---
title: "Membranes and Membrane Transport"
book: "University Biology — Year 1"
subject: biology
language: en
chapter: 7
exercises: 12
source: https://one-course.com/books/biology/3/en/chapter/7-membranes-and-membrane-transport
---

# Chapter 7 — Membranes and Membrane Transport

Drop a red blood [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) into pure water and it swells and bursts within a second; drop it into strong salt and it shrivels. Put it back into plasma and it keeps its biconcave shape for four months, pumping sodium out and potassium in every second of that time. The membrane that does this is five nanometres thick — a film of lipid two molecules deep, threaded with proteins — and it is the boundary across which every exchange of [Chapter 1](https://one-course.com/books/biology/3/en/chapter/1-the-organism-a-system-in-interaction-with-its-environment#ch-b1-organism-environment) finally happens. This chapter describes the membrane’s structure, the physics of what crosses it unaided, the proteins that carry, pump and [channel](#def-b1-membranes-transport-transporters) the rest, the electrical potential this traffic produces, and the vesicles that move what no protein can.

## 7.1 The fluid mosaic

**Definition 7.1 (The plasma membrane).**

The *plasma membrane* — and every membrane of the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) — is a *lipid bilayer* about $5\,\mathrm{nm}$ thick: two sheets of phospholipids ([Chapter 9](https://one-course.com/books/biology/3/en/chapter/9-lipids#ch-b1-lipids)) with their hydrophobic tails facing each other and their polar heads facing the two aqueous sides, with cholesterol between the tails, and *membrane proteins* embedded in it or attached to it. In the *fluid mosaic* model (Singer and Nicolson, 1972) the bilayer is a two-dimensional fluid in which lipids and proteins diffuse laterally, and the proteins are a mosaic of independent units, not a continuous coat. The two faces differ: sugars attached to lipids and proteins face outward only (the *glycocalyx*), and some phospholipids are confined to one leaflet.

![The fluid mosaic. Two leaflets of phospholipids, tails inward, with cholesterol among them; integral proteins spanning the bilayer (a channel with its pore, a carrier), a peripheral protein on one face, and sugar chains on the outer face only.](https://one-course.com/images/onecourse/chapters/biology-3/b1-membranes-transport/fig-7881b5ea9581.svg)

*The [fluid mosaic](#def-b1-membranes-transport-membrane). Two leaflets of phospholipids, tails inward, with cholesterol among them; [integral proteins](#def-b1-membranes-transport-proteins) spanning the bilayer (a [channel](#def-b1-membranes-transport-transporters) with its pore, a carrier), a peripheral protein on one face, and sugar chains on the outer face only.*

![A red blood cell, a platelet and a lymphocyte under the scanning electron microscope (colourised). The red cell’s biconcave shape is held by a protein mesh under its membrane; it survives four months of squeezing through capillaries narrower than itself. Image: National Cancer Institute, public domain.](https://one-course.com/images/onecourse/chapters/biology-3/b1-membranes-transport/img-be58ad120a56.jpg)

*A red blood [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell), a platelet and a lymphocyte under the scanning [electron microscope](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#prop-b1-cell-unit-of-life-microscopes) (colourised). The red [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell)’s biconcave shape is held by a protein mesh under its membrane; it survives four months of squeezing through capillaries narrower than itself. Image: National Cancer Institute, public domain.*

**Proposition 7.2 (Membranes are bilayers and fluids).**

The [plasma membrane](#def-b1-membranes-transport-membrane) is exactly two lipid molecules thick, and its components move within its plane.

**Evidence.** Gorter and Grendel (1925) extracted the lipids of a known number of red blood [cells](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) and spread them as a monolayer on water: the area was twice the [cells](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell)’ total surface, so the membrane is a bilayer. Electron microscopy shows every membrane as two dark lines (the stained heads) around a pale core (the tails), $5\,\mathrm{nm}$ in all. Frye and Edidin (1970) fused a mouse [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) with a human [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) whose surface proteins had been labelled with dyes of two colours: after forty minutes at $37\,{}^{\circ}\mathrm{C}$ the two colours were completely intermixed over the hybrid [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell), and not at $4\,{}^{\circ}\mathrm{C}$, where the lipid is nearly solid. Bleaching a spot of a fluorescent [membrane protein](#def-b1-membranes-transport-membrane) with a laser and watching the fluorescence return as unbleached molecules diffuse in (FRAP) measures the lateral diffusion: $1\,\text{µ}\mathrm{m}^{2}$ in a few seconds for lipids, slower for proteins, and zero for proteins anchored to the [cytoskeleton](https://one-course.com/books/biology/3/en/chapter/6-functional-organization-of-the-eukaryotic-cell#def-b1-eukaryotic-cell-cytoskeleton). ∎

**Definition 7.3 (Membrane proteins).**

*Integral* [membrane proteins](#def-b1-membranes-transport-membrane) span the bilayer with one or more hydrophobic helices and can only be extracted with detergents; *peripheral* proteins are bound to one face. By function: *transporters* ([channels](#def-b1-membranes-transport-transporters), carriers, pumps), *receptors* that bind a signal outside and act inside, *enzymes*, *anchors* linking the [cytoskeleton](https://one-course.com/books/biology/3/en/chapter/6-functional-organization-of-the-eukaryotic-cell#def-b1-eukaryotic-cell-cytoskeleton) to the [extracellular matrix](https://one-course.com/books/biology/3/en/chapter/4-animal-body-plans-and-tissues#def-b1-body-plans-tissues-connective), and *recognition* proteins bearing the sugars by which [cells](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) identify one another. Proteins are half the mass of a typical membrane and nearly all of its function.

