These first two problems are designed to get you to think in terms of steady-state and transient situations, which might occur in a variety of physiological contexts. They assume a basic knowledge of energy metabolism and enzyme kinetics, which is summarized on pages 70 to 90 of "Eckert Animal Physiology" by Randall et al.

1. Consider a tissue that contains (ATP+ADP)= 5mM, and (phosphocreatine + creatine) = (PCr+Cr)=40 mM. It uses oxidative metabolism of glucose to convert ADP to ATP, which is then utilized for various cell processes. Assume that the production of ATP is regulated like a simple enzyme reaction, with KM = 0.5 mM ADP, and Vmax = 200 mM of ATP produced/min.

a) How much "oxygen debt" is incurred in going from a resting level of metabolism (ATP hydrolysis) of 5 mM/min to an active level of 100 mM/min? What is meant here by "oxygen debt" is: How much oxidative metabolism (producing 6 ATP per O2) will be required to restore the original steady state equilibrium if the ATP usage suddenly drops from the high level to the low level? In effect, this represents energy that was drawn from the existing concentrations of PCr and ATP to establish the high level of output, and is then replenished when the tissue drops back down to the initial low level.
Use a value of 20 for the equilibrium constant for

ADP + PCr --> ATP + Cr.

After calculating your answer in mM of oxygen, make a conversion to "measurable" units: ml of oxygen per gm of tissue. (Assume 1 gm tissue = 0.8 ml of "solution", and T = 30 C.)

b) Part (a) tells us the magnitude of the oxygen debt, but we might also want to know the recovery time. We could ask: If the ATP usage changes back to the resting level instantaneously, how long will it take for the "oxygen debt" to be 90% "repaid"?
An accurate analysis of this question is not trivial, and is probably best done by numerical simulation. If you don't want to work that hard, just use the approximation that the recovery follows an exponential time course. So once you figure out the initial rate, the rest is easy.

2. Now let's be slightly more realistic and consider that the tissue has the capacity for anaerobic metabolism of glucose to lactate to produce ATP, in addition to its aerobic metabolism. Assume the same total ADP+ATP and Cr+PCr concentrations as in Problem 1. Assume that the initial glycolytic steps have a maximum capacity for utilizing glucose at 300 mM glucose/min, producing 600 mM of ATP/min, with a KM of 0.5 mM ADP. The oxidative steps can handle up to 10 mM/min of pyruvate, to produce 180 mM of ATP/min (again with KM = 0.5 mM ADP). Assume that the usage of ATP by the tissue can also be described by simple enzyme kinetics, with a maximum rate of ATP usage of 500 mM/min, a KM for substrate ATP of 0.2 mM, and a KI for competitive inhibition by ADP of 0.4 mM. (These are realistic KM and KI values for muscle.)
(You will need to use the standard equation involving KM and KI, which is not given in a very useful form in the Eckert text.
Reaction rate V = Vmax/{1+(1+[I]/KI)KM/[S]}).

a) What is the maximum metabolic rate that can be reached by this tissue, if excess pyruvate is converted to lactic acid and allowed to accumulate? What is the rate of lactic acid accumulation? (A graphical solution is acceptable, and may be a good way to understand how this system is working.)
b) Accumulation of lactic acid in the tissues will slow things down. Let's assume that the accumulation of lactic acid causes the glycolysis rate to be reduced to the point where there is a balance between lactic acid production by glycolysis and lactic acid consumption by oxidative metabolism. Assume that lactic acid accumulation has inhibited the use of ATP, by reducing the Vmax for ATP usage to 200 mM/min. What is the steady-state rate of metabolism under these conditions?
This situation is a gross oversimplification of reality, which makes the mathematics easier. You can imagine that a real attempt to model these situations would at least have to introduce functional relationships between rates and lactic acid concentration, and consider the varying concentration of lactic acid. It is also unlikely that the values of KM for both metabolic reactions would be the same. In real muscles, the ratio between anaerobic and aerobic metabolic rates is usually larger than the ratio of about 3.0 calculated here.
c) Now describe quantitatively what happens when the tissue goes from a resting level of metabolism (with no lactic acid accumulation) using 5 mM ATP/min to its maximum steady state rate, calculated in (a). Estimate the total "oxygen debt" after 1 minute of activity at this maximal rate. (A possible answer is 54.2 mM O2; this is based on the assumption that the increased level of ATP usage is entirely supported by draw-down of ATP and PCr until these reach their new steady state levels.)
Does this calculation underestimate the true "oxygen debt" associated with the transition to a high activity level? Is this a major error?

