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Basic Principles of Transport

Robert I. Macey, Teresa F. Moura

10.1002/cphy.cp140106

Source: Supplement 31: Handbook of Physiology, Cell Physiology

Originally published: 1997

Published online: January 2011

Full Article on Wiley Online Library



Abstract

The sections in this article are:

1 Thermodynamics
1.1 Maximal Work Is Attained on Reversible Paths
1.2 Changes in Free Energy Are Obtained from Chemical Potentials: Free Energy per Mole
1.3 Solutes in Phase Equilibrium
1.4 Osmotic Equilibrium
1.5 Chemiosmotic Coupling
2 Diffusion
2.1 Flux Is Proportional to the Concentration Gradient: Fick's First Law
2.2 Conservation of Matter: Fick's Second Law
2.3 Progress of Diffusion: Mean Square Displacement = 2 Dt
2.4 Randomly Diffusing Molecules Spread Out in a Normal Distribution
2.5 The Diffusion Front, Where Solute Depletion Ends and Accumulation Begins, Moves as 2nDt
2.6 Diffusion Transients in Thin Membranes Are Very Rapid
2.7 Permeation through Membranes Takes Place in At Least Three Steps
2.8 The Exchange Time for Filling or Emptying of Cells Equals V/PA
2.9 In Simple Exponential Processes, the Time Constant = the Mean Residence Time
2.10 Cytoplasmic Diffusion Transients Are Rapid
2.11 Unstirred Layers Can Be a Significant Barrier
2.12 Membrane Diffusion Is Assumed to Be Rate‐Limiting for Plasma Membranes
2.13 Selective Permeability of Lipid Bilayers Is Determined Primarily by Solubility
3 Water Transport
3.1 Water Transport Can Be Driven by Three Different Gradients
3.2 In Lipid Membranes Water Transport Occurs by Solubility‐Diffusion Mechanism, with Pf = Posm = Pd
3.3 Osmotic Gradients Generate Hydraulic Pressure Gradients in Aqueous Channels
3.4 In Narrow Channels the Ratio Posmotic/Pdiffusion = the Number of Water Molecules Contained within the Channel
3.5 Coupling of Solute and Solvent Transport Is Described by the Kedem‐Katchalsky Equations
4 Ionic Diffusion
4.1 Diffusion with Superimposed Drift Due to External Forces
4.2 Ions Transported by Simple Diffusion Follow the Ussing Flux Ratio Relation
4.3 Bulk Solutions Carry No Net Charge
4.4 The Constant Field Is a Convenient Idealization
4.5 Conductance Depends on Ionic Concentrations
4.6 Permeability Ratios Can Be Measured by Changes in Membrane Potential
4.7 An Electrogenic Pump Contributes to Ψm
4.8 Channels Can Be Incorporated into the Nernst‐Planck Formulation
5 Energy Barriers
5.1 The Born Energy Estimates the Work Required to Transfer an Ion from One Medium to Another
5.2 Born Energy, Image Forces, Dipole Potentials, and Hydrophobic Interactions Contribute to the Energy Barriers of Lipid Bilayers
5.3 Solvation Energies Are Important Determinants of Channel Accessibility
5.4 Surface Potentials Modify the Transmembrane Potential as Well as Local Ion Concentrations
5.5 Transport across Energy Barriers
5.6 Eyring Rate Theory: Rate Constants Depend on Ψ
5.7 Kinetic Approaches
6 Channels
6.1 Single‐Occupancy Channels with Binding Sites Show Saturation Kinetics
6.2 Competition, Unidirectional Flux, and the Ussing Flux Ratio
6.3 Single‐Occupancy Channels: Voltage Dependence in Symmetric Channels
6.4 Single‐Occupancy‐Channel Results Can Be Generalized to N Sites
6.5 Multiple Occupancy
7 Simple Carriers
7.1 Net Flux
7.2 Unidirectional Flux
7.3 Equilibrium at the Boundaries
7.4 Rate‐Limiting Steps at the Boundaries
7.5 Energy‐driven Simple Carrier Systems
8 Cotransport
8.1 Thermodynamics: Cotransport Can Move Solutes “Uphill”
8.2 Kinetic Description
8.3 General Net Flux
8.4 Unidirectional Fluxes
8.5 Kinetics of Simultaneous Binding of mA and nB Resembles Simple Carrier Kinetics When Carrier Concentrations Are Replaced with the Product AmBn
8.6 Relations between the Net Flux Equation Parameters and the Michaelis‐Menten—type Parameters for 1:1 Stoichiometry
9 Countertransport
9.1 Thermodynamics
9.2 Kinetic Models
9.3 Ping‐Pong Model
9.4 Sequential Model
9.5 Generalization to m‐n Stoichiometry for Simultaneous Binding
10 Fluctuating Barriers: Channels and Carriers
10.1 Channel Transport Properties Depend on the Rate of Transition between the Conformations
10.2 Channels with Fluctuating Barriers Do Not Show Michaelis‐Menten Kinetics
10.3 Channels with Fluctuating Barriers Can Show Carrier Kinetics
Figure 1. Figure 1.

A linear spring attached to a weightless pan containing sand rises a distance h when sand is removed.
Figure 2. Figure 2.

Two ways of depicting a cyclic reaction. After reacting with membrane carrier Y on the inner surface of the membrane, solute C is pumped out of the cell. Energy is supplied by the substrate S, which is degraded to P.
Figure 3. Figure 3.

Lattice model of diffusion. The solute at position xi is “caged” by surrounding water molecules. It may advance to position xi+1 when it acquires sufficient thermal energy to break out of its cage. Energy barriers separating the solute from adjacent positions are illustrated, with average distance between activation energy peaks of height ΔGo‡ given by Λ. J represents net flux from xi to xi+1.
Figure 4. Figure 4.

Common types of symmetry encountered in diffusion problems: A. spherical symmetry; B. cylindrical symmetry; C. planar symmetry.
Figure 5. Figure 5.

Concentration profiles, described by Equation 71, show the fate of solute molecules that originate (time t = to = 0) within the thin shaded rectangle as they diffuse through the solution. The y axis represents the concentration at each position. As time progresses (e.g., at t = t1) the solutes diffuse in both directions, depleting the rectangle and creating a bell‐shaped (Gaussian) curve. The curve flattens more and more with time (e.g., t = t2) until at t = ∞ the concentration profile becomes uniformly flat.
Figure 6. Figure 6.

Plots illustrating time course of diffusion from a source originally located at x = 0. The top plot is similar to Figure 5. The second and third plots illustrate the corresponding flux and solute accumulation respectively (n=1; see text). The shaded area distinguishes regions of solute accumulation from regions of depletion.
Figure 7. Figure 7.

Concentration profiles within a homogeneous membrane following sudden addition of solute to side “in” on the left. The concentration on side “o” at x > d is maintained at zero. The y axis represents concentration, the x axis distance. At time t = 0 solute is added at concentration C = Cin to the left‐hand side. At that instant there is no solute within the membrane: the concentration profile is represented by the black line. As time progresses, the membrane fills with solute and the concentration profile quickly approaches the steady state shown by the straight line at t = ∞. Intermediate profiles are represented by the curves labeled t1 and t2. For most practical purposes, the steady state is reached “instantly” (see text).
Figure 8. Figure 8.

A symmetric homogeneous membrane is represented by a series of energy barriers. Interfacial barriers (barriers 0 and m + 1) are encountered by the solute upon entering or leaving the membrane. Internal barriers (1 to m) are encountered while passing through the membrane.
Figure 9. Figure 9.

Effects of unstirred layers on permeability. The illustration, consisting of three “membranes” in series, shows the concentration profiles through each of these. Concentration is plotted on the y axis, distance on the x axis. P1 and P2 represent the permeabilities of the two unstirred layers; Pm represents the permeability of the membrane.
Figure 10. Figure 10.

A plot of nonelectrolyte permeability in egg phosphati‐dylcholine‐decane bilayers as a function of their partition coefficient in hexadecane redrawn from the paper of Walter and Gutknecht 138. Numbers in the figure correspond to the solutes as follows: 1. water; 2. hydrofluoric acid; 3. ammonia; 4. hydrochloric acid; 5. formic acid; 6. methylamine; 7. formamide; 8. nitric acid; 9. urea; 10. thiocyanic acid; 11. acetic acid; 12. ethylamine; 13. ethanediol; 14. acetamide, 15. propionic acid; 16. 1,2‐propanediol; 17. glycerol; 18. butyric acid; 19. 1,4‐butanediol; 20. benzoic acid; 21. hexanoic acid; 22. salicylic acid; 23. codeine.
Figure 11. Figure 11.

Three ways of measuring water flow through a membrane. A. A hydraulic piston pushing on side “in” forces water through the membrane to side “o.” A permeable mechanical support is provided to prevent mechanical deformation of the membrane. B. An impermeable solute on side “o” draws the water toward it. C. Flux of labeled water placed on side “in.”
Figure 12. Figure 12.

A membrane water channel forms a continuum bridging the two bathing solutions. Both side “in” and the channel contain pure water, while side “o” contains an impermeable solute. Rapid equilibrium at the interfaces requires continuity of the chemical potential for water, but the impermeable solute on side “o” creates a discontinuity in its concentration at the “o” interface. These boundary conditions are satisfied when there is an equal and opposite discontinuity in hydrostatic pressure at the “o” interface. Panel A: Plot of hydrostatic pressure profile through the membrane. The pressure drop across the membrane creates a hydraulic flow of water through the membrane even though both bathing media are at atmospheric pressure. Panel B: Equilibrium (J = 0) ensues when the cryptic pressure drop within the membrane is wiped out by the application of a hydrostatic pressure (P – Patm) = ΔII = RTΣCi to side “o.”
Figure 13. Figure 13.

Efflux of solute out of an elementary volume where motion is constrained to the x direction. If the solute velocity = dx/dt, then all solutes contained within the box defined by dxdydz at time t will have passed through the shaded plane in time t + dt.
Figure 14. Figure 14.

Concentration and electrical potential profiles through a membrane.
Figure 15. Figure 15.

Normalized I vs. Ψm plot obtained from Equation 182 for various values of Co while Cin is held fixed. Values for Co/Cin are indicated on the graph. Note that the slope; (conductances) of all curves approach the same limiting value (determined by Equation 186) at infinite Ψm.
Figure 16. Figure 16.

A model channel illustrating the continuum approach with a macroscopic dielectric constant. Channel cross‐sectional area, a(x), varies from the wide‐mouthed vestibule on the left to a narrow, negatively charged, aperture that approximates the size of an ion on the right. The dotted lines show equipotential lines generated by the channel charge; the solid lines show lines of force.

Redrawn from Dani and Levitt 30
Figure 17. Figure 17.

Born energy (dashed lines) for three different ionic radii are obtained from Equation 219. Corresponding Born energy corrected for “image forces” and shown as solid lines are obtained from Equation 221 using the first 20 terms of the series.
Figure 18. Figure 18.

Lipid bilayer energy barriers for a hydrophobic ion with radius = 4.2 Å. Top panel: The upper trace shows a composite consisting of the Born image plus neutral hydrophobic energies. The two lower curves of plot A, “ + Dipole” and “—Dipole,” are plots of dipole energies for cations and anions respectively. Bottom panel shows the composite sum of these energies, one for anions, the other for cations. The negative dipole potential lowers the energy barrier for anions, making them very permeable.

Redrawn from Flewelling and Hubbel 41
Figure 19. Figure 19.

Energy‐barrier profile of gramicidin channel constructed from molecular dynamics simulations by Roux and Karplus. Simulations were accomplished by dividing the channel into four sections labeled I to IV (lower figure), corresponding to sections separated by open circles in energy profile (upper figure). Significance of the different sections: II—transition from bulk water to single file solvation; I—the beginning of single‐file transport; III—single‐file transport; IV—intermolecular region.

