Random Movements in Space and Time

  • Ronald W. Shonkwiler
  • James Herod
Part of the Undergraduate Texts in Mathematics book series (UTM)


Many biological phenomena, at all levels of organization, can be modeled by treating them as random processes, behaving much like the diffusion of ink in a container of water. In this chapter, we discuss some biological aspects of random processes, namely, the movement of oxygen across a human placenta. While these processes might seem to be quite different at first glance, they actually act according to very similar models. We begin with a description of biological membranes, structures that regulate the movement of material into, out of, and within the functional compartments of a cell. At the core of a membrane is a layer of water-repelling molecules. This layer has the effect of restricting the free transmembrane movement of any substance that is water soluble, although water itself can get past the layer. The transmembrane movement of the normal water-soluble compounds of cellular metabolism is regulated by large biochemical molecules that span the membrane. They are called permeases, or transport proteins. Permeases have the ability to select the materials that cross a membrane. Other membranes anchor critical cellular components that promote chemical reactions through catalysis. A human fetus requires oxygen for its metabolic needs. This oxygen is obtained from its mother, who breathes it and transfers it via her blood to the placenta, an organ that serves as the maternal–fetal interface. Because the blood of mother and child do not mix, material exchange between them must take place across a group of membranes. The chemical that transports the oxygen is hemoglobin, of which there are at least two kinds, each exhibiting a different strength of attachment to oxygen molecules. Further, chemical conditions around the hemoglobin also affect its attachment to oxygen. The conditions at the placenta are such that there is a net transmembrane movement of oxygen from maternal hemoglobin to fetal hemoglobin. This chapter also serves as an introduction to the discussions of the blood vascular system of Chapter 9, of biomolecular structure of Chapter 8, and of HIVin Chapter 10.


Random Walk Root Mean Square Random Movement Maternal Blood Fetal Blood 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References and Suggested Further Reading

  1. [1] Membrane Structure:
    W. S. Beck, K. F. Liem, and G. G. Simpson, Life: An Introduction to Biology, 3rd ed., Harper–Collins, New York, 1991.Google Scholar
  2. [2] Membrane Transport:
    E. K. Yeargers, Basic Biophysics for Biology, CRC Press, Boca Raton, FL, 1992.Google Scholar
  3. [3] Diffusion:
    H. C. Berg, Random Walks in Biology, Princeton University Press, Princeton, NJ, 1993.Google Scholar
  4. [4] Diffusion:
    R. K. Hobbie, Intermediate Physics for Medicine and Biology, 2nd ed., Wiley, New York, 1988, 65.Google Scholar
  5. [5] Diffusion In Biology:
    J. D. Murray, Mathematical Biology, Springer-Verlag, New York, 1989.MATHGoogle Scholar
  6. [6] Fluid Resistance:
    S. I. Rubinow, Introduction to Mathematical Biology, Wiley, New York, 1975.MATHGoogle Scholar
  7. [7] Diffusion Across A Slab:
    D. L. Powers, Boundary Value Problems, Academic Press, New York, 1979.MATHGoogle Scholar
  8. [8] Oxygen Dissociation Curves, Fetal Blood:
    W. T. Keeton and J. L. Gould, Biological Science, 5th ed., Norton, New York, 1993.Google Scholar
  9. [9] Epidemiology:
    J. P. Fox, C. E. Hall, and L. R. Elveback, Epidemiology: Man and Disease, Macmillan, New York, 1970.Google Scholar
  10. [10] Epidemiology And Disease:
    J. P. Krier and R. F. Mortenson, Infection, Resistance, and Immunity, Harper and Row, New York, 1990.Google Scholar
  11. [11] Placenta:
    F. C. Battaglia and G. Meschia, An Introduction to Fetal Physiology, Academic Press/Harcourt Brace Jovannovich, New York, 1986.Google Scholar
  12. [12] Placenta:
    K. S. Comline, K. W. Cross, G. S. Dawes, and P. W. Nathanielsz, Foetal and Neonatal Physiology, Cambridge University Press, Cambridge, UK, 1973.Google Scholar
  13. [13] Placenta:
    R. E. Forster II, Some principles governing maternal–foetal transfer in the placenta, in Foetal and Neonatal Physiology, Cambridge University Press, Cambridge, UK, 1973, 223–237.Google Scholar
  14. [14] Placenta:
    A. Costa, M. L. Costantino, and R. Fumero, Oxygen exchange mechanisms in the human placenta: Mathematical modelling and simulation, J. Biomed. Engrg., 14 (1992), 85–389.Google Scholar
  15. [15] Placenta:
    H. Bartels, W. Moll, and J. Metcalfe, Physiology of gas exchange in the human placenta, Amer. J. Obstetrics Gynecol., 84 (1962), 1714–1730.Google Scholar
  16. [16] Placenta:
    J. Metcalfe, H. Bartels, and W. Moll, Gas exchange in the pregnant uterus, Physiol. Rev., 47 (1967), 782–838.Google Scholar
  17. [17] Placenta:
    A. Guettouche, J. C. Challier, Y. Ito, Y. Papapanayotou Cherruault, and A. Azancot-Benisty, Mathematical modeling of the human fetal arterial blood circulation, Internat. J. Biomed. Comput., 31 (1992), 127–139.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York 2009

Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations