Physical Biology of the Cell

Rob Phillips, Jane Kondev, Julie Theriot

Chapter 12

The Mathematics of Water - all with Video Answers

Educators

Chapter Questions

Problem 1

Bacteria use swimming to seek out food. Imagine that the bacterium is in a region of low food concentration. For the bacterium to profit from swimming to a region with more food, it has to reach there before diffusion of food molecules makes the concentrations in the two regions the same. Here we find the smallest distance that a bacterium needs to swim so it can outrun diffusion.
(a) Make a plot in which you sketch the distance traveled by a bacterium swimming at a constant velocity $v$ as a function of time $t,$ and the distance over which a food molecule will diffuse in that same time. Indicate on the plot the smallest time and the smallest distance that the bacterium needs to swim to outrun diffusion.
(b) Make a numerical estimate for these minimum times and distances for an $E$ coli swimming at a speed of $30 \mu \mathrm{m} / \mathrm{s}$. The diffusion constant of a typical food molecule is roughly $500 \mu \mathrm{m}^{2} / \mathrm{s}$
(c) Estimate the number of ATP molecules the bacterium must consume (hydrolyze) per second in order to travel at this speed, assuming that all of the energy usage goes into overcoming fluid drag. The amount of energy released from one ATP molecule is approximately $20 k_{\mathrm{B}} T .$ Note that the bacterial flagellar motor is actually powered by a proton gradient and this estimate focuses on the ATP equivalents associated with overcoming fluid drag.

Sana Riaz

Sana Riaz

Numerade Educator

Problem 2

In this problem, we examine the hydrostatic pressure in a fluid at rest.
(a) Consider a small fluid element $\Delta x \times \Delta y \times \Delta z$ at rest. Write down the balance of forces on the fluid element due to the fluid pressure $p$ and the gravitational pull of the Earth. Show that this leads to the differential equation
\[
\nabla p=\rho \mathbf{g}
\]
where $g$ is the acceleration due to gravity.
(b) Solve the differential equation derived in (a) assuming a uniform fluid density. Show that the pressure in the fluid is given by $p(z)=p_{0}-\rho g z,$ where the z-axis is in the direction opposite to g.
(c) Estimate the atmospheric pressure. (Hint: Look up the density of air and make a reasonable guess for the height of the atmosphere.)
(d) If you raise your arm above your head as the doctor is measuring your blood pressure, how much will the measurement change compared with when you keep your arm level with the heart. Try it next time you are at the doctor's officel

Sana Riaz

Numerade Educator

Problem 3

(a) Use the Internet to search for "Emiliana huxleyi bloom" and enjoy the images from space of the ocean blooms produced by these coccolithophores. One of our favorite pictures can be found on the book's website.
(b) By examining a scanning electron micrograph of Emiliana huxleyl, estimate its size and mass.
(c) Assuming that the density of a hypothetical cell is
1.3 times that of water, work out its average height from the bottom of a beaker filled with air and filled with water at room temperature. Plot the average height as a function of the cell radius.
(d) Use the result from (c) to deduce a length scale at which the effects of gravity and thermal forces are comparable. Consider the gravitational energy released when a spherical particle drops over a distance equal to its radius, and compare it with the thermal energy.

Sana Riaz

Numerade Educator

Problem 4

(a) At high Reynolds number, the drag force on a moving object becomes largely independent of the viscosity of the fluid. Given that the drag force in this case is a function of the fluid density $\rho,$ the object size $R,$ and its speed $v,$ use dimensional analysis to obtain a formula for the drag force.
(b) From a sheet of paper, cut out two circles, one with a radius twice as large as the other, and using scissors and some Scotch tape turn them into two cones as shown in Figure $12.19(\mathrm{A}) .$ Stand on a chair and drop the two cones from the same height, pointy end down, as shown in Figure $12.19(\mathrm{B}) .$ You will observe that after a short period, both cones reach a terminal speed. Which cone is moving faster? Explain your observation based on the formula derived in (a). (Hint: Derive a formula for the terminal speed as a function of the cone size by balancing the drag force and the force of gravity.)