Redrawn from Roux and Karplus 120
Figure 20. Figure 20.

Electrical potential profiles in the presence of surface charge. Surface‐charge density on inner surface at left is assumed larger than the corresponding charge density on outer surface at the right. The fast rise in the potential on entering the membrane from either side “in” or “o” is due to a small dipole potential. Notice that the internal transmembrane potential difference is larger than the measured membrane potential.
Figure 21. Figure 21.

Illustration of a simple channel with two binding sites. Each channel can exist in three states: an empty channel Yoo, a channel with C on the inside Yco, and a channel with C on the outside Yoc. The upper triangular sketch shows the transitions among the states. The lower “reaction” scheme shows the reversible transport of Cin to Co.
Figure 22. Figure 22.

Reaction scheme illustrating a single‐occupancy channel and two binding sites, where two solutes, C and B, compete. See Fig. 21 for details.
Figure 23. Figure 23.

Normalized plot of I vs. Ψm (Equation 263) with Cin = Co = K. Values of λ/k−in are indicated in the graph. Channel saturation becomes more evident at lower voltages the larger the λ/k−in (translocation rate compared to dissociation rate). Dotted straight lines show the corresponding linear conductance predicted by corresponding constant‐field equations.
Figure 24. Figure 24.

Reaction scheme illustrating a single‐occupancy channel with n + 2 binding sites. See Figure 21 for details.
Figure 25. Figure 25.

Illustration of a simple carrier where a mobile component Y within the membrane facilitates permeation. Beginning on side “in” the solute Cin combines with an empty reactive carrier site that is exposed on side “in” and is designated as Yoin. The reaction forms a carrier‐solute complex Ycin. Ycin then undergoes a translocation step (conformation change) that exposes the reactive site to side “o,” where it is designated as Yco. The complex Yco then dissociates, releasing Co to side “o.” The free carrier Yoo with the reactive site still facing side “o” will then complete the cycle by translocating its reactive site to side “in,” where it again becomes Yoin.
Figure 26. Figure 26.

Cotransport random carrier (m = n = 1). The cycle begins on side “in” where Ain or Bin combine with its reactive site on component Yooin, forming the complex YAoin or YoBin. This complex reacts with the other solute, forming the ternary complex YABin. YABin undergoes a translocation step, exposing the reactive sites to side “o,” and becomes YABo. YABo dissociates, releasing in a random order Ao or Bo to side “o,” going through the intermediate complexes YAoo or YoBo, before becoming totally in the undissociated form Yooo. Yooo will then translocate them to side “in,” completing the cycle. The association‐dissociation reactions at each boundary are assumed to be in equilibrium. At these boundaries only three out of the four cyclic dissociation constants, Keq, are independent. This is reflected in the terms containing α1 and α2 (see discussion of Equations 331, 332, and 333).
Figure 27. Figure 27.

Reaction cycle used to derive Equation 335. See text.
Figure 28. Figure 28.

Cotransport random: Families of Lineweaver‐Burk plots of 1/JA‐influx versus 1/Ao for increasing values of Bo indicated by the curved arrow. All plots intersect in the second or third quadrant, depending on the relative values of α1 and the product (1+k−1X) (1+k2X). To interpret plots, recall that for each value of Bo, the plot shows a straight line with a y‐axis intercept equal to 1/JmaxA‐influx and an x‐axis intercept equal to −1/KMA‐influx a: α1 < (1 + k−1X) (1+k2X). From Equations 352 and 353, yA = yB > 0 and the intersection lies in the second quadrant… From the changes in the x and y intercepts, it follows that JmaxA‐influx increases with Bo while KMA‐influx decreases. b: If α1 > (1+k−1X) (1+k2X) then yA = yB < 0. The intersection lies below the x axis, in the third quadrant. Both JmaxA‐influx and KMA‐influx will increase with Bo. c: If α1 = (1+k−1X) (1+k2X) then yA = yB = 0. The intersection lies on the x axis. In addition, JmaxA‐influx increases with increasing Bo, while KMA‐influx remains constant, independent of Bo.
Figure 29. Figure 29.