## 7.2 Crossing the membrane without help

**Proposition 7.4 (Permeability of the bilayer).**

A pure [lipid bilayer](#def-b1-membranes-transport-membrane) is freely permeable to small non-polar molecules ($\mathrm{O_2}$, $\mathrm{CO_2}$, $\mathrm{N_2}$, steroids), fairly permeable to small uncharged polar molecules (water, urea, ethanol), poorly permeable to larger polar molecules (glucose, amino acids), and practically impermeable to ions ($\mathrm{Na^+}$, $\mathrm{K^+}$, $\mathrm{Cl^-}$, $\mathrm{H^+}$) and to macromolecules. The permeability spans twelve orders of magnitude, from $10^{-2}\,\mathrm{cm}/\mathrm{s}$ for water to $10^{-14}\,\mathrm{cm}/\mathrm{s}$ for $\mathrm{Na^+}$: a hydrophobic core $3\,\mathrm{nm}$ thick lets through what dissolves in oil and stops what carries a charge.

**Theorem 7.5 (Fick’s law of diffusion).**

The net flux $J$ of a solute across a membrane of area $S$ (moles per second) is proportional to the concentration difference across it:

$$
J = P\,S\,(c_{\text{out}} - c_{\text{in}}),
$$

where the *permeability coefficient* $P$ (in $\mathrm{m}/\mathrm{s}$) lumps together the solute’s diffusion coefficient in the membrane, its solubility in the lipid and the membrane’s thickness ($P = DK/\ell$). Diffusion needs no energy and always runs down the gradient; it is *passive transport*.

**Proof.** Inside the membrane the solute diffuses down a linear concentration profile from $Kc_{\text{out}}$ to $Kc_{\text{in}}$ ($K$ the partition coefficient between lipid and water); Fick’s first law in the bulk, $J/S = -D\,\dd c/\dd x$, gives $J/S = DK(c_{\text{out}} -
c_{\text{in}})/\ell$. ∎

**Definition 7.6 (Osmosis, water potential).**

*Osmosis* is the net diffusion of water across a membrane that lets water through but not the solutes, from the solution where water is more concentrated (fewer solutes) to the one where it is less. It is described by the *water potential* $\Psi$, the chemical potential of water expressed as a pressure, zero for pure water at atmospheric pressure:

$$
\Psi = \Psi_s + \Psi_p, \qquad \Psi_s = -RTc_s ,
$$

where $\Psi_s$, the *solute potential*, falls with the total solute concentration $c_s$ (in osmoles per litre, the van ’t Hoff law) and $\Psi_p$, the *pressure potential*, is the hydrostatic pressure above atmospheric. Water moves from higher to lower $\Psi$. At $25\,{}^{\circ}\mathrm{C}$, $RT = 2.48\,\mathrm{MPa}\,\mathrm{L}/\mathrm{mol}$: a $0.3\,\mathrm{osmol}/\mathrm{L}$ solution has $\Psi_s = -0.74\,\mathrm{MPa}$. A solution is *isotonic* to a [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) when no net water moves, *hypotonic* when water enters, *hypertonic* when it [leaves](https://one-course.com/books/biology/3/en/chapter/3-functional-organization-of-a-flowering-plant#def-b1-flowering-plant-organization-organs).

![Red blood cells in an isotonic solution (biconcave discs), in a hypotonic one (swollen to spheres, about to burst) and in a hypertonic one (shrunken and crenated). Water follows its potential; the cell has no wall to resist.](https://one-course.com/images/onecourse/chapters/biology-3/b1-membranes-transport/img-b8494a06b9e5.jpg)

*Red blood [cells](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) in an [isotonic](#def-b1-membranes-transport-osmosis) solution (biconcave discs), in a hypotonic one (swollen to spheres, about to burst) and in a hypertonic one (shrunken and crenated). Water follows its potential; the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) has no wall to resist.*

![Plasmolysis: onion epidermis in a strong salt solution. The protoplast of each cell has lost water and pulled away from the rigid wall, which keeps its shape. In water the protoplast would swell back against the wall and stop, turgid.](https://one-course.com/images/onecourse/chapters/biology-3/b1-membranes-transport/img-a08960e10eb4.jpg)

*Plasmolysis: onion epidermis in a strong salt solution. The protoplast of each [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) has lost water and pulled away from the rigid wall, which keeps its shape. In water the protoplast would swell back against the wall and stop, turgid.*

**Example 7.7 (A red cell in three solutions).**

Plasma is $0.3\,\mathrm{osmol}/\mathrm{L}$: $\Psi_s = -0.74\,\mathrm{MPa}$ on both sides, no net flow. In pure water ($\Psi = 0$) the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell), at $-0.74\,\mathrm{MPa}$ inside, takes up water until its membrane, which can stretch only a few percent, ruptures. In $0.6\,\mathrm{osmol}/\mathrm{L}$ salt it loses water until its inside is as concentrated, at half its volume. A plant [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) in pure water does not burst: as water enters, the wall is stretched and $\Psi_p$ rises until $\Psi_p = -\Psi_s$, the [water potential](#def-b1-membranes-transport-osmosis) inside is zero, and the flow stops with the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) turgid at $0.74\,\mathrm{MPa}$.

**Method 7.8 (Water-potential bookkeeping).**

1. Convert every solute concentration to osmoles (a salt that dissociates into two ions counts twice) and compute $\Psi_s =  -RTc_s$ .
2. Add the pressure term: $\Psi_p = 0$ for a solution in an open vessel or an animal [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) , positive for a turgid plant [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) , negative for water under tension in a [xylem](https://one-course.com/books/biology/3/en/chapter/3-functional-organization-of-a-flowering-plant#def-b1-flowering-plant-organization-tissues) vessel.
3. Water flows toward the lower $\Psi$ . Equilibrium is reached when the two $\Psi$ are equal: by dilution of the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) ’s contents (animal [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) ), or by a rise of pressure (plant [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) ).
4. Read the sign of the result as the direction of flow, and its size as the driving force; the flow rate also depends on the membrane’s water permeability, raised a hundredfold by [aquaporin](#def-b1-membranes-transport-transporters) [channels](#def-b1-membranes-transport-transporters) .

