154 CATALYZING INQUIRY
- The total number of genes involved is N, a number of order 30,000.
- The number of genes that regulate a given target is a constant (call it K) for all regulated genes; K
is a small integer. - The regulatory signal associated with a connection or the expression of a gene is either on or off.
(In fact, almost certainly it is not just the fact of a connection between genes that influences regulation,
but rather the nature of that connection as a continuous time-varying value such as a molecular concen-
tration over time.) - Every gene is governed by the same transition rule (i.e., a Boolean function) that specifies its state
(on or off) as a function of the activities of its K inputs at the immediately earlier time. - The regulatory network operates synchronously (and, by implication, kinetics are unimportant).
- Secondary effects on genetic regulation arising from the nondigital characteristics of DNA (such
as methylation) can be neglected. - The genes that regulate and genes that are regulated (which may overlap) are connected at
random.
Box 5.9 provides more details about this model. Because the model treats all genes as identical (i.e.,
all obey the same transition rule) and assigns connections between genes at random, it obviously lacks
fidelity to any specific genome and cannot predict the biological phenomenology of any specific organ-
ism. Yet, it may provide insight into biological order that emerges from the structure of the genetic
regulatory network itself.
Simulations of the operation of this model yielded interesting behavior, which depends on the
values of N and K. For K = 1 or K > 5, the behavior of the network exhibits little interesting order, where
order is defined in terms of fixed cycles known as attractors. If K = 1, the networks are static, with the
number of attractors exponential in the size of the network and the cycle length approaching unity. If
K > 5, there are few attractors, and it is the cycle length that is exponential in the size of the network.
However, for K = 2, the network does exhibit order that has potential biological significance—both the
number of attractors and the cycle length are proportional to N1/2.^75
What might be the biological significance of these results?
- The trajectory of an attractor through its successive states would reflect the fact that, over time,
different genes are expressed in a biological organism. - The fact that there are multiple attractors within the same genome suggests that multiple biologi-
cal structures might exist, even within the same organism, corresponding to the genome being in one of
these attractor states. An obvious candidate for such structures would be multiple cell types. That is,
this analysis suggests that a cell type corresponds to a given state cycle attractor, and the different
attractors to the different cell types of the organism. Another possibility is that different but similar
attractors correspond to cells in different states (e.g., disease state, resting state, perturbed state). - The fact that an attractor is cyclic suggests that it may be related to cyclic behavior in a biological
organism. If cell types can be identified with attractors, the cyclic trajectory in phase space of an
attractor may correspond to the cell cycle in which a cell divides. - States that can be moved from one trajectory (for one attractor) to another trajectory (and another
attractor) by changing a single state variable are not robust and may represent the phenomenon that
small, apparently minor perturbations to a cell’s environment may kick it into a different state. - The square root of the number of genes in the human genome (around 30,000) is 173. Under the
assumption of K = 2 scaling, this would correspond to the number of cyclic attractors and thus to the
number of cell types in the human body. This is not far from the number of cell types actually observed
(^75) A. Bhattacharjya and S. Liang, “Power-Law Distributions in Some Random Boolean Networks,” Physical Review Letters
77(8):1644, 1996.