We analyze a class of ordinary differential equations representing a simplified model of a genetic network. In this network, the model genes control the production rates of other genes by a logical function. The dynamics in these equations are represented by a directed graph on an n-dimensional hypercube (n-cube) in which each edge is directed in a unique orientation. The vertices of the n-cube correspond to orthants of state space, and the edges correspond to boundaries between adjacent orthants. The dynamics in these equations can be represented symbolically. Starting from a point on the boundary between neighboring orthants, the equation is integrated until the boundary is crossed for a second time. Each different cycle, corresponding to a different sequence of orthants that are traversed during the integration of the equation always starting on a boundary and ending the first time that same boundary is reached, generates a different letter of the alphabet. A word consists of a sequence of letters corresponding to a possible sequence of orthants that arise from integration of the equation starting and ending on the same boundary. The union of the words defines the language. Letters and words correspond to analytically computable Poincaré maps of the equation. This formalism allows us to define bifurcations of chaotic dynamics of the differential equation that correspond to changes in the associated language. Qualitative knowledge about the dynamics found by integrating the equation can be used to help solve the inverse problem of determining the underlying network generating the dynamics. This work places the study of dynamics in genetic networks in a context comprising both nonlinear dynamics and the theory of computation.

1.
S. A.
Fodor
, “
Massively parallel genomics
,”
Science
277
,
393
395
(
1997
).
2.
R.
Sapolsky
,
L.
Hsie
,
A.
Berno
,
G.
Ghandour
,
M.
Mittmann
, and
J. B.
Fan
, “
High-throughput polymorphism screening and genotyping with high-density oligonucleotide arrays
,”
Biomol. Eng.
14
,
187
192
(
1999
).
3.
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer-Verlag, New York, 1990).
4.
D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding (Cambridge University Press, Cambridge, 1995).
5.
J. E. Hopcroft and J. D. Ullman, Introduction to Automata Theory, Languages, and Computation (Addison-Wesley, Reading, MA, 1979).
6.
M. Sipser, Theory of Computation (PWS, Boston, 1997).
7.
L. Blum, F. Cucker, M. Shub, and S. Smale, Complexity and Real Computation (Springer-Verlag, New York, 1997).
8.
W. S.
McCulloch
and
W.
Pitts
, “
A logical calculus of the ideas immanent in nervous activity
,”
Bull. Math. Biophys.
5
,
115
133
(
1943
).
9.
S. A.
Kauffman
, “
Metabolic stability and epigenesis in randomly constructed genetic networks
,”
J. Theor. Biol.
22
,
437
467
(
1969
).
10.
J. J.
Hopfield
, “
Neurons with graded responses have collective computational properties like those of two-state-neurons
,”
Proc. Natl. Acad. Sci. U.S.A.
81
,
3088
3092
(
1984
).
11.
H. T. Siegelmann, Neural Networks and Analog Computation: Beyond the Turing Limit (Birkhauser, Boston, 1999).
12.
R.
Thomas
, “
Boolean formalization of genetic control circuits
,”
J. Theor. Biol.
42
,
563
585
(
1973
);
R. Thomas and R. D’Ari, Biological Feedback (Chemical Rubber Company, Boca Raton, FL, 1990).
13.
S. A. Kauffman, Origins of Order: Self-Organization and Selection in Evolution (Oxford University Press, Oxford, 1993).
14.
L.
Glass
and
S. A.
Kauffman
, “
Cooperative components, spatial localization and oscillatory cellular dynamics
,”
J. Theor. Biol.
34
,
219
237
(
1972
).
15.
L.
Glass
and
S. A.
Kauffman
, “
The logical analysis of continuous, nonlinear biochemical control networks
,”
J. Theor. Biol.
39
,
103
129
(
1973
).
16.
L.
Glass
, “
Combinatorial and topological methods in nonlinear chemical kinetics
,”
J. Chem. Phys.
63
,
1325
1335
(
1975
).
17.
L.
Glass
and
J. S.
Pasternack
, “
Prediction of limit cycles in mathematical models of biological oscillations
,”
Bull. Math. Biol.
40
,
27
44
(
1978
).
18.
L.
Glass
and
J. S.
Pasternack
, “
Stable oscillations in mathematical models of biological control systems
,”
J. Math. Biol.
6
,
207
223
(
1978
).
19.
T.
Mestl
,
E.
Plahte
, and
S. W.
Omholt
, “
Periodic solutions in systems of piecewise-linear differential equations
,”
Dyn. Stab. Syst.
