted in a full model of Ca\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{2+}$$\end{document}2+ handling by mitochondria, our description of the pore reproduces observations in mitochondrial suspensions. Moreover, the model predicted the occurrence of hysteresis in the switching between the two modes, upon addition and removal of free Ca\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{2+}$$\end{document}2+ in the extra-mitochondrial medium. Stochastic simulations then confirmed that the pore can undergo transient openings resembling those observed in intact cells.

Computational Modeling of In Vitro Swelling of Mitochondria: A Biophysical Approach

Swelling of mitochondria plays an important role in the pathogenesis of human diseases by stimulating mitochondria-mediated cell death through apoptosis, necrosis, and autophagy. Changes in the permeability of the inner mitochondrial membrane (IMM) of ions and other substances induce an increase in the colloid osmotic pressure, leading to matrix swelling. Modeling of mitochondrial swelling is important for simulation and prediction of in vivo events in the cell during oxidative and energy stress. In the present study, we developed a computational model that describes the mechanism of mitochondrial swelling based on osmosis, the rigidity of the IMM, and dynamics of ionic/neutral species. The model describes a new biophysical approach to swelling dynamics, where osmotic pressure created in the matrix is compensated for by the rigidity of the IMM, i.e., osmotic pressure induces membrane deformation, which compensates for the osmotic pressure effect. Thus, the effect is linear and reversible at small membrane deformations, allowing the membrane to restore its normal form. On the other hand, the membrane rigidity drops to zero at large deformations, and the swelling becomes irreversible. As a result, an increased number of dysfunctional mitochondria can activate mitophagy and initiate cell death. Numerical modeling analysis produced results that reasonably describe the experimental data reported earlier.

Simple kinetic model of mitochondrial swelling in cardiac cells

Mitochondria play an important role in both cell survival and cell death. In response to oxidative stress, they undergo opening of non-selective permeability transition pores (PTP) in the inner mitochondrial membrane. Sustained PTP opening triggers mitochondrial swelling due to increased colloidal osmotic pressure in the matrix accompanied by mitochondrial membrane depolarization and ATP hydrolysis. Mitochondrial swelling is the major factor leading to mitochondria-mediated cell death through both apoptosis and necrosis. Hence, precise estimation of the threshold parameters of the transition of reversible swelling to irreversible swelling is important for understanding the mechanisms of PTP-mediated cell death as well as for the development of new therapeutic approaches targeting the mitochondria under pathological conditions. In this study, we designed a simple kinetic model of the Ca²⁺ -induced mitochondrial swelling that describes the mechanisms of transition from reversible to irreversible swelling in cardiac mitochondria. Values of kinetic parameters calculated using parameter estimation techniques that fit experimental data of mitochondrial swelling with minimum average differences between the experimental data and model parameters. Overall, this study provides a kinetic model verified by data simulation and model fitting that adequately describes the dynamics of mitochondrial swelling.

Different approaches to modeling analysis of mitochondrial swelling

Mitochondria are critical players involved in both cell life and death through multiple pathways. Structural integrity, metabolism and function of mitochondria are regulated by matrix volume due to physiological changes of ion homeostasis in cellular cytoplasm and mitochondria. Ca²⁺ and K⁺ presumably play a critical role in physiological and pathological swelling of mitochondria when increased uptake (influx)/decreased release (efflux) of these ions enhances osmotic pressure accompanied by high water accumulation in the matrix. Changes in the matrix volume in the physiological range have a stimulatory effect on electron transfer chain and oxidative phosphorylation to satisfy metabolic requirements of the cell. However, excessive matrix swelling associated with the sustained opening of mitochondrial permeability transition pores (PTP) and other PTP-independent mechanisms compromises mitochondrial function and integrity leading to cell death. The mechanisms of transition from reversible (physiological) to irreversible (pathological) swelling of mitochondria remain unknown. Mitochondrial swelling is involved in the pathogenesis of many human diseases such as neurodegenerative and cardiovascular diseases. Therefore, modeling analysis of the swelling process is important for understanding the mechanisms of cell dysfunction. This review attempts to describe the role of mitochondrial swelling in cell life and death and the main mechanisms involved in the maintenance of ion homeostasis and swelling. The review also summarizes and discusses different kinetic models and approaches that can be useful for the development of new models for better simulation and prediction of in vivo mitochondrial swelling.

High resolution respirometry analysis of polyethylenimine-mediated mitochondrial energy crisis and cellular stress: Mitochondrial proton leak and inhibition of the electron transport system

Polyethylenimines (PEIs) are highly efficient non-viral transfectants, but can induce cell death through poorly understood necrotic and apoptotic processes as well as autophagy. Through high resolution respirometry studies in H1299 cells we demonstrate that the 25kDa branched polyethylenimine (25k-PEI-B), in a concentration and time-dependent manner, facilitates mitochondrial proton leak and inhibits the electron transport system. These events were associated with gradual reduction of the mitochondrial membrane potential and mitochondrial ATP synthesis. The intracellular ATP levels further declined as a consequence of PEI-mediated plasma membrane damage and subsequent ATP leakage to the extracellular medium. Studies with freshly isolated mouse liver mitochondria corroborated with bioenergetic findings and demonstrated parallel polycation concentration- and time-dependent changes in state 2 and state 4o oxygen flux as well as lowered ADP phosphorylation (state 3) and mitochondrial ATP synthesis. Polycation-mediated reduction of electron transport system activity was further demonstrated in 'broken mitochondria' (freeze-thawed mitochondrial preparations). Moreover, by using both high-resolution respirometry and spectrophotometry analysis of cytochrome c oxidase activity we were able to identify complex IV (cytochrome c oxidase) as a likely specific site of PEI mediated inhibition within the electron transport system. Unraveling the mechanisms of PEI-mediated mitochondrial energy crisis is central for combinatorial design of safer polymeric non-viral gene delivery systems.

Mitochondria, motor neurons and aging

While the role of mitochondria in aging has been well characterized, their involvement in motor neuron aging remains poorly understood. Thus, we performed an exhaustive ultrastructural study of mitochondria in motor neurons from aged rats that revealed dramatic alterations in the mitochondria of axon terminals at neuromuscular junctions, characterized by swelling, mitochondrial fusion and the presence of megamitochondria. These alterations were not observed in ventral horn motor neurons in the spinal cord of aged rats, which were only altered by the appearance of electron-dense bodies in the dilated matrix cristae. Using X-ray microanalytical techniques we demonstrated the presence of calcium in these bodies, suggesting Ca(2+) overload. Moreover, in motor neurons from aged rats, cytochrome c and activated caspase 3 were detected in the cytoplasm of axon terminals at neuromuscular junctions, factors implicated in the apoptosis. Active caspase 3 was also found in the nucleus, soma and axons of aged alpha motor neuron neurons, where it mainly associated with microtubules. The colocalization of dynein and cleaved caspase 3 in neuromuscular junctions is strongly suggestive of the retrograde transport of apoptotic factors to the soma. These results are consistent with the early stages of degeneration in neuromuscular junctions during aging, which is followed by dying back. Given that aging is a key risk factor for neurodegenerative diseases, such as amyotrophic lateral sclerosis (ALS), the identification of age-related motor neuron degeneration initiated at the distal end of the axon may provide a new therapeutic target for early intervention.