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MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Department of Electrical Engineering and Computer Science,

Department of Mechanical Engineering,

Division of Bioengineering and Environmental Health,

Harvard-MIT Division of Health Sciences and Technology

Quantitative Physiology: Cells and Tissues

2.791J/2.794J/6.021J/6.521J/BE.370J/BE.470J/HST.541J

Homework Assignment #1 Issued: September 8, 2004

Due: September 16, 2004

Reading

Lecture 2 Volume 1: 3.1-3.1.3 3.2-3.2.2

Lecture 3 Volume 1: 3.1.4-3.1.5 3.5-3.5.2.1

Lecture 4 Volume 1: 3.6-3.6.1.2 3.7-3.7.2

AnnouncementsHomework will consist ofExercises and Problems. Exercises are generally more conceptual and

require less number crunching. Exercises often require writing sentences to explain what you have

learned. Written solutions should be submitted for both Exercises and Problems.

Exercise 1. According to the random walk model, solute molecules move and thereby diffuse be

cause of collisions with water molecules. Solute collisions with other solute molecules are gener

ally ignored under the assumption that the water molecules vastly outnumber the solute molecules.

To get a feeling for the validity of this assumption, and to appreciate the number of particles and

spatial scales involved, consider the diffusion of potassium ions in the cytoplasm of a red blood

cell. Assume that the volume of the cell is 90 fL, and that the concentration of potassium ions in

the cytoplasm is 150 mmol/L.

Part a. Estimate the number of potassium ions in the cytoplasm of the cell.

Part b. Estimate the average distance between potassium ions in the cell.

Part c. Estimate the number of water molecules in the cytoplasm of the cell.

Part d. Estimate the average distance between water molecules in cytoplasm.

Part e. Determine the ratio of water molecules to potassium ions in cytoplasm.

Exercise 2. At a junction between two neurons, called a synapse, there is a 20 nm cleft that

separates the cell membranes. A chemical transmitter substance is released by one cell (the presynaptic cell), diffuses across the cleft, and arrives at the membrane of the other (post-synaptic)

cell. Assume that the diffusion coefficient of the chemical transmitter substance is D = 5106cm2/s. Make a rough estimate of the delay caused by diffusion of the transmitter substanceacross the cleft. What are the limitations of this estimate? Explain.

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Exercise 3. The time course of one-dimensional diffusion of a solute from a point source in space

and time has the form

cn(x, t) = no

4Dtex

2/4Dt,

where no is the number of moles of solute per unit area placed at x = 0 at t = 0. cn(x, t) iscomputed as a function of time for locationsxaand xb, and shown in the following figure.

t

xa

xb

cn

(x,t)

Isxa> xb or isxa < xb? Explain.

Problem 1. A general solution to a first-order linear differential equation with constant coefficients

can be written as

n(t) =n+ (n0

n)et/.

a) Determine the slopem0=dn/dtatt= 0in terms of the constantsn,n0and .

b) If this slope were extended for t > 0 (i.e., ifn(t) = n0+ m0t), for what value oft willn(t) =n?

c) Plotn(t)and n(t)whenn0= 10,n= 10, and= 1.

Problem 2. Four solutions to the differential equation

dx(t)

dt +Ax(t) +B = 0

are shown in the following plot.

0 5 10

0

3

3

Part a. Find values ofA and B that are consistent with curvea. Are these values unique? If not,find a second set of constants that are consistent.

Part b. Repeat part a for curveb. Compare these results to those of part a. Explain similarities anddifferences.

Part c. Repeat part a for curve c. Compare these results to those of parts a and b. Explainsimilarities and differences.

Part d. Repeat part a for curved. Determine all possible values ofAandB for this case.

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Problem 3. Assume that nomol/cm2 of sucrose (with a diffusion constant D0.5105cm2/s)

are placed in a trough of water at a point x= 0 at time t= 0. Assume that the concentration ofsucrose is a function of xand tonly.

2a) Show that for any fixed point xp the maximum concentration occurs at time tm =xp/2D.

b) How long does it take for the concentration to reach a maximum at x= 1 cm?

Problem 4. The following figure illustrates a cascaded system of two water tanks. Water flows out

of the first tank and into the second at a rate r1(t), and out of the second tank at a rate r2(t).

h1 r1

h2r2

(t) (t)

(t)(t)

The rates of flow out of the tanks are proportional to the heights of the water in the tanks: r1(t) =k1h1(t) and r2(t) = k2h2(t), where k1 and k2are each 0.02 m

2/minute. The height of tank 1 is 1 m

and that of tank 2 is 2 m. The cross-sectional area of tank 1 is A1 = 4 m2 and that of the second

tank is A2 = 2 m2. At time t= 0, tank 1 is full and tank 2 is empty.

a. If the height of water in tank 2 ever exceeds the height of the tank (2 m), the water will

overflow. Will the water ever overflow? Explain.

b. Set up a system of differential equations to determine h2(t). Solve the equations to determinean expression for h2(t).

c. At what time does the water in tank 2 reach its peak? What will be the maximum height of

water ever achieved in tank 2?

d. At what time will the water stop flowing out of tank 1? Explain your answer in mathematical

terms and then in physical terms.

e. If both tanks were full at t= 0, would the second tank ever overflow? Explain.

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Problem 5. Transport by diffusion tends to move solutes from regions of high concentration to

regions of low concentration, thereby making the spatial distribution of solute more uniform. Thus

diffusion is often associated with mixing. However, diffusion can also be used to separate solutes in

a mixture. Such separation is important as a mechanism to purify mixtures of biological materials

(such as mixtures of DNA fragments) and is currently being exploited in microfabricated systems

for biological and medical analysis (which we will look at in greater detail in the Microfluidics

Laboratory Project).Assume that a mixture of two molecular species is loaded into a long and narrow channel filled

with water. Let x represent distance in the the longitudinal direction and assume that n0 moleculesof solute A and n0 molecules of solute B are loaded into location x=0at time t=0. Assume thatsolutes A and B have different molecular weights and that their diffusivities are DA =10

7 cm2/s

and DB =4107 cm2/s, respectively. The following figure illustrates how the two species tend

to separate as they diffuse.

0

x (

i

10 10

m)Co

ncentratons

ofA

andB

Part a. As time elapses, the number of molecules of solute A that remain in a test region within

the 10m of the starting position (i.e., 10m< x

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Reading

Lecture 5 Volume 1: 3.8-3.8.5

Lecture 6 Volume 1: 4.1-4.3.2.3 4.4-4.5.1.2

Lecture 7 Volume 1: 4.7-4.7.1.2

Lecture 8 Volume 1: 4.7.2-p.230 Fig.4.26 Fig.4.28 4.8.2-4.8.3

Exercise 1. Describe the dissolve-diffuse theory for diffusion through cellular membranes.

Exercise 2. Two time constants are involved in two-compartment diffusion through a membrane:

the steady-state time constant of the membrane (ss) and the equilibrium time constant for the two

compartments (eq). Without the use of equations, describe these two time constants.

Exercise 3. A solute n

diffuses through a membrane that separates two compartments that have

different initial concentrations. The concentrations in the two compartments as a function of time,a bcn(t)and cn(t), are shown in the following figure.

t

c

a

n(t)

c

b

n(t)

The volumes of the two compartments are Va and Vb. Is Va >Vb or is Va

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concentration of 0.1 mol/L. Based on these measurements, de Vries determined the correct formula

for raffinose. Which formula would you choose and why would you choose it?

Problem 1. Consider diffusion through a thin membrane that separates two otherwise closed com

partments. As illustrated in Figure 1, the membrane and both compartments have cross sectional

areas A = 1cm2. Compartment 1 has length L1 = 50cm, compartment 2 has length L2 = 10

c( )c1(t) c2(t)

2

W 4 cm

L1

x,t

Area=1cm

=10

=50cm L2 =10cm

Figure 1: Two compartments separated by a membrane.

cm, and the membrane thickness is W = 104 cm. Assume that (1) the compartments containsugar solutions and that both compartments are well stirred so that the concentration of sugar in

compartment 1 can be written as c1(t)and that in compartment 2 can be written as c2(t); (2) the

concentration of sugar in the membrane can be written as c(x,

t), where x

represents distancethrough the membrane; (3) the diffusivity of sugar in the membrane is Dsugar =10

5 cm2/s and

the membrane:water partition coefficient km:w is 1; (4) the concentration of sugar in the membrane

has reached steady state at time t=0and that c1(0)=1mol/L and c2(0)=0mol/L.

a) Compute the flux of sugar through the membrane at time t=0, s(0).

b) Compute the final value of concentration of sugar in compartment 1, c1().

c) Let eq characterize the amount of time required to reach equilibrium. What would happen

to eq if the diffusivity of sugar in the membrane were doubled? Explain your reasoning.

