The growth rate of senile plaques is determined by the competition between the rate of deposition of free Aß aggregates into plaques and the autocatalytic production of free Aß aggregates.

The formation of amyloid beta (Aß) deposits (senile plaques) is one of the hallmarks of Alzheimer's disease (AD). This study investigates what processes are primarily responsible for their formation. A model is developed to simulate the diffusion of amyloid beta (Aß) monomers, the production of free Aß aggregates through nucleation and autocatalytic processes, and the deposition of these aggregates into senile plaques. The model suggests that efficient degradation of Aß monomers alone may suffice to prevent the growth of senile plaques, even without degrading Aß aggregates and existing plaques. This is because the degradation of Aß monomers interrupts the supply of reactants needed for plaque formation. The impact of Aß monomer diffusivity is demonstrated to be small, enabling the application of the lumped capacitance approximation and the derivation of approximate analytical solutions for limiting cases with both small and large rates of Aß aggregate deposition into plaques. It is found that the rate of plaque growth is governed by two competing processes. One is the deposition rate of free Aß aggregates into senile plaques. If this rate is small, the plaque grows slowly. However, if the rate of deposition of Aß aggregates into senile plaques is very large, the free Aß aggregates are removed from the intracellular fluid by deposition into the plaques, leaving insufficient free Aß aggregates to catalyze the production of new aggregates. This suggests that under certain conditions, Aß plaques may offer neuroprotection and impede their own growth. Additionally, it indicates that there exists an optimal rate of deposition of free Aß aggregates into the plaques, at which the plaques attain their maximum size.

Effects of Time-Dependent Adenosine Triphosphate Consumption Caused by Neuron Firing on Adenosine Triphosphate Concentrations in Synaptic Boutons Containing and Lacking a Stationary Mitochondrion.

The precise mechanism behind the supply of adenosine triphosphate (ATP) to approximately half of the presynaptic release sites in axons that lack a stationary mitochondrion is not fully understood. This paper presents a mathematical model designed to simulate the transient ATP concentration in presynaptic en passant boutons. The model is utilized to investigate how the ATP concentration responds to increased ATP demand during neuronal firing in boutons with a stationary mitochondrion and those without one. The analysis suggests that neuron firing may cause oscillations in the ATP concentrations, with peak-to-peak amplitudes ranging from 0.06% to 5% of their average values. However, this does not deplete boutons lacking a mitochondrion of ATP; for physiologically relevant values of model parameters, their concentration remains approximately 3.75 times higher than the minimum concentration required for synaptic activity. The variance in average ATP concentrations between boutons containing a stationary mitochondrion and those lacking one ranges from 0.3% to 0.8%, contingent on the distance between the boutons. The model indicates that diffusion-driven ATP transport is rapid enough to adequately supply ATP molecules to boutons lacking a stationary mitochondrion.

Numerical modeling of senile plaque development under conditions of limited diffusivity of amyloid-ß monomers.

This paper introduces a new model to simulate the progression of senile plaques, focusing on scenarios where concentrations of amyloid beta (Aß) monomers and aggregates vary between neurons. Extracellular variations in these concentrations may arise due to limited diffusivity of Aß monomers and a high rate of Aß monomer production at lipid membranes, requiring a substantial concentration gradient for diffusion-driven transport of Aß monomers. The dimensionless formulation of the model is presented, which identifies four key dimensionless parameters governing the solutions for Aß monomer and aggregate concentrations, as well as the radius of a growing Aß plaque within the control volume. These parameters include the dimensionless diffusivity of Aß monomers, the dimensionless rate of Aß monomer production, and the dimensionless half-lives of Aß monomers and aggregates. A dimensionless parameter is then introduced to evaluate the validity of the lumped capacitance approximation. An approximate solution is derived for the scenario involving large diffusivity of Aß monomers and dysfunctional protein degradation machinery, resulting in infinitely long half-lives for Aß monomers and aggregates. In this scenario, the concentrations of Aß aggregates and the radius of the Aß plaque depend solely on a single dimensionless parameter that characterizes the rate of Aß monomer production. According to the approximate solution, the concentration of Aß aggregates is linearly dependent on the rate of monomer production, and the radius of an Aß plaque is directly proportional to the cube root of the rate of monomer production. However, when departing from the conditions of the approximate solution (e.g., finite half-lives), the concentrations of Aß monomers and aggregates, along with the plaque radius, exhibit complex dependencies on all four dimensionless parameters. For instance, under physiological half-life conditions, the plaque radius reaches a maximum value and stabilizes thereafter.

Numerical and Analytical Simulation of the Growth of Amyloid-ß Plaques.

Numerical and analytical solutions were employed to calculate the radius of an amyloid-ß (Aß) plaque over time. To the author's knowledge, this study presents the first model simulating the growth of Aß plaques. Findings indicate that the plaque can attain a diameter of 50 µm after 20 years of growth, provided the Aß monomer degradation machinery is malfunctioning. A mathematical model incorporates nucleation and autocatalytic growth processes using the Finke-Watzky model. The resulting system of ordinary differential equations was solved numerically, and for the simplified case of infinitely long Aß monomer half-life, an analytical solution was found. Assuming that Aß aggregates stick together and using the distance between the plaques as an input parameter of the model, it was possible to calculate the plaque radius from the concentration of Aß aggregates. This led to the "cube root hypothesis," positing that Aß plaque size increases proportionally to the cube root of time. This hypothesis helps explain why larger plaques grow more slowly. Furthermore, the obtained results suggest that the plaque size is independent of the kinetic constants governing Aß plaque agglomeration, indicating that the kinetics of Aß plaque agglomeration is not a limiting factor for plaque growth. Instead, the plaque growth rate is limited by the rates of Aß monomer production and degradation.

Lewy body radius growth: The hypothesis of the cube root of time dependency.

This paper presents a model for the growth of Lewy bodies (LBs), which are pathological hallmarks of Parkinson's disease (PD). The model simulates the growth of classical LBs, consisting of a core and a halo. The core is assumed to comprise lipid membrane fragments and damaged organelles, while the halo consists of radiating alpha-synuclein (α-syn) fibrils. The Finke-Watzky model is employed to simulate the aggregation of lipid fragments and α-syn monomers. Analytical and numerical exploration of the governing equations yielded approximate solutions applicable for larger times. The application of these approximate solutions to simulate LB radius growth led to the discovery of the cube root hypothesis, which posits that the LB radius is proportional to the cube root of its growth time. Sensitivity analysis revealed that the LB radius is unaffected by the kinetic rates of nucleation and autocatalytic growth, with growth primarily regulated by the production rates of lipid membrane fragments and α-syn monomers. The model indicates that the formation of large LBs associated with PD is dependent on the malfunction of the machinery responsible for the degradation of lipid membrane fragments, α-syn monomers, and their aggregates.

Why slow axonal transport is bidirectional - can axonal transport of tau protein rely only on motor-driven anterograde transport?

Slow axonal transport (SAT) moves multiple proteins from the soma, where they are synthesized, to the axon terminal. Due to the great lengths of axons, SAT almost exclusively relies on active transport, which is driven by molecular motors. The puzzling feature of slow axonal transport is its bidirectionality. Although the net direction of SAT is anterograde, from the soma to the terminal, experiments show that it also contains a retrograde component. One of the proteins transported by SAT is the microtubule-associated protein tau. To better understand why the retrograde component in tau transport is needed, we used the perturbation technique to analyze how the full tau SAT model can be simplified for the specific case when retrograde motor-driven transport and diffusion-driven transport of tau are negligible and tau is driven only by anterograde (kinesin) motors. The solution of the simplified equations shows that without retrograde transport the tau concentration along the axon length stays almost uniform (decreases very slightly), which is inconsistent with the experimenal tau concentration at the outlet boundary (at the axon tip). Thus kinesin-driven transport alone is not enough to explain the empirically observed distribution of tau, and the retrograde motor-driven component in SAT is needed.

Effect of mitochondrial circulation on mitochondrial age density distribution.

