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Understanding bistability in complex enzyme-driven reaction networks

  1. Gheorghe Craciun * , † ,
  2. Yangzhong Tang ‡ , and
  3. Martin Feinberg ‡ , § ,
  1. *Mathematical Biosciences Institute, 231 West 18th Avenue, and
  2. ‡Departments of Chemical Engineering and
  3. §Mathematics, 140 West 19th Avenue, Ohio State University, Columbus, OH 43210; and
  4. †Departments of Mathematics and Biomolecular Chemistry, University of Wisconsin, Madison, WI 53706

  1. Communicated by Avner Friedman, Ohio State University, Columbus, OH, April 6, 2006 (received for review February 1, 2006)

Abstract

Much attention has been paid recently to bistability and switch-like behavior that might be resident in important biochemical reaction networks. There is, in fact, a great deal of subtlety in the relationship between the structure of a reaction network and its capacity to engender bistability. In common physicochemical settings, large classes of extremely complex networks, taken with mass action kinetics, cannot give rise to bistability no matter what values the rate constants take. On the other hand, bistable behavior can be induced in those same settings by certain very simple and classical mass action mechanisms for enzyme catalysis of a single overall reaction. We present a theorem that distinguishes between those mass action networks that might support bistable behavior and those that cannot. Moreover, we indicate how switch-like behavior results from a well-studied mechanism for the action of human dihydrofolate reductase, an important anti-cancer target.

Footnotes

  • To whom correspondence should be addressed. E-mail: feinberg.14{at}osu.edu

  • Author contributions: G.C. and M.F. designed research; G.C., Y.T., and M.F. performed research; G.C., Y.T., and M.F. contributed analytic tools; and G.C. and M.F. wrote the paper.

  • ‖There are many combinations of parameters that give rise to bistability. Fig. 1 was drawn for the following choice (see Supporting Appendix): k 1 = 93.43, k 2 = 2539, k 3 = 481.6, k 4 = 1,183, k 5 = 1,556, k 6 = 121,192, k 7 = 0.02213, k 8 = 1,689, k 9 = 85,842, ξS1 = 1, ξΣ2 = 1, ξP = 1, F S1 = 2,500, F S2 = 1,500. All initial conditions were chosen such that c E + c ES1 + c ES2 + c ES1S2 = 1. Thereafter, the reactions themselves conserve the total amount of enzyme. Ho and Li (12) have done some rudimentary bifurcation studies.

  • ↵**We mean here the differential equations formulated for the homogeneous cell, as in the passage from the network in Eq. 3 to Eq. 4 , having the largest number of free parameters, i.e., with all species permitted to pass through the cell boundary. When, as in the toy cell example, certain species are entrapped or, more precisely, do not give rise to terms of the form −ξA c A in the model differential equations, the theorem nevertheless remains true with a technical modification: In this case, the hypothesis serves to deny the capacity of a network to engender two nondegenerate stoichiometrically compatible positive steady states (22). In subsequent discussion of Table 1, then, preclusion of multiple steady states should be understood to carry, implicitly, this technical qualification.

  • Conflict of interest statement: No conflicts declared.

  • Abbreviations:

    Abbreviations:

    SR,
    species-reaction;
    c-pair,
    complex pair;
    DHFR,
    dihydrofolate reductase.

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