Biological Membranes & Transport

Membrane Fluidity

Membrane fluidity is a critical property that affects all membrane functions -- transport, signaling, cell division, and membrane fusion. Fluidity depends on several factors:

• Temperature: Higher temperature increases fluidity by providing more kinetic energy for lipid motion
• Fatty acid chain length: Shorter chains interact less via van der Waals forces, increasing fluidity
• Degree of unsaturation: Cis double bonds introduce kinks that prevent tight packing, increasing fluidity
• Cholesterol content: Dual role -- at high temperatures, cholesterol reduces fluidity by restricting phospholipid movement; at low temperatures, it prevents crystallization by disrupting regular packing

Lipid Motions in Membranes

Lipid molecules undergo several types of motion within the bilayer, differing dramatically in their rates:

• Lateral diffusion: Rapid movement within the plane of the membrane. A phospholipid can diffuse $\sim 2\;\mu\text{m/s}$, traversing the length of a bacterial cell in about 1 second.
• Rotational diffusion: Spinning around the lipid's long axis; very fast.
• Flexion: Bending of the acyl chains; fast and increases toward the bilayer center.
• Transverse diffusion (flip-flop): Movement from one leaflet to the other. Extremely slow without enzyme assistance ($t_{1/2} \sim$ hours to days). Catalyzed by flippases (inward, ATP-dependent), floppases (outward, ATP-dependent), and scramblases (bidirectional, energy-independent).

Leaflet Asymmetry

The asymmetric localization of PS on the inner leaflet is biologically significant: exposure of PS on the outer leaflet is an early marker of apoptosis, recognized by phagocytes (the "eat me" signal). PS externalization is also required for blood coagulation (provides a negatively charged surface for clotting factor assembly).

Cholesterol in Membranes

Cholesterol constitutes up to $\sim 25$ mol% of the eukaryotic plasma membrane. Its rigid steroid ring inserts between phospholipid acyl chains with the hydroxyl group oriented toward the aqueous interface. Cholesterol has a dual effect on fluidity:

• At physiological/high temperatures: restricts acyl chain motion, reducing fluidity and permeability
• At low temperatures: prevents the close packing of acyl chains, inhibiting crystallization (gel phase formation)

This dual action broadens the phase transition, effectively making the membrane resistant to temperature fluctuations -- a property known as the condensing effect.

Lipid Rafts

Lipid rafts are dynamic, nanoscale membrane microdomains enriched in cholesterol and sphingolipids. The saturated acyl chains of sphingolipids pack tightly with cholesterol, forming a liquid-ordered ($L_o$) phase that is more structured than the surrounding liquid-disordered ($L_d$) phospholipid bilayer. Lipid rafts concentrate specific signaling proteins (GPI-anchored proteins, Src-family kinases) and are involved in signal transduction, membrane trafficking, and pathogen entry.

Integral (Transmembrane) Proteins

Integral membrane proteins span the lipid bilayer one or more times. They require detergents or organic solvents for extraction. Two major structural motifs are observed:

• $\alpha$-Helical bundles: Single or multiple transmembrane $\alpha$-helices with hydrophobic side chains facing the lipid interior. Examples: G protein-coupled receptors (GPCRs, 7 TM helices), receptor tyrosine kinases (1 TM helix), ion channels
• $\beta$-Barrels: Antiparallel $\beta$-sheets rolled into a cylindrical barrel. Found exclusively in the outer membranes of Gram-negative bacteria, mitochondria, and chloroplasts. Examples: porins, $\text{OmpF}$

A single transmembrane $\alpha$-helix typically consists of 20--25 hydrophobic residues, sufficient to span the $\sim 30$ $\text{\AA}$ hydrophobic core of the bilayer.

Peripheral Proteins

Peripheral membrane proteins are associated with the membrane surface through electrostatic interactions and hydrogen bonds with integral proteins or lipid head groups. They can be released by changes in pH or ionic strength (no detergent required). Examples include cytochrome $c$ (inner mitochondrial membrane), spectrin and ankyrin (erythrocyte cytoskeleton), and many signaling proteins.

