Math 121 - Calculus for Biology I Spring Semester, 2004 Quotient Rule
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1. Hemoglobin Affinity for O₂
2. Quotient Rule
3. Graphing Example
4. Worked Examples
5. Dissociation Curve for Hemoglobin
6. Mitotic Model
7. References

• The quotient rule is developed to examine the maximum rate of change in O₂ affinity by hemoglobin
• It is used for graphing of rational functions - locating minima and maxima

Hemoglobin Affinity for O₂

• Hemoglobin is the most important molecule in erythrocytes (red blood cells)
• It has evolved to carry O₂ from the lungs and remove CO₂ from the tissues
• For humans, the hemoglobin molecule consists mainly of two a and two b polypeptide chains
• Each polypeptide chain contains a porphyrin ring with iron near the active binding site
• The four polypeptide chains fold into a quaternary structure that has evolved to very efficiently bind up to four molecules of O₂

• Oxygen is required by all of our cells
• Hemoglobin uses cooperative binding to effectively load and unload O₂ molecules
• In cooperative binding, the binding of one molecule facilitates the binding of one or more other molecules
• Cooperative binding is often seen where a steep dissociation curve is needed
• It is a variant of the Michaelis-Menten velocity curve with a characteristic S-shape
• The protein has more of an on/off function
• The steepness in the dissociation curve is needed for effective O₂ exchange
  • A small partial pressure difference in the concentration of O₂ results in easy unloading of O₂ at the tissues
  • In the lungs, the O₂ readily loads onto the hemoglobin molecules
  • A different dissociation curve allows the removal of CO₂

• Ranney and Sharma [1] give the kinetic dissociation curve for hemoglobin
• The dissociation curve for hemoglobin is highly sensitive to pH, temperature, and other factors
• Oxygen affinity is expressed by a dissociation function that measures the percent of hemoglobin in the blood saturated with O₂ as a function of the partial pressure of O₂
• The fraction of hemoglobin saturated with O₂ satisfies the function

• y is the fraction of hemoglobin saturated with O₂
• P is the partial pressure of O₂ measured in torrs
• The Hill coefficient n represents the number of molecules binding to the protein
• K is the binding equilibrium constant
• Hemoglobin has a Hill coefficient of 2.7-3.2 though it can bind cooperatively up to 4 molecules of O₂
• Experimental measurements show that the values of n and K are 3 and 19,100, respectively
• Below is a graph of the O₂ saturation curve

• Where the dissociation curve is steepest, the O₂ binds and unbinds to hemoglobin over the narrowest changes in partial pressure of O₂
• This allows the most efficient exchange of O₂ in the tissues
• The steepest part of the dissociation curve is where the derivative is at its maximum
• This is the point of inflection
• The curve is defined by a rational function, so we need a quotient rule to find its derivative

Quotient Rule

• Let f(x) and g(x) be two differentiable functions
• The quotient rule for finding the derivative of the quotient of these two functions is given by

• f '(x) and g'(x) are the derivatives of the respective functions
• In words, the quotient rule says that the derivative of the quotient is "the bottom times the derivative of the top minus the top times the derivative of the bottom all over the bottom squared."

Example: Consider the function

Sketch a graph of this function.

Solution:

Intercepts

• The y-intercept is given by y = f(0) = -1/2
• The x-intercept solves f(x) = 0
  • Set the numerator equal to zero

• The x-intercept is x = 1

Asymptotes

• The vertical asymptotes are when the denominator is zero, so

• The vertical asymptotes are x = -1 and x = 2
• The horizontal asymptote examines f(x) for large values of x
  • The largest exponents in the numerator are both 2
  • For large x, f(x) behaves like x²/ x² = 1
  • The horizontal asymptote is y = 1

Extrema

• The derivative uses the quotient rule

• The critical points are found by setting the derivative equal to zero
• Set the numerator equal to zero or

• The critical points are x_c = 1 and x_c = 5
• Evaluating the function f(x) at these critical points
  • Local maximum at (1, 0)
  • Local minimum at (5, 8/9)