Recent publications report that although the mitochondria population in an axon can be quickly replaced by a combination of retrograde and anterograde axonal transport (often within less than 24 hours), the axon contains much older mitochondria. This suggests that not all mitochondria that reach the soma are degraded and that some are recirculating back into the axon. To explain this, we developed a model that simulates mitochondria distribution when a portion of mitochondria that return to the soma are redirected back to the axon rather than being destroyed in somatic lysosomes. Utilizing the developed model, we studied how the percentage of returning mitochondria affects the mean age and age density distributions of mitochondria at different distances from the soma. We also investigated whether turning off the mitochondrial anchoring switch can reduce the mean age of mitochondria. For this purpose, we studied the effect of reducing the value of a parameter that characterizes the probability of mitochondria transition to the stationary (anchored) state. The reduction in mitochondria mean age observed when the anchoring probability is reduced suggests that some injured neurons may be saved if the percentage of stationary mitochondria is decreased. The replacement of possibly damaged stationary mitochondria with newly synthesized ones may restore the energy supply in an injured axon. We also performed a sensitivity study of the mean age of stationary mitochondria to the parameter that determines what portion of mitochondria re-enter the axon and the parameter that determines the probability of mitochondria transition to the stationary state. The sensitivity of the mean age of stationary mitochondria to the mitochondria stopping probability increases linearly with the number of compartments in the axon. High stopping probability in long axons can significantly increase mitochondrial age.

Mitochondrial transport in symmetric and asymmetric axons with multiple branching junctions: a computational study.

Mitochondrial aging has been proposed to be involved in a variety of neurodegenerative disorders, such as Parkinson's disease. Here, we explore the impact of multiple branching junctions in axons on the mean age of mitochondria and their age density distributions in demand sites. The study examined mitochondrial concentration, mean age, and age density distribution in relation to the distance from the soma. We developed models for a symmetric axon containing 14 demand sites and an asymmetric axon containing 10 demand sites. We investigated how the concentration of mitochondria changes when an axon splits into two branches at the branching junction. Additionally, we studied whether mitochondrial concentrations in the branches are affected by what proportion of mitochondrial flux enters the upper branch versus the lower branch. Furthermore, we explored whether the distributions of mitochondrial mean age and age density in branching axons are affected by how the mitochondrial flux splits at the branching junction. When the mitochondrial flux is unevenly split at the branching junction of an asymmetric axon, with a greater proportion of the flux entering the longer branch, the average age of mitochondria (system age) in the axon increases. Our findings elucidate the effects of axonal branching on the mitochondrial age.

Mitochondrial Transport in Symmetric and Asymmetric Axons with Multiple Branching Junctions: A Computational Study.

We explore the impact of multiple branching junctions in axons on the mean age of mitochondria and their age density distributions in demand sites. The study looked at mitochondrial concentration, mean age, and age density distribution in relation to the distance from the soma. We developed models for a symmetric axon containing 14 demand sites and an asymmetric axon containing 10 demand sites. We examined how the concentration of mitochondria changes when an axon splits into two branches at the branching junction. We also studied whether mitochondria concentrations in the branches are affected by what proportion of mitochondrial flux enters the upper branch and what proportion of flux enters the lower branch. Additionally, we explored whether the distributions of mitochondria mean age and age density in branching axons are affected by how the mitochondrial flux splits at the branching junction. When the mitochondrial flux is split unevenly at the branching junction of an asymmetric axon, with a greater proportion of the flux entering the longer branch, the average age of mitochondria (system age) in the axon increases. Our findings elucidate the effects of axonal branching on mitochondria age. Mitochondria aging is the focus of this study as recent research suggests it may be involved in neurodegenerative disorders, such as Parkinson's disease.

ATP diffusional gradients are sufficient to maintain bioenergetic homeostasis in synaptic boutons lacking mitochondria.

Previous work on mitochondrial distribution in axons has shown that approximately half of the presynaptic release sites do not contain mitochondria, raising the question of how the boutons that do not contain mitochondria are supplied with ATP. Here, we develop and apply a mathematical model to study this question. Specifically, we investigate whether diffusive transport of ATP is sufficient to support the exocytic functionality in synaptic boutons which lack mitochondria. Our results demonstrate that the difference in ATP concentration between a bouton containing a mitochondrion and a neighboring bouton lacking a mitochondrion is only approximately 0.4%, which is still 3.75 times larger than the ATP concentration minimally required to support synaptic vesicle release. This work therefore suggests that passive diffusion of ATP is sufficient to maintain the functionality of boutons which do not contain mitochondria.