Lipid-Anchored Proteins

These proteins are covalently attached to lipid molecules that insert into the bilayer:

• GPI anchors: Glycosylphosphatidylinositol links protein to the outer leaflet. Examples: alkaline phosphatase, CD59, prion protein
• Palmitoylation: Thioester linkage of palmitate (C16:0) to cysteine; reversible. Targets proteins to the inner leaflet
• Myristoylation: Amide linkage of myristate (C14:0) to N-terminal glycine; co-translational and irreversible
• Prenylation: Thioether linkage of farnesyl (C15) or geranylgeranyl (C20) isoprenoid to C-terminal cysteine. Targets proteins to inner leaflet. Examples: Ras GTPases

Simple Diffusion: Fick's First Law

The rate of diffusion of a solute across a membrane is described by Fick's first law:

where $J$ is the flux (mol/cm$^2$/s), $D$ is the diffusion coefficient, and $dC/dx$ is the concentration gradient. The negative sign indicates that net flux occurs from high to low concentration.

For membrane transport, this is often expressed in terms of a permeability coefficient $P$:

where $K$ is the partition coefficient (lipid/water solubility ratio) and $\Delta x$ is the membrane thickness. Small nonpolar molecules ($\text{O}_2, \text{CO}_2, \text{N}_2$) cross freely. Small polar molecules ($\text{H}_2\text{O}$, urea, glycerol) cross slowly. Ions and large polar molecules are essentially impermeant.

Osmosis and Tonicity

Osmosis is the net movement of water across a semipermeable membrane from a region of lower solute concentration to higher solute concentration. The osmotic pressure ($\Pi$) is given by the van't Hoff equation:

where $i$ is the van't Hoff factor (number of particles per formula unit),$M$ is molarity, $R$ is the gas constant, and $T$ is absolute temperature.

• Isotonic: No net water movement; cell maintains normal volume
• Hypotonic: Water enters the cell; cell swells (lysis in erythrocytes = hemolysis)
• Hypertonic: Water leaves the cell; cell shrinks (crenation in erythrocytes)

Aquaporins are integral membrane proteins that form selective water channels, dramatically increasing water permeability (up to $3 \times 10^9$ water molecules/s per channel). AQP1 was discovered by Peter Agre (Nobel Prize, 2003). At least 13 aquaporin isoforms exist in humans.

Channel Proteins

Channels form aqueous pores through the membrane, allowing selected ions or small molecules to flow at rates approaching the diffusion limit ($\sim 10^7 - 10^8$ ions/s). Channels are typically gated -- they open and close in response to specific stimuli:

• Voltage-gated: Open in response to changes in membrane potential (e.g., $\text{Na}^+$, $\text{K}^+$, $\text{Ca}^{2+}$ channels in neurons)
• Ligand-gated: Open upon binding of a specific ligand (e.g., nicotinic acetylcholine receptor, GABA$_\text{A}$ receptor)
• Mechanosensitive: Open in response to mechanical stress or membrane deformation

Ion selectivity is achieved through selectivity filters -- narrow constrictions in the channel pore lined with specific residues. The $\text{K}^+$ channel selectivity filter (GYG motif) discriminates $\text{K}^+$ over $\text{Na}^+$ by a factor of $\sim 10{,}000$, despite $\text{Na}^+$ being smaller (Roderick MacKinnon, Nobel Prize 2003).

Transporter Proteins (Carriers)

Transporters bind their substrate on one side of the membrane, undergo a conformational change, and release the substrate on the other side. Transport rates are much slower than channels ($\sim 10^2 - 10^4$ molecules/s). The kinetics follow Michaelis-Menten behavior:

where $V_{\max}$ is the maximum transport rate and $K_m$ (or $K_t$ for transport) is the substrate concentration at half-maximal velocity.

Glucose Transporters (GLUTs)

• GLUT1: Ubiquitous, particularly in erythrocytes and blood-brain barrier. $K_m \approx 1$ mM (always active since blood glucose $\approx 5$ mM)
• GLUT2: Liver, pancreatic $\beta$-cells, small intestine. $K_m \approx 15$ mM (acts as glucose sensor)
• GLUT3: Neurons. $K_m \approx 1.4$ mM (ensures constant glucose supply to brain)
• GLUT4: Skeletal muscle, adipose tissue. Insulin-dependent -- translocated to membrane upon insulin signaling. $K_m \approx 5$ mM

Energetics of Ion Transport

The free energy change for transporting an ion across a membrane includes both the concentration gradient and the electrical potential difference:

where $R$ is the gas constant, $T$ is temperature, $z$ is the ion charge, $F$ is Faraday's constant (96,485 J/V/mol), and $\Delta\psi$ is the membrane potential. The first term accounts for the chemical gradient; the second for the electrical gradient. Together they define the electrochemical gradient.