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Problem 2. A thin membrane and a thick membrane, that are otherwise identical, are used to

separate identical solutions of volume V

= 1 cm3 (Figure 2). All the membrane surfaces facingthe solutions have area A

= 1 cm2. The thin membrane has thickness ds = 104cm; the thick

membrane has thickness dl = 1 cm. For t

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Problem 3. Glucose is dripped at a constant rate R = 2mol/s into a bath that contains 1012

identical red blood cells, as shown in the following figure.

Assume that each red blood cell has a volume = 25 (VC m)3 and a surface area AC =80 (m)2

,

and that neither of these changes over the time interval considered in this problem. Assume that

the volume of the bath is 1 L, and that the bath is well stirred. (You may assume that the amount

of water dripped into the bath is negligibly small.) The concentration of glucose in the bath, cb(t),if found to increase as a function of time t, as shown in the following plot.

cb(t) (mmol/L)

5

t (s)00 300 600 900

Part a. Is the following logic True or False?

The flux of glucose through each of the cell membranes cannot be constant over the

time 0 < t < 900s, because if it were, the concentration cb(t)would be a linearfunction of time.

If the truthfulness of this statement cannot be determined from the information provided, describe

what additional information is needed.

Part b.Determine the flux of glucose through the membrane of each cell at time t =

900

s.Use our normal convention that outward flux (i.e., flux leaving the cell) is positive and inward

flux is negative. Determine the numerical value (or numerical expression) and units. If you

cannot determine the numerical value from the information provided, describe what additional

information is needed.

Part c. Determine the flux of glucose through the membrane of each cell at time t = 0s. Useour normal convention that outward flux (i.e., flux leaving the cell) is positive and inward flux

is negative. Determine the numerical value (or numerical expression) and units. If you cannot

determine the numerical value from the information provided, describe what additional information

is needed.

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Problem 4. All cells are surrounded by a cell membrane. The cytoplasm of most cells contains a

variety of organelles that are also enclosed within membranes. Assume that a spherical cell with

radius R=50m contains a spherical organelle called a vesicle, with radius r= 1m, as shown

in the following figure.

i l icl

i m

i l ll

i m

spher ca ves e

rad us r = 1

spher ca ce

rad us R = 50

bath

Assume that the membranes surrounding the cell and vesicle are uniform lipid bilayers with iden

tical compositions and the same thickness d=10nm. Assume that solute Xis transported across

both the cell and vesicle membrane via the dissolve and diffuse mechanism. Assume that X

dissolves equally well in the bath and in the aqueous interiors of the vesicle and cell. Assume that

the solute Xdissolves 100 times less readily in the membrane (i.e., the partitioning coefficient is

0.01). Assume the diffusivity of Xin the membranes is 107cm2/s.Initially, the concentration of Xis zero inside the cell and inside the vesicle. At time t=0, the

cell is plunged into a bath that contains Xwith concentration 1 mmol/L.

a) Estimate the time that is required for the concentration of X

in the cell to reach 0.5 mmol/L.

Find a numerical value or explain why it is not possible to obtain a numerical value with the

information that is given.

b) Estimate the time that is required for the concentration of X

in the vesicle to reach0.5 mmol/L. Find a numerical value or explain why it is not possible to obtain a numeri

cal value with the information that is given.

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Problem 5. As your first assignment at Tinyfluidics Inc., you are asked to design a microfluidic

device that will remove small molecules from a sample of fluid that contains both large molecules

and small molecules. After some thinking, you design the laminar flow device shown below.

L

W

in

in

le in fild

buffer

buffer

samp

waste out

waste out

trate out

The sample is injected in a port with width . The sample flow is surrounded by buffers injected

on both sides of the sample. The combined flow then passes through a channel that has width W

and length L after which the fluids are separated into a desired filtrate output (in a channel of width

d) and two waste outputs. Assume that the fluid moves with the same speed v in all parts of the

microfluidic device (although this is not generally true, it is a convenient starting point). Assume

that L0.

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Reading



Lecture 10 Volume 1: 6.4.2-6.4.3 6.6-6.7.4

Lecture 11 Volume 1: 7.2.1 7.2.3 7.2.4.1 7.4

Announcements

Exam 1 will be held on Thursday, October 7, 2004 from 7:30 PM to 9:30 PM

The exam is closed-book: notes on both sides of one 8111sheet of paper2

may be used for reference. Calculators may be used, but computers and wireless

devices may not be used.

There will be no recitations on the day of the exam.

Exercise 1. It is known that the membrane of a certain type of cell is highly permeable to water,

but relatively impermeable to L-glucose, sodium ions, and chloride ions. When the cell is removed

from interstitial fluids and placed in a 150 mmol/L NaCl solution, the cell neither shrinks nor

swells.

a) Would the cell shrink, swell, or remain at constant volume if placed in 150 mmol/L solution

of L-glucose? Explain.

b) Would the cell shrink, swell, or remain at constant volume if placed in a 300 mmol/L solution

of L-glucose? Explain.

Exercise 2. If Equation 4.68 in volume 1 of the text is multiplied by A(t), the result can be writtenas

dVi(t) =A(t)LVRT (t)C

iCo (t) .dt

Using no mathematical formulas or equations, describe the meaning of of this equation in a few

well chosen English sentences.

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Exercise 3. The following figure shows schematic diagrams of two cells that have the same volume

but quite different shapes.

One cell (left panel) is spherical, the other is approximately cylindrical but contains a large number

of microvilli. The cell membranes have the same hydraulic conductivity to water. If the two cells

are subjected to the same decrease in extracellular osmolarity, which cell swells more rapidly?

Explain.

Exercise 4. Consider the simple, symmetric, four-state carrier model. For each of the following

E, NESi , Noconditions, find NE

i , No ES, and S. Explain the physical significance of each of youranswers.

a) = 0.

b) = 0.

c) K

= 0.

i oExercise 5. Consider the simple, symmetric, four-state carrier model when c = cS

= 0. SketchS

the carrier density in each of its four states as a function of /. Give a physical interpretation ofyour results.

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Problem 1. A volume element with constant cross-sectional area Ahas rigid walls and is dividedinto two parts by a rigid, semipermeable membrane that is mounted on frictionless bearings so that

the membrane is free to move in the x-direction as shown in the following figure.

c1 c2

0 x 10(cm)

The semipermeable membrane is permeable to water but not to the solutes (glucose or NaCl or

CaCl2). At t = 0, solute 1 is added to side 1 to give an initial concentration of c1(0)and solute2 is added to side 2 to give an initial concentration of c2(0). Concentrations are specified as thenumber of millimoles of glucose or NaCl or CaCl2 per liter of solution. The initial position of

the membrane is x(0). For each of the following parts, find the final (equilibrium) values of themembrane position x(), and the concentrations, c1()and c2().

1 2(a.) cglucose(0)=0; cglucose(0)=10; x(0)=5.

1 2c(b.) glucose(0)=30; cglucose(0)=70; x(0)=7.

1 2(c.) cglucose(0)=20; cglucose(0)=10; x(0)=3.

1 2(d.) cglucose(0)=30; cNaCl(0)=20; x(0)=4.

2(e.) 1 (0)=20; cN

aCl

(0)=30; x(0)=3.cCaCl2

Problem 2. A spherical cell has a freely distensible membrane that is permeable to solute A, im

permeant to solute B, and permeable to water. The cell contains NI

moles of impermeant solutes,

and is allowed to equilibrate in a bath in which the concentration of A is zero and the concentration

of B is c1. The bath is large compared to the cell. The cell volume in this bath is V0. At t=0, the1cell is moved to a bath that contains equal concentrations of A and B, cA =cB = 2c1.

a. Is the new bath hyper-, hypo-, or iso-osmotic with the cell at t

=

0?

b. If the hydraulic conductivity of the cell is LV

, what is the rate of increase of the volume,

dV(t)/dtjust after the cell is moved (i.e., at t=0+)?

c. What is the equilibrium volume of the cell in terms of V0? Explain briefly.

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Problem 3. The following figure shows the design of a miniature pump that can be implanted in

the body to deliver a drug. No batteries are required to run this pump!

Rigid, Frictionless,semipermeable impermeable

Chamber 1 Chamber 2

Drug

0.7 cm(solute) (drug)

orifice

membrane piston

3 cm

The pump contains two cylindrical chambers filled with incompressible fluids: the two chambers

together have a length of 3 cm and a diameter of 0.7 cm. Chamber 1 is filled with a solution whose

concentration is 10 mol/L; the osmolarity of this solution greatly exceeds that of body fluids.