Absorption rate governs cell transduction in dry macroporous scaffolds.

Developing the next generation of cellular therapies will depend on fast, versatile, and efficient cellular reprogramming. Novel biomaterials will play a central role in this process by providing scaffolding and bioactive signals that shape cell fate and function. Previously, our lab reported that dry macroporous alginate scaffolds mediate retroviral transduction of primary T cells with efficiencies that rival the gold-standard clinical spinoculation procedures, which involve centrifugation on Retronectin-coated plates. This scaffold transduction required the scaffolds to be both macroporous and dry. Transduction by dry, macroporous scaffolds, termed "Drydux transduction," provides a fast and inexpensive method for transducing cells for cellular therapy, including for the production of CAR T cells. In this study, we investigate the mechanism of action by which Drydux transduction works through exploring the impact of pore size, stiffness, viral concentration, and absorption speed on transduction efficiency. We report that Drydux scaffolds with macropores ranging from 50-230 µm and with Young's moduli ranging from 25-620 kPa all effectively transduce primary T cells, suggesting that these parameters are not central to the mechanism of action, but also demonstrating that Drydux scaffolds can be tuned without losing functionality. Increasing viral concentrations led to significantly higher transduction efficiencies, demonstrating that increased cell-virus interaction is necessary for optimal transduction. Finally, we discovered that the rate with which the cell-virus solution is absorbed into the scaffold is closely correlated to viral transduction efficiency, with faster absorption producing significantly higher transduction. A computational model of liquid flow through porous media validates this finding by showing that increased fluid flow substantially increases collisions between virus particles and cells in a porous scaffold. Taken together, we conclude that the rate of liquid flow through the scaffolds, rather than pore size or stiffness, serves as a central regulator for efficient Drydux transduction.

Dynein Dysfunction Prevents Maintenance of High Concentrations of Slow Axonal Transport Cargos at the Axon Terminal: A Computational Study.

Here, we report computational studies of bidirectional transport in an axon, specifically focusing on predictions when the retrograde motor becomes dysfunctional. We are motivated by reports that mutations in dynein-encoding genes can cause diseases associated with peripheral motor and sensory neurons, such as type 2O Charcot-Marie-Tooth disease. We use two different models to simulate bidirectional transport in an axon: an anterograde-retrograde model, which neglects passive transport by diffusion in the cytosol, and a full slow transport model, which includes passive transport by diffusion in the cytosol. As dynein is a retrograde motor, its dysfunction should not directly influence anterograde transport. However, our modeling results unexpectedly predict that slow axonal transport fails to transport cargos against their concentration gradient without dynein. The reason is the lack of a physical mechanism for the reverse information flow from the axon terminal, which is required so that the cargo concentration at the terminal could influence the cargo concentration distribution in the axon. Mathematically speaking, to achieve a prescribed concentration at the terminal, equations governing cargo transport must allow for the imposition of a boundary condition postulating the cargo concentration at the terminal. Perturbation analysis for the case when the retrograde motor velocity becomes close to zero predicts uniform cargo distributions along the axon. The obtained results explain why slow axonal transport must be bidirectional to allow for the maintenance of concentration gradients along the axon length. Our result is limited to small cargo diffusivity, which is a reasonable assumption for many slow axonal transport cargos (such as cytosolic and cytoskeletal proteins, neurofilaments, actin, and microtubules) which are transported as large multiprotein complexes or polymers.

Research of Mitochondrial Function, Structure, Dynamics and Intracellular Organization.

Mitochondria have been recognized as the energy (in the form of ATP)-producing cell organelles, required for cell viability, survival and normal cell function [...].

Computation of the mitochondrial age distribution along the axon length.