Cotransport sequential. Families of Lineweaver‐Burk plots illustrating graphical criteria for an ordered sequence mechanism for influx with Ao always binding first. a: Plot of 1/JA‐influx versus 1/Ao for increasing values of Bo indicated by the curved arrow. As Bo → ∞, the slope → 0. b: Plot of 1/JB‐influx versus 1/Bo for increasing values of Ao indicated by the curved arrow. All lines intersect on the y axis.
Figure 30. Figure 30.

Countertransport carrier ping‐pong mechanism. Beginning on side “in” the solute Ain or Bin combine with the reactive site of component Yooin. The complex YAin or YBin will then undergo a translocation step, moving the reactive site to side “o.” The complex then dissociates, releasing Ao or Bo to side “o.” The undissociated carrier with the reactive site still facing side “o” will then either translocate its reactive site in the free (Yooo) or bound form (YAo or YBo) to side “in.” Reactions of the solute A and B with the reactive site of component Y are assumed to be in equilibrium.
Figure 31. Figure 31.

Countertransport sequential model with m = n = 1. When A binds to oinY•o or ooY•in, it forms the complex AinY•o or AoY•in. This complex will then bind B on the empty reactive site located on the opposite side of the membrane, forming AinYBo or AoYBin. On the other hand, if B binds first, then the complex oinYBo or ooYBin would be formed before AoYBo or AoYBin. Only three out of the four cyclic dissociation constants, Keq, are independent (see Equations 48 and 49). For each step, the dissociation constant is located closest to the dissociation arrow.
Figure 32. Figure 32.

Countertransport ping‐pong. Lineweaver‐Burk plot for the influx of a solute A for different concentrations of B. a: Plot of stimulation of the influx of A by the counter solute (Bin) on the trans side of the membrane where it binds to the free carrier when there is no slippage, b: Plot of a competitive inhibition. Influx of A is decreased by Bo, which competes for the same carrier site on the same side “o” (cis side) of the membrane.
Figure 33. Figure 33.

Reaction scheme for single‐site channel with energy barriers that can fluctuate between two conformations, Y and Y*. The dashed arrows are redundant; they have been drawn to facilitate an overview of all transport pathways.
Figure 34. Figure 34.

Two plots of flux vs. concentration for singly occupied dynamic channels, with different Po obtained from Equation 394. Values of the parameters are shown on each curve. Neither curve is a rectangular hyperbola. Note the maximum in the upper plot.
Figure 35. Figure 35.

Operation of cycle (Equation 390) shows how channel with very large fluctuating barriers behaves like conventional carrier.


Figure 1.

A linear spring attached to a weightless pan containing sand rises a distance h when sand is removed.



Figure 2.

Two ways of depicting a cyclic reaction. After reacting with membrane carrier Y on the inner surface of the membrane, solute C is pumped out of the cell. Energy is supplied by the substrate S, which is degraded to P.



Figure 3.

Lattice model of diffusion. The solute at position xi is “caged” by surrounding water molecules. It may advance to position xi+1 when it acquires sufficient thermal energy to break out of its cage. Energy barriers separating the solute from adjacent positions are illustrated, with average distance between activation energy peaks of height ΔGo‡ given by Λ. J represents net flux from xi to xi+1.



Figure 4.

Common types of symmetry encountered in diffusion problems: A. spherical symmetry; B. cylindrical symmetry; C. planar symmetry.



Figure 5.

Concentration profiles, described by Equation 71, show the fate of solute molecules that originate (time t = to = 0) within the thin shaded rectangle as they diffuse through the solution. The y axis represents the concentration at each position. As time progresses (e.g., at t = t1) the solutes diffuse in both directions, depleting the rectangle and creating a bell‐shaped (Gaussian) curve. The curve flattens more and more with time (e.g., t = t2) until at t = ∞ the concentration profile becomes uniformly flat.



Figure 6.