3) Assume that a motor enzyme operates by forming cross-bridges with the following three properties:
a) At equilibrium (velocity V=0), each cross-bridge exerts an average force of 5 pN.
b) Quick releases cause a decrease in force, with each cross-bridge acting like a simple linear elastic element, with a force constant of 1 pN/nm.
c) After a quick release, the force recovers towards its equilibrium value by a simple exponential recovery process, with a rate constant k = 1000/sec.
Calculate the shape of the force-velocity curve for this system, the maximum sliding velocity, and the maximum power output per cross-bridge. If the maximum possible work that can be generated by hydrolysis of 1 ATP molecule is 60 pN nm, what is the maximum amount of sliding per ATP for each motor enzyme molecule at the velocity that gives maximum power output? If a muscle is made of a a mixture of fibers with different values of k, what will be the force-velocity relationship for the composite muscle?

4) Assume that Hill's equation (see page 363-364 of EAP.)

(P+a)(V+b) = b(P0+a)

is a valid description of the force (P) -- velocity (V) behavior of muscle. (Note that b = a Vmax/P0)
Consider a muscle that contains equal numbers of fast and slow muscle fibers.
For the fast fibers, P0 = 1, Vmax = 10, a = 0.2
For the slow fibers, P0 = 1, Vmax = 4, a = 0.2
Compute the force -- velocity curve for this composite muscle. Does it satisfy Hill's equation?
What if the values of a for the two fiber types are not the same?
What do you conclude about the interpretation of experimentally measured force-velocity curves?

5) Consider a vertebrate skeletal muscle with the following parameters:
Sarcomere length 2.4 Ám.
Myosin filament length 1.61 Ám, with a bare zone that is 0.15 Ám long.
Actin filaments begin to overlap if the sarcomere length is reduced to 2.0 Ám.
Three myosin molecules at 14.3 nm intervals along the myosin filament.
Distance between center of myosin filament and center of adjacent actin filaments is 24 nm.
a) calculate the concentration of myosin molecules in the myofibril (moles/liter).
b) calculate the number of myosin molecules that can act in parallel to exert force on 1 Ám2 of muscle cross-section. (Note that the number of myosin heads is twice this.)
c) calculate the concentration of troponin C molecules in the myofibril.
Use these results in the following problems.

The problems in the next group deal primarily with diffusion; the handout on diffusion should be the starting point for obtaining the information and methods that you need.

6) Consider a cylindrical myofibril with a diameter of 2.0 Ám. It is excited to make twitch contractions during which it shortens by about 25% L0, using about 5 ATP per myosin molecule.
If the surface of the myofibril is maintained at [ATP]=5 mM and [ADP]=~0, what is the maximum possible rate of contraction (twitches per second)? Use 0.15 x 10-5 cm2/sec for the ATP diffusion coefficient.
a) Use an approximate approach. Assume that the [ATP] at the center of the myofibril is 1 mM, and that the ATP and ADP concentrations do not influence the behavior of the muscle. This reduces the problem to a simple problem of diffusion with constant chemical reaction, in cylindrical coordinates.
A more complete analysis, using values of the KM for ATP and the KI for ADP given in Problem 2, is more difficult. You might think about how to deal with this problem.
b) Now consider a more realistic situation where phosphocreatine (PCr) is also available, with concentrations of 16 mM PCr and 4 mM creatine maintained at the surface of the myofibril. Make a rough estimate of how this will change the answer.
A complete analysis could be done numerically, as done for spermatozoa by Tombes et al.
c) Now repeat your calculations assuming a diameter of 20 Ám for the myofibril.
What conclusions do you reach from these calculations?


