# PROBLEM SET 1.2. KINETICS AND DIFFUSION

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1.PS2.1

PROBLEM SET 1.2. KINETICS AND DIFFUSION

1. A. The empirical formula of glucose is C6H12O6. W hat is its m olecular weight?

B. Isotonic glucose is 5% glucose w:v. How much glucose would we need to make 100 mL of

isotonic glucose?

2. A. You need to make 250 mL of a stock solution of 0.1 M Na2 ATP. Its formula weight is 605.2

g mol-1. How much Na2 ATP should you weigh out?

B. Your advisor is skeptical of your abilities. He wants you to check out the 0.1 M ATP solution

and tells you to do it spectrophotometrically. Spectrophotometry relies on the different

abilities of chemicals to absorb light of specific wavelengths. A diagram of a

spectrophotometer is shown below.

Fig. 1.PS2.1. Light path in a single-beam

spectrophotometer. The view is from above.

Light from a source is collimited (making a

narrow beam) and passed through a

monochromator that selects a narrow band

of wavelength of light to be passed through

the sample. A photomultiplier tube (PMT)

detects the light and measures its intensity.

Comparison of this intensity, I, to the

intensity when the sample is missing, I0,

allows calculation of the absorbance.

Absorbance is recorded with time or as a

function of wavelength.

At particular wavelengths, chemicals absorb light according to their chemical structure and their

concentration. The law governing the absorption of light is the Beer-Lambert Law:

A = , C d

where A is the absorbance, , is a constant that depends on the chemical and typically varies with the

wavelength of light - it is the molar extinction coefficient and is in units of M-1 ; C is the concentration of

the chemical (in M) and d is the path length. The molar extinction coefficient is defined for a path length of

1 cm. The absorbance is defined as

A = log ( I0 / I )

where I0 is the incident light intensity and I is the transmitted light inten sity.

Your advisor tells you that ,259 = 15.4 x 103 M-1; this is the molar extinction coefficient of ATP at a

wavelength of incident light of 259 nm. He tells you to make a dilution of the stock by taking 25 :L of the

stock solution and diluting it to 100 mL. W hat absorbance do you expect of the final diluted solution, if you

made it up correctly, at 8 = 259 nm?

1.PS2.2

3. A. The molecular weight of ryanodine is 493.54 g mol-1. You want to make 10 mL of a 10 mM

stock solution. How much ryanodine should you weigh out?

B. You make a dilution of the 10 mM ryanodine stock by pipetting 10 :L of the stock solution to

a 10 mL volumetric flask and adding water to the mark. You measure the absorbance as a

function of wavelength (against a water blank, using a standard 1 cm path length optical cell)

and find a peak at 271 nm with an absorbance of 0.179. W hat is ,271 for ryanodine? (See

1.PS2 problem #2 for a discussion of the Beer-Lampert Law and a definition of the molar

extinction coefficient).

4. A. Magnesium chloride has a formula of MgCl2 C 6H2O. W hat is its formula weight?

B. You desire to make 1 L of 0.1 MgCl2 solution. How much MgCl2 C 6H2O should you weigh

out?

C. You need to make 25 mL of a 25 mM solution of MgCl2. How much of the 0.1 M stock

solution do you add to the 25 mL volumetric flask?

5. The extracellular fluid volume varies with the size of the person. Suppose in an individual we

determine that the ECF is 14 L. The average [Na+] in the ECF is about 143 mM.

A. What is the total amount of Na+ in the ECF, in moles? In grams?

B. Suppose this person works out and sweats 1.5 L with an average [Na+] of 50 mM. During

this time the urine output is 30 mL with an average [Na+] of 600 mM. How much Na+ is lost

during the workout?

C. If the person does not drink fluids at all during the workout, what will be the [Na+] in the

plasma at the end of the workout? Assume that all of the fluid in the sweat and urine

originated from the ECF.

6. The body normally produces about 2 g of creatinine per day. The amount varies with individuals and

is approximately proportional to the muscle mass. It is excreted through the kidneys according to

Urinary excretion of creatinine = GFR x Plasma concentration of creatinine, where GFR is an

abbreviation for “glomerular filtration rate”. If the GFR is 120 mL min-1, what is the plasma

concentration of creatinine at steady-state? Hint: assume the body is at steady state with respect

to creatinine.

7. Just before noon, your plasma glucose concentration was 100 mg dL-1. This plasma glucose is

approximately evenly distributed among 3.5 L of plasma and 10.5 L of interstitial fluid that comprises

your 14 L of ECF. Glucose is readily distributed in both compartm ents. You drink a can of soda that

contains 35 g of glucose.

A. How much would you blood glucose rise if all the glucose in the soda was absorbed and none

of it was metabolized?

B. Given that post-prandial (after eating) increases in blood glucose amount to maybe 40 mg

dL-1, depending on the meal, over a period of an hour, give a crude estimate of the rate of

glucose uptake by the peripheral tissues. Assume that the meal contains 100 g of

carbohydrates and all of it is absorbed in one hour.

1.PS2.3

8. The association reaction for Ca2+ and EGTA (a chemical that binds Ca2+) is written as

Ca2+ + EGTA W CaCEGTA

Under defined and particular conditions of temperature and ionic mixture, the association constant

was determined to be KA = 2.52 x 106 M-1. In a chemical mixture, 400 :M EGTA was included

and the free [Ca2+] determined by a Ca2+-selective electrode was found to be 4 x 10-7 M. Assuming

that there are no other binding agents for Ca2+, what is the total [Ca2+] in the mixture?

9. 2,4-dinotrophenyl acetate decomposes in alkaline solution with a pseudo-first order rate constant of

11.7 s-1 at 25°. It is a “pseudo” first order rate constant because it depends on the pH.

A. If the initial concentration of DNPA is 1 mM, what is its concentration after 15 s?

B. At what tim e is the concentration reduced to 0.5 mM (that is, what is the half-life of the

reaction)?

C. After 5 min of reaction, what is the concentration of DNPA.

10. The following data were obtained for the rate of the Mg-Ca-ATPase activity of vesicles of cardiac

sarcoplasmic reticulum as a function of temperature. W hat can you tell about the activation energy?

Temperature

(°C)

ATPase rate

(:mol m in-1mg-1)

6.9 0.068

11.5 0.138

15.8 0.300

19.8 0.568

20.2 0.585

25.6 1.236

26.1 1.154

31.0 2.238

34.8 3.030

39.2 4.220

1.PS2.4

11. Superoxide reduces cytochrome C in the reaction

Cyt CCFe3+ + O2

- Y Cyt CCFe2+ + O2

Where Cyt CCFe3+ is the oxidized form and Cyt CCFe2+ is the reduced form of cytochrome C. The

reaction can be followed spectrophotometrically at 550 nm. The extinction coefficient for the reduced

form of Cytochrome C is ,RED = 2.99 x 104 M-1 and the extinction coefficient for the oxidized form ,OX

= 0.89 x 104 M-1 (Massey, V. The microestimation of succinate and the extinction coefficient of

cytochrome C. Biochim. Biophys. Acta 34:255-256 (1959)). See 1.PS2 problem #2 for a discussion

of extinction coefficients and spectrophotometry.

When xanthine oxidase converts xanthine to uric acid, it produces superoxide that can be measured

using cytochrom e C reduction. The following data were obtained for A550:

Time (min) A550

0 0.1326

1 0.1478

2 0.1637

3 0.1791

4 0.1941

5 0.2073

6 0.2202

A. Calculate the rate of cytochrome C reduction.

B. The xanthine oxidase was added in 75:L of 6.5 mg XO per mL into a 3 mL reaction mixture.

Calculate the specific activity of cytochrome C reduction (mols of cytochrome C reduced per

min per mg of XO protein.)

12. You suspect you are anemic and your physician orders some tests. He finds that your hemoglobin

is 13g %. The molecular weight of hemoglobin is 66,500 g mol-1.

A. What is the concentration of hemoglobin in molar in your blood?

B. Each hemoglobin binds 4 oxygen molecules. If the hemoglobin is saturated with oxygen,

what is the concentration of O2 bound to Hb, in molar?

C. Convert the answer in B to volume using the ideal gas equation, PV = nRT where T is the

absolute temperature, R = 0.082 L atm mol-1 °K-1, V is the volume that we seek and P = 1

atm. The conditions for volume of gas are usually STPD - standard temperature and

pressure, dry. The standard temperature is 0 °C and pressure is 1 atm.

1.PS2.5

13. The rate of ATP hydrolysis by ATPases can be followed by the coupled enzyme assay shown below:

Fig. 1.PS2.2 ATP hydrolysis of pyruvate kinase converts

phosphoenolpyruvate to pyruvic acid. This is coupled by lactic

dehydrogenase to the conversion of pyruvic acid to lactic acid

and conversion of NADH to NAD+. The progress of the

reaction can be followed spectrophotometrically by the change

in absorbance of NADH.