Plots illustrating time course of diffusion from a source originally located at x = 0. The top plot is similar to Figure 5. The second and third plots illustrate the corresponding flux and solute accumulation respectively (n=1; see text). The shaded area distinguishes regions of solute accumulation from regions of depletion.



Figure 7.

Concentration profiles within a homogeneous membrane following sudden addition of solute to side “in” on the left. The concentration on side “o” at x > d is maintained at zero. The y axis represents concentration, the x axis distance. At time t = 0 solute is added at concentration C = Cin to the left‐hand side. At that instant there is no solute within the membrane: the concentration profile is represented by the black line. As time progresses, the membrane fills with solute and the concentration profile quickly approaches the steady state shown by the straight line at t = ∞. Intermediate profiles are represented by the curves labeled t1 and t2. For most practical purposes, the steady state is reached “instantly” (see text).



Figure 8.

A symmetric homogeneous membrane is represented by a series of energy barriers. Interfacial barriers (barriers 0 and m + 1) are encountered by the solute upon entering or leaving the membrane. Internal barriers (1 to m) are encountered while passing through the membrane.



Figure 9.

Effects of unstirred layers on permeability. The illustration, consisting of three “membranes” in series, shows the concentration profiles through each of these. Concentration is plotted on the y axis, distance on the x axis. P1 and P2 represent the permeabilities of the two unstirred layers; Pm represents the permeability of the membrane.



Figure 10.

A plot of nonelectrolyte permeability in egg phosphati‐dylcholine‐decane bilayers as a function of their partition coefficient in hexadecane redrawn from the paper of Walter and Gutknecht 138. Numbers in the figure correspond to the solutes as follows: 1. water; 2. hydrofluoric acid; 3. ammonia; 4. hydrochloric acid; 5. formic acid; 6. methylamine; 7. formamide; 8. nitric acid; 9. urea; 10. thiocyanic acid; 11. acetic acid; 12. ethylamine; 13. ethanediol; 14. acetamide, 15. propionic acid; 16. 1,2‐propanediol; 17. glycerol; 18. butyric acid; 19. 1,4‐butanediol; 20. benzoic acid; 21. hexanoic acid; 22. salicylic acid; 23. codeine.



Figure 11.

Three ways of measuring water flow through a membrane. A. A hydraulic piston pushing on side “in” forces water through the membrane to side “o.” A permeable mechanical support is provided to prevent mechanical deformation of the membrane. B. An impermeable solute on side “o” draws the water toward it. C. Flux of labeled water placed on side “in.”



Figure 12.

A membrane water channel forms a continuum bridging the two bathing solutions. Both side “in” and the channel contain pure water, while side “o” contains an impermeable solute. Rapid equilibrium at the interfaces requires continuity of the chemical potential for water, but the impermeable solute on side “o” creates a discontinuity in its concentration at the “o” interface. These boundary conditions are satisfied when there is an equal and opposite discontinuity in hydrostatic pressure at the “o” interface. Panel A: Plot of hydrostatic pressure profile through the membrane. The pressure drop across the membrane creates a hydraulic flow of water through the membrane even though both bathing media are at atmospheric pressure. Panel B: Equilibrium (J = 0) ensues when the cryptic pressure drop within the membrane is wiped out by the application of a hydrostatic pressure (P – Patm) = ΔII = RTΣCi to side “o.”



Figure 13.

Efflux of solute out of an elementary volume where motion is constrained to the x direction. If the solute velocity = dx/dt, then all solutes contained within the box defined by dxdydz at time t will have passed through the shaded plane in time t + dt.



Figure 14.

Concentration and electrical potential profiles through a membrane.



Figure 15.

Normalized I vs. Ψm plot obtained from Equation 182 for various values of Co while Cin is held fixed. Values for Co/Cin are indicated on the graph. Note that the slope; (conductances) of all curves approach the same limiting value (determined by Equation 186) at infinite Ψm.



Figure 16.