Chamber 2 is filled with the drug solution. The two chambers are separated by a frictionless, mass

less, and impermeable piston. The piston moves freely and supports no difference in hydraulicpressure between the chambers; the piston allows no transport of water, solute or drug between

chambers. The pump walls are rigid, impermeable and cylindrical with an orifice at one end for

delivering the drug and a rigid, semipermeable membrane at the other end. The orifice diameter is

sufficiently large that the hydraulic pressure drop across this orifice is negligible and sufficiently

small so that the diffusion of drug through the orifice is also negligible. The semipermeable mem

brane is permeable to water only, and not permeable to the solute. Assume that T =300K.

a) Provide a discussion of 50 words or fewer for each of the following:

i) What is the physical mechanism of drug delivery implied by the pump design?

ii) What is(are) the source(s) of energy for pumping the drug?

iii) Assume there is an adequate supply of drug in the pump for the lifetime of the im

planted subject and that it is necessary to provide a constant rate of drug delivery.

Which fundamental factors limit the useful lifetime of this pump in the body?

b) When implanted in the body, the pump delivers the drug at a rate of 1 L/h. Find the valueof the hydraulic conductivity, LV

, of the semipermeable membrane.

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Problem 4. A large fraction of the molecules in a cell membrane are phospholipids, which have a

hydrophilic head and hydrophobic tails. When purified phospholipids are added to a saline solu-

tion, the phospholipids self assemble into a variety of stable structures, one of which is spherical

and is called a liposome.

Liposome

Bath

Liposomes have saline interiors and exteriors that are separated

by a phospholipid bilayer, much like a biological cell. Liposomes

can be used as artificial cells to test theories about membrane trans-port.

Assume that a liposome is created in a solution that contains

200 mmol/L of a solute Ithat cannot permeate the phospholipid bi-

layer. Assume that the inner solution initially contains 200 mmol/L

of solute I and the initial radius is30m. Assume that the thicknessof the bilayer is much smaller than the diameter of the liposome.

Assume that water can permeate the phospholipid bilayer, and that hydraulic pressure gradients

across the bilayer can be ignored. Assume that the temperature is 300 K.

Assume that the liposome is transfered at time t = 0 to a bath that contains 200 mmol/L of

solute A and 100 mmol/L of solute B. The liposome initially shrinks but then swells and reachesan ultimate radius of approximately38m as shown in the following plots. The left plot shows thetime course for the first 1000 seconds. The right plot shows just the first 100 seconds of the same

data.

0 500 1000

25

30

35

40

Radius(m)

Radius(m)

Time (seconds) Time (seconds)

0 50 100

28

29

30

Part a. Determine the asymptotic value of concentration cI(t)of solute I in the liposome as timeincreases, i.e., determine

limt

cI(t).

Provide a numerical value (or numerical expression) with units. If it is not possible to determine

the concentration from the information provided, explain why.Part b. Indicate which, if either, of the solutes A and B can permeate through the artificial mem-

brane? If it is not possible to determine this information, explain why.

Part c. Determine the hydraulic conductivity LV of the artificial membrane. Provide a numer-

ical value (or numerical expression) with units. If it is not possible to determine the hydraulic

conductivity from the information provided, explain why.

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Problem 5. A large fraction of the molecules in a cell membrane are phospholipids, which have

a hydrophobic head and hydrophilic tails. When purified phospholipids are added to a saline

solution, the phospholipids self assemble into a variety of stable structures, one of which is called

a liposome.

Liposome

Bath

Liposomes have saline interiors and exteriors that are separated by a phospholipid bilayer, much

like a biological cell, as illustrated above. Liposomes can be used as artificial cells to test the

ories about membrane transport. Assume that a liposome is created in a solution that contains

100 mmol/L of a solute Athat cannot permeate the phospholipid bilayer. Assume that water canpermeate the phospholipid bilayer, and that the liposome comes to equilibrium with a volume of

1 pL (1012 L) of internal solution containing A with concentration 100 mmol/L.The liposome is then transferred to one of the following solutions

solution 1: 100 mmol/L of A plus 10 mmol/L of B solution 2: 90 mmol/L of A plus 10 mmol/L of B solution 3: 100 mmol/L of B

where Bis a solute that can permeate the phospholipid bilayer. Both A and Bare nonelectrolytes,and the baths are large compared to the volume of the liposome. Assume that transport of water

and transport of the permeant solute Bare independent of each other, i.e., water transport does noteffect transport of Band vice versa.

(a.) Calculate the equilibrium volume of the liposome in solution 1. Discuss your result briefly.

(b.) Calculate the equilibrium volume of the liposome in solution 2. Discuss your result briefly.

(c.) Calculate the equilibrium volume of the liposome in solution 3. Discuss your result briefly.

Problem 6. A monosaccharide, M, is known to be transported through a cell membrane by acarrier so that

oci(t) c=max oK+ci(t) K+c

o

where c

i

(t)

is the intracellular concentration of M, c is the external concentration of M,

is theoutward flux of M (mol/cm2s) and max

is the maximum flux with which the carrier system

is capable of transporting M. The area of the cell, A, is 106cm2, and K is 100 mmol/L. Thefollowing experiment is performed: the cell initially contains zero moles of M, and at t = 0thecell is placed in an isotonic solution containing a concentration of Mequal to co (constant), where

oc K. The internal concentration of M is found to bei oc(t)=c (1et/), t0

where =100s. The volume of the cell remained roughly constant at 1010 mL throughout theexperiment. Determine max.

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Homework Assignment #4 Issued: October 8, 2004

Due: Thursday October 14, 2004

Reading

Lecture 15 Volume 1: 7.5


Announcements

This homework assignment is smaller than average to give you time to work on your lab reports.

First drafts of your lab reports are due Friday, October 15, 2004 at 10:00 AM. Bring 3 copies.

One will be reviewed by the technical staff. One will be reviewed by the writing staff. One will

be reviewed by a peer student group. You and your partner will be assigned to review the report of

another student group. All reviews are due Tuesday October 19, 2004 when they will be discussed

at the Writing Clinic, to be held at 7:30 PM.

Exercise 1. Define electroneutrality and briefly explain its physical basis.

Exercise 2. Define the Nernst equilibrium potential and briefly explain its physical basis.

Problem 1. Two compartments of a fluid-filled chamber are separated by a membrane as shown in

the following figure.

I

V

1 mmol/L NaCl0.1 mmol/L KCl

0.1 mmol/L NaCl

1 mmol/L KCl

Compartment 1

Membrane

Compartment 2

+

The area of the membrane is 100 cm2

and the volume of each compartment is 1000 cm3

. Thesolution in compartment #1 contains 1 mmol/L NaCl and 0.1 mmol/L KCL. The solution in com

partment #2 contains 0.1 mmol/L NaCl and 1 mmol/L KCL. The temperatures of the solutions are

24C. The membrane is known to be permeable to a single ion, but it is not known if that ion issodium, potassium, or chloride. Electrodes connect the solutions in the compartments to a battery.

The current Iwas measured with the battery voltage V =0and was found to be I=1mA.

a) Identify the permeant ion species. Explain your reasoning.

b) Draw an equivalent circuit for the entire system, including the battery. Indicate values for

those components whose values can be determined.

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c) Determine the current I that would result if the battery voltage were set to 1 volt. Explainyour reasoning.

Problem 2. Three compartments are separated from each other by semi-permeable membranes, as

illustrated in the following figure.

Vm

Im

+

cNa

= 100

cK= 0

cCl= 100

cNa

= 20

cK= 80

cCl= 100

cNa

= 10

cK= 0

cCl= 10

compartment 1 compartment 2 compartment 3

permeant to permeant toNa+only Clonly

Each compartment contains well-stirred solutions of sodium, potassium, and chloride ions, with

concentrations indicated in the figure (in mmol/L). The membrane between compartment 1 and 2

is permeant to sodium ions only, and its specific electrical conductivity GNa is 5 mS/cm2. The

membrane between compartment 2 and 3 is permeant to chloride ions only, and its specific electri

cal conductivityGCl is 2 mS/cm2. Both membranes have areasA=10cm2. The temperature T issuch that RT/(Floge)=60mV.

a) Sketch an electrical circuit that represents the steady-state relation between current and volt

age for the three compartments. Label the nodes that correspond to compartments 1, 2, and

3. Include the switch in your sketch. Label Im, Vm, and the conductances.

b) Let V1 and V2 represent the steady-state potentials in compartments 1 and 2 with referenceto compartment 3 when the switch is open. Calculate numerical values for V1 and V2.

c) Compute the steady-state value of the current Im when the switch is closed.

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Due: October 21, 2004

Reading

Lecture 17 Volume 2: Chapter 1

Lecture 18 Volume 2: 2.1-2.4.2

Lecture 19 Volume 2: 2.4.3-2.5

Announcements

The recitation on October 19 is cancelled so that we can meet that evening for a Writing Clinic. The

Writing Clinic will be heldfrom 7:30 to 9:30 PM. Please return your written critiquesof your

peers laboratory report at that time. You will also receive critiques from the technical and

writing staffs.