Dissociation Curve for Hemoglobin

• The dissociation curve for O₂ with hemoglobin shown above uses the specific function

• Compute the derivative using the quotient rule

• Below is the graph of this derivative

• The maximum derivative occurs at about P_O2 = 21 torrs
• The exact value of the maximum derivative uses the second derivative equal to zero
• The second derivative is

• The last expression requires some algebra or Maple
• The second derivative is equal to zero when either P = 0 or 9550^1/3 = 21.22
• The point of inflection occurs at P = 21.22 with y(P) = 0.333 or about 1/3 of the hemoglobin is saturated by O₂

Mitotic Model

• Multicellular organisms
  • First cell grow exponentially (Malthusian growth)
  • Cell growth regulated to develop particular patterns and shapes
  • Cells differentiate into organs with specific functions
  • Adult organisms maintain a constant number of cells
• Mitosis is the process of cellular division
• Cancer is the breakdown of control in cellular division
• How does a cell recognize when it should divide?
  • Cells must recognize their neighboring environment of other cells
  • For example, a skin cell obviously needs to undergo mitosis when either wear or damage of the skin requires replacement cells

• The regulation of mitosis is very important
• One controversial biochemical theory (late 1960s) was that cells communicated with neighboring cells by tissue-specific inhibitors known as chalones
• Chalones are released by cells and diffuse in the environment
• With sufficient quantities of chalones, cells are inhibited from undergoing mitosis

• Theory assumes that chalones bind specifically to certain proteins involved in mitosis
• The chalones inactivate the mitotic proteins, leaving the cell in a quiescent state
• The inhibition process of effector molecules binding to a protein is often modeled using a Hill function
• Let P_n represent a cell density at a particular time n (P_n+1 is the next time period)
• An appropriate mathematical model for the cell density
  • With mitotic divisions and cell loss
  • Dependent on the cellular density with inhibition

• b and c are parameters that fit the data based on chalone kinetics
• The function f(P_n) is known as an updating function
• When the cell density P_n is very low, then the denominator of the model is insignificant

P_n+1 = 2P_n,

• For low density the population doubles in each time period (discrete Malthusian growth model)

• Consider the specific mitotic model

• Determine what the cell density is at equilibrium
• Graph the updating function f(P_n)
• Give some biological interpretations

Equilibria of the Mitotic Model

• At equilibrium, the population density is the same at all time intervals

P_n+1 = P_n = P_e

• Substituting into the model

• The equilibria are P_e = 0 or P_e = 100
  • First equilibrium is the trivial equilibrium (no cells exist)
  • The second equilibrium is the preferred density of cells in a particular tissue

Graphing the Mitotic Updating Function

• To graph f(P_n), examine the intercepts, asymptotes, and any extrema
• The only intercept is (0, 0), the origin
• The denominator is always positive, so no vertical asymptotes
• The power of P_n in the denominator is 4, which exceeds the power of P_n in the numerator
• There is a horizontal asymptote at y = 0.

• To find any extrema, differentiate f(P_n)

• Setting this derivative equal to zero

• The maximum of the updating function occurs at (75.98, 113.98)
• Below is a graph of the updating function

• The graph shows that the greatest production of cells occurs at a cell density of 75.98
• At a cell density of P_n = 100, the production equals the number dying - the model is at equilibrium
• For fairly high density, this model predicts a toxic effect from the crowding
• This gives a major die-off so that the next time period has a very low density
• This model is very simplistic, but it does demonstrate some of the important concepts behind biochemical inhibition

References

[1] W. S. Bullough and E. B. Laurence (1968) Chalones and cancer, Nature 220, 134-135.

[2] H. M. Ranney and V. Sharma, Structure and function of hemoglobin, in Willliam's Hematology, eds. E. Beutler, M. A. Lictman, B. S. Coller, T. J. Kipps, 5th Edition, 1995, p. 417-425.