Sana Riaz

Numerade Educator

Problem 5

A molecule of DNA (length $L$ ) is attached at both ends to beads via the two strands of the double helix, so it is torsionally constrained. Assume that you have twisted one bead relative to the other by a total angle $\phi,$ and you keep the molecule under tension to prevent it from supercoiling. Now you attach a small bead of radius $R$ to both the strands and break one of the strands below the tiny bead, so that the molecule can unwind by rotating around the single bond swivel in the unbroken strand as shown in Figure $10.38 .$ To understand this experiment, we need to consider the drag forces on the rotating bead.
(a) Show that the drag coefficient for a sphere of radius $R$ rotating at angular velocity es is given by $K \eta R^{3}$, where $K$ is a numerical factor. For rotational motion, the drag coefficient relates the angular velocity to the frictional torque. Use the approach that led to the Stokes formula in the chapter. The idea is to say that the size of the viscous stresses is $v / R=\omega$ and the area over which they act is $R^{2}$. Combine these two to get the force, and then the torque.
(b) When DNA is highly twisted, the torque as a function of twist angle $\phi$ is constant at a value of roughly $\tau=33 \mathrm{pNnm}$ Use the data shown in Figure 12.20 to determine the numerical factor $K$ in the drag coefficient, in multiples of $\pi$
(c) Write an expression for the angular velocity of the small bead in terms of the viscosity $\eta$, the radius of the bead $R$ the constant of torsional stiffness $C,$ and the length and the number of whole extra turns in the DNA molecule, $N$, where $N=\phi / 2 \pi,$ Estimate the angular velocity of an $R=400 \mathrm{nm}$ bead if the length of DNA is $14.8 \mathrm{kb}$ and it has been twisted by 50 extra turns. The torsional stiffness of DNA is approximately $400 \mathrm{pN} \mathrm{nm}^{2}$, Express your answer in revolutions per second.
(d) Use the results obtained in (a) and (b) to determine the rotational drag coefficient of a sphere rotating around its center of mass. Use the fact that the rotational motion of the bead in the experiment can be decomposed into translational motion of the center of mass and rotation around the center of mass.

Sana Riaz

Numerade Educator

Problem 6

(a) E.coli swims at about $20 \mu \mathrm{m} / \mathrm{s}$ by rotating a bundle of helical flagella. If the motors were to turn 10 times faster than normal, what would their swimming speed be? If their fluid environment were made 10 times more viscous, but the motors were to turn at the same rate, what would the swimming speed be? How does the power output of the motor change in these two hypothetical situations?
(b) Two micron-sized spheres, one made of silver and the other gold, sediment (that is, fall under gravity) in a viscous fluid. The silver sphere has twice the radius of the gold one. Which sediments faster?
(c) The left ventricle of the human heart expels about $50 \mathrm{cm}^{3}$ of blood per heartbeat. Assuming a pulse rate of I heartbeat per second and a diameter of the aorta of about $2 \mathrm{cm},$ what is the mean velocity of blood in the aorta? What is the Reynolds number?
(d) What is the Reynolds number of a swimming bacterium? A tadpole? A blue whale?

Caroline Jones

Numerade Educator

Problem 7

Proteins and other macromolecules can be separated by size using centrifugation. The idea is to spin a sample containing proteins of different size in solution. The spinning produces a centrifugal force per unit mass $g_{c},$ which leads to diffusion with a drift velocity that depends on the protein size. We assume that a protein in the sample can be approximated as a ball of radius $R$
(a) Following the discussion in the chapter, fill in the steps leading up to the formula for the drift velocity of the protein as a function of its radius (Equation 12.45 ),
$v_{\text {drift }}=\frac{2\left(\rho_{\text {protein }}-\rho_{\text {solvent }}\right) g_{c} R^{2}}{9_{\eta}}$
where $\rho_{\text {protein }}$ and $\rho_{\text {solvent }}$ are the densities of the protein and the solvent and $\eta$ is the solvent viscosity.
(b) Estimate the drift velocity for hemoglobin in water in an ultracentrifuge with $g_{c} \approx 10^{5} g$, where $g=10 \mathrm{m} / \mathrm{s}^{2}$ is the acceleration of freely falling objects in Earth's gravitational field. Assume a typical protein density of $1.2 \mathrm{g} / \mathrm{cm}^{3}$
(c) We would like to separate two similar proteins, having the same density, $\rho^{\prime}=1.35 \mathrm{g} / \mathrm{cm}^{3} .$ They have diameters of $4 \mathrm{nm}$ and $5 \mathrm{nm}$, respectively. The two protein species start out mixed together in a thin layer at the top of a $1 \mathrm{cm}$ long centrifuge tube. How large should the centrifuge acceleration $q_{c}$ be so that the two proteins are separated before they drift to the end of the tube?

Sana Riaz

Numerade Educator