We briefly review spatially homogeneous mechanistic mathematical models describing the interactions between a malignant tumor and the immune system. We begin with the simplest (single equation) models for tumor growth and proceed to consider greater immunological detail (and correspondingly more equations) in steps. This approach allows us to clarify the necessity for expanding the complexity of models in order to capture the biological mechanisms we wish to understand. We conclude by discussing some unsolved problems in the mathematical modeling of cancer-immune system interactions.

Myogenin and MyoD are proteins that bind to the regulatory regions of a battery of skeletal muscle genes and can activate their transcription during muscle differentiation. We have recently found that both proteins interact with the enhancer of the nicotinic acetylcholine receptor (nAChR) a subunit, a gene that is regulated by innervation. This observation prompted us to study if myogenin and MyoD transcript levels are also regulated by skeletal muscle innervation. Using Northern blot analysis, we found that MyoD and myogenin mRNA levels begin to decline at embryonic day 17 and attain adult levels in muscle of newborn and 3-week-old mice, respectively. In contrast, nAChR mRNAs are highest in newborn and 1-week-old mouse muscle and decline thereafter to reach adult levels in 3-week-old mice. To determine if the downregulation of myogenin and MyoD mRNA levels during development is due to innervation, we quantitated message levels in adult calf muscles after denervation. We found that in denervated muscle myogenin and MyoD mRNAs reach levels that are approximately 40-and 15-fold higher than those found in innervated muscle. Myogenin mRNAs begin to accumulate rapidly between 8 and 16 hr after denervation, and MyoD transcripts levels begin to increase sharply between 16 hr and 1 day after denervation. The increases in myogenin and MyoD mRNA levels precede the rapid accumulation of nAChR a-subunit transcripts; receptor mRNAs begin to accumulate significantiy after 1 day of denervation. The effects of denervation are specific because skeletal a-actin mRNA levels are not affected by denervation. In addition, we found that the repression of myogenin and MyoD expression by innervation is due, at least in part, to "electrical activity." Direct stimulation of soleus muscle with extracellular electrodes repressed the increase of myogenin and MyoD transcripts after denervation by 4-to 3-fold, respectively. In view of these results, it is interesting to speculate that myogenin and/or MyoD may regulate a repertoire ofskeletal muscle genes that are down-regulated by electrical activity.Development of skeletal muscle cells is characterized by a series ofevents that include commitment, differentiation, and maturation. Myoblasts arise from the commitment of pluripotential mesodermal cells to the myogenic lineage. The myoblasts proliferate and later differentiate and fuse to form multinucleated embryonic myotubes. Differentiation is characterized by the transcriptional activation of a battery of muscle-specific genes coding for metabolic enzymes, contractile proteins, ion channels, and neurotransmitter receptors (1-4

We present previously undescribed spatial group patterns that emerge in a one-dimensional hyperbolic model for animal group formation and movement. The patterns result from the assumption that the interactions governing movement depend not only on distance between conspecifics, but also on how individuals receive information about their neighbors and the amount of information received. Some of these patterns are classical, such as stationary pulses, traveling waves, ripples, or traveling trains. However, most of the patterns have not been reported previously. We call these patterns zigzag pulses, semizigzag pulses, breathers, traveling breathers, and feathers.nonlocal hyperbolic system ͉ signal reception ͉ spatial pattern ͉ zigzag P attern formation is one of the most studied aspects of animal communities. Here we present 10 complex spatial patterns that emerge in a one-dimensional mathematical model used to describe the formation and movement of animal groups.Some of the most remarkable examples of patterns observed in animal groups are related to the behavior displayed by these groups (1). Stationary aggregations formed by resting animals, migrating herds of ungulates, zigzagging flocks of birds, and milling schools of fish are only a few of the patterns. To understand the underlying mechanisms, scientists use mathematical models to simulate these observed biological patterns. The most spectacular examples of group patterns shown by numerical simulations are obtained with individual-based models: swarms, tori, and polarized groups (2, 3). A second mathematical modeling approach is based on continuum models, which are usually described by partial differential equations. In many areas, the continuum models have been successful at deducing conditions that give rise to biological patterns [e.g., morphogenesis (4)], even in one spatial dimension (5). However, this has not been the case for animal grouping models. The one-dimensional continuum models that investigate animal aggregations fail to account for the multitude of complex patterns that one can observe in nature. Generally, the patterns exhibited by these models are simple: local parabolic models do not support traveling waves (6), and nonlocal parabolic models can give rise to stationary pulses (7) or to traveling waves, provided that diffusion is density-dependent (8). Hyperbolic models give rise to ripples (9) and aggregations (9, 10). Considering that one-dimensional models have not explained the complexity of the patterns observed in biological systems, scientists have directed their attention toward two-dimensional models. The results are more complex [e.g., ripples (11), stationary aggregations (7), vortex-like groups (12), patches of aligned individuals (13,14)], but they still cannot account for the multitude of observed patterns.One possible reason for this failure is that the assumptions considered by these models do not fully describe the social interactions between individuals governing group formation. More precisely, these models consider that...

We construct and analyze a nonlocal continuum model for group formation with application to self-organizing collectives of animals in homogeneous environments. The model consists of a hyperbolic system of conservation laws, describing individual movement as a correlated random walk. The turning rates depend on three types of social forces: attraction toward other organisms, repulsion from them, and a tendency to align with neighbors. Linear analysis is used to study the role of the social interaction forces and their ranges in group formation. We demonstrate that the model can generate a wide range of patterns, including stationary pulses, traveling pulses, traveling trains, and a new type of solution that we call zigzag pulses. Moreover, numerical simulations suggest that all three social forces are required to account for the complex patterns observed in biological systems. We then use the model to study the transitions between daily animal activities that can be described by these different patterns.