7) Now lets try to analyse the diffusion of Ca++ ions in and out of the myofibril to activate and deactivate contraction. Assume that the inactive state corresponds to [Ca++]<=10-7 M, with almost no calcium bound to troponin C (TnC), and that the active state corresponds to [Ca++] = 2 x 10-5 M, with 2 Ca++ ions bound by each TnC molecule. From the TnC content, you can calculate how much Ca++ must enter the myofibril to achieve activation. If the periphery of the myofibril is suddenly exposed to a high Ca++ concentration, how long will it take to activate contraction?
This is not a steady-state diffusion problem, and the presence of TnC makes this problem different from Problem (6). Here is a way to get an approximate answer:
If you ignore the effect of TnC on diffusion, you can answer the following question: If the concentration at the surface of the myofibril is suddenly raised to and maintained at a concentration C0, how long will it take for the necessary amount of Ca++ ions to enter the myofibril? You can look up the solution to this problem in Crank -- see Fig. 5.7. Suppose C0 is set at 0.025 mM. The curve labelled 0 (why this one?) in this figure tells us that the average internal concentration of Ca++ will reach 0.020 mM, or 80% of its final value, when sqrt(Dt/a2) = 0.46, where a is the radius of the myofibril. An accepted value for the diffusion coefficient for Ca++ ions in cytoplasm is 0.2 x 10-5 cm2/sec, somewhat less than the value for a simple aqueous solution (0.5 x 10-5).
a) Calculate values of t for both a=1 Ám and a=10 Ám, for the case where the effect of TnC is ignored.
b) How will the presence of TnC affect these estimates?
c) What conclusion can be reached from this approximate analysis?
This is very approximate, and not very satisfying. A better estimate can be generated by numerical integration of this diffusion problem. Divide the cylinder into n cylindrical shells, with thickness dx=a/n. The diffusion flux into the ith shell from the shell outside it will be 2 D(ai/n)(C[i+1]-C[i])/dx. The diffusion flux out through the inside surface of the ith shell will be 2 D(a(i-1)/n)(C[i]-C[i-1])/dx. The rate of increase of the amount of Ca++ in the ith shell will be the difference between these two fluxes. For a very short time interval, dt, assume that the change in Ca++ content in the ith shell will be dt*(flux difference)/(volume of the shell). If there are no Ca++ binding proteins, this tells us the change in Ca++ concentration in this shell. We need to compute the concentration vs. time for each shell. The effect of TnC can be introduced in a very simple manner, assuming that the Ca++ binding sites are independent and equivalent, and equilibrate very rapidly. In this case,
total Ca++ = C + S/(1+K/C),
where C is the free Ca++ concentration, S is the concentration of TnC Ca binding sites, and K is the equilibrium constant for binding to TnC sites. We need to solve this quadratic equation for C, in order to obtain changes in free Ca++ concentration to put into the diffusion expressions.
A C++ program for these computations is given below. You should be able to get it to run without problems, as long as you use a sufficiently short time step. With the values given, dt should be 0.001 msec, or less. The stability criterion is approximately 2Ddt/dx2 <1. This is a very unsophisticated program, which runs slowly because it does only a very primitive adjustment of the size of the time steps to fit the needs of the computation. First confirm that it reproduces the results in (a) for the case where there is no TnC. (You should always examine the effects of using different dt and n values, when doing this kind of simulation.)
You could improve this model by allowing for finite rates for reaction between Ca and TnC, and for a finite shell of external space on the outside of the myofibril.
d) Now try to develop your own analysis of the time required for inactivation of a muscle when the Ca++ concentration at the surface of the myofibrils is reduced to 5 x 10-8 M. How can you redesign this system so that this time is more reasonable?



//****************
//program for calculating diffusion into a cylinder with internal binding sites
//
#include <math.h> // or #include <fp.h> if using a PowerPC Mac (also, use double instead of float)
#include <iostream.h>