The progress of the reaction can be followed by A340. The extinction coefficient of NAD+ at 340 nm

is negligible. The exctinction coeffcient of NADH at 340 nm is 6.2 x 103 M-1. See 1.PS2.2 problem

#2 for a discussion of extinction coefficients and spectrophotometry. In one reaction, the

concentration of Ca-ATPase was 0.22 mg mL-1 and A340 was 0.65 at t=0 min and 0.455 at t=2.0 min.

What is the activity of the Ca-ATPase in units of :mol m in-1 mg-1?

14. Show by representative calculations that Stirling’s formula

is a good approximation for n!. Use n=1,2,3,4,5

15 Show that the equation

obeys Fick’s Second Law of Diffusion.

1.PS2.6

16. The intestinal enterocytes form a covering over the intestinal lining which, to the first approximation,

can be considered to be a plane. Assuming no binding or sequestration within the cell, what is the

estimated time of diffusion of Ca2+ across the intestinal enterocyte? The length of the enterocyte is

20 :m and assume that the effective diffusion coefficient of Ca2+ is about 0.4 x 10-5 cm2s-1.

17. Table 1.PS2.1 lists the diffusion coefficients and the molecular weight of a variety of proteins. W hat

relationship can you deduce between the size and the diffusion coefficients of these soluble proteins?

(Hint: regress ln D against ln Mr.). Is the relationship you found consistent with the Stokes-Einstein

equation?

Table 1.PS2.1 Diffusion coefficients and Mr for a variety of proteins

Protein Molecular Weight D x 107 (cm2s-1)

milk lipase 6,600 14.5

Metallothionein 9,700 12.4

Cytochrome C 12,000 12.9

Ribonuclease 12,600 13.1

Myoglobin 16,890 11.3

Chymotrypsinogen 23,200 9.5

Carbonic anhydrase 30,600 10.0

Peroxidase II 44,050 6.8

Albumin 68,500 6.1

Lactoperoxidase 92,620 6.0

Aldolase 149,100 4.6

18. The free diffusion coefficient of oxygen in aqueous solutions is about 1.5 x 10-5 cm2 s-1. If the diffusion

distance between air and blood is 0.5 :m, about how long is the diffusion time?

19. Suppose a soluble protein has a molecular weight of 45 kDa and a density of 1.06 g cm-3. Suppose

further that the viscosity of the cytoplasm has a viscosity of 0.005 Pa s (about five times that of water -

there is debate about the visc osity of cytoplasm with numbers varying from 0.001 to over .1 Pa s).

A. Estimate the diffusion coefficient for the protein in the cytoplasm at 37°C

B. If the protein were synthesized in the cell body, or soma, of a neuron in the spinal cord, about

how long would it tak e to diffuse to the axon terminal 75 cm away?

1.PS2.7

20. Diffusion coefficients in cytoplasm has been estimated by a technique of photobleaching recovery.

In this technique, an area of the cytoplasm is irradiated with light to photobleach a fluorescent probe.

Recovery of fluorescence in the region is achieved by diffusion of unbleached probes from adjacent

areas of the cytoplasm. The translational diffusion coefficient can be estimated from the half-time of

fluorescent recovery. (Axelrod, D., et al., Mobility measurements by analysis of fluorescence

photobleaching recovery kinetics. Biophys. J. 16:1055-1069 (1976)). This technique was applied to

estim ate the relative viscosity of cytoplasm and nucleoplasm by microinjecting fluorescein

isothiocyanate-labeled dextrans of varying molecular sizes and measuring the fluorescence

photobleaching recovery (Lang, I., et al., Molecular mobility and nucleoplasmic flux in hepatoma cells.

J. Cell Biol. 102:1183-1190 (1986)). These authors obtained the following data:

Probe Molecular

Weight

Equivalent

radius

D in Dilute

solution

D in cytoplasm D in nucleoplasm

(kD) (nm) D is in units of 10-6 cm2s-1

FD20 17.5 3.30 0.651 0.080 ---

FD40 41.0 4.64 0.463 0.044 0.069

FD70 62.0 5.51 0.390 0.029 0.056

FD150 156.9 9.07 0.237 0.015 0.036

A. Plot D against 1/a, where a is the molecular radius, for each of the solutions. From the

Stokes-Einstein relation, you would expect the resulting curves to pass through the origin of

zero diffusion coefficient with infinite radius. Do the curves extrapolate back in this way?

Why or why not?

B. Regardless of the intercept, the slope of the plot from part A ought to be related to the

viscosity of the medium. Use the slopes to estimate the relative viscosity of the dilute

solution, cytoplasm and nucleoplasm.