We describe a compartmental model of mitochondrial transport in axons, which we apply to compute mitochondrial age at different distances from the soma. The model predicts that at the tip of an axon that has a length of 1 cm, the average mitochondrial age is approximately 22 h. The mitochondria are youngest closest to the soma and their age scales approximately linearly with distance from the soma. To the best of the authors' knowledge, this is the first attempt to predict the spatial distribution of mitochondrial age within an axon. A sensitivity study of the mean age of mitochondria to various model parameters is also presented.

The Complex Interplay between Mitochondria, ROS and Entire Cellular Metabolism.

Besides their main function for energy production in form of ATP in processes of oxidative phosphorylation (OxPhos), mitochondria perform many other important cellular functions and participate in various physiological processes that are congregated. For example, mitochondria are considered to be one of the main sources of reactive oxygen species (ROS) and therefore they actively participate in the regulation of cellular redox and ROS signaling. These organelles also play a crucial role in Ca2+ signaling and homeostasis. The mitochondrial OxPhos and their cellular functions are strongly cell/tissue specific and can be heterogeneous even within the same cell, due to the existence of mitochondrial subpopulations with distinct functional and structural properties. However, the interplay between different functions of mitochondria is not fully understood. The mitochondrial functions may change as a response to the changes in the cellular metabolism (signaling in). On the other hand, several factors and feedback signals from mitochondria may influence the entire cell physiology (signaling out). Numerous interactions between mitochondria and the rest of cell, various cytoskeletal proteins, endoplasmic reticulum (ER) and other cellular elements have been demonstrated, and these interactions could actively participate in the regulation of mitochondrial and cellular metabolism. This review highlights the important role of the interplay between mitochondrial and entire cell physiology, including signaling from and to mitochondria.

Effects of axon branching and asymmetry between the branches on transport, mean age, and age density distributions of mitochondria in neurons: A computational study.

We report a computational study of mitochondria transport in a branched axon with two branches of different sizes. For comparison, we also investigate mitochondria transport in an axon with symmetric branches and in a straight (unbranched) axon. The interest in understanding mitochondria transport in branched axons is motivated by the large size of arbors of dopaminergic neurons, which die in Parkinson's disease. Since the failure of energy supply of multiple demand sites located in various axonal branches may be a possible reason for the death of these neurons, we were interested in investigating how branching affects mitochondria transport. Besides investigating mitochondria fluxes between the demand sites and mitochondria concentrations, we also studied how the mean age of mitochondria and mitochondria age densities depend on the distance from the soma. We established that if the axon splits into two branches of unequal length, the mean ages of mitochondria and age density distributions in the demand sites are affected by how the mitochondria flux splits at the branching junction (what portion of mitochondria enter the shorter branch and what portion enter the longer branch). However, if the axon splits into two branches of equal length, the mean ages and age densities of mitochondria are independent of how the mitochondria flux splits at the branching junction. This even holds for the case when all mitochondria enter one branch, which is equivalent to a straight axon. Because the mitochondrial membrane potential (which many researchers view as a proxy for mitochondrial health) decreases with mitochondria age, the independence of mitochondria age on whether the axon is symmetrically branched or straight (providing the two axons are of the same length), and on how the mitochondria flux splits at the branching junction, may explain how dopaminergic neurons can sustain very large arbors and still maintain mitochondrial health across branch extremities.

An analytical solution simulating growth of Lewy bodies.

This paper reports a minimal model simulating the growth of a Lewy body (LB). To the best of our knowledge, this is the first model simulating LB growth. The LB is assumed to consist of a central spherical core, which is composed of membrane fragments and various dysfunctional intracellular organelles, and a halo, which is composed of alpha-synuclein (α-syn) fibrils. Membrane fragments and α-syn monomers are assumed to be produced in the soma at constant rates. The growth of the core and the halo are simulated by the Finke-Watzky model. Analytical (closed-form) solutions describing the growth of the core and the halo are obtained. A sensitivity analysis in terms of model parameters is performed.

Bidirectional, unlike unidirectional transport, allows transporting axonal cargos against their concentration gradient.