void main()
{
const pi =3.14159; // actually, pi cancels out
int steps = 10000;
float dt;
float time;
cout<<"Enter value for total diffusion time (msec): ";
cin>>time;
time/=1000; // convert to sec
cout<<"Enter value for time step (msec): ";
cin>>dt;
dt/=1000; //convert to sec
steps=time/dt;
dt=dt/10; // starting value
float r;
cout<<"Enter value for radius of myofibril (Ám) ";
cin>>r;
r=0.0001*r; // convert to cm
float S=0.0000; // concentration of TnC Ca binding sites
cout<<"Enter value for concentration of TnC Ca binding sites (M): ";
cin>>S;
const float K=0.000001; // TnC ca binding constant
const float D=0.000002; //diffusion constant in cm2/sec
const int n=40; //number of intervals dividing the radius
float dx=r/n;
float C[n+2]; // free Ca++ use intervals from 1 to n
float T[n+1]; // total Ca++
float Cext;
cout<<"Enter value for external Ca concentration at time 0 (M) ";
cin>>Cext;
C[n+1]=Cext;
float Cflux = 2*D*pi*r/(dx*n);
float Cvol = pi*r*r/(n*n);
float flux, volume, b,c;
float fluxin,fluxout;
for(int i=0;i<=n;i++){
C[i]=0.0;
T[i]=0.0;}
float totalT=0.0;
time=0.0;
for(int t=1;t<=steps;t++){
if (t==50) dt*=5.0;
if (t==500) dt*=2.0;
time+=dt;
fluxin=0.0;
for(int i=1;i<=n;i++){
fluxout=fluxin; // from previous time step
fluxin=i*(C[i+1]-C[i]);
flux=Cflux*(fluxin-fluxout);
volume =Cvol*(i*i-(i-1)*(i-1));
T[i]+=dt*flux/volume;
// C*C +(K+S-T)*C -KT = 0 quadratic equation
b=K+S-T[i];
c=-K*T[i];
C[i]=0.5*(-b+sqrt(b*b-4*c));
if (t==steps && !(i%(n/10))){
cout<<i<<" : "<<C[i]<<" : "<<T[i]<<" CaTnC= "<<(T[i]-C[i])<<endl;}
}
totalT+=dt*Cflux*fluxin; //flux into cylinder so far
}
totalT/=(pi*r*r); // total Ca concentration in cylinder
cout<<"total time = "<<1000*time<<" msec"<<endl;
cout<<"total Ca entry (as M concentration) = "<<totalT<<endl;
totalT=0;
for(i=1;i<=n;i++){
totalT+=(T[i]-C[i])*(i*i-(i-1)*(i-1));}
totalT=totalT/(n*n);
cout<<"avg [TnC-bound Ca] = "<<totalT<<endl;
totalT=0;
for(i=1;i<=n;i++){
totalT+=(C[i])*(i*i-(i-1)*(i-1));}
totalT=totalT/(n*n);
cout<<"avg free [Ca++] = "<<totalT<<endl;
}

Now that you see how easy this is, you should feel comfortable about using this sort of numerical analysis for all kinds of problems. In many cases this is referred to as "compartmental analysis".
For a different approach to numerical analysis of diffusion problems, see Appendix B of H. Berg's book, "Random Walks in Biology".

8) ATP is generated in mitochondria located at the basal end of a sperm flagellum, and diffuses along the flagellum. Consider a flagellum that is using ATP uniformly along the length at a rate of 45,000 ATP molecules per second per Ám of length.

a) If the flagellum is a membrane-bounded cylinder with a length of 40 Ám, and the difference in ATP concentration between the base and tip of the flagellum is 8 mM, what must be the diameter of the flagellum in order to obtain an adequate diffusion flux of ATP? Use a value of 0.15 x 10-5 cm2/sec for the diffusion coefficient. (For the purposes of this problem, neglect any contribution of energy from dephosphorylation of ADP or phosphcreatine.)

b) Many mammalian spermatozoa have tapered tails. We might expect that this would optimize diffusion, since the required flux past any point along the length decreases along the length. However, if we simply think about optimizing diffusion, we quickly realize that the optimal solution is an infinite diameter. We need another parameter to optimize. Let's assume that we also want to minimize the cost of the membrane, by minimizing the surface area of the flagellar membrane. We could ask: What function of diameter vs. length will minimize the membrane surface area and still allow ATP to diffuse at the necessary rate? (An additional constraint is that the minimum diameter required to enclose the axoneme is 0.2 Ám.)
If you can find an answer to this question, fine. Otherwise, try a simpler question: If the flagellar diameter decreases linearly along the length, is it possible to supply the ATP and have less membrane surface area than for the case of uniform diameter examined in (a)?
The equation that you need can be obtained by starting with the assumption that at any point x along the length, the flux of ATP must be equal to q'(S-x), where q' is the rate of use of ATP per unit length, and S is the total length of the flagellum. You should be able to show that if b is the radius at the base and a is the radius at the tip of the flagellum, the required concentration difference is