A model channel illustrating the continuum approach with a macroscopic dielectric constant. Channel cross‐sectional area, a(x), varies from the wide‐mouthed vestibule on the left to a narrow, negatively charged, aperture that approximates the size of an ion on the right. The dotted lines show equipotential lines generated by the channel charge; the solid lines show lines of force.

Redrawn from Dani and Levitt 30


Figure 17.

Born energy (dashed lines) for three different ionic radii are obtained from Equation 219. Corresponding Born energy corrected for “image forces” and shown as solid lines are obtained from Equation 221 using the first 20 terms of the series.



Figure 18.

Lipid bilayer energy barriers for a hydrophobic ion with radius = 4.2 Å. Top panel: The upper trace shows a composite consisting of the Born image plus neutral hydrophobic energies. The two lower curves of plot A, “ + Dipole” and “—Dipole,” are plots of dipole energies for cations and anions respectively. Bottom panel shows the composite sum of these energies, one for anions, the other for cations. The negative dipole potential lowers the energy barrier for anions, making them very permeable.

Redrawn from Flewelling and Hubbel 41


Figure 19.

Energy‐barrier profile of gramicidin channel constructed from molecular dynamics simulations by Roux and Karplus. Simulations were accomplished by dividing the channel into four sections labeled I to IV (lower figure), corresponding to sections separated by open circles in energy profile (upper figure). Significance of the different sections: II—transition from bulk water to single file solvation; I—the beginning of single‐file transport; III—single‐file transport; IV—intermolecular region.

Redrawn from Roux and Karplus 120


Figure 20.

Electrical potential profiles in the presence of surface charge. Surface‐charge density on inner surface at left is assumed larger than the corresponding charge density on outer surface at the right. The fast rise in the potential on entering the membrane from either side “in” or “o” is due to a small dipole potential. Notice that the internal transmembrane potential difference is larger than the measured membrane potential.



Figure 21.

Illustration of a simple channel with two binding sites. Each channel can exist in three states: an empty channel Yoo, a channel with C on the inside Yco, and a channel with C on the outside Yoc. The upper triangular sketch shows the transitions among the states. The lower “reaction” scheme shows the reversible transport of Cin to Co.



Figure 22.

Reaction scheme illustrating a single‐occupancy channel and two binding sites, where two solutes, C and B, compete. See Fig. 21 for details.



Figure 23.

Normalized plot of I vs. Ψm (Equation 263) with Cin = Co = K. Values of λ/k−in are indicated in the graph. Channel saturation becomes more evident at lower voltages the larger the λ/k−in (translocation rate compared to dissociation rate). Dotted straight lines show the corresponding linear conductance predicted by corresponding constant‐field equations.



Figure 24.

Reaction scheme illustrating a single‐occupancy channel with n + 2 binding sites. See Figure 21 for details.



Figure 25.

Illustration of a simple carrier where a mobile component Y within the membrane facilitates permeation. Beginning on side “in” the solute Cin combines with an empty reactive carrier site that is exposed on side “in” and is designated as Yoin. The reaction forms a carrier‐solute complex Ycin. Ycin then undergoes a translocation step (conformation change) that exposes the reactive site to side “o,” where it is designated as Yco. The complex Yco then dissociates, releasing Co to side “o.” The free carrier Yoo with the reactive site still facing side “o” will then complete the cycle by translocating its reactive site to side “in,” where it again becomes Yoin.



Figure 26.

Cotransport random carrier (m = n = 1). The cycle begins on side “in” where Ain or Bin combine with its reactive site on component Yooin, forming the complex YAoin or YoBin. This complex reacts with the other solute, forming the ternary complex YABin. YABin undergoes a translocation step, exposing the reactive sites to side “o,” and becomes YABo. YABo dissociates, releasing in a random order Ao or Bo to side “o,” going through the intermediate complexes YAoo or YoBo, before becoming totally in the undissociated form Yooo. Yooo will then translocate them to side “in,” completing the cycle. The association‐dissociation reactions at each boundary are assumed to be in equilibrium. At these boundaries only three out of the four cyclic dissociation constants, Keq, are independent. This is reflected in the terms containing α1 and α2 (see discussion of Equations 331, 332, and 333).