## 7.3 Transport proteins

**Definition 7.9 (Channels, carriers, pumps).**

*Channels* are [integral proteins](#def-b1-membranes-transport-proteins) with a water-filled pore through which a specific ion or small molecule diffuses down its gradient at up to $10^8$ per second; many are *gated*, opening in response to a voltage, a ligand or a mechanical force. *Aquaporins* are water channels. *Carriers* bind their solute on one face, change conformation, and release it on the other, a thousand times a second at most; the glucose transporter of the red [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) is one. Channels and carriers working down a gradient perform *facilitated diffusion*, passive but selective and saturable. *Pumps* are carriers that couple the transport of a solute against its gradient to the hydrolysis of ATP: *primary active transport*. The *sodium–potassium pump* of animal [cells](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) exports three $\mathrm{Na^+}$ and imports two $\mathrm{K^+}$ per ATP; the *proton pump* of plant, fungal and bacterial membranes exports $\mathrm{H^+}$.

![Five ways across. Simple diffusion through the lipid; a channel and a carrier (facilitated diffusion, down the gradient); the sodium–potassium pump (primary active transport, paid in ATP); a symporter that uses the sodium gradient the pump built to drag glucose uphill (secondary active transport).](https://one-course.com/images/onecourse/chapters/biology-3/b1-membranes-transport/fig-27db8edfadc9.svg)

*Five ways across. Simple diffusion through the lipid; a [channel](#def-b1-membranes-transport-transporters) and a carrier ([facilitated diffusion](#def-b1-membranes-transport-transporters), down the gradient); the [sodium–potassium pump](#def-b1-membranes-transport-transporters) ([primary active transport](#def-b1-membranes-transport-transporters), paid in ATP); a [symporter](#prop-b1-membranes-transport-secondary) that uses the sodium gradient the pump built to drag glucose uphill ([secondary active transport](#prop-b1-membranes-transport-secondary)).*

**Proposition 7.10 (Secondary active transport).**

A gradient built by a pump is a store of energy that other carriers spend: a *symporter* moves a solute uphill by coupling it to an ion moving downhill in the same direction (the sodium–glucose transporter of the intestine and kidney; the proton–sucrose transporter of the phloem), an *antiporter* couples it to an ion moving the opposite way (the sodium–calcium exchanger). In animal [cells](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) the currency is the $\mathrm{Na^+}$ gradient; in plants, fungi and bacteria the $\mathrm{H^+}$ gradient.

**Example 7.11 (Kinetics tell the mechanism).**

The flux of glucose into a red [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) rises with the outside concentration, then levels off at a maximum, like an enzyme ([Chapter 13](https://one-course.com/books/biology/3/en/chapter/13-enzymes-and-biochemical-catalysis#ch-b1-enzymes)): a carrier, saturable, with a $K_m$ of about $1.5\,\mathrm{mmol}/\mathrm{L}$; a similar sugar, mannose, competes for it. The flux of urea rises in proportion to its concentration without limit: simple diffusion. The flux of glucose into an intestinal [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) stops when sodium is removed from the lumen, or when the pump is poisoned with ouabain: [secondary active transport](#prop-b1-membranes-transport-secondary).

## 7.4 The membrane potential

**Definition 7.12 (Membrane potential).**

Every living [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) holds an electrical potential difference across its [plasma membrane](#def-b1-membranes-transport-membrane), the *membrane potential* $V_m = V_{\text{in}} - V_{\text{out}}$, negative inside: $-70\,\mathrm{mV}$ in a [neuron](https://one-course.com/books/biology/3/en/chapter/4-animal-body-plans-and-tissues#def-b1-body-plans-tissues-nervous), $-90\,\mathrm{mV}$ in a muscle fibre, $-120\,\mathrm{mV}$ or more in a plant [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell). It arises because the membrane is selectively permeable to ions whose concentrations differ across it.

**Theorem 7.13 (The Nernst equation).**

An ion of charge $z$ at concentrations $c_{\text{out}}$ and $c_{\text{in}}$ is at equilibrium across the membrane — no net flux, though the membrane is permeable to it — when the potential equals its *[equilibrium potential](#thm-b1-membranes-transport-nernst)*

$$
E_{\text{ion}} = \frac{RT}{zF}\,\ln\frac{c_{\text{out}}}{c_{\text{in}}}
= \frac{61.5\,\mathrm{mV}}{z}\,\log_{10}\frac{c_{\text{out}}}{c_{\text{in}}}
\quad (37\,{}^{\circ}\mathrm{C}).
$$

For $\mathrm{K^+}$ at $5\,\mathrm{mmol}/\mathrm{L}$ outside and $140\,\mathrm{mmol}/\mathrm{L}$ inside, $E_K = 61.5\log(5/140) = -89\,\mathrm{mV}$; for $\mathrm{Na^+}$ at $145\,$ outside and $12\,$ inside, $E_{Na} = +67\,\mathrm{mV}$.

**Proof.** Moving one mole of the ion from outside to inside changes the free energy by the sum of a concentration term and an electrical term:

$$
\Delta G = RT\ln\frac{c_{\text{in}}}{c_{\text{out}}} + zFV_m .
$$

At equilibrium $\Delta G = 0$, which gives $V_m = (RT/zF)\ln(c_{\text{out}}/c_{\text{in}})$. Converting to base-ten logarithms and inserting $R = 8.314\,\mathrm{J}\,\mathrm{mol}^{-1}\,\mathrm{K}^{-1}$, $T = 310\,\mathrm{K}$, $F = 96\,485\,\mathrm{C}/\mathrm{mol}$ gives the numerical form. ∎

**Proposition 7.14 (Origin of the resting potential).**

The resting membrane is far more permeable to $\mathrm{K^+}$ (through open potassium [channels](#def-b1-membranes-transport-transporters)) than to $\mathrm{Na^+}$; $\mathrm{K^+}$ leaks out down its concentration gradient, leaving the inside negative, until the electrical pull back nearly balances the leak. The resting potential therefore lies close to $E_K$, pulled a little toward $E_{Na}$ by the small sodium leak (the Goldman equation weights each ion’s Nernst term by its permeability). The pump sustains the gradients that the leaks would otherwise dissipate; it also contributes a few millivolts directly, since it moves three charges out for two in.