q'S2 [b2ln(b/a) - b(b-a)]
--------------------------------
p Db2 (b-a)2

instead of

0.5 q' S2
----------------,
p Db2

which is the expression for the case of b=a. The comparison of the required concentration differences, for values of b and a that give the same surface area for the tapered cylinder and the uniform cylinder, tells you whether or not a tapered flagellum is better. By trial and error (or a more elegant method) you can find values of b and a that will give the target C of 8 mM.

9) Some kinds of spermatozoa are inhibited by CO2. It has been suggested that the CO2 produced by metabolism might therefore be a major factor in maintaining spermatozoa in an inactive state during storage. With sea urchin spermatozoa, experiments have shown that exposure to a mixture of 10% CO2 in either air or oxygen completely inhibits sperm motility and inhibits about 75% of the respiratory uptake of oxygen.
Consider spermatozoa that are stored in a cylindrical sperm duct at a concentration of 1011 sperm cells per cm3. The diameter of the sperm duct is 1 mm.

a) Can CO2 accumulate to inhibitory levels if the spermatozoa are respiring at 25% of the normal rate of 10-18 moles of O2/sec/spermatozoon? Use a value of 1.6 x 10-5 cm2/sec for the diffusion coeficient for CO2 in water.
As a start, assume that the sperm duct is surrounded by an unstirred cylinder of sea water with a radius of 3 mm, and that at greater radii, the environment is more or less stirred so that the CO2 concentration is very low. You need a formula that gives the steady state flux through the cylinder of sea water when its outer radius (b=3 mm) is at 0 concentration and its inner radius (a=0.5 mm) is at a concentration C= minimum concentration needed for inhibition. You should be able to derive this formula easily, or look it up in the library (an equally valuable exercise, but probably more time-consuming). You can then find the mM concentration of CO2 at the outer boundary of the sperm duct. However, to interpret this, you need to relate the CO2 concentration to the CO2 partial pressure. Standard values for the solubility of CO2 in pure water are 0.05 mM/mm Hg at 20oC and 0.03 mM/mm Hg at 37oC.
b) How much does the conclusion that you can draw from this calculation depend on the rough assumption that was made about the value of b?
c) How much does the conclusion depend on the diameter of the sperm duct?

d) There is an additional complication: the dissolved CO2 is in equilibrium with its hydrate, carbonic acid, and carbonic acid dissociates:
CO2 + H2O <--> H+ + HCO3-.
The equilibrium constant, K, for this reaction is about 5 x 10-7 for pure water at 20oC and 8 x 10-7 M for plasma at 38oC:
K=[H+][HCO3-]/[CO2].
This tells you that this reaction increases the amount of CO2 in the solution by only a tiny amount.
But what about the fact that in this problem we are dealing with sea water, which already contains 2.3 mM HCO3- ? What will be different if we dissolve CO2 in sea water instead of pure water?
What if we dissolve CO2 in blood plasma instead of pure water or sea water?

10. Consider a tissue where up to 80% of the oxygen can be removed from the blood flowing through the capillaries. Use a simple Krogh cylinder model, and equations 7.1 and 7.12 of Weibel, without worrying about the limitations of these equations.
Assume that the capillaries have diameters of 8.0 Ám and lengths of 0.80 mm. Assume that the hydrostatic pressure difference along the length of the capillaries is 20 mm Hg, and that the mean oxygen partial pressure in the capillaries is 40 mm Hg. Use a value of 2 centipoise for the "viscosity" of blood.
Calculate and plot curves showing:
a) % oxygen extraction from the blood passing through the capillaries, and
b) Number of open capillaries required per square mm of tissue cross-section,
as functions of the tissue metabolic rate, from 0.001 ml O2 per minute per cm3 up to the maximum attainable rate. (For simplicity, assume that the tissue cross-section equals the sum of the Krogh cylinder cross-sections, without worrying about the packing problem.)
What major factors have been ignored in this analysis? How much error will this introduce? (Reading Chapter 12 of Weibel will probably help you answer this question.)
Are there values of capillary diameter and length that would be more optimal for this tissue?