Figure 27.

Reaction cycle used to derive Equation 335. See text.



Figure 28.

Cotransport random: Families of Lineweaver‐Burk plots of 1/JA‐influx versus 1/Ao for increasing values of Bo indicated by the curved arrow. All plots intersect in the second or third quadrant, depending on the relative values of α1 and the product (1+k−1X) (1+k2X). To interpret plots, recall that for each value of Bo, the plot shows a straight line with a y‐axis intercept equal to 1/JmaxA‐influx and an x‐axis intercept equal to −1/KMA‐influx a: α1 < (1 + k−1X) (1+k2X). From Equations 352 and 353, yA = yB > 0 and the intersection lies in the second quadrant… From the changes in the x and y intercepts, it follows that JmaxA‐influx increases with Bo while KMA‐influx decreases. b: If α1 > (1+k−1X) (1+k2X) then yA = yB < 0. The intersection lies below the x axis, in the third quadrant. Both JmaxA‐influx and KMA‐influx will increase with Bo. c: If α1 = (1+k−1X) (1+k2X) then yA = yB = 0. The intersection lies on the x axis. In addition, JmaxA‐influx increases with increasing Bo, while KMA‐influx remains constant, independent of Bo.



Figure 29.

Cotransport sequential. Families of Lineweaver‐Burk plots illustrating graphical criteria for an ordered sequence mechanism for influx with Ao always binding first. a: Plot of 1/JA‐influx versus 1/Ao for increasing values of Bo indicated by the curved arrow. As Bo → ∞, the slope → 0. b: Plot of 1/JB‐influx versus 1/Bo for increasing values of Ao indicated by the curved arrow. All lines intersect on the y axis.



Figure 30.

Countertransport carrier ping‐pong mechanism. Beginning on side “in” the solute Ain or Bin combine with the reactive site of component Yooin. The complex YAin or YBin will then undergo a translocation step, moving the reactive site to side “o.” The complex then dissociates, releasing Ao or Bo to side “o.” The undissociated carrier with the reactive site still facing side “o” will then either translocate its reactive site in the free (Yooo) or bound form (YAo or YBo) to side “in.” Reactions of the solute A and B with the reactive site of component Y are assumed to be in equilibrium.



Figure 31.

Countertransport sequential model with m = n = 1. When A binds to oinY•o or ooY•in, it forms the complex AinY•o or AoY•in. This complex will then bind B on the empty reactive site located on the opposite side of the membrane, forming AinYBo or AoYBin. On the other hand, if B binds first, then the complex oinYBo or ooYBin would be formed before AoYBo or AoYBin. Only three out of the four cyclic dissociation constants, Keq, are independent (see Equations 48 and 49). For each step, the dissociation constant is located closest to the dissociation arrow.



Figure 32.

Countertransport ping‐pong. Lineweaver‐Burk plot for the influx of a solute A for different concentrations of B. a: Plot of stimulation of the influx of A by the counter solute (Bin) on the trans side of the membrane where it binds to the free carrier when there is no slippage, b: Plot of a competitive inhibition. Influx of A is decreased by Bo, which competes for the same carrier site on the same side “o” (cis side) of the membrane.



Figure 33.

Reaction scheme for single‐site channel with energy barriers that can fluctuate between two conformations, Y and Y*. The dashed arrows are redundant; they have been drawn to facilitate an overview of all transport pathways.



Figure 34.

Two plots of flux vs. concentration for singly occupied dynamic channels, with different Po obtained from Equation 394. Values of the parameters are shown on each curve. Neither curve is a rectangular hyperbola. Note the maximum in the upper plot.



Figure 35.

Operation of cycle (Equation 390) shows how channel with very large fluctuating barriers behaves like conventional carrier.

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Robert I. Macey, Teresa F. Moura. Basic Principles of Transport. Compr Physiol 2011, Supplement 31: Handbook of Physiology, Cell Physiology: 181-259. First published in print 1997. doi: 10.1002/cphy.cp140106