**Evidence.** In a squid axon the resting potential follows $E_K$ when the outside potassium is varied over a wide range (a straight line of slope $58\,\mathrm{mV}$ per decade at high concentrations), and departs from it at low concentrations exactly as a small sodium permeability predicts. Blocking the pump with ouabain [leaves](https://one-course.com/books/biology/3/en/chapter/3-functional-organization-of-a-flowering-plant#def-b1-flowering-plant-organization-organs) the potential almost unchanged for minutes, then lets it decay over hours as the gradients run down: the potential is a diffusion potential, the pump its long-term support. ∎

![Resting potential against external potassium. At high potassium the membrane behaves as a potassium electrode and follows the Nernst line; at low potassium the small sodium leak pulls it above E_K.](https://one-course.com/images/onecourse/chapters/biology-3/b1-membranes-transport/fig-e8fac9a383b3.svg)

*Resting potential against external potassium. At high potassium the membrane behaves as a potassium electrode and follows the Nernst line; at low potassium the small sodium leak pulls it above $E_K$.*

**Example 7.15 (Where the pump’s energy goes).**

Pumping one $\mathrm{Na^+}$ out against $12\,$ to $145\,\mathrm{mmol}/\mathrm{L}$ and $-70\,\mathrm{mV}$ costs $RT\ln(145/12) + F\times 0.070 = 6.4 + 6.8 =
13.2\,\mathrm{kJ}/\mathrm{mol}$; three of them, $40\,\mathrm{kJ}$, plus two $\mathrm{K^+}$ in at $RT\ln(140/5) - F\times 0.070 = 8.6 - 6.8 =
1.8\,\mathrm{kJ}/\mathrm{mol}$ each: $43\,\mathrm{kJ}$ per cycle against the $50\,\mathrm{kJ}$ of one ATP under cellular conditions. The pump runs near its thermodynamic limit, and it consumes a third of a resting animal’s ATP, two thirds in the brain.

## 7.5 Bulk transport

**Proposition 7.16 (Vesicular transport).**

Macromolecules and particles cross the [plasma membrane](#def-b1-membranes-transport-membrane) without passing through it: by *exocytosis*, a vesicle fuses with the membrane and empties outward; by *[endocytosis](https://one-course.com/books/biology/3/en/chapter/6-functional-organization-of-the-eukaryotic-cell#def-b1-eukaryotic-cell-endocytosis)*, the membrane engulfs material into a vesicle ([Chapter 6](https://one-course.com/books/biology/3/en/chapter/6-functional-organization-of-the-eukaryotic-cell#ch-b1-eukaryotic-cell)). Both require ATP, both move membrane as well as contents, and both preserve the sidedness of the membrane. The three routes across a membrane — through the lipid, through a protein, inside a vesicle — are selective in three ways: by solubility, by molecular recognition, by the choice of what is engulfed.

**Example 7.17 (Cholesterol delivery).**

Cholesterol travels in the blood inside protein-coated particles too large for any carrier. A [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) needing cholesterol displays receptors for the particle; particle and receptor gather in a coated pit, which pinches off, and the vesicle delivers the particle to a [lysosome](https://one-course.com/books/biology/3/en/chapter/6-functional-organization-of-the-eukaryotic-cell#def-b1-eukaryotic-cell-endomembrane) where the cholesterol is freed. A person whose receptors are defective cannot clear the particles: blood cholesterol doubles and arteries clog in early adulthood. One [membrane protein](#def-b1-membranes-transport-membrane), one disease.

## 7.6 Exercises

**Exercise 7.1 ★.**

Describe the [fluid mosaic model](#def-b1-membranes-transport-membrane) in four sentences: the lipids, the proteins, the fluidity, the asymmetry.

**Solution of Exercise 7.1.**

A bilayer of phospholipids, tails inward, with cholesterol, forms a $5\,\mathrm{nm}$ sheet. Proteins are embedded across it or attached to one face, as separate units. Lipids and most proteins diffuse laterally in the plane, which is a two-dimensional fluid. The two leaflets differ in lipid composition, and the sugar chains are on the outer face only.

**Exercise 7.2 ★.**

Rank $\mathrm{O_2}$, glucose, $\mathrm{Na^+}$, water and ethanol by their permeability through a pure [lipid bilayer](#def-b1-membranes-transport-membrane), and explain the order.

**Solution of Exercise 7.2.**

$\mathrm{O_2}$ (small, non-polar) $>$ ethanol (small, weakly polar) $>$ water (small, polar) $\gg$ glucose (large, polar) $\gg$ $\mathrm{Na^+}$ (charged, hydrated). Permeability falls with polarity, size and above all charge, because the solute must dissolve in the hydrophobic core.

**Exercise 7.3 ★.**

Compute the [solute potential](#def-b1-membranes-transport-osmosis) of a $0.1\,\mathrm{mol}/\mathrm{L}$ sucrose solution and of a $0.1\,\mathrm{mol}/\mathrm{L}$ NaCl solution at $25\,{}^{\circ}\mathrm{C}$.

**Solution of Exercise 7.3.**

Sucrose: $-2.48\times 0.1 = -0.25\,\mathrm{MPa}$. NaCl dissociates into two ions, $0.2\,\mathrm{osmol}/\mathrm{L}$: $-0.50\,\mathrm{MPa}$.

**Exercise 7.4 ★.**

Compute the Nernst potential of $\mathrm{Cl^-}$ at $120\,\mathrm{mmol}/\mathrm{L}$ outside and $4\,\mathrm{mmol}/\mathrm{L}$ inside, at $37\,{}^{\circ}\mathrm{C}$. Is chloride at equilibrium in a [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) at $-89\,\mathrm{mV}$?

**Solution of Exercise 7.4.**

$E_{Cl} = (61.5/(-1))\log(120/4) = -61.5\times 1.48 = -91\,\mathrm{mV}$. At $-89\,\mathrm{mV}$ chloride is within $2\,\mathrm{mV}$ of equilibrium: it is distributed passively.

**Exercise 7.5 ★★.**

Gorter and Grendel used $4.7 \times 10^{9}$ red [cells](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) of surface $100\,\text{µ}\mathrm{m}^{2}$ each and obtained a lipid monolayer of $0.92\,\mathrm{m}^{2}$. Compute the ratio of monolayer area to [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) surface and conclude.

**Solution of Exercise 7.5.**

[Cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) surface $4.7 \times 10^{9}\times 10^{-10}\,\mathrm{m^2} =
0.47\,\mathrm{m}^{2}$; ratio $0.92/0.47 = 1.96 \approx 2$: the lipid covers the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) twice, so the membrane is a bilayer.

**Exercise 7.6 ★★.**

A plant [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) has a [solute potential](#def-b1-membranes-transport-osmosis) of $-0.9\,\mathrm{MPa}$ and a [pressure potential](#def-b1-membranes-transport-osmosis) of $0.5\,\mathrm{MPa}$. It is placed in a solution of [solute potential](#def-b1-membranes-transport-osmosis) $-0.6\,\mathrm{MPa}$ in an open dish. Which way does water move, and what is the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell)’s [pressure potential](#def-b1-membranes-transport-osmosis) at equilibrium (assume its [solute potential](#def-b1-membranes-transport-osmosis) does not change)?

**Solution of Exercise 7.6.**

[Cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) $\Psi = -0.9 + 0.5 = -0.4\,\mathrm{MPa}$; solution $-0.6\,\mathrm{MPa}$: water [leaves](https://one-course.com/books/biology/3/en/chapter/3-functional-organization-of-a-flowering-plant#def-b1-flowering-plant-organization-organs) the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell). Equilibrium when $\Psi_{\text{cell}} = -0.6$: $\Psi_p = -0.6 + 0.9 = 0.3\,\mathrm{MPa}$; the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) loses some [turgor](https://one-course.com/books/biology/3/en/chapter/6-functional-organization-of-the-eukaryotic-cell#def-b1-eukaryotic-cell-plantcell) but stays turgid.

**Exercise 7.7 ★★.**

Glucose uptake by a [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) is measured at increasing external concentrations: $1\,\mathrm{mmol}/\mathrm{L}\text{, }2\,\mathrm{mmol}/\mathrm{L}\text{, }5\,\mathrm{mmol}/\mathrm{L}\text{, }10\,\mathrm{mmol}/\mathrm{L}\text{ and }20\,\mathrm{mmol}/\mathrm{L}$ give $0.4\,\text{µ}\mathrm{mol}/\mathrm{min}\text{, }0.67\,\text{µ}\mathrm{mol}/\mathrm{min}\text{, }1.0\,\text{µ}\mathrm{mol}/\mathrm{min}\text{, }1.25\,\text{µ}\mathrm{mol}/\mathrm{min}\text{ and }1.43\,\text{µ}\mathrm{mol}/\mathrm{min}$. Show that the uptake saturates, estimate the maximum rate and the concentration giving half of it, and name the mechanism.

**Solution of Exercise 7.7.**

Doubling the concentration from 10 to 20 raises the rate by only $15\,\%$: saturation. Plotting $1/v$ against $1/c$ (or noting that $v = 1.0$ at $5\,\mathrm{mmol}/\mathrm{L}$ and about $1.67$ at infinite $c$) gives $V_{\max} \approx 1.7\,\text{µ}\mathrm{mol}/\mathrm{min}$ and half-maximum at about $3\,\mathrm{mmol}/\mathrm{L}$: [facilitated diffusion](#def-b1-membranes-transport-transporters) by a carrier.

**Exercise 7.8 ★★.**

Explain why the resting potential of a [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) is close to $E_K$ and not to $E_{Na}$, and what would happen to it if sodium [channels](#def-b1-membranes-transport-transporters) suddenly opened.

**Solution of Exercise 7.8.**

At rest the membrane’s permeability is dominated by open potassium [channels](#def-b1-membranes-transport-transporters); potassium leaks out until the inside is negative enough to hold it, i.e. near $E_K$. If sodium [channels](#def-b1-membranes-transport-transporters) opened, the permeability would be dominated by sodium and the potential would swing toward $E_{Na}$, about $+60\,\mathrm{mV}$ — the action potential.

**Exercise 7.9 ★★.**

The sodium–glucose [symporter](#prop-b1-membranes-transport-secondary) carries two $\mathrm{Na^+}$ per glucose. With $\mathrm{Na^+}$ at $145\,\mathrm{mmol}/\mathrm{L}$ outside and $12\,$ inside and $V_m = -60\,\mathrm{mV}$, compute the free energy available from the two sodium ions and the maximum glucose concentration ratio (inside/outside) the [symporter](#prop-b1-membranes-transport-secondary) can build at $37\,{}^{\circ}\mathrm{C}$.

**Solution of Exercise 7.9.**

Per mole $\mathrm{Na^+}$ in: $RT\ln(12/145) + F(-0.060) = -6.4 - 5.8
= -12.2\,\mathrm{kJ}$; two: $-24.4\,\mathrm{kJ}$. Glucose uphill costs $RT\ln(c_{\text{in}}/c_{\text{out}})$; equality gives $\ln(c_{\text{in}}/c_{\text{out}}) = 24\,400/2577 = 9.5$, a ratio of about $13\,000$.

**Exercise 7.10 ★★★.**

A [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) is cooled to $4\,{}^{\circ}\mathrm{C}$. Predict, with reasons, the effects on: membrane fluidity, simple diffusion of $\mathrm{O_2}$, the pump, the ion gradients over hours, the [membrane potential](#def-b1-membranes-transport-potential), and the volume of the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell). (Consider that the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell)’s proteins exert an osmotic pull that the sodium gradient normally balances.)