Exercise 1. Active ion transport is said to have a direct and an indirect effect on the resting

potential of a cell. Define both effects and discuss the distinction between the two effects.

Exercise 2. Describe the distinctions between the following terms that refer to ion transport across

a cellular membrane: electrodiffusive equilibrium, steady-state, resting conditions, and cellular

quasi-equilibrium.

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0

Exercise 3. Figure 1 shows measurements of the resting potential of a glial cell for different values

of extracellular potassium concentration (left panel). These measurements are to be interpreted in

terms of the network model shown in the right panel of the same figure. Assume that co =150N a

Mud puppy

Slope:

glial cell

59 mV/decade

+

+

Im

Ip

Ip

K

IaKI

a

G GK

VK

Na

Na

Na

Restingpotential(m

V)

20 +

40

60 Vom

80VNa

100

120

0.1 1 10 100co

K(mmol/L)

Figure 1: Measurements and model of electrical responses of a glial cell.

immol/L, c = 15mmol/L and that the external solution is maintained isotonic with the cyto-Na plasm by controlling impermeant solutes. Assume that sodium and potassium concentrations are

o aconstant, except for cK, and that the pump system, which consists of Ia and IK, is nonelectro-N a

genic.

a) Consider only the region for which the data are well fit by the straight line of slope 59

mV/decade. Indicate whether the following statements are true or false and give a brief

reason for each answer.

i Im =

0.ii) Vo VK.m

iii) GNa GK.

iv) VNa > VK.iv) cK =100mmol/L.a avi) I =IN a.K

p pvii) I =IN a.K

aviii) I =GN a(Vo VN a).N a m

b) It is proposed that deviation of the data from the straight line for the lowest co is a result ofKa change in GK that occurs when Vo < 110

mV. For the data shown, is this a reasonablemhypothesis? Does it require that GK for V

o = 125mV is larger or smaller than GK formVo > 100mV? Explain.m

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Exercise 4. The ionic concentrations of a uniform isolated cell are given in the following table.

Concentration (mmol/L)

Inside Outside

Potassium 150 15

Sodium 15 150

An electrode is inserted into the cell and connected to a current source so that the current throughthe cell membrane is Im. The steady-state voltage across the cell membrane Vm is determined as afunction of the current as shown in the following figure.

+

+

+

+

(mV)

0

40

0.40

Vm

Vm

VmIm

Im

IK

I

V VK

G GK

Im (nA)

Na

Na

Na

Assume that: (1) the cell membrane is permeable to only K+ and Na+ ions; (2) the Nernst equi

librium potentials are Vn = (60/zn)log10(cn/cion)(mV); (3) ion concentrations are constant; (4)

active transport processes make no contribution to these measurements.

a) Determine the equilibrium potentials for sodium and potassium ions, VN a and VK.

b) What is the resting potential of the cell with these ionic concentrations?

c) With the current Im adjusted so that Vm =VK, what is the ratio of the sodium current to thetotal membrane current, IN a/Im?

d) What is the total conductance of the cell membrane Gm =GN a +GK?

e) Determine GNa and GK.

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Problem 1. The resting membrane potential, Vo , of two uniform, isolated cells is measured as amofunction of the external concentration of potassium, cK, with the sodium concentration held fixed

on oat its normal value, cN a, and then as a function of the external sodium concentration, cNa , withthe potassium concentration held fixed at its normal value, con. The results for these two cells areKshown in the following figure.

Cell 1 Cell 2

150

100

50

0

50

Slope: 60 mV/decade Slope: 48 mV/decadeRestingpotential(mV)

150

100

50

0

50

1 10 100 1000 1 10 100 1000co

(mmol/L) co (mmol/L)K K

Slope: 12 mV/decadeSlope: 0 mV/decadeRestin

gpotential(mV)

150

100

50

0

50

150

100

50

0

50

1 10 100 1000 1 10 100 1000o ocN

a

(mmol/L) cN

a

(mmol/L)

You may assume that for each cell: (1) external solutions are isotonic; (2) the membranes are

impermeable to ions other than potassium and sodium; (3) the internal concentrations of potassium

and sodium are maintained constant by a non-electrogenic active transport mechanism; (4) the

total membrane conductance is 10 nS; (5) the normal resting potential is 60mV; (6) the internal

concentration of sodium is 20 mmol/L.

G

a) For each cell, determine the total conductance of the membrane to potassium and to sodium,

K and GNa , respectively. If either value is indeterminate from the information given, de

scribe what additional information would be needed.

b) For each cell, determine the internal and external concentrations of potassium and the ex

ternal concentration of sodium at the normal resting potential (60 mV). If the value isindeterminate from the information given, describe what additional information would be

needed.

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c) For each cell, determine the simplest equivalent electric network model that relates the de

pendence of the resting potential of the cell on the ion concentrations. Indicate the values of

all elements in the network.

Problem 2. The membrane of a cell is known to be permeable to Na+ and K+ only, with

passive ionic conductances of GN a = 106 S/cm2 and GK = 10

3 S/cm2, and contains a

(Na+ K+)ATPase active transport mechanism that transports 3 molecules of Na+ outward and

2 molecules of K+ inward through the membrane for each molecule of ATP split into ADP and

phosphate. In the experiment shown in Figure 2, the cell is kept at a temperature of 24C. For

tt

t

60 60

80

Injectionof Perfusion

sodiumions withouabain

V

om

Figure 2: Effect of sodium injection and ouabain on resting potential.

t < t1, the cell is in its normal resting state with resting potential Vo 60mV, and resting ionm

concentrations shown in the following table.

Ion Concentration (mmol/L)

Internal External

Sodium 15 135

Potassium 156 15

At t=t1, Na+ is injected rapidly into the cell interior, without the passage of any current through

the membrane, to double the intracellular sodium concentration. When the injection is completed,

the membrane potential hyperpolarizes to approximately 80

mV. At t

=

t2 ouabain, a blockerof the active transport mechanism, is applied to the cell and the membrane potential returns to

approximately its normal resting value.

a) Is the active transport mechanism electrogenic or nonelectrogenic? Explain!

b) Does the active transport mechanism contribute appreciably to the resting potential for t

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Problem 3. The membrane of a cell contains an active transport mechanism that pumps three

sodium ions out of the cell for every two potassium ions that it pumps into the cell. The membrane

also supports the passive transport of sodium and potassium ions, but is impermeant to all other ions

and is impermeant to water. The sodium conductivity is 105S/cm2and the potassium conductivityis 104S/cm2. The cell is allowed to come to steady state and its membrane potential is 52.5mV.The Nernst equilibrium potential for sodium is 60mV and the Nernst equilibrium potential for

60

3

potassium is mV. The net outward current density due to active transport is A/cm2

.8

a) Draw an electrical circuit to represent ionic transport across the membrane of this cell. In

clude labels for each of the 6 numbered parameters provided in the problem statement.

b) Is the cell at rest? If yes, prove that it is at rest. If no, explain why not.

c) Is the cell in quasi-equilibrium? If yes, prove that it is at quasi-equilibrium. If no, explain

why not.

d) Is the active transport mechanism electrogenic? Explain.

Problem 4. Consider the model of a cell shown in the following figure.

+

+

+

Vom

Im

Ip

IpK

IaKI

a

G GK

V VK+

IpCl

GCl

VCl

Na

Na

Na

Na

The cell has channels for the passive transport of sodium, potassium, and chloride as well as a

pump that actively transports sodium out of the cell and potassium into the cell. The pump ratio

is Ia K = 1.5. The following table shows the intracellular and extracellular concentrations,Nernst equilibrium potentials, and conductance ratios for sodium and potassium. Some informa

tion is also given for chloride; blank entries represent unknown quantities.

N a/Ia

cin con Vn Gn/GK

(mmol/L) (mV)Na+ 10 140 +68 0.1K+ 140 10 68 1Cl 150 1

The cell also contains impermeant intracellular ions. Assume that the cell is in equilibrium at t=0,i.e., assume that at t= 0the cell has reached a condition for which all solute concentrations, thecell volume, and the membrane potential are constant.

a) Choose one of the following statements and explain why it is true.

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i) The cell resting potential depends on GCl .

ii) The cell resting potential depends on VCl .

iii) The cell resting potential depends on both GCl and VCl.

iv) The cell resting potential does not depend on GCl .

v) The cell resting potential does not depend on VCl .

vi) The cell resting potential does not depend on eitherG

Cl or VCl .

b) Determine Vo.m

c) At t=0, the external concentration of chloride is reduced from 150 mmol/L to 50 mmol/Lby substituting an isosmotic quantity of an impermeant anion for chloride. Assume that the

concentrations of sodium and potassium both inside and outside the cell remain the same

and that the volume of the cell does not change.

i) Determine Vom(0+), the value of the membrane potential immediately after the change

in solution. You may ignore the effect of the membrane capacitance.

ii) Determine Vom(), the value of the membrane potential after the cell has equilibrated.i

iii) Determine cCl (

), the intracellular chloride concentration after the cell has equilibrated.

iv) Give a physical interpretation of your results in i), ii), and iii).

v) Discuss the validity of the assumptions that the sodium and potassium concentrations

in the cell are constant and that the volume does not change.