10
,
179
193
(
1995
).
20.
T.
Mestl
,
C.
Lemay
, and
L.
Glass
, “
Chaos in high dimensional neural and gene networks
,”
Physica D
98
,
33
52
(
1996
).
21.
R.
Edwards
, “
Analysis of continuous-time switching networks
,”
Physica D
146
,
165
199
(
2000
).
22.
R.
Edwards
and
L.
Glass
, “
Combinatorial explosion in model gene networks
,”
Chaos
10
,
691
704
(
2000
).
23.
H. H.
McAdams
and
L.
Shapiro
, “
Circuit simulation of genetic networks
,”
Science
269
,
650
656
(
1995
).
24.
D.
Thieffry
and
R.
Thomas
, “
Dynamical behavior of biological regulatory networks, II. Immunity control in bacteriophage lambda
,”
Bull. Math. Biol.
57
,
277
295
(
1995
).
25.
J.
Reinitz
and
D. H.
Sharp
, “
Mechanism of eve stripe formation
,”
Mech. Dev.
49
,
133
158
(
1995
).
26.
G. Marnellos and E. Mjolsness, “A gene network approach to modeling early neurogenesis in drosophila,” in Pacific Symposium on Biocomputing ’98, edited by R. B. Altman, A. K. Dunker, L. Hunter, and T. E. Klein (World Scientific, Singapore, 1998), pp. 30–41.
27.
L.
Mendoza
and
E.
Alvarez-Buylla
, “
Dynamics of the genetic regulatory network for Arabidopsis thaliana flower morphogenesis
,”
J. Theor. Biol.
193
,
307
319
(
1998
).
28.
B. J.
Meyer
and
M.
Ptashne
, “
Gene regulation at the right operator (OR) of bacteriophage λ. III. λ repressor directly activates gene transcription
,”
J. Mol. Biol.
139
,
195
205
(
1980
).
29.
J. P.
Crutchfield
, “
The calculi of emergence: Computations, dynamics, and induction
,”
Physica D
75
,
11
54
(
1994
).
30.
C.
Moore
and
P.
Lakdawala
, “
Queues, stacks, and transcendentality at the transition to chaos
,”
Physica D
135
,
24
40
(
2000
).
31.
L.
Glass
and
R.
Young
, “
Structure and dynamics of neural network oscillators
Brain Res.
179
,
207
218
(
1979
);
D. T. Kaplan and L. Glass, Understanding Nonlinear Dynamics (Springer-Verlag, New York, 1995), pp. 70–73.
32.
S. Liang, S. Fuhrman, and R. Somogyi. “REVEAL, A general reverse engineering algorithm for inference of genetic network architectures,” in Pacific Symposium on Biocomputing ’98, edited by R. B. Altman, A. K. Dunker, L. Hunter, and T. E. Klein (World Scientific, Singapore, 1998), vol. 3, pp. 18–29.
33.
T. Akutsu, S. Miyano, and S. Kuhara, “Identification of genetic networks from a small number of gene expression patterns under the Boolean network model,” in Pacific Symposium on Biocomputing ’99, edited by R. B. Altman, A. K. Dunker, L. Hunter, and T. E. Klein (World Scientific, Singapore, 1999), vol. 4, pp. 17–24.
34.
T. Akutsu, S. Miyano, and S. Kuhara, “Algorithms for inferring qualitative models of biological networks,” in Pacific Symposium on Biocomputing 2000, edited by R. B. Altman, A. K. Dunker, L. Hunter, and T. E. Klein (World Scientific, Singapore, 2000), vol. 5, pp. 290–301.
35.
C.-H.
Yuh
,
H.
Bolouri
, and
E. H.
Davidson
, “
Genomic cis-regulatory logic: Experimental and computational analysis of a sea urchin gene
,”
Science
279
,
1896
1902
(
1998
).
36.
T.
Mestl
,
R. J.
Bagley
, and
L.
Glass
, “
Common chaos in arbitrarily complex feedback networks
,”
Phys. Rev. Lett.
79
,
653
656
(
1997
).
37.
M. B.
Elowitz
and
S.
Leibler
, “
A synthetic oscillatory network of transcriptional regulators
,”
Nature (London)
403
,
335
338
(
2000
).
38.
T. S.
Gardner
,
C. R.
Cantor
, and
J. J.
Collins
, “
Construction of a genetic toggle switch in Escherichia coli
,”
Nature (London)
403
,
339
342
(
2000
).
This content is only available via PDF.
You do not currently have access to this content.