**Solution of Exercise 7.10.**

Fluidity falls (tails pack, the bilayer approaches a gel); oxygen diffusion slows but continues; the pump, an enzyme, nearly stops; the leaks continue, so over hours sodium enters and potassium [leaves](https://one-course.com/books/biology/3/en/chapter/3-functional-organization-of-a-flowering-plant#def-b1-flowering-plant-organization-organs) and the gradients decay; the potential falls toward zero as the gradients fade; with sodium entering and the proteins’ osmotic pull no longer balanced, water enters and the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) swells — cold-stored [cells](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) and [organs](https://one-course.com/books/biology/3/en/chapter/2-functional-organization-of-a-mammal#def-b1-mammal-organization-organ) swell for exactly this reason.

**Exercise 7.11 ★★★.**

Frye and Edidin’s fused [cells](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) showed complete mixing of surface proteins at $37\,{}^{\circ}\mathrm{C}$ in $40\,\mathrm{min}$, none at $4\,{}^{\circ}\mathrm{C}$, and none at $37\,{}^{\circ}\mathrm{C}$ when ATP synthesis was blocked — the last result was later shown to be wrong. Say what each result would imply about the mechanism of mixing, and why the correct version (mixing does not need ATP) matters for the [fluid mosaic model](#def-b1-membranes-transport-membrane).

**Solution of Exercise 7.11.**

Mixing at $37\,{}^{\circ}\mathrm{C}$ but not at $4\,{}^{\circ}\mathrm{C}$ implies a temperature-dependent process — diffusion in a fluid that becomes solid in the cold. No mixing without ATP would imply an active, energy-driven redistribution (motors, vesicle cycling) rather than diffusion. The corrected result, that mixing needs no ATP, is what the [fluid mosaic model](#def-b1-membranes-transport-membrane) requires: proteins move by thermal diffusion in a two-dimensional fluid, without the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) doing work.

**Exercise 7.12 ★★★.**

“The [membrane potential](#def-b1-membranes-transport-potential) is a by-product of the ion gradients, not something the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) builds directly.” Discuss in a paragraph, with the [Nernst equation](#thm-b1-membranes-transport-nernst), the pump, and the number of ions actually needed to charge the membrane (a capacitance of $1\,\text{µ}\mathrm{F}/\mathrm{cm}^{2}$).

**Solution of Exercise 7.12.**

The potential is the value at which the electrical force balances the diffusion of the most permeant ion, given by Nernst; the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) sets the gradients (by the pump) and the permeabilities (by its [channels](#def-b1-membranes-transport-transporters)), and the potential follows. A membrane of $1\,\text{µ}\mathrm{F}/\mathrm{cm}^{2}$ at $-70\,\mathrm{mV}$ carries $7 \times 10^{-8}\,\mathrm{C}/\mathrm{cm}^{2}$, about $4 \times 10^{11}$ ions per square centimetre — for a $20\,\text{µ}\mathrm{m}$ [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) some $10^5$ ions, against $10^{10}$ potassium ions inside: a negligible fraction of the ions, moving a negligible distance, charges the capacitor. The gradients are the store; the potential is the reading on it, and the pump’s direct contribution (three charges out for two in) is a few millivolts.

## 7.7 Problem: The Enterocyte

**Problem 7.1.**

Weekend problem — the cell that pulls glucose out of the gut: pump, symporter, carrier and the Nernst equation, ending on the highest glucose ratio the cell can build

An intestinal epithelial [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) (enterocyte) is $25\,\text{µ}\mathrm{m}$ tall and $5\,\text{µ}\mathrm{m}$ wide; its apical membrane faces the lumen, its basolateral membrane the blood. Its [cytosol](https://one-course.com/books/biology/3/en/chapter/6-functional-organization-of-the-eukaryotic-cell#def-b1-eukaryotic-cell-organelle) holds $12\,\mathrm{mmol}/\mathrm{L}$ of $\mathrm{Na^+}$ and $140\,\mathrm{mmol}/\mathrm{L}$ of $\mathrm{K^+}$; the lumen and the blood hold $145\,\mathrm{mmol}/\mathrm{L}$ of $\mathrm{Na^+}$ and $5\,\mathrm{mmol}/\mathrm{L}$ of $\mathrm{K^+}$. The [membrane potential](#def-b1-membranes-transport-potential) is $-60\,\mathrm{mV}$. Take $T = 310\,\mathrm{K}$, $R = 8.314\,\mathrm{J}\,\mathrm{mol}^{-1}\,\mathrm{K}^{-1}$, $F = 96\,500\,\mathrm{C}/\mathrm{mol}$, $RT/F = 26.7\,\mathrm{mV}$, and $\Delta G_{\text{ATP}} = -50\,\mathrm{kJ}/\mathrm{mol}$ in the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell).

**Part I — The gradients.**

1. Compute the Nernst potentials of $\mathrm{Na^+}$ and $\mathrm{K^+}$ .
2. Which ion is nearer equilibrium at $-60\,\mathrm{mV}$ ? What does this say about the resting permeabilities?
3. Compute the free energy change for one mole of $\mathrm{Na^+}$ entering the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) , and for one mole of $\mathrm{K^+}$ leaving.
4. Compute the free energy cost of one pump cycle ( $3\,\mathrm{Na^+}$ out, $2\,\mathrm{K^+}$ in) and compare with the energy of one ATP. What is the efficiency?
5. The pump is on the basolateral membrane only. Explain why this placement is necessary for the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) to move glucose from lumen to blood.

**Part II — Glucose uphill.** The apical membrane carries a [symporter](#prop-b1-membranes-transport-secondary) taking in one glucose with two $\mathrm{Na^+}$; the basolateral membrane carries a glucose carrier ([facilitated diffusion](#def-b1-membranes-transport-transporters)).