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2.791J/2.794J/6.021J/6.521J/BE.370J/BE.470J/HST.541J


Due: October 28, 2004

Reading

Lecture 20 Volume 2: 4.1-4.1.2.3 4.2-4.2.2.2

Lecture 21 Volume 2: 4.2.3-4.2.3.1

Lecture 22 Volume 2: 4.2.3-4.2.3.2

Announcements

Laboratory reports are DUE Monday October 25 at noon. You should submit one copy of the

final draft of the report. You should include the following items in an appendix: final proposal,

copy of your critique of a peers first draft, peer critique of your first draft, critique of your first

draft by writing program, critique of your first draft by technical staff, photocopies of notes taken

during original lab session. Note that there is a SEVERE LATENESS PENALTY. The grade for

a late report will be multiplied by a lateness factor

t/72L= 0.3et/4 + 0.7e

where tis the number of hours late. The lateness factor is plotted below. Notice that the maximum

grade for a report that is more than ONE DAY LATE is less than 50%.

8 1The exam is closed-book: notes on both sides of two

Lateness

factorL

1.0

0.5

0.0one day

1 10 100

Time t past deadline (hours)

This lateness factor is applied regardless of the reason for the lateness, except for health related

problems or personal problems certified by the Deans office. Specifically, lateness due to computer

and/or printer problems is not exempt.

Exam 2 will be held on Thursday, November 4 from 7:30 PM to 9:30 PM

11 sheets of paper may be2

used for reference. Calculators may be used, but computers and wireless devices may not be used.

There will be no recitations on the day of the exam.

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Exercise 1. Give aphysical explanation for why the conduction velocity is larger in larger diameter

fibers, if all other factors are the same.

Exercise 2. Give aphysical explanation of the meaning of Equation 2.18 (in volume 2 of the text)

without the use of equations.

Exercise 3. Let the function f(z, t) represent a solution to the wave equation. This solution isshown in the following figure as a function of time t at the position z= 0.

0 1 20

1

f(0

)

,t

t(ms)

Notice that f(0, t) is non-zero for 0

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Problem 2. A fine platinum wire with a resistance per unit length of 130 /cm is inserted inside aportion of a squid axon as illustrated below.

V1(t) V2(t)

iial Wi

z

Axonact on

potent re

(cm)

z1 z2

The wire is so thin that its volume can be ignored. The axon (500 m diameter) is electrically stimulated to produce a propagated action potential traveling in the +zdirection. The action potentialis recorded at two intracellular sites: V1(t)is recorded at z =z1 and V2(t)is recorded at z =z2.The distance between the stimulus electrode (not shown) and z1 is 2 cm. Results are shown in thefollowing figure.

0

V1(t) V2(t)+50

50

100Intracellularpotential

(mV)

0 5 10 15 20 25

Time (ms)

The resistivity of the axoplasm of this axon is 23 -cm. The resistance per unit length of theexternal solution is 1.2 /cm. The wire begins at some location between z1 and z2, but the exactposition of the beginning is not known and should not be used in any of your calculations.

a) Determine the instantaneous speed of the action potential as its peak passes the point z=z1.

b) Determine the instantaneous speed of the action potential as its peak passes the point z=z2.

c) Sketch the extracellular potential as a function of space (z) that results at the time that thepeak of the action potential passes the point z=z1. Include distances z14< z < z1+4cm.Indicate the scale for the y axis. Describe the important features of this plot.

d) Sketch the extracellular potential as a function of space (z) that results at the time that thepeak of the action potential passes the point z=z2. Include distances z24< z < z2+4cm.Indicate the scale for the y axis. Describe the important features of this plot.

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Problem 3. In measurement 1, a cylindrical unmyelinated axon of radius ais placed in a largevolume of isotonic sea water and the conduction velocity of the action potential is measured to

be 1. The osmotic pressure of the sea water is then doubled by increasing the salt concentration,and the axon is allowed to come to osmotic equilibrium. Then, in measurement 2, the conduction

velocity is measured to be 2. Assume that during these experiments

the number of ions transported through the axon membrane is negligible compared to the

number of ions inside the axon, the volume of the axon is negligible compared to the volume of the bath,

the specific electric properties of the membrane (the capacitance per unit area and the con

ductance per unit area) are the same in both measurements,

the length of the axon remains unchanged, and

the external resistance per unit length of the axon, ro, is negligible compared to the internalresistance per unit length of the axon.

Part a. Let ri1 and ri2represent the internal resistance per unit length of the axon in measurements1 and 2, respectively. Determine the numerical value (or a numerical expression) for the ratio of

these resistances:ri2

/ri1. Briefly explain your reasoning.

Part b. Determine the numerical value (or a numerical expression) for the ratio of speeds: 2/1.Briefly explain your reasoning.

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Homework Assignment #7 Issued: November 8, 2004

Due: November 18, 2004

Reading


Lecture 24 Volume 2: 4.4.8

Lecture 25 Volume 2: 4.4.1

Lecture 26 Volume 2: 3.1-3.2.1 3.3-3.4.2.1

Lecture 27 Preparing an oral presentation

Lecture 28 Volume 2: 3.4.2.4-3.4.3.1 3.4.3.3-3.5


Lecture 30 Volume 2: 5.3-5.7

Lecture 31 Volume 2: 6.1-6.1.1 6.4-6.4.1.5 6.2-6.2.2 6.5-p407

Announcements

There is no homework assignment due this week, because of Veterans day. This assignment is due

on the Thursday after Veterans day (November 18).

First Drafts of your HH project is due on November 19 (one day after this homework is due!).

The first draft should consist of hardcopies of 3-5 technical slides of the type you plan to show

during your final oral presentation. Each slide should be accompanied by a separate sheet of paper

that lists approximately three bullet points that you will discuss in your presentation of this slide.

The first draft should also contain a 1 page extended abstract for your HH project. The abstract

should summarize your findings and explain why they are interesting.

Please take advantage of THIS week to work on your project plus the HH portion of this

homework assignment!

Exercise 1. The following assertions apply to responses calculated according to the Hodgkin-

Huxley model in response to a step of membrane potential applied at t = 0. For each assertion,state if it is true or false and explain your answer.

a) The leakage conductance is constant.

b) The sodium conductance is discontinuous at t=0.

c) The potassium conductance is discontinuous at t=0.

d) The leakage current is constant.

e) The sodium current is discontinuous at t=0.

f) The potassium current is discontinuous at t=0.

g) The factors n(t), m(t), and h(t)are discontinuous at t=0.

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h) The time constants n, m, and h are discontinuous at t= 0.

i) The steady-state values n, m, and h

are discontinuous at t= 0.

Exercise 2. Figure 1 shows the relation between the membrane potential and the membrane current

density during a propagated action potential as computed from the Hodgkin-Huxley model. The

2

Vm

(mV)

Jm

(mA/cm

)

0

Jm

Vm

0.4

40

080

0.41 2 3 4

Time (ms)

Figure 1: Relation of membrane potential and membrane current density during a propagated

action potential. The dotted vertical line marks the time of occurrence of the peak of the action

potential.

membrane current density consists of an initial outward current followed by an early inward current

whose peak occurs before the peak in the action potential.

a) The initial outward current is due primarily to which of the following:

i) an ionic current carried by sodium ions.

ii) an ionic current carried by potassium ions.

iii) an ionic current carried by chloride ions.

iv) an ionic current carried by calcium ions.

v) a capacitance current.

b) The early inward current is due primarily to which of the following:

i) an ionic current carried by sodium ions.

ii) an ionic current carried by potassium ions.

iii) an ionic current carried by chloride ions.iv) an ionic current carried by calcium ions.

v) a capacitance current.

c) Before the peak of the action potential, the membrane potential increases from its resting

value whereas the membrane current density is first outward (increasing and then decreasing)

and then reverses polarity to become inward (decreasing and then increasing again). Discuss

this complex relation between membrane potential and current. In particular, explain how

the Hodgkin-Huxley model accounts for the fact that the current can be both inward and

outward during an interval of time when the membrane potential is depolarizing.

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Exercise 3. Does the time constant of a cylindrical cell depend on its dimensions? Does the space

constant of a cylindrical cell depend on its dimensions?