Primary Active Transport: P-type ATPases

P-type ATPases form a phosphorylated intermediate during the transport cycle (hence "P-type"). The most important examples:

• $\text{Na}^+/\text{K}^+$-ATPase (sodium-potassium pump): Transports 3 $\text{Na}^+$ out and 2 $\text{K}^+$ in per ATP hydrolyzed. This electrogenic pump generates a net outward current, contributing to the resting membrane potential. It consumes $\sim 25\%$ of total cellular ATP (up to 70% in neurons). Inhibited by cardiac glycosides (ouabain, digoxin).
• $\text{Ca}^{2+}$-ATPase (SERCA): Sarcoplasmic/endoplasmic reticulum $\text{Ca}^{2+}$-ATPase pumps 2 $\text{Ca}^{2+}$ into the SR lumen per ATP, maintaining cytosolic $[\text{Ca}^{2+}] \sim 100$ nM (vs. $\sim 1$ mM in SR). Critical for muscle relaxation. Regulated by phospholamban in cardiac muscle.
• $\text{H}^+/\text{K}^+$-ATPase: Found in gastric parietal cells. Secretes $\text{H}^+$ into the stomach lumen, creating pH $\sim 1$ (a million-fold concentration gradient). Target of proton pump inhibitors (omeprazole/Prilosec).

Secondary Active Transport

Secondary active transport harnesses the energy stored in an ion gradient (usually $\text{Na}^+$ in animal cells, $\text{H}^+$ in bacteria and plants) to drive the transport of another solute against its gradient:

• Symporters (cotransporters): Both solutes move in the same direction. Example: SGLT1 ($\text{Na}^+$-glucose symporter) in intestinal epithelium -- 2 $\text{Na}^+$ + 1 glucose transported inward per cycle
• Antiporters (exchangers): Solutes move in opposite directions. Example: $\text{Na}^+/\text{H}^+$ exchanger (NHE) -- regulates intracellular pH; $\text{Na}^+/\text{Ca}^{2+}$ exchanger (NCX) -- 3 $\text{Na}^+$ in for 1 $\text{Ca}^{2+}$ out (important in cardiac myocytes)

The Nernst Equation

The Nernst equation calculates the equilibrium potential for a single ion species -- the membrane potential at which there is no net flux of that ion:

where $R = 8.314$ J/(mol$\cdot$K), $T = 310$ K (37$°$C),$z$ is the ion valence, and $F = 96{,}485$ C/mol.

The Goldman-Hodgkin-Katz (GHK) Equation

The resting membrane potential depends on the contributions of all permeable ions, weighted by their relative permeabilities. The Goldman equation accounts for this:

where $P_K$, $P_{Na}$, and $P_{Cl}$ are the membrane permeabilities for each ion. Note that $\text{Cl}^-$ concentrations are inverted (inside in numerator, outside in denominator) because of its negative charge.

At rest, the membrane is much more permeable to $\text{K}^+$ than to $\text{Na}^+$(typical ratio $P_K : P_{Na} : P_{Cl} \approx 1 : 0.04 : 0.45$), so the resting potential is dominated by $\text{K}^+$ and lies close to $E_K$.

• Typical resting $V_m$ (nerve): $\approx -70$ mV
• Typical resting $V_m$ (skeletal muscle): $\approx -90$ mV
• Typical resting $V_m$ (cardiac muscle): $\approx -85$ mV

During an action potential, a transient increase in $P_{Na}$ (voltage-gated $\text{Na}^+$ channels open) drives $V_m$ toward $E_{Na}$ ($\approx +67$ mV), causing depolarization. Subsequent opening of voltage-gated $\text{K}^+$ channels restores$V_m$ toward $E_K$ (repolarization).