6. Compute the free energy released by two $\mathrm{Na^+}$ entering the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) .
7. Write the free energy needed to move one glucose from the lumen (concentration $c_L$ ) to the [cytosol](https://one-course.com/books/biology/3/en/chapter/6-functional-organization-of-the-eukaryotic-cell#def-b1-eukaryotic-cell-organelle) ( $c_C$ ).
8. At the [symporter](#prop-b1-membranes-transport-secondary) ’s limit the two are equal. Compute the maximum ratio $c_C/c_L$ .
9. If the lumen falls to $0.1\,\mathrm{mmol}/\mathrm{L}$ of glucose late in absorption, what cytosolic concentration can the [symporter](#prop-b1-membranes-transport-secondary) still maintain?
10. Blood glucose is $5\,\mathrm{mmol}/\mathrm{L}$ . Explain how glucose [leaves](https://one-course.com/books/biology/3/en/chapter/3-functional-organization-of-a-flowering-plant#def-b1-flowering-plant-organization-organs) the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) into the blood without any further energy.
11. Why is the [symporter](#prop-b1-membranes-transport-secondary) placed on the apical and the carrier on the basolateral membrane, and not the reverse?
12. Oral rehydration solutions for cholera contain glucose and salt. Explain, from the [symporter](#prop-b1-membranes-transport-secondary) , why the glucose makes the salt — and the water — be absorbed.

**Part III — Counting the traffic.** The [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) absorbs $2 \times 10^{-15}\,\mathrm{mol}$ of glucose per second.

13. Compute the $\mathrm{Na^+}$ entering per second through the [symporters](#prop-b1-membranes-transport-secondary) , and the number of pump cycles per second needed to export it.
14. Compute the ATP spent per second on this export, and the fraction of the glucose absorbed that would have to be respired to pay for it (about 30 ATP per glucose).
15. Each [symporter](#prop-b1-membranes-transport-secondary) cycles $50\,$ times per second. How many [symporters](#prop-b1-membranes-transport-secondary) does the apical membrane need? The apical membrane is $25\,\text{µ}\mathrm{m}^{2}$ with microvilli multiplying it by $20\,$ : compute the [symporter](#prop-b1-membranes-transport-secondary) density per square micrometre.
16. The $\mathrm{Na^+}$ that enters brings water by [osmosis](#def-b1-membranes-transport-osmosis) . Two $\mathrm{Na^+}$ and one glucose, with the accompanying $\mathrm{Cl^-}$ , are four osmoles; the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) keeps its osmolarity at $0.3\,\mathrm{osmol}/\mathrm{L}$ . Compute the volume of water that must follow the solutes per second, and the time it would take to double the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) ’s volume if the water did not leave through the basolateral membrane.
17. Compute the number of water molecules absorbed per glucose molecule ( $18\,\mathrm{g}/\mathrm{mol}$ , density $1\,\mathrm{g}/\mathrm{mL}$ ).

**Part IV — Charging the membrane.** The membrane is a capacitor of $1\,\text{µ}\mathrm{F}/\mathrm{cm}^{2}$.

18. Compute the area of the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) ’s membrane (take a box $25\,\text{µ}\mathrm{m}$ by $5\,\text{µ}\mathrm{m}$ by $5\,\text{µ}\mathrm{m}$ without microvilli) and its capacitance.
19. Compute the charge needed to hold $-60\,\mathrm{mV}$ and the number of monovalent ions it represents.
20. Compute the number of excess ions per square micrometre of membrane that this represents.
21. Compute the number of $\mathrm{K^+}$ ions in the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) , and the fraction that must leave to charge the membrane. Comment.
22. The pump moves a net charge out on each cycle. Using question 13, compute the current it carries and the change in potential it would produce in one second if nothing else moved. Why does the potential not in fact run away?
23. A drug blocks the pump. Predict, in order, the changes in the sodium gradient, the glucose absorption, the [membrane potential](#def-b1-membranes-transport-potential) and the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) volume.
24. A drug blocks the potassium [channels](#def-b1-membranes-transport-transporters) instead. Predict the change in the resting potential.
25. State the result: the maximum glucose concentration ratio the enterocyte can build, and the two [membrane proteins](#def-b1-membranes-transport-membrane) that make it possible.