Exercise 4. For each of the following statements, assume that the electrical properties of a patch

of the membrane of the cell can be represented as a parallel resistance and capacitance. Assume

that the cell has a cylindrical shape with a radius that is small compared to the length of the cell.Determine if each assertion is true or false and give a reason for your choice.

a) For an electrically small cell, the membrane potential in response to a step of current through

the membrane is an exponential function of time.

b) If a step of current is applied through one part of the membrane of an electrically small cell,

the resulting changes in membrane potential will be constant along the length of the cell for

all times after the step.

c) For an electrically large cell, the steady-state value of the membrane potential in response to a

step of current applied through the membrane at one position along the cell is an exponential

function of longitudinal position along the cell.

d) For an electrically large cell, the steady-state value of the membrane potential in response to

a step of current applied through the membrane at one position along the cell is a Gaussian

function of position along the cell.

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Problem 1. The Hodgkin-Huxley model was used to compute propagated action potentials for

default values of the parameters and for 4 test cases. The following plots show the spatial depen

dence of membrane potential that results 1 ms after the stimulus current was applied at z= 0.

60 60

Membranepotential

Membranepoten

tial

(mV)

(mV)

A B

C

Membranepotential

Membranepoten

tial

(mV)

(mV)

40 4020 20

0 0

20 20

40 40

60 60

80 80

0 1 2 3 0 1 2

Distance (cm) Distance (cm)

60 60 D40 4020 20

0 0

20 20

40 40

60 60

80 80

0 1 2 3 0 1 2

Distance (cm) Distance (cm)

Each of AD shows a plot with two curves: the thin gray curve was obtained for default parameters

and the thick black curve was obtained for one of the tests cases. In each test case, a single

parameter was changed from its default value. Default values of the axon characteristics are as

follows length: 3 cm, radius: 0.0238 cm, cytoplasm resistivity: 35.4 cm, extracellular specificresistance: 0 /cm.

a. Which of AD shows results when intracellular sodium concentration was reduced fromicNa

= 50 to 25 mmol/L. Explain.

b. Which of AD shows results when the maximum potassium conductance GK was increased

from 36 to 72 mS/cm2

? Explain.

c. Which of AD shows results when cytoplasmic resistivity was decreased from 35.4 to

30 cm? Explain.

4

3

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Problem 2. Ionic currents are calculated for a space-clamped squid giant axon using the Hodgkin-

Huxley model, with all parameters set to their default values (as listed on page 191 of volume 2

of the text) except for one parameter. For timest < 0, the membrane potential is held at 75mV

and the model is at steady state. Att = 0, the membrane potential is stepped to +5mV and the

resulting ionic current density is computed. Results for six calculations are shown in the following

figure.

0 1000

Time (ms)

IonicCurrentDensity

(mA/cm

2)

2000

2

1

0

1

2

J2

J3

J5

J4

J6

J1

Part a. Which curve represents the ionic current that results when all parameters have default

values exceptcoNa

= 50mmol/L? Explain.

Part b. Which curve represents the ionic current that results when all parameters have default

values exceptcoK

= 400mmol/L? Explain.

Part c. Which curve represents the ionic current that results when all parameters have default

values exceptGNa = 0? Explain.

Part d. Which curve represents the ionic current that results when all parameters have default

values exceptGK = 0? Explain.

Problem 3. Space-clamped responses of the Hodgkin-Huxley model were calculated with the

leakage conductance set to zero, GL = 0, and with all other parameters set to their default val-

ues. The model was stimulated with a pulse of membrane current density of duration 0.5 ms; the

amplitude was varied. The following figure shows the response for an amplitude of 40A/cm2.

500

400

300

200

100

0

100

0 0.5 1 1.5 2 2.5 3

VmJion

Jm

mV

andA/c

m2

Time (ms)

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For each amplitude of the pulse of membrane current density, the ionic current density Jion andthe membrane potential Vm were determined at t = 0.7ms, where t = 0marks the onset of thecurrent pulse. The relation between Jion at t = 0.7ms and Vm at t = 0.7ms that results whenthe pulse amplitude is varied is shown in the following figures. The left panel shows results for a

broad range of membrane potential, from 80to +60mV. The right panel shows results for thenarrower range from 76to 62mV.

400

300

200

100

0

0.2

0.1

0

Jion

(t

.

A/cm

2

)

=

07ms)(

806040 20 0 20 40 60 76 74727068666462

Vm(t=0.7ms)(mV)

Part a. Determine the value of the resting potential to within 1 mV. Explain your choice.

Part b. If the current stimulus is such that Vm(t) =66mV at t= 0.7ms, does the membranepotential increase with time or decrease with time? Explain.

Part c. Determine the value of the threshold potential to within 1 mV. Explain your choice.

Part d. Between the resting potential and this threshold potential, determine whether |JNa|>|JK|,|JNa |=|JK|, or |JNa| |JK|, |JNa | = |JK|, or

|JNa |

>ro.

b) Describe a method by which the data in Figure 4 could be analyzed to estimate the current

densityKm flowing out of an internode. Apply your method to determine whether current is

flowing into or out of the internode between nodes 5 and 6 at t0 =0.75ms.

Problem 4. Although there is considerable scatter, the ratio of the inner diameter dto outer diameter Dof the layer of myelin that encircles a myelinated fiber tends to be about 0.74, as shown inthe following figure, where every symbol represents measurements of d/Dand dfor a differentfiber.

0.9

vagus nerve

sciatic nerve

d/D

0.85

0.8

0.75

0.7

0.65

0.6

0.55

0.5

Dd

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5d(m)

The point of this problem is to investigate the hypothesis that this relation between d

and D

resultsfrom an evolutionary optimization of the cable model.

Part a. Assume that the myelinated part (the internode) of a myelinated fiber can be represented by

the cable model. Assume the myelin can be represented by a homogeneous electrical material with

resistivity mand permittivity m. Assume the intracellular conductor is a homogeneous conductorwith resistivity w. Assume that the extracellular conductor has negligible resistance. Determinean expression for the space constant C of this model in terms of the inner diameter dand the outerdiameter Dof the layer of myelin. Hint: The radial resistance of a cylindrical shell is given below.

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Dd

o

+

V

lnR = =I 2L d

Part b. The expression derived in part a is plotted below

0.6

0.4

C/o

0.2

0

0 0.5 1

d/D

where o =D m/8w. Thus, if Dis fixed (i.e., if the axon is constrained to fit into a constantvolume), then o is a constant, and C is maximum when d0.6D.

b1) Explain in physical terms why the space constant gets smaller as the value of ddecreasesbelow 0.6D.

b2) Explain in physical terms why the space constant gets smaller as the value of d

increasesabove 0.6D.

Part c. The value of d/Dthat maximizes the space constant of the cable model is remarkably closeto the ratio of 0.74 seen experimentally. Nevertheless, it is smaller. One possible reason why it is

smaller is that we ignored the resistance of the outer conductor ro. How would the space constantsdependence on d/D change if the resistance of a thin layer of saline (thickness = 0.1D) wereincluded in the calculation. Make a plot that contains both the old relation (shown in the previous

plot) and the new relation. Briefly describe how the addition of the outer resistance changes the

predicted space constant.

D

L

d

V D

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Homework Assignment #9 Issued: December 2, 2004

This homework assignment will not be collected.

Exercise 1. Explain the origin of gating current.

Exercise 2. State whether each of the following are true or false and give a reason for your answer.

a) Tetrodotoxin blocks the flow of potassium through the sodium channel.

b) The macroscopic sodium current recorded by an electrode in a cell is a sum of the single-

channel sodium currents that flow through single sodium channels.

c) The macroscopic sodium current recorded by an electrode in a cell is the average of the

single-channel sodium currents that flow through single sodium channels.

d) Ionic and gating currents give identical information about channel kinetic properties.

Exercise 3. Explain why the gating current is outward in response to a depolarization independent

of the sign of the charge on the gate.

Exercise 4. List 4 distinct properties shown by ionic currents measured from single voltage-gatedion channels.

1

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Problem 1. The voltage across a membrane patch is stepped from Vo to Vf at t= 0and single-m mchannel ionic currents are recorded as a function of time. Typical records at 6 different values of

Vf are shown in the following figure.m

Vfm

(mV)00

10

201.4

0

401.8

60

2.2

80

100

Current(pA) 0

0

2.60

3

0 8Time (ms)

a) Is the open-channel voltage-current characteristic of this channel linear or nonlinear?

b) What is the conductance of the open channel?

c) What is the equilibrium (reversal) potential for this channel?

d) It is proposed that this channel is the voltage-gated sodium channel responsible for sodium-

activated action potentials. Discuss this suggestion.

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Problem 3. This problem deals with the relation of current to voltage for single ion channels.