11. Consider an animal that has a heart rate of 80/min and a stroke volume of 100 ml. The systolic pressure is 120 mm Hg, and the filling pressure is 20 mm Hg (see EAP Fig. 12-12). What is the ratio between the oxygen consumed by the heart and the amount of oxygen transported by the blood?

12. Using the numbers in the table "Dimensions of human airway model 'A' ", calculate W(z), the viscous work generated by the flow through each generation (0-16) of conducting airways. To make the problem simpler, assume that the conducting airways do not change volume during respiration (this is not really true, except for the largest tracheal tubes). Assume that the respiratory cycle involves inhalation to the volumes shown in the table, and exhalation to a point where the volume beyond generation 16 is half the volume given by the table, at a rate of 30 cycles per minute. Plot the results as W(z) vs. z. (If you do this by hand, calculating for every other value of z is good enough.)
What conclusions can you draw from this plot?
If a disease causes a decrease in airway diameter, where will this be most serious?
Think about the problem of redesigning the lung to reduce the respiratory work with the minimal increase in total volume.
Calculate (approximately) the ratio between the total viscous work and the amount of energy that is available from metabolism, assuming that 27% of the oxygen is extracted from the volume of air taken into generations 17-23 and the alveoli.
You can use the following value for the dynamic viscosity of air in laminar flow: 1.8 x 10-4 poise.

13. Let's examine the assumption that O2 diffusion in the acini is too rapid to be a limiting factor in pulmonary gas exchange, and that air in each acinus can be considered to be "thoroughly mixed". Note that this assumption is implicitly discarded in Fig. 12.19 of Weibel.
Consider the standard lung described by Weibel, with a volume of 4.8 L at 75% maximum inflation, under conditions of relatively shallow respiration, with the lung volume cycling between 2.4 L and 2.9 L at 15 cycles per minute. Because the dead space of airway generations 0 to 17 totals about 0.2 L, the volume of fresh air drawn into the acini is only 0.3 L, for a total acini volume of 2.7 L. In the worst case, if the volumes of airway generations 18, 19 and 20 do not change when the lung contracts, the total volume of these 3 generations is 0.3 L, and fresh air will only get to the beginning of generation 21. A more reasonable estimate might be that some fresh air will be delivered into generation 21, but this doesn't really change the situation much. Transport of O2 through airway generations 21-23 to the alveoli must occur by diffusion . How can we model this diffusion problem?
Here is an approximate model: assume that at time 0, the O2 concentration of a small sphere is C0. The O2 diffuses out of the small sphere into an infinite medium where the initial concentration is 0.75*C0. How long will it take for 90% of the extra O2 in the inner sphere to diffuse out into the outer medium? This is only an approximation because in the lung we are not dealing with diffusion into an infinite medium, but into a bounded medium which could be approximated by a sphere with a volume that is 9 times the volume of the inner sphere. If the surface of the outer sphere is not allowed to increase above 0.75*C0, the rate of diffusion will be increased. Diffusion of 90% of the extra oxygen out of the inner sphere will give a reasonably uniform distribution of oxygen concentration. Section 3.2 of Crank tells you how to solve problems of this type, and Fig. 3.1 tells you that the diffusion time will be less than a2/D. What is an appropriate value for a, the radius of the inner sphere? For two spheres with a volume ratio of 9:1, the ratio of radii is 2.08:1. The total distance through airway generations 21,22,23 to the alveolar surface is about 1.9 mm, and the total path length through the acinus is estimated as about 5 mm, so 3 mm seems like a reasonable value for a. Using D for O2 = 0.2 cm2/sec, we find that t is about 0.5 sec, comfortably less than the 4 sec respiratory cycle time.
Now develop your own analysis of the situation during near maximal respiration, cycling between volumes of 2.4 L and 4.8 L at 40 cycles per minute. (This will require a rather different approach to the diffusion problem, but using a spherical model is still a good idea.)