1Physiology 472/572 - 2009 - Quantitative modeling of biological systems Lecture 5: Membrane transport Functions and properties of the cell membrane • separates extracellular and intracellular compartments Extracellular Intracellular - Na+ 142 mM 10 mM - Amino acids 0.03% 0.2% - pH 7.4 7.0 - electrical potential 0 -85 mV • controls interactions of cell with other cells and environment • many transport and signaling functions • consists of a lipid bilayer with embedded proteins • phospholipid molecules have a hydrophilic 'head' and a hydrophobic 'tail' • permeable to non-polar molecules, which diffuse through membrane (O2, CO2, alcohol) • almost impermeable to highly polar molecules (including ions and water) • transport of polar molecules requires special transporters Passive diffusion through a membrane • partition coefficient K = (concentration in membrane)/(concentration in solution) • membrane with diffusivity D, thickness Δx • Flux ΔxCCDKΔxKCKCDJ2121−=−= • Depends on both solubility and diffusivity • Membrane is not actually homogeneous through its thickness2Membrane transporters - channels and carriers • Channel is a hole or pore, 0.5 - 1 nm diameter - usually carries small solutes, by passive diffusion - transport is rapid, 106 - 109 ions per second - permeability may be modulated, e.g., voltage gated channels • Carrier moves solutes by a conformational change - usually carries larger solutes - transport is slower, 102 - 104 molecules per second - may be passive (i.e., facilitated) or active (driven by ATP or other substrates) A simple model for facilitated (carrier-mediated) transport • Solute A moves from side I to side II • Carrier X has states I and II • Assumptions - A can only cross as AX - X is membrane bound - A + X ⇌ AX equilibrates rapidly: [AI][XI] = k [AXI] and [AII][XII] = k [AXII] - the net rate of XI → XII is D' ([XI] - [XII]) - the net rate of AXI → AXII is JA = D' ([AXI] - [AXII]) - system is in steady state - [XI] + [XII] + [AXI] + [AXII] = XT , the total amount of carrier3 • Analysis - at steady state the rates of XI → XII and AXI → AXII must add to zero so D'([XI] - [XII] + [AXI] - [AXII]) = 0, i.e., [XI] + [AXI] = [XII] + [AXII] = XT/2 - use [XI] = k [AXI]/[AI] and solve to get k][A][A2X][AXIITI+= - similarly k][A][A2X][AXIIIITII+= • Result ⎟⎟⎠⎞⎜⎜⎝⎛+−+=′=k][A][Ak][A][AV])[AX-]([AX DJIIIIIImaxIIA where 2XDTmax′=V - simpler case k[A][A]VJmaxA+= is known as Michaelis-Menten kinetics - good description of many carrier or enzyme-mediated processes - Vmax is the maximum rate, approached for high substrate levels - k is the Michaelis constant, the concentration at which the rate is half of
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