**Solution of Problem 7.1.**

**1.** $E_{Na} = 26.7\ln(145/12) = +66\,\mathrm{mV}$; $E_K =
26.7\ln(5/140) = -89\,\mathrm{mV}$. **2.** $\mathrm{K^+}$ ($29\,\mathrm{mV}$ away) is nearer than $\mathrm{Na^+}$ ($126\,\mathrm{mV}$ away): the membrane is much more permeable to $\mathrm{K^+}$. **3.** $\mathrm{Na^+}$ in: $RT\ln(12/145) + F(-0.060) = -6.4 -
5.8 = -12.2\,\mathrm{kJ}/\mathrm{mol}$. $\mathrm{K^+}$ out: $RT\ln(5/140) -
F(-0.060) = -8.6 + 5.8 = -2.8\,\mathrm{kJ}/\mathrm{mol}$. **4.** Cost $= 3\times 12.2 + 2\times 2.8 = 42.2\,\mathrm{kJ}$ per cycle against $50\,\mathrm{kJ}$ per ATP: efficiency $84\,\%$. **5.** The pump keeps cytosolic sodium low; the [symporter](#prop-b1-membranes-transport-secondary) on the apical face uses the inward sodium gradient to pull glucose in from the lumen; the pump must be on the other face so that the sodium it exports goes to the blood, not back into the lumen, which would leave the lumen’s sodium to recycle and the net movement zero. **6.** $2\times 12.2 = 24.4\,\mathrm{kJ}/\mathrm{mol}$ of glucose. **7.** $\Delta G = RT\ln(c_C/c_L)$ (glucose is uncharged). **8.** $\ln(c_C/c_L) = 24\,400/2577 = 9.47$; $c_C/c_L =
1.3 \times 10^{4}$. **9.** Up to $1300\,\mathrm{mmol}/\mathrm{L}$ in principle: the ratio is never reached because glucose [leaves](https://one-course.com/books/biology/3/en/chapter/3-functional-organization-of-a-flowering-plant#def-b1-flowering-plant-organization-organs) through the basolateral carrier; in practice the [cytosol](https://one-course.com/books/biology/3/en/chapter/6-functional-organization-of-the-eukaryotic-cell#def-b1-eukaryotic-cell-organelle) stays near $5\text{ to }20\,\mathrm{mmol}/\mathrm{L}$. **10.** The [cytosol](https://one-course.com/books/biology/3/en/chapter/6-functional-organization-of-the-eukaryotic-cell#def-b1-eukaryotic-cell-organelle) (above $5\,\mathrm{mmol}/\mathrm{L}$) is more concentrated than the blood; the carrier lets glucose diffuse down that gradient into the blood, passively. **11.** Reversed, the [symporter](#prop-b1-membranes-transport-secondary) would pump glucose from the blood into the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) and the carrier would let it leak into the lumen: the gut would secrete glucose. Polarity of the transporters is the direction of absorption. **12.** The [symporter](#prop-b1-membranes-transport-secondary) takes up sodium only with glucose; glucose in the lumen therefore drives sodium absorption through it even when the cholera toxin blocks other routes; chloride follows sodium electrically and water follows the salt osmotically. **13.** $4 \times 10^{-15}\,\mathrm{mol}/\mathrm{s}$ of $\mathrm{Na^+}$, i.e. $2.4 \times 10^{9}$ ions per second; at three per cycle, $8 \times 10^{8}$ cycles per second. **14.** $8 \times 10^{8}$ ATP per second $= 1.3 \times 10^{-15}\,\mathrm{mol}/\mathrm{s}$; at 30 ATP per glucose, $4.4 \times 10^{-17}\,\mathrm{mol}/\mathrm{s}$ of glucose, $2.2\,\%$ of the glucose absorbed. **15.** $1.2 \times 10^{9}$ glucose per second at 50 per [symporter](#prop-b1-membranes-transport-secondary): $2.4 \times 10^{7}$ [symporters](#prop-b1-membranes-transport-secondary); membrane $500\,\text{µ}\mathrm{m}^{2}$: $48\,000$ per square micrometre, i.e. one per $20\,\mathrm{nm}$ square — a membrane packed with transporters. **16.** Osmoles entering: $4\times2 \times 10^{-15}\,\mathrm{mol}/\mathrm{s} =
8 \times 10^{-15}\,\mathrm{osmol}/\mathrm{s}$; at $0.3\,\mathrm{osmol}/\mathrm{L}$, water $= 2.7 \times 10^{-14}\,\mathrm{L}/\mathrm{s} = 27\,\text{µ}\mathrm{m}^{3}/\mathrm{s}$. [Cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) volume $25\times 5\times 5 = 625\,\text{µ}\mathrm{m}^{3}$: doubled in $23\,\mathrm{s}$. The water [leaves](https://one-course.com/books/biology/3/en/chapter/3-functional-organization-of-a-flowering-plant#def-b1-flowering-plant-organization-organs) as fast as it enters, across the basolateral membrane, and this is how the gut absorbs water. **17.** $2.7 \times 10^{-14}\,\mathrm{L}/\mathrm{s}$ of water is $1.5 \times 10^{-12}\,\mathrm{mol}/\mathrm{s}$; per glucose ($2 \times 10^{-15}\,\mathrm{mol}/\mathrm{s}$): about 750 water molecules. **18.** $S = 2(25\times 5 + 25\times 5 + 5\times 5) =
550\,\text{µ}\mathrm{m}^{2} = 5.5 \times 10^{-6}\,\mathrm{cm}^{2}$; $C = 5.5 \times 10^{-12}\,\mathrm{F}$. **19.** $Q = CV = 5.5\times 10^{-12}\times 0.060 =
3.3 \times 10^{-13}\,\mathrm{C}$; $/1.6 \times 10^{-19} = 2.1 \times 10^{6}$ ions. **20.** $2.1 \times 10^{6}/550 = 3800$ ions per square micrometre, one excess charge per $16\,\mathrm{nm}$ square of membrane. **21.** $\mathrm{K^+}$ in the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell): $0.140\,\mathrm{mol/L}\times
6.25 \times 10^{-13}\,\mathrm{L}\times6 \times 10^{23} = 5.3 \times 10^{10}$ ions; the charge needs $0.004\,\%$ of them: the concentrations are unchanged by charging the membrane. **22.** Net charge per cycle $+e$ out: current $8 \times 10^{8}\times
1.6 \times 10^{-19} = 1.3 \times 10^{-10}\,\mathrm{A}$; in one second the membrane would gain $1.3 \times 10^{-10}\,\mathrm{C}$, i.e. $\Delta V = Q/C = 24\,\mathrm{V}$. It does not, because every charge pumped out is immediately balanced by ions flowing back through [channels](#def-b1-membranes-transport-transporters) (mainly $\mathrm{K^+}$ leaking in less, or $\mathrm{Na^+}$ re-entering): the membrane’s conductance shorts the pump’s current, and the pump adds only millivolts. **23.** Cytosolic sodium rises over minutes; the [symporter](#prop-b1-membranes-transport-secondary) loses its driving force and glucose absorption stops; the potential decays toward zero as the gradients run down; sodium and water enter and the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) swells. **24.** With the potassium leak blocked, the sodium leak dominates: the potential moves toward $E_{Na}$, becoming much less negative (depolarisation). **25.** A ratio of about $13\,000$ ([cytosol](https://one-course.com/books/biology/3/en/chapter/6-functional-organization-of-the-eukaryotic-cell#def-b1-eukaryotic-cell-organelle) over lumen), set by two sodium ions per glucose at $-60\,\mathrm{mV}$; made possible by the [sodium–potassium pump](#def-b1-membranes-transport-transporters) (basolateral) and the sodium–glucose [symporter](#prop-b1-membranes-transport-secondary) (apical).