Assume that conduction through an open ion channel is governed by the equation

I=(VmVe),

VwhereI

is current through a single open channel,

is the conductance of a single open channel,m is the membrane potential across the channel, and Ve is the equilibrium (reversal) potential for

the channel. For each of the channels in this problem, assume that = 25 pS and Ve = 20 mV.

a) The membrane potential Vm and the average single-channel current iobtained from threedifferent single channels (A, B, and C) are shown in Figure 1. Both the membrane potential

Channel A Channel B Channel C

t t

t60 60

20 20 +10

+80

t t

t

1

0.50.50.75 -0.2

+0.3

Vm(t

i(t

Vm(t

i(t

Vm(t

i(t

) (mV)

) (pA)

) (mV)

) (pA)

) (mV)

) (pA)

Figure 1: Average single-channel currents.

and current are plotted on a time scale such that the changes appear instantaneous and only

the final values of these variables can be discerned in the plots; i.e., the kinetics are not

shown. For each of these channels, answer the following questions and explain your answers:

i) Is this channel voltage-gated for the illustrated depolarization?

ii) Is the channel activated (opened) or inactivated (closed) by the illustrated depolariza

tion?

b) Assume that each voltage-gated channel contains one two-state gate where is the timeconstant of transition between states. For each of the channels, sketch the time course of i(t)on a normalized time scale t/. Clearly show the current near t= 0.

Problem 4. Three three-state voltage-gated channels (channels a, b, and c) have the kinetic dia

gram and state occupancy probabilities shown in Figure 2. These channels have the same voltagedependent rate constants and the same equilibrium potential which is +40 mV. For the membrane

potential shown, the channels are in state 1 with probability 1 for t0. The channels differ only in their state conductances and state gating charges asshown in Figure 3. Denote the expected values of the single-channel random variables as follows:

the conductance as ga(t), gb(t), and gc(t); the ionic currents as ia(t), ib(t), and ic(t); the gatingcharges as qa(t), qb(t), and qc(t); the gating currents as iga(t), igb(t), and igc(t).

a) Which of the waveforms shown in Figure 4 best represents gb(t)? Explain.

b) Which of the waveforms shown in Figure 4 best represents gc(t)? Explain.

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5

1

10

1

0

0.5

1

0 0.5 1 1.5

State

occupancy

probability

Vm(t)

S1 S2 S3

x1(t)

x2(t)

x3(t)

Time (ms)

Figure 2: State diagram and occupancy probabilities for a three-state channel. The state occupancy

probabilities for states S1, S2, and S3 are x1(t), x2(t), and x3(t), respectively.

Channel a

5

1

10

1

5

1

10

1

Channel b

1

1

2

2 3

3

S1 S2 S3

S1 S2 S3

Q1 = 1

Q1

Q2

Q2

Q3

Q3

= 10

= 0

= 0

= 10 = 0

= 0

= 0

= 0

= 0

= 0

= 1

Channel c

5

1

10

1

1

2

3

S1

S2

S3

Q1 Q2 Q3

= 0 = 0 = 10

= 1 = 0 = 0

Figure 3: State diagrams of three three-state channel models. The models differ in state conduc

tances and state gating charge but not in rate constants.

c) Which of the waveforms shown in Figure 4 best represents iga(t)? Explain.

d) Which of the waveforms shown in Figure 4 best represents igc(t)? Explain.

e) Which of these channel models exhibits activation followed by inactivation of the ionic cur

rent? Explain.

f) Which of these channel models exhibits an ionic current that does not inactivate? Explain.

g) Which of these channel models represents a channel that closes on depolarization? Explain.

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t tVm(t)Vm(t)

w1(t)

w2(t)

w3(t)

w4(t)

w5(t)

w6(t)

w7(t)

w8(t)

Figure 4: Waveforms of responses. The horizontal axis corresponds to w(t) = 0, and the verticalaxis to t= 0.

Problem 5. Figure 5 shows a model of a voltage-gated ion channel with one three-state gate plus

representative single-channel ionic and gating current records.

a) Assume that the voltage-current characteristic of the channel is the same for states 1 and 3

and is linear. Determine the open channel conductance and equilibrium (reversal) potentialfor this channel.

b) The ionic current trace shown in Figure 5 has three non-zero segments. Determine which

state the gate is in during each non-zero segment. Explain your reasoning.

c) Figure 6 illustrates the dependence of the steady-state probability that the channel will be

in each of its three states on the membrane potential. Let iss represent the average valueof the ionic current that results after steady-state conditions are reached in a voltage clamp

experiment in which Vm is held constant. Assume that the experiment is repeated for anumber of different values of membrane potential Vm. Plot the relation between iss and Vm.Describe the important features of your plot.

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state 1

t

(ms)

+50

50

inside

outside

+Vm (mV)

state 2 i(pA)

t (ms)

32

1

inside

outside

+

state 3

t (ms)

inside

outside

+

ig (pA)

Figure 5: Channel with one three-state gate. The left panels illustrate the three states: states 1 and

3 are open states, state 2 is a closed state. The right panels illustrate the responses of the channel

to a step in membrane potential Vm(t) at time t= 0 (top right) which gives rise to the ionic currenti(t) and gating current ig(t) illustrated in the middle right and lower right panels, respectively.

100 1000

0.5

1

x1 x2 x3

Membrane potential (mV)

Figure 6: Steady-state probabilities for a channel with one three-state gate. x1, x2, and x3represent the steady-state probabilities of being in state 1, state 2, and state 3, respectively, as a

function of membrane potential.

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MASSACHUSETTS

INSTITUTE

OF

TECHNOLOGY

Department

of

Electrical

Engineering

and

Computer

Science,

Department

of

Mechanical

Engineering,

Division

of

Bioengineering

and

Environmental

Health,

Harvard-MIT

Division

of

Health

Sciences

and

Technology

QuantitativePhysiology:CellsandTissues

2.791J/2.794J/6.021J/6.521J/BE.370J/BE.470J/HST.541J

Hodgkin-HuxleyProject

This

theoretical

project

is

intended

to

provide

an

opportunity

to

learn

about

the

complex

behav

iors

that

can

be

exhibited

by

the

Hodgkin-Huxley

model

and

to

compare

behaviors

of

the

model

with

behaviors

of

electrically

excitable

cells.

We

have

two

simulations.

The

first

models

a

space-

clamped

axon.

That

model

generatesmembraneactionpotentials. Thesecondmodelsanaxon

without

space-clamp

(although

the

second

model

can

simulate

a

space

clamp

since

the

longitudi

nalresistancescanbesettozero). Thesecondmodelcanproducepropagatedactionpotentials.

Thesecondmodelcanbeused toexploreawiderrangeofphenomena than thefirst. Thefirst

model

is

faster

and

simpler

than

the

second.

Thus

both

models

are

useful,

and

your

project

can

use

eitherorbothofthemodels.

StudentsareSTRONGLYencouraged toworkinpairs,however, individualprojectswillbe

approvedifthereareextenuatingcircumstances.Ifapairofstudentscollaborateonaprojectthey

shouldsubmitasingleproposalandasinglereportwhichidentifiesbothmembersoftheteamand

gives

both

email

addresses.

Proposals

should

be

submitted

via

the

form

available

on theMIT

server. Proposalswillbereturnedassoonaspossiblesothatstudentscanrevisethem. Onlythe

final,

accepted

proposal

will

be

given

a

grade.

Thedemonstrationprojectperformedinlectureontheeffectoftemperaturecannotbethebasis

ofastudentproject.

Practicalconsiderationsinthechoiceofatopic

ProjectscaninvolvealmostanyofthepropertiesoftheHodgkin-Huxleymodel.However,toavoid

projectswhoseaimsarevague(e.g.,IwouldliketounderstandhowtheHodgkin-Huxleymodel

works)theproposedprojectshouldbeintheformofaspecificandtestablehypothesis.Projects

thatinvolvemonthsofcomputationshouldobviouslybeavoided.Theamountofcomputationtime

shouldbeexplicitlytakenintoaccountinplanningaproject.Forexample,anyprojectthatinvolves

measuringthethresholdofoccurrenceofanactionpotentialformanydifferentparametervaluesis

boundtobeverytimeconsuming,becausedeterminingthethresholdforasinglesetofparameters

itself

involves

many

computations.

The

task

is

to

choose

a

physiological

property

of

the

excitation

of

the

action

potential

that

is

of

interest,

and

then

to

define

a

specific,

feasible

project.

Choiceoftopics

Topics

can

involve

comparing

predictions

of

the

Hodgkin-Huxley

model

with

measurements

on

cells.

For

example,

the

text

contains

data

on

the

effects

of

many

external

parameters

(e.g.,

ionic

concentrations,cell type)onactionpotentials. Aprojectmightinvolvereadingtheoriginalpa

persthatdescribesuchmeasurements(someweremadebeforetheHodgkin-Huxleymodelwas

formulated),and testingthehypothesisthatthesemeasurementsare(orarenot)consistentwith

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the

Hodgkin-Huxley

model.

Similarly,

a

project

might

involve

examining

the

effect

of

some

phar

macological

substance

on

measurements

of

the

action

potential

and

testing

the

hypothesis

that

the

substance

produces

its

effect

by

changing

one

or

another

parameter

of

the

model.

These

projects

will

require

some

reading

of

original

literature

which

is

often

difficult

and

usually

time

consuming.

However,

such

a

project

can

lead

to

a

very

rewarding

educational

experience.

Alternatively,

the

projectmightinvolveapurelytheoreticaltopicinwhichsomepropertyofthemodelisexplained

in

terms

of

its

underlying

structure.

This

type

of

project

does

not

necessarily

involve

reading

theoriginalliterature.

Examplesofhypotheses

1.

Hypothesis

The

effect

of

temperature

on

the

conduction

velocity

of

the

squid

giant

axon

can

be

fit

by

the

Hodgkin-Huxley

model.

Articles

in

the

literature

should

be

consulted

for

this

project:

Chapman,R.A.(1967).Dependenceontemperatureoftheconductionvelocityofthe

actionpotentialofthesquidgiantaxon.J.Physiol.213:1143-1144.

Easton,D.M.andSwenberg,C.E.(1975).Temperatureandimpulsevelocityingiant

axon

of

squidloligopealei.Am.J.Physiol.229:1249-1253.

2. HypothesisWhentwoactionpotentialsareelicited,onejustafteranother, thevelocity

ofthesecondisslowerthanthevelocityofthefirstactionpotential. Thisphenomenonis

predicted

by

the

Hodgkin-Huxley

model.

Articles

in

the

literature

should

be

consulted

for

this

project:

George,S.A.,Mastronarde,D.N.,andDubin,M.W.(1984).Prioractivityinfluences

thevelocityofimpulsesinfrogandcatopticnervefibers.BrainRes.304:121-126.

3.

Hypothesis

The

threshold

current

for

eliciting

an

action

potential

with

an

intracellular

electrode

is

higher

for

a

space-clamped

than

for

an

unclamped

model

of

an

axon.

4. HypothesisIncreasingthemembranecapacitancewilldecreasetheconductionvelocity.

5.

Hypothesis

Increasing

the

membrane

conductance

(by

scaling

all

the

ionic

conductances)

will

increase

the

conduction

velocity.

6. HypothesisIncreasingtheexternalconcentrationofsodiumwillincreasetheconduction

velocity.

7.

Hypothesis

Increasing

the

external

concentration

of

potassium

will

increase

the

conduc

tion

velocity.

8. HypothesisIncreasingtheexternalconcentrationofcalciumwillincreasetheconduction

velocity.

9.

Hypothesis

Increasing

the

temperature

will

increase

the

conduction

velocity.

10. HypothesisThedifferenceinwaveformoftheactionpotentialofafrognodeofRanvier

andofasquidgiantaxon(Figure1.9 involume2of the text)canbe reproducedby the

Hodgkin-Huxley

model

of

a

squid

giant

axon

by

a

change

in

temperature.

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11.

Hypothesis

The

membrane

capacitance

determines

the

time

course

of

the

rising

phase

of

the

action

potential.

Increasing

the

membrane

capacitance

decreases

the

rate

of

increase

of

the

rising

phase

of

the

action

potential.

12.

Hypothesis

The

falling

phase

of

the

action

potential

(repolarization)

can

occur

in

the

absence

of

a

change

in

potassium

conductance.

13.

Hypothesis

Increasing

the

temperature

sufficiently

blocks

the

occurrence

of

the

action

potential

because

the

membrane

time

constant

limits

the

rate

at

which

the

membrane

vari

ables

can

change

and

prevents

the

difference

in

time

course

of

the

sodium

and

potassium

activation

which

is

responsible

for

initiation

of

the

action

potential.

14.

Hypothesis

The

initiation

of

the

action

potential

is

independent

of

the

potassium

conduc

tance.

15.

Hypothesis

The

prolonged

plateau

of

the

cardiac

muscle

action

potential

can

be

accounted

for

by

the

Hodgkin-Huxley

model

with

a

potassium

conductance

that

has

a

slow

activation.

16.

Hypothesis

The

effect

of

tetraethylammonium

chloride

(TEA)

on

the

action

potential

of

the

squid

giant

axon

can

be

modelled

with

the

Hodgkin

Huxley

model

by

decreasing

Kn

and

increasing

Kh.Articlesintheliteratureshouldbeconsultedforthisproject:

Armstrong,C.M.(1966). TimecourseofTEA+-inducedanamalousrectificationin

squid

giant

axons.J.Gen.Physiol.50:491-503.

Armstrong,C.M.andBinstock,L.(1965).Anomalousrectificationinthesquidgiant

axon

injected

with

tetraethylammonium

chloride.J.Gen.Physiol.48:859-872.

Tasaki,I.andHagiwara,S.(1957).Demonstrationoftwostablepotentialstatesinthe

squidgiantaxonundertetraethylammoniumchloride.J.Gen.Physiol.40:859-885.

17.

Hypothesis

The

shape

of

the

action

potential

in

the

presence

of

tetraethylammonium

chloride(TEA)canbeaccountedforbytheHodgkin-Huxleymodelwithareducedmaximum

valueofthepotassiumconductance. Articlesintheliteratureshouldbeconsultedforthis

project:

Armstrong,C.M.(1966). TimecourseofTEA+-inducedanamalousrectificationin

squidgiantaxons.J.Gen.Physiol.50:491-503.

Armstrong,C.M.andBinstock,L.(1965).Anomalousrectificationinthesquidgiant

axoninjectedwithtetraethylammoniumchloride.J.Gen.Physiol.48:859-872.

Tasaki,I.andHagiwara,S.(1957).Demonstrationoftwostablepotentialstatesinthe

squid

giant

axon

under

tetraethylammonium

chloride.

J.

Gen.

Physiol.

40:859-885.

18.

Hypothesis

Increasing

the

external

calcium

concentration

will

block

the

occurrence

of

the

actionpotentialbecausethiswillreducethedifferenceinthetimeconstantofsodiumand

potassiumactivationwhichisresponsiblefortheinitiationoftheactionpotential.

19.

Hypothesis

Increasing

the

external

concentration

of

potassium

will

decrease

the

refractory

period;decreasingthisconcentrationwilllengthentherefractoryperiod.

20.

Hypothesis

Increasing

the

external

concentration

of

sodium

will

decrease

the

refractory

period;decreasingthisconcentrationwilllengthentherefractoryperiod.

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21.

Hypothesis

Absolute

and

relative

refractory

periods

are

decreased

by

increasing

the

rate

constants

for

sodium

inactivation

and

for

potassium

activation.

22. HypothesisRepolarizationcannotoccurifthepotassiumactivationrateconstantiszero.

23. HypothesisThethresholdoftheactionpotentialtoabriefpulseofcurrentdecreasesas

theexternalpotassiumcurrentisincreased.

24.

The

Hodgkin-Huxley

model

with

default

parameters

does

not

exhibit

accommodation.

Hy

pothesis

Accommodation

occurs

if

the

leakage

conductance

is

increased.

25. TheHodgkin-Huxleymodelwithdefaultparametersdoesnotexhibitaccommodation.Hy

pothesisAccommodationoccursifthepotassiumconductanceisincreased.

26. HypothesisIncreasingtheleakageequilibriumpotentialwillblocktheactionpotential.

27.

Hypothesis

The

effect

of

the

changes

in

concentration

of

sodium

ions

on

the

action

po

tential

of

the

giant

axon

of

the

squid

can

be

accounted

for

by

the

Hodgkin-Huxley

model.

Articles

in

the

literature

should

be

consulted

for

this

project:

HodgkinA.L.andKatz,B.(1949).Theeffectofsodiumionsontheelectricalactivity

of

the

giant

axon

of

the

squid.J.Physiol.108:37-77.

Baker,P.F.,Hodgkin,A.L.,andShaw,T.I.(1961).Replacementoftheprotoplasmof

agiantnervefibrewithartificialsolutions.Nature190:885-887.

28.

Hypothesis

In

response

to

rectangular

pulses

of

current,

the

rheobase

of

the

strength-

duration

relation

increases

as

temperature

increases.

29. HypothesisAnincreaseintemperatureresultsinadecreaseinthedurationoftherefrac

toryperiod.

30. HypothesisThethresholdmembranepotentialatwhichtheHodgkin-Huxleymodelpro

ducesanactionpotentialinresponsetoabriefpulseofcurrent isequal to themembrane

potentialforwhichthelinearizedHodgkin-Huxleyequationshaveunstableeigenvalues.

31.

Application

of

a

long-duration

constant

current

to

the

Hodgkin-Huxley

model

produces

a

train

of

action

potentials.

Hypothesis

The

frequency

of

the

action

potentials

increases

withincreasingcurrentamplitude.

32.

Application

of

a

long-duration

constant

current

to

the

Hodgkin-Huxley

mod

6021 fall 2004

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