Even though most axonal cargos are synthesized in the soma, the concentration of many of these cargos is larger at the presynaptic terminal than in the soma. This requires transport of these cargos from the soma to the presynaptic terminal or other active sites in the axon. Axons utilize both bidirectional (for example, slow axonal transport) and unidirectional (for example, fast anterograde axonal transport) modes of cargo transport. Bidirectional transport seems to be less efficient because it requires more time and takes more energy to deliver cargos. In this paper, we studied a family of models which differ by the modes of axonal cargo transport (such as anterograde and retrograde motor-driven transport and passive diffusion) as well as by the presence or absence of pausing states. The models are studied to investigate their ability to describe axonal transport against the cargo concentration gradient. We argue that bidirectional axonal transport is described by a higher-order mathematical model, which allows imposing cargo concentration not only at the axon hillock but also at the axon terminal. The unidirectional transport model allows only for the imposition of cargo concentration at the axon hillock. Due to the great lengths of the axons, anterograde transport mostly relies on molecular motors, such as kinesins, to deliver cargos synthesized in the soma to the terminal and other active sites in the axon. Retrograde transport can be also motor-driven, in which case cargos are transported by dynein motors. If cargo concentration at the axon tip is higher than at the axon hillock, retrograde transport can also occur by cargo diffusion. However, because many axonal cargos are large or they assemble in multiprotein complexes for axonal transport, the diffusivity of such cargos is very small. We investigated the case of a small cargo diffusivity using a perturbation technique and found that for this case the effect of diffusion is limited to a very thin diffusion boundary layer near the axon tip. If cargo diffusivity is decreased in the model, we show that without motor-driven retrograde transport the model is unable to describe a high cargo concentration at the axon tip. To the best of our knowledge, our paper presents the first explanation for the utilization of seemingly inefficient bidirectional transport in neurons.

Analysis of Mitochondrial Function, Structure, and Intracellular Organization In Situ in Cardiomyocytes and Skeletal Muscles.

Analysis of the function, structure, and intracellular organization of mitochondria is important for elucidating energy metabolism and intracellular energy transfer. In addition, basic and clinically oriented studies that investigate organ/tissue/cell dysfunction in various human diseases, including myopathies, cardiac/brain ischemia-reperfusion injuries, neurodegenerative diseases, cancer, and aging, require precise estimation of mitochondrial function. It should be noted that the main metabolic and functional characteristics of mitochondria obtained in situ (in permeabilized cells and tissue samples) and in vitro (in isolated organelles) are quite different, thereby compromising interpretations of experimental and clinical data. These differences are explained by the existence of the mitochondrial network, which possesses multiple interactions between the cytoplasm and other subcellular organelles. Metabolic and functional crosstalk between mitochondria and extra-mitochondrial cellular environments plays a crucial role in the regulation of mitochondrial metabolism and physiology. Therefore, it is important to analyze mitochondria in vivo or in situ without their isolation from the natural cellular environment. This review summarizes previous studies and discusses existing approaches and methods for the analysis of mitochondrial function, structure, and intracellular organization in situ.

Model of drug delivery to populations composed of two cell types.

The rate of drug delivery to cells and the subsequent rate of drug metabolism are dependent on the cell membrane permeability to the drug. In some cases, tissue may be composed of different types of cells that exhibit order of magnitude differences in their membrane permeabilities. This paper presents a brief review of the components of the tissue scale three-compartment pharmacokinetic model of drug delivery to single-cell-type populations. The existing model is extended to consider tissue composed of two different cell types. A case study is presented of infusion mediated delivery of doxorubicin to a tumor that is composed of a drug reactive cell type and of a drug resistive cell type. The membrane permeabilities of the two cell types differ by an order of magnitude. A parametric investigation of the population composition is conducted and it is shown that the drug metabolism of the low permeability cells are negatively influenced by the fraction of the tissue composed of the permeable drug reactive cells. This is because when the population is composed mostly of drug permeable cells, the extracellular space is rapidly depleted of the drug. This has two compounding effects: (i) locally there is simply less drug available to the neighboring drug resistant cells, and (ii) the depletion of the drug from the extracellular space near the vessel-tissue interface leaves less drug to be transported to both cell types farther away from the vessel.