130 CATALYZING INQUIRY
concentration would have to be modeled discretely. As more of it is synthesized, the concentration
becomes high enough that a continuous approximation is justified and is then more efficient for simu-
lation and analysis.
The point at which this switch is made is dependent not just on copy number but also on where in the
dynamical state space the system resides. If the system is near a bifurcation point, small fluctuations may be
significant. Theories of how to accomplish this dynamic switching are lacking. As models grow more
complex, different parts of the system will have to be modeled with different mathematical representations.
Also, as models from different sources begin to be joined, it is clear that different representations will be
used. It is critical that the theory and applied mathematics of hybrid dynamical systems be developed.
5.3.3 Multiscale Models,
Multiscale models describe processes occurring at many time and length scales. Depending on the
biological system of interest, the data needed to provide the basis for a greater understanding of the
system will cut across several scales of space and time. The length dimensions of biological interest
range from small organic molecules to multiprotein complexes at 100 angstroms to cellular processes at
1,000 angstroms to tissues at 1-10 microns, and the interaction of human populations with the environ-
ment at the kilometer scale. The temporal domain includes the femtosecond chemistry of molecular
interactions to the millions of years of evolutionary time, with protein folding in seconds and cell and
developmental processes in minutes, hours, and days. In turn, the scale of the process involved (e.g.,
from the molecular scale to the ecosystem scale) affects both the complexity of the representation (e.g.,
molecule base, concentration based, at equilibrium or fully dynamic) and the modality of the represen-
tation (e.g., biochemical, genetic, genomic, electrophysiological, etc.).
Consider the heart as an example. The macroscopic unit of interest is the heartbeat, which lasts
about a second and involves the whole heart of 10 cm scale. But the cardiac action potential (the
electrical signal that initiates myocellular contractions) can change significantly on time scales of milli-
seconds as reflected in the appropriate kinetic equations. In turn, the molecular interactions that under-
lie kinetic flows occur on time scales on the order of femtoseconds. Across such variation in time scales,
it is not feasible to model 10^15 molecular interactions in order to model a complete heartbeat. Fortu-
nately, in many situations the response with the shorter time scale will converge quickly to equilibrium
or quasi-steady-state behavior, obviating the need for a complete lower-level simulation.^30
For most biological problems, the scale at which data could provide a central insight into the
operation of the whole system is not known, so multiple scales are of interest. Thus, biological models
have to allow for transition among different levels of resolution. A biologist might describe a protein as
a simple ellipsoid and then in the next breath explain the effect of a point mutation by the atomic-level
structural changes it causes in the active site.^31
Identifying the appropriate ranges of parameters (e.g., rate constants that govern the pace of chemi-
cal reactions) remains one of the difficulties that every modeler faces sooner or later. As modelers know
well, even qualitative analysis of simple models depends on knowing which “leading-order terms” are
to be kept on which time scales. When the relative rates are entirely unknown—true of many biochemi-
cal steps in living cells—it is hard to know where to start and how to assemble a relevant model, a point
that underscores the importance of close dialogue between the laboratory biologist and the mathemati-
cal or computational modeler.
Finally, data obtained at a particular scale must be sufficient to summarize the essential biological
activity at that scale in order to be evaluated in the context of interactions at greater scales of complexity.
The challenge, therefore, is one of understanding not only the relationship of multiple variables operat-
ing at one scale of detail, but also the relationship of multivariable datasets collected at different scales.
(^30) A.D. McCulloch and G. Huber, “Integrative Biological Modelling in Silico,” pp. 4-25 in ‘In Silico’ Simulation of Biological
Processes No. 247, Novartis Foundation Symposium, G. Bock and J.A. Goode, eds., John Wiley & Sons Ltd., Chichester, UK, 2002.
(^31) D. Endy and R. Brent, “Modeling Cellular Behavior,” Nature 409(6818):391-395, 2001.