Detailed Solution:
1. Structure of the Cell Membrane (Fluid Mosaic Model):
Proposed by S.J. Singer and G.L. Nicolson in 1972, the fluid mosaic model describes the plasma membrane as a dynamic, fluid-like structure embedded with a "mosaic" of proteins and carbohydrates.

• Phospholipid Bilayer: The fundamental structural matrix. Phospholipids are amphipathic. The hydrophilic (water-loving) phosphate heads face the aqueous environment inside (cytosol) and outside the cell. The hydrophobic (water-fearing) fatty acid tails face inward, avoiding water.
• Proteins:
  • Integral (Transmembrane) Proteins: Span across the entire lipid bilayer. They act as channels, carriers, and receptors.
  • Peripheral Proteins: Attached only to the inner or outer surface of the membrane. They act as enzymes and help shape the cell cytoskeleton.
• Cholesterol: Dispersed between phospholipids in animal cells. It acts as a bidirectional buffer for membrane fluidity (preventing it from becoming too rigid in cold or too fluid in heat).
• Carbohydrates: Attached to proteins (glycoproteins) or lipids (glycolipids) on the extracellular surface. They form the glycocalyx, essential for cell-cell recognition and immune response.

2. Functions of the Cell Membrane:

• Selective Permeability: Regulates the entry and exit of molecules. Small nonpolar molecules (like $O_2, CO_2$) diffuse easily, while large or charged molecules (like glucose, $Na^+$) require specialized protein channels.
• Compartmentalization: Encloses the cellular contents, separating the internal metabolic processes from the external environment.
• Signal Transduction: Receptor proteins on the membrane bind to extracellular ligands (like hormones) and transmit the signal into the cell to trigger a specific biological response.
• Cellular Recognition: Glycoproteins act as identification tags, allowing the immune system to distinguish self-cells from foreign pathogens.

Detailed Solution:
1. Structure of Mitochondria:
Mitochondria are double-membrane-bound organelles often referred to as the "powerhouses" of the cell.

• Outer Membrane: Smooth, highly permeable, containing porin proteins that allow the passage of ions and small molecules.
• Inner Membrane: Highly impermeable and selectively permeable. It is deeply folded into structures called Cristae. These folds significantly increase the surface area available for housing the complexes of the Electron Transport Chain (ETC) and ATP synthase.
• Intermembrane Space: The narrow region between the inner and outer membranes. During cellular respiration, protons ($H^+$) are pumped into this space, creating the proton-motive force necessary for ATP generation.
• Mitochondrial Matrix: The dense fluid enclosed by the inner membrane. It contains enzymes required for the Citric Acid Cycle (Krebs cycle), mitochondrial DNA (mtDNA), and 70S ribosomes.

2. Functions of Mitochondria:

• ATP Production (Oxidative Phosphorylation): The primary function. Through the Krebs cycle and the Electron Transport Chain, mitochondria generate the vast majority of cellular ATP by oxidizing glucose derivatives.
• Regulation of Apoptosis: Mitochondria act as the control center for programmed cell death (apoptosis). In response to cellular stress, they release Cytochrome C into the cytosol, which activates the caspase cascade, leading to controlled cell death.
• Calcium Ion Storage: They act as transient storage buffers for intracellular calcium ($Ca^{2+}$), working alongside the Endoplasmic Reticulum to maintain calcium homeostasis, which is vital for muscle contraction and nerve signaling.
• Thermogenesis: In brown adipose tissue (brown fat), mitochondrial uncoupling proteins (UCP1 / Thermogenin) allow protons to re-enter the matrix without passing through ATP synthase. The energy is dissipated as heat instead of ATP, providing non-shivering thermogenesis to keep infants and hibernating animals warm.

Detailed Solution:
Muscle contraction is fundamentally driven by the interaction between two primary cytoskeletal proteins: Actin (thin filaments) and Myosin (thick filaments). The mechanism is explained by the Sliding Filament Theory.

1. Structural Roles:

• Actin (Thin Filament): Consists of a double-helical polymer of G-actin globular proteins. The actin filament possesses specific binding sites for myosin heads. In a relaxed muscle, these binding sites are physically blocked by two regulatory proteins: Tropomyosin and Troponin.
• Myosin (Thick Filament): A motor protein shaped like a golf club, featuring a long tail and a globular head. The myosin head possesses an ATP-binding site and ATPase activity (ability to hydrolyze ATP), as well as an actin-binding site.

2. The Mechanism of Contraction (Cross-Bridge Cycle):

• Calcium Influx: A nerve impulse triggers the release of $Ca^{2+}$ ions from the sarcoplasmic reticulum into the sarcoplasm.
• Exposure of Binding Sites: $Ca^{2+}$ binds to Troponin, causing a conformational change that pulls Tropomyosin away from the myosin-binding sites on the actin filament.
• Cross-Bridge Formation: The energized myosin head (which has already hydrolyzed ATP into ADP + $P_i$) binds firmly to the exposed site on the actin filament, forming a cross-bridge.
• The Power Stroke: Upon binding, the myosin head releases the inorganic phosphate ($P_i$) and ADP. This causes the myosin head to pivot forcefully, pulling the actin filament towards the center of the sarcomere (the M-line). This physical sliding generates the muscle contraction.
• Cross-Bridge Detachment: A new molecule of ATP binds to the myosin head. This binding breaks the bond between myosin and actin, detaching the cross-bridge.
• Reactivation of Myosin Head: The myosin head hydrolyzes the newly bound ATP into ADP and $P_i$, utilizing the released energy to "re-cock" the head back into its high-energy starting position, ready for the next cycle.

Detailed Solution:
1. Overview of Neurodegenerative Diseases:
Neurodegenerative diseases represent an umbrella term for a range of conditions that primarily affect the neurons in the human brain. These diseases are characterized by the progressive and irreversible loss of structure, function, and ultimately the death of neurons. Because neurons generally do not reproduce or replace themselves, the damage is permanent.
Examples include:

• Alzheimer's Disease (AD): Characterized by memory loss and cognitive decline. Pathologically marked by amyloid-beta plaques and tau tangles.
• Parkinson's Disease (PD): Primarily a movement disorder caused by the death of dopaminergic neurons in the substantia nigra. Characterized by tremors and rigidity, marked by Lewy body accumulations ($\alpha$-synuclein).
• Huntington's Disease (HD): A genetic disorder resulting in motor and cognitive decline due to a mutated huntingtin protein.

2. The Tau Hypothesis (Specific to Alzheimer's Disease):
The Tau hypothesis is one of the leading pathophysiological models explaining the neuronal death seen in Alzheimer's disease.

• Normal Function of Tau: In a healthy neuron, the cytoskeleton is supported by microtubules, which act as "railway tracks" for the transport of nutrients, organelles (like mitochondria), and neurotransmitters down the axon. Tau is a highly soluble Microtubule-Associated Protein (MAP). Its primary function is to bind to these microtubules, stabilizing them and keeping the tracks straight and functional.
• Hyperphosphorylation: In Alzheimer's disease, the Tau proteins undergo abnormal and excessive chemical modification known as hyperphosphorylation (addition of too many phosphate groups).
• Detachment and Collapse: Due to hyperphosphorylation, the Tau proteins undergo a conformational shape change and lose their affinity for the microtubules. They detach from the microtubule tracks. Without the stabilizing presence of Tau, the microtubules quickly disintegrate and collapse. This immediately halts axonal transport, starving the neuron of essential nutrients and disrupting synaptic communication.
• Formation of Neurofibrillary Tangles (NFTs): The detached, hyperphosphorylated Tau proteins clump together inside the neuron body, forming highly insoluble aggregates known as Neurofibrillary Tangles (NFTs). These tangles further physically obstruct intracellular functions.
• Cell Death: The combination of the collapsed transport system and the toxic accumulation of intracellular NFTs ultimately triggers apoptosis, leading to neuronal death and the subsequent brain atrophy characteristic